Essays on Insurance-Linked Capital Market Instruments,Solvency Measurement, and Insurance Pricing
DISSERTATIONof the University of St.Gallen,
School of Management,Economics, Law, Social Sciences
and International Affairsto obtain the title of
Doctor of Philosophy in Management
submitted by
Alexander Braun
from
Germany
Approved on the application of
Prof. Dr. Hato Schmeiser
and
Prof. Dr. Manuel Ammann
Dissertation no. 3870
Braun Druck & Medien GmbH, Tuttlingen, 2011
The University of St. Gallen, School of Management, Economics, Law,
Social Sciences and International Affairs hereby consents to the printing
of the present dissertation, without hereby expressing any opinion on the
views herein expressed.
St. Gallen, May 13, 2011
The President:
Prof. Dr. Thomas Bieger
To my dear parents/Meinen lieben Eltern
Marliese & Herbert Braun
Acknowledgements
The dissertation at hand would not have been feasible without the con-
tinuous encouragement and support of a number of people, both before
and during the time of its writing.
To begin with, I wish to express my sincere gratitude to my super-
visor, Prof. Dr. Hato Schmeiser, who always offered valuable advice and
suggestions and created an excellent research environment with excep-
tional working conditions. In addition, I thank my co-supervisor, Prof.
Dr. Manuel Ammann, for his constructive feedback and his thoughtful
comments, which helped me to further improve this dissertation.
Moreover, I am grateful to my co-authors as well as my colleagues of
the Institute of Insurance Economics of the University of St. Gallen for
inspiring discussions, exciting joint research projects, and an extraordi-
nary pleasant working atmosphere. They all contributed to the unique
experience that I enjoyed during my doctoral studies.
Finally, I would like to wholeheartedly thank my long-time girlfriend,
my brother, my grandmother, and my parents. They are the most impor-
tant pillars in my life and accompanied me through highs and lows over
all those years. In particular, this achievement would be inconceivable
without the generosity, devotion, and unremitting support from my dear
parents, to whom I owe so much.
St. Gallen, July 2011
Alexander Braun
Vorwort
Die vorliegende Dissertation ware ohne die andauernde Forderung und
Unterstutzung einer Reihe von Personen sowohl vor als auch wahrend
ihrer Entstehungsphase nicht realisierbar gewesen.
Zunachst gilt mein aufrichtiger Dank meinem Referenten und Be-
treuer Prof. Dr. Hato Schmeiser, der mir stets wertvolle Ratschlage und
Anregungen gab und eine herausragende Forschungsumgebung mit ex-
zellenten Arbeitsbedingungen schuf. Ausserdem danke ich meinem Ko-
referenten Prof. Dr. Manuel Ammann fur sein konstruktives Feedback
und seine durchdachten Kommentare, die mir dabei halfen, diese Disser-
tation noch weiter zu verbessern.
Ferner bin ich meinen Koautoren und meinen Kollegen am Institut
fur Versicherungswirtschaft der Universiat St. Gallen fur inspirierende
Diskussionen, spannende gemeinsame Forschungsprojekte und eine aus-
serordentlich angenehme Arbeitsatmosphare dankbar. Sie alle trugen da-
zu bei, dass mein Doktoratsstudium zur einmaligen Erfahrung wurde.
Schliesslich mochte ich von ganzem Herzen meiner langjahrigen Freun-
din, meinem Bruder, meiner Grossmutter und meinen Eltern danken.
Sie sind die wichtigsten Saulen in meinem Leben und haben mich in all
den Jahren durch Hohen und Tiefen begleitet. Insbesondere ware das
Erreichte undenkbar ohne die Grosszugigkeit, Hingabe und unablassige
Unterstutzung meiner lieben Eltern, denen ich so viel zu verdanken
habe.
St. Gallen im Juli 2011
Alexander Braun
Outline iii
Outline
I Performance and Risksof Open-End Life Settlement Funds 1
II Pricing Catastrophe Swaps:A Contingent Claims Approach 59
III Solvency Measurementof Swiss Occupational Pension Funds 119
IV Stock vs. Mutual Insurers:Who Does and Who Should Charge More? 163
Curriculum Vitae 221
iv Contents
Contents
Contents iv
List of Figures vii
List of Tables ix
Summary xi
Zusammenfassung xiii
I Performance and Risksof Open-End Life Settlement Funds 1
1 Introduction 2
2 Market overview and fund business model 52.1 The U.S. life settlement market: an overview . . . . . . . 52.2 Closed-end vs. open-end life settlement funds . . . . . . . 72.3 The anatomy of open-end life settlement funds . . . . . . 11
3 Empirical analysis 163.1 Data and sample selection . . . . . . . . . . . . . . . . . . 163.2 The return distribution of open-end life settlement funds 223.3 Performance measurement and correlation analysis . . . . 263.4 Analysis of individual funds . . . . . . . . . . . . . . . . . 28
4 Risks of open-end life settlement funds 364.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Valuation risk . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Longevity risk . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Liquidity risk . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 Policy availability risk . . . . . . . . . . . . . . . . . . . . 424.6 Operational risks . . . . . . . . . . . . . . . . . . . . . . . 434.7 Credit risk . . . . . . . . . . . . . . . . . . . . . . . . . . 444.8 Changes in regulation and tax legislation . . . . . . . . . 45
5 Summary and conclusion 47
6 Appendix 486.1 Index descriptions (from the providers) . . . . . . . . . . 486.2 Performance measures . . . . . . . . . . . . . . . . . . . . 51
References 53
Contents v
II Pricing Catastrophe Swaps:A Contingent Claims Approach 59
1 Introduction 60
2 Literature review 62
3 Catastrophe swaps 64
3.1 Contract design . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Market development . . . . . . . . . . . . . . . . . . . . . 67
3.3 Areas of application . . . . . . . . . . . . . . . . . . . . . 68
3.4 Accounting and regulation . . . . . . . . . . . . . . . . . 69
3.5 Comparison to other risk transfer instruments . . . . . . 70
4 Pricing model 70
4.1 Risk-neutral valuation of catastrophe derivatives . . . . . 72
4.2 Pricing catastrophe swaps ex-ante . . . . . . . . . . . . . 74
4.3 Pricing catastrophe swaps in the loss reestimation phase . 78
5 Empirical analysis 87
5.1 Severity distributions for natural disasters in the U.S. . . 87
5.2 Derivation of implied Poisson intensities . . . . . . . . . . 94
5.3 The stochastic process of implied Poisson intensities . . . 101
6 Summary and conclusion 109
7 Appendix: The market price of cat risk 110
References 112
III Solvency Measurementof Swiss Occupational Pension Funds 119
1 Introduction 120
2 The particularities of the Swiss pension system 123
3 The model framework 125
4 The traffic light approach 131
5 Implementation and calibration 132
5.1 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
vi Contents
6 Sensitivity analysis 1426.1 Equity allocation . . . . . . . . . . . . . . . . . . . . . . . 1436.2 Asset concentration . . . . . . . . . . . . . . . . . . . . . 1456.3 Misestimation of liabilities . . . . . . . . . . . . . . . . . 1476.4 Coverage ratio . . . . . . . . . . . . . . . . . . . . . . . . 1486.5 Lowest acceptable coverage ratio . . . . . . . . . . . . . . 1506.6 Exchange rate risk . . . . . . . . . . . . . . . . . . . . . . 151
7 Supervisory review and actions 154
8 Notes on a potential introduction in Switzerland 156
9 Conclusion 158
References 159
IV Stock vs. Mutual Insurers:Who Does and Who Should Charge More? 163
1 Introduction 164
2 Literature review 166
3 Empirical analysis 171
4 Model framework 1754.1 Stock insurer claims structure . . . . . . . . . . . . . . . 1754.2 Mutual insurer: full equity participation . . . . . . . . . . 1774.3 Mutual insurer: partial equity participation . . . . . . . . 1874.4 Claims structure relationships . . . . . . . . . . . . . . . 192
5 Numerical analysis 1965.1 Option pricing formulae . . . . . . . . . . . . . . . . . . . 1965.2 The impact of recovery option and equity participation . 1995.3 Premium, safety level, and equity capital . . . . . . . . . 203
6 Economic implications 209
7 Conclusion 213
References 215
Curriculum Vitae 221
List of Figures vii
List of Figures
Performance and Risksof Open-End Life Settlement Funds
1 Stylized mechanics of an open-end life settlement fund . . 122 Life settlements in comparison to other asset classes
(12/2003–06/2010) . . . . . . . . . . . . . . . . . . . . . . 253 Individual life settlement funds in comparison
(01/2007–06/2010) . . . . . . . . . . . . . . . . . . . . . . 30
Pricing Catastrophe Swaps:A Contingent Claims Approach
4 Illustration of the barrier option pricing approach . . . . . 865 Natural disaster losses in the U.S. (1900–2005) . . . . . . 916 Histograms of normalized natural disaster losses . . . . . 927 Monthly implied intensities (08/2005–09/2010) . . . . . . 998 Intensity factor scores and out-of-sample forecast example 1049 ACF and PACF for the hurricane factor . . . . . . . . . . 10610 ACF and PACF for the earthquake factor . . . . . . . . . 107
Solvency Measurementof Swiss Occupational Pension Funds
11 Sensitivity analysis: equity allocation . . . . . . . . . . . . 14412 Sensitivity analysis: asset concentration . . . . . . . . . . 14613 Sensitivity analysis: misestimation of liabilities . . . . . . 14914 Sensitivity analysis: coverage ratio . . . . . . . . . . . . . 14915 Sensitivity analysis: minimum coverage ratio . . . . . . . 15116 Sensitivity analysis: actual and minimum coverage ratio . 15217 Sensitivity analysis: FX hedging . . . . . . . . . . . . . . 153
Stock vs. Mutual Insurers:Who Does and Who Should Charge More?
18 Stock insurer equity and policyholder payoff . . . . . . . . 17819 Mutual insurer default put option payoff . . . . . . . . . . 18120 Mutual insurer recovery option payoff . . . . . . . . . . . 18221 Mutual insurer equity payoff: full participation . . . . . . 18522 Mutual insurer policyholder stake payoff . . . . . . . . . . 18623 Mutual insurer (expected) equity payoff . . . . . . . . . . 190
viii List of Figures
24 Theoretical premium comparison . . . . . . . . . . . . . . 19325 Equity-premium combinations:
full equity participation and no recovery option . . . . . . 20426 Equity-premium combinations:
full equity participation and recovery option . . . . . . . . 20727 Equity-premium combinations:
partial equity participation and recovery option . . . . . . 208
List of Tables ix
List of Tables
Performance and Risksof Open-End Life Settlement Funds
1 Closed-end vs. open-end life settlement funds . . . . . . . 92 Life settlement funds in the original dataset . . . . . . . . 19
3 Sample details . . . . . . . . . . . . . . . . . . . . . . . . 224 Descriptive statistics: index return distributions
(12/2003–06/2010) . . . . . . . . . . . . . . . . . . . . . . 275 Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . 296 Descriptive statistics: fund return distributions
(01/2007–06/2010) . . . . . . . . . . . . . . . . . . . . . . 327 Summary statistics: 14 fund return distributions . . . . . 34
Pricing Catastrophe Swaps:A Contingent Claims Approach
8 Catastrophe risk transfer instruments in comparison . . . 719 Descriptive statistics for (non-zero) disaster losses . . . . . 9010 Estimates: lognormal, Burr, and Pareto distribution . . . 95
11 Estimates: Weibull, gamma, and exponential distribution 9512 Descriptive statistics: time series of cat swap prices . . . . 97
13 Descriptive statistics: implied intensity time series . . . . 10014 Correlation matrices and factor loadings . . . . . . . . . . 10015 Hurricane intensity factor: unit root tests and estimates . 108
16 Earthquake intensity factor: unit root tests and estimates 108
Solvency Measurementof Swiss Occupational Pension Funds
17 Input parameters for the sample funds in 2007 . . . . . . 135
18 Input parameters for the sample funds in 2008 . . . . . . 13519 Asset allocations of the sample funds in 2007 . . . . . . . 13620 Asset allocations of the sample funds in 2008 . . . . . . . 136
21 Annualized means and standard deviations . . . . . . . . 13822 Correlation matrix . . . . . . . . . . . . . . . . . . . . . . 138
23 One-year default probabilities for different rating classes . 13924 Probabilities and test outcomes for 2007 . . . . . . . . . . 14125 Probabilities and test outcomes for 2008 . . . . . . . . . . 141
26 Parameters for a representative pension fund . . . . . . . 143
x List of Tables
Stock vs. Mutual Insurers:Who Does and Who Should Charge More?
27 Descriptive statistics of the data . . . . . . . . . . . . . . 17328 Estimation results . . . . . . . . . . . . . . . . . . . . . . 17629 Input parameters and values for the stock insurer . . . . . 20030 Impact of recovery option and equity participation . . . . 20131 Impact of the maximum amount of additional premiums . 203
Summary xi
Summary
This dissertation consists of four parts, each of which comprises anindividual research paper. The first two parts pertain to the field ofinsurance-linked capital market instruments. The paper ”Performanceand Risks of Open-End Life Settlements” is a comprehensive analysisof open-end funds, which exclusively invest in so-called senior life set-tlements, that is, U.S. life insurance policies traded in the secondarymarket. It comprises an explanation of the business model of open-endlife settlement funds, an empirical analysis of their return distributions,a performance measurement, and an in-depth risk analysis. Althoughthe empirical results suggest that life settlement funds offer attractivereturns paired with low volatilities and are virtually uncorrelated withestablished asset classes, investors need to be aware of latent risk factorssuch as liquidity, longevity and valuation risks, which are largely notreflected by the examined historical data. Yet, these aspects should betaken into account in order not to overestimate the performance of thisasset class.
The second research paper, ”Pricing Catastrophe Swaps: A Contin-gent Claims Approach”, centers around the catastrophe (cat) swap, afinancial instrument through which natural disaster risks can be trans-ferred between counterparties. It begins with a discussion of the typicalcontract design, the current state of the market, and major areas of ap-plication. In addition, a two-stage option-theoretic pricing approach isproposed, which distinguishes between the main risk drivers ex-ante andduring the loss reestimation phase. Catastrophe occurrence is modeled asa doubly stochastic Poisson process (Cox process) with mean-revertingOrnstein-Uhlenbeck intensity. Moreover, by fitting various parametricdistributions to historical loss data from the U.S., the heavy-tailed Burrdistribution is found to be the most adequate representation for lossseverities. The pricing model is then applied to market quotes for hur-ricane and earthquake contracts to derive implied Poisson intensities.Since a first order autoregressive process provides a good fit to the re-sulting time series, its continuous-time limit, the Ornstein-Uhlenbeckprocess should be well suited to represent the dynamics of the Poissonintensity in a cat swap pricing model.
With the research paper ”Solvency Measurement of Swiss Occupa-tional Pension Funds”, the focus in the third part of the dissertationis on solvency measurement in the occupational pension system. Based
xii Summary
on the combination of a stochastic pension fund model and a trafficlight signal approach, a solvency test for occupational pension funds inSwitzerland is proposed. Being designed as a regulatory standard model,the set-up is intentionally kept parsimonious and, assuming normally dis-tributed asset returns, a closed-form solution can be derived. Despite itssimplicity, the framework comprises the essential risk sources needed insupervisory practice. Due to its ease of calibration, it is additionally wellsuited for the fragmented Swiss market, keeping costs of solvency testingat a minimum. To illustrate its application, the model is calibrated andimplemented for a small sample of ten Swiss pension funds. Moreover,a sensitivity analysis is conducted to identify important drivers of theshortfall probabilities for the traffic light conditions.
Finally, the fourth and last part of the dissertation contains the re-search paper ”Stock vs. Mutual Insurers: Who Does and Who ShouldCharge More?”, which is an empirical and theoretical analysis of therelationship between the premiums of insurers in the legal form of stockand mutual companies. An evaluation of panel data for the German mo-tor liability insurance sector does not provide indications that mutualscharge significantly higher premiums than stock insurers. Subsequently,a comprehensive model framework for the arbitrage-free pricing of in-surance contracts is employed to compare stock and mutual insurancecompanies with regard to the three central magnitudes premium size,safety level, and equity capital. Although, from a normative perspective,there are certain circumstances in which the premiums of stock and mu-tual insurers should be equal, these situations would generally requirethe mutual to hold comparatively less capital. This being inconsistentwith the empirical results, it appears that the observed insurance pricesare not arbitrage-free.
Zusammenfassung xiii
Zusammenfassung
Diese Dissertation besteht aus vier Teilen, die jeweils eine in sich ge-schlossene Forschungsarbeit enthalten. Kapitel eins und zwei sind in dasThemenfeld der versicherungsgebundenen Kapitalmarktinstrumente ein-zuordnen. Die Arbeit
”Performance and Risks of Open-End Life Settle-
ments“ stellt eine umfassende Analyse offener Fonds dar, die in am Zweit-markt gehandelte US-amerikanische Lebensversicherungen, sogenannteSenior Life Settlements, investieren. Sie umfasst eine Erlauterung desGeschaftsmodells und der Funktionsweise offener Lebensversicherungs-fonds, eine empirische Analyse ihrer Renditeverteilungen, eine Perfor-mancemessung sowie eine detaillierte Risikoanalyse. Obwohl die empi-rischen Ergebnisse darauf hindeuten, dass offene Lebensversicherungs-fonds attraktive Renditen bei geringer Volatilitat bieten und praktischunkorreliert mit etablierten Anlageklassen sind, sollten sich Investorenlatenter Risikofaktoren wie beispielsweise Liquiditats-, Langlebigkeits-und Bewertungsrisiken bewusst sein, welche sich grosstenteils nicht inden untersuchten historischen Daten widerspiegeln. Eine Berucksichtig-ung dieser Risiken ist jedoch erforderlich, um die Performance dieserAnlageklasse nicht zu uberschatzen.
Im Mittelpunkt der zweiten Forschungsarbeit”Pricing Catastrophe
Swaps: A Contingent Claims Approach“ steht der Katastrophen-Swap,ein Finanzinstrument, welches eine Ubertragung der Risiken von Natur-katastrophen zwischen Marktteilnehmern ermoglicht. Die Ausfuhrungenbeginnen mit einer Erlauterung der ublichen Vertragsgestaltung, der ge-genwartigen Marktlage sowie der wichtigsten Anwendungsgebiete. Da-ruber hinaus wird ein zweistufiger Bewertungsansatz vorgeschlagen, derzwischen den Hauptrisikotreibern ex-ante und wahrend der Aktualisie-rung der zugrunde liegenden Verlustschatzungen unterscheidet. Das Auf-treten von Katastrophen wird mit einem doppelt stochastischen Poisson-Prozess (Cox-Prozess) modelliert, dessen Intensitat einem Ornstein-Uhlenbeck-Prozess folgt. Durch das Anpassen verschiedener parametri-scher Verteilungsfunktionen an historische US-amerikanische Verlustda-ten kann des Weiteren die endlastige Burr-Verteilung als angemessensteReprasentation der Verlustschwere identifiziert werden. Anschliessendwird das Bewertungsmodell auf Marktpreise von Orkan- und Erdbeben-kontrakten angewendet, um implizite Poisson-Intensitaten abzuleiten.Da ein autoregressiver Prozess erster Ordnung die resultierenden Zeitrei-hen gut abbildet, sollte sein zeitstetiges Aquivalent, der Ornstein-Uhlen-
xiv Zusammenfassung
beck-Prozess, zur Darstellung der Dynamik der Poisson-Intensitaten ineinem Bewertungsmodell fur Katastrophen-Swaps gut geeignet sein.
Mit der Forschungsarbeit”A Traffic Light Approach to Solvency Mea-
surement of Swiss Occupational Pension Funds“ liegt der Fokus im drit-ten Teil der Dissertation auf der Solvenzmessung in der beruflichen Vor-sorge. Es wird ein Solvenztest fur Schweizer Vorsorgeeinrichtungen vor-geschlagen, welcher auf der Kombination eines stochastischen Pensions-kassenmodells mit einem Ampelsignalansatz basiert. Im Sinne eines regu-latorischen Standardmodells wird der Aufbau bewusst einfach gehalten.Zudem kann unter der Annahme normalverteilter Anlagerenditen eine ge-schlossene Losung abgeleitet werden. Trotz der relativ geringen Komple-xitat deckt das System die wesentlichen in der aufsichtlichen Praxis erfor-derlichen Risikoquellen ab. Aufgrund der einfach durchzufuhrenden Ka-librierung ist es zusatzlich gut zur Anwendung im fragmentierten Marktfur Schweizer Vorsorgeeinreichtungen geeignet und halt die Kosten derSolvenzregulierung so gering wie moglich. Zur Veranschaulichung derAnwendung des Modells wird es mittels einer kleinen Stichprobe vonzehn Vorsorgeeinrichtungen kalibriert und umgesetzt. Daruber hinauswird eine Sensitivitatsanalyse durchgefuhrt, um wichtige Einflussfakto-ren der Unterschreitungswahrscheinlichkeiten fur die Ampelbedingungenzu identifizieren.
Der vierte und letzte Teil der Dissertation schliesslich beinhaltetdie Forschungsarbeit
”Stock vs. Mutual Insurers: Who Does and Who
Should Charge More?“, welche eine empirische und theoretische Analy-se des Zusammenhangs zwischen den Pramien von Versicherern in derRechtsform der Aktiengesellschaft und des Versicherungsvereins auf Ge-genseitigkeit darstellt. Eine Auswertung von Paneldaten aus dem Bereichder deutschen Kfz-Haftpflichtversicherung liefert keinerlei Anhaltspunk-te dafur, dass Versicherungsvereine signifikant hohere Pramien berechnenals Aktiengesellschaften. Im Anschluss wird ein umfassender Modellrah-men fur die arbitragefreie Bewertung von Versicherungsvertragen ein-gesetzt, um Versicherer in Form von Aktiengesellschaften und Vereinenhinsichtlich der drei zentralen Grossen Pramienhohe, Sicherheitsniveauund Eigenkapital zu vergleichen. Obwohl es aus normativer Sicht be-stimmte Umstande gibt, in denen die Pramien einer Aktiengesellschaftund eines Versicherungsvereins auf Gegenseitigkeit gleich sein sollten,ware es hierfur erforderlich, dass der Verein vergleichsweise weniger Ka-pital vorhalt. Da dies den empirischen Ergebnissen widerspricht, scheinendie beobachteten Versicherungspreise nicht arbitragefrei zu sein.
1
Part I
Performance and Risks of
Open-End Life Settlement
Funds
Abstract
In this paper, we comprehensively analyze open-end funds dedicated to
investing in U.S. senior life settlements. We begin by explaining their
business model and the roles of institutions involved in the transactions
of such funds. Next, we conduct the first empirical analysis of life settle-
ment fund return distributions as well as a performance measurement,
including a comparison to other asset classes. Since the funds contained
in our dataset cover a large fraction of this relatively young segment
of the capital markets, representative conclusions can be derived. Even
though the empirical results suggest that life settlement funds offer at-
tractive returns paired with low volatility and are virtually uncorrelated
with other asset classes, we find latent risk factors such as liquidity,
longevity and valuation risks. Since these risks did generally not mate-
rialize in the past and are hence largely not reflected by the historical
data, they cannot be captured by classical performance measures. Thus,
caution is advised in order not to overestimate the performance of this
asset class.1
1Alexander Braun, Nadine Gatzert, and Hato Schmeiser (2009), Performance andRisks of Open-End Life Settlement Funds, Working Papers on Risk Management and
Insurance, No. 73. This paper has been accepted for publication in the Journal of
Risk and Insurance.
2 I Life Settlement Funds
1 Introduction
In the secondary market for life insurance, policyholders sell their con-
tracts to life settlement providers, which usually pass them on to in-
vestors or, in some cases, hold them on their own balance sheet. Such
transactions are termed ”life settlements”. The payment to the selling
policyholder is above the surrender value of the life insurance policy of-
fered by the primary insurer. The investor continues to pay premiums
until the contract is either resold or until it matures due to death or
reaching a fixed term and, in turn, receives the associated payoff. The
life settlement asset class, which emerged towards the end of the last
century, is not entirely new. Larger volumes of life insurance policies,
primarily those of terminally ill AIDS patients, had already been traded
in the so-called viatical settlements market of the 1980s. Most recently,
however, the asset class has begun to attract increasing attention from
the capital markets, since its return characteristics of low volatility and
virtually no correlation with other asset classes are appealing to a wide
range of investors. In addition, several Wall Street banks explore ways
to enter this business.
Since life settlements are a rather young asset class, literature on the
topic is still scarce and mainly practitioner-oriented. One of the early
analyses of the life settlement industry was provided by Giacalone (2001),
followed by Doherty and Singer (2002), who discuss benefits and welfare
gains arising from the secondary market for life insurance policies. Fur-
thermore, Kamath and Sledge (2005) review the characteristics of the
market for U.S. life settlements and the main drivers of its growth. While
Ingraham and Salani (2004), Freeman (2007) and Leimberg et al. (2008)
describe the decision making and due diligence process, McNealy and
Frith (2006) focus on the sourcing process for life settlements and point
out major product-flow constraints. In addition, Ziser (2006) and Smith
and Washington (2006) focus on transactional aspects, such as the di-
versification of life settlement portfolios in order to reduce risks. Seitel
(2006) and Seitel (2007) examine the industry from an institutional in-
vestor’s and a life settlement provider’s viewpoint, respectively. Other
1 Introduction 3
studies of market development, size, participants, regulatory environ-
ment, and future prospects include Moodys (2006), Conning & Company
(2007), and Ziser (2007). Moreover, a special report by Fitch Ratings
(2007) identifies selected risks associated with the market. Casey and
Sherman (2007) discuss whether life settlements should be regarded as
a security, Gatzert et al. (2009) analyze the effects of a secondary mar-
ket on the surrender profits of life insurance providers and Katt (2008)
discusses direct sales without intermediaries. Finally, Gatzert (2010) pro-
vides a comprehensive overview and discussion of benefits and risks of
the secondary markets for life insurance in the U.K., Germany, and
the U.S.
Apart from these publications, which focus on the market conditions
and their implications, the literature has presented other topics related
to life settlements. Regulation and tax aspects are reviewed by Doherty
and Singer (2003), Kohli (2006), and by Gardner et al. (2009). A study
by Deloitte (2005) features an actuarial analysis of the value generated
for the seller in a life settlement transaction. Russ (2005) examines the
quality of life expectancy estimates and Milliman Inc. (2008) offers in-
sights on mortality experience for two U.S. providers. Further publica-
tions include Zollars et al. (2003) and Mason and Singer (2008) who
address the valuation of life settlements. Perera and Reeves (2006) and
Stone and Zissu (2007) explore the sensitivity of life settlement returns to
life expectancy estimates and possibilities of risk mitigation, respectively.
Finally, Stone and Zissu (2006) as well as Ortiz et al. (2008) consider
the securitization of life settlements, a likely and natural future direc-
tion for the asset class when considering that the agencies A.M. Best
and DBRS have already provided their views on rating methodologies
for such transactions (see A.M. Best, 2009; DBRS, 2008).
To the best of our knowledge, no empirical analysis of investment
return characteristics and performance for the life settlement asset class
has been conducted in the literature yet. In addition, a comprehensive
analysis of its risks from an investor’s perspective is still missing. A ma-
jor reason for the lack of empirical work in this context is the scarcity
4 I Life Settlement Funds
of publicly available data on life settlement transactions. In the last few
years, however, a growing number of open-end funds exclusively dedi-
cated to investing in U.S. life settlements has emerged. These funds
determine their portfolio values on a monthly basis, thus providing the
possibility for a performance analysis based on time series data. Con-
sequently, in this paper, we contribute to the literature by conducting
the first empirical analysis of life settlement fund return distributions, a
general performance measurement, and a comparison to established as-
set classes. In addition, we put the empirical results into perspective by
extensively elaborating on the risks associated with open-end life settle-
ment funds. Our dataset has been provided by AA-Partners, a private
consulting firm specialized in U.S. life settlements, and is, in its entirety,
not publicly available. Since the dataset largely covers this segment
of the capital markets, we believe it to be a unique opportunity to gain
early insights into the return characteristics of this rather new asset class.
The remainder of the paper is structured as follows. In Section 2 we
give a brief overview of the secondary market for life insurance in the
U.S. and discuss key aspects of the structure and business model of life
settlement funds, which are essential to an understanding of their risk
profile. Section 3 is the empirical section, comprising the examination
of the funds’ return characteristics, the performance measurement and
the correlation analysis both on an aggregate level and for the individ-
ual funds in the dataset. A discussion of the risks associated with life
settlements in general and open-end life settlement funds in particular
is presented in Section 4. Finally, in Section 5 we conclude.
2 Market overview and fund business model 5
2 Life settlements: market overview
and fund business model
2.1 The U.S. life settlement market: an overview
Not many countries have a secondary market for life insurance policies,
because of the dependence on a sufficiently large primary market and
available target policies. The primary market in the U.S. represents the
largest life insurance market worldwide, making up 24.17 percent of the
global premium volume in 2007 (see SwissRe, 2007). In particular, ap-
proximately 160 million individual life insurance policies are currently
in force, with an aggregate face amount of more than 10 trillion USD
(see ACLI, 2007).2 According to the U.S. Individual Life Insurance Per-
sistency Study 2009 by the Life Insurance Marketing and Research As-
sociation (LIMRA) and the Society of Actuaries (SOA), this figure can
be broken down into 51.8 percent whole life, 23.9 percent term life, 14.5
percent universal life, and 9.8 percent variable universal life policies (see
LIMRA, 2009).
In the U.S. senior life settlement market, life insurance policies of in-
sureds above the age of 65 with below-average life expectancy—typically
2-12 years—and impaired health are purchased.3 Traded target policies
mainly include lifelong contracts with death benefit payment such as uni-
versal or whole life insurance, with universal life being by far the largest
segment.4 These contracts differ in their premium payment method,
which may be an important criterion with regard to the attractiveness
2For comparison, the U.S. equity market capitalization as of June 2010 is 12.4trillion USD (source: S&P), U.S. government bond notional outstanding as of Au-gust 2010 amounts to 8.4 trillion USD (source: U.S. Treasury), U.S. corporate bondnotional outstanding as of Q1/2010 is 7.2 trillion USD (source: Securities Industryand Financial Markets Association), global hedge fund assets under management asof Q2/2010 amount to 1.5 trillion USD (source: Credit Suisse Asset Management),and global commodity derivative notional outstanding as of December 2009 is 2.4trillion USD (source: Bank for International Settlements).
3This is in contrast to the life settlement markets in the U.K. or Germany, whereendowment contracts with a fixed maturity are traded. For an overview of the sec-ondary market for life insurance, see Gatzert (2010).
4In the partial study of Life Policy Dynamics LLC (LPD) (2007a,b), the share ofuniversal life among purchased policies is approximately 80-85 percent.
6 I Life Settlement Funds
for investors. While whole life contracts have constant level premiums,
universal life policies offer the possibility of flexible premium payments
as long as the cash value (policyholder’s reserve) remains positive. When
selling the policy to a life settlement provider, the policyholder receives a
payment that exceeds the surrender value but is less than the death ben-
efit. The provider determines the offer price by subtracting the present
value of expected future costs from the present value of expected future
benefits associated with the contract. The actual amount depends in
large part on the insureds estimated life expectancy. Thus, an impor-
tant yield driver from the investors perspective is the quality of the life
expectancy estimates provided by medical underwriters. Life settlement
providers commonly sell the policies on to investors who continue to pay
the premiums necessary to keep the policy in force and, in turn, receive
the death benefit (face value) when the insured person dies.5 Hence,
while the payment amount—the face value of the policy—is known when
a policy is purchased, the payment date is stochastic. The shorter the
insured lives after having sold the policy, the higher the return for the
investor, since only few premiums have to be paid and the death benefit
is received earlier.
In line with the large primary market, the U.S. life settlement market
has ample potential.6 In its Data Collection Report 2006, the Life In-
surance Settlement Association (LISA) published data it collected from
11 life settlement providers, which were estimated to represent about 50
percent of the industry. Those figures show that the annual death bene-
fits settled increased by around 65 percent from 3.9 billion USD in 2005
to more than 6.4 billion USD in 2007, and the number of settled policies
rose by 54 percent from 2,025 to 3,138 (see LISA, 2008). Other market
estimates include Conning & Company (2007) (5.5 billion USD in 2005;
6.1 billion USD in 2006) and Kamath and Sledge (2005) (total market
size: 13 billion USD in 2005).
5According to a special report by Moodys (2006), life settlements are primarilypurchased by institutional investors.
6The market volume is commonly reported in terms of the aggregated face valueof purchased policies.
2.2 Closed-end vs. open-end life settlement funds 7
2.2 Closed-end vs. open-end life settlement funds
The first life settlement funds appeared between 2002 and 2004, offer-
ing investors access to this relatively young asset class, which emerged
during the late 1990s in the aftermath of the fading market for viatical
settlements.7 For most investors, an investment through funds is sig-
nificantly more attractive and convenient than a direct purchase of the
underlying life insurance policies due to diversification benefits and the
reliance on professional expertise to determine the portfolio composition.
In addition, the complex acquisition process of a life insurance policy, in-
cluding legal requirements and transaction costs, are a major constraint
to direct investments.
Thus, the popularity of funds investing in U.S. life settlements has
grown continuously in recent years. During this period, two types of such
funds have evolved.8 Closed-end life settlement funds in the legal form of
limited partnerships with a fixed maturity strongly resemble structures
that are well-known from other illiquid asset classes such as investments
in real estate, aircraft or ships. In these cases, the fund management
company or a special-purpose subsidiary typically acts as the fund’s gen-
eral partner, while investors participate in the fund as limited partners.
The fund shares are therefore virtually an entrepreneurial equity holding
for which a premature redemption is not intended. These closed-end life
settlement funds are domiciled in the country of their primary investor
base, which are currently mainly Germany, the U.K., Ireland, and Lux-
embourg (see, e.g., Seitel, 2006; Moodys, 2006). They follow a classical
buy-and-hold investment style, generally do not use leverage and have
a rather moderate fee schedule, comparable to common mutual funds,
where the manager receives a fixed percentage of the assets under man-
7Viatical settlements are life insurance contracts of terminally ill policyholders,which are sold in the secondary market. The viaticals business surged during theAIDS epidemic in the late 1980s (see, e.g., Fitch Ratings, 2007.
8The information in this section is largely based on offering memorandums aswell as marketing material of a number of funds, which in some cases was publiclyavailable on their websites, whereas in other cases was received upon request. Webelieve that the typology we offer adequately captures the key characteristics andmain differences of these investment vehicles.
8 I Life Settlement Funds
agement.9 Another distinctive feature is the liquidity reserve most of
them build up from subscription payments in order to handle liquidity
risks arising from a lack of cash inflows after the final close of the fund.
Money returning from maturing policies is usually distributed to the
limited partners instead of reinvested. Closed-end life settlement funds
provide an annual report on their operations but refrain from delivering
portfolio valuations on a regular basis.
In contrast to their closed-end counterparts, open-end life-settlement
funds are perpetual and generally offer ongoing subscriptions and re-
demptions in either monthly or quarterly intervals. Liquidity from an
investor’s point of view is usually restricted by notice periods between
30 and 90 days, lock-ups of up to 3 years, and so-called gates: limits
on the amount which can be withdrawn in a given period. This type of
life settlement fund is almost exclusively domiciled in offshore banking
places and thus features a variety of legal forms consistent with local par-
ticularities. Active trading of the portfolio and leverage is possible and
the fee structure is hedge fund-like with management fees of one to two
percent and performance fees of up to twenty percent for which in some
cases hurdle rates and high water marks apply. The death benefit pro-
ceeds from matured policies are almost exclusively reinvested in order to
acquire new life settlement assets, whereas distributions to investors are
rather exceptional. Taking these characteristics into account, together
with targeted absolute returns of between eight and fifteen percent p.a.,
these funds have structural similarities to hedge funds.10 Open-end life
settlement funds provide valuations on a regular basis. Since the sec-
ondary market for life insurance policies is not as large and developed
as other capital markets, the underlying of life settlement funds is es-
sentially illiquid. Accordingly, a marking-to-market of their portfolios is
usually not possible and the need for mark-to-model valuation mecha-
nisms arises. On each valuation date, the funds employ their valuation
9A few exceptions exist with regard to these characteristics. Those resemble pri-vate equity funds, combining a limited partnership structure domiciled in an offshorebanking location with the possibility to actively trade policies as well as performancefees and leverage.
10Due to their specific underlying, however, life settlement funds are long-only.
2.2 Closed-end vs. open-end life settlement funds 9
Type Closed-end Open-end
Domicilecountry of primaryinvestor base
offshorebanking locations
Legal form limited partnerships depends on domicile
Regulationsubject tonational regulation
virtually unregulated
Maturity fixed perpetual
Subscriptions not after final closeongoing(usually monthly)
Redemptions at maturityongoing(monthly or quarterly)
Lock-Up period n/a up to 3 years
Notice period n/a 30 - 90 days
Redemption limits n/a common
Investment style buy-and-hold active trading possible
Leverage none possible
Fee schedulefixed percentageof capital
management/performancefees; hurdle rates; highwater marks
Liquidity reserve common not common
Death benefits distribution reinvestment
Valuations annual report on a monthly basis
Table 1: Closed-end vs. open-end life settlement funds
10 I Life Settlement Funds
methodology in order to determine the net asset value (NAV) of their
portfolio, i.e., the value of their assets less the value of their liabilities,
which then forms the basis for subscriptions and redemptions of fund
shares.11 As a result, time series of monthly NAVs for open-end life
settlement funds exist and can be used to conduct an empirical per-
formance analysis. Table 1 summarizes the main structural differences
between closed-end and open-end life settlement funds.
Although closed-end and open-end life settlement funds are currently
quite common, it is uncertain whether both of these formats will prevail
throughout the next decade. From their emergence until they become
established, asset classes usually traverse an evolutionary process with
regard to their wrapping, beginning with rather illiquid structures such
as closed-end funds and successively migrating to more liquid ones as
the market grows larger, more transparent, and increasingly standard-
ized. The advent of derivatives as well as securitization are commonly
seen as indications of a maturing asset class. Against this background,
industry experts expect open-end funds to dominate the life settlement
market in the future. Early signs of this development are already be-
coming apparent: there are a number of initiatives to promote standard-
ization, transparency and the diffusion of information pertaining to life
settlement transactions. One example is the Institutional Life Markets
Association (ILMA), which was founded by institutional investors such
as Credit Suisse, Goldman Sachs, and Mizuho International PLC.12 Fur-
thermore, according to AA-Partners, half a dozen new open-end funds
are currently being prepared for launch. In contrast to that, there seems
to be humble activity in the closed-end segment.
11Funds can either apply the ”investment method” or the ”fair value method” forthe ongoing valuation of their life insurance policies. The choice needs to be made onan instrument-by-instrument basis and is binding for the entire term of the contract.These methods will be described in further detail in Section 4.2.
12For more information refer to www.lifemarketsassociation.org.
2.3 The anatomy of open-end life settlement funds 11
2.3 The anatomy of open-end life settlement funds
To interpret empirical results for open-end life settlement funds and ana-
lyze their risk profile, it is of critical relevance that one first understands
their mechanics. To the best of our knowledge, neither the structure nor
the business model of open-end life settlement funds has been compre-
hensively described before. Consequently, the remainder of this section
explains how these funds operate. For the sake of clarity it is organized
based on the roles of the various involved parties. A stylized representa-
tion of an open-end life settlement fund is depicted in Figure 1.
As any other collective investment scheme, life settlement funds de-
pend on so-called trustees, i.e., certain institutions which hold their prop-
erty and facilitate their transactions. Hence, before entering business,
the fund management company needs to appoint a custodian (deposi-
tary) in its country of domicile. The primary function of the custodian
is to hold the fund’s assets. In general, the custodian administers any liq-
uid assets, such as government bonds or cash and assigns the safekeeping
of life settlements to a sub-custodian in the United States. Furthermore,
the custodian is responsible for the administration of the fund shares
(units), for receiving and holding application money, and for redistribut-
ing funds to investors in the course of redemptions.
Whenever life insurance policies are acquired, the custodian transfers
the necessary amount of money to the sub-custodian, which, in turn,
uses it to settle transactions. With regard to policy purchases, the sub-
custodian also serves as an escrow agent, facilitating the acquisition by
retaining the payment for the respective life settlement in an escrow
account while the transfer documents are sent to the insurance company
in order to change ownership rights and beneficiaries. Once the amended
documents have been returned by the insurance company, the money is
released to the seller. The original life insurance contracts as well as
the transfer and assignment documents are subsequently held by the
sub-custodian on behalf of the fund. Whenever due, regular premiums
are paid by the sub-custodian. In some cases, the sub-custodian is also
12
ILifeSettlementFunds
Secondary market regulatory environment and U.S. tax legislation
Insureds
Life settlement fund(open-end)
Liquid assets*
Premium reserve
Other third party services
Life settlementportfolio
Medicalunderwriters
Original
beneficiaries
Servicer
Lifesettlementproviders**
Lifeinsurancecompanies Auditor Actuarial Legal
advisors advisors
Investors’funds
Sub-custodian
(Escrow agent)
Custodian
(Depositary)
Investors
Banks
Leverage
Liquidity
* Note that in case the fund retains any liquid assets such as government bonds or cash, those are usually held by the custodian.** Some policy sellers are represented by life settlement brokers, who negotiate with several life settlement providers to obtain the best offer.
Figure 1: Stylized mechanics of an open-end life settlement fund
2.3 The anatomy of open-end life settlement funds 13
responsible for building up a premium reserve account for the fund in
order to be able to mitigate potential liquidity shortages.
Medical underwriters review the medical records of the insureds and,
based on the information contained therein, prepare mortality profiles
that comprise a summary of the medical conditions, a mortality sched-
ule, and an estimation of the life expectancy for each insured.13 For this
purpose, they assess how certain characteristics and medical conditions
affect the insureds mortality relative to a ”standard” or reference mor-
tality (see A.M. Best, 2009). The outcome is a specific multiplier (also
called mortality rating) which modifies the reference mortality. Method-
ologies for the derivation of the multiplier as well as standard mortality
tables depend on the medical underwriters. However, within the last
few years, many medical underwriters have opted for the Valuation Ba-
sic Tables (VBT), which are prepared by a task force of the Society of
Actuaries (SOA).14 These tables include mortality rates for ages up to
ninety years over time horizons from one to twenty-five years, which
have been derived from historical data and are differentiated according
to simple characteristics such as smoking status and gender.15 Although
life expectancy estimates have systematically increased over the last few
years, the figures provided by different medical underwriters for the same
lives can vary substantially, implying a potential for misestimation (see
A.M. Best, 2009; Gatzert, 2010). This has implications both for the
pricing of life settlements and for the fund returns. Consequently, some
funds seek to mitigate the impact of misestimation by demanding at
least two life expectancy estimates and then applying the longer one or
a (weighted) average of the two.
The servicer (tracking agent) performs a wide variety of supporting
services in the context of premium and claims administration for the pool
13The four largest medical underwriters in the U.S. life settlement market are 21stServices, AVS, EMSI, and Fasano (see Russ, 2005; Gatzert, 2010).
14See www.soa.org.15Mortality rates are commonly denoted by qx (where x stands for the current
age of the group under consideration) and measure the number of deaths per 1000individuals of a population in a certain time period (typically one year).
14 I Life Settlement Funds
of lives in the portfolio.16 Its goal is to ensure a smooth and on-time
transfer of legal paperwork, notifications, and cash flows. The servicer
notifies the trustees and provides them with disbursement instructions
for the regular premium payments and maintains close contact with the
insurance company to obtain the latest information on developments of
each policy (e.g., cash surrender values). Moreover, it is responsible for
ordering the policyholder’s medical records and life expectancy estima-
tions from the medical examiners and then archiving them. Another key
duty is the tracking of the insured, i.e., the maintenance of registers with
their contact details as well as the verification of their life/death status.
For this purpose, the servicer relies on routines which resemble those em-
ployed in consumer loan servicing such as subscribed database services,
mailings, and telephone calls. In addition, it matches social security
numbers to death indices on a regular basis. Whenever the servicer be-
comes aware of the death of a policyholder, it immediately informs the
fund manager and the trustees and obtains the death certificate. After
the signed insurance claim package has been provided by the trustee, the
servicer forwards it to the insurance company and follows up until the
claim is paid so as to facilitate the prompt collection of death benefits.
Life settlement providers (life settlement companies) source life in-
surance contracts from policyholders or licensed brokers in order to pass
them on to the funds. For this purpose, the funds usually set certain
investment criteria, which reflect the cornerstones of their portfolio di-
versification approach. Life settlement providers can also act as invest-
ment advisors, pitching life settlements to the manager and participat-
ing in the policy-picking and portfolio structuring process. Whereas
some funds rely on a so-called single-source approach, thus exclusively
collaborating with just one life settlement company, others deliberately
maintain business relationships with several. Such a multi-source ap-
proach is meant to improve the funds’ access to life settlement assets,
especially in times of greater product-flow constraints or less active mar-
kets (see, e.g., McNealy and Frith, 2006). An important aspect to be
16Note that the average fund portfolio in our dataset comprises 193 lives, while themaximum number of lives in a portfolio is 567 (see Table 2 in Section 3.1).
2.3 The anatomy of open-end life settlement funds 15
considered with regard to life settlement providers are their incentives
to act in the interest of investors. Since their fees are paid upfront and
generally depend on number and volume of the policies rather than their
long-term investment performance, the degree of diligence that can be
expected from life settlement providers during the acquisition process
is questionable.17 More specifically, to increase their chance of prevail-
ing in the competitive bidding process for a policy they could, e.g., be
tempted to avoid medical underwriters which issue rather conservative
life expectancy estimates, since those would be associated with a lower
offer price. Once acquired, the policy is then resold by the life settlement
provider to the fund whose investors ultimately have to bear the risk of
a misestimated life expectancy.
In addition, the general mechanics of open-end life settlement funds
are usually complemented by third-party service providers. Auditors ad-
vise on accounting and tax implications, inspect the funds’ balance sheets
and income statements, and issue annual reports with their opinion of
the funds’ financial situation. Moreover, actuarial advisors assist with
the pricing of transactions as well as the valuation of life settlements in
the portfolio and review actuarial models used by the funds. Similarly,
legal advisors offer counseling with regard to the legal form, draft all
the contracts, and ensure the completeness of documentation packages
in addition to compliance with the applicable legislation and regulation.
Banks are involved either by providing medium- to longer-term debt fi-
nancing, which some funds use to leverage their investments, or through
a liquidity facility, which is commonly employed to bridge life settlement
purchases or premium payments in the absence of other cash inflows.
Finally, life insurance companies originally issued the policies and must
be notified about the transfer of ownership. They continue to receive
the premiums after the sale has been completed and pay out the death
benefits to the fund’s sub-custodian after the insured has passed away.
17This is quite similar to the incentive problem that ultimately led to the demiseof the U.S. subprime market where the common practice of instantly selling-on ini-tiated mortgages to third parties, such as investment banks (originate-to-distribute),created a lack of long-term financial incentives and instigated originators to an ex-treme relaxation of lending standards.
16 I Life Settlement Funds
3 Empirical analysis
3.1 Data and sample selection
We obtained our data on open-end life settlement funds from AA-Partners
AG, a Zurich-based investment boutique specialized in this asset class.18
Analogously to providers of hedge fund data, AA-Partners maintains an
extensive network in the life settlement industry, through which it is
in a position to collect performance data directly from fund managers.
Using a variety of sources, it carries out regular cross-checks and veri-
fications of its fund database to ensure correct classification, reliability
and representativeness. The original dataset comprises monthly NAVs of
17 open-end funds, which, according to AA-Partners, largely cover this
market.19 Each fund is USD denominated, subject to an independent
audit conforming to international standards and almost all are purely
dedicated to investing in U.S. senior life settlements, i.e., mixed strategy
funds are excluded.20 In our view, this dataset is a valid opportunity for
an empirical analysis, as we are not aware of any other sources of such
comprehensive time series data for life settlement funds.
Table 2 provides additional information with regard to inception, size,
fee structure, and liquidity profile of the fund shares.21 While the oldest
fund in the dataset began operations in late 2003, other funds emerged
just as recently as 2007/2008. This suggests that the asset class has
gradually attracted the attention of the investment industry throughout
the last decade. Interestingly, the funds are quite different in size as
reflected by investment volumes, number of policies in the portfolio, and
the sum of face values. This can be an important factor with regard to
potential policy availability issues which will be discussed in Section 4.5.
18AA-Partners acts as an independent third party advisor with regard to investmentsolutions for U.S. life settlements. Its main services include investment advice relatedto open-end funds, valuation of life settlement portfolios, market research, and datacollection. See www.aa-partners.ch for more information.
19The dataset consists of single funds. To our knowledge, life settlement fund offunds do currently not exist.
20However, one fund has a minor position in U.K. endowment policies and anotherone holds a small fraction of viatical settlements.
21For confidentiality reasons the fund names have been substituted with numbers.
3.1 Data and sample selection 17
While there is some variation in the fee structures, most funds seem to
charge a management fee of around two percent and a performance fee
of around twenty percent. The majority of life settlement funds in the
dataset offers subscriptions and redemptions on a monthly basis with a
notice period of 30 days. Furthermore, several funds partially protect
themselves against excessive cash outflows by imposing redemption gates
and lock-up periods on their investors.
Since the market is still in an early stage of its development, not all of
the funds feature time series of sufficient length for statistical inference.
To capture the risks and returns of open-end life settlement funds as
comprehensively as possible, we have created a custom index, beginning
with the oldest fund, which appeared in December 2003. Whenever the
inception date of another fund is reached, it is added to the index and
whenever the return time series of a fund ceases prematurely (e.g., due to
suspended reporting or liquidation), it drops out of the index. The index
time series ends in June 2010 and comprises 79 monthly returns. At any
point in time, the returns of all index constituents are equally weighted.22
A further analysis of the individual funds will be conducted in Section
3.4. In addition to the custom life settlement index, we have selected
broad indices as representatives for various other asset classes in order to
conduct performance comparisons and correlations analyses.23 In this
context, the U.S. stock market is represented by the S&P 500 while
the FTSE U.S. Government Bond Index as well as the DJ Corporate
Bond Index have been selected as proxies for the respective bond mar-
kets. Furthermore, the HFRI Fund Weighted Composite Index serves as
a broad measure for the hedge fund universe, while real estate returns
are provided through the S&P/Case-Shiller Home Price Index (Compos-
22Note that this approach of calculating the index assumes an investor with a naıvediversification approach, assigning the same target portfolio weight to all availablelife settlement funds at any point in time. We believe this procedure to be an ade-quate way of reflecting the development of an open-end life settlement fund portfoliobetween 2003 and 2010.
23Indices are sufficiently diversified portfolios. Thus, an analysis based on indicesis well suited to examine the risk-return profile of aggregate asset classes.
18 I Life Settlement Funds
ite of 20).24 Finally, the S&P GSCI, a recognized measure of general
commodity price movements, is used as indicator for the global com-
modity markets. The selection is completed by the S&P Listed Private
Equity Index. Since congruent time series are required for our analysis,
the scarcity of available life settlement fund data constrains the choice
of time period and return interval for the other asset classes. Hence,
monthly index returns from December 2003 to June 2010 have been col-
lected for those as well.25 Wherever available, total return indices have
been used to account for coupons and dividends, which would otherwise
not be reflected in prices. Table 3 summarizes the sample characteristics.
As with hedge fund data, our sample suffers from certain biases,
which have to be considered when interpreting the empirical results in
the following section.26 Self-selection bias arises from the rather opaque
nature of the funds which, in contrast to mutual funds, are not obliged to
disclose return data to the public. This bias is likely to be particularly
large if non-reporting funds significantly underperform their reporting
counterparts. However, we are aware of 17 funds that essentially make
up the market. Of these 17 funds, only two suspended their reporting
during the time period under consideration. Hence, we consider this bias
not to be material.27 In addition, survivorship bias arises when funds,
which ceased to exist, are not included in a database. If these funds ter-
minated operations as a result of poor performance, the available data
is likely to overstate historical returns and understate risk. AA-Partners
knows of three funds, which were shut down, but have never been part
of their database.28 Apart from that, two of the 17 funds in our dataset
24We deliberately chose the S&P/Case-Shiller Index instead of publicly listed RealEstate Investment Trust (REIT) indices, since the latter are strongly influenced bygeneral stock market dynamics and due to this noisiness only partly reflect the per-formance of the true underlying real estate assets. This phenomenon with regard toREITs has been described by Giliberto (1993) and Ling et al. (2000).
25The data has been downloaded from the Bloomberg database.26For a more detailed discussion of these biases, see L’Habitant (2007).27Self-selection bias cannot be quantified as the returns for non-reporting funds
remain unobservable.28Due to the over-the-counter character of the market for open-end life settlement
funds, data collection is a very challenging and time consuming task. Even insti-tutions with extensive connections into the life settlement industry, such as AA-Partners, are unable to obtain return data in certain cases.
3.1
Data
and
sam
ple
selection19
Fund 100 Fund 101 Fund 102 Fund 103 Fund 104 Fund 105
Currency USD USD USD USD USD USD
Inception Apr 03 June 06 Aug 03 June 07 Nov 05 Dec 03
Volume (mn) 385 950 428 102 466 62
Sum of facevalues (mn)
770 2367 720 362 619 108
No. of policies 261 567 447 183 406 65
Management fee 0.75% 1.95% 2.00% 2.00% 1.50% 1.50%
Performance fee n/a 20.00% n/a 20% 75% n/a
Hurdle Rate n/a 10.00% n/a 9.00% 8.00% n/a
Style passive passive passive passive passive passive
Subscriptions monthly monthly monthly monthly monthly monthly
Redemptions monthly monthly monthly monthly monthly monthly
Notice Period 30 days 30 days 30 days 90 days 45 days 30 days
Redemption fees n/a
year 2: 4%year 3: 4%year 4: 3%year 5: 3%
10%,decreasingby 0.33%per month
year 2: 8%year 3: 7%year 4: 4%nil after
5%,decreasingby 1%per year
year 1: 7%year 2: 5.5%year 3: 4%year 4: 2.5%year 5: 1%
Lock-up Period n/a 1 year n/a 1 year n/a n/a
Redemptionlimits (gates)
10% ofoutstandingshares p.a.
n/a20% ofoutstandingshares p.a.
30(60)% ofinvestmentin year 2(3)
10% of sharesper redemptiondate
10% of sharesper redemptiondate
Table 2: Life settlement funds in the original dataset
20
ILifeSettlementFunds
Fund 201 Fund 202 Fund 203 Fund 204 Fund 205 Fund 208
Currency USD USD USD USD USD USD
Inception Jan 05 July 06 Feb 05 March 04 Dec 04 Nov 06
Volume (mn) 31 494 unknown unknown 10 57
Sum of facevalues (mn)
98 unknown unknown unknown 45 283
No. of policies 126 unknown unknown unknown 40 113
Management fee 1.50% 2% 1.75% 1.75% 1.50% 1.25%
Performance fee n/a n/a 25% 20% 20% 15%
Hurdle Rate n/a n/a 8% 6% n/a 7%
Style passive passive passive passive passive passive
Subscriptions weekly monthly monthly monthly monthly monthly
Redemptions weekly monthly Monthly monthly quarterly monthly
Notice Period 30 days 30 days 30 days 30 days 60 days 90 days
Redemption fees
5%,decreasingby 1%per year
7% untilyear 8,4% after
deferredsales charge(5 years)
year 1: 8.6%year 2: 8.6%year 3: 7.5%year 4: 6%year 5: 5%
3%
17.5%,decreasingby 2.5%per year
Lock-up Period n/a 3 years 1 year 6 months n/a n/a
Redemptionlimits (gates)
n/a 20% p.a. 20% p.a. 10% p.a. n/a 20% p.a.
Table 2: Life settlement funds in the original dataset - continued
3.1
Data
and
sam
ple
selection21
Fund 210 Fund 212 Fund 216 Fund 217 Fund 514
Currency USD USD USD USD USD
Inception July 04 Dec 07 Jan 07 Jan 08 June 06
Volume (mn) 100 8 43 5 178
Sum of facevalues (mn)
178 20 130 20 344
No. of policies 242 17 58 8 179
Management fee 0.30% 1.25% 2.00% 2.00% 0%
Performance fee n/a 10% 20% 20% 30%
Hurdle Rate n/a 10% 8% n/a 6.5%
Style passive passive active passive passive
Subscriptions monthly monthly monthly monthly monthly
Redemptions monthly monthly quarterly quarterly monthly
Notice Period 30 days 30 days 90 days 90 days 90 days
Redemption fees
year 1: 3%year 2: 2%year 3: 1%nil after
n/a2% afterfirst year
n/a8%, decreasingby 1.6%per year
Lock-up Period n/a n/a 1 year 1 year n/a
Redemptionlimits (gates)
n/a 5% p.a.10% oftotal assets p.a.
10% oftotal assets p.a.
20% p.a.
Table 2: Life settlement funds in the original dataset - continued
22 I Life Settlement Funds
Observed variables8 indices(see Appendix for additional information)
Selection criterionBroad market indices, i.e., diversified portfolios(as representative as possible for each asset class)
Return interval monthly returns
Sample period12/2003–06/2010(79 data points)
Source of dataAA-Partners AG for life settlement fundsBloomberg for indices of other asset classes
Table 3: Sample details
are currently being liquidated and the final proceeds to investors are
unknown at this time. According to AA-Partners, those liquidation pro-
ceeds can be expected to be considerably smaller than the last NAV
published by the funds. These considerations imply that survivorship
bias could, to some extent, be an issue in the context of our empirical
analysis. Since the return time series of those funds which were not in-
cluded in the database as well as the liquidation proceeds for the two
terminated funds are not available to us, it is not possible to measure and
consequently explicitly control for survivorship bias. Finally, illiquidity
bias is an issue with regard to life settlement funds. Life settlements are
highly illiquid assets. Thus, a marking-to-market is difficult due to the
absence of regularly quoted market prices. Accordingly, the fund man-
agers have considerable flexibility when determining NAVs, which they
could use to smooth monthly returns. This bias is of major importance
as we will see in the detailed risk analysis of the funds in Section 4.
3.2 The return distribution
of open-end life settlement funds
Based on the fact that the main underlying risks are biometric in nature
rather than originating from the broader capital markets, academics and
practitioners have repeatedly emphasized that life settlements should
offer attractive returns paired with a conservative risk profile and are
3.2 The return distribution of open-end life settlement funds 23
uncorrelated with other asset classes (see, e.g., Stone and Zissu, 2007).
In order to verify this, we conduct the first empirical analysis of this
asset class.29 We begin with a characterization of the empirical return
distributions, which forms the basis for subsequent comparisons. Figure
2 plots the performance of all previously mentioned asset classes, ex-
cept commodities and private equity, between December 2003 and June
2010.30 All time series have been indexed to 100 at the beginning of the
period under consideration, thus reflecting the development of the value
of a hypothetical investment of 100 USD over time.
At a first glance, the graph of open-end life settlement funds looks
excellent. It dominates both bond indices at almost every point in time
and has only been exceeded by stocks and real estate until the subprime
crisis in the U.S. struck in summer 2007 and spread into the global capital
markets in 2008. Over the whole period, only a hedge fund investment
would have yielded a higher value. These observations are also reflected
in the figures characterizing the return distribution, which can be found
Table 4. The portfolio of life settlement funds represented by our cus-
tom index exhibits generally respectable positive returns and very low
volatility. Furthermore, it has only suffered a comparatively moderate
drawdown31 during the financial crisis of 2007 to 2009. With the substan-
tial quantity of 37.30 percent, life settlement funds generated the third
highest total return of all analyzed asset classes from December 2003
to June 2010. Only hedge funds (45.90 percent) and government bonds
(37.38 percent) provided higher total returns over this period. Apart
from corporate bonds, which yielded a mere 2.00 percent, the remaining
asset classes even exhibited negative total returns. An investment in
stocks, for example, would have lost 2.60 percent of its original value.
29To the best of our knowledge, the scarcity of NAV data did not allow for anyearlier empirical analysis.
30The S&P GSCI as well as the S&P Listed Private Equity Index with their com-paratively high volatility have been excluded from this figure in order to enhance thereadability. Please refer to Table 4 for the respective data.
31That is, a loss incurred over a certain time period.
24 I Life Settlement Funds
Studying the means of the monthly return distributions reveals a
similar pattern. With 0.40 percent (4.85 percent p.a.), open-end life
settlement funds had a higher mean return than all other asset classes
except for hedge funds (0.50 percent; 5.98 percent p.a.) and government
bonds (0.41 percent; 4.91 percent p.a.). While private equity (0.37 per-
cent; 4.44 percent p.a.) and commodities (0.25 percent, 2.95 percent p.a.)
also exhibited positive mean returns over the period under consideration,
those of the remaining asset classes were close to zero. Moreover, life set-
tlement funds were by far the least volatile investment, as represented by
their return standard deviation of 0.66 percent (2.28 percent p.a.). Even
government bond returns with a standard deviation of 1.10 percent (3.80
percent p.a.) were almost twice as volatile, let alone stocks, commodi-
ties and private equity, where the multiplier is more than six, eleven,
and thirteen, respectively. Maximum and minimum returns are furthest
apart for the asset classes with the highest volatilities, i.e., private equity,
commodities, and stocks, while the empirical return distribution for life
settlements merely spans 5.94 percent from a maximum of 2.79 percent
to a minimum of -3.15 percent.
The remarkable impression provided by the portfolio of life settlement
funds is further bolstered by taking into account the small number of
negative returns: only 9 during the whole examination period of 79
months (see row 11 of Table 4). All remaining asset classes experienced
many more negative months, ranging from 26 to 33. However, the life
settlement fund return distribution exhibits the comparatively largest
negative skewness (-1.97) and positive excess kurtosis (12.66), implying
a long and heavy left tail. These values for the third and fourth moments
lead to an exceptionally high Jarque-Bera test statistic (578.55), meaning
the null hypothesis of normality has to be rejected on all reasonable
significance levels.32
32Although for almost all other asset classes, the null hypothesis under the Jarque-Bera test is rejected on the one percent significance level as well, their test statisticsare considerably smaller.
3.2
The
return
distrib
ution
ofop
en-en
dlife
settlemen
tfu
nds
25
6080
100
120
140
160
time
index
lev
el
2004 2005 2006 2007 2008 2009 2010
Life Settlement Fund IndexS&P 500FTSE U.S. Government Bond IndexDJ U.S. Corporate Bond IndexHFRI Fund Weighted Composite IndexS&P/Case−Shiller Home Price Index
Figure 2: Life settlements in comparison to other asset classes (12/2003–06/2010)
26 I Life Settlement Funds
3.3 Performance measurement
and correlation analysis
To elaborate on the risk return profile of open-end life settlement funds,
we apply four common performance measures.33 Apart from the proba-
bly most classic performance measure in finance literature, the Sharpe
Ratio, we calculate the Sortino Ratio, the Calmar Ratio and the Excess
Return on Value at Risk (VaR) for the asset classes under consider-
ation.34 Based on these indicators, we establish a rank order for all
asset classes with positive excess returns.35 The results are displayed
in the lower part of Table 4. With a Sharpe Ratio of 0.3327, life settle-
ment funds clearly rank first with a considerable distance to the second-
ranked government bonds (0.2039). Hedge funds (0.1589), private equity
(0.0211), and commodities (0.0079) on ranks 3, 4, and 5 also feature a
positive Sharpe Ratio, which, however, in the latter case is close to zero.
Negative Sharpe Ratios for the remaining investment alternatives reflect
their poor performance over the analyzed time period, falling short of a
possible investment at the risk-free rate. Looking at their Sortino Ra-
tio of 0.4580 and Excess Return on VaR of 0.2889, we gather the same
picture: life settlement funds outperformed the runners-up government
bonds and hedge funds by far.36 The performance ranking based on the
Calmar Ratio is a slight exception. With a value of 0.0695, the life settle-
ment fund index ends up on the second position, just behind government
bonds (Calmar Ratio of 0.0813).
Certainly, the choice of the time period for the analysis—including
the financial crisis 2008/2009—negatively influences the image of almost
all established asset classes. Nevertheless, two important factors should
33The definitions for these performance measures can be found in the Appendix.34The average 1-month U.S. Treasury Bill rate between December 2003 and June
2010 has been used as a proxy for the risk-free interest rate rf . The figures can beaccessed on www.ustreas.gov. With regard to the Sortino Ratio, we choose rf as thethreshold return τ . The Excess Return on VaR is based on the 95 percent VaR.
35The applied performance measures are not meaningful for negative excess returnssince, in that case, a higher value of the risk measure in the denominator leads to abetter result (less negative ratio).
36Our results are in line with the findings of Eling and Schuhmacher (2007) in thatthe different performance measures lead to almost the same rank order.
3.3
Perfo
rmance
measu
remen
tan
dcorrelation
analy
sis27
LSFI S&P 500 FTSEUSGBI
DJ CBI HFRIFWI
S&P/CSHPI
S&PGSCI
S&PLPEI
Total return 37.30% -2.60% 37.38% 2.00% 45.90% -0.84% -4.97% -1.90%
Mean return 0.40% 0.07% 0.41% 0.04% 0.50% 0.00% 0.25% 0.37%
annualized 4.85% 0.78% 4.91% 0.53% 5.98% -0.03% 2.95% 4.44%
Standard deviation 0.66% 4.39% 1.10% 1.95% 1.97% 1.27% 7.77% 8.74%
annualized 2.28% 15.20% 3.80% 6.75% 6.84% 4.40% 26.91% 30.26%
Maximum 2.79% 9.39% 3.24% 7.63% 5.15% 1.99% 19.67% 30.54%
Minimum -3.15% -16.94% -2.75% -6.43% -6.84% -2.79% -28.20% -30.33%
Skewness -1.97 -1.08 -0.21 0.11 -1.11 -0.50 -0.64 -0.43
Excess kurtosis 12.66 2.48 0.75 4.05 2.65 -0.61 1.41 3.91
Jarque-Bera test 578.55 35.74 2.45 54.18 39.41 4.58 11.92 52.83
*** *** - *** *** - *** ***
Negative months 9 30 26 33 26 39 33 27
SHR (rank) 0.33 (1) -0.03 0.20 (2) -0.07 0.16 (3) -0.15 0.01 (5) 0.02 (4)
SOR (rank) 0.46 (1) -0.03 0.33 (2) -0.10 0.22 (3) -0.18 0.01 (5) 0.03 (4)
CAR (rank) 0.07 (2) -0.01 0.08 (1) -0.02 0.05 (3) -0.07 0.00 (5) 0.01 (4)
ERVaR (rank) 0.29 (1) -0.01 0.20 (2) -0.06 0.12 (3) -0.08 0.01 (5) 0.01 (4)
Indices: Life settlement funds index (LSFI); Standard & Poor’s 500 (S&P 500); FTSE U.S. Government Bond Index (FTSE USGBI);Dow Jones Corporate Bond Index (DJ CBI); HFRI Fund Weighted Composite Index (HFRI FWI); Standard & Poor’s/Case-ShillerHome Price Index (S&P/CS HPI); Standard & Poor’s Goldman Sachs Commodities Index (S&P GSCI); Standard & Poor’s ListedPrivate Equity Index (S&P LPEI). Significance levels: *** = 1%, ** = 5%, * =10%. Performance measures: Sharpe Ratio (SHR);Sortino Ratio (SOR); Calmar Ratio (CAR); Excess Return on VaR (ERVaR).
Table 4: Descriptive statistics: index return distributions (12/2003–06/2010)
28 I Life Settlement Funds
be considered. First, as mentioned in Section 3.1, the choice of time
period was not arbitrary but determined by the availability of data for
the life settlement fund market. Second, the rather weak performance of
some of the indices representing the other asset classes under considera-
tion underscores even more strongly how extraordinary these empirical
observations for life settlements are. This finding should trigger addi-
tional questions as to why this asset class has seemingly been able to
withstand the major dislocations in the world’s capital markets.
Finally, to complete the empirical analysis on the portfolio basis, we
examine the correlation structure between life settlement funds and the
other indices in our sample. Table 5 displays the correlation matrix as
well as the significance levels for the correlation t-test. Only one of the
tested Bravais-Pearson correlation coefficients between the returns on the
custom life settlement fund index and the other indices turned out to be
statistically significant. In particular, life settlement fund and corporate
bond returns seemed to be negatively correlated during our examination
period. Overall, it appears as if life settlements rightly have the repu-
tation of being virtually uncorrelated with other asset classes.37 To put
further emphasis on this result, we provided the correlation coefficients
among the remaining asset classes. Apart from two exceptions involving
corporate bonds, those are all significantly different from zero. Particu-
larly, all correlations of the HFRI Fund Weighted Composite Index with
the traditional asset classes are highly significant, raising doubts about
the suitability of hedge funds as a means for portfolio diversification. Life
settlement funds, on the contrary, seem to offer excellent diversification
qualities.
3.4 Analysis of individual funds
Due to the extraordinary performance of the life settlement fund index
revealed in the previous section, we deem it necessary to conduct further
analyses on a disaggregate level. Thus, we examine return distributions
and performance for the individual life settlement funds in the sample.
37To be more precise, we cannot reject the null hypothesis that life settlementreturns are uncorrelated with the returns of the other asset classes.
3.4
Analy
sisofin
div
idual
funds
29
LSFI S&P500
FTSEUSGBI
DJ CBI HFRIFWI
S&P/CSHPI
S&PGSCI
S&PLPEI
(I) (II) (III) (IV) (V) (VI) (VII) (VIII)
(I) 1.0000 -0.1231 -0.0414 -0.2683 ** -0.0606 -0.1679 -0.0292 -0.0834
(II) 1.0000 -0.2575 ** 0.3487 *** 0.8015 *** 0.2982 ** 0.3877 *** 0.8779 ***
(III) 1.0000 0.3639 *** -0.3795 *** -0.2681 ** -0.2361 * -0.2228 *
(IV) 1.0000 0.3603 *** 0.0427 0.1057 0.2794 **
(V) 1.0000 0.2712 ** 0.5866 *** 0.7665 ***
(VI) 1.0000 0.2442 * 0.3149 ***
(VII) 1.0000 0.4124 ***
(VIII) 1.0000
Indices: Life settlement funds index (LSFI); Standard & Poor’s 500 (S&P 500); FTSE U.S. Government Bond Index (FTSE USGBI);Dow Jones Corporate Bond Index (DJ CBI); HFRI Fund Weighted Composite Index (HFRI FWI); Standard & Poor’s/Case-ShillerHome Price Index (S&P/CS HPI); Standard & Poor’s Goldman Sachs Commodities Index (S&P GSCI); Standard & Poor’s ListedPrivate Equity Index (S&P LPEI). Significance levels: *** = 1%, ** = 5%, * =10%.
Table 5: Correlation Matrix
30
ILifeSettlementFunds
7080
9010
011
012
013
0
time
fund lev
el
2007 2008 2009 2010
Fund 101Fund 204Fund 205
Figure 3: Individual life settlement funds in comparison (01/2007–06/2010)
3.4 Analysis of individual funds 31
To ensure congruent time series, we selected the period from January
2007 until June 2010.38 This enables us to include as many funds from
the original dataset as possible, while still retaining a total of 42 monthly
returns in the time series. As a consequence, we removed three funds,
which did not yet exist in January 2007. Also note that due to vari-
ous reasons (see fund status in Table 6) the time series for some of the
remaining 14 funds stop before June 2010. The results for each fund
are reported in Table 6, Table 7 provides some summary statistics, and
Figure 3 displays the development of an investment of 100 USD in each
of the life settlement funds over the considered time period. While we
observe a solid growth in value for most funds, there are some exceptions
that differ from the pack.
In particular, we notice that Fund 202 and Fund 216 experienced a
comparatively larger number of negative months and Fund 101, Fund
204, as well as Fund 205 exhibited a large drawdown. The magnitude of
this remarkable negative monthly return is -18.97 percent, -16.68 percent,
and an enormous -51.12 percent for Fund 101, Fund 204, and Fund 205,
respectively. As a result, the return volatilities (standard deviations)
of 3.05 percent, 3.48 percent, and 9.36 percent for these three funds
are much higher than the average of 1.41 percent and the variation in
maximum and minimum returns as well as skewness and excess kurtosis
across all individual funds appears substantial (see Table 7). The highest
maximum return of 9.41 percent (Fund 204) in one month compares to a
mere 0.76 percent for Fund 514. More alarming for investors, however, is
the discrepancy in minimum returns. While those are equal to or greater
than zero for 7 of the 14 funds and Fund 102 still generated 0.54 percent
in its worst month, the previously mentioned devastating drawdown of
Fund 205 (-51.12 percent) marks the lower bound of the range. Industry
experts point out a variety of explanations for the sudden collapse in
38For the fund performance figures to be comparable, they need to be calculatedbased on congruent time periods. Although the chosen period is relatively short, ithelps to understand two important questions: Does the performance of certain indi-vidual funds considerably differ from the results we observed for the index (portfolioof funds) in the previous section? Did the financial crisis have an impact on individualfunds (this did not really seem to be the case on the aggregate level)?
32
ILifeSettlementFunds
Fund100
Fund101
Fund102
Fund104
Fund105
Fund201
Fund202
Fund status active active active active active merged active
Sample size (months) 42 42 42 42 42 28 42
Total return 31.85% 0.96% 37.27% 35.97% 32.49% n/a 7.11%
Mean return 0.66% 0.07% 0.76% 0.73% 0.67% 0.46% 0.17%
annualized 7.94% 0.89% 9.09% 8.81% 8.08% 5.47% 1.99%
Standard deviation 0.52% 3.05% 0.10% 0.14% 0.55% 0.58% 0.71%
annualized 1.81% 10.56% 0.35% 0.47% 1.90% 2.02% 2.47%
Maximum 2.62% 2.05% 0.92% 1.05% 3.95% 3.03% 2.26%
Minimum -0.94% -18.97% 0.54% 0.46% 0.25% 0.00% -1.57%
Skewness 1.02 -6.22 -0.34 0.37 5.46 3.42 0.27
Excess kurtosis 6.13 39.76 -0.97 0.32 32.45 14.46 0.98
Negative months 1 3 0 0 0 0 15
SHR (rank) 1.01 (6) -0.02 6.14 (2) 4.38 (5) 0.98 (7) 0.55 (9) 0.04 (10)
SOR (rank) n/a -0.02 n/a n/a n/a n/a 0.06 (1)
CAR (rank) n/a -0.00 n/a n/a n/a n/a 0.02 (1)
ERVaR (rank) n/a -0.22 n/a n/a n/a n/a 0.04 (1)
Performance measures: Sharpe Ratio (SHR); Sortino Ratio (SOR); Calmar Ratio (CAR); Excess Return on VaR (ERVaR).
Table 6: Descriptive statistics: individual fund return distributions (01/2007–06/2010)
3.4
Analy
sisofin
div
idual
funds
33
Fund203
Fund204
Fund205
Fund208
Fund210
Fund216
Fund514
Fund status suspendedreporting
liquidated suspendedreporting
active active active active
Sample size (months) 20 32 38 42 42 42 42
Total return n/a n/a n/a 31.14% 33.74% 2.05% 29.43%
Mean return 0.45% -0.10% -1.89% 0.65% 0.69% 0.05% 0.62%
annualized 5.41% -1.16% -22.71% 7.77% 8.34% 0.59% 7.39%
Standard deviation 0.46% 3.48% 9.36% 0.11% 0.11% 0.43% 0.07%
annualized 1.61% 12.05% 32.42% 0.37% 0.37% 1.49% 0.24%
Maximum 1.70% 9.41% 2.68% 0.88% 0.88% 1.10% 0.76%
Minimum -0.12% -16.68% -51.12% 0.47% 0.40% -1.57% 0.44%
Skewness 1.67 -3.04 -4.70 0.65 -0.70 -0.66 -0.78
Excess Kurtosis 2.37 19.12 22.78 -0.17 0.62 4.50 0.51
Negative months 1 4 8 0 0 20 0
SHR (rank) 0.6761 (8) -0.0669 -0.2167 4.8178 (4) 5.1965 (3) -0.2027 6.7865 (1)
SOR (rank) n/a -0.0886 -0.2259 n/a n/a -0.2442 n/a
CAR (rank) n/a -0.0139 -0.0397 n/a n/a -0.0555 n/a
ERVaR (rank) n/a -0.1773 -0.3397 n/a n/a -0.2429 n/a
Performance measures: Sharpe Ratio (SHR); Sortino Ratio (SOR); Calmar Ratio (CAR); Excess Return on VaR (ERVaR).
Table 6: Descriptive statistics: individual fund return distributions (01/2007–06/2010) - continued
34 I Life Settlement Funds
Mean Standarddeviation
Maximum Minimum
Mean return 0.29% 0.69% 0.76% -1.89%
Standard deviation 1.41% 2.53% 9.36% 0.07%
Maximum return 2.38% 2.25% 9.41% 0.76%
Minimum return -6.32% 14.41% 0.54% -51.12%
Skewness -0.26 3.00 5.46 -6.22
Excess Kurtosis 10.21 13.40 39.76 -0.97
Table 7: Summary statistics: 14 fund return distributions
the NAVs of Fund 101, 204, and 205. The introduction of the 2008 VBT
tables by the Society of Actuaries (SOA) is certainly an important deter-
minant in this regard. In comparison to the 2001 release, which had been
widely applied in the life settlement industry, life expectancies associated
with the new tables are generally longer. In some cases, these differences
necessitated substantial policy devaluations. Another important factor
is the turmoil in the wake of the financial crisis, which significantly inten-
sified in September 2008 after the bankruptcy of Lehman Brothers and
the AIG bail-out. Due to the great extent of uncertainty in the capital
markets, many open-end life settlement fund investors began to redeem
their fund shares. In combination with a lack of subscriptions, these
excessive redemptions resulted in a liquidity shortage for some funds.
Particularly those with a substandard cash management were suddenly
forced to sell policies at fire sale prices in order to avoid complete dis-
tress. Finally, those funds, which opted for fair value accounting of life
settlements, had to substantially write down their assets to reflect the
changed market environment in late 2008 and early 2009.39
39The different valuation methods and their consequences for the evolution of fundNAVs over time are explained in more detail in the following section.
3.4 Analysis of individual funds 35
Since notable discrepancies between individual funds seem to exist,
careful selection of the fund manager can be crucial. This finding is
supported by the four performance measures we discussed earlier.40 For
instance, we observe negative Sharpe Ratios for the Funds 101, 204, 205,
and 216, implying an average monthly return below the risk-free rate. In
addition, the positive Sharpe Ratios of the remaining funds range from
6.7865 down to 0.0419, a figure that is worse than those for government
bonds, corporate bonds, and hedge funds over the same time period.41
Furthermore, it should be noted that the current status of four of the
analyzed funds is an alarming sign. Fund 203 and Fund 205 suspended
their reporting during the period under consideration, Fund 201 was
merged with Fund 103 (which had been excluded from the analysis in
this section due to its short time series and is currently being liquidated),
and Fund 204 has been terminated. Consequently, the performance fig-
ures derived from the available data for these funds can be expected to
be still upward biased.42
Overall, according to the empirical analysis of the life settlement
index return profile, the asset class indeed appears to be an interest-
ing investment opportunity, offering solid returns comparable to those
provided by government bonds, complemented by an extraordinary low
volatility as well as virtually no correlation with other asset classes. Nev-
ertheless, an examination on the individual fund instead of the index level
revealed anomalies. Although half of the funds under consideration did
not experience a single negative month and, even for the weaker perform-
40The average 1-month U.S. Treasury Bill rate between January 2007 and June2010 has been used as a proxy for the risk-free interest rate. Note that for mostfunds, Sortino Ratios are unavailable since returns did extremely rarely or not at alldrop below the threshold, i.e., the risk-free rate. Thus, the Lower Partial Momentin the denominator is either very close to or exactly zero and the ratio consequentlymeaningless or not defined. Additionally, Calmar Ratios have been omitted wheneverthe lowest return in the series was positive (or negative but very close to zero),rendering a drawdown-based measure pointless. Finally, for those life settlementfunds with no more than a single negative return, an informative 95 percent VaRcannot be derived and therefore Excess Returns on VaR are not available.
41Sharpe Ratios (01/2007 - 06/2010) of the other asset classes for comparisonpurposes: stocks: -0.1057; government bonds: 0.4760; corporate bonds: 0.0851; hedgefunds: 0.0741; real estate: -0.6212; commodities: -0.0466; private equity: -0.0875.
42This was already pointed out in the discussion of potential biases in Section 3.1.
36 I Life Settlement Funds
ers such an occasion appears to be rare relative to the established asset
classes, a negative month—if it actually occurs—can in fact cause a seri-
ous (Fund 101 and Fund 205) or even fatal drawdown (Fund 204). While
the observed performance of life settlement funds could be a result of the
market being inefficient and providing arbitrage opportunities because
many investors have not yet discovered the asset class’ attractiveness,
a more likely explanation is that considerable risks embedded in those
funds are largely not reflected in historical performance data. Therefore,
we will conduct an in-depth risk analysis in the following section, taking
into account the structural insights that we elaborated on in Section 2.3.
4 Risks of open-end life settlement funds
4.1 Overview
During the recent financial crisis, investments with attractive returns
and presumably low risk, such as higher rated tranches of so-called sub-
prime residential mortgage-backed securities turned out to be very risky,
whereas those risks, which finally materialized, had not been reflected by
ex ante risk analyses.43 In combination with our empirical results, this
raises a degree of suspicion. Hence, in the following section, we focus on
latent risks associated with the asset class and, in particular, open-end
life settlement funds.44 Since most of the risks can hardly be quanti-
fied, one needs to rely on a comprehensive qualitative risk analysis. The
discussion offers an explanation for the observed unusual performance of
open-end life settlement funds. We identify the following key risk drivers
in descending order of their severity, as determined by their expected
43See, e.g., studies by the Senior Supervisors Group (2008), the Financial StabilityForum (2008), and the International Institute of Finance (2008).
44Note that most of these risks arise from the characteristics of the underlyinglife settlement assets. Consequently, closed-end life settlement funds as describedin Section 2.2 are generally exposed to them as well. Due to structural differences,however, closed-end funds are better equipped to cope with some of the risk factorsexplained in this section. Liquidity reserves, the absence of leverage, and the factthat investors are locked in until maturity, e.g., mitigate the impact of liquidity risks.Similarly, closed-end funds are less likely to run into policy availability issues andthe associated pricing pressure, since, after the so-called ramp-up period, they do notneed to permanently acquire new policies.
4.2 Valuation risk 37
detrimental impact on an investment in life settlement funds: valuation
risk, longevity risk, liquidity risk, policy availability risk, operational
risk, credit risk, and changes in regulation and tax legislation.
4.2 Valuation risk
The most severe risk factor associated with life settlement funds is ar-
guably valuation risk. As described in Section 2.2, the valuation of a life
settlement portfolio is commonly conducted on a mark-to-model basis.
This means that due to the absence of objective market values, fund
shares are dealt based on model values determined by the fund manage-
ment, even though it is not clear whether the assets can in fact be sold
at those values. In addition, not all models are reviewed by an actuarial
advisor, implying the necessity of a profound actuarial know-how of the
fund management.
The Financial Accounting Standards Board (FASB) guidelines for
life settlements distinguish two valuation approaches: the investment
method and the fair value method (see FASB, 2006). While, in both
cases, initial measurement is based on the purchase (transaction) price,
the NAV development of a life settlement fund materially depends on
the methodology used for subsequent measurement. The purchase price
is agreed upon by the counterparties of a life settlement transaction.
Through the life settlement provider, the fund typically submits an offer
to the policyholder, which he or she can accept or reject. The offer price
is commonly calculated as the present value of expected future payoffs
less the present value of expected premium payments and other costs.
However, the discount rate in this context is not derived from a term
structure but determined by the internal rate of return the fund aims
to achieve on the investment, which is generally a function of its cost
of capital (see, e.g., Zollars et al., 2003). The key factor in determining
the expected cash flows from a life insurance contract is an insureds life
expectancy. After the initial examination, further life expectancy esti-
mates are carried out at each fund’s own discretion.
38 I Life Settlement Funds
When using the investment method, the initial recognition of the pol-
icy in the books is given by the purchase price plus initial direct costs
(legal costs, commissions paid, etc.). Further valuation has to be con-
ducted by capitalizing any continuing costs such as premiums to keep the
policy in force. Gains may only be recognized in case of a policy resale
or in case of the insured’s death, and are then given by the difference be-
tween the sales proceeds or the death benefit payment and the carrying
amount of the life settlement contract. In contrast, a loss must be rec-
ognized for impairments, i.e., if there is updated information available,
indicating that the expected policy payoff does not suffice to cover the
carrying amount of the contract plus all projected undiscounted future
premiums. This can occur if an increase in the life expectancy becomes
evident or if the creditworthiness of the primary insurer deteriorates sub-
stantially. As an alternative, the FASB proposes the fair value method,
where the initial value of a life settlement investment is also determined
by the purchase price and, after that, ongoing valuation is based on the
fair value, i.e., the sales price that the asset is likely to achieve in the mar-
ket less transactions costs, with value changes being directly recognized
in periodic earnings. However, due to the illiquid nature of life settle-
ments, a mark-to-market approach is typically not practicable. Thus,
it is prevalent to estimate fair values by marking-to-model. Since these
valuation models are based on extensive assumptions and there is little
oversight as to their validity, the fair value method implies a largely sub-
jective assessment.
Overall, the solid performance that could be observed in Section 3 is
likely to be all but a mere by-product of the accounting oriented valua-
tion methodology for life settlements implied by the widespread invest-
ment method. This approach leaves room for large price movements only
if death benefits are received or life expectancy estimates are renewed
and differ significantly from the original ones. In all other cases, one
should observe an almost linear growth path. Hence, it is quite likely
that most funds, which displayed more stable returns over the considered
time period, tend to avoid the fair value method. However, although life
settlements are acquired at a large discount of their face value and the
4.3 Longevity risk 39
purchase price tends to understate the fair value on the transaction date,
the investment method still involves the risk of an incorrect purchase
price due to model errors or misestimated life expectancy (see Perera
and Reeves, 2006). If the whole industry would be obliged to dispose
the investment method and switch to fair values, life settlement fund re-
turns would probably become considerably more volatile than suggested
by our empirical observations over the last years. Moreover, the fair
value method is associated with a further pitfall. Since fund managers
are in a position to change their mark-to-model estimation methodol-
ogy over time, they could on the one hand smooth returns and on the
other hand evaluate fund shares at fire sale prices in the case of exten-
sive redemptions by investors. Based on these considerations, erroneous
valuation is, as already mentioned in Section 3.4, a likely cause for the
major drawdowns in the time series of Fund 101, 204, and 205.
4.3 Longevity risk
Another key risk factor is longevity risk, which describes the possibil-
ity that the insured lives longer than originally expected. To measure
the sensitivity of senior life settlement portfolios to changes in mortality
rates and longevity risk, also called life extension risk, Stone and Zissu
(2006) propose to use a life expectancy duration. The more the actual
lifetime exceeds the expected lifetime, the less valuable the policy be-
comes for the fund and its investors. The reason is that initial pricing
assumptions turned out to be incorrect in that premium payments have
to be made longer and the death benefit is received later than expected.
Longevity risk is particularly important in its systematic form, i.e., if the
life expectancy of the whole portfolio is simultaneously prolonged. The
discovery of a cure or a mitigating treatment for a common illness, e.g.,
implies a substantial increase in the correlation between those lives in
the portfolio, which had been suffering from that particular disease (see
Perera and Reeves, 2006). To cope with longevity risk, life settlement
funds diversify their portfolios across different types of diseases, purchase
insurance coverage (if available) or employ innovative risk management
tools such as longevity swaps.
40 I Life Settlement Funds
Assessing the quality of life expectancy estimates is challenging and
results are rarely disclosed to the public. According to Milliman Inc.
(2008), which examined the mortality experience data of two providers
gained from filings with the Texas Department of Insurance, the actual
number of deaths recorded from 2004 to 2006 was only 60 percent of those
that had been expected. This provides an indication of the fundamen-
tal longevity risk that is inherent in life settlement portfolios. Realized
investor returns in this case are likely to be considerably smaller than
originally projected. In line with these findings, A.M. Best (2009) de-
scribed how five year old portfolios showed signs that the life expectancy
estimates had historically been too short and that since 2005, medical
underwriters have issued more conservative ones. Furthermore, as in-
dicated by industry experts, some of the largest medical underwriters,
which have been able to steadily expand their influence in the market,
seem to systematically underestimate life expectancies.
Thus, the importance of longevity risk should not be misjudged, par-
ticularly against the background of the potential incentive problems of
life settlement providers, which were discussed in Section 2.3. It should
be of central interest to investors with which medical underwriters fund
managers cooperate and whether they require more than one life ex-
pectancy estimate to be at least partially protected against major errors
in medical underwriting. Taking these considerations into account, it
may well be that a large number of insureds in the portfolios of Fund
101, 204, and 205 turned out to live much longer than initially expected,
forcing the funds to realize substantial losses on the respective life set-
tlement assets.
4.4 Liquidity risk
After the initial sale of fund shares, there are in principle two sources
of cash inflows on the fund level—new subscriptions and death benefit
payments—neither of which occur on a regular basis or are easy to fore-
cast. In addition, open-end funds typically reinvest death benefits in
order to purchase new policies or use them to pay due premiums. Some
4.4 Liquidity risk 41
fund managers maintain a position in liquid assets, a reserve account,
or can draw on short-term debt financing through a liquidity facility.45
Cash outflows, in contrast, occur on a regular basis due to premium
payments, redemptions, and potentially interest plus repayment in case
the fund is leveraged. In combination with the illiquid nature of the
underlying, this implies that life settlement funds are fairly vulnerable
to becoming liquidity strained. The consequences for investors could be
devastating. If a fund falls short of sufficient cash to cover due redemp-
tions, it has no choice but to sell off assets to make up for the missing
amount unless a reserve account has been set up or short-term debt fi-
nancing is attainable. Then again, the fund will probably not be able
to sell life settlements from its portfolio at an acceptable value at short
notice due to the mediocre permanent trading activity in the market
as well as the complexity and length of the transactions. Moreover, a
distressed life settlement fund is highly likely to default on the ongo-
ing premium payments of at least some of its policies, causing them to
lapse. Evidently, these risks increase disproportionately with the degree
of leverage applied by a life settlement fund since it also has to bear
the debt service. The same is true if policies are premium financed, i.e.,
if the fund takes out loans to fund premium payments. As with hedge
funds, some life settlement funds partially protect themselves against the
problem of illiquid assets and excessive redemptions by imposing lock-up
periods, gates, and redemption fees. As a last resort, most fund man-
agers reserve the right to suspend redemptions. While these measures
reduce liquidity risk at the fund level, they clearly hamper liquidity of
the fund shares at the investors’ level and should thus be carefully fac-
tored into an investment decision if one does not want to find his money
locked into a life settlement fund in major distress. Inevitably, the ma-
jor dislocations in the capital markets during the peak of the financial
crisis in 2008 have led to an imbalance between subscriptions and re-
demptions, which revealed severe liquidity issues of a number of funds.
This is another likely cause for the observed losses of Fund 101, 204, and
205.
45The reader is referred to the structural overview in Section 2 to identify thesesources of liquidity.
42 I Life Settlement Funds
4.5 Policy availability risk
Along with valuation, longevity, and liquidity risk, there is also availabil-
ity risk and competitive pricing pressure, because the secondary market
is limited by the size of the primary market as well as the number of avail-
able target policies. Evidently, the identification of suitable policies is a
critical success factor for an investment in life settlements (see Moodys,
2006). In addition, funds will have to consider the number of contracts
and the policy mix in their portfolios, including different types of diseases
and different primary insurers to diversify risks. Target policies typically
satisfy specific criteria such as a reduced policyholder life expectancy of
on average 113 months, a high face value of on average 1.8 million USD,
and a policyholder age of approximately 76 years. Moreover, ideally the
insured would have otherwise surrendered the policy (see Milliman Inc.,
2008). Such contracts are not plentiful. According to Moodys (2006),
around one percent of the permanent policies in force in the U.S. market
match the characteristics commonly targeted by life settlement funds.
This is one reason for the fact that only about fifteen to twenty-five per-
cent of the policies presented to life settlement providers are actually
purchased (see, e.g., McNealy and Frith, 2006). Other reasons include
the inability of the policyholder to qualify for renewed coverage and the
failure of the transaction partners to agree on the purchase price.
It is imperative to take these potential availability constraints into
account, since the supply-demand-situation on the life settlement mar-
ket substantially influences acquisition prices. Problems for the funds
can occur if a large inflow of capital into the asset class is not met by
a sufficient supply of adequate policies or if the market activity in gen-
eral freezes. The resulting competitive pressure implies a reduction in
achievable returns due to higher purchase prices. Furthermore, even for
fund managers which have performed well to date, there may be adverse
changes in the portfolio composition if the number of valuable life settle-
ment investment opportunities noticeably decreases. In such a scenario,
it is of importance whether a fund runs a single or multi-source approach
with regard to life settlement providers since those fund managers with
access to a larger number of life insurance policies are likely to be in a
4.6 Operational risks 43
better position when supply in general is short. Apart from that, fund
size can play an important role because smaller funds may find it easier
to source enough policies that fit their investment criteria and offer an
attractive risk-return perspective. Larger funds, on the contrary, could
face situations where they need to relax their policy picking standards to
be able to invest all of their investor money. Since market participants
have not reported any supply shortages during the last two years, it is
rather unlikely that Fund 101, 204, and 205 experienced drawdowns due
to constrained policy availability. Nevertheless, investors should bear
this potential risk in mind.
4.6 Operational risks
Among the less severe but still noteworthy risk factors are operational
risks: insured fraud risk, litigation or legal risks, and operational risks
originating from third-party service providers. Insured fraud risk could
mean a misrepresentation of one’s health status in order to achieve a
higher price for the policy. In rare cases the policyholder also might not
disclose all original beneficiaries or fraudulently sell the same policy to
multiple buyers. Furthermore, insureds may use sales proceeds to im-
prove their living standard and medical care, which can increase their
life expectancy and, in turn, reduce investor returns.
Litigation and legal risks can arise due to the high complexity of con-
tractual agreements, despite the fact that sales processes are becoming
increasingly standardized. Life insurance companies may possibly con-
test the policy and refuse to pay the death benefit, e.g., due to lack of
insurable interest. In addition, payments are typically held back if the
insureds body is missing. This can be done by insurers for up to seven
years (see Perera and Reeves, 2006). Furthermore, former beneficiaries
could initiate lawsuits, accusing life settlement firms of unethical sales
practice or invalid transfer with the intent to claim the death benefit
for themselves. As a consequence, the payment may be substantially de-
layed or not transferred at all. In such a case, legal expenses may even
exceed the return from the policy.
44 I Life Settlement Funds
Further operational risks arise from the reliance on third-party ser-
vice providers. The tracking agent, for instance, might fail to service the
policy properly such that the insured’s death is reported late or he cannot
be located posthumously, thus delaying the collection of death benefits.
However, most servicers are insured up to some amount against such
operational risks. Another important risk factor with respect to the in-
volved third parties is fraud. In particular, life settlement providers may
collude with brokers in order to discourage competitive bids. In 2006,
one of the largest life settlement companies, Coventry First, was sued by
New York Attorney General Eliot Spitzer and accused of bid-rigging to
keep policy purchase prices low. The provider was believed to have made
secret payments to life settlement brokers in exchange for which they al-
legedly suppressed competitive bids from other life settlement companies.
The lawsuit was settled in October 2009 with Coventry First paying an
additional 1.4 million USD to policyholders to adequately compensate
them for the appropriate market value of their life insurance policies. Fur-
thermore, the company agreed to pay 10.5 million USD to the state of
New York to end the litigation. As a corollary of this settlement, no fine
or penalty was issued against Coventry. Another prominent case is Mu-
tual Benefits Corporation, which was alleged to have made substantial
misrepresentations to investors in its marketing material, prospectuses,
as well as through its network of sales people and failed to disclose focal
information over several years. In particular, life expectancy estimates
for a large number of its policies were fraudulently assigned at the discre-
tion of its directors. As a consequence, around 90 percent of the policies
needed to be maintained significantly beyond their life expectancy esti-
mates, inflicting high losses on investors. In the particular cases of Fund
101, 204, and 205, however, we deem it unlikely that losses occurred due
to a manifestation of operational risk factors.
4.7 Credit risk
Life settlement funds also face credit risk due to a potential default of pri-
mary insurers. Although such a credit event was thought to be virtually
impossible before the financial crisis, the AIG bail-out in 2008 provides
4.8 Changes in regulation and tax legislation 45
evidence that the default of an insurer, no matter what ultimately causes
it, can be an issue. Yet, since the average rating of the insurance com-
panies in the portfolios of our sample funds is ”AA” and policyholders’
claims rank most senior in the case of insolvency, we believe that credit
risk has been irrelevant at least in the past with regard to the problems
of Fund 101, 204, and 205. In addition, in the unlikely case of an insurer
default, there are still state-dependent insurance guarantee funds in the
U.S., which provide protection to policy owners.46
4.8 Changes in regulation and tax legislation
Finally, there is a risk of adverse amendments to regulatory frameworks
and tax legislation. Until recently, regulation of the U.S. life settle-
ment market was partially lax and inconsistent (see Fitch Ratings, 2007).
While this has changed, regulation still varies by state. Few states do
not regulate transactions at all, other states regulate viatical transac-
tions but not senior life settlements, and still others require that brokers
and providers be licensed (see Gatzert, 2010). One often discussed prob-
lem in the United States is stranger-originated (or investor-initiated) life
insurance (STOLI), as it contradicts the principle of insurable interest
which had already been established in the early 19th century before it
was confirmed by the U.S. Supreme Court in 1911 in Grigsby v. Russell
(see Katt, 2008). The principle of insurable interest distinguishes insur-
ance from speculation. It was designed to protect the insured, since, if
allowed to purchase insurance on the lives of strangers, the holder of
the policy has a financial interest in the death of the insured. The main
feature of a STOLI process is that the policy is not initiated by the policy-
holder, but by an investor or third-party lender who provides the insured
with cash to cover the premium payments and ultimately receives the
death benefit (see, e.g, Fitch Ratings, 2007; Ziser, 2007; Gatzert, 2010).
STOLI must be distinguished from the common practice of non-recourse
46However, in most cases an insurance guarantee fund would probably not coverthe full death benefit of the policies due to the high face values in the case of seniorlife settlements. In addition, to the best of our knowledge, there has not been aprecedent yet. Hence, it is not clear from a legal point of view, whether an insuranceguarantee fund would need to pay for life settlement fund investors.
46 I Life Settlement Funds
premium financing, which allows policyholders who do have an insurable
interest to fund their premium payments with a loan that is collateral-
ized by the insurance policy (see Freedman, 2007).
To introduce transparency and clear rules in the market, the National
Association of Insurance Commissioners (NAIC) proposed the Viatical
Settlements Model Act, which would ban life settlements of non-recourse
premium financed policies during the first five years of the contract (see,
e.g, Fitch Ratings, 2007; Ziser, 2007; Gatzert, 2010). In November 2007,
the National Conference of Insurance Legislators (NCOIL) introduced
the Life Settlement Model Act, which does not include the five-year
ban proposed by the NAIC, but explicitly defines STOLI as a fraudu-
lent life settlements act (see NCOIL, 2007). In addition, the NCOIL
proposal prohibits premium financing companies from owning or being
financially involved in policies they finance (see Gatzert, 2010). The frag-
ile legal status of STOLI appears to have an impact on the demand by
institutional investors in that they generally avoid purchasing premium
financed policies (see Beyerle, 2007). Overall, both proposals by NAIC
and NCOIL are still criticized and may be refined, thus implying ongo-
ing uncertainty in respect to the regulatory treatment of life settlements
(see, e.g., Freedman, 2007). Another risk factor relates to tax legislation.
As Fitch Ratings (2007) points out, the absence of insurable interest be-
tween policyholder and beneficiary may affect tax advantages associated
with life insurance. Moreover, in 2009 the U.S. Internal Revenue Service
(IRS) determined that death benefit payments to foreign life settlement
investors will be subject to withholding tax. Although these aspects
distinctly affect the market’s legal environment, we believe that adverse
changes in regulation and tax legislation did not cause the distress of
Fund 101, 204, and 205.
5 Summary and conclusion 47
5 Summary and conclusion
We comprehensively analyze open-end funds dedicated to U.S. senior life
settlements, explaining their business model and the roles of institutions
involved in the transactions of such funds. In addition, we contribute to
the literature by conducting the first empirical analysis of life settlement
fund return distributions as well as a performance measurement, includ-
ing a comparison to established asset classes. Since the funds contained
in our dataset largely cover this young segment of the capital markets,
representative conclusions can be derived. Based on these findings, we
elaborate on the risk profile of life settlement assets in general and open-
end life settlement funds in particular.
Although our empirical results suggest that life settlements generally
offer attractive returns paired with low volatility and are uncorrelated
with other asset classes, we find substantial latent risks associated with
the funds, such as liquidity, longevity and valuation risks. Since these did
generally not materialize in the past and are hence largely not reflected
by the historical data, they cannot be captured by classical performance
measures. Therefore, investors should not be misled by a superficial first
impression of the asset class. Caution is advised and the expected re-
turn on life settlement funds should be regarded as a compensation for
investors who decide to bear those risks.
It is advisable to perform extensive due diligence on life settlement
funds, focusing on valuation methodology, cash management, asset pipe-
lines as well as business partners. Wherever possible, independent third
parties such as auditors and rating agencies can be involved for cross-
checking and to deliver additional information such that the investor is
able to balance the expected returns against a comprehensive qualitative
assessment of latent risks before deciding on the portfolio weight he
would like to allocate to life settlement funds. Nonetheless, our results
also illustrated that—within reasonable limits—life settlements certainly
provide a suitable means for diversification as they seem to be genuinely
uncorrelated with the broader capital markets.
48 I Life Settlement Funds
6 Appendix
6.1 Index descriptions (from the providers)
- Life Settlement Funds Index:
A custom index of open-end life settlement funds. This index is an
equally weighted portfolio consisting of all available funds at any
point in time. The aim of the index is to track the development of
a portfolio of life settlement funds between 12/2003 and 06/2010
as adequately as possible.
Bloomberg Ticker: -
Further information: -
- S&P 500:
The S&P 500 is widely regarded as the best single measure of
the U.S. equities market and includes 500 leading companies in
the major industries of the U.S. economy. Although the S&P 500
focuses on the large cap segment of the market, with approximately
75 percent coverage of U.S. equities, it is also well suited to assess
the total market.
Bloomberg Ticker: SPX <Index> <Go>
Further information: www.standardandpoors.com
- FTSE U.S. Government Bond Index:
FTSE Global Government Bond Indices comprise central govern-
ment debt from 22 countries, denominated in the domestic currency
or Euros for Eurozone countries. These are total return indices,
taking into account the price changes as well as interest accrual
and payments of each bond.
Bloomberg Ticker: FGGVUSP5 <Index> <Go>
Further information: www.ftse.com/Indices
6.1 Index descriptions (from the providers) 49
- DJ Corporate Bond Index:
The Dow Jones Corporate Bond Index is an equally weighted bas-
ket of 96 recently issued investment-grade corporate bonds with
laddered maturities. The objective of this index is to capture the
return of readily tradable, high-grade U. S. corporate debt.
Bloomberg Ticker: DJCBP <Index> <Go>
Further information: www.djindexes.com/mdsidx
- HFRI Fund Weighted Composite Index:
The HFRI Monthly Indices are designed to reflect hedge fund in-
dustry performance by means of equally weighted composites of
constituent funds. They range from the industry-level view of the
HFRI Fund Weighted Composite Index, which encompasses over
2000 funds, to the increasingly specific-level of sub-strategy classi-
fications. Fund of Funds are not included.
Bloomberg Ticker: HFRIFWI <Index> <Go>
Further information: www.hedgefundresearch.com
- S&P/Case-Shiller Home Price Index (Composite of 20):
The S&P/Case-Shiller Home Price Indices are designed to gauge
the value growth of residential real estate in various regions across
the United States. The underlying methodology to measure house
price movements has been developed in the 1980s and is still con-
sidered to be the most accurate way to measure this asset class.
Bloomberg Ticker: SPCS20 <Index> <Go>
Further information: www.standardandpoors.com
50 I Life Settlement Funds
- S&P GSCI (USD, Total Return):
The S&P GSCI provides investors with a reliable and publicly avail-
able benchmark for investment performance in the commodity mar-
kets. The index is designed to be tradable, readily accessible to
market participants, and cost efficient to implement. The S&P
GSCI is widely recognized as the leading measure of general com-
modity price movements and inflation in the world economy.
Bloomberg Ticker: SPGSCITR <Index> <Go>
Further information: www.standardandpoors.com
- S&P Listed Private Equity Index (USD, Total Return):
In the last few years increasing numbers of private equity businesses
have listed on stock exchanges to meet investor requirements for
liquidity and transparency. The S&P Listed Private Equity In-
dex is comprised of 30 leading listed private equity companies that
meet size, liquidity, exposure, and activity requirements. It is de-
signed to provide tradable exposure to the leading publicly listed
companies in the private equity space.
Bloomberg Ticker: SPLPEQTR <Index> <Go>
Further information: www.standardandpoors.com
6.2 Performance measures 51
6.2 Performance measures
The Sharpe Ratio (see Sharpe, 1966) is given by
Sharpe Ratioi =µi − rfσi
, (1)
where µi is the average monthly return on asset i, rf is the risk
free monthly interest rate and σi represents the standard deviation of
monthly returns. The Sharpe Ratio has often been criticized because
of its apparent inability to capture all characteristics of non-normal re-
turn distributions. Thus it is viewed as a misleading indicator for the
risk return profile of certain investments (see, e.g., Amin and Kat, 2003).
Consequently, complementary performance indicators utilize alterna-
tive risk measures in order to avoid the alleged problems associated with
the Sharpe Ratio. One of these measures is the Sortino Ratio (see Sortino
and Van Der Meer, 1991), which employs the Lower Partial Moment of
order 2 (LPM2) instead of the standard deviation, i.e.,
Sortino Ratioi =µi − τ
√
LPM2i(τ). (2)
The nth order LPM for asset i is defined as:
LPMni(τ) =1
T
T∑
t=1
max [τ − rit, 0]n.
In general, Lower Partial Moments quantify risk through negative
deviations from a certain threshold return τ (e.g., the mean return, the
risk free interest rate or 0). The order n governs the weighting for this
downside risk and should therefore be higher, the more risk averse the
investor is (see Fishburn, 1977).
Other modern performance measures are based on drawdown, i.e.,
the loss incurred over a certain time period. The Calmar Ratio, which
has become common among practitioners, particularly in the context of
hedge fund performance measurement, is given by:
52 I Life Settlement Funds
Calmar Ratioi =µi − rf−MDi
. (3)
It relates the excess return over the risk free interest rate to the
maximum drawdown MDi, which represents the lowest return over the
period under consideration. Since the lowest return is usually negative,
the formula for the Calmar ratio contains a minus sign in the denomina-
tor, turning it into a positive risk figure.
Finally, performance measures can also be based on Value at Risk
figures. The Value at Risk for an asset i (V aRi) is the loss over a certain
period, which is not exceeded with a prespecified probability (1−α), i.e.,
the α-quantile of the return distribution under consideration. One such
indicator is Excess Return on Value at Risk:
Excess Return on VaRi =µi − rfV aRi
. (4)
References 53
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59
Part II
Pricing Catastrophe Swaps:
A Contingent Claims
Approach
Abstract
In this paper, we comprehensively analyze the catastrophe (cat) swap,
a financial instrument which has attracted little scholarly attention to
date. We begin with a discussion of the typical contract design, the
current state of the market, as well as major areas of application. Sub-
sequently, a two stage contingent claims pricing approach is proposed,
which distinguishes between the main risk drivers ex-ante and during the
loss reestimation phase. Catastrophe occurrence is modeled as a doubly
stochastic Poisson process (Cox process) with mean-reverting Ornstein-
Uhlenbeck intensity. In addition, we fit various parametric distributions
to normalized historical loss data for hurricanes and earthquakes in the
U.S. and find the heavy-tailed Burr distribution to be the most adequate
representation for loss severities. Applying our pricing model to market
quotes for hurricane and earthquake contracts, we derive implied Poisson
intensities which are subsequently condensed into a common factor for
each peril by means of exploratory factor analysis. Further examining
the resulting factor scores, we show that a first order autoregressive pro-
cess provides a good fit. Hence, its continuous-time limit, the Ornstein-
Uhlenbeck process should be well suited to represent the dynamics of
the Poisson intensity in a cat swap pricing model.47
47Alexander Braun (2010), Pricing Catastrophe Swaps: A Contingent Claims Ap-proach, Working Papers on Risk Management and Insurance, No. 78.
60 II Catastrophe Swaps
1 Introduction
Since the early 1990s, insurance and reinsurance companies have been
using innovative financial instruments to lay off natural disaster risk in
the capital markets. To date the most popular of these alternative risk
transfer tools is the catastrophe (cat) bond, a security which pays regu-
lar coupons to the investor unless a catastrophic event occurs, leading to
full or partial loss of principal. The principal is held by a special purpose
vehicle (SPV) in the form of highly-rated securities and paid out to the
hedging (re)insurer to cover its losses if the trigger condition, which has
been defined in the bond indenture, is fulfilled. In addition to cat bonds,
catastrophe derivatives can be employed to access the capital markets.
In 1992 the Chicago Board of Trade (CBOT) initiated exchange-traded
catastrophe futures and options based on its own loss index (see Swiss
Re, 2009). Due to humble trading activity, these contracts were soon
replaced by options based on Property Claim Services (PCS) indices, a
measure for catastrophe losses in nine geographical regions of the United
States. However, PCS-options, which paid off for the buyer in case the
underlying index exceeded the strike price, were eventually also discon-
tinued in 2000 (see Cummins and Weiss, 2009). Despite their earlier
demise, catastrophe derivatives with an exclusive focus on U.S. hurricane
risk have recently been re-launched by several exchanges. Catastrophe
event-linked futures (ELFs), which are co-offered by Deutsche Bank and
the Insurance Futures Exchange (IFEX), feature a binary payoff contin-
gent on regional PCS losses.48 Similar contracts are listed on the Eu-
ropean Exchange (EUREX). Apart from insurance futures, the Chicago
Mercantile Exchange (CME) and the New York Mercantile Exchange
(NYMEX) provide catastrophe options as well. While the former refer-
ence the so-called CME Hurricane Index,49 NYMEX options settle based
on the Re-Ex index by Gallagher Re (see Cummins, 2008).
48IFEX contacts are traded on the Chicago Climate Futures Exchange. Refer tothe IFEX website for more information.
49This index, which had been developed by the reinsurance intermediary Carvill,was formerly known as the Carvill Hurricane Index (CHI). In 2009, CME grouppurchased the rights and renamed it to CME Hurricane Index. See www.artemis.bm.
1 Introduction 61
Although there is a growing body of literature on catastrophe bonds,
futures and options, it seems that another innovative risk transfer in-
strument of increasing importance for risk managers and investors has
been neglected so far: the catastrophe swap. Cat swaps are over-the-
counter (OTC) contracts, allowing (re)insurers to tap additional risk
capacity by synthetically passing a portion of their insurance risk on to
a counterparty. The latter could be an investor, who, in turn, gains
unfunded exposure to natural disaster risk. As indicated by industry ex-
perts, catastrophe swaps have been steadily gaining ground over the past
few years and due to recent progress with regard to contract documenta-
tion and standardization, the market outlook is fairly promising. Early
references to catastrophe swaps can be found in Borden and Sarkar (1996)
and Canter et al. (1997), who mentioned the instrument in their articles
on insurance-linked securities. Furthermore, Cummins (2008) and Cum-
mins and Weiss (2009) briefly describe the general mechanism behind a
catastrophe swap contract. Apart from these publications, however, the
instrument has, to the best of our knowledge, not attracted scholarly
attention. This paper is intended to fill this gap.
The remainder of the paper is structured as follows. In Section 2, we
briefly review the extant literature on the pricing of catastrophe-linked
instruments. Furthermore, Section 3 contains a discussion of the main
characteristics of cat swap contracts and an overview of the current state
of the market as well as major areas of application. A two-stage approach
for the pricing of catastrophe swaps ex-ante and in the loss reestimation
phase is presented in Section 4. In Section 5, we fit different parametric
distributions to normalized historical hurricane and earthquake loss data
for the U.S. to select an adequate loss severity distribution. Subsequently,
the pricing model is applied to back out implied Poisson intensities from
cat swap market data which we condense into a common factor for each
peril by means of exploratory factor analysis. The resulting factor score
times series are then used to estimate a first order autoregressive process
and evaluate its fit, thereby shedding some light on the adequacy of a
mean-reverting process for the Poisson intensity in our cat swap model.
Finally, in Section 6, we conclude.
62 II Catastrophe Swaps
2 Literature review
While no article has been devoted to potential pricing approaches for
catastrophe swaps yet, a few authors have discussed the pricing of cat
bonds, futures, and options. In this regard, models based on three dif-
ferent theoretical foundations have been brought forward.50 First of all,
within their empirical examination of cat bond prices, Lane (2000) as well
as Lane and Mahul (2008) apply actuarial pricing methodology, thereby
acknowledging the resemblance of catastrophe-linked capital market in-
struments to traditional reinsurance.
In contrast to that, utility-based approaches are centered around the
notion that insurance markets are generally incomplete, implying that it
is not possible to find a unique equivalent martingale measure by merely
ruling out arbitrage opportunities. Embrechts and Meister (1997) pro-
vide a generic discussion of catastrophe futures pricing in a utility max-
imization context. Furthermore, Aase (1999) treats catastrophe risk as
systematic and resorts to a partial equilibrium framework with constant
absolute risk aversion to derive pricing formulae for cat futures, caps,
call options, and spreads. Similarly, Cox and Pedersen (2000) derive a
pricing approach for cat bonds in an incomplete markets setting based
on equilibrium pricing theory and time separable utility. Christensen
and Schmidli (2000) introduce an exponential utility model for cat fu-
tures which includes loss reporting lags. Amending his earlier work on
cat derivatives pricing by employing a Markov model for the dynamics
of underlying, Aase (2001) proposes a competitive equilibrium approach
which assumes constant relative risk aversion of the representative agent.
In addition, Young (2004) computes the indifference price of cat bonds
based on exponential utility investor preferences. Utility-based pricing
of cat bonds is also considered by Egami and Young (2008). Moreover,
Dieckmann (2009) proposes a dynamic equilibrium model for cat bonds
with an external habit process as in Campbell and Cochrane (1999).
50Galeotti et al. (2009) provide an empirical comparison of some of these ap-proaches.
2 Literature review 63
However, the majority of papers on the pricing of cat bonds and
derivatives proposes preference free no-arbitrage frameworks. Cummins
and Geman (1994, 1995) value cat futures and call spreads with an Asian
option approach, assuming a jump-diffusion process with constant jump
amplitude for the claim dynamics. Besides, Chang et al. (1996) develop
a cat option model based on a stochastic time change linked to insurance
futures transactions, allowing them to convert a compound Poisson into
a pure diffusion process for which risk-neutral valuation is readily appli-
cable and a parsimonious closed formula can be derived. Similarly, by
means of stochastic time change and Laplace transform, Geman and Yor
(1997) present a semi-analytical solution for the price of cat options on a
loss index which follows a jump-diffusion process. Having priced simple
cat bonds under the Black and Scholes (1973) assumptions in the first
section of their paper, Louberge et al. (1999) subsequently consider a
compound Poisson process in combination with a simple binomial model
for the interest rate. Another no-arbitrage pricing model for cat bonds
built upon a compound Poisson process is presented by Baryshnikov et al.
(2001). Lee and Yu (2002) additionally contemplate default risk of the
cat bond issuer as well as issues of moral hazard and basis risk, adopt-
ing a structural credit model with stochastic interest rates as in Cox
et al. (1985). Furthermore, Bakshi and Madan (2002) provide a closed-
form solution for (PCS) cat option prices based on the assumption that
losses follow a mean-reverting Markov process with one-sided jumps. A
compound doubly stochastic Poisson process (Cox process) is used by
Burnecki and Kukla (2003) to value zero-coupon and coupon cat bonds
and by Dassios and Jang (2003) to model stop-loss reinsurance contracts
and cat derivatives. Muermann (2003) assumes a compound Poisson loss
process and values cat derivatives relative to observed premiums of in-
surance contracts on the same underlying risks. Moreover, in his model
for options on a PCS index, Schmidli (2003) distinguishes between catas-
trophe occurrence and loss development period, which are governed by
a compound Poisson process and a Geometric Brownian Motion, respec-
tively. A barrier option framework for the price of a cat bond is proposed
by Vaugirard (2003a,b, 2004), who assumes a jump-diffusion process for
the underlying physical index and stochastic interest rates based on a Va-
64 II Catastrophe Swaps
sicek (1977) model. In addition, Cox et al. (2004) consider the valuation
of double trigger catastrophe put options when losses are generated by a
compound Poisson process and Jaimungal and Wang (2006) generalize
their work by incorporating stochastic interest rates. While Lee and Yu
(2007) apply insights from their earlier work on cat bonds to reinsurance
contracts, Biagini et al. (2008) use Fourier transform to derive an ana-
lytical solution for the price of an option with catastrophe occurrence
and loss development period. Muermann (2008) applies a cat call option
model based on a compound Poisson process for the underlying loss index
to extract the market price of insurance risk from market quotes of traded
cat derivatives. Chang et al. (2008, 2010) generalize their concept from
the mid-1990s from a complete market continuous-time to an incomplete
market discrete-time framework to price Asian-style cat options with a
doubly-binomial model. Additionally, they consider stochastic Poisson
intensities described by a mean-reverting Ornstein-Uhlenbeck process
and reduce the computational effort through a stochastic time change
from calendar to claim time. Hardle and Cabrera (2010) price a hybrid
cat bond for earthquakes, assuming a doubly stochastic Poisson process
for the flow of catastrophic events. Finally, Wu and Chung (2010) employ
a doubly stochastic Poisson process with Ornstein-Uhlenbeck intensity
in combination with a Cox et al. (1985) model for the interest rate and
the framework of Jarrow and Yu (2001) for counterparty default risk to
price catastrophe bonds, futures, and options.
3 Catastrophe swaps
3.1 Contract design
Catastrophe swaps are financial instruments through which natural dis-
aster risk can be transferred between counterparties. In a typical con-
tract, the protection buyer (fixed payer) agrees to make periodic pre-
mium payments to the protection seller (floating payer) in exchange for
a predetermined binary compensation payment51 contingent on the oc-
51This is usually the full notional value. Alternatively, the payoff profile can belinearly increasing in the underlying losses within the layer between an attachmentlevel and a cap.
3.1 Contract design 65
currence of a trigger event (covered event) in the covered territory (see,
e.g., Swiss Re, 2006). While the covered territory defines the country
or geographic region in which a catastrophe has to strike in order to
be relevant under the swap transaction, a trigger event is determined
by the so-called reference peril (reference catastrophe), the associated
reference losses (reference amount), and the contract’s thresholds. The
term reference peril means the type of disaster which is covered under
the swap, e.g., wind storms including named hurricanes. Whenever such
a catastrophe occurs, the appointed loss report provider assigns a serial
number to it and publishes an initial estimate of the resulting insurance
industry losses. This loss estimate is subsequently refreshed on a regular
basis, with the final loss report usually being released no later than six
months after the event. An important characteristic in this regard is
that the reference losses from different natural disasters are not aggre-
gated but tracked separately. Hence, the trigger mechanism relates to
the losses of individual catastrophes, not a sum of losses. Cat swaps typ-
ically exhibit two thresholds: an event and a slightly higher acceleration
threshold.52 If, during the term of the contract, a final loss estimate
for a reference peril reaches the event threshold (attachment level), it
results in an immediate payoff to the protection buyer and the subse-
quent termination of the contract. Similarly, the payoff under the swap
is triggered instantaneously by an interim loss estimate in excess of the
acceleration threshold. Finally, the protection buyer receives a payoff at
maturity if an interim loss estimate is equal to or higher than the event
threshold.
A concrete example for cat swap contracts offered in current market
practice are ”Deutsche Bank Event Loss Swaps” (ELS).53 ELS for U.S.
wind storms, i.e., hurricanes and tornadoes, have been launched in late
2006 and are available with thresholds of USD 20 billion, USD 30 billion
or USD 50 billion, while the attachment levels for earthquake-based con-
tracts can be set at USD 10 billion and USD 15 billion. The standard
52The acceleration threshold is commonly set ten percent above the event threshold.See ISDA documentation template.
53This information is based on a press release by Deutsche Bank.
66 II Catastrophe Swaps
maturity is one calendar year and notional amounts are staggered in lots
of USD 5 million. Similar standardized contracts for U.S. wind and earth-
quake events called ”Swiss Re Natural Catastrophe Swaps” (SNaCSTM)
have been launched by Swiss Re (see Swiss Re, 2009).
For U.S. transactions, the Property Claim Services (PCS) division of
Insurance Services Office, Inc. (ISO) acts as loss report provider. PCS
has access to a nationwide network of industry representatives, claim
departments and adjusters, insurance agents, meteorologists, and pub-
lic authorities through which it gathers loss information. In January
2009, several leading firms in the insurance industry have founded the
European index provider PERILS AG, which collects insurance data and
provides a benchmark measure for losses caused by natural catastrophes
in Europe.54
An alternative to the above mentioned contract structure is a trans-
action format termed pure risk swap (or portfolio swap). In a pure risk
swap, two (re)insurance companies exchange uncorrelated catastrophe
risk exposures from their existing books in order to improve portfolio di-
versification and potentially reduce regulatory capital requirements (see
Bruggeman, 2007). Thereby, insurers whose business is locally concen-
trated in an area which is particularly susceptible to natural disasters
can replace a portion of their core risk with another type of peril that
they may not be able to access directly. Pure risk swaps can be executed
through intermediaries, via the web-based Catastrophic Risk Exchange
(CATEX) or directly in the OTC market (see Mutenga and Staikouras,
2007; Cummins, 2008). Like standard catastrophe swaps, these con-
tracts are usually set up such that the present values of the two swap
sides exactly balance and there are no up-front payments between the
counterparties. Instead, money under the swap is only exchanged in
case of a qualifying event. This requires an alignment of the triggers as
well as precise risk modeling in order to match expected losses through
54Founding shareholders of PERILS AG include Allianz SE, AXA, AssicurazioniGenerali, Groupama, Guy Carpenter, Munich Re, Partner Re, Swiss Re, and ZurichFinancial Services. Fore more information refer to the company website.
3.2 Market development 67
the configuration of the terms and conditions of the contract. A promi-
nent risk swap example is a 2003 transaction in which Mitsui Sumitomo
Insurance swapped USD 100 million of Japanese typhoon risk against
USD 50 million of North Atlantic hurricane risk and USD 50 million of
European windstorm risk with Swiss Re (see Cummins, 2008). Since
pure risk swaps reference the counterparties’ insurance portfolios, they
are indemnity-based contracts and not financial instruments. Pure risk
swaps are not the focus of this paper.
3.2 Market development
Although cat bonds and insurance derivatives have been around for al-
most two decades, the market for catastrophe swaps is very young and,
due to its OTC character, information on transactions is currently largely
anecdotal (see Cummins and Weiss, 2009). Nevertheless, industry ex-
perts claim that the market size is increasing rapidly (see, e.g., Cum-
mins, 2008) and the World Economic Forum (2008) estimates that cat
swaps, together with industry loss warranties (ILWs), currently account
for about USD 10 billion in outstanding notional. In May 2009, the
International Swaps and Derivatives Association (ISDA) released a doc-
umentation template for catastrophe swap transactions referencing U.S.
windstorm events.55 The goal of these standardized definitions for key
terms is to reduce uncertainty, improve liquidity and transparency, and
encourage growth in the market. ISDA documentation for other refer-
ence catastrophes, such as California earthquakes, is already planned.
The introduction of ISDA standards is an important step with regard
to the development of the catastrophe swap market as well as the ac-
ceptance of the instrument among investors and is expected to result in
increasing trading volumes (see Swiss Re, 2009). To date, swap counter-
parties are mainly insurance and reinsurance companies. Yet, as for in-
surance futures, new participants, such as investment banks, hedge funds,
and other institutional investors, could soon be encouraged to establish
themselves as market makers.56 Illiquidity of swaps has been cited as
55This template can be accessed on the ISDA website.56See IFEX website for additional information on the insurance futures market
structure.
68 II Catastrophe Swaps
a significant shortcoming relative to tradable catastrophe securities and
insurance options (see, e.g., Cummins and Weiss, 2009). However, in-
creasing standardization and the introduction of ISDA standards now
also enables swap counterparties to assign a contract to other investors
in order to close out their position, thus enhancing liquidity.
3.3 Areas of application
By no-arbitrage reasoning, catastrophe swaps should behave similar to
cat bonds, i.e., they should share their properties of comparatively high
yields and immaterial return correlation with other asset classes.57 Con-
sequently, from an investor’s perspective catastrophe swaps are an at-
tractive means to gain synthetic exposure to natural disaster risk, i.e.,
without requiring to fund the purchase of a cat bond. In addition, it is
not necessary for the protection buyer to actually hold a book of insur-
ance contracts, nor does the protection seller need to have the status of
a regulated insurance entity to be eligible as swap counterparty. There-
fore, apart from hedging insurance risks, catastrophe swaps could also be
applied for investment purposes. An example are negative basis trades
between cat swaps and bonds. This risk-arbitrage strategy, which is com-
mon in the credit markets, aims at exploiting price discrepancies between
the cash and derivative instrument. If a cat bond spread is sufficiently
larger than the spread on an adequately matching catastrophe swap, i.e.,
if the basis is negative, a positive carry can be locked-in by buying the
bond and simultaneously buying protection under the swap agreement.58
The idea is that the occurrence of an event should trigger both instru-
ments such that the loss on the bond is (at least partially) compensated
by the payoff from the swap. Another potential field of application for
cat swaps are synthetic Collateralized Debt Obligations (CDOs) of catas-
trophic risks. In general, a CDO is a securitization of a pool of assets.
These assets are purchased and held by an SPV, which funds the transac-
57These characteristics of cat bond returns have been documented by several au-thors, see, e.g., Litzenberger et al. (1996), Bantwal and Kunreuther (2000), GuyCarpenter (2008) or Cummins and Weiss (2009).
58In analogy to the credit markets, the basis can be defined as catastrophe swapminus catastrophe bond spread.
3.4 Accounting and regulation 69
tion through the issuance of rated security tranches of differing seniority.
In contrast to a so-called true sale or cash CDO structure, a synthetic
CDO does not involve the physical transfer of assets. Instead, the SPV
gains risk exposure by selling protection under swap contracts. While of-
fering substantial risk transfer capacity to (re)insurers, cat CDOs enable
investors to take a position in a diversified portfolio of natural disaster
risks by selecting a tranche which matches their specific risk appetite. A
concrete example is the USD 200 million transaction ”Fremantle 2007-I”
arranged by ABN AMRO, which featured three classes of notes rated
AAA, BBB+ and BB− by Fitch Ratings and utilized cat swaps to trans-
fer the risk to the SPV. For a further discussion of insurance risk CDOs
refer to Forrester (2008).
3.4 Accounting and regulation
In contrast to ILWs, catastrophe swaps are financial instruments, not
insurance contracts. Their regulatory and accounting treatment is un-
ambiguous: Under International Financial Reporting Standards (IFRS)
as well as U.S. GAAP, pure index contracts, such as catastrophe swaps,
are not eligible for reinsurance accounting and consequently do not influ-
ence the underwriting result. Instead, they have to be accounted for at
fair value, which can lead to elevated volatility in the income statement
of the (re)insurer, since technical liabilities are currently not marked to
market (see, e.g., World Economic Forum, 2008). In addition, the cur-
rent Solvency framework in Europe as well as the National Association
of Insurance Commissioners (NAIC) regulation in the U.S. only accept
instruments with an indemnity trigger (i.e., instruments without basis
risk) as (re)insurance contracts. However, it appears that under the
upcoming Solvency II, which is currently still being refined, all instru-
ments which accomplish an effective economic risk transfer could result
in regulatory capital relief (see, e.g., Swiss Re, 2009). In the U.S., on
the contrary, new reserving rules, employing a more economic stance
with regard to risk mitigation instruments, are currently not envisioned.
Klein and Wang (2009) believe this to be a major impediment for U.S.
insurers to use swaps on a larger scale.
70 II Catastrophe Swaps
3.5 Comparison to other risk transfer instruments
Although they do not include an indemnity trigger, catastrophe swaps
are economically largely equivalent to ILWs. In solely referencing indices,
they are not subject to issues of asymmetric information and moral haz-
ard and, as a result, do not require an intensive underwriting process.
Thus, they can serve as cost-effective substitutes for traditional reinsur-
ance contracts. Very much like ILWs and most cat bonds, catastrophe
swaps do expose the protection buyer to basis risk if the transaction
is aimed at hedging a specific portfolio of underwritten insurance con-
tracts. Basis risk arises as the industry index is typically not perfectly
correlated with the losses which the (re)insurer suffers on his book of
business. Cat swaps are highly standardized and avoid the structural
complexities and costs associated with the issuance of full-fledged insur-
ance securitizations, such as setting up an SPV and entering a total
return swap. Consequently, from the perspective of a (re)insurance com-
pany, they are simple to initiate and can be executed much more rapidly
than a cat bond transaction. Compared to ILWs, cat swaps bear the
additional potential of becoming more liquidly tradable once the market
fully takes off. Swaps in general are unfunded transactions and thus by
design not fully collateralized. However, since fluctuations in a contract’s
mark-to-market value will effect regular margin calls between the swap
partners, counterparty default risk for the protection buyer is limited.
Table 1 summarizes the main points of this comparison.
4 Pricing model
Below we introduce a two-stage pricing framework for catastrophe swaps.
Ex-ante the main risk drivers of the instrument are the random num-
ber of natural disasters, their timing, and the associated final loss esti-
mates.59 Apart from determining the fair cat swap spread before the
occurrence of a catastrophe, however, market participants will also need
to value their contracts in the special situation when a catastrophe has
already struck and an initial loss estimate has been published. In this
59Refer to Section 3.1.
4P
ricing
model
71
CatastropheSwaps
Cat Bonds ILWsReinsuranceContracts
triggers industry index
indemnity-basedindustry indexpure parametricparametric indexmodeled loss
double trigger:industry indexindemnity trigger
indemnity-based
moral hazard nonenone if pureindex trigger
low high
basis risk highnone if pureindemnity trigger
high none
standardization very high low high low
transaction cost low high low high
counterpartydefault risk
partiallycollateralized
negligible(collateralized)
collateralizationpossible
collateralizationpossible
accountingtreatment
financialinstrument
dependson trigger
reinsurance reinsurance
Table 8: Catastrophe swaps, cat bonds, ILWs, and reinsurance contracts in comparison
72 II Catastrophe Swaps
case, which will be termed loss reestimation phase in the following, the
timing of the catastrophe and an approximate range for the correspond-
ing losses are known. Yet, there is still uncertainty as to the further
development of the loss estimates until the final loss report or maturity.
Consequently, we distinguish between the pricing of cat swaps before a
catastrophe (ex-ante) and during the reestimation phase.60 Both model
components will be developed in a continuous-time contingent claims
framework without bid-ask spreads, transaction costs, short-selling con-
straints, taxes, or other market frictions.
4.1 Risk-neutral valuation of catastrophe derivatives
Since catastrophe swaps are not insurance contracts, it seems adequate
to price them with financial rather than actuarial approaches.61 Based
on the no-arbitrage principle modern option pricing theory as constituted
by Black and Scholes (1973) as well as Merton (1973) has established the
preference free pricing of derivative instruments under the risk-neutral
(equivalent martingale) measure. The absence of arbitrage opportunities
in the capital markets implies the existence of such an equivalent mar-
tingale measure. For a single unique equivalent martingale measure to
arise, however, markets also need to be complete, meaning that all contin-
gent claims are replicable through available securities (see Harrison and
Kreps, 1979). Hence, the general incompleteness of insurance markets
prevents the uniqueness of the equivalent martingale measure. Neverthe-
less, the literature on catastrophe derivatives and bonds is dominated by
contingent claims valuation frameworks, irrespective of the non-traded
underlying and the fact that natural disaster risk cannot be hedged with
traditional securities.62 Different solutions have been proposed to tackle
the ambiguity with regard to the change of measure.
60Practitioners call exchange traded catastrophe instruments ”live cat” beforea catastrophic event and ”dead cat” during the loss reestimation phase. Seewww.theifex.com
61Although a discussion with industry experts indicated that actuarial pricing cur-rently seems to prevail in practice.
62Refer to the literature review in Section 1.
4.1 Risk-neutral valuation of catastrophe derivatives 73
Some authors assume that market completeness is preserved due to
the existence of other tradable and sufficiently liquid instruments which
are driven by the same source of randomness.63 Consequently, repli-
cating portfolios can be formed and a unique risk-neutral measure is
obtainable by recovering a market price of catastrophe risk from ob-
served quotes.64 While early attempts by CBOT to establish a liquid
market for exchange-traded catastrophe derivatives have failed, several
exchanges have recently re-introduced various types of contracts. Among
these are options and futures on the CME hurricane index and futures on
the PCS indices (see Swiss Re, 2009). Although the market is still juve-
nile, its long-term outlook is promising.65 Alternatively, investors could
turn to OTC catastrophe derivatives or cat bonds in order to mimic
movements of cat swap positions. In addition, as argued by Muermann
(2003), insurance and reinsurance contracts permit indirect trading in
the underlying catastrophe risk. Lastly, Cummins and Geman (1995)
and Vaugirard (2003a,b, 2004) mention that certain energy, commodity,
and weather derivatives could be suitable to track continuous changes in
catastrophe losses, since geological and meteorological determinants of
insurance claims impact the value of these instruments, too.
Another common approach to specify a unique pricing measure de-
spite an incomplete markets set-up dates back to Merton (1976). Anal-
ogous to his reasoning, natural disasters can be treated as unsystematic
shocks to the overall economy which are fully diversifiable, implying
risk-neutrality of the market participants.66 Thus, there is no cat risk
premium and model parameters under the physical and equivalent mar-
tingale measure are identical. This stance is supported by the empir-
63Examples are Cummins and Geman (1994, 1995), Chang et al. (1996), Gemanand Yor (1997), Baryshnikov et al. (2001), Muermann (2003), Vaugirard (2003a,b,2004), Muermann (2008), Chang et al. (2008, 2010).
64A brief illustration of the reasoning behind this proceeding is given in AppendixA. Concrete examples for arbitrage portfolios in the context of PCS options can befound in the empirical study of Balbas et al. (1999), who demonstrate the generalapplicability of financial theory to the sphere of catastrophe derivatives.
65However, due to currently still restricted trading volumes and liquidity, thesemay not be ideal instruments to approximate an instantaneous riskless portfolio yet.
66This stance is adopted in Bakshi and Madan (2002), Lee and Yu (2002), Coxet al. (2004), Jaimungal and Wang (2006), as well as Lee and Yu (2007).
74 II Catastrophe Swaps
ical studies of Hoyt and McCullough (1999) as well as Cummins and
Weiss (2009), who provide evidence for the zero-beta characteristics of
catastrophe-linked instruments.
Finally, the issue of incomplete markets can be overcome by selecting
a particular change of measure such as the well-known Esscher trans-
form (see Gerber and Shiu, 1996).67 Another such distortion operator
for the risk-neutral valuation of insurance contracts has been introduced
by Wang (2000).
We follow Biagini et al. (2008) as well as Wu and Chung (2010) in
not further discussing the choice of martingales and the change from
the physical measure P to the risk-neutral measure Q. Instead, we will
assume that Q has been predetermined according to one of the above-
mentioned alternatives and directly proceed to a risk-neutral formula-
tion of our model framework. The equivalent martingale measure Q is
restricted to the class which only corrects parameters, while stochastic
processes and distributions retain the same characteristics as under P.
4.2 Pricing catastrophe swaps ex-ante
The first, more general component of our pricing approach is meant to
be applied before a natural disaster has occurred in the covered territory.
As mentioned above, the major uncertainty in this phase of a cat swap
transaction centers around the stochastic number and timing of events
during the term of the contract as well as the ultimate losses they cause.
Hence, we will focus on these underlying sources of randomness, while
abstracting from any uncertainty surrounding the reestimation process
of claims. Let (Ω,F ,Q) denote a probability space with the set of all
possible outcomes Ω, a filtration F for the relevant subsets of Ω, and the
equivalent probability measure Q. In line with the prevailing practice
in the literature on catastrophe derivatives, we assume that the number
67Examples are Embrechts and Meister (1997), Schmidli (2003), and Dassios andJang (2003).
4.2 Pricing catastrophe swaps ex-ante 75
of natural disasters nt,t+∆t in any time interval (t, t + ∆t] is Poisson
distributed with intensity λt,t+∆t:
nt,t+∆t ∼ P (λt,t+∆t) , ∀t ∈ (0, T − ∆t], (5)
where λt,t+∆t =∫ t+∆t
tλ(u)du. Furthermore, we follow Chang et al.
(2008, 2010) as well as Wu and Chung (2010) and allow for cyclicality
in the occurrence of catastrophes by incorporating a stochastic Poisson
intensity, which is assumed to adhere to the mean-reverting Ornstein-
Uhlenbeck process
dλ(t) = κ (µλ − λ(t)) dt+ σλdWQλ (t), (6)
with mean reversion rate κ, long-term mean µλ, volatility of the inten-
sity σλ, and a standard Brownian motion dWQλ (t). Thus, the arrival
of natural disasters is governed by a doubly stochastic Poisson process
(Cox process), i.e., two-stage randomization procedure: the Ornstein-
Uhlenbeck process in Equation (6) generates the intensity for the Pois-
son distribution of nt,t+∆t. Intuitively this makes particularly sense for
periodic climate patterns such as the El Nino phenomenon which recurs
on average in five year intervals or the annual Atlantic hurricane season
in the U.S. from June to November. However, it also seems suitable to
capture such as the typical clustering of earthquakes. We aim to provide
some empirical support for this assumption in Section 5.3.
Each catastrophe i is associated with a stochastic final loss estimate,
represented by positive independent and identically distributed (i.i.d.)
random variables Yi with distribution function FY (x). We further as-
sume that nt,t+∆t and Yi are stochastically independent and that there is
no time delay between the occurrence of the catastrophe and the issuance
of the final loss report.68 Consequently, the aggregate final loss estimates
68It is straightforward to extend the model with a deterministic or stochastic wait-ing time between occurrence of the catastrophe and the issuance of the final lossreport. We abstain from this additional layer of complexity as it is neither empha-sized in the literature nor central to the cat swap pricing problem.
76 II Catastrophe Swaps
due to natural disasters in any time interval (t, t+ ∆t] can be expressed
as compound Poisson process with expected value λt,t+∆tEQ(Yi):
Lt,t+∆t =
nt,t+∆t∑
i=1
Yi, ∀t ∈ (0, T − ∆t]. (7)
Recall from Section 3.1 that the payoff of a cat swap transaction is
triggered immediately when the final loss estimate for a single natural
disaster equals or exceeds the event threshold. Hence, in contrast to
the usual procedure for other cat instruments, we must not aggregate
losses from different events over the whole term of the contract. Instead,
each final loss estimate is separately compared to the event threshold at
the time of its occurrence. Since the instrument terminates prematurely
if a trigger event has been identified, its timing is crucial for valuation
purposes and a pricing model needs to capture path dependency. We
achieve this by sequentially reevaluating the loss process in Equation (7)
for infinitesimally small time steps dt from t = 0, i.e., the outset of the
contract until its maturity t = T :
lim∆t→0
Lt,t+∆t = Lt,t+dt ≡ dLt, ∀t ∈ (0, T − ∆t]. (8)
Consequently, instead of one process for the whole term, we generate
a series of compound Poisson processes. Under this set-up, arrivals in
non-overlapping intervals are independent, the probability of exactly one
catastrophe per time step is approximately λt,t+dt, and the probability
of more than one catastrophe in a marginal interval of length dt is vir-
tually zero.69
As described in Section 2, a cat swap consists of a fixed leg, which
comprises the stream of regular premiums by the protection buyer, and a
floating leg, which is the compensation payment by the protection seller
contingent on a trigger event. Swap pricing generally entails the separate
69Alternatively the the occurrence of a catastrophe at each time step could bemodeled as a Bernoulli trial, implying a binomially distributed sum. With regard tototal number of events until maturity this differentiation is less relevant, since, for alarge number of trials, the Binomial converges to the Poisson distribution.
4.2 Pricing catastrophe swaps ex-ante 77
valuation of each leg in a transaction, with the goal of balancing their
present values the through the fair spread scat:
PVfloating = PVfixed(scat). (9)
We define the first passage time (or stopping time) τ for the series of
compound Poisson processes as the earliest instant in which a final loss
estimate is equal to or higher than the event threshold ET :
τ ≡ inft | Lt,t+dt ≥ ET. (10)
Consider a catastrophe swap contract with maturity T , notional N ,
and a payoff which is determined as a preset percentage α of the no-
tional. Assume that a trigger event can occur at any given point in time
and causes an immediate payoff under the swap contract.70 Then the
following is an expression for the present value of the floating leg:
PVfloating = EQ0 [e−rταN1τ≤T ]. (11)
Here, 1τ≤T is the indicator function which equals 1 if τ ≤ T and 0
otherwise. In addition, r is the instantaneous risk-free rate, i.e., the term
structure is assumed to be flat and deterministic. Although relatively
recent publications in the context of cat bond valuation have introduced
stochastic interest rates based on the well-known term structure models
of Vasicek (1977) or Cox et al. (1985) (see, e.g, Lee and Yu, 2002; Vaugi-
rard, 2003a,b; Wu and Chung, 2010), we decide to dismiss this possibility
in favor of computational efficiency. Since catastrophe swap contracts
are exclusively available with one year maturities, the effect of random
changes in the risk-free term structure should have a notably lesser im-
pact than on medium or longer-term instruments. Consequently, this
assumption does not severely influence our results.71
70Since we model the final loss estimate for each catastrophe, we do not need toallow for a development period of the losses after τ . However, through this assumptionwe do abstract from delays in payment collection.
71Young (2004) argues that term structure models are appropriate for instrumentswith a maturity of more than one year.
78 II Catastrophe Swaps
Furthermore, the fixed leg of a cat swap consists of regular premiums
as well as an accrual payment in case the first passage time does not
coincide with a premium payment date:
PVfixed(scat) = PVpremiums(scat) + PVaccrual(scat). (12)
Given a final loss estimate has not exceeded the event threshold, the
buyer of cat swap protection makes payments of scatN∆ti on premium
dates ti, where i= 1,...,n, ∆ti is the length of a premium period (ti−ti−1),
and scat is the fair spread we are looking for. Consequently, we can value
the premium part of the fixed leg as follows:
PVpremiums(scat) =
n∑
i=1
e−rtiEQ0 [scatN∆ti1τ>ti ]. (13)
Finally, the present value of the expected accrual payment for the
time period since the last premium date (τ − ti−1) can be expressed as:
PVaccrual(scat) = EQ0 [e−rτscatN(τ − ti−1)1ti−1≤τ≤ti ]. (14)
Evidently, pricing the cat swap involves solving a first passage time
problem. Due to the compound Poisson process in Equation (7), however,
a closed-form solution for the first passage time cannot be derived (see
Kou and Wang, 2003). Therefore, one needs to resort to Monte Carlo
techniques for an estimation of the fair spread scat.
4.3 Pricing catastrophe swaps
in the loss reestimation phase
After a cat swap has been entered, its mark-to-market value will fluc-
tuate in accordance with the occurrence of natural disasters as well as
the development of their associated loss estimates. More specifically, a
particularly sharp increase in value should be observed when interim
loss estimates approach the contract’s thresholds. However, the ex-ante
pricing approach introduced above does not reflect this sensitivity of
the instrument with regard to interim loss reports. Therefore, it is less
4.3 Pricing catastrophe swaps in the loss reestimation phase 79
suitable to price cat swaps during the reestimation phase, i.e., after a
catastrophe has occurred and an initial loss estimate has been issued.
At that time it is still unclear what the final loss estimate will be and
whether the preset acceleration threshold will be exceeded by an interim
loss estimate before maturity. In the remainder of this section, we aim
to capture this uncertainty with regard to the reestimation of losses by
means of a parsimonious barrier option framework under which closed-
form expressions for the cat swap spread can be derived. Assume that,
under the risk-neutral measure Q, the dynamics of the interim loss esti-
mates Li(t) referenced by the catastrophe swap are adequately described
by a Geometric Brownian Motion:
dLi(t)
Li(t)= rdt+ σdWQ
Li(t), (15)
with volatility σ and a standard Q-Wiener process dWQ
Li(t). The choice
of a diffusion process for the development of catastrophic loss estimates
is common in the literature on cat bond and derivative pricing (see, e.g.,
Bakshi and Madan, 2002; Schmidli, 2003; Biagini et al., 2008). The Ge-
ometric Brownian motion in particular has been applied by quite a few
authors to model the accrual of losses and unpredictability in reporting
over time.72 Apart from that, it ensures analytical tractability and the
resulting lognormally distributed estimates are in line with the empirical
findings of Levi and Partrat (1991) and Burnecki et al. (2000) for PCS
loss data.
We define the distance to the acceleration threshold AT (≥ ET ) at
time t as follows:
DAT (t) =AT
Li(t). (16)
Similarly, we will refer to the ratio of ET to the loss estimate Li(t)
as the distance to the event threshold:
DET (t) =ET
Li(t). (17)
72Examples are Cummins and Geman (1994, 1995), Geman and Yor (1997), Lou-berge et al. (1999), Gatzert et al. (2007), and Wu and Chung (2010).
80 II Catastrophe Swaps
Again, we need to match both legs of the swap transaction through scat:
PVfloating = PVfixed(scat). (18)
We begin with the floating leg. Assume that at time s (0 ≤ s ≤T ) an initial or interim loss estimate Li(s) for a specific catastrophe
is available. Although Li(s) is unknown at the outset of a contract,
i.e., in t = 0, it is deterministic during the reestimation phase. The
interim loss process determined by Equation (15) starts at Li(s) and
the protection buyer receives a fixed payment whenever an interim loss
estimate Li(t) during the remaining term of the contract breaches the
acceleration threshold for the first time, i.e., when DAT (t) hits unity.
This payoff profile equals a binary up-and-in one-touch barrier option
on the interim loss estimates. Define the first passage time τd for the
diffusion process as:73
τd ≡ inft | Li(t) ≥ AT. (19)
From Rubinstein and Reiner (1991a,b) we know that in the present
setting, the following analytic expression for the first passage time den-
sity applies:
h(τd) =ln(DAT (s))√
2πστ3/2d
exp
−1
2
(
− ln(DAT (s)) + (r − σ2
2 )τd
σ√τd
)2
. (20)
Now, recall from Section 3.1, that a compensation payment αN by
the protection seller can also be due at maturity T in case an interim loss
estimate is equal to or higher than the event threshold, i.e., Li(T ) ≥ ET
which implies DET (T ) ≤ 1. Thus, ET can be interpreted as the strike
price of a binary European-type call option. However, in order not to
price certain states twice, this option also needs to include a knock-out
73Due to the design of typical catastrophe swap contracts with a binary payoff assoon as the acceleration threshold is reached, we do not need to allow for a develop-ment period of the losses after τd has occurred. Again, we do abstract from delays inpayment collection. Also note that the model ignores so-called extension thresholdswhich, if exceeded by the interim loss estimates, cause a prespecified extension of thecontract’s maturity to allow for a further accumulation of losses.
4.3 Pricing catastrophe swaps in the loss reestimation phase 81
feature so that it lapses whenever Li(t) ≥ AT , i.e., when the payoff
from the previously discussed one-touch option is triggered. To see this
consider the case where Li(T ) = AT ≥ ET . Here both the up-and-
in one-touch and a simple binary European call option would pay off.
Hence, in order to value the floating leg, we need to combine the up-
and-in one-touch (UAIone touch) with a binary up-and-out call (UAOcall)
option which pays off if and only if Li(T ) ≥ ET and the acceleration
threshold has not been hit during the term of the contract:
PVfloating = UAIone touch + UAOcall. (21)
Applying results from Rubinstein and Reiner (1991a,b), the price of
the up-and-in one touch (binary) in t = s can be expressed as:
UAIone touch =
∫ T
s
αNe−rτdh(τd)dτd
= αN
∫ T
s
e−rτdh(τd)dτd
= αNQ(T ),
(22)
with
Q(u) =
∫ u
s
e−rτdh(τd)dτd = DAT (s)a+bΦ(d1) +DAT (s)a−bΦ(d2),
where s < u, Φ(x) is the standard normal cumulative distribution func-
tion (cdf), and
a =r
σ2− 1
2, b =
√
(r − σ2
2 )2 + 2rσ2
σ2,
d1 =− ln(DAT (s)) − bσ2u
σ√u
,
d2 =− ln(DAT (s)) + bσ2u
σ√u
.
82 II Catastrophe Swaps
Furthermore, the up-and-out cash-or-nothing call (binary) in t = s
is worth (see Rubinstein and Reiner, 1991a,b):
UAOcall = αN (B1(T ) −B2(T ) +B3(T ) −B4(T )) (23)
with
B1(T ) = e−rTΦ(x1 − σ√T ), B2(T ) = e−rTΦ(x2 − σ
√T ),
B3(T ) = e−rTDAT (s)2aΦ(−y1 + σ√T ),
B4(T ) = e−rTDAT (s)2aΦ(−y2 + σ√T ),
and
x1 =− ln(DET (s)) + (a+ 1)σ2T
σ√T
, x2 =− ln(DAT (s)) + (a+ 1)σ2T
σ√T
,
y1 =ln(AT 2/(Li(s)ET )) + (a+ 1)σ2T
σ√T
,
y2 =ln(DAT (s)) + (a+ 1)σ2T
σ√T
.
The fixed leg again comprises the stream of spread payments and
an accrual which accounts for the fact that the contract can be triggered
in between two scheduled premium dates:
PVfixed(scat) = PVpremiums(scat) + PVaccrual(scat). (24)
The protection buyer pays scatN∆ti on the remaining premium dates ti(s < ti ≤ T ) with ∆ti being the length of a premium period (ti − ti−1).
Defining the survival probability of the contract from time s to time
u (i.e., the probability that a trigger event does not occur before u) as
4.3 Pricing catastrophe swaps in the loss reestimation phase 83
(1 −H(u)) with H(u) =∫ u
sh(τd)dτd, the stream of premium payments
has the following value in t = s:
PVpremiums(scat) =
n∑
i=1
scatN∆tie−rti (1 −H(ti))
= scatN
n∑
i=1
∆tie−rti (1 −H(ti))
= scatNΣpremiums,
(25)
where
Σpremiums =
n∑
i=1
∆tie−rti (1 −H(ti)),
and (see, e.g., Vaugirard, 2003b)
H(u) =
∫ u
s
h(τd)dτd = DAT (s)2aΦ(z1) + Φ(z2),
with
z1 =− ln(DAT (s)) − (r − σ2
2 )u
σ√u
, z2 =− ln(DAT (s)) + (r − σ2
2 )u
σ√u
.
Moreover, the present value in t = s of the expected accrual payment
can be expressed as:
PVacc(scat) =
n∑
i=1
∫ ti
ti−1
scatN(τd − ti−1)e−rτdh(τd)dτd
= scatN
[n∑
i=1
∫ ti
ti−1
τde−rτdh(τd)dτd −
n∑
i=1
ti−1
∫ ti
ti−1
e−rτdh(τd)dτd
]
= scatN
[n∑
i=1
(J(ti) − J(ti−1)) −n∑
i=1
ti−1(Q(ti) −Q(ti−1))
]
= scatN
[
J(T ) −n∑
i=1
ti−1(Q(ti) −Q(ti−1))
]
= scatNΣaccrual, (26)
84 II Catastrophe Swaps
where
Σaccrual =
[n∑
i=1
(J(ti) − J(ti−1)) −n∑
i=1
ti−1 (Q(ti) −Q(ti−1))
]
,
and (see, e.g, Gil-Bazo, 2006)
J(u) =
∫ u
s
τde−rτdh(τd)dτd
=− ln(DAT (s))
bσ2
(DAT (s)a+bΦ(d1) −DAT (s)a−bΦ(d2)
).
Inserting (21) to (26) into Equation (18), the fair cat swap spread
can be calculated as follows:
PVfloating = PVfixed(scat)
(UAIone touch + UAOcall) = scatN (Σpremiums + Σaccrual)
scat =(UAIone touch + UAOcall)
N (Σpremiums + Σaccrual). (27)
The decision for a switch from the ex-ante to this barrier option pric-
ing approach must depend on the size of the loss estimate Li(s). If it
is too low, the subsequent reestimation process does not constitute a
major risk driver and the instrument will be worthless under the barrier
option approach. This is due to the fact that small Li(s) are associated
with large distances to the thresholds, which implies low probabilities of
the up-and-in one touch being triggered and the up-and-out call ending
up in the money. Consequently, Li(s) needs to be sufficiently close to
ET so that a substitution of the model is sensible. As a rule of thumb
we suggest to switch whenever the spread under the barrier option ap-
proach associated with a current loss estimate Li(s) is at least equal to
the spread under the ex-ante approach. This criterion is illustrated in
Figure 4(a) by means of a simple numerical example. It is based on a
4.3 Pricing catastrophe swaps in the loss reestimation phase 85
cat swap contract with ET = 20 bn, AT = 22 bn, and up-front premium
instead of regular spread payments, i.e., we simply price the floating leg
as shown in Equation (21). The following parameter values have been
assumed: σ = 0.2, r = 0.02, α = 1, N = 1, and λt,t+∆t = λ = 2. The
dotted line in the center shows the prices from the ex-ante approach for
different T , while the solid line at the top and the dashed line at the bot-
tom represent prices from the barrier option approach for a current loss
estimate of Li(s) = 18 bn and Li(s) = 17 bn, respectively. We observe
that for Li(s) = 17 bn the prices from the ex-ante and the barrier option
approach are quite similar across all T . For Li(s) = 18 bn, in contrast,
the barrier option approach already results in higher cat swap spreads
than the ex-ante approach, which is insensitive to the level of initial and
interim loss estimates. Therefore, in this situation, we would suggest to
consider a substitution of the pricing model if an initial or interim loss
estimate is about 15 percent below the event threshold ET . Finally, to
complete our discussion of cat swap valuation during the reestimation
phase, in Figure 4(b) we have plotted the sensitivity of the up-front pre-
mium and its option components with regard to Li(t) for a remaining
time to maturity of T = 0.5 and the above-mentioned parameter values.
Evidently, an increase in Li(t), i.e., a shrinking distance to ET and AT
inflates the cat swap spread exponentially since it results in a higher
probability of a payoff to the protection buyer.
86 II Catastrophe Swaps
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
time to maturity T (years)
up−
fron
t pre
miu
m
Barrier option approach for Li(s) = 18 bnEx−ante approach
Barrier option approach for Li(s) = 17 bn
(a) Barrier option vs. ex-ante pricing approach
15 16 17 18 19 20 21 22
0.0
0.1
0.2
0.3
0.4
interim loss estimate Li(t)
up−
fron
t pre
miu
m
Cat swapUp−and−in one touchUp−and−out Call
(b) Sensitivity with regard to interim loss estimates
Figure 4: Illustration of the barrier option pricing approach
5 Empirical analysis 87
5 Empirical analysis
5.1 Severity distributions
for natural disasters in the U.S.
In this section we want to select a distribution for the severity of final
losses Yi, which is a crucial component of our ex-ante pricing model
(see Section 4.2). The class of heavy-tailed distributions is particularly
relevant in the context of catastrophe-related claims, since it allows to
properly account for the low frequency high severity character of natu-
ral disasters by assigning comparatively large probabilities to extreme
losses.74
Although some authors have applied distributions with lighter tails,
such as the gamma and exponential distribution,75 extant empirical ev-
idence indicates that those are outperformed by heavy-tailed distribu-
tions. Levi and Partrat (1991), e.g., estimate the lognormal, Pareto,
and exponential distribution based on U.S. hurricane losses between 1954
and 1986 and conclude that the former provides the best fit. Their re-
sult is confirmed by the work of Burnecki et al. (2000), who examine the
time series of the quarterly national PCS loss index from 1950 to 2000,
fitting lognormal, Pareto, Burr, and gamma distributions. Consistent
with this empirical evidence, the lognormal distribution remains by far
the most common choice for the modeling of catastrophe loss amounts in
the literature.76 In contrast to that, Milidonis and Grace (2008) analyze
catastrophe loss data for the state of Florida between 1949 and 2004
and find that the lognormal is outperformed by the Pareto as well as the
74Heavy-tailed distributions have a density function which converges to zero moreslowly than an exponential function, i.e., compared to the exponential distributionthey exhibit more probability mass in the tails. Typical examples are the lognormal,Weibull (with a shape parameter < 1), Pareto, and Burr distribution. For a moredetailed discussion of this class of distributions see Bryson (1974).
75Bakshi and Madan (2002) as well as Dassios and Jang (2003), e.g., use the expo-nential distribution and Aase (1999) as well as Jaimungal and Wang (2006) employthe gamma distribution.
76Examples are Chang et al. (1996), Louberge et al. (1999), Schmidli (2003), Leeand Yu (2002), Burnecki and Kukla (2003), Vaugirard (2003a,b, 2004), Lee and Yu(2007), as well as Wu and Chung (2010).
88 II Catastrophe Swaps
Burr distribution. Based on a likelihood ratio test, they subsequently
decide to employ the latter. Since their samples differ in terms of time
period, geographic focus, and loss normalization method, there is some
ambiguity as to which of these previous empirical studies best suits our
purpose. The article of Milidonis and Grace (2008) is the most recent
and at least partially covers the last decade which has been particularly
prominent with regard to Atlantic hurricane activity. However, it exclu-
sively focuses on the state of Florida while Levi and Partrat (1991) and
Burnecki et al. (2000) use national data. In addition, despite their larger
dataset, Milidonis and Grace (2008) only report estimation results for
the shorter time series from 1990 to 2004, thus ignoring once in a century
events such as the 1906 San Francisco Earthquake or the Great Miami
Hurricane of 1926. Yet, on a normalized scale, damages caused by these
extreme incidents have been shown to considerably exceed those of more
recent natural disasters.77
Hence, we aim to conduct an empirical investigation ourselves in or-
der to select a distribution which is capable of adequately capturing the
tail characteristics of catastrophe losses. We use the datasets of Pielke
Jr. et al. (2008) and Vranes and Pielke Jr. (2009), who, in their recent
empirical work, normalize estimates of economic losses from all major
U.S. hurricanes and earthquakes between 1900 and 2005 to 2005 Dollars.
They have published their results in extensive appendices to the articles,
thereby providing a reliable basis for the estimation of catastrophe loss
distributions.78 Their normalization methodology is based on inflation,
wealth, and population growth in the affected areas and has proven its
capability to effectively adjust historical loss data for societal factors.
The inflation and wealth adjustment are based on the implicit gross do-
mestic product price deflator and the fixed assets and consumer durable
goods statistic, respectively. Both magnitudes are available from the U.S.
77In 2005 Dollars, the 1906 San Francisco Earthquake and the 1926 Great MiamiHurricane would have caused losses of USD 284 billion and USD 161 billion, respec-tively (see Pielke Jr. et al., 2008; Vranes and Pielke Jr., 2009). This compares to USD116 billion for Katrina, which was the second most severe hurricane in U.S. history.
78Note, however, that economic losses regularly exceed insured losses. As a con-sequence, our fitted loss distributions must be viewed as quite conservative for thepurpose of pricing of cat instruments.
5.1 Severity distributions for natural disasters in the U.S. 89
Bureau of Economic Analysis (BEA). Moreover, the population adjust-
ment is conducted with county-level statistics from the U.S. Department
of Census. Figure 5 shows the time series of normalized annual U.S.
hurricane and earthquake losses. The corresponding histograms can be
found in Figure 6. While the histogram for earthquake losses is a little
more pointy than for hurricanes, the shape of both indicates an asym-
metrical distribution with a particularly heavy right tail. Hence, we will
fit the lognormal, Burr, Pareto, and Weibull distribution to the data
by obtaining estimates for the parameters of their respective cumulative
distribution function (cdf):
- lognormal cdf for µ ∈ R and σ > 0
F (x) =
∫ x
0
1√2πσu
e−(lnx−µ)2
2σ2 = Φ
(lnx− µ
σ
)
, (28)
- Burr cdf with a (shape1), b (shape2), and c (scale) > 0 (also called
Burr XII distribution)
F (x) = 1 −(
1 +(x
c
)b)−a
, (29)
- Pareto cdf with a (shape) and c (scale) > 0 (also called Pareto
type II or Lomax distribution)79
F (x) = 1 −(
1 +x
c
)−a
, (30)
- Weibull cdf with a (shape) and c (scale) > 0
F (x) = 1 − e−( xc)a . (31)
In addition, the gamma and exponential distribution will be consid-
ered for comparison purposes:
79Note that this is a special case of the Burr XII distribution with shape parameterb = 1. The Pareto distribution is often used in a generalized form with a third, so-called location parameter which allows for a lower limit above zero. We waive thatgenerality here.
90 II Catastrophe Swaps
- gamma cdf with a (shape), c (scale), and β = 1/c (rate) > 0
F (x) =
∫ x
0
ua−1e−u/c
Γ(a)cadu, (32)
- exponential cdf for β (rate) > 0
F (x) = 1 − e−βx. (33)
While some of these distributions are only defined for strictly positive
values, our dataset also includes years without damages. To tackle this
issue we follow Burnecki et al. (2000), remove the characteristic spike
at zero (see histograms), and estimate the cdfs on the subset of positive
observations. Table 9 provides some descriptive statistics with regard to
the empirical distribution of non-zero losses.
Hurricane losses (USD bn)
max. 161.3311 quantiles
min. 0.0190 0% 0.0190
mean 12.1362 25% 0.4003
median 3.0555 50% 3.0555
s.d. 24.7117 75% 13.2370
skewness 3.7786 100% 161.3311
excess kurtosis 16.7313
Earthquake losses (USD bn)
max. 283.7353 quantiles
min. 0.0019 0% 0.0019
mean 9.0238 25% 0.0526
median 0.1707 50% 0.1707
s.d. 41.2054 75% 1.3028
skewness 6.1166 100% 283.7350
excess kurtosis 37.5699
Table 9: Descriptive statistics for (non-zero) disaster losses (1900–2005)
5.1 Severity distributions for natural disasters in the U.S. 91
1900 1920 1940 1960 1980 2000
050
100
150
Year
Nor
mal
ized
hurr
ican
e lo
sses
(U
SD
billion
)
(a) Normalized annual hurricane losses
1900 1920 1940 1960 1980 2000
050
100
150
Year
Nor
malize
d e
arth
quake
loss
es (
USD
billion
)
(b) Normalized annual earthquake losses
Figure 5: Natural disaster losses in the U.S. (1900–2005)
92 II Catastrophe Swaps
Normalized hurricane losses (USD billion)
Fre
quen
cy
0 2 4 6 8 10
01
23
4
(a) Histogram of normalized hurricane losses
Normalized earthquake losses (USD billion)
Fre
quen
cy
0 2 4 6 8 10
01
23
4
(b) Histogram of normalized earthquake losses
Figure 6: Histograms of normalized natural disaster losses
5.1 Severity distributions for natural disasters in the U.S. 93
In order to assess the fit of the above parametric distributions, we
apply the Kolmogorov-Smirnov and the Anderson-Darling test, which
are based on a comparison of the empirical distribution function F (x) =1n
∑ni=1 1xi≤x and the theoretical distribution function F (x, θ), where θ
represents a vector of parameter estimates. The null hypothesis for both
tests is that the sample at hand comes from the specified distribution
(H0: F (x) = F (x, θ)). Although the chi-square goodness of fit test is
also very common, it will not be considered due to its sensitivity to the
binning of the data and its low power for small sample sizes. Since empir-
ical samples such as our dataset of cat losses generally contain a rather
small amount of extreme observations, it is questionable whether the
chi-square test would reveal a severe misfit of the tail. The Kolmogorov-
Smirnov test, in contrast, is more suitable for small samples. In addition,
the Anderson-Darling test, a modification of the Kolmogorov-Smirnov
test, is one of the globally most powerful goodness of fit tests (see Levi
and Partrat, 1991). It puts a higher weight on the tail of the distribution
and is therefore the most adequate statistic for our purpose.
Tables 10 and 11 contain maximum likelihood estimation (MLE) re-
sults for the cdf parameters as well as the Kolmogorov-Smirnov (KSn)
and Anderson-Darling (ADn) goodness of fit test statistics and their cor-
responding p-values. The lower the respective test statistic (the higher
the p-value), the better the fit of the distribution. As expected, the
exponential distribution is rejected on all reasonable significance levels.
While the gamma distribution does very poorly with respect to the earth-
quake dataset (p-values < 0.01), it fits the hurricane losses surprisingly
well. In fact, its ADn value for the earthquake dataset is even smaller
than that of the more heavy-tailed Pareto distribution. Furthermore,
the lognormal distribution seems to be a reasonable choice for both sam-
ples, although it is outperformed in the tail by the Burr and the Weibull
distribution for hurricane losses and by the Burr and the Pareto distri-
bution for earthquake losses (see respective ADn values). Overall, the
Burr distribution exhibits the lowest Anderson-Darling statistics for both
datasets, leading us to conclude that it is the most suitable candidate
for the severity of natural disaster damages. Note that this confirms the
94 II Catastrophe Swaps
empirical results of Milidonis and Grace (2008). Thus, in the context of
the following analysis, we adopt a Burr loss severity distribution with
the parametrizations as shown in Table 10.
5.2 Derivation of implied Poisson intensities
The vast majority of authors opts for a simple homogeneous Poisson
process to represent the arrival of claims due to catastrophes.80 The
adequacy of this choice has been underlined by the empirical studies of
Levi and Partrat (1991) and Milidonis and Grace (2008), who test the
goodness of fit of the Poisson distribution with ISO/PCS data and find it
to be superior to the alternative binomial distribution. Hence, the choice
of a general Poisson distribution for the frequency of natural disasters
seems hardly questionable. However, more advanced modeling frame-
works have been based on the time-inhomogeneous Poisson process81 or
the doubly stochastic Poisson process (Cox process).82 Just recently,
e.g., Chang et al. (2010) as well as Wu and Chung (2010) suggested to
employ a mean-reverting intensity process, which we also adopted within
our ex-ante model framework in Section 4.2. Since, to the best of our
knowledge, there is little empirical evidence to support these dynamics
for the Poisson intensity as of yet, in this section we employ our model to
derive intensities implied by cat swap market quotes as a basis for a time
series analysis. In analogy to implied volatilities in option markets we
define the implied intensity as the fixed value λt,t+∆t = λ ∀t ∈ (0, T−∆t]
which, if used in our cat swap pricing framework, generates a theoretical
spread equal to the observed market spread.
80See Cummins and Geman (1994, 1995), Chang et al. (1996), Embrechts andMeister (1997), Geman and Yor (1997), Aase (1999), Louberge et al. (1999), Chris-tensen and Schmidli (2000), Bakshi and Madan (2002), Lee and Yu (2002), Vaugirard(2003a,b, 2004), Cox et al. (2004), Jaimungal and Wang (2006), Lee and Yu (2007),as well as Muermann (2008).
81See Embrechts and Meister (1997), Schmidli (2003), and Biagini et al. (2008)82Christensen and Schmidli (2000), Basu and Dassios (2002), and Dassios and
Jang (2003), e.g., use a Cox process with gamma, lognormal, and shot noise intensity,respectively
5.2
Deriva
tion
ofim
plied
Poisson
inten
sities95
lognormal Burr Pareto
Earthquake Hurricane Earthquake Hurricane Earthquake Hurricane
µ -1.3778 0.8064 a 0.4027 4.8358 a 0.4602 0.6620
σ 2.5835 2.1346 b 1.1018 0.5886 c 0.0503 1.1300
c 0.0426 70.1560
KSn 0.1218 0.0683 KSn 0.0712 0.0784 KSn 0.0703 0.0926
p-value 0.4038 0.7699 p-value 0.9420 0.6101 p-value 0.9472 0.3982
ADn 0.8275 0.6403 ADn 0.3264 0.4958 ADn 0.3451 1.5995
p-value 0.4610 0.6102 p-value 0.9167 0.7506 p-value 0.9004 0.1545
Table 10: Parameter estimates and test statistics for the lognormal, Burr, and Pareto distribution
Weibull gamma exponential
Earthquake Hurricane Earthquake Hurricane Earthquake Hurricane
a 0.3557 0.5252 a 0.2050 0.3909 β 0.1108 0.0824
c 0.9774 6.3610 β 0.0227 0.0322
KSn 0.1673 0.0787 KSn 0.2440 0.0993 KSn 0.6546 0.3084
p-value 0.1025 0.6054 p-value 0.0037 0.3159 p-value 0.0000 0.0000
ADn 1.9962 0.5264 ADn 4.3858 1.4651 ADn 81.275 27.4791
p-value 0.0925 0.7196 p-value 0.0057 0.1851 p-value 0.0000 0.0000
Table 11: Parameter estimates and test statistics for the Weibull, gamma, and exponential distribution
96 II Catastrophe Swaps
Due to their OTC character, market quotes for catastrophe-linked
instruments are generally scarce and hardly publicly available. Yet, we
obtained ILW quotes from the BMS pricing grid which is published by
the Thomson Reuters Insurance Linked Securities community on a reg-
ular basis. Where possible, the figures have been cross-checked with
expert judgment based on various sources as well as direct market in-
telligence. Interviews with industry practitioners revealed that, from
a pricing perspective, ILWs and cat swaps are currently not differenti-
ated, which confirms the suitability of ILW premiums for our research
purpose. The dataset comprises time series of monthly up-front prices
from August 2005 to September 2010 for U.S. hurricane and earthquake
contracts with one year maturities and event thresholds of USD 20 bn,
USD 25 bn, USD 30 bn, USD 40 bn, as well as USD 50 bn. In case of a
trigger event each contract pays off its full notional. Table 12 contains
some descriptive statistics for the time series. We notice that, for each
attachment level, the prices of hurricane contracts exhibit a higher mean
and standard deviation than those of earthquake contracts. To acquire
protection against hurricane losses in excess of USD 20 bn, e.g., a protec-
tion buyer needed to pay an average up-front premium of 26.80 percent
of the notional between 08/2005 and 09/2010. The USD 20 bn earth-
quake contract, in contrast, was available for an average price of 14.33
percent. Furthermore, while the minimum premiums for corresponding
hurricane and earthquake contracts differ only slightly, the maximums
for the former are considerably larger. Finally, both hurricane and earth-
quake contracts are on average more expensive for lower thresholds.
In the following, we employ the the ex-ante pricing framework from
Section 4.2 to back out implied intensity time series from the market
quotes.83 Recall that the model has been formulated under the risk-
neutral measure Q. Thus, for it to be applicable to real data, we need
to determine a change of measure as discussed in Section 4.1. Due to
extant empirical support for the zero beta characteristics of cat instru-
83Note that a similar proceeding is applied by Hardle and Cabrera (2010), who backout the implied intensity for a Mexican cat bond and compare it to figures derivedfrom the reinsurance market as well as historical data.
5.2 Derivation of implied Poisson intensities 97
Hurricane contracts
attachment 20 bn 25 bn 30 bn 40 bn 50 bn
mean 0.2680 0.2203 0.1870 0.1437 0.1170
s.d. 0.0509 0.0467 0.0452 0.0386 0.0324
max. 0.3750 0.3167 0.2667 0.2083 0.1750
min. 0.1200 0.0800 0.0575 0.0450 0.0350
Earthquake contracts
attachment 20 bn 25 bn 30 bn 40 bn 50 bn
mean 0.1433 0.1159 0.0924 0.0786 0.0676
s.d. 0.0239 0.0231 0.0197 0.0162 0.0146
max. 0.2000 0.1750 0.1500 0.1200 0.1000
min. 0.1000 0.0800 0.0600 0.0500 0.0425
Table 12: Descriptive statistics: time series of cat swap prices
ments, we assume that catastrophe risk is unsystematic. As a result, the
model parameters under Q remain the same as under the physical mea-
sure P. Since a closed-form solution for the cat swap price is unavailable,
our calculations are based on Monte Carlo simulations. For this pur-
pose, we discretize the model and evaluate the underlying loss process of
Equation (7) for a total of 10,000 sample paths, each one consisting of
252 trading days per year (i.e., ∆t = 1/252). The intuition is that, while
a catastrophe can occur on any day, the official declaration of a trigger
event, i.e., a final loss estimate in excess the of event threshold, and
the resulting transfer of cash flows will only take place on trading days.
Furthermore, the risk free interest rate is the monthly yield on 1-year
U.S. T-Bills84 and, as suggested by our results in the previous section,
we employ a Burr distribution with the parametrizations from Table 10
for the respective loss severities Yi of hurricanes and earthquakes. The
proceeding to capture the implied intensities works as follows: instead
of the stochastic process of Equation (6), we assume a deterministic an-
nual λ which corresponds to an intensity of λ∆t per trading day. Then
84The rates can be accessed on www.ustreas.gov.
98 II Catastrophe Swaps
we embed the valuation framework into a one-dimensional optimization
algorithm which searches for λ by recalculating the model price until it
matches the monthly market quote for each contract. In doing so, we
extract ten time series of annualized implied intensities, i.e., one for each
hurricane and earthquake contract. The results are displayed in Figure
7. Table 13 contains some descriptive statistics.
From a theoretical perspective, the market implied intensities for each
type of peril at any point in time should not vary across event thresholds
(attachment levels), since all contracts are driven by the same catastro-
phes. Yet, there seem to be slight differences in the cross sections: we
observe a tendency for the means and standard deviations of hurricane
implied intensities to increase and those of earthquake implied intensi-
ties to decrease with the attachment level (see Table 13). Unreported
results of Welch’s t-test indicate that the differences in all pairs of means
of the implied intensities for hurricane contracts are insignificant.85 For
earthquake contracts, in contrast, the means of the implied intensity
time series seem to differ significantly, suggesting that the model might
be somewhat less suitable to capture the characteristics of cat swaps on
earthquake risk. From the sample of Pielke Jr. et al. (2008) we derive
a historical number of 1.95 U.S. hurricanes per year between 1900 and
2005. Similarly, records of the Insurance Information Institute show that
an average of 1.80 hurricanes per year made landfall between 1990 and
2009.86 These figures lie well within the overall range of implied inten-
sities for each hurricane cat swap contract in Table 13. In contrast to
that, historical experience from the Vranes and Pielke Jr. (2009) dataset
indicates that, between 1900 and 2005, 0.76 major earthquakes occurred
in the U.S. per year. This figure is at the lower bound of the ranges of
implied intensities we derived from the market prices of the earthquake
contracts, thus again alluding to some refinement potential.
85Welch’s test allows to compare the means of two samples without assuming thattheir variances are equal. The null hypothesis is equality of the means.
86Refer to www.iii.org.
5.3 The stochastic process of implied Poisson intensities 99
time
2006
20072008
20092010
event threshold
20
25
30
3540
4550
implied
inten
sity 01234
5
(a) Hurricane contracts
time
2006
20072008
20092010
event threshold
20
25
30
3540
4550
implied
inten
sity 01234
5
(b) Earthquake contracts
Figure 7: Monthly implied intensities (08/2005–09/2010)
100
IICatastropheSwaps
Hurricane contracts Earthquake contracts
attachment 20 bn 25 bn 30 bn 40 bn 50 bn 20 bn 25 bn 30 bn 40 bn 50 bn
mean 2.1221 2.0840 2.0951 2.1742 2.3011 1.5346 1.4059 1.3077 1.1472 1.0675
s.d. 0.4718 0.5009 0.5658 0.6355 0.7113 0.3627 0.3126 0.3084 0.2472 0.2370
max. 3.2618 3.2778 3.2561 3.4430 3.8274 2.7309 2.3719 2.1101 1.6771 1.5221
min. 0.8609 0.7055 0.5829 0.5870 0.6346 0.9860 0.8704 0.7806 0.7361 0.6423
JB p-value 0.7438 0.4999 0.3289 0.8600 0.9867 0.0070 0.0528 0.1715 0.0709 0.0859
KSn p-value 0.3339 0.5438 0.4606 0.2829 0.6632 0.2900 0.2316 0.3923 0.0432 0.0635
ADn p-value 0.4789 0.6039 0.4283 0.3647 0.7672 0.3956 0.4169 0.5682 0.1104 0.0790
Table 13: Descriptive statistics and p-values of normality tests for the implied intensity time series
Hurricane contracts Earthquake contracts
attachment 20 bn 25 bn 30 bn 40 bn 50 bn 20 bn 25 bn 30 bn 40 bn 50 bn
20 bn 1.0000 1.0000
25 bn 0.9734 1.0000 0.9525 1.0000
30 bn 0.9546 0.9606 1.0000 0.9126 0.9446 1.0000
40 bn 0.9672 0.9679 0.9667 1.0000 0.8975 0.8996 0.9122 1.0000
50 bn 0.9125 0.9237 0.9370 0.9645 1.0000 0.7639 0.7712 0.8188 0.8789 1.0000
loading 0.9775 0.9802 0.9767 0.9914 0.9578 0.9609 0.9765 0.9645 0.9377 0.8260
Table 14: Correlation matrices and factor loadings for the implied intensity time series
5.3 The stochastic process of implied Poisson intensities 101
5.3 The stochastic process
of implied Poisson intensities
Having considered the cross-sectional characteristics of the implied in-
tensities, we now want to focus on our actual research goal: the time
series analysis. More specifically, we aim to determine whether a mean-
reverting Ornstein-Uhlenbeck type process is an adequate assumption
for the dynamics of cat swap implied intensities. A first indication is
provided through Figure 7. We observe a clear sign of cyclicality: im-
plied intensities seem to adhere to some sort of wave pattern over time.
Since an Ornstein-Uhlenbeck process is the continuous-time limit of a
discrete-time first order autoregressive process, i.e., an AR(1), we could
now simply apply the Box-Jenkins methodology to every single implied
intensity time series and assess the fit of different models. Instead, how-
ever, we choose a slightly more efficient way to tackle the issue. Table
14 shows the correlation matrices for the implied intensity time series
of hurricane and earthquake cat swaps from 08/2005 through 09/2010.
Evidently, the implied intensities for contracts on the same type of peril
are highly correlated,87 suggesting that there is at least one common
underlying driver which can be revealed by means of exploratory factor
analysis (EFA). EFA is a statistical technique that describes the covari-
ance (correlation) structure of observed random variables in terms of a
smaller number of latent variables called factors. Once we have identi-
fied these factors for the two perils and derived their respective factor
scores, we can focus our analysis on their time series instead of those for
the individual contracts. The following is an analytical representation
of the general EFA model:
X = Λξ + δ (34)
where X is the vector of observed variables (indicators), Λ represents
the matrix of factor loadings, ξ is the vector of latent variables (factors),
and δ stands for the vector of unique factors (residuals). Applying matrix
algebra, one can derive the covariance matrix Σ implied by the model:
Σ = ΛΦΛ′ + Ψδ. (35)
87All correlations are significant on the one percent level.
102 II Catastrophe Swaps
with Φ being the covariance matrix of the factors and Ψδ being the covari-
ance matrix of the residuals. The parameters (factor loadings and resid-
ual variances) for the EFA model are determined by means of maximum
likelihood estimation (MLE) such that the model implied covariance (cor-
relation) matrix fits its empirically observed counterpart as closely as
possible. Standard EFA assumes multivariate normality of the indicator
variables. In order to check this prerequisite the well-known Jarque-
Bera test as well as the previously introduced Kolmogorov-Smirnov and
Anderson-Darling tests have been applied to the implied intensity time
series (see Table 13 for the respective p-values). Since all test results but
two are insignificant on the five per cent level, i.e., we cannot reject the
null hypothesis that the sample has been drawn from a normal distribu-
tion, we reason that multivariate normality is given.88 The adequacy of
our sample for an EFA is underlined by a Kaiser-Mayer-Olkin (KMO)
Measure of 0.88 for the hurricane and 0.86 for the earthquake implied
intensities. While 0 < KMO < 1, KMO > 0.5 indicates that an EFA can
be performed and if KMO > 0.8, the sample is particularly well suited
for the analysis (see Kaiser, 1974). Furthermore, conducting Bartlett’s
test of sphericity, we reject the null hypothesis of all pairwise correlations
being zero on all reasonable significance levels with a χ2 test statistic of
679.05 for hurricanes and 474.57 for earthquakes.89 Consequently, we
can proceed and apply EFA to the sample.
Initial factor extraction is conducted by means of principal compo-
nents analysis, which provides as many factors as there are indicator vari-
ables, i.e., five for each peril. We find that, for the hurricane contracts,
the first factor explains 96.23 percent of the variance of the implied in-
tensity series. Similarly, for the earthquake contracts, it explains 90.10
percent of the variance. Thus, as previously suspected, a one-factor solu-
tion is an adequate choice with regard to the dimensionality of the model.
The last row of Table 14 contains the factor loadings we obtained for the
88A vector of random variables is multivariate normally distributed if all of itselements follow a univariate normal distribution.
89Note that Bartlett’s test also requires multivariate normality.
5.3 The stochastic process of implied Poisson intensities 103
one-factor EFA based on the previously generated correlation matrices.90
Apart from one exception all factor loadings are higher than 0.90, un-
derlining a strong influence of the common factor for each peril on the
implied intensity time series for the individual contracts.91 In addition,
factor score estimates ξ have been computed by means of the so-called
regression method, which employs the sample covariance matrix Σ and
the estimated factor loadings matrix Λ as follows:
ξ = Λ′Σ−1X. (36)
The time series of the standardized factor scores for hurricane and
earthquake contracts are plotted in Figure 8(a). As for the individual
implied intensities we observe a clear cyclical pattern. Having the time
series of the factor scores at hand, it is now straightforward to test which
stochastic process provides a satisfactory fit to the data. Before estimat-
ing the parameters, one commonly preselects reasonable model specifi-
cations through the patterns observed in the autocorrelation function
(ACF) and partial autocorrelation function (PACF) of the time series,
which have been plotted in Figures 9 and 10. Both the factor scores for
hurricane and for earthquake contracts exhibit the theoretical character-
istics of an AR(1): an ACF which successively decays towards zero and
a PACF with a significant spike at the first lag while all other lags are
insignificant. Since the spike in the PACF is positive, we can expect a
positive coefficient in the autoregressive process.92 The high value of the
partial autocorrelation at lag one suggests that the process is near inte-
grated or might have a unit root. Hence, we conduct the Dickey-Fuller
unit root test with asymptotic and small sample (MacKinnon) critical
values as well as the Philipps Perron test.93 In contrast to the Dickey-
Fuller test, the Philipps Perron test is robust to serial correlation and
heteroskedasticity in the error terms of the test regression. In addition,
90EFA is commonly performed with standardized variables. Hence, all indicatorvariables are demeaned and divided by their standard deviation before the analysis.
91Note that the factor loadings can be interpreted as correlation coefficients betweenindicator variables and factors.
92In the absence of a negative autoregressive coefficient, the oscillating decline inthe ACF could be a sign for seasonality.
93Since the factor score series are standardized, we do not include an intercept inthe equation of the test regression.
104 II Catastrophe Swaps
−4
−2
02
4
time
stan
dar
diz
ed fac
tor
scor
e
2006 2007 2008 2009 2010
hurricane contractsearthquake contracts
(a) Time series of factor scores (standardized)
−4
−2
02
4
time
fact
or s
core
fore
cast
2009 2010
actual valueAR(1) forecast
(b) One month forecasts for the hurricane factor
Figure 8: Intensity factor scores and out-of-sample forecast example
5.3 The stochastic process of implied Poisson intensities 105
the KPSS test for stationarity is considered. The p-values for these tests
with a lag length of one can be found in the left part of Tables 15 and
16. Since all of the unit root tests are statistically significant at least on
the five per cent level (i.e., we reject the null hypothesis of a unit-root)
and the KPSS stationarity tests are insignificant, we conclude that there
does not seem to be a unit root problem.
We now estimate an AR(1) on the hurricane and earthquake factor
score series. Tables 15 and 16 contain the results. For comparison pur-
poses we have also included an AR(2). Generally, if a model is suitable
to capture the pattern inherent in a time series, its residuals should be
white noise. To test the null hypothesis of independent residuals, we
calculate the Ljung-Box (LB) Q-statistic with a lag of 3 and 10.94 Since
these tests turn out to be insignificant (p-values > 0.1000), the residuals
of both the AR(1) and the AR(2) should not be autocorrelated. Due
to their different parameter specifications, the in-sample performance of
these models is compared by means of two common goodness of fit cri-
teria: the Akaike information criterion (AIC) and the Schwarz Bayesian
criterion (SBC) which incorporates a relatively larger penalty term for
the number of parameters. For both criteria the model with the smaller
value is considered superior. Although the AR(1) appears slightly worse
under the AIC, it is associated with a better SBC for the hurricane and
earthquake factor. In combination with the fact that the second coeffi-
cient of the AR(2) is insignificant on the five per cent level, this leads us
to decide in favor of the AR(1).95 Consequently, the Ornstein-Uhlenbeck
process, which is the continuous-time equivalent of the AR(1) seems to
be an adequate choice for the intensity dynamics in a cat swap pricing
model. In addition, it could be considered for out-of-sample forecasts
of implied intensities and, in turn, cat swap spreads (see Figure 8(b)
for an example). Before such an application, however, one should con-
duct further analyses of the short and long term forecasting performance.
94It is common to conduct the Ljung-Box test for a short and a long lag length.95Unreported results for pure moving average (MA), higher order AR, and com-
bined ARMA models indicate their inferiority due to insignificant coefficients, non-white noise residuals, or worse AIC/SBC values.
106 II Catastrophe Swaps
10 20 30 40
−0.
50.
00.
51.
0
Lag
AC
F
(a) Autocorrelation function (ACF)
10 20 30 40
−0.5
0.0
0.5
1.0
Lag
Part
ial A
CF
(b) Partial autocorrelation function (PACF)
Figure 9: ACF and PACF for the hurricane intensity factor
5.3 The stochastic process of implied Poisson intensities 107
10 20 30 40
−0.
50.
00.
51.
0
Lag
AC
F
(a) Autocorrelation function (ACF)
10 20 30 40
−0.5
0.0
0.5
1.0
Lag
Part
ial A
CF
(b) Partial autocorrelation function (PACF)
Figure 10: ACF and PACF for the earthquake intensity factor
108
IICatastropheSwaps
p-values for unit root tests AR(1) AR(2)
Dickey-Fuller 0.0075 coefficients 0.9463 1.1249 -0.2008
MacKinnon 0.0031 p-value 0.0000 0.0000 0.0606
Phillips-Perron 0.0000 LB Q(3)/Q(10) p-value 0.1124 0.2835 0.2293 0.4341
KPSS (stationarity test) > 0.1000 AIC/SBC 67.6732 71.9275 67.3087 73.6901
Abbreviations: AR(1): first order autogregressive process; AR(2): second order autoregressive process; LB Q(t): Ljung-Box Q-statisticfor lag t; AIC: Akaike information criterion; SBC: Schwarz Bayesian criterion.
Table 15: Hurricane intensity factor: unit root tests and model estimation results
p-values for unit root tests AR(1) AR(2)
Dickey-Fuller 0.0241 coefficients 0.8751 1.0470 -0.2027
MacKinnon 0.0089 p-value 0.0000 0.0000 0.0548
Phillips-Perron 0.0000 LB Q(3)/Q(10) p-value 0.5643 0.5577 0.5715 0.3637
KPSS (stationarity test) > 0.1000 AIC/SBC 94.4151 98.6694 93.9122 100.2936
Abbreviations: AR(1): first order autogregressive process; AR(2): second order autoregressive process; LB Q(t): Ljung-Box Q-statisticfor lag t; AIC: Akaike information criterion; SBC: Schwarz Bayesian criterion.
Table 16: Earthquake intensity factor: unit root tests and model estimation results
6 Summary and conclusion 109
6 Summary and conclusion
In this paper, we contribute to the literature through a comprehensive
analysis of the catastrophe swap, a relatively new financial instrument
which has attracted little scholarly attention to date. We begin with a
brief discussion of the typical contract design, the current state of the
market, as well as major areas of application. Subsequently, a two-stage
contingent claims pricing approach for catastrophe swaps is proposed,
which distinguishes between the main risk drivers ex-ante and during
the loss reestimation phase. The occurrence of catastrophes is modeled
as a doubly stochastic Poisson process with mean-reverting Ornstein-
Uhlenbeck intensity. In addition, we fit various parametric distributions
to normalized historical loss data for hurricanes and earthquakes in the
U.S. and find the heavy-tailed Burr distribution to be the most adequate
representation for loss severities. Applying our ex-ante pricing model to
market quotes for hurricane and earthquake contracts, we then derive
implied intensities which are subsequently condensed into a common
factor for each peril by means of exploratory factor analysis. Further
examining the resulting factor scores, we show that an AR(1) provides a
good fit. Hence, its continuous-time limit, i.e., the Ornstein-Uhlenbeck
process should be well suited to represent the dynamics of the Poisson
intensity in a cat swap pricing model.
Future research could be centered around refinements of the pricing
model such as the inclusion of stochastic interest rates which is reason-
able in case longer term cat swap contracts begin to be traded. More-
over, it would be interesting to use implied intensities from cat swap
transactions in valuation models for other, less standardized and liquid
catastrophe-linked instruments such as cat bonds or even reinsurance.
In doing so, one could ensure consistent pricing across different markets
for catastrophe risk and eliminate potential arbitrage opportunities. Fi-
nally, the AR(1) could be applied to produce implied intensity and cat
swap spread forecasts, the accuracy of which would need to be assessed
relative to natural competitors such as the random walk or an AR(1)
with seasonality.
110 II Catastrophe Swaps
7 Appendix: The market price of cat risk
In the case of derivatives on non-traded underlyings market completeness
can be preserved if other liquidly tradable instruments exist, which are
driven by the same source of risk. Hence, replicating portfolios can be
formed and a unique risk-neutral measure is obtainable by recovering a
market price of risk from observed quotes. Below, we briefly recapitulate
the reasoning of Hull (2008) for the simple case where the underlying
process is a Geometric Brownian Motion. Consider two derivatives G1(ξ)
and G2(ξ) on the variable ξ. Suppose the dynamics of ξ, which by itself
is not a traded asset, are adequately described through the following
diffusion process, with drift a and volatility b:
dξt = aξtdt+ bξtdWt. (37)
where dWt is a standard Wiener process. Further, assume that G1(ξ)
and G2(ξ) adhere to the processes:
dG1(ξt) = µ1G1(ξt)dt+ σ1G1(ξt)dWt, (38)
dG2(ξt) = µ2G2(ξt)dt+ σ2G2(ξt)dWt, (39)
such that the Wiener process dWt, which originates from the dynamics of
the underlying, is the only source of uncertainty affecting the derivative
prices. By buying σ2G2(ξt) of the first derivative and selling σ1G1(ξt) of
the second derivative, an investor could now form a portfolio Ω, which
would be instantaneously risk-free (i.e., dWt is eliminated):
Ω(ξt) = σ2G2(ξt)G1(ξt) − σ1G1(ξt)G2(ξt). (40)
Using (2) and (3), the marginal change in value of this portfolio can
be expressed as follows:
dΩ = σ2G2(ξt)dG1 − σ1G1(ξt)dG2
= [σ2G2(ξt)µ1G1(ξt) − σ1G1(ξt)µ2G2(ξt)] dt.(41)
7 Appendix: The market price of cat risk 111
Consequently, this portfolio must yield the risk-free interest rate over
the next marginal time period:
dΩ = rΩdt. (42)
Inserting (4) and (5) and eliminating G(ξ)1G(ξ)2dt, this is equivalent
to σ2µ1 − σ1µ2 = rσ2 − rσ1, or
µ1 − r
σ1=µ2 − r
σ2= λξ. (43)
λξ is the market price of risk for the underlying ξ. If the no-arbitrage
principle holds, λξ at any given point in time has to be the same for any
derivative dependent on ξ, regardless of its specification. The market
price of risk gauges the risk-return tradeoff for financial instruments
based on ξ. Multiplied by the volatility (i.e., the quantity of risk) of the
asset under consideration, it represents the risk premium over and above
the risk-free rate, which investors require to hold this asset: µ− r = λξσ.
112 II Catastrophe Swaps
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119
Part III
Solvency Measurement of
Swiss Occupational Pension
Funds
Abstract
In this paper, we combine a stochastic pension fund model with a traffic
light approach to solvency measurement of occupational pension funds in
Switzerland. Assuming normally distributed asset returns, a closed-form
solution can be derived. Despite its simplicity, we believe the model
comprises the essential risk sources needed in supervisory practice. Due
to its ease of calibration, it is well suited for a regulatory application
in the fragmented Swiss market, keeping costs of solvency testing at a
minimum. We calibrate and implement the model for a small sample
of ten Swiss pension funds in order to illustrate its application and the
derivation of traffic light signals. In addition, a sensitivity analysis is
conducted to identify important drivers of the shortfall probabilities for
the traffic light conditions. Although our analysis concentrates solely
on Switzerland, the approach could also be applied to similar pension
systems.96
96Alexander Braun, Przemys law Rymaszewski, and Hato Schmeiser (2010), A Traf-fic Light Approach to Solvency Measurement of Swiss Occupational Pension Funds,Working Papers on Risk Management and Insurance, No. 74. This paper has beenpublished in: Geneva Papers on Risk and Insurance - Issues and Practice, 2011,36(2):254-282.
120 III Pension Fund Solvency
1 Introduction
The recent crisis in the global financial markets hit not only banks and
insurers but also the pension fund industry. The resulting underfunding
of a large number of pension schemes triggered a discussion about the
rearrangement of prudential regulation and supervision for occupational
pension funds in Switzerland. The obligatory character of occupational
pension plans for the majority of Swiss employees, the large volume in-
vested through them (according to the Swiss Federal Statistical Office,
in 2008 the aggregated book value of assets was approximately equal
to the Swiss GDP), as well as significant social costs linked to poten-
tial insolvencies demonstrate that this debate is not exclusively political.
Instead, a solvency test for pension funds is of considerable relevance
to employees, employers, and pensioners. Supervision and regulation of
pensions in Switzerland is currently conducted at the cantonal level (see,
e.g., Gugler, 2005). The main task of these regulators is to ensure that
the pension funds comply with the legal requirements. Besides, they re-
ceive the annual reports and the report of an independent occupational
pension expert, whose duty is the valuation of a fund’s technical liabili-
ties. The expert also examines whether or not a fund is able to cover its
liabilities. Comprehensive solvency regulation, however, is not present
for occupational pension funds in Switzerland, although banks and in-
surance companies have to adhere to Basel II and the Swiss Solvency
Test (SST), respectively (see, e.g., Eling et al., 2008).
This paper is an attempt to address this issue. We suggest an effi-
cient solvency test for occupational pension funds, providing condensed
information for the stakeholder groups instead of prescribing regulatory
capital. For this purpose, we adopt a model for pension funds under
stochastic rates of return and combine it with a traffic light approach,
allowing an efficient comparison of the risks inherent in different funds
as well as a comprehensible communication of results of the solvency
test. This signal based approach can be used not only to support the
supervisory process, but also to facilitate an increased level of market
discipline. However, more transparency within the pension fund market
1 Introduction 121
can intensify the latter only if insureds are both capable of interpreting
the received signal and of taking actions as a consequence of the infor-
mation they receive.
The literature with regard to stochastic pension fund modeling has
been strongly influenced by the work of O’Brien (1986, 1987) and
Dufresne (1988, 1989, 1990). While the former proposes a continuous-
time approach, the early model of Dufresne operates in a discrete-time
environment. This original discrete-time model has subsequently been
applied and extended in several papers. Haberman (1992) introduces
time delays with regard to additional contributions for unfunded liabili-
ties and, in a consecutive paper, Haberman (1993a) examines the effects
of changes in the valuation frequency for the pension fund’s assets and
liabilities. Furthermore, Zimbidis and Haberman (1993) use the model
with time delays to derive expectations and variances for fund and con-
tribution level distributions. In two additional publications, Haberman
(1993b, 1994) drops the assumption of independent and identically dis-
tributed (iid) asset returns in favor of a first-order autoregressive process
and utilizes the model to compare different pension funding methods. In
contrast to the discrete-time focus of the majority of papers, Haberman
and Sung (1994) present and employ a continuous-time model to simulta-
neously minimize an objective function for contribution rate and solvency
risk. Haberman (1997) reverts to a discrete-time version with iid asset
returns and analyses funding approaches to control contribution rate risk
of defined benefit pension funds. Cairns (1995) extends previous work
by turning to the fund’s asset allocation strategy as a means of control-
ling funding level variability. In a later paper, Cairns (1996) presents
a pension fund model in continuous-time with continuous adjustments
to the asset allocation and contribution rate. A similar model but with
stochastic benefit outgo is discussed in Cairns (2000), while Cairns and
Parker (1997) apply a discrete-time approach and compare the effect of
a change from iid to autoregressive returns on the variability of funding
level and contribution rates. Finally, Bedard and Dufresne (2001) show
that the dependence of successive rates of return can have a considerable
effect on the model results in a multi-period setting.
122 III Pension Fund Solvency
The model we present is based on the discrete-time framework which
has been frequently employed in the literature in order to analyze issues,
such as contribution rate risk or behavior of the funding level over time.
However, it has not been previously considered in the context of solvency
measurement. We adapt the model as to capture the particularities as-
sociated with the occupational pension fund system in Switzerland and
demonstrate that its simplicity and ease of calibration are advantages
for an application as a regulatory standard model in this fragmented
market. The model enables us to estimate shortfall probabilities which
are then funneled into a traffic light approach in order to send a signal to
stakeholder groups, which carries condensed information about a fund’s
financial strength and is straightforward to interpret, even for less sophis-
ticated claim holders. Although the scope of our analysis is limited to
Switzerland, both the model itself and the insights from its application
can be transferred to similar pension systems.
The remainder of this paper is organized as follows. Section 2 sets
the stage with a brief introduction to the particularities of Switzerland’s
occupational pension fund system. The stochastic pension fund model
which forms the basis for the proposed solvency test is presented in
Section 3, while Section 4 explains the traffic light approach to solvency
measurement. Section 5 comprises an exemplary calibration of the model
and illustrates its application by computing shortfall probabilities and
deriving the traffic lights for a small sample of ten Swiss pension funds. A
sensitivity analysis is then conducted in Section 6 in order to identify im-
portant drivers of the shortfall probabilities for the traffic light conditions.
Section 7 focuses on the supervisory review process. Some additional
considerations concerning a potential implementation in Switzerland are
provided in Section 8. Finally, in Section 9, we conclude.
2 The particularities of the Swiss pension system 123
2 The particularities
of the Swiss pension system
The Swiss pension system comprises three pillars. The first pillar is
earnings-related and embedded in the public social security scheme; the
second pillar relates to the mandatory occupational pension fund sys-
tem;97 the third pillar consists of additional benefits that need to be
accumulated individually (see, e.g., see, e.g., Brombacher Steiner, 1999;
OECD, 2009). Our paper focuses on the second pillar which is governed
by the Swiss occupational pension law (abbreviated in German: BVG
and BVV2). At the heart of the second pillar, which, apart from retire-
ment pensions, also provides widow(er) and invalidity pensions, are the
occupational pension funds (in German: Vorsorgeeinrichtungen).
The vast majority of occupational pension funds in Switzerland takes
the legal form of private trusts, where the employees have a right of par-
ity participation in the administrative council (Art. 55 BVG).98 Apart
from single-employer pension funds, which are run exclusively for the em-
ployees of one company, the specific structure of the Swiss economy with
many small and medium-sized businesses necessitates so-called multi-
employer pension funds (in German: Sammeleinrichtungen; see Swiss
Federal Statistical Office, 2009). This relieves small businesses from the
burden of setting up their own pension fund, because they can join a
multi-employer fund which bundles the occupational pension schemes
of several independent firms.99 A change of pension fund can only be
completed by the employer with the agreement of the majority of em-
ployees. The second pillar is covered by a guarantee fund (in German:
Sicherheitsfonds BVG), with the main purpose of subsidizing schemes
with an adverse age structure and guaranteeing the obligatory payments
of defaulted funds.
97Participation in the occupational pension system is mandatory for all employeesof age 18 or older who earn a minimum annual salary of 20’520 CHF (Art. 7 BVG).
98Pension funds of the federation, cantons, and municipalities are institutions underpublic law.
99Employers are obliged to either establish a firm-specific pension fund or to joinmulti-employer fund with the consent of their employees (Art. 11 BVG).
124 III Pension Fund Solvency
Compulsory pension contributions are based on the so-called coordi-
nated salary100 (in German: koordinierter Lohn) of the employee and the
employer has to bear at least half of each installment (Art. 8 and Art. 66
BVG).101 These regular payments are credited to a pension account (in
German: Altersguthaben) and at least compounded with an obligatory
minimum rate of return (currently 2 percent). Once the insured reaches
the retirement age of 65 for men or 64 for women (Art. 13 BVG), the
obligatory pension annuity is calculated by multiplying the annuity con-
version rate, which is currently 6.8 percent, with the final balance of
the pension account (Art. 14 BVG).102 The Swiss Federal Council (in
German: Der Schweizerische Bundesrat) determines both the minimum
interest rate and the conversion rate at two- and ten-year intervals, re-
spectively.103 In general, Swiss occupational pension funds can be set
up either as defined contribution or as defined benefit plans.
One important aspect of the occupational pension fund system in
Switzerland is that funds are legally allowed to temporarily operate with
a deficit of assets relative to liabilities (Art. 65c BVG). Such an under-
funding of liabilities is indicated by the coverage ratio, i.e., the proportion
of the market value of assets over technical liabilities, falling below 100
percent (Art. 44 BVV2). However, the tolerance of a temporary under-
funding is strictly linked to the condition that a pension fund continues
its ongoing obligatory pension payments and takes action to restore full
coverage within an adequate time horizon. In addition, the pension fund
has to promptly inform the regulator, the employer, the employees, and
the pensioners about the magnitude and causes of the asset shortage as
well as countermeasures that have been initiated. The pension fund has
to eliminate the deficit itself as the guarantee fund can merely intervene
100Currently the coordinated salary is the part of an employee’s annual incomebetween 23’940 and 82’080 CHF.101Voluntary payments in excess of the compulsory contributions are possible.102The pension funds can provide annuities over and above the obligatory level.103When determining the minimum interest rate, the Swiss Federal Council takes
into account the recent development of the returns of marketable investments, witha particular focus on government bonds, corporate bonds, equities, and real estate(Art. 15 BVG). Mortality improvements are accounted for through an adjustment ofthe conversion rate.
3 The model framework 125
in case of insolvency (Art. 65d BVG). For this purpose, the fund can
raise additional contributions from the employer and the employees to
rectify the deficit. If and only if all other actions prove insufficient, the
fund is allowed to go below the obligatory minimum interest rate by up
to 0.5 percent for no longer than 5 years.
3 The model framework
We suggest building a solvency framework for occupational pension funds
around underfunding probabilities, at the center of which we need a
stochastic pension fund model. While advanced internal models could
be allowed for the supervision of pension funds with sophisticated risk
management know-how and processes, the requirements of a regulatory
standard model suggest an approach that concentrates on the most es-
sential risk drivers. The complexity of such a standard model should be
kept within adequate limits so that the introduction of the solvency reg-
ulation does not cause an unjustifiably large increase in personnel and
infrastructure cost, especially for smaller occupational pension funds.
Apart from that, a properly developed simple model is capable of cap-
turing the main determinants of pension fund activity (see Cairns and
Parker, 1997). Moreover, the feasibility of the whole concept depends on
sufficient data being available for calibration. This is more likely to be
the case for an approach which entirely relies on observable variables such
as accounting figures. With these considerations in mind, we decide in
favor of a discrete-time model that ensures universal applicability, cost-
efficient implementation, and straightforward calibration.104 The model
we present is based on the work of Cairns and Parker (1997). In the fol-
lowing, we adapt it to the specific characteristics of occupational pension
funds in Switzerland and combine it with a traffic light approach for the
assessment of shortfall probabilities in order to construct a pragmatic
solvency test.
104Equivalent formulations in continuous time can be found in the literature (see,e.g., Cairns, 1996).
126 III Pension Fund Solvency
Consider a one-period evaluation horizon and continuous compound-
ing. If the occupational pension fund is assumed to have a stationary
membership and all cash flows are exchanged at the beginning of the
period, the asset process of the pension fund can be described as follows:
A1 = exp(r1) (A0 + C0 −B0) , (44)
where
- A1: stochastic market value of the assets in t = 1,
- r1: stochastic return on the assets between t = 0 and t = 1,
- A0: assets in t = 0,
- C0: contributions for the period between t = 0 and t = 1,
- B0: benefit payments for the period between t = 0 and t = 1.
The aggregated asset return consists of normally distributed returns
for each asset class in the fund’s portfolio:
r1 =
n∑
i=1
wiri, (45)
with ri ∼ N (µi, σi) , ∀i ∈ 1, . . . , n, where
- wi: portfolio weight for asset class i,
- ri: return of asset class i between t = 0 and t = 1,
- n: number of asset classes in the portfolio.
Note that for some asset classes, the assumption of normally dis-
tributed returns is merely an approximation (see, e.g., Officer, 1972).
However, it will enable us to derive a closed-form solution, which we
consider a very valuable aspect of a standard solvency model.
Since occupational pension funds commonly have a large pool of em-
ployees and pensioners, their liabilities are fairly well diversified and
consequently relatively stable. Hence, the crucial source of risk is consti-
tuted by a pension fund’s asset allocation and a deterministic approach
for the liabilities is justifiable. In general, the value of the life insurance
3 The model framework 127
liabilities in t = 0 is calculated as the present value of expected future
payments to those insured less the present value of expected premium
inflows. These cash outflows are estimated actuarially, taking into ac-
count the age structure and mortality profile of the fund as well as the
targeted rate of return, which needs to be equal to or greater than the
obligatory minimum. Although an actuarial technical interest rate is
commonly used in this context, it is more adequate to apply the current
interest rate term structure. Therefore, we incorporate the market value
of the liabilities into our model and define the corresponding yield as
the valuation rate of interest iv. Issues resulting from a potential mises-
timation of the pension liabilities will be addressed in Section 6. If the
liabilities are assumed to be continuously compounded at iv, we have
the following relationship:
L1 = exp(iv) (L0 +RC0 −B0) , (46)
where
- L1: market value of the liabilities in t = 1,
- iv: interest rate for the valuation of the liabilities,
- L0: market value of the liabilities in t = 0,
- RC0: regular contributions for the period between t = 0 and t = 1.
The assumptions of normally distributed asset returns and determin-
istic liabilities could be relaxed by resorting to numerical solutions, e.g.,
via a Monte-Carlo simulation framework. In that case, many different
distributional assumptions and dependency structures could be incorpo-
rated. Similarly, a numerical solution would allow the introduction of
a longer time horizon and intermediate time steps or a continuous-time
framework.105
The contributions between t = 0 and t = 1, C0, consist of two distinct
elements:
C0 = RC0 +AC0, (47)
105See Buhlmann (1996) for the calculation of ruin probabilities in a similar context,applying a multi-dimensional geometric Brownian motion for the asset dynamics.
128 III Pension Fund Solvency
with
AC0 = αmax [L0 −A0, 0] , 106 (48)
where
- AC0: additional contributions between t = 0 and t = 1 for the
recovery of a deficit in t = 0,
- α: fraction of the deficit in t = 0, which will be covered between
t = 0 and t = 1.
At the beginning of each period due additional contributions are
determined based on the current deficit of assets relative to liabilities.
Hence, AC0 also accounts for additional contributions remaining from
prior deficits. Consider, e.g., a deficit in t = −1. The resulting addi-
tional contribution AC−1 will increase the value of the assets in t = 0,
A0, which then forms the basis for the calculation of AC0. Therefore,
if AC−1 together with the development of the assets and liabilities be-
tween t = −1 and t = 0 was sufficient to eliminate the deficit, there will
be no need for further additional contributions and AC0 will be zero.
Additional contributions are subject to two restrictions. First of all,
α ≥ αmin =1
θ, (49)
which implies
ACmin0 = αmin max [L0 −A0, 0] , (50)
where
- θ: maximum number of years for the elimination of the deficit (set
by the regulator),
- αmin: minimum fraction of the deficit in t = 0, which needs to be
covered between t = 0 and t = 1.
The restriction in Inequality (49) implies that deficits have to be
eliminated within an adequate time horizon (see Section 2), which will
106Note that a negative value of L0 −A0 implies a positive fluctuation reserve or apositive amount of uncommitted funds.
3 The model framework 129
be set by the regulator through the choice of θ.107 As a consequence,
additional contributions in the period under consideration must not fall
below a certain minimum, ACmin0 , as defined in (50), since otherwise
the elimination of the deficit would take too long. Intuitively, the fewer
years available for the fund to restore its coverage ratio at least to unity,
the higher the scheduled additional contributions for each year have to
be. Furthermore,
A0 ≥ Amin0 = βL0, (51)
which implies
ACmax0 = max [L0 − βL0, 0] = max [(1 − β)L0, 0] , (52)
with
- β: lowest acceptable coverage ratio Amin0 /L0 (set by the regulator).
Excessive additional contributions are disputable, since they transfer
the investment risk from the pensioners to the employees and employers.
Accordingly, Inequality (51) accounts for the fact that deficits can only be
healed by means of additional contributions up to a certain amount. For
instance, consider a case in which the value of assets falls to zero. Clearly,
a restructuring of the pension fund is not feasible in this case. Hence, in
order to protect those insured from having to pay an unacceptably large
amount of additional contributions into a pension fund in major distress,
we define a lower limit for the assets Amin0 (a fixed percentage of the pen-
sion fund’s liabilities), which puts a cap on additional contributions per
period. This amount, termed ACmax0 , is defined in Equation (52) and
based on β, i.e., the lowest coverage ratio acceptable by the regulator.
Usually, 0 ≤ β ≤ 1, and the lower β, the higher the maximum amount of
additional contributions that can be charged by the pension fund in any
given period.108 If the assets fall below the threshold Amin0 , the fund will
ceteris paribus be unable to rectify the deficit within a single period. In
107In Switzerland this time period is not legally defined. In current practice, however,a five-year span seems to have emerged as convention.108Note that theoretically β could also exceed one. In such a case, additional con-
tributions would be ruled out by our model framework. To see this, refer to Equa-tion (52).
130 III Pension Fund Solvency
addition, if ACmin0 exceeds ACmax
0 , which is theoretically possible, par-
ticularly for high values of α (low values of θ) and β, the pension fund
faces an existential funding problem, since it would be required to collect
a larger amount of additional contributions than it is actually allowed to.
Under the above assumptions, the assets at the end of the evaluation
period are log-normally distributed with:
E[
A1
]
= E [exp (r1) (A0 + C0 −B0)]
= exp
(
E [r1] +var [r1]
2
)
(A0 + C0 −B0) ,(53)
and
var[
A1
]
= var [exp (r1) (A0 + C0 −B0)]
= (A0 + C0 −B0)2exp
(2E [r1] + var [r1]
)
·(exp (var [r1]) − 1
). (54)
Hence, in order to calculate the first two central moments, which
entirely determine the asset distribution in t = 1 under the assumption of
normally distributed returns, estimates for E [r1] and var [r1] are required.
Using Equation (45), mean and variance for the returns of the aggregated
asset portfolio can be calculated in the following manner:
E [r1] = E
[n∑
i=1
wiri
]
=
n∑
i=1
wiE [ri] (55)
and
var [r1] = σ2r1 = var
[n∑
i=1
wiri
]
=
n∑
i=1
n∑
j=1
wiwjρri,rjσriσrj , (56)
where ρri,rj denotes the correlation coefficient between the returns of
asset class i and j.
4 The traffic light approach 131
4 The traffic light approach
There are several ways to implement a solvency framework. The regu-
lator could, for example, prescribe that each pension fund needs to set
aside regulatory capital based on the outcome of a solvency test. Such
an approach is common in the banking and insurance industries. In the
case of occupational pension funds, however, which do not posses equity
capital, this is rather problematic as the funds would need to build up
reserves from contributions. On the other hand, pension funds have the
possibility to demand additional contributions from employers and em-
ployees, which is similar to authorized equity capital of corporations that
can be drawn in predefined cases. The risk of not being able to raise
this capital when needed is negligible, since it resembles a tax levied by
the government. Thus, we believe that the prescription of regulatory
capital is not the most suitable approach for pension funds. Instead, our
proposal is oriented towards early alert. For solvency measurement pur-
poses, we combine the previously introduced pension fund model with a
concept akin to a value-at-risk framework and funnel the results into a
so-called traffic light approach.
As discussed in the previous section, the model delivers a determinis-
tic value for the liabilities at the end of the analyzed period. Using this
value as a threshold in conjunction with the asset distribution, we can
derive shortfall probabilities for the pension fund under consideration.
These probabilities could be compared to reference probabilities ψ, e.g.,
default rates from rating agency data, in order to generate a signal for
the regulator and the insured. Various categorizations for such a signal
are conceivable. As a straightforward solution, we suggest the following:
- green:
Pr(
A1 ≤ L1
)
≤ ψ, (57)
- yellow:
Pr(
A1 +ACmax1 ≤ L1
)
≤ ψ, (58)
132 III Pension Fund Solvency
- red:
Pr(
A1 +ACmax1 ≤ L1
)
> ψ, (59)
where ACmax1 denotes the maximum amount of additional contributions
which can be charged by the pension fund in t = 1. ACmax1 is deter-
ministic, since it is based on the value of the liabilities in t = 1.109 If
the probability of underfunded liabilities in t = 1 is smaller than the
preset reference probability ψ, the pension fund is assigned a green light.
In addition, if the assets and the maximum additional contributions in
t = 1 are only insufficient to cover the liabilities with a probability lower
than ψ, the light is yellow. In this case the fund is able to suppress the
probability of underfunded liabilities in t = 1 below the reference prob-
ability through its option of charging additional contributions. Finally,
the red light comes up if the probability that the assets plus ACmax1 fall
short of the liabilities exceeds ψ.
5 Implementation and calibration
5.1 Input data
A major advantage of the model is its low implementation cost due to
the use of readily available data. In this section, we illustrate that even
for smaller occupational pension funds with less sophisticated risk man-
agement techniques in place, it should be straightforward to calibrate
and implement the model. For the purpose of calibration, we rely on
accounting figures from the funds’ annual reports. In practice, pension
funds and regulators would be able to use superior data from their man-
agement accounting and financial planning units or databases. As such
internal data is not available to us, we deem annual reports to be the
most adequate and reliable source. Note that this approach is subject to
certain limitations. As defined in Section 3, a solvency test for pension
funds should theoretically be based on market values of assets and lia-
bilities. This is in line with the latest developments in risk management
109Alternatively, different reference probabilities could be chosen for all three condi-tions.
5.1 Input data 133
practice as well as supervisory frameworks for the insurance sector (Sol-
vency II and the SST). Yet, figures derived from annual reports are, in
general, not consistent with market values. In particular, reported pen-
sion liabilities are commonly valued using a technical interest rate, i.e.,
an actuarial rate instead of the prevailing term structure. Consequently,
we substitute iv in Equation (46) with the technical interest rate itec ap-
plied by each fund. The appropriate level of the technical interest rate
is currently controversially discussed in Switzerland. More specifically,
some pension funds seem to be reluctant to reduce it as to reflect the
low interest rate environment which resulted from the financial crisis
2007/2008, implying an even greater discrepancy between market and
book values of the liabilities. Nonetheless, we believe that the following
illustration of the proposed solvency framework offers useful insights.
Tables 17 to 20 show the parameter values we collected for ten occu-
pational pension funds in Switzerland.110 It is important to note that
coverage ratios, assets, liabilities, technical interest rates, and portfolio
weights for 2007 and 2008 have been extracted from annual reports of
the same year. In contrast to that, 2008 and 2009 figures have been used
for contributions and benefits of 2007 and 2008, respectively, assuming
that the funds can perfectly forecast these magnitudes at the beginning
of the period.111 Since the market values of the funds’ assets could not
be directly obtained from their annual reports, they have been estimated
by multiplying the reported coverage ratios (A0/L0) with the book val-
ues of the liabilities. Furthermore, we decided to conduct the analysis
based on seven broad asset classes. Tables 19 and 20 contain the port-
folio weights each fund assigns to the these asset classes.112 Note that
the asset allocation of some pension funds is fairly concentrated. The
implications of this issue together with the effect of insufficient diversi-
110The funds were made anonymous.111This proceeding has been chosen since our model treats contributions and benefits
as deterministic (see Section 3). If the funds are unable to produce reliable forecasts,however, the model could be revised by incorporating benefits and contributions asstochastic variables.112If deemed necessary, the solvency test could be based on a more detailed catego-
rization of the asset side.
134 III Pension Fund Solvency
fication within the subportfolio for each asset class will be addressed in
Section 6.2. While the market environment in 2007 was still relatively
stable, the 2008 figures reflect the major turbulences caused by the global
financial crisis. Thus, this dataset enables us to apply the solvency test
in two different economic settings. In addition, we included single as
well as multi-employer funds to further increase the informative value of
our calculations.
It could be discussed whether the parameters for the asset class return
distributions should be preset by the regulator, thereby reducing discre-
tionary competencies to a minimum. However, taking into account the
ease of estimation and regulatory verification of these parameter values,
we suggest they should be determined by the pension funds themselves.
Therefore, means, volatilities, and pair-wise correlations for the return
distributions of the seven asset classes have been estimated from capital
market time series data. To this end, we have chosen broad indices as
representatives for each asset class.113 The S&P U.S. Treasury Bond
Index and the SBI Swiss Government Bond Index have been selected as
proxies for the international and Swiss government bond markets, respec-
tively. International equities are represented by the MSCI World, while
the Swiss Market Index (SMI) is employed for the Swiss equity market.
Real estate returns are provided through the Rued Blass Swiss REIT
Index and the HFRI Fund Weighted Composite Index serves as a broad
measure for the alternative investments universe. Finally, the Swiss 3M
Money Market Index is used as an indicator for the development of cash
holdings. Distribution moments as well as a correlation matrix based
on monthly returns for these indices from January 1997 to December
2007 are exhibited in Tables 21 and 22. Based on the the simplifying
assumption that the pension funds can perfectly hedge exchange rate
fluctuations at a negligible cost, we have not converted the time series
of the three U.S. Dollar denominated indices into Swiss Francs. Since
hedging foreign currency investments against exchange rate risk is very
113Wherever available, total return indices have been used to account for couponsand dividends.
5.1
Input
data
135
CHF mn Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10
A0/L0 111% 103% 104% 130% 115% 104% 110% 116% 107% 102%
A0 11’591.90 3’136.98 247.65 14’585.39 16’996.47 1’173.10 1’498.20 6’582.35 34’703.20 13’589.43
L0 10’415.01 3’048.57 237.67 11’176.54 14’792.40 1’130.16 1’360.16 5’688.67 32’524.09 13’309.92
C0 1’175.49 260.93 50.42 631.57 750.00 179.18 61.21 1’200.24 1’360.62 891.41
RC0 919.13 249.60 50.42 631.57 750.00 79.45 61.21 1’200.24 1’183.44 891.41
AC0 256.36 12.33 0.00 0.00 0.00 99.72 0.00 0.00 177.18 0.00
B0 787.37 201.88 15.58 837.47 797.20 87.57 90.87 711.13 2’359.56 952.76
itec 3.75% 4.00% 3.00% 4.00% 4.00% 3.00% 3.50% 3.50% 4.00% 4.00%
Table 17: Input parameters for the sample funds in 2007
CHF mn Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10
A0/L0 100% 88% 85% 105% 97% 88% 86% 98% 96% 88%
A0 10’934.19 2’866.88 230.28 11’712.17 14’822.70 1’109.74 1’174.94 6’207.89 30’290.15 11’704.35
L0 10’923.27 3’257.81 270.60 11’186.41 15’265.40 1’261.07 1’373.40 6’317.17 31’611.52 13’285.30
C0 951.45 275.08 47.18 615.18 720.80 168.87 53.28 1’000.23 1’357.44 983.90
RC0 937.62 261.17 47.18 615.18 720.80 106.50 48.56 1’000.23 1’357.44 983.90
AC0 13.83 13.91 0.00 0.00 0.00 62.37 4.72 0.00 0.00 0.00
B0 816.38 213.94 13.39 759.69 787.90 95.05 108.06 782.97 2’113.04 882.45
itec 3.75% 4.00% 3.50% 4.00% 4.00% 3.00% 3.50% 3.50% 3.50% 3.50%
Table 18: Input parameters for the sample funds in 2008
136
IIIPensionFund
Solvency
% Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10
Bonds (intl.) 9.7 14.4 33.5 13.2 18.3 23.0 8.9 12.3 23.5 15.3
Bonds (CH) 24.1 23.9 8.1 24.1 15.6 32.0 8.9 34.6 46.3 30.5
Stocks (intl.) 15.8 12.7 21.9 30.6 17.4 16.6 34.3 20.1 13.2 9.2
Stocks (CH) 4.4 19.0 6.2 7.8 11.2 13.6 5.5 15.9 7.3 18.3
Real Estate 11.4 15.4 6.5 9.0 23.8 2.5 25.0 12.7 5.4 10.3
Alternatives 9.7 4.0 10.3 12.5 4.3 6.9 7.7 2.2 0.0 8.1
Cash 25.0 10.7 13.4 2.9 9.4 5.4 9.6 2.2 4.3 8.4
Table 19: Asset allocations of the sample funds in 2007
% Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10
Bonds (intl.) 9.2 15.8 15.0 22.4 5.2 22.7 6.4 13.5 19.4 18.5
Bonds (CH) 33.4 28.0 29.8 31.8 31.7 29.5 10.0 37.3 53.1 37.0
Stocks (intl.) 8.5 11.9 18.4 15.9 11.0 14.4 25.1 13.2 12.6 6.7
Stocks (CH) 2.4 13.4 5.1 3.7 9.2 12.5 5.1 11.7 6.9 13.5
Real Estate 10.8 16.6 7.8 9.1 28.1 7.4 30.5 14.8 6.1 10.5
Alternatives 17.1 2.4 7.7 10.2 4.5 9.2 7.5 4.7 0.0 8.3
Cash 18.6 11.9 16.4 6.9 10.2 4.4 15.4 4.7 2.0 5.6
Table 20: Asset allocations of the sample funds in 2008
5.2 Results 137
common114 in the asset management industry and the trading costs for
foreign exchange (FX) futures and options are relatively small,115 we
believe this to be an acceptable approach for our purpose. The effect of
imperfect FX hedging will be considered in Section 6.6.
5.2 Results
In order to be able to interpret the shortfall probabilities with the traffic
light approach presented in Section 4, we need to determine reference
probabilities. A straightforward approach is to refer to historic default
rate data as commonly collected and maintained by the large rating
agencies. Consequently, we suggest constructing probability intervals
for rating categories based on default rate experience. The regulator
could then set a minimum target rating for occupational pension funds,
which is linked to the threshold probability.
In the following we use global corporate cumulative default rates from
1981 to 2008 provided by Standard & Poor’s (2009) and establish inter-
vals for the one-year default probabilities as shown in Table 23. While
the use of specific default rates for the investment industry in general
or the pension fund market segment in particular would be preferable,
we need to rely on the rather high-level data available to us. Neverthe-
less, in case of an introduction in practice, it would be advisable for the
regulator to cooperate with rating agencies in order to access a more pre-
cise database. As a reasonable minimum target rating for pension funds,
we propose the lowest investment grade rating category: BBB. Partic-
ipation in the occupational pension fund system in Switzerland is not
voluntary.116 In addition, the volume invested through contributions of
employers and employees is significant. Therefore, occupational pension
114As an example, consider an investment in a foreign currency denominated gov-ernment bond. If left completely unhedged, this would be an outright speculation onthe exchange rate, as the returns in the investor’s home currency will be dominatedby exchange rate movements, implying that the asset does not exhibit the typicalcharacteristics of a government bond.115Flat fees for FX futures trades at the Chicago Mercantile Exchange (CME), the
largest regulated FX marketplace worldwide, can be as low as 0.11 USD, dependingon membership and volume. For more information see http://www.cmegroup.com.116Refer to Section 2.
138
IIIPensionFund
Solvency
S&P USTI SIX SBI MSCI WO SIX SMI RBREI HFRI SMMI
Currency USD CHF USD CHF CHF USD CHF
Mean return (%) 5.83 3.63 7.02 8.50 4.98 10.32 1.60
Volatility (%) 3.71 3.51 14.04 17.22 7.67 7.19 0.29
Indices: S&P US Treasury Bond Index (S&P USTI); Swiss Government Bond Index (SIX SBI); MSCI World (MSCI WO); Swiss MarketIndex (SIX SMI); Rued Blass Real Estate Index (RBREI); HFRI Fund Weighted Composite Index (HFRI); Swiss 3M Money MarketIndex (SMMI).
Table 21: Annualized means and standard deviations for the seven asset classes
S&P USTI SIX SBI MSCI WO SIX SMI RBREI HFRI SMMI
S&P USTI 1.00
SIX SBI 0.57 1.00
MSCI WO -0.27 -0.22 1.00
SIX SMI -0.30 -0.17 0.74 1.00
RBREI 0.00 0.21 0.27 0.26 1.00
HFRI -0.19 -0.16 0.77 0.50 0.22 1.00
SMMI 0.21 0.12 -0.21 -0.14 -0.07 -0.14 1.00
Indices: S&P US Treasury Bond Index (S&P USTI); Swiss Government Bond Index (SIX SBI); MSCI World (MSCI WO); Swiss MarketIndex (SIX SMI); Rued Blass Real Estate Index (RBREI); HFRI Fund Weighted Composite Index (HFRI); Swiss 3M Money MarketIndex (SMMI).
Table 22: Correlation matrix for the seven asset classes
5.2 Results 139
% AAA AA A BBB BB B
lower bound 0.00 0.03 0.08 0.24 0.99 4.51
upper bound 0.03 0.08 0.24 0.99 4.51 25.67
Table 23: One-year S&P default probabilities for different rating classes
funds bear much responsibility for an individual’s retirement provisions.
Against this background, their financial strength should be demanded to
be investment grade. Otherwise the uncertainty for those insured would
be considerable, while they are not free to entrust their money with other
financial institutions of their choice. Moreover, from the perspective of
regulators and financial market participants, it would be very difficult to
argue why pension funds should be allowed a notably lower credit quality
than other financial institutions such as banks or insurance companies.
Having determined the reference probability ψ to be 0.99 percent
(lower bound of BBB), we can now run the model calculations and inter-
pret the results. For each fund, the probabilities for the traffic light condi-
tions in 2007 and 2008 as well as the associated test outcomes (pass/fail)
are presented in Tables 24 and 25, respectively. The calculations for the
yellow condition have been conducted based on a β of 0.95 and 0.90, i.e.,
we limited the maximum additional contributions per period to 5 and 10
percent of the liabilities.117 First, we observe that four out of ten funds
fail the green condition in 2007, although none has underfunded liabili-
ties at the outset (the lowest coverage ratio among the sample funds in
2007 was 102 percent, see Table 17). When inspecting the coverage ratio
these four funds actually reported in 2008 (see Table 18), we find that
all of them in fact suffer from underfunded liabilities, ranging from 2 to
an alarming 15 percent. A fund with a 15 percent deficit of assets rela-
tive to liabilities is in a serious state, since, even for the lower β of 0.90,
it cannot be restructured through additional contributions in a single
period. Taking into account that the current convention in Switzerland
117Recall the definition of β from Equation (52). As discussed, it is ultimately upto the regulator to set a value for β.
140 III Pension Fund Solvency
is a maximum of 5 years to eliminate the deficit, the fund needs addi-
tional contributions of at least 3 percent per year. These are already
close to the 5 percent upper limit which we applied in the analyses for
the yellow condition, underscoring the severity of this situation. Hence,
a failure of the green condition is only acceptable in exceptional cases
and should instantly trigger heightened attention from the supervisor as
well as those insured. In this context it should also be emphasized, that
a need for refinements to the traffic light approach is not automatically
constituted by the fact that Fund 3 ends up with a deficit in excess of
ACmax1 in 2008, although it was assigned a yellow light in the previous
period. The proposed solvency test is exclusively based on probabilities.
Therefore, by all means, a fund can pass one or both traffic light con-
ditions and still end up with substantial unfunded liability at the end
of the period. Being assigned a green or yellow light only means that
the probability for the respective event is sufficiently low. However, if
similar discrepancies are detected for in the context of a comprehensive
quantitative impact study prior to the introduction of the solvency test,
its overall configuration and calibration could be reconsidered.
The second point we learn from Table 24 is that a β of 0.95 is more
than enough to compress the probabilities for the yellow condition to very
low levels for almost all funds. Evidently, this effect is even stronger for
β = 0.90. While the probabilities are virtually zero for the financially
sounder pension funds, even those which did not conform to the green
condition seem to be able to comply with the yellow condition without
difficulties. This illustrates an important point, which had already been
mentioned in Section 4: the option to demand additional contributions
implies that a large part of the pension funds’ investment risk is ulti-
mately borne by employees and employers. Consequently, in case of
a practical implementation of this approach, the supervisory authority
should carefully determine the upper limit on additional contributions.
A further insight we gain from Table 17 is that merely comparing the
coverage ratios, as currently done in supervisory practice in Switzerland,
is generally insufficient to capture the risk profile of pension funds. To
5.2
Resu
lts141
% Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10
green condition 0.01 18.36 7.29 0.00 0.10 0.07 3.28 0.24 0.70 21.42
pass fail fail pass pass pass fail pass pass fail
yellow condition 0.00 3.02 0.47 0.00 0.00 0.00 0.45 0.01 0.00 3.06
β = 0.95 pass fail pass pass pass pass pass pass pass fail
yellow condition 0.00 0.18 0.01 0.00 0.00 0.00 0.03 0.00 0.00 0.13
β = 0.90 pass pass pass pass pass pass pass pass pass pass
Table 24: Probabilities and test outcomes for the traffic light conditions in 2007
% Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund 8 Fund 9 Fund 10
green condition 29.46 99.44 99.96 4.36 66.93 83.99 99.32 48.31 81.83 99.71
fail fail fail fail fail fail fail fail fail fail
yellow condition 1.23 91.07 97.74 0.09 23.06 43.52 94.09 10.94 26.29 92.20
β = 0.95 fail fail fail pass fail fail fail fail fail fail
yellow condition 0.00 53.69 72.41 0.00 2.41 8.34 72.87 0.65 1.19 50.28
β = 0.90 pass fail fail pass fail fail fail pass fail fail
Table 25: Probabilities and test outcomes for the traffic light conditions in 2008
142 III Pension Fund Solvency
see this, compare Fund 3 and Fund 6. Both are characterized by an
equal coverage ratio of 104 percent in 2007 (see Table 19). However,
only Fund 6 passes the green condition of our proposed solvency test
(see Table 24). The reason is simple: a comparison of the coverage ratio
does not take into account differences in asset allocation and the option
to charge additional contributions.
Finally, when examining the results of the solvency test for 2008 in
Table 25, we notice that the financial crisis has strongly influenced the
condition of the pension funds in our sample. Since all funds except
Fund 1 and Fund 4 already exhibit underfundings at the beginning of
the period, their probabilities for both the green and yellow condition
have increased considerably. As a result, not a single pension fund is
able to pass the green condition and only one (Fund 4) passes the yellow
condition based on a β of 0.95. Although the analyzed sample is rather
small, this illustrates that the financial crisis has left the Swiss pension
fund sector in a dramatic situation.
6 Sensitivity analysis
In this section, we explore the main drivers of the shortfall probabilities
for the traffic light approach. These are relevant for the regulator in
various ways, including the political discussion about the state of the
Swiss occupational pension fund sector, the supervisory determination
of model variables, and the decision about measures in case a pension
fund fails the green or yellow condition of the solvency test. We base the
analysis on a standard (representative) pension fund, the input data for
which can be found in Table 26.118 This data is mainly based on 2007
average figures from the Swisscanto (2008) pension fund survey, compris-
ing 265 occupational pension funds in Switzerland, and has been comple-
mented and cross-checked with annual report data from our sample.119
The standard pension fund under consideration is financially sound at
118Unless noted otherwise, β has been set to 0.95.119The Swisscanto series of surveys is published on an annual basis and contains
representative data with regard to the structure, performance, capitalization andportfolio allocation of the participating funds.
6.1 Equity allocation 143
Input parameters Asset Allocation
A0/L0 110% Bonds (intl.) 13%
A0 11’000 Bonds (CH) 27%
L0 10’000 Stocks (intl.) 18%
C0 1’000 Stocks (CH) 10%
RC0 1’000 Real Estate 15%
AC0 - Alternatives 7%
B0 750 Cash 10%
itec 4%
Table 26: Parameters for a representative pension fund
the beginning of the period with a coverage ratio of 110 percent and a
fairly balanced asset allocation.
6.1 Equity allocation
The first sensitivity we examine is related to the proportion of equities
in the pension fund’s portfolio. In this context we proceed as follows:
from the original asset allocation in Table 26, we calculate the weight
of each asset class with regard to the remaining part of the portfolio if
stocks (international and Swiss) are excluded. As an example, consider
the category alternative investments: aside from stocks, the remaining
asset classes together make up 72 percent of the portfolio, 7 percent of
which are alternative investments. Consequently, alternative investments
are assigned a ”residual” weight of 7/72 = 9.7 percent for the analysis.
In the same fashion, we get 18.1 percent for international bonds, 37.5
percent for Swiss bonds, 20.8 percent for real estate, 9.7 percent for al-
ternatives, and 13.9 percent for cash. We then successively calculate
the shortfall probabilities associated with the traffic light conditions for
an increasing portfolio weight of stocks, beginning with zero and ending
with the legal limit of 50 percent. In every case, the percentage is equally
shared between Swiss and international equities. For each allocation, the
remainder of the portfolio is distributed among the other asset classes
according to the previously calculated residual weights.
144 III Pension Fund Solvency
0.1 0.2 0.3 0.4 0.5
0.00
0.01
0.02
0.03
0.04
portfolio weight allocated to equities
shor
tfal
l pro
bab
ility
green conditionyellow conditionreference probabilitygreen (original parameters)yellow (original parameters)
Figure 11: Sensitivity analysis: equity allocation
Figure 11 shows the results. As one would expect, the shortfall prob-
abilities generally increase in the portfolio weight assigned to equities.
The probabilities associated with the pension fund’s original portfolio
composition as shown in Table 26 are represented through a point and a
triangle on the curves at the 0.28 position,120 while the threshold prob-
ability of 0.99 percent has been indicated by the dotted horizontal line.
In its current state this average pension fund evidently passes the green
condition with ease. On the one hand, we observe that the increase of
the probability curve for the green condition is quite strong, revealing
a critical portfolio weight for equities of 0.34, i.e., well below the legal
limit of 0.5. On the other hand, Figure 11 reveals that the fund would in
no case fail the yellow condition, even for the highest possible allocation
to stocks. Hence, as already suspected in the previous section, allowing
additional contributions up to 5 percent of the liabilities within a single
120Note that these points are in fact slightly off the curve since at the position 0.28the curve has been calculated with 0.14 allocated to international and 0.14 allocatedto Swiss stocks, while the original asset allocation shows 0.18 and 0.1, respectively.
6.2 Asset concentration 145
year bears a considerable potential to suppress the shortfall probabilities.
These results have an important implication for the regulator. One of
the prevalent regulatory actions in case of a failure of the green traffic
light condition should be an in-depth analysis of the portfolio compo-
sition of the respective pension fund with a focus on the more volatile
asset classes such as equities. This could be followed by a dialog between
the fund and the regulator to agree on an optimization of the portfolio
to lower the probability of failing the green condition while still retaining
reasonable return potential.
6.2 Asset concentration
In Section 5.1 we calibrated the model based on indices (well-diversified
portfolios), representing various asset classes. This approach implicitly
assumes that pension funds adequately diversify their investments within
the subportfolio of each asset class. In practice, a basic degree of diversi-
fication should be ensured, since pension funds have to obey mandatory
limits on their asset positions. The equity portfolio, e.g., cannot account
for more than 50 percent of a fund’s total assets. In addition, within this
equity portfolio, the maximum investment per individual stock (domes-
tic or international) is currently limited to 5 percent of the total assets.
Yet, with its calibration relying on indices, the solvency test might not be
well suited for an application to pension funds which hold insufficiently
diversified subportfolios. Thus, we briefly illustrate the impact of con-
centration issues within subportfolios, using domestic equity holdings as
an example.
For this purpose, we form an equally weighted portfolio (naıve di-
versification), consisting of an increasing number of stocks which are
drawn from the constituents of the SMI Index. First, the portfolio only
contains a single stock. Additional stocks are then successively added
in random order until the portfolio contains a total of eight stocks.121
For each step, we recalculate mean and volatility of the domestic equity
121The final portfolio consists of the following equities, mentioned in the sequencein which they have been added: Credit Suisse Group, Adecco SA, Roche Holding AG,Holcim Ltd., SGS SA, Nestle SA, Swatch Group AG, and Swiss Re.
146 III Pension Fund Solvency
1 2 3 4 5 6 7 8
number of stocks in domestic stock portfolio
shor
tfal
l pro
bab
ility
0.00
00.
005
0.01
00.
015
0.02
0
yellow conditiongreen conditionreference probability
Figure 12: Sensitivity analysis: asset concentration
portfolio (Table 21, column 4) as well as the correlations with the other
asset classes (Table 22, line 4 and column 4) and recalibrated the model
accordingly. The resulting shortfall probabilities for the solvency test
are illustrated in Figure 12, together with the original case based on the
complete SMI (20 stocks). As expected, the shortfall probabilities tend
to decrease with a rising number of equally weighted stock holdings in
the portfolio, i.e., with a decreasing asset concentration.122 More specif-
ically, for the relatively small number of six equities in the portfolio,
the shortfall probabilities are already fairly close to those of the original
case with the SMI as domestic equity portfolio. Therefore, only very
high degrees of asset concentration in the subportfolios should result in
a notable distortion of the proposed solvency test. However, if a pension
fund naıvely diversifies its holdings over at least half a dozen stocks, the
122Note that while we observe a general decrease in the shortfall probability for agrowing number of stocks in the portfolio, it can sometimes slightly increase when anew stock is added, depending on the order of inclusion. This effect occurs due tochanges on the overall correlation structure in the portfolio.
6.3 Misestimation of liabilities 147
use of indices for calibration purposes seems to be a valid approach. To
further underscore this, note that the standard pension fund underlying
the sensitivity analyses in this section has a total of 18 (international)
+ 10 (domestic) = 28 percent invested in equities (see Table 26). Taking
into account the legal limit of 5 percent per individual stock, this implies
that the fund needs to have at least 28/5 = 5.6 ≈ 6 different stocks in
its portfolio. Similarly, consider a hypothetical pension fund which in-
vested the legal maximum of 50 percent of its portfolio in equities. As
a result, its equity holdings would need to consist of a minimum of 50/5
= 10 different stocks. Nevertheless, if for some reason a subportfolio is
extremely concentrated, the model should be recalibrated accordingly.
6.3 Misestimation of liabilities
Another interesting question centers around the valuation of liabilities.
As explained in Sections 4 and 5, the model at the heart of our approach
to measuring pension fund solvency treats the liabilities as deterministic
and relies on input figures which are reported by the pension funds them-
selves. A current discussion in the Swiss pension fund system revolves
around the technical interest rate, which serves as a discount rate for the
stochastic future cash outflows in the context of an actuarial valuation of
the liabilities. It has repeatedly been stated that many funds hesitated
to lower their technical interest rate in lockstep with the development
of the term structure, thereby understating the present value of their
technical liabilities (see, e.g., Swisscanto, 2008). In addition, despite var-
ious hedging techniques broadly applied in practice (see, e.g., Mao et al.,
2008; Yang and Huang, 2009), there is a remaining uncertainty about fu-
ture mortality improvements and their modeling. Obviously, a potential
misestimation of the liabilities will have consequences for the results of
the proposed solvency test. Thus, Figure 13 displays the sensitivity of
the shortfall probabilities with regard to the estimation error of the tech-
nical liabilities. We observe a pattern similar to the effect of a change
in the portfolio weight of stocks examined in Section 6.1. Again, the
graph comprises a point and a triangle, representing the shortfall prob-
abilities for the original value of the liabilities. In the area to the left of
148 III Pension Fund Solvency
these points, where the liabilities are found to have been overestimated
its liabilities, the actual shortfall probabilities rapidly decline towards
zero. The opposite is true for an underestimation, however. If the liabil-
ities were a mere 1.6 percent higher than originally estimated, the fund
would already breach the threshold for condition green. Beyond that
estimation error, the increase of the probabilities becomes even steeper.
Again, the whole curve for the yellow condition lies below the reference
probability. A practical insight associated with these results is that the
supervisory review should include an in-depth analysis of the method-
ology, assumptions, and database which the pension funds employ to
estimate their liabilities. In case the supervisor has reasons to doubt the
precision of the estimates, the outcome of the solvency test would have
to be adjusted.
6.4 Coverage ratio
Next, we consider the probabilities’ sensitivity to the coverage ratio of the
pension fund at the beginning of the period. Figure 14 shows the results
for coverage ratios varying from 1.1 down to 0.85. Again, the values of
0.3803 percent for the green condition and 0.0085 percent for the yellow
condition associated with the original coverage ratio of 1.1 (see Table 26)
are represented by a point and a triangle on the curves.123 The results
for coverage ratios over and above 1.1 are not particularly interesting
as the shortfall probabilities quickly become very small. Similarly, we
observe that for coverage ratios of below 0.9, the probabilities are very
close to 1 and thus far beyond any reasonable threshold. Some more
attention should be devoted to the region between 0.9 and 1.1. Just
below 1.1, both curves initially exhibit a slightly negative slope, which
then sharply increases in magnitude below 1.05 for the curve representing
the green and below 1.0 for the curve representing the yellow condition.
This is an important result: pension funds with a coverage ratio of below
1.05 are relatively likely to end up with underfunded liabilities at the end
of the period. In addition, if their liabilities are just about covered at
the beginning of the period, even the probability for failing the yellow
123Due to their small difference relative to the scale chosen for the overall graph,these points appear as one.
6.4 Coverage ratio 149
−0.04 −0.02 0.00 0.02 0.04
0.00
0.01
0.02
0.0
30.0
4
estimation error for technical liabilities in t = 1
shor
tfal
l pro
bab
ility
green conditionyellow conditionreference probabilitygreen (original parameters)yellow (original parameters)
Figure 13: Sensitivity analysis: misestimation of liabilities
0.85 0.90 0.95 1.00 1.05 1.10
0.0
0.2
0.4
0.6
0.8
1.0
coverage ratio in t = 0
shor
tfall p
robab
ility
green conditionyellow conditiongreen (original parameters)yellow (original parameters)
Figure 14: Sensitivity analysis: coverage ratio
150 III Pension Fund Solvency
condition grows to levels where it begins to be perceivable. This has
important implications for the Swiss occupational pension fund system.
In particular, the common practice of letting pension funds continue their
business with dramatically underfunded liabilities without a specifically
tight supervisory review and careful amendments to their overall strategy
has to be considered inadequate from the point of view of modern risk
management and solvency regulation principles.
6.5 Lowest acceptable coverage ratio
Furthermore, we examine the sensitivity of the shortfall probability for
the yellow condition with regard to β, i.e., the lowest coverage ratio
acceptable by the supervisor. The results are depicted in Figure 15.
Consistent with our previous analyses, the curve begins at a β of 0.95
which is associated with a near zero probability (0.0085 percent) of an
underfunding after additional contributions (marked by a triangle). For
an increasing β, however, we observe a non-linear rise in the probabil-
ity. A β of 0.97, for example, is already associated with a probability of
0.046 percent, which is more than five times the above value. When β
approaches 1, i.e., additional contributions are ruled out, the probability
reaches the value of 0.3803 percent associated with the green condition
(marked by a point). This suggests that the impact of each percentage
of additional contributions allowed to fix deficits is relatively strong.
Since both the current and lowest acceptable (minimum) coverage
ratio have a strong influence on the shortfall probability for the yellow
condition, we finally want to consider their joint impact in order to assess
which combinations have counterbalancing or strengthening effects (see
Figure 16). A very important observation is, that for β below approx-
imately 0.96, the yellow condition becomes rapidly less binding, even
if we assume that the fund already begins the period with a relatively
weak coverage ratio of around 1. For a β of 0.90, the shortfall probabil-
ity for condition yellow is virtually negligible until the coverage reaches
0.95 where it begins to rise sharply. In contrast, if β is set to 1 (no
additional contributions allowed), even a relatively small underfunding
6.6 Exchange rate risk 151
leads to large shortfall probabilities, which reach 100 percent around the
coverage ratio of 0.90. Thus, as previously suspected, the regulator’s
choice of β has a crucial impact on the bindingness of the yellow traffic
light condition. Simply allowing pension funds with a low coverage ratio
to draw on large amount of additional contributions per period provides
them with a convenient means to continue business without significant
revisions to their asset or risk management practices. This somewhat
contradicts the purpose of a solvency test and essentially means that
premium payers subsidize pensioners, an effect which is generally not
intended within the second pillar of the Swiss pension system.
0.95 0.96 0.97 0.98 0.99 1.00
0.00
00.
001
0.00
20.
003
0.00
4
lowest acceptable coverage ratio
shor
tfal
l pro
bab
ility
green (original parameters)yellow (original parameters)
Figure 15: Sensitivity analysis: minimum coverage ratio
6.6 Exchange rate risk
In Section 5.1 we mentioned that the U.S. Dollar denominated indices
have not been converted to Swiss Francs for the model calibration. For
this to be an adequate approach, pension funds would need to hedge
out major exchange rate fluctuations in their asset portfolios at an im-
152
IIIPensionFund
Solvency
coverage ratio in t = 0
0.90
0.95
1.00
1.05
lowes
t accep
table co
vera
ge ra
tio
0.90
0.92
0.94
0.96
0.98
1.00
shortfall p
robab
ility
0.2
0.4
0.6
0.8
Figure 16: Sensitivity analysis: actual and minimum coverage ratio
6.6 Exchange rate risk 153
1 0.75 0.5 0.25 0
FX hedged fraction of U.S. Dollar denominated portfolio
shor
tfal
l pro
bab
ility
0.00
00.
010
0.02
00.0
30yellow conditiongreen condition
reference probability
Figure 17: Sensitivity analysis: FX hedging
material cost. In this section, we relax the assumption of a perfectly
FX hedged portfolio and analyze the impact of exchange rate risk on the
shortfall probabilities. To this end, we convert the time series of the three
U.S. Dollar denominated indices124 to Swiss Francs and compute the as-
sociated returns. For each index, we then calculate weighted averages
of the returns of the original time series (U.S. Dollars) and the returns
of the converted time series (Swiss Francs), applying weights of 100, 75,
50, 25, and 0 percent. These weights are meant to reflect the percentage
of the foreign currency denominated portfolio which has been hedged
against exchange rate risk.125 Accordingly, a 100 percent weight on the
returns of the U.S. Dollar index time series reflects a situation where the
whole portfolio is immune to exchange rate fluctuations, whereas a 100
percent weight on the returns of the Swiss Franc converted time series
implies no FX hedging activity at all. For all combinations in between,
the exchange rate risk is assumed to be partially hedged. Subsequently,
means, volatilities, correlation matrix, and the resulting shortfall proba-
bilities are recalculated for each case (see Figure 7 for the results). For
154 III Pension Fund Solvency
both the green and yellow condition the shortfall probabilities expect-
edly rise with the exchange rate exposure. If foreign investments in the
pension fund’s portfolio remain entirely unhedged, the probability for
the green condition turns out to be more than seven times higher than
in the case of a perfect FX hedge. However, we also see that the pension
fund would still be assigned a green light if it protects only half of its
foreign asset holdings against exchange rate risk. In addition, the yellow
condition is passed in every case, even without any FX hedging activi-
ties. From these insights we conclude that a model calibration based on
foreign currency denominated indices should be valid, as long as pension
funds hedge a large part of their foreign asset portfolio against exchange
rate fluctuations. If this is not the case, a model recalibration should be
requested and monitored by the supervisory authority.
7 Supervisory review and actions
In analogy to Basel II and the SST, the approach we introduced and
illustrated throughout the previous sections should be embedded into
a comprehensive supervisory review process. As part of this process,
occupational pension funds should be obliged to report and comment
on certain key figures resulting from the application of the supervisory
model in regular intervals. This quantitative solvency report could be
accompanied by a qualitative judgment of risks which are not explicitly
covered by the model framework, such as credit and operational risk.
In order to react properly to the risk and solvency situation of pension
funds, the regulator should possess a variety of competencies. According
to the degree of compliance with the traffic light conditions, a certain
catalog of measures could be decided. For pension funds which are as-
124These are the S&P U.S. Treasury Bond Index, the MSCI World, and the HFRIFund Weighted Index. See Table 21.125While this is a rather general analysis, abstracting from a detailed characteri-
zation of the associated transactions with regard to strategy, timing, instruments,volumes, strike prices, etc., we believe it to be satisfactory in this context. A moreelaborate treatment of FX hedging issues in the asset management industry is beyondthe scope of this paper.
7 Supervisory review and actions 155
signed a green light, the regulator could stick to periodic reviews focused
on the adequacy of the regulatory standard model. As illustrated in Sec-
tion 6.2, a recalibration could become necessary in certain cases.
If a pension fund hands in a regulatory report with a yellow light, it
should be subjected to closer scrutiny. This could, for instance, comprise
a comprehensive check-up of the fund’s assets, liabilities, liquidity, and
cash flow profile with a particular focus on valuation methodologies and
assumptions. In addition, such funds could be put on a regulatory watch
list, resulting in a shortened reporting interval. The requirement to de-
sign a concept for financial restructuring is also a potential measure to
be imposed on funds in the yellow category. Such a concept would need
to cover the asset and liability side, demonstrating how a solid solvency
situation can be restored through a combination of portfolio adjustments
as well as capital replenishment by means of additional contributions. In
any case, the regulator would have to ensure that the lower and upper
limit for additional contributions is obeyed.
If a fund is assigned a red light, more drastic consequences would be
necessary. These could comprise constraints to the management’s ability
to choose its asset allocation with the aim of preventing the fund from
incurring additional investment risks. Otherwise the problem of ”gam-
bling for resurrection” could arise, meaning that the fund management
tries to rescue the institution through large bets. Furthermore, the regu-
lator should be authorized to issue directives to the management of red
light funds. Moreover, the inclusion of additional contributors should
be suspended until the fund has been restored to an acceptable level of
solvency. This protects prospective fund members from the excessive
subsidization of current pensions through their contributions. Finally,
the regulator should have the ability to replace the board and fund man-
agement of highly distressed pension funds with a special administrator.
Beyond that, rules with regard to the publication and dispersion of
these easily interpretable solvency signals could increase transparency,
and, given the receivers can appropriately react to the information, pro-
156 III Pension Fund Solvency
mote market discipline. Hence, apart from the supervisory authority,
receivers of the signal should be employers, employees, transaction part-
ners, and the general public. Due to a reduction of information asymme-
tries, pension funds with an abnormally high shortfall probability would
thus have to face public scrutiny and reactions of their business partners.
8 Some notes on a potential introduction
in Switzerland
An important organizational requirement for pension funds which would
arise from a concrete introduction of the solvency test is the recruit-
ment of personnel with an adequate background for the application and
maintenance of stochastic pension fund models. Further requirements
relate to the necessary infrastructure for running the model, including
databases and software. In order to align the fund manager’s interest
with that of the insured, the former should benefit from the prevention
of yellow signals. This could, for example, be achieved by linking his
variable compensation to a combination of fund performance and traffic
light signals.
As explained in Section 2, Swiss occupational pension funds take
the legal form of private trusts, which have very limited possibilities
of self-supervision. A corporation, in contrast, has bodies such as the
board and annual meeting, which serve supervisory purposes. Conse-
quently, the introduction of a regulatory framework for occupational
pension funds could be complemented with a fundamental reformation
of the legal forms they can adopt. The recommended traffic light ap-
proach would then receive additional disciplinary weight through board
and shareholders of the corporation as receivers of the signal.
The degree of market discipline emanating from the traffic light ap-
proach strongly depends on its familiarity to stakeholder groups and
the expected consequences of bad signals such as the potential threat of
many insured wanting to change their pension fund. However, employees
8 Notes on a potential introduction in Switzerland 157
are currently not free in their choice, which greatly reduces this sort of
pressure. Therefore it needs to be discussed whether the introduction of
the solvency regulation should be accompanied by a liberalization of the
market itself, enabling a free choice of the pension fund. The downside
would be, that the situation of an already distressed fund could further
deteriorate in case a large number of insureds wants to redeem their
holdings. Nevertheless, we believe that more flexibility in this regard is
warranted and would be an important step towards an efficient regula-
tion of Swiss occupational pension funds.
Finally, the regulator could conceive of establishing higher barriers to
entry for pension schemes. These could, for example, be fit and proper
conditions for the individuals managing the pension fund, as common
for employees in other branches of the financial services industry such
as banking. Participation in the Swiss pension fund market is currently
not tied to specific criteria. Setting prerequisites would likely lead to a
consolidation, reducing the current number of funds from approximately
2’500 in 2008126 to a number which can be supervised more efficiently.
126See Swiss Federal Statistical Office under http://www.bfs.admin.ch.
158 III Pension Fund Solvency
9 Conclusion
We adopt a stochastic pension fund model and combine it with a traf-
fic light approach for solvency measurement purposes. The calibration
and implementation of the model with a small sample of ten pension
funds illustrates its application for the computation of probabilities and
derivation of traffic light signals. The model adequately captures the
particularities associated with the occupational pension fund system in
Switzerland. Due to its efficiency and ease of calibration it is well suited
as a regulatory standard model in this very fragmented market, keeping
costs of the solvency test at a minimum, even for small pension funds
with less sophisticated risk management know-how and infrastructure.
In addition, the sensitivity analysis identifies important drivers of the
shortfall probabilities and can thus assist the regulator with regard to
specific decisions associated with the configuration of the framework.
However, some questions remain in respect to model design and cali-
bration. First, we did not explicitly account for credit risk in the fund’s
asset portfolio. Therefore, the supervisory authority should exercise ad-
ditional care with regard to solvency test results for pension funds with a
relatively high proportion of default-able instruments, such as corporate
bonds, in their portfolio. Second, an incorporation of stochastic liabil-
ities and different statistical distributions for the modeled asset classes
could be discussed, although a departure from the associated assump-
tions would necessitate a switch from the closed-form to a numerical
solution. Third, portfolio diversification and foreign currency exposure
have to be borne in mind as critical factors with regard to the proposed
calibration procedure. Finally, a practical implementation would need
to be preceded by a comprehensive quantitative impact study for the
majority of Swiss pension funds. Overall, we consider this straightfor-
ward framework to be an adequate first step towards a state-of-the-art
solvency regulation of occupational pension funds in Switzerland.
References 159
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163
Part IV
Stock vs. Mutual Insurers:Who Does and Who ShouldCharge More?
Abstract
In this paper, we empirically and theoretically analyze the relationship
between the insurance premium of stock and mutual companies. Eval-
uating panel data for the German motor liability insurance sector, we
do not find evidence that mutuals charge significantly higher premiums
than stock insurers. If at all, it seems that stock insurer policies are
more expensive. Subsequently, we employ a comprehensive model frame-
work for the arbitrage-free pricing of stock and mutual insurance con-
tracts. Under the chosen set-up, the formulae for the premium and the
present value of the equity of a stock insurer are nested in our more
general model. Based on a numerical implementation of the framework,
we then compare stock and mutual insurance companies with regard
to the three central magnitudes premium size, safety level, and equity
capital. Although we identify certain circumstances under which the mu-
tual’s premium should be equal to or smaller than the stock insurer’s,
these situations would generally require the mutual to hold less capital
than the stock insurer. This being inconsistent with our empirical results,
it appears that policies offered by stock insurers are overpriced relative
to policies of mutuals. While our analysis focuses on the insurance con-
text, the insights can be transferred to other industries where mutual
companies are an established legal form.127
127Alexander Braun, Przemys law Rymaszewski, and Hato Schmeiser (2011), Stockvs. Mutual Insurers: Who Does and Who Should Charge More?, Working Papers on
Risk Management and Insurance, No. 87.
164 IV Stock vs. Mutual Insurers
1 Introduction
Private insurance firms in many insurance markets can be organized ei-
ther as mutual or stock insurance companies. Similar to policyholders of
a stock insurance company, those of a mutual insurer are obliged to pay
the insurance premium which, in turn, entitles them to an indemnity
payment contingent on the occurrence of a loss. Apart from that, how-
ever, several important differences between these two legal forms exist
(see, e.g., Smith and Stutzer, 1990). First of all, in contrast to stock in-
surers, mutuals are in fact owned by their policyholders. By paying the
respective premium, the buyer of a mutual policy becomes a so-called
member, which is economically equivalent to acquiring a policyholder
and an equityholder stake in the firm.128 As a result, those insured by a
mutual are usually granted direct or indirect participation in the admin-
istrative bodies and should thus be able to exert influence on business
decisions. To establish a similar position, policyholders of stock insur-
ance companies would need to acquire ownership rights by purchasing
the company’s common stock. Unlike the shareholders of a stock insurer,
however, members of a mutual cannot simply sell their equity stake. This
is due to the fact that, in practice, it is not explicitly differentiated from
the policyholder stake and a secondary market does not exist. Hence,
the only way to fully realize the value of the equity are liquidation or
demutualization of the company, which would need to be enacted col-
lectively by a majority of the members.129 A further difference to stock
insurers is, that mutual members can expect occasional premium refunds
if the company is profitable. These payouts are economically akin to the
dividends a stock insurer distributes to its shareholders. Finally, stock
insurance companies cannot draw on their policyholders to recover fi-
nancial deficits, whereas the membership in a mutual insurer might be
associated with the obligation to make additional premium payments
contingent on the firm being in financial distress. These additional pre-
miums are virtually authorized capital, i.e., equity which has not been
128Rasmusen (1988) describes rights and obligations resulting from a membershipin savings and loan associations, credit unions, and mutual savings banks.129In the course of a demutualization, the insurer changes its legal form and is
transformed into a stock company.
1 Introduction 165
paid in yet (see, e.g., Mayers and Smith, 1988). Since the legal form de-
termines these rights and obligations associated with the purchase of an
insurance contract, it should ceteris paribus result in different arbitrage-
free prices for policies, covering identical claims.
While there is a large body of literature, dealing with various aspects
of mutual and stock companies, to the best of our knowledge, there has
not yet been a rigorous empirical and theoretical analysis of the rela-
tionship between the premium of stock and mutual insurers. Therefore,
in this paper, we want to shed some light on this research question by
evaluating panel data for the German motor liability insurance sector.
In addition, we contribute to the literature by employing a contingent
claims model framework to consistently price stock and mutual insurance
contracts. For this purpose, we split the arbitrage-free mutual insurance
premium into an ownership and policyholder stake, both of which are
further decomposed into distinct option-theoretic building blocks. The
model explicitly takes into account the restricted ability of members to
realize the value of their equity stake as well as the mutuals’ right to
charge additional premiums in times of financial distress, which will be
termed recovery option in the course of this paper. Under the chosen
set-up, the formulae for the premium and the present value of the equity
of a stock insurer are nested in our more general model. Moreover, we
derive conditions, under which the premiums of a stock and a mutual
insurance company should theoretically be equal. Finally, combining our
empirical and theoretical results, we are able to derive relevant economic
implications. While we apply our model within the insurance context, its
insights can be transferred to other industries where mutual companies
are an established legal form such as credit unions and pension funds.
The remainder of this paper is organized as follows. Section 2 con-
tains a comprehensive overview of previous literature on issues surround-
ing stock and mutual insurance companies. In Section 3, we apply panel
data methodology to provide some empirical evidence with regard to the
relationship between the premiums of stock and mutual insurers. Aim-
ing to explain these empirical results by means of normative theory, in
166 IV Stock vs. Mutual Insurers
Section 4 we develop our contingent claims model framework, beginning
with the simple and well-established case of the stock insurance com-
pany. Subsequently, we consider a mutual insurer with recovery option
and fully realizable equity, before formally describing the general case
with partial participation in future equity payoffs. Section 5 comprises
a comprehensive numerical analysis which forms the basis for our nor-
mative findings. In Section 6, we integrate our empirical and theoretical
results and discuss relevant economic implications. Finally, in Section 7,
we conclude.
2 Literature review
The literature comparing stock and mutual insurance companies has
predominantly dealt with agency issues associated with the legal form.
Coase (1960) argues that the ownership structure of a company, which
is determined by property rights constituting the discretionary power of
control, is relevant only in the presence of transaction costs. This is due
to the fact that conflicts of interest between different stakeholders may
arise and entail costs, which depend on the extent of discretion as well
as established control mechanisms. Ownership structure is identified as
one possible means of control. In this spirit, Mayers and Smith (1981)
develop a positive theory on insurance contracting, extending the fun-
damental work of Jensen and Meckling (1976) on agency theory. They
analyze incentives resulting from the different ownership arrangements
of stock and mutual insurers and discuss two kinds of potential conflicts
between parties brought together in an insurance firm. On the one hand,
asymmetric information and the call option-like payoff profile associated
with the shareholder position in a stock insurer imply that the equity
value increases with the risk inherent in the company.130 At the same
time, however, riskier assets are detrimental to the position of the poli-
cyholders, giving rise to the so-called owner-policyholder conflict.
130The notion that the equity stake in a company can be interpreted as a call optionon its assets, struck at the face value of the liabilities, was introduced by Merton(1974).
2 Literature review 167
Against this background, the company’s owners will seek to estab-
lish efficient sanction mechanisms, ensuring that the management acts
in their interest. Consequently, agency costs occur and impair economic
efficiency compared to a setting without transaction costs. Since owners
and policyholders within a mutual insurance company coincide, agency
costs can be reduced.131 On the other hand, stock insurers provide more
efficient sanction mechanisms to tackle the so-called owner-management
conflict, which results from diverging incentives between shareholders
and company executives. In addition to being held responsible by the
organizational bodies of the insurance company, which are controlled by
its owners, poorly performing executives of a stock insurer must fear
market discipline such as, for instance, hostile takeovers.132 The reason
is that, in contrast to a mutual insurer, the equity of a stock insurer is
freely tradable and not linked to a particular insurance policy. Hence,
agency costs resulting from the so-called owner-manager conflict can be
expected to be higher for mutuals.133 Assuming that a large number
of decision makers (owners) cannot coordinate as efficiently as single en-
tity or individual, the costs of control can be expected to rise with the
granularity of the equity stake. While the majority of shares of publicly
listed corporations are frequently owned by large blockholders, only a
marginal fraction of the ownership rights is allocated to each member of
a mutual firm. Thus, internal sanction mechanisms are likely to be more
effective for stock than for mutual insurers. Accordingly, from the poli-
cyholder perspective, the optimal choice of legal form should depend on
the trade-off between agency costs arising from the owner-policyholder
and the owner-manager conflict. Therefore, Mayers and Smith (1981,
1988, 1994) argue that stock firms should be more prevalent in activi-
ties that involve significant managerial discretion, since, in this context,
potential owner-manager conflicts are most severe (also see Pottier and
131Also see Garven (1987). Similar incentives can be achieved by including partici-pation rights in the stock insurance contracts (see, e.g., Garven and Pottier, 1995).132The new owner normally exchanges the board of the company (see, e.g., Mayers
and Smith, 1988).133Fama and Jensen (1983a,b) argue that a further mechanism to control manage-
ment is the fact that assets of all mutual financial institutions need to be redeemed ondemand of their members. However, we assent to the arguments raised by Smith andStutzer (1990), who suggest that this is not the case within the insurance context.
168 IV Stock vs. Mutual Insurers
Sommer, 1997). In contrast to that, mutuals should theoretically prevail
in the long-term lines of business that are usually encumbered with a
more significant owner-policyholder conflict potential, such as the life
insurance sector (see Hansmann, 1985; Mayers and Smith, 1988).
A number of empirical articles support the previously explained
agency-theoretic considerations. Lamm-Tennant and Starks (1993) pro-
vide evidence for the owner-policyholder conflict by showing that stock
insurers are generally riskier than mutual insurance companies. This
is coherent with the results of Lee et al. (1997), who analyze both le-
gal forms in the context of insurance guaranty funds. Furthermore, the
greater potential for the owner-manager conflict in mutuals is illustrated
by Greene and Johnson (1980), who conduct a survey in which they
analyze policyholder awareness of the rights resulting from the owner-
ship stake in a mutual insurance company. Compared to the holders of
publicly traded stock, members of the analyzed mutual companies were
less aware of their voting rights and appeared to exercise less control.
Similarly, Wells et al. (1995) find that, in contrast to managers of stock
insurers, those of mutuals have a higher free cash flow at their disposal,
implying a greater opportunity to waste cash on unprofitable invest-
ments. Further evidence for the owner-manager conflict in the context
of mutual and stock insurers is provided by Mayers and Smith (2005),
who document that mutual company charters are more likely to contain
provisions which limit the range of operating policies of the firm. Zou
et al. (2009) observe that, probably owing to their inferior management-
control mechanisms, mutuals tend to pay significantly lower dividends
than stock insurers. Finally, analyzing data from the property-liability
insurance sector, He and Sommer (2010) find that, compared to stock
insurers, the board of mutuals generally comprises a larger fraction of
outside directors. They argue that additional monitoring through out-
side directors is necessary since ownership and control in mutuals are
separated to a greater extent, thus increasing agency costs arising from
the owner-manager conflict.134
134The owner-manager conflict in the context of mutual and stock banks has, e.g.,been considered by Gropper and Hudson (2003) who provide evidence for considerableexpense-preference behavior in mutual savings and loans associations based on a U.S.wide sample.
2 Literature review 169
Another major strand of literature deals with changes in the le-
gal form of an insurer. Fletcher (1966) as well as Mayers and Smith
(1986) focus on mutualization issues. However, much more research has
been conducted on the demutualization process. A survey by Fitzger-
ald (1973) identifies economic pressure as the main reason for the con-
version of small property-liability insurers into stock companies, while
Viswanathan and Cummins (2003) view access to capital as a major
driver for demutualization. Furthermore, Carson et al. (1998) find the
level of free cash flow to be significantly related to the probability of mu-
tual firms transforming into stock companies. Zanjani (2007) analyzes
macroeconomic and regulatory conditions under which mutual insurance
companies have been formed in order to explain the observed evolution of
the whole U.S. life insurance industry from the mutual towards the stock
insurer form. He concludes that tight state regulation did not coincide
with a demise of the mutual form. Instead, a general rise in founding
capital requirements seems to have harmed mutuals due to their very
limited access to external funding. Moreover, Erhemjamts and Leverty
(2010) argue that the incentive to demutualize differs by the type of con-
version: full demutualization versus mutual holding company. Finally,
in their empirical study of U.S. life insurers, McNamara and Rhee (1992)
find that increased efficiency seems to be an important reason for demu-
tualization.
The question of efficiency differences between stock and mutual firms
has been further examined by several other authors. Spiller (1972)
finds evidence that ownership structure is a determinant of performance.
While Jeng et al. (2007) present mixed results with regard to efficiency
improvements implied by changes of the legal form, Cummins et al.
(1999) find mutuals to be less cost-efficient.135 Furthermore, in their
study based on Spanish insurance market data, Cummins et al. (2004)
identify differences in efficiency between stocks and mutuals only for
small mutual insurance companies. Harrington and Niehaus (2002) fo-
135Iannotta et al. (2007) conducted a similar study for the banking industry. Con-trolling for company characteristics as well as geography and time, they find thatmutual banks are less profitable than stock banks. Moreover, they provide evidencefor a higher loan quality among mutuals compared to stock and public sector banks.
170 IV Stock vs. Mutual Insurers
cus on dissimilarities concerning capital structure, which may result from
the costs of raising new capital and Viswanathan (2006) finds initial pub-
lic offerings of mutuals to be significantly underpriced. The latter result
is confirmed by Lai et al. (2008).
Besides agency-theoretic considerations, (de)mutualization, and effi-
ciency implied by the legal form, various other topics related to stock
and mutual insurers have been explored in the literature. Differences in
the contractual structure of policies offered by mutual and stock insurers
are examined by Smith and Stutzer (1990, 1995). They argue that infor-
mation asymmetries rather than agency problems are the major determi-
nant for the types of contracts offered by mutuals. The parallel existence
of different legal forms of insurance companies is justified, amongst oth-
ers, by self-selection of those insured. In addition, Cass et al. (1996)
consider how a Pareto optimal risk allocation can be achieved through
mutual insurance in the presence of individual risk. Ligon and Thistle
(2005) point out that issues arising from asymmetric information can
restrict the size of mutual institutions. Using an equilibrium model in
which mutuals can exclusively offer fully participating policies, Friesen
(2007) show that stock companies can only provide partially participat-
ing insurance when their shareholders require premiums that ensure a
fair return on equity. Finally, Laux and Muermann (2010) demonstrate
that, by linking policies to the provision of capital, mutuals can resolve
free-rider and commitment issues faced by stock insurers.
3 Empirical analysis 171
3 Empirical analysis
In this section, we want to empirically investigate whether the legal form
of an insurance company is a determinant of the premium it charges. To
ensure comparability, the insurance product under consideration needs
to be as homogeneous as possible. Therefore, our sample is based on
annual accounting figures for the German motor vehicle liability insur-
ance sector.136 The data has been obtained from Hoppenstedt, a major
provider of company information for a wide variety of industries of the
German economy. To ensure consistency, we have carried out cross-
checks with the annual reports of the respective insurers. The sample
consists of 99 stock and 14 mutual insurers for which repeated obser-
vations over a differing number of time periods between 2000 and 2006
are available. Hence, we are working with unbalanced panel data, cov-
ering 532 and 87 firm years for stock and mutual insurance companies,
respectively. Table 27 contains some descriptive statistics on the panel
dataset. We measure the price of insurance by means of the average an-
nual gross premium (AvPrem), which is obtained by dividing the total
annual premium volume in the motor liability business line of each firm
by the respective number of contracts.137 Within the analysis, we control
for various additional factors which are likely to influence the insurance
price. The average annual loss (AvLoss), defined as the amount of losses
in the motor insurance line divided by the number of contracts, is used
as a proxy for underwriting risk. In a similar manner, the average an-
nual costs of the motor liability business line (AvCosts) are employed to
account for differences in the efficiency of the companies. Furthermore,
we include the equity ratio (EqR), i.e., the book value of equity divided
136Specialty insurers have been excluded.137An alternative measure for the insurance price is the economic premium ratio
(EPR) which has been suggested by Winter (1994) and is frequently used in theliterature (see, e.g., Gron, 1994; Cummins and Danzon, 1997; Phillips et al., 2006).For a given business line of an insurer, the EPR is the ratio of premium revenuesnet of expenses and policyholder dividends relative to the estimated present value oflosses (see Phillips et al., 2006). Since, in the subsequent chapters, we are interestedin the mutual premium which includes an equityholder and a policyholder stake,policyholder dividends cannot be excluded. In addition, our data does not cover line-specific estimates for the present value of losses. Hence, we control for underwritingrisk by incorporating average annual losses into our regression equations.
172 IV Stock vs. Mutual Insurers
by the book value of the assets, as well as the log total premium volume
in a given year (LTP ) to control for capital structure and size effects.
The Lagrange multiplier (pooling) test, conducted in line with
Gourieroux et al. (1982), suggests significant cross-sectional and time
effects in our data.138 In this case, the pooled ordinary least squares
(OLS) estimator is known to be inefficient: it does not fully exploit the
information inherent in panel datasets (see, e.g., Petersen, 2008). In-
stead, more sophisticated models are needed to make the most effective
use of our data. Based on the Hausman test (see Hausman, 1978) with a
χ2 test statistic of 483.70 and four degrees of freedom, we reject the ran-
dom effects (RE) model. A likely reason for this outcome are significant
correlations between unit-specific components and regressors, implying
an inconsistent RE (and pooled OLS) estimator. While a fixed effects
(FE) model with unit-specific intercept terms could handle this sort of
correlation, the so-called FE within estimator is based on a transforma-
tion of the regression equation into deviations from individual means
and is thus incapable of capturing the impact of time-invariant variables
(see, e.g., Wooldridge, 2010). This a serious issue since our analysis is
focused on the legal form, which, if at all, changes very rarely.
Therefore, we decide to apply the Hausman-Taylor estimator, an in-
strumental variables approach combining characteristics of FE and RE
models (see Greene, 2007; Verbeek, 2008). It is capable of handling
correlations between independent variables and unobserved unit-specific
effects and enables us to estimate coefficients for time-invariant regres-
sors. Consider the following linear regression equation:
AvPremit = µ+ β1AvLossit + β2AvCostsit + β3EqRit
+ β4LTPit + β5Stocki + ui + ǫit. (60)
where µ is the intercept and Stocki is a time-invariant dummy variable
representing the legal form of insurer i, which is set to one for stock
and zero for mutual companies. The ui are N − 1 (here: 112) unit-
specific fixed effects and ǫit denotes the independent and identically dis-
138We compute a χ2 test statistic of 2, 134.13, with two degrees of freedom.
3E
mpirica
lanaly
sis173
Panel A: Pooled stocks and mutuals
abbreviation mean std. dev. minimum maximum skewness kurtosis
Average annual premium AvPrem 256.6744 57.8718 110.8053 577.1687 0.5238 5.5277
Average annual loss AvLoss 225.6424 63.3354 54.6619 514.9426 0.7426 4.5793
Average annual costs AvCosts 40.4934 23.1798 5.8454 287.5176 3.1525 26.7360
Equity ratio EqR 0.2244 0.1127 0.0387 0.7432 1.3999 5.5725
Log-Total premiums LTP 19.0946 1.4775 13.4902 22.9681 -0.2079 2.8635
Panel B: Mutuals
abbreviation mean std. dev. minimum maximum skewness kurtosis
Average annual premium AvPrem 206.7204 40.2694 149.1075 327.2924 1.1263 3.9781
Average annual loss AvLoss 180.2888 45.4460 110.6300 371.1519 2.4661 10.3262
Average annual costs AvCosts 27.1455 13.3401 5.8454 53.9230 -0.0729 1.9150
Equity ratio EqR 0.2798 0.1253 0.1138 0.5804 0.8573 2.7524
Log-Total premiums LTP 19.3349 1.2750 16.2262 21.1092 -0.7238 3.3359
Panel C: Stocks
abbreviation mean std. dev. minimum maximum skewness kurtosis
Average annual premium AvPrem 264.8436 56.2097 110.8053 577.1687 0.4961 6.3869
Average annual loss AvLoss 233.0593 62.7851 54.6619 514.9426 0.6260 4.8106
Average annual costs AvCosts 42.6762 23.7181 9.3966 287.5176 3.2660 26.9900
Equity ratio EqR 0.2154 0.1079 0.0387 0.7432 1.5321 6.5753
Log-Total premiums LTP 19.0553 1.5054 13.4902 22.9681 -0.1359 2.8267
Descriptive statistics for the variables which enter the empirical analysis. In Panel A, all available data has beenpooled, whereas Panel B and C refer to the separate subsamples of mutual and stock insurers. The underlyingcurrency is Euro.
Table 27: Descriptive statistics of the data
174 IV Stock vs. Mutual Insurers
tributed error term. In order to estimate this model, Hausman and Tay-
lor (1981) propose the following instruments: exogenous regressors, i.e.,
those explanatory variables that are uncorrelated with the unit-specific
effects, are their own instruments. In addition, endogenous time-varying
and time-invariant regressors are instrumented by their own individual
means (over time) and those of the exogenous time-varying regressors,
respectively.139 Hence, the analysis requires at least as many exogenous
time-varying as there are endogenous time-invariant regressors, i.e., one
in our case. Based on a correlation test between the above explanatory
variables and their unit-specific components from a fixed effects model,
we identify EqR as exogenous.
An alternative three-stage procedure for estimation of time-invariant
variables in panel data models named fixed effects vector decomposition
(FEVD) has been proposed by Plumper and Troeger (2007). Originated
in the empirical political science literature, FEVD quickly gained popu-
larity among researchers in various fields. Although the authors provided
Monte Carlo simulation results to underline the apparent favorable char-
acteristics of their estimator, it has recently been severely criticized. In
particular, FEVD standard errors have been shown to be systematically
too small and the estimator is inconsistent if time-invariant variables are
correlated with unit-specific effects (see Breusch et al., 2010 and Greene,
2010). Despite these major shortcomings, we decide to additionally ap-
ply this method for comparison purposes.
Table 28 contains the estimation results for the Hausman-Taylor ap-
proach, the FEVD procedure, as well as a simple FE model.140 Apart
from EqR, all time-varying regressors seem to be key determinants
of the insurance premium, since they are associated with statistically
significant coefficients for each of the three estimators. For the time-
invariant variable Stock, in contrast, we get diverging results. While
the Hausman-Taylor estimator does not indicate a significant difference
139A more detailed treatment of the Hausman-Taylor estimator is beyond the scopeof this paper. The reader is referred to advanced panel data texts such as Hsiao(2002), Baltagi (2005), and Wooldridge (2010).140The heteroskedasticity and autocorrelation consistent (HAC) covariance matrices
of Andrews (1991) as well as Driscoll and Kraay (1998) have been applied.
4 Model framework 175
in the average premium of stock and mutual insurers, the FEVD coef-
ficient suggests that mutuals tend to charge less. Taking into account
the above-mentioned limitations of FEVD, we are evidently more confi-
dent in the Hausman-Taylor estimate. For our purpose, however, it is
sufficient to conclude that observed premiums are either approximately
equal, or stock insurer policies tend to be more expensive. To put it
differently, we do not find evidence that mutuals charge higher premi-
ums than stock insurers. Throughout the remainder of this paper we
want to adopt a normative stance and explore whether this empirical
phenomenon is consistent with fair insurance prices as suggested by con-
tingent claims theory.
4 Model framework
In this section we present a general contingent claims model framework
for insurance companies based on the seminal work of Merton (1974)
as well as Doherty and Garven (1986). Assume that the firm runs for
a single period and all stakes are paid in full at the outset. The econ-
omy is characterized by perfect capital markets, i.e., there are no bid-ask
spreads, transaction costs, short-selling constraints, taxes or other mar-
ket frictions. We begin with the relatively simple case of the stock in-
surance company (Section 4.1), which is then incrementally generalized
to include the specifics of mutual insurers. In Section 4.2, we introduce
the recovery option, i.e., the right to demand additional payments in
times of financial distress. Similarly, in Section 4.3, we further extend
our model by allowing for incomplete participation of members in the
mutual’s equity payoffs.
4.1 Stock insurer claims structure
Equity stake
An insurance firm in the legal form of a corporation (stock insurer) is
bankrupt, if the market value of the assets A1 available at the end of the
period is insufficient to cover its claims costs (losses) L1, i.e., A1 < L1.
176 IV Stock vs. Mutual Insurers
Hausman-Taylor FEVD Procedure Fixed Effects Model
(Intercept) -213.4151*** -237.3012*** —
(-2.6692) (-12.1466)
AvLoss 0.3420*** 0.3469*** 0.3420***
(15.4295) (9.9042) (10.9533)
AvCosts 0.6053*** 0.5994*** 0.6053***
(7.3825) (6.1891) (3.9955)
EqR 20.0231 15.7489* 20.0231
(1.0095) (1.9075) (0.5184)
LTP 19.2463*** 18.7959*** 19.2463***
(7.0319) (17.3699) (7.3742)
Stock -3.9429 33.7803*** —
(-0.0470) (14.7292)
Coefficients and t-statistics (in parentheses) for Hausman-Taylor estimator, theFEVD procedure, and the standard FE model. The average annual premium(AvPrem) is regressed on the following set of explanatory variables: averageannual losses (AvLoss), average annual costs (AvCosts), equity ratio (EqR),and logged total premium (LTP ). Hausman-Taylor and FEVD additionallyinclude the time-invariant variable legal form (Stock). ***, **, and * denotestatistical significance on the 1, 5, and 10 percent confidence level.
Table 28: Estimation results
Due to the limited liability of the owners, the equity in t = 1 is worth
zero in this case. Therefore, the payoff profile of the equity stake equals
that of a European call option on the company’s assets, struck at the
value of the claims. Hence, the present value of the equity of a publicly
traded stock insurer EC0, which is a function a parameter set P, can be
expressed as follows
ECS0 = e−rEQ
0 [max (A1 − L1; 0)]
= e−rEQ0 (A1 − L1) +DPOS
0 , (61)
where EQ0 denotes the conditional expectation in t = 0 under the risk-
neutral measure Q, r is the riskless interest rate, and P contains the
relevant parameters for any specific option pricing framework.141 The
call option payoff is equivalent to a long position in the assets and a
141Under the Black and Scholes (1973) model, e.g., the parameter set P wouldcontain the initial value of the assets, the level of claims costs (i.e., the option’s strikeprice), the asset volatility, the risk-free interest rate, as well as the time to maturity.
4.2 Mutual insurer: full equity participation 177
short position in the claims costs (A1 − L1) plus the value of the so-
called default put option of the stock insurer (DPOS). To see this refer to
Figure 18. The default put option is a proxy for the expected bankruptcy
cost and therefore a measure for the safety level of the firm from the
policyholder perspective (see Doherty and Garven, 1986). Its present
value DPOS0 = DPOS
0 (P) is equal to
DPOS0 = e−rEQ
0 [max (L1 −A1; 0)] . (62)
Policyholder stake
If the stock insurer is solvent at time t = 1, the insurance company fully
indemnifies policyholders for their incurred losses. In case of bankruptcy,
however, policyholders only receive the part of their claims which is
covered by the remaining market value of the assets in t = 1. Based on
this payoff profile, the present value of the policyholder stake and thus
the fair premium πS0 of a stock insurer, P S
0 = P S0 (P), is:
P S0 = πS
0 = e−rEQ0 (L1) −DPOS
0 . (63)
The first term represents the present value of expected future claims
costs and corresponds to a default-free insurance premium. The second
term is the value of default put option. This relation implies that stock
insurers with a higher (lower) default risk should charge lower (higher)
premiums πS0 . In the absence of arbitrage, the contribution of the equi-
tyholders and policyholders in t = 0 will be equal to ECS0 and P S
0 = πS0 ,
respectively, implying that the purchase of each stake is associated with
a net present value of zero. The insurance company then invests the sum
A0 = ECS0 + πS
0 in the capital markets.
4.2 Mutual insurer claims structure:
full participation in equity payoff
Equity stake
As discussed in Section 1, one important aspect in which mutuals may
differ from stock insurance companies is their potential right to demand
178
IV
Stock
vs.MutualInsurers
ECS1
L1
PS1
A1 − L1
L1 A1
450
DPOS1
Figure 18: Payoff to the equityholders ECS1 (solid line) and policyholders P S
1 (dotdashed line) of a stock insurancecompany in t = 1. The dashed lines illustrate the elements of the replicating portfolio (A1 − L1 and DPOS
1 ).
4.2 Mutual insurer: full equity participation 179
additional premiums in times of financial distress. Provided a mutual
insurer exhibits such a recovery option and its members fully participate
in the payoff profile of the equity, the present value of the mutual’s equity
stake, ECMf0 , can be expressed as
ECMf0 = e−rEQ
0 (A1 − L1) +RO0 +DPOM0 , (64)
where RO0 equals the present value of the recovery option and DPOM0
denotes the present value of the default put option of the mutual insurer.
Comparing Equations (61) and (64), we notice that these two option
components replace DPOS0 . Due to the recovery option, the default put
option of the mutual insurer ceteris paribus differs from its stock insurer
counterpart (see Figure 19 for a graphical illustration). In particular,
the mutual insurer remains solvent as long as the recovery option has
not been fully exhausted. Accordingly, the assets in t = 1 have to fall
under a lower default threshold X = L1−Cmax than for the stock insurer
before bankruptcy is declared and the remaining assets are distributed
among those members with valid claims. Cmax denotes the upper limit
on additional payments which can be charged through the recovery op-
tion.142 Formally, the present value of the mutual insurer’s default put
option, DPOM0 = DPOM
0 (P , Cmax), is defined as
DPOM0 = POX
0 +BPO0 (65)
where
POX0 = e−rEQ
0
(POX
1
)= e−rEQ
0 [max (X −A1; 0)] , (66)
and
BPO0 = e−rEQ0 (Cmax1A1<X) . (67)
1 is the indicator function, which equals one if A1 < X and zero
otherwise. POX0 is a simple European put option with strike price X
and BPO0 is a cash-or-nothing binary put option which reflects the fact
that, in the instance in which the mutual insurer becomes insolvent, the
assets will have already dropped below the claims by an amount of Cmax.
142Cmax is usually defined in a company’s charter. In our model, it can be easilyadjusted to account for members’ potential default risk or reluctance to pay additionalpremiums.
180 IV Stock vs. Mutual Insurers
In other words, in case of a mutual insurer bankruptcy, losses on the poli-
cyholder stake will be at least Cmax. By comparing the respective payoff
profiles in Figure 19, we notice that generally POX0 ≤ DPOM
0 ≤ DPOS0 .
In addition, the smaller Cmax, the more valuable DPOM0 and, in the spe-
cial case of Cmax = 0 (i.e., X = L1), we get POX0 = DPOM
0 = DPOS0
and BPO0 = 0.
Figure 20, depicts the payoff profile for two different specifications of
the recovery option. We define the standard (basic) recovery option as
one which allows to raise no more than the exact amount of the missing
capital. Its present value, RO0 = RO0(P , Cmax), can be expressed as
RO0 = DPOS0 −DPOM
0
= DPOS0 − POX
0 −BPO0. (68)
and thus equals a long position in DPOS0 and a short position in DPOM
0 .
To put it differently, instead of the stock insurer’s default put option, the
owners of a mutual insurer hold a combination of the recovery option
and the default put option of the mutual, implying that the value DPOS0
is perfectly decomposed into RO0 and DPOM0 , i.e., DPOS
0 = RO0 +
DPOM0 . Consequently, the equity of the stock and the mutual insurer
do not differ in value. However, ceteris paribus mutual members enjoy
a higher safety level of their policies since the probability that their
insurance claims in t = 1 are paid in full is greater than for the stock firm.
Intuitively, the recovery option works as follows: whenX ≤ A1 ≤ L1, i.e.,
if the assets in t = 1 fall below the claims by an amount less than Cmax
such that the recovery option is sufficient to rectify the deficit, L1−A1 is
demanded from policyholders. This is exactly enough additional capital
to eliminate the shortage. Note that the lower Cmax, the less valuable
RO0 and for Cmax = 0, RO0 is worthless. In contrast to that, Cmax =
L1 is associated with the maximum value of the recovery option, while
the default put option of the mutual insurer has no value in this case.
Therefore, Cmax determines how the value of the stock insurer default
put option is split into DPOM0 and RO0.
4.2
Mutu
alin
surer:
full
equity
particip
ation181
L1 A1
45
L1
X
DPOM1
0
Cmax
X
BPO1
DPOS1
POX1
Cmax
Figure 19: Mutual insurer default put option payoff in t = 1 (DPOM1 , solid line). The dashed lines indicate the
elements of the replicating portfolio (BPO1 and POX1 ). For comparison purposes, the dotdashed line illustrates
the default put option of a stock insurer with an identical claim structure (DPOS1 ).
182 IV Stock vs. Mutual Insurers
L1 A1
45
arctan(λ)
RO1(P, Cmax, λ > 1)
RO1(P, Cmax, λ = 1)
L1
Cmax
−Cmax
X X⋆
λDPOS1
−λPOX⋆
1
−BPO1
DPOS1
−POX1
0
Cmax
1
λCmax
Figure 20: Mutual insurer recovery option payoff in t = 1: RO1(P ,Cmax, λ = 1) (bold dotdashed line) and RO1(P , Cmax, λ > 1) (boldsolid line). The thin dotdashed and dashed lines illustrate the respectivereplicating portfolios.
4.2 Mutual insurer: full equity participation 183
Theoretically, a distressed mutual insurer might be allowed to collect
more than just the missing capital from its members, implying that the
firm can build up a reserve. By adjusting Equation (68) we can extend
our model framework to account for this special case, which will be called
excess of loss recovery option. The following is a more general expression
for the present value of the recovery option, RO0 = RO0(P , Cmax, λ),
RO0 = λDPOS0 − λPOX⋆
0 −BPO0, (69)
where
POX⋆
0 = e−rEQ0
(
POX⋆
1
)
= e−rEQ0 [max (X⋆ −A1; 0)] , (70)
with X⋆ = L1 − 1λC
max and λ ∈ [1;∞). Consequently, in the gen-
eral case, the recovery option is a position of λ units of DPOS long,
λ units of POX⋆
short, and BPO short. The parameter λ constitutes
a straightforward charging rule and denotes the multiple of additional
payments over the deficit. For λ = 2, e.g., the mutual is able to charge
policyholders twice the deficit and build up a reserve from the surplus.
The impact of λ on the recovery option payoff profile is illustrated
in Figure 20. Intuitively, the higher this multiple, the lower the dis-
tance between L1 and X⋆, the steeper the slope of RO1 in this interval,
and the smaller the amount by which the assets have to fall below the
claims so that the mutual will simply collect Cmax.143 Analogously to
POX, POX⋆
is a European put option with strike X⋆, which depends
on λ. Clearly, if λ = 1 we have POX⋆
0 = POX0 . In Figure 20, we
see that RO1(P , Cmax, λ > 1) > RO1(P , Cmax, λ = 1), which implies
RO0(P , Cmax, λ > 1) > RO0(P , Cmax, λ = 1).
To sum up, if the recovery option is designed to simply eliminate a
given deficit (λ = 1), the payoff profile of the overall equity stake of a
143For λ → ∞, the slightest deficit will induce the mutual insurer to charge addi-tional payments of Cmax.
184 IV Stock vs. Mutual Insurers
mutual insurer will ceteris paribus be the same as for the stock insurer.
However, if the recovery option is specified so that the mutual can charge
multiples of a given deficit (λ > 1), its equity stake will be relatively more
valuable. This can be easily seen by comparing the payoff profile for the
equityholders of a mutual insurer with excess of loss recovery option
(Figure 21) to that plain call option shape we saw for the stock insurer
in Figure 18. In any case, the value of the equity stake ECMf0 depends
not only on the parameter set P but also on the specific characteristics
of the recovery option as represented by Cmax and λ.
Policyholder stake
Consistent with its equity, we define the present value of the policyholder
stake of a mutual insurer as
PM0 = e−rEQ
0 (L1) −RO0 −DPOM0 . (71)
Again, e−rEQ0 (L1) is the fair insurance premium without default risk
and instead of DPOS0 we have a short position in the combination of the
recovery option and the default put option of the mutual. If, at the end
of the period, the assets have fallen below the claims costs L1 but not
the mutual’s default threshold X, i.e., X < A1 ≤ L1, the policyholder
stake of the mutual insurance company is associated with an equal or
a higher financial loss than that of a stock insurer. This is due to the
fact that the mutual charges λ(L1 − A1) through the recovery option,
while the insolvency of the stock insurer results in a policyholder deficit
of L1−A1. Therefore, generally DPOS0 ≤ RO0+DPOM
0 and PM0 ≤ PS
0 .
More specifically, the policyholder stake of a mutual insurance company
is less valuable than that of an otherwise identical stock insurer when
it contains a recovery option with λ > 1 (see Figure 22). Similarly,
we know from Equation (64) that the present value of the equity stake
increases for more expensive recovery options. Hence, an excess of loss
recovery option essentially redistributes value from the policyholder to
the equity stake. If λ = 1, in contrast, we have DPOS0 = RO0 +DPOM
0
and consequently PS0 = PM
0 (refer to Equation (68)).
4.2
Mutu
alin
surer:
full
equity
particip
ation185
ECMf1
A1 − L1
L1 A1
4545
arctan(λ)
arctan(λ− 1)
RO1DPOM
1
Cmax
X X⋆0
Zone I Zone II Zone III
Figure 21: Mutual insurer equity payoff (full participation) in t = 1 (ECMf1 ) for λ > 1 (solid line). The dashed
lines indicate the elements of the replicating portfolio.
186
IV
Stock
vs.MutualInsurers
PM1
ECMf1
L1
L1
A1
45
arctan(λ) − 45
DPOM1
+ RO1
Cmax
X X⋆0
Zone I Zone II Zone III
Figure 22: Mutual insurer policyholder stake payoff in t = 1 (PM1 ) for λ > 1 (solid line). The payoff profile of the
default put option (DPOM1 ), the recovery option (RO1), and the equity stake given full participation (ECMf
1 )are drawn as dashed lines for comparison purposes.
4.3 Mutual insurer: partial equity participation 187
Arbitrage-free premium
Since, through the purchase of a policy in a mutual company, one ac-
quires the equity and the policyholder stake at the same time, the fair
premium of a mutual insurer must comprise the arbitrage-free price of
both components. Accordingly, we define ΠM0 = ΠM
0 (P , Cmax, λ) as
ΠM0 = PM
0 + ECMf0
= e−rEQ0 (L1) −RO0 −DPOM
0︸ ︷︷ ︸
policyholder stake
+ e−rEQ0 (A1 − L1) +RO0 +DPOM
0︸ ︷︷ ︸
equityholder stake
= e−rEQ0 (A1) . (72)
Thus, if members fully participate in the equity payoffs, purchasing a
policy from a mutual insurer is equivalent to acquiring a position the
company’s assets. Policyholders of an otherwise identical stock insurer
additionally would have to buy the common stock of the company in
order to establish the same payoff profile.
4.3 Mutual insurer claims structure:
partial participation in equity payoff
Equity stake
There is generally no secondary market for ownership stakes in mutual
insurance companies. As a consequence, payoffs from the equity stake of
a mutual insurer and thus its present value crucially depend on the pre-
mium refund policy of the management and the ability of the members
to prompt an initial public offering (IPO) or break-up of the company.
Let α be the payout (premium refund) ratio and pL the probability of
demutualization or liquidation of the company.144 The impact of these
144Under agency-theoretic considerations the firm’s management generally has apreference to retain as much capital in the company as possible. This aspect of theso-called owner-manager conflict lowers the premium refund ratio α. Furthermore, incontrast to a corporation, there are no blockholders in a mutual insurer. Therefore,pL will depend on the members’ ability to coordinate an agreement on the demutu-alization or liquidation of the firm.
188 IV Stock vs. Mutual Insurers
parameters on the payoff profile of the equity stake depends on the zones
indicated in Figures 21, 22, and 23 which are determined by the realiza-
tions of assets and claims in t = 1. If A1 < X, i.e., the assets have
fallen below the default threshold (Zone I), the mutual is insolvent, bro-
ken up and the remaining assets are distributed to its members. Thus,
the equity stake is worthless and neither α nor pL are relevant in Zone
I. Furthermore, if X < A1 < L1 (Zone II) the mutual insurer exercises
the recovery option to charge additional payments (via the policyholder
stake). It is safe to assume that a mutual in financial distress will re-
frain from premium refunds, implying that members can only fully re-
alize the equity payoff via an IPO or the liquidation of the company.
Hence, in Zone II only pL has an influence on the present value of the
equity. Finally, in Zone III, where the company is solvent and does
not need to exercise the recovery option, members receive the whole eq-
uity value with probability pL, or a premium refund of α(A1 −L1) with
probability (1 − pL). We summarize these two cases in the parameter
γ = pL + (1 − pL)α, which can be interpreted as the expected value of
the equity stake in Zone III, normalized to unity. Since α ∈ [0; 1] and
pL ∈ [0; 1], we get γ ∈ [0; 1]. Under this set-up, the present value of a
mutual’s equity stake in the general case (recovery option and partially
realizable equity), ECM0 = ECM
0 (P , Cmax, λ, pL, α), can be described as
follows:
ECM0 = 0
︸︷︷︸
Zone I
+ pL
[
(λ− 1)DPOS0 − λPOX⋆
0 + POX0
]
︸ ︷︷ ︸
Zone II
+ γe−rEQ0 [max (A1 − L1; 0)]
︸ ︷︷ ︸
Zone III
= γ[
e−rEQ0 (A1 − L1) +DPOS
0
]
+ pL
(
λDPOS0 − λPOX⋆
0 + POX0 −DPOS
0
)
= γe−rEQ0 (A1 − L1) + γDPOS
0
+ pL(RO0 +DPOM
0 −DPOS0
)
= γe−rEQ0 (A1 − L1) − (pL − γ)DPOS
0
+ pL(RO0 +DPOM
0
). (73)
4.3 Mutual insurer: partial equity participation 189
For a graphical verification of this expression refer to Figure 23. As
mentioned previously, the equity payoff in the interval [0, X] (Zone I) is
zero. Furthermore, the payoff profile between X and L1, i.e., in Zone
II, is characterized by an asymmetric butterfly spread, consisting of
(λ − 1) units of DPOS0 long, λ units of POX⋆
0 short and one unit of
POX0 long.145 Finally, the equity payoff in Zone III is equal to a long
stake in γ units of a simple call option on the assets with strike price
L1. Recalling Equation (61), we realize that this call option is exactly
the one describing the equity value of a stock insurance company. As
also illustrated in Figure 23, the consideration of the parameters pLand γ, which were introduced above, results in a flattening of the pay-
off of the mutual insurer’s equity stake in Zones II and III. In the ab-
sence of arbitrage, members of a mutual insurance company anticipate
that they can only partially access future cash flows arising from the
equity stake, implying a reduction of its present value. The difference
between ECMf0 and ECM
0 – represented by the shaded area in Figure 23
– is the discount in the present value of the equity stake resulting from
the incomplete participation of the current members in its future pay-
off. In our contingent claims framework, this ”non-realizable” equity,
ECMn0 = ECMn
0 (P , Cmax, λ, pL, α), has a price in t = 0 equal to
ECMn0 = ECMf
0 − ECM0
= e−rEQ0 (A1 − L1) +RO0 +DPOM
0︸ ︷︷ ︸
equity given full participation
− pL(RO0 +DPOM
0
)
− γe−rEQ0 (A1 − L1)
+ (pL − γ)DPOS0
realizable equity
= (1 − γ) e−rEQ0 (A1 − L1) + (pL − γ)DPOS
0
+ (1 − pL)(RO0 +DPOM
0
). (74)
In order to comprehensively understand the effect of pL, α, and, in
turn, γ, on the value of the equity stake of the mutual insurance company,
we consider two special cases. First of all, the expression for the present
145Note that for λ = 1, we have a standard put option butterfly spread.
190
IV
Stock
vs.MutualInsurers
ECM1
ECMf1
ECMn1
L1 A1
45
arctan(λ− 1)
arctan[pL(λ− 1)]
arctan(γ)
X X⋆
0
Zone I Zone II Zone III
Figure 23: Mutual insurer (expected) equity payoff in t = 1 in case of partial equity participation (ECM1 ) for
λ > 1, pL, and α < 1 (solid line). The dashed line illustrates the equity stake given full participation (ECMf1 )
for comparison purposes.
4.3 Mutual insurer: partial equity participation 191
value of the equity stake under full participation, i.e., Equation (64), is
nested in the more general Equation (73). The exact payoff profile we
saw in Figure 21 (ECMf1 ) in the previous section can only be realized
in the special case of pL = 1, i.e., full participation in the equity payoff
stream in all three zones. To see this note that pL = 1 directly results
in γ = 1 such that Equation (73) becomes Equation (64) and Equa-
tion (74) collapses to zero: there is no non-realizable equity component.
Apart from that, pL < 1 and α = 1 also results in γ = 1: the mutual
distributes the whole equity to its members when it is solvent, but when-
ever the recovery option is exercised, there are no premium refunds and
participation in the equity payoff is contingent on the probability of liq-
uidation pL. Consequently, the first term in Equation (74) disappears
and the remainder reduces to (1 − pL)(RO0 +DPOM
0 −DPOS). This
means that only the excess value of the recovery and default put option
of a mutual over the default put option of a stock insurer constitutes
non-realizable equity.146 While, in this case, the payoff profile in Zone
III is the same as for full equity participation, we get a flatter curve in
Zone II. Overall, in the arbitrage-free setting, realizable equity ECM0 and
non-realizable equity ECMn0 will always sum up to ECMf
0 , while pL and
α govern the size of these components relative to each other.
Policyholder stake
While participation in the future cash flows of the equity stake might be
limited, there are no such restrictions associated with the policyholder
stake. In other words, the present value of the policyholder stake re-
mains the same as in Section 4.2, comprising the default-free premium
e−rEQ0 (L1) as well as a short position in the recovery and the default
put option of the mutual. Thus, we have the same expression as in
Equation (71):
PM0 = e−rEQ
0 (L1) −RO0 −DPOM0 . (75)
146In case there is no such excess value, i.e., for λ = 1, the non-realizable equity iszero and the equity stake of the mutual equals that of the stock insurer.
192 IV Stock vs. Mutual Insurers
Arbitrage-free premium
As explained at the end of Section 4.2, the arbitrage-free premium of
a mutual insurer must comprise the present values of both the equity
and the policyholder stake. In the general case of partial participation
in the equity payoff, however, the price of the equity stake splits into a
realizable and a non-realizable component. Yet, the overall level of the
mutual insurance company premium remains unchanged and equals the
expected discounted value of the firm’s assets in t = 1. Consequently, in
the general case, we replace Equation (72) with an alternative expression
for ΠM0 = ΠM
0 (P , Cmax, λ):
ΠM0 = PM
0 + ECM0 + ECMn
0
= e−rEQ0 (L1) −RO0 −DPOM
0︸ ︷︷ ︸
policyholder stake
+ γe−rEQ0 (A1 − L1)
− (pL − γ)DPOS0
+ pL(RO0 +DPOM
0
)
realizable equity stake
+ (1 − γ) e−rEQ0 (A1 − L1)
+ (pL − γ)DPOS0
+ (1 − pL)(RO0 +DPOM
0
)
non-realizable equity
= e−rEQ0 (A1) . (76)
4.4 Claims structure relationships
Below we briefly illustrate the theoretical impact of recovery option and
limited participation in equity payoffs on the premium of a mutual in-
surer relative to a comparable stock insurer. Imagine two insurance firms
with the exact same underlying assets and claims: one is founded as a
corporation and the other one adopts the legal form of a mutual. Fig-
ure 24 depicts the relationship between the claim structures of these two
companies in four distinct cases, characterized by different configurations
of recovery option and equity participation. As defined in Equation (63),
the marginal premium charged by a stock insurer equals the value of its
4.4
Cla
ims
structu
rerelation
ship
s193
stock insurer mutual insurer
PS0
πS0
ECS0
ΠM0
πM0
πM0
RO0 + DPOM0
−DPOS0
case
equityparticipation
excess of lossrecoveryoption
PM0
ECMf0
I
fullγ = 1
noλ = 1
PM0
ECMn0
ECM0
II
partialγ < 1
noλ = 1
PM0
ECMf0
III
fullγ = 1
yesλ > 1
PM0
ECMn0
ECM0
IV
partialγ < 1
yesλ > 1
Figure 24: Theoretical premium comparison
194 IV Stock vs. Mutual Insurers
policyholder stake. The premium of the mutual insurer corresponding
to this reference case, however, depends on the appointed setting.
In Case I, the mutual insurance company is either not allowed to
charge additional premiums at all (i.e., Cmax = 0, which results in
DPOS0 = DPOM
0 )147 or the amount of additional premiums is restricted
to the actual deficit L1 − A1 (i.e., λ = 1, which results in DPOS0 =
DPOM0 + RO0)
148. In addition, the equity stake of the mutual is fully
realizable (pL = 1 and, hence, γ = 1). Comparing Equation (63) and
(71), we see that under these circumstances P S0 = PM
0 : there is no dif-
ference between the value of the policyholder stakes of a stock and a
mutual insurer. Moreover, comparing Equation (61) and (64), we no-
tice that both equity stakes have the same value, i.e., ECS0 = ECMf
0 .
Due to the fact that the equity of the mutual can be entirely realized
and there are no additional contributions in excess of a loss, the rights
of mutual members are economically identical to those of the combined
policyholder and ownership stake of the stock insurer. In other words,
since policyholders and owners coincide, the position in a mutual could
be replicated by simply purchasing both an insurance contract and an
appropriate amount of shares of the stock insurer. Hence, the aggregate
premium ΠM0 charged by a mutual should equal the premium of a stock
insurer, πS0 , plus the value of its equity ECS
0 .
In Case II, the mutual insurer’s company charter excludes additional
premiums in excess of a loss (λ = 1). However, its equity stake can-
not be fully realized (γ < 1). Since, in an arbitrage-free market, ra-
tional individuals anticipate this, the ownership stake of the mutual
insurer is separated into a realizable and a non-realizable component
and prospective mutual members are generally not willing to provide
the latter. Consequently, the mutual premium is now πM0 , i.e., ΠM
0 net
of the non-realizable equity ECMn0 . The full premium ΠM
0 can only be
demanded if members are being compensated for ECMn0 , e.g., through a
binding right to payments from future policyholders upon the beginning
147Refer to Equations (65) to (68).148Refer to Equation (68).
4.4 Claims structure relationships 195
of their membership in the mutual.149 It is important to note that, in
any case, the non-realizable equity needs to be paid in for the company
to be founded at all. This is due to the fact that less initial equity than
ECMf0 is associated with a lower expected payoff in t = 1. In anticipation
of this consequence, individuals will further reduce their willingness to
pay, eventually reaching an equilibrium where the value of both stakes
is zero. In this situation, the mutual insurance company cannot find
customers if it charges a positive arbitrage-free premium, since every
insurance policy would be associated with a negative net present value.
Thus, if the members do not provide ECMn0 , an external third party such
as a founding capital provider, whose capital repayment is contractually
guaranteed, would need to step in instead.
Case III represents the claims structure of the mutual if its equity is
fully realizable (pL = 1 and, hence, γ = 1) and its recovery option allows
to charge additional premiums over and above the actual loss (λ > 1).
Due to the excess of loss recovery option, the value of the equity position
increases and the value of the policyholder position decreases compared
to the stock insurer (and Case I) by an amount equal to the difference
between RO0 + DPOM0 and DPOS
0 . The higher λ, the bigger the shift
between both stakes. Since the recovery option solely redistributes value
between the stakes, the overall amount of assets within the company is
unchanged. Therefore, the overall mutual premium remains equal to ΠM0 .
Finally, the combined effect of partially realizable equity (γ < 1) and
excess of loss recovery option (λ > 1) is illustrated in Case IV. Again,
the equity stake splits into ECMn0 plus ECM
0 and πM0 denotes the full
mutual premium less the present value of the non-realizable equity. In
contrast to Case II, however, both equity components are slightly more
expensive, since value is shifted from the policyholder to the equity stake
via the excess of loss recovery option. As before, a non-zero arbitrage-
149In a multiperiod framework such compensation payments could be conducted atthe end of each period. For instance, the current members (from t = 0 to t = 1) wouldneed to receive the right to be paid an amount of ECMn
1in t = 1 by the members of
the following period (t = 1 to t = 2). This right is worth ECMn0
= e−rEQ0
(
ECMn1
)
today.
196 IV Stock vs. Mutual Insurers
free solution can only be achieved if the non-realizable equity is paid in
as well. Hence, more expensive recovery options (for a higher λ), are
ceteris paribus associated with a more valuable (non-realizable) equity
and a lower πM0 .
5 Numerical analysis
In this section, we concretely describe assets and claims costs as con-
tinuous-time stochastic processes and present closed-form solutions for
the various option prices on which our model framework is based. Sub-
sequently, we provide a brief numerical example to further illustrate the
model mechanics as well as the effect of recovery options and equity par-
ticipation on the premium of a mutual insurer. In addition, based on the
numerical implementation of our model framework, we derive normative
insights with regard to feasible combinations of premium, safety level,
and capital structure of stock and mutual insurance companies.
5.1 Option pricing formulae
Suppose that assets are traded continuously in time and that the term
structure of interest rates is flat and deterministic. The insurance com-
panies’ assets are assumed to be stochastic and their dynamics is mod-
eled by the following Geometric Brownian Motion under the risk-neutral
measure Q:
dAt
At= rdt+ σAdWQ
At, (77)
where the drift is given by the risk-free interest rate r and σA denotes the
volatility of assets and dWAt is a standard Wiener process under Q.150
150In this set-up, asset returns are normally distributed. While, in most cases, this ismerely an approximation of the empirically observed distributions (see, e.g., Officer,1972; Akgiray and Booth, 1988; Lau et al., 1990), it simplifies matters by allowing usto apply closed-form solutions. Since insurance companies tend to hold a considerablefraction of bonds in their investment portfolios, an alternative set-up could includeterm structure models (see, e.g., Vasicek, 1977; Cox et al., 1981).
5.1 Option pricing formulae 197
The insurer’s claims are assumed to be deterministic.151 Under these
assumptions, closed-form solutions for the present values of the various
European options described in Section 4 are available; see Black and
Scholes (1973). In line with the one-period model from Section 4, the
present value of the stock insurer default put option, DPOS0 , can be
computed as follows:
DPOS0 = e−rEQ
0
(DPOS
1
)= e−rEQ
0 [max (L1 −A1; 0)]
= e−rL1Φ(−d1) −A0Φ(−d2), (78)
where Φ(x) is the cumulative distribution function of the standard nor-
mal distribution and
d1 =ln(A0/L1) + r − σ2
A/2
σA,
d2 =ln(A0/L1) + r + σ2
A/2
σA.
In addition, the present value of the put option POX0 in Equation
(66), which is one of the two building blocks of the default put option of
a mutual insurer (DPOM), can be calculated using the following formula:
POX0 = e−rEQ
0
(POX
1
)= e−rEQ
0 [max (X −A1; 0)]
= e−rXΦ(x1) −A0Φ(x2)
= e−r(L1 − Cmax)Φ(−x1) −A0Φ(−x2), (79)
where
x1 =ln [A0/(L1 − Cmax)] + r − σ2
A/2
σA,
x2 =ln [A0/(L1 − Cmax)] + r + σ2
A/2
σA.
151This decision is made for reasons of computational simplicity. Since the modelframework in Section 4 has been deliberately kept on a general level, different as-sumptions for the asset and claims dynamics as well as associated option-pricingframeworks can be applied without loss of generality.
198 IV Stock vs. Mutual Insurers
The second building block of the DPOM is a cash-or-nothing binary
put option which pays Cmax if A1 < X and zero otherwise. Rubinstein
and Reiner (1991) show that the price of this option is equal to:
BPO0 = e−rCmaxΦ(−x1). (80)
Using Equations (79) and (80), the formula for the present value of
the default put option of the mutual insurer, DPOM0 , can be derived:
DPOM0 = POX
0 +BPO0
= e−r(L1 − Cmax)Φ(−x1) −A0Φ(−x2) + e−rCmaxΦ(−x1)= e−rL1Φ(−x1) −A0Φ(−x2). (81)
This formula somehow resembles Equation (78), which describes the
price of the default put option of a stock insurer. Yet, the probabili-
ties with which the parameters e−rL1 and A0 are weighted differ. To
grasp the intuition behind this, recall from Section 4.2 that the assets
A1 have to fall below the threshold X before the default put option of
the mutual insurer is in the money. Contingent on A1 < X, however,
the payoff profiles of DPOM and DPOS are congruent (refer back to
Figure 19): in the area A1 < X, both options pay L1 −A1. As a result,
the formula for DPOM0 includes e−rL1 and A0, but weighted with the
probabilities Φ(−x1) and Φ(−x2) instead of Φ(−d1) and Φ(−d2).
Finally, to calculate the value of the recovery option in the general
case (i.e., for λ > 1), we additionally need the closed-form solution for
the put option POX⋆
0 . Following the same rationale as above, we get
POX⋆
0 = e−rEQ0
(
POX⋆
1
)
= e−rEQ0 [max (X⋆ −A1; 0)]
= e−rX⋆Φ(z1) −A0Φ(z2)
= e−r(L1 −1
λCmax)Φ(−z1) −A0Φ(−z2), (82)
with
z1 =ln[A0/(L1 − 1
λCmax)
]+ r − σ2
A/2
σA,
5.2 The impact of recovery option and equity participation 199
z2 =ln[A0/(L1 − 1
λCmax)
]+ r + σ2
A/2
σA.
Combining Equations (78), (80), and (82), the value of the recovery
option can be expressed as:152
RO0 = λDPOS0 − λPOX⋆
0 −BPO0,
= λ[e−rL1Φ(−d1) −A0Φ(−d2)
]
− λ[e−r(L1 −1
λCmax)Φ(−z1) −A0Φ(−z2)]
− e−rCmaxΦ(−x1)= λe−rL1Φ(−d1) − λA0Φ(−d2) − λe−rL1Φ(−z1)
+ λA0Φ(−z2) + e−rCmaxΦ(−z1) − e−rCmaxΦ(−x1)= λ
e−rL1 [Φ(−d1) − Φ(−z1)] −A0 [Φ(−d2) − Φ(−z2)]
+ e−rCmax [Φ(−z1) − Φ(−x1)] . (83)
5.2 The impact of recovery option
and participation in equity payoff
Having determined asset and claims dynamics as well as the associated
option pricing formulae, the equity and policyholder stake of mutual and
stock insurance companies can now be valued. Table 29 contains the ba-
sic input parameters used in our numerical examples and the resulting
present values (PVs) for the stock insurer.
The first three columns of Table 30 illustrate the impact of the recov-
ery option in a mutual insurance company with full participation in the
equity payoff stream (pL = 1 and α = 1).153 For λ = 1, i.e., no excess
of loss recovery option, the value of the default put option of the stock
insurer DPOS0 (0.2481) perfectly splits into RO0 (0.2463) and DPOM
0
(0.0018). In addition, equity ECMf0 (30.2481) and policyholder stake PM
0
152Note that [Φ(−d1) − Φ(−z1)] is Pr(X⋆ < A1 < L1) and [Φ(−z1) − Φ(−x1)] isPr(X < A1 < X⋆).153These numerical results correspond to Case I and III in Section 4.4.
200 IV Stock vs. Mutual Insurers
A0 100 initial value of the assets
L0 70 initial value of the liabilities
σA 0.20 volatility of the asset returns
r 0.03 risk free rate
DPOS0
0.2481 PV of the stock insurer’s default put option
ECS0
30.2481 PV of the stock insurer’s equity
PS0
= πS0
69.7519 PV of the stock insurer’s policyholder claims
Table 29: Input parameters and values for DPOS0 , ECS
0 , and P S0
(69.7519) of the mutual are worth the same as those of the stock insurer
shown in Table 29: although the two companies differ in terms of le-
gal form, they are economically identical in this case. Since, through
a membership in the mutual, one acquires both stakes, the mutual pre-
mium (ΠM0 = 30.2481 + 69.7519 = 100) equals the present value of the
assets. For an increasing λ, however, we observe a non-linear growth in
RO0, resulting in a value of 0.2708 in case 110 percent of a deficit can
be demanded from mutual members, i.e., λ = 1.1. In this case, the sum
RO0+DPOM0 (0.2726) is almost ten percent higher than DPOS
0 (0.2481).
Furthermore, in Table 30 we see that the excess of loss recovery option
redistributes value from the policyholder to the equityholder stake since
PM0 falls and ECMf
0 , ECMn0 , as well as ECM
0 rise in λ. Analogously to
the default put option of the stock insurer, DPOM0 measures the safety
level of a mutual insurance company’s policyholder stake. As DPOM0
remains the same (0.0018) for all values of λ and is always lower than
DPOS0 (0.2481), the mutual insurer with recovery option has a higher
safety level than the otherwise identical stock insurer.
The three columns in the center of Table 30 show the case where
mutual members partially participate in the equity payoff (pL = 0.1 and
α = 0.1).154 Again, for λ = 1, we have RO0+DPOM0 = DPOS
0 = 0.2481.
This time, however, the total equity value ECMf0 (30.2481) splits into
a realizable component ECM0 (5.7471) and a non-realizable component
ECMn0 (24.5010). The former is considerably lower than the latter, since
154See Case II and IV in Section 4.4.
5.2
The
impact
ofrecovery
option
and
equity
particip
ation201
pL = 1, α = 1 pL = 0.1, α = 0.1 pL = 0, α = 0
λ 1.00 1.05 1.10 1.00 1.05 1.10 1.00 1.05 1.10
DPOM0
0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018
RO0 0.2463 0.2586 0.2708 0.2463 0.2586 0.2708 0.2463 0.2586 0.2708
DPOM0
+ RO0 0.2481 0.2604 0.2726 0.2481 0.2604 0.2726 0.2481 0.2604 0.2726
ECM0
30.2481 30.2604 30.2726 5.7471 5.7484 5.7496 0.0000 0.0000 0.0000
ECMn0
0.0000 0.0000 0.0000 24.5010 24.5120 24.5230 30.2481 30.2604 30.2726
ECMf0
30.2481 30.2604 30.2726 30.2481 30.2604 30.2726 30.2481 30.2604 30.2726
PM0
69.7519 69.7396 69.7274 69.7519 69.7396 69.7274 69.7519 69.7396 69.7274
ΠM0
100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000
Table 30: Impact of recovery option and equity participation given Cmax = 25
202 IV Stock vs. Mutual Insurers
the figures are based on a fairly low premium refund rate and probabil-
ity of liquidation. As in the previous case, a rise in λ implies a more
expensive recovery option. The associated value redistribution reduces
PM0 and increases both components of the equity stake. Consistent with
our choice of pL and α, however, ECMn0 absorbs a relatively larger share.
Although the ten percent likelihood of liquidation assumed for this nu-
merical example probably has to be considered relatively high from a
real world perspective, the realizable equity stake has already become
quite small. Consequently, even lower values for pL, which are perfectly
conceivable, would result in a situation where virtually the whole eq-
uity is attributed to the non-realizable component. As explained in
Section 4.4, the capital in such a case would need to be provided by
a third party, since, under the arbitrage-free framework applied, mutual
members would not be prepared to incur a negative net present value
investment. Again, DPOM0 = 0.0018 < DPOS
0 = 0.2481 for all λ. There-
fore, as in the previous example, the mutual insurer’s policyholder stake
exhibits a higher safety level than that of the stock insurer. The last
three columns of Table 30 contain the numerical results when the equity
stake is not realizable at all (pL = 0 and α = 0). Obviously, in this case,
the whole equity value is attributed to the non-realizable component.
Apart from λ, the maximum amount of additional premiums Cmax is
a key determinant of the recovery option value and has a direct impact on
the safety level of the firm. Table 31 illustrates that a recovery option
does not exist if Cmax = 0.155 Instead, the default put option of the
mutual insurance company is exactly the same as for a stock insurer
(DPOM0 = DPOS
0 = 0.2481). The higher Cmax, i.e., the less binding
the upper limit on additional payments, the more valuable becomes the
recovery option. In addition, an increase in Cmax simultaneously results
in a decline of DPOM0 , implying an improving safety level. For Cmax =
40, we get RO0 = 0.2726 and the mutual’s default put option is (almost)
worthless because the value of the assets in t = 1 would have to drop by
more than 40 below the value of the claims for it to be in the money. As
in Table 30, the decrease in PM0 due to the incremental growth in RO0
155See Figure 20 in Section 4.2.
5.3 Premium, safety level, and equity capital 203
Cmax 0 10 20 30 40
DPOM0 0.2481 0.1016 0.0103 0.0002 0.0000
RO0 0.0000 0.1604 0.2612 0.2727 0.2729
DPOM0 +RO0 0.2481 0.2620 0.2715 0.2729 0.2729
ECM0 5.7471 5.7485 5.7495 5.7496 5.7496
ECMn0 24.5010 24.5135 24.5221 24.5233 24.5233
ECMf0 30.2481 30.2620 30.2716 30.2729 30.2729
PM0 69.7519 69.7380 69.7284 69.7271 69.7271
ΠM0 100.0000 100.0000 100.0000 100.0000 100.0000
Table 31: Impact of the maximum amount of additional contributions(Cmax) given partial participation in the equity payoffs of the mutualfirm and an excess of loss recovery option: pL = 0.1, α = 0.1, λ = 1.1
is counterbalanced by an increased value of the equity stake (realizable
and non-realizable component).
5.3 Stock vs. mutual insurers:
premium, safety level, and equity capital
In the following, we compare a stock and a mutual insurer with identical
underlying assets and claims with regard to the three central magnitudes
premium size, safety level, and equity capital, considering cases with and
without recovery option as well as full and partial participation in equity
payoffs. Again, the calculations have been based on the parameter values
in Table 29. While other configurations would change the magnitude of
the observed effects, their direction remains the same.
We begin with the case where the mutual insurer does not have a
recovery option (Cmax = 0) and its equity stake can be fully realized
by the members. In Figure 25, the arbitrage-free mutual and stock in-
surer premiums have been plotted against the value of the respective
equity stakes. Under the arbitrage-free framework used, both curves
must start at zero. Let us first look at the solid curve, which rep-
204
IV
Stock
vs.MutualInsurers
0 5 10 15 20 25
6065
7075
80
85
EC0S, EC0
Mf
P0S=
π0S, P
0M, Π
0M
Curves:Π0
M (Mutual premiums)L0 (PV of claims costs)L0 −DPO0
M (Safety levels of mutuals with RO)P0
M= P0
S = π0S (PV of policyholder stakes)
Points:Π0
M= L0
Figure 25: Equity-premium combinations for full equity participation and no recovery option
5.3 Premium, safety level, and equity capital 205
resents equity-premium-combinations for the stock insurer and equity-
policyholder-stake-combinations for the mutual insurer. As the amount
of initial equity capital is raised, the stock insurer premium converges
towards the present value of the claims costs L0 (represented by a dot-
ted horizontal line).156 For any amount of equity capital, the distance
between L0 and the solid curve equals the present value of the stock
insurer’s default put option (DPOS0 ), which, in this case, is identical
to that of the mutual insurer (DPOM0 ) since it is assumed that the lat-
ter does not have a recovery option. The vertical dotted line is meant
to serve as a concrete example. As a consequence, if they are identi-
cally capitalized, mutual and stock insurer offer contracts with the same
safety level. In addition, more equity capital is associated with a de-
cline in DPOS0 (= DPOM
0 ) due to the fact that a larger equity buffer
reduces the likelihood of the assets dropping below the claims costs at
the end of the period. The dashed curve represents premiums of the
mutual insurer. Since members have to purchase both stakes, it lies
strictly above the solid curve. Thus, in the absence of a recovery option,
if both companies hold the same amount of equity capital and members
of the mutual insurer can fully participate in its equity stake, then they
should be charged higher premiums than the policyholders of the stock
insurer. Another relevant observation is related to the point where the
ΠM0 -curve intersects the L0-line (marked by a small circle). If the mutual
insurer holds more initial equity capital than associated with this point,
its premium must be strictly higher than that of the stock insurer, no
matter how well capitalized the latter is. This is due to the fact that the
πS0 -curve converges to but never exceeds L0.
Next, we introduce a basic recovery option (Cmax > 0, λ = 1), while
still allowing for full participation in the equity payoffs of the mutual
insurer. As discussed in Section 4.2, the recovery option enables mutual
insurers to stay solvent and satisfy all claims, even if their equity cap-
ital is fully exhausted. More specifically, a mutual insurer is bankrupt
156In Section 4.1 we explained that the fair stock insurance premium equals thepresent value of the policyholder stake, i.e., πS = PS
0. Besides, L0 is the default-free
premium.
206 IV Stock vs. Mutual Insurers
only if the deficit of assets relative to liabilities exceeds the limit on ad-
ditional premiums (Cmax), which implies DPOS0 = RO0 + DPOM
0 or
DPOM0 < DPOS
0 . This is illustrated in Figure 26, where we now have
an additional dotdashed curve, reflecting the safety levels of the mutual
insurer. While DPOS0 is still represented by the distance from L0 to
the solid curve, the distance between L0 and the dotdashed curve equals
DPOM0 . Since the dotdashed lies strictly above the solid curve, the mu-
tual insurer with recovery option exhibits a strictly better safety level
than the identically capitalized stock insurer. In other words, the mu-
tual insurer with recovery option needs less equity capital to achieve the
same safety level as the stock insurer.157 Furthermore, analogously to
Figure 25, the mutual must charge a higher premium than the stock in-
surer if it holds more equity capital than associated with the intersection
of the ΠM0 -curve and the L0-line. Consequently, safety level and premium
of a well-capitalized insurance company should be higher if it adopts the
legal form of a mutual. In contrast to the results in Figure 25, however,
we now find capitalizations for which the premium of the mutual can be
equal to or lower than that of the stock insurer with an identical safety
level. To see this, we focus on the intersection between the dashed (ΠM0 )
and dotdashed curve (L0 − DPOM0 ), which has been highlighted by a
black dot. If the mutual insurance company holds precisely this much
equity capital, it exhibits the same safety level and charges the same
premium as the stock insurer with the amount of equity capital which
corresponds to the black triangle.158 Right of the black dot, the mutual
charges more and left of the black dot it charges less than the stock in-
surer with the same safety level.
In Figure 27, we account for limited participation in the equity payoff
stream of the mutual insurer by splitting its capital into the realizable
and the non-realizable component.159 However, as explained in Sec-
157Intuitively, the recovery option can be interpreted as an equity substitute. Hence,in most jurisdictions mutual insurers can—to some extent—account for their recoveryoption when calculating solvency capital charges.158To find the latter, follow an imaginary horizontal line from the black dot to the
right until it reaches the πS0
-curve.159The non-realizable equity is calculated based on pL = 0.1 and α = 0.1.
5.3
Prem
ium
,safety
level,an
deq
uity
capital
207
0 5 10 15 20 25
6065
7075
8085
EC0S, EC0
Mf
P0S=
π0S, P
0M, Π
0MCurves:Π0
M (mutual premiums in PV terms)L0 (PV of claims costs)L0 −DPO0
M (safety levels of mutuals with RO)P0
M= P0
S = π0S (PV of policyholder stakes)
Points:Π0
M= L0 −DPO0
M Π0
M= L0
Figure 26: Equity-premium combinations for full equity participation and recovery option
208
IV
Stock
vs.MutualInsurers
0 5 10 15 20 25
6065
7075
80
85
EC0S, EC0
Mf
P0S=
π0S, P
0M, π
0M
Curves:π0
M (Mutual premiums)L0 (PV of claims costs)L0 −DPO0
M (safety levels of mutuals with RO)P0
M= P0
S = π0S (PV of policyholder stakes)
Points:π0
M= L0 −DPO0
M π0
M= L0
Figure 27: Equity-premium combinations for partial equity participation and recovery option
6 Economic implications 209
tion 4.4, both components need to be paid in for the company to be able
to begin business. Therefore, the x-axis is still based on the full value of
the mutual’s equity ECMf0 and the dotdashed curve, representing safety
levels of the mutual insurer, is unaffected by this change. In contrast
to that, however, the non-realizable equity is excluded from the mutual
premium, meaning that πM0 instead of ΠM
0 is shown on the y-axis. The
intuition behind this proceeding is that members are either compensated
by an amount equal to the present value of the non-realizable equity, or
the latter is provided by a third party, e.g., a founding capital provider.
Since, for each amount of initial equity capital, the mutual premium is
now lower than in the case of full equity participation (Figures 25 and
26), the πM0 -curve has a smaller slope than the ΠM
0 -curve (plotted in light
grey). Hence, for a decreasing probability of liquidation pL, the premi-
ums of the mutual insurance company converge to those of the identically
capitalized stock insurer as the non-realizable equity is not borne by the
members. Besides, the dashed curve (ΠM0 ) now intersects the dotted line
(L0) further to the right such that we have a broader range of capitaliza-
tions, which allow the mutual to match the premium of the stock insurer.
Similarly, the intersection between the dashed (πM0 ) and the dotdashed
curve (L0 −DPOM0 ) has been shifted to the right, implying a larger set
of stock insurer capital structures for which the mutual is able to provide
less expensive policies with the same safety level.160
6 Economic implications
Due to competition in insurance markets one might expect the premi-
ums of stock and mutual companies not to differ significantly (see, e.g.,
Mayers and Smith, 1988). This view is partially supported by the empir-
ical evidence we presented in Section 3. Despite the different results for
two common estimators we were able to conclude that, in any case, mu-
tuals do not charge higher premiums than stock firms. If at all, it seems
that stock insurer policies are more expensive. We can now combine
160The black triangle, representing that particular capitalization of the stock insurerfor which the mutual can chose to match both its safety level and its premium, isnow outside the scale of Figure 27.
210 IV Stock vs. Mutual Insurers
these empirical findings with the normative results from the previous
section to derive economic implications with regard to the relationship
of stock and mutual insurer premiums. To begin with, we sum up un-
der which specific circumstances the contingent claims model framework
supports the equality of premiums.
First of all, in the absence of a recovery option and if members can
fully participate in a mutual insurer’s equity payoffs, its premium can
only be similar to that of a stock insurer when its capitalization and
safety level are very low (Figure 25). However, such a scenario is un-
likely to occur in practice since, in most jurisdictions, solvency regulation
frameworks ensure a minimum safety level for insurance companies.161
If the equity of a mutual insurer without recovery option is only partially
realizable, i.e., its premium curve in Figure 25 becomes flatter, there are
more likely to be capitalizations which allow the mutual to charge the
same or a lower premium than the stock insurer, while still conforming
to the applicable solvency standards. By holding less equity capital than
the stock insurer, however, the mutual would also maintain a compara-
tively lower safety level.
Secondly, even in the presence of a recovery option, the mutual com-
pany with fully realizable equity is only able to match or undercut the
prices of the stock insurer when featuring less initial equity capital. Yet,
despite the generally smaller equity buffer, the mutual’s safety level could
be lower than, similar to, or even higher than that of the stock insurer
with the same premium, depending on its capitalization.162 Again, the
practical relevance of this scenario depends on the lower limit for the
safety level as established by the applicable solvency regulation. How-
ever, due to the recovery option, it is less likely that all capital structures
which enable mutuals to charge less than stock insurers are ruled out.
161Within our model framework, a minimum safety level is equivalent to an upperlimit on the present value of the default put option. Hence, it could be reflected inFigures 25 to 27 by means of a vertical line, the area to the left of which would notbe admissible under the prevailing solvency standards.162Consider the area around the black dot in Figure 26.
6 Economic implications 211
Finally, reconsider the situation where the equity of mutuals with
recovery option is only partially realizable (Figure 27). As before, of-
fering policies for the same or a lower premium than the stock insurer
requires that the mutual commands less equity. However, in this case
it will be more likely that the mutual also complies with the respective
solvency standards since for any given capital structure, a larger fraction
of non-realizable equity is associated with a lower mutual premium.163
As mentioned in Section 1, there are generally no liquid secondary mar-
kets for ownership stakes of mutuals. Consequently, the non-realizable
equity should be rather large, leading us to believe that this might the
most relevant case from a practical perspective.
To sum up, while our arbitrage-free model does not generally exclude
the possibility of the mutual premium being lower than the stock insurer
premium, in any case, such an outcome would require the mutual to
hold less equity capital than the stock insurer. Within the empirical
analysis, however, we explicitly controlled for capital structure effects
as well as other premium determinants such as underwriting risk and
administration costs. Thus, it appears that the empirically observed
prices are not arbitrage-free in the sense of the applied contingent claims
approach. In other words, from a normative perspective, policies offered
by stock insurers seem to be overpriced relative to policies of mutuals.
Since this situation is not a theoretical equilibrium, it can only prevail
due to further factors which are exogenous to our model. One such
aspect might be that we consider stakes in present value terms while
observed mutual premiums are quoted as up-front cash flows, i.e., net of
the recovery option value which can be viewed as an ex-post premium
component. However, due to its rather low value compared to the overall
mutual premium (see numerical analysis in Section 5), it is safe to assume
that the recovery option has a minor impact on the results. Moreover,
the deviation from the theoretical premium relationship could be caused
by superior marketing and sales efforts of stock companies. Although
163Recall that, relative to Figure 26, the black dot and triangle in Figure 27 areshifted to the right.
212 IV Stock vs. Mutual Insurers
this might be a reason for the persistence of economic rents, its impact
is difficult to assess in the absence of empirical work on the subject.
Another point to be taken into account is that asymmetric informa-
tion can be an important issue in insurance markets. As explained in
Section 4.4, the notion of perfectly informed individuals which underlies
our contingent claims context implies that a mutual insurance company
would not be able to attract customers if its premium includes the present
value of non-realizable equity. Yet, in a situation where prospective mu-
tual members are unaware of economic differences associated with the
legal form of insurance companies, they are unable to correctly assess
the value of a policy. Therefore, the deviation from our arbitrage-free
results might occur because mutual members do not have enough infor-
mation or are not financially literate enough to determine the fair price
of both stakes included in the mutual premium. Asymmetric information
could lead to a scenario in which individuals actually pay for all or part
of the non-realizable equity without being compensated in some form.
Evidently, this would imply a transfer of wealth to an unknown group
of future profiteers such as, e.g., a generation of policyholders which
participates in the liquidation or demutualization of the firm. However,
such a violation of the no-arbitrage condition does not need to be re-
curring. Since most of the mutual insurance companies in our sample
are rather old and well-capitalized firms (see Table 27), wealth transfers
could have taken place in the past. Some of the affected individuals
might have already left the company without adequate compensation.
Current members benefit from this development as an accumulation of
equity reserves through violations of the no-arbitrage condition in the
past would imply that mutuals are now able to offer policies for a lower
premium than stock insurers. Alternatively, wealth transfers could also
persist between the policyholders and owners of the stock insurance com-
panies, implying that the former overpay for their contracts. Finally, a
combination of these sorts of wealth transfers within stock and mutual
organizations is conceivable.
7 Conclusion 213
7 Conclusion
In this paper, we empirically and theoretically analyze the relationship
between the insurance premium of stock and mutual companies. Eval-
uating panel data for the German motor liability insurance sector, we
do not find evidence that mutuals charge significantly higher premiums
than stock insurers. If at all, it seems that stock insurer policies are more
expensive. Subsequently, we employ a comprehensive model framework
for the arbitrage-free pricing of stock and mutual insurance contracts.
Based on a numerical implementation of our model, we then compare
stock and mutual insurance companies with regard to the three central
magnitudes premium size, safety level, and equity capital. Although
we identify certain circumstances under which the mutual’s premium
should be equal to or smaller than the stock insurer’s, these situations
would generally require the mutual to hold less capital than the stock
insurer. This being inconsistent with our empirical results, it appears
that policies offered by stock insurers are overpriced relative to policies
of mutuals.
Although various reasons for the observed deviation of our empiri-
cal and theoretical results are conceivable, we believe a violation of the
no-arbitrage principle due to asymmetric information to be the most
plausible explanation. Therefore, we argue that the documented dis-
crepancies are an indicator for likely wealth transfers between different
stakeholder groups of mutual and stock companies. A more detailed
identification of the size and direction of these wealth transfers could be
an interesting avenue for future research. Since such an analysis would
need to be based on a separate consideration of the different stakes, our
contingent claims model framework is well suited for an application in
this context. On the empirical side, however, more detailed insurance
company information would be required. Another interesting research
question centers around the coexistence of stock and mutual insurance
companies. Our normative results could be a starting point for a further
consideration of this topic. As previously discussed, an arbitrage-free
market implies that rational individuals would not be willing to pay for
214 IV Stock vs. Mutual Insurers
the non-realizable component of the equity stake. Hence, we suggested
that mutual companies can only come into existence if, e.g., their initial
members are granted the right to compensation payments for the non-
realizable equity by future member generations or if a third party acts
as founding capital provider. Since both alternatives are rarely observed
in practice, it would be interesting to explore other possibilities which
enable mutuals to coexist with stock companies.
References 215
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Curriculum Vitae
Personal InformationName: Alexander Braun
Date of Birth: 13th of April 1981
Place of Birth: Tuttlingen, Germany
Nationality: German
Education02/2009–present University of St. Gallen (HSG), Switzerland
Doctoral Studies in Finance
07/2010–08/2010 University of Michigan, Ann Arbor, United States
Summer Program in Quantitative Research Methods
10/2003–05/2007 University of Mannheim, Germany
Diplom-Kaufmann (Majors: Banking and Finance, Marketing)
02/2004–12/2004 Monash University, Melbourne, Australia
Study Abroad Program (Major: International Management)
10/2001–08/2003 University of Mannheim, Germany
Pre-Diploma in Management and Intercultural Studies
09/1991–06/2000 Immanuel-Kant-Gymnasium, Tuttlingen, Germany
Abitur (A-Levels)
Work Experience
02/2009–present Institute of Insurance Economics
University of St. Gallen, Switzerland
Project Manager and Research Associate
07/2007–09/2008 Lehman Brothers Ltd., London, United Kingdom
Senior Analyst (full-time), Capital Markets Division
02/2006–04/2006 Deutsche Bank AG, Frankfurt am Main, Germany
Intern, Global Markets
09/2005–01/2006 SAP Deutschland AG & Co. KG, Walldorf, Germany
Working Student (part-time), Marketing Analysis
01/2005–04/2005 SAP Deutschland AG & Co. KG, Walldorf, Germany
Intern, Strategic Marketing and Project Management
09/2003–10/2003 BASF AG, Ludwigshafen, Germany
Intern, Management Accounting
05/2001–07/2001 Karl Storz GmbH & Co. KG, Tuttlingen, Germany
Intern, Accounting and Product Management