Valuation of Catastrophe Reinsurance Contract with Considering Equity
Depression-The application of Hybrid CAT Bond Issuing
Chih-Chen Hsu* Juei-Hsiang Chen
Abstract
This study develops a contingent claim framework designed to evaluate
reinsurance contracts of proposed-hybrid catastrophe (CAT) bond to scrutinize
reinsurance companies as to how they reduce default risk and increase reinsurance
contract value by issuing hybrid CAT bonds. Pauline and Henri (2009) proposed the
concept of hybrid instruments. The study alone with this concept to design and to
price a variation of hybrid catastrophe bond that combines CAT bond with a
catastrophe equity put option. Such a bond possesses instrument characteristics of pre-
and post-loss financing that will better provide channels for risk transfer. Results
showed that changes, basis risks, trigger levels, and catastrophe risks inherent in
reinsurance contract and default risk premium value are related to the initial capital
structure of the reinsurance company. Under the premise that instruments are set the
same, even with the consideration of basis risk, the issuing of hybrid CAT bonds is
comparable to that of CAT bonds in that it can further reduce default risk premium
and increase reinsurance contract value.
Keywords: Contingent framework, Hybrid CAT Bond, Reinsurance contract
JEL Classification: C15, C73, G12, G22
Hsu: Assistant Professor of Finance, National Chung Cheng University, Cha-Yi 62100, Taiwan(R.O.C.), Email:
[email protected] . Chen: Mater student of Finance department, National Chung Cheng University, Cha-Yi
62100, Taiwan(R.O.C.), Email: [email protected] . Corresponding author: Chih-Chen Hsu.
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1. Introduction
Significant losses from catastrophic events in recent years pose a serious threat to
reinsurance companies, resulting in problems of increasing liability and capital
shortage. The cause of such problems is mainly due to the increase in the frequency of
catastrophic events and the mounting losses these events incurred. They lead to
problems of insufficient collection of reinsurance premiums or lack of sufficient risk
aversion by the insurance companies. At this time, reinsurance companies are fully
exposed to this type of catastrophe risk. Therefore, how to reduce reinsurance
company exposure to the impact of this type of catastrophe risk, and how to further
enhance the value of reinsurance contract, will be the subject of this study.
Catastrophic events are low-frequency high-loss by nature, resulting in not only
economic but also human casualties. In recent years, natural catastrophes (hurricanes,
floods, earthquakes, and tsunamis) have been intensely reported by the news media.
Events such as the Indonesian tsunami in 2004, Hurricane Katrina in 2005, and
311-earthquake in Japan in 2011 have all resulted in serious human casualties and
economic losses in their respective countries. Outcomes that accompany a catastrophe
incident are often unknown, and the amount of compensation is uncertain, thus
catastrophe events are difficult to predict.
Traditionally, property insurance companies, in their attempt to avoid
responsibility for huge losses caused by catastrophic events that can lead to a default,
often transfer these catastrophe risks by purchasing reinsurance contracts from
reinsurance companies. According to property and casualty insurance companies, the
actual size of claims may cover all or part of a loss. This type of contract can enable a
property and casualty insurance company, when faced with huge catastrophe loss
claims, to obtain additional capital injections, thereby transfer these catastrophe risks
the property insurance company has to face. The theory, however, from regional or
national perspective, catastrophe events represent systemic risk. The financial impact
of catastrophes can be dispersed by the existing international reinsurance market,
which does not need to issue any catastrophe link securities. Yet, the development of
the market does not follow that, which is expected from Pauline and Henri (2009). In
recent years, a single catastrophic event (earthquake or typhoon) can result in 50 to
100 billion dollars of damage. The amount of catastrophe loss at this level can put a
huge pressure on the underwriting capacity of the entire insurance industry, and poses
a serious threat to the credit risk of many reinsurance companies (Cummins, et al,
2002). Once the catastrophic event has occurred, reinsurance companies must fulfill
compensation obligations under the terms of reinsurance contract, which will
immediately increase their liabilities (Duan and Yu, 2005), causing the likelihood that
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these reinsurance companies have insufficient capital on hand to withstand a
catastrophic event of such a magnitude. If a certain number of market participants
were to withdraw from the market until after the injection of new capital, then this
type of catastrophe risk can be transferred. Recent reinsurance market research
showed that, due to the limited capacity of international reinsurance market, the needs
of insurance market are no longer being fulfilled (Froot, 1999 and 2001; Harrington
and Niehaus, 2003).
However, catastrophic events represent negative news. In an efficient market,
stocks will instantly reflect the operating conditions of reinsurance companies, and
catastrophic events that cause the loss of equity value of reinsurance companies will
cause their stock price to immediately fall. Studies mentioned in literature that were
done by Sprecher and Pertl (1983), and Davidson, Chandy, and Cross (1987), on the
impact of catastrophe events on stock price of insurance companies, showed that there
is a negative correlation between the value of these companies and the magnitude of
the losses. Reilly and Drzycimski (1973) found that major world events, once they
occur, tend to be reflected in the stock price straight away. Anderson and Cross (1990)
tracked the impact of California earthquake of October 17, 1989 on the stock prices of
the real estate industry and found that the market treats quakes as new information,
which is then reflected in the stock prices of real estate companies located in and
around San Francisco area as negative returns. Reinhold (1995) also found significant
negative price reactions to Hurricane Andrew being reflected in property insurance
company stocks. However, just because a catastrophic event has occurred, it does not
necessarily mean that the effect on insurance company stock prices is negative, and
that the share price of insurance companies must fall. Shelor, Anderson, and Cross
(1992), having monitored market reaction to the stocks of property and casualty
insurance companies after an earthquake, discovered an effect different from the
negative ones experienced by the aforementioned real estate companies. The price
effect on property and casualty insurance industry stocks was actually positive. The
tendency for insurance company stocks to rise following catastrophe events further
points to investors confidence (positive effect) on future demands for insurance
relative to share price decline (negative effect) of insurance company stocks caused
by catastrophic loss. Aiuppa, Carney, and Krueger (1993) also examined the effect of
earthquakes on the value of property insurance company stocks, and found them to be
similarly positive. Various views of others have also been expressed in literature on
whether stocks can really reflect the true equity value of reinsurance companies.
In considering the problem of a lack of sufficient means on the part of
reinsurance, companies to pay for catastrophic claims, both government and market
sectors have proposed various solutions to address the problem. The government part
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involves the proposed government funded CAT reinsurance program or the issuing of
government CAT bonds, such as CAT bonds issued by the Government of Mexico
(Wolfgang and Brenda, 2010). The market sector part involves the use of financial
innovation to create numerous insurance event-linked products, such as CAT options,
CAT bonds, CAT equity put options, and CAT exchanges. These developments are all
designed to allow reinsurance companies to provide more effective risk transfer
channels.
Insurance event-linked securities are a type of structured product that link
financial claims to an insurance event (Cummins and Philip, 2000). Their most
important function is to transfer catastrophe risk borne by the insurance company to
the capital market. These products have very many types, ranging from CAT bonds,
CAT options, or capital-CAT exchange types. Once catastrophe occurs, the equity
value of reinsurance companies will go down. Past methods of increasing capital
(opening public offering or cash replenishment) were not able to return the capital of
reinsurance companies back to the original level. However, the insurance
event-linked securities can provide even more additional capital-raising channels.
Next, we will introduce CAT bonds, introduce CAT equity put (CatEPut), and
provide a description of hybrid CAT bonds as proposed in this thesis study.
CAT bonds are a kind of insurance-linked security, also known as event-linked
bonds, belonging to pre-loss debt financing. When a specific catastrophe event occurs,
CAT bonds will fulfill the obligation to pay. Although it was only until the late 1990s
before CAT bond market finally took off to a slow start, it has now become a stable
source of underwriting for insurance and reinsurance companies. The CAT bond
market has experienced continued steady growth and has set new records of market
issuing volume in 2005, 2006, and 2007 (Cummins, 2008). CAT bonds carry full
guarantees, which eliminate anxiety over credit risk of CAT bonds. Since correlation
between catastrophe and stock market return rate is low, from risk diversification
point of view, CAT bonds are more admired and respected by investors (Litzenberger,
et al, 1996).
Catastrophe equity put (CatEPut) option is part of equity financing, which refers
to the right to sell equity based on previously agreed strike price. The option buyer
puts up the option money to give consent to purchase, in the event of catastrophe, the
CatEPut offer for the company's equity based on a price agreed in advance. After the
catastrophe, when reinsurance companies are in need of capital injection, CatEPut
allows reinsurance companies to obtain any additional capital injection at an
advantageous price. Therefore, CatEPut option is designed to provide a type of
catastrophe risk transfer channel for reinsurance companies. In addition, any
additional issue of preferred share will dilute existing shareholders' equity value.
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This study, based on the idea of hybrid CAT bond suggested by Pauline and
Henry (2009), essentially proposed to combine two single type products, namely, CAT
bond and CatEPut option, into a hybrid type product. The design of hybrid CAT bond
type product is aimed to not only transfer catastrophe risk faced by the reinsurance
companies to the capital market, but also provide downside risk protection for the
equity value. It also has a dual natured pre-loss and post-loss financing. The issuing of
hybrid CAT bonds will increase the issuance volume in the market.
The focus of most of the relevant studies in literature was on single product
assessments such as reinsurance contracts, CAT bonds, and CatEPut options, with
additional focus on raising the value of catastrophe reinsurance contracts by
reinsurance companies using CAT bonds (Lee and Yu, 2007). The contribution of this
study are as follows: 1) actually created a hybrid CAT bond from a combination of
CAT bond and CatEPut option; 2) established contingent claims framework to
examine how reinsurance companies were able to lower credit risk and raise the value
of catastrophe reinsurance contracts by issuing hybrid CAT bonds; and 3) discovered
that, relative to CAT bonds, hybrid CAT bonds are positioned to raise reinsurance
contract value to a greater extent. Specifically, we will explore, in the process of
setting up assets, liability, interest rate, and catastrophe loss variations, how insurance
linked securities issuance changed the capital structure of reinsurance companies and
further affected the valuation of insurance contracts. Then, we will make individual
comparisons to discover which insurance-linked security issued by the reinsurance
companies is more favorable. The structure of this study is further explained below.
Part II will describe the process of dynamic in the assets, interest rates, liabilities, and
catastrophe losses of reinsurance companies. Part III will show in detail the profit
functions of reinsurance contracts under different issuance scenarios of reinsurance
companies. Part IV will show the resulting values generated by numerical analysis,
and discuss and compare the values presented. Part V will present the conclusion of
this study.
2. Insurance Company Model Setting
The study adopted the structural approach pioneered by Merton (1974 and 1977)
to value the corporate debt contract, allowing us to link together various concerns
about reinsurance companies such as asset value, debt value, capital structure, default
risk, and financial claims. In addition, the study also followed the structured model
adopted by Cummins (1988), Duan et al (1995), and Duan and Yu (2005), taking into
account the dynamic process of interest rate and liability, and observed how
reinsurance companies were able to transfer catastrophe risks, which they themselves
6
are facing, to the capital market through the issuing of insurance-linked products.
The study considered two kinds of insurance-linked products, CAT bonds, and
hybrid CAT bonds. Here, we first consider the case of CAT bonds issued by
reinsurance companies. Issuing of CAT bonds can transfer catastrophe risk to the
capital market so that the underwriting capacity of reinsurance companies themselves
can be enhanced (Lee and Yu, 2007). CAT bond can be issued through reinsurance
companies, or through special purpose companies (SPC). To focus attention on the
weight of reinsurance contract, the study assumed that CAT bonds are issued through
a SPC, and that the fair price of CAT bonds are also assumed to be consistent with
results obtained by Litzenberger (1996), Lee and Yu (2002), and Vaugirard (2003).
Recent studies found that reinsurance companies, even with capital injections
obtained through the issuing of catastrophe bonds, are not completely able to actually
transfer their own assumed catastrophe risks (Cummins, Lewis, Phillips (2002), Nell
and Richter (2004). Catastrophic events will increase the liability of reinsurance
companies (Duan and Yu, 2005) because, with everything else being equal,
catastrophic events will further reduce the equity value of reinsurance companies. The
study assumes that the market is efficient. That is, the market equity value of a
reinsurance company will fully reflect the fact that catastrophe losses lead to decline
in equity value. To solve the risk of decline in the value of equity, we will consider in
this section the issuing of CatEPuts by reinsurance companies to transfer risk to
capital market. CatEPuts does not need to be issued by SPC, and assumes that the fair
price of CatEPut portion is consistent with the reasonable results obtained by Lin et al
(2009) and Chang et al (2011).
The aforementioned single product type is limited by market size, and the issue
amount is subject to restrictions. To enhance the effectiveness of products, the concept
of hybrid products has emerged to transfer catastrophe risk even further that,
according to the development of financial markets in the past, suggests that the selling
of such complex hybrid products by financial intermediaries is feasible (Pauline and
Henry, 2009). Based on this concept, the study put CAT bond and CatEPut option
together to create a type of hybrid CAT bond. To focus attention on the weight of
reinsurance contract, the study assumes that this type of hybrid CAT bond is issued
through SPC, where the fair price of CAT bonds and CatEPut options are consistent
with those mentioned above.
The capital structure of reinsurance companies and catastrophic loss thresholds
are important factors used to value reinsurance contracts, which fully explains the
dynamic process of reinsurance company assets, liabilities, and catastrophic losses
that corresponds to the changing process of real world probability measure (P) and
risk neutral probability measure (Q).
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2.1 Asset dynamic process
The study used the traditionally assumed asset dynamic process as being a
logarithmic diffusion process to further incorporate the effect of interest rate on the
value of assets into the model. Since a larger portion of asset investment portfolio of
reinsurance company is usually fixed income assets (interest rate sensitive assets)
(Duan and Yu, 2005; Lee and Yu, 2007), this type of model setting is more apt at
describing the change process of asset value of reinsurance companies.
,t V V t V V t
dV Vdt Vdr VdWµ φ σ= + + (1)
Where,
t
V : Value of asset of reinsurance companies in period t
t
r : Value of risk-free interest rate in period t
,V t
W : Credit risk of reinsurance companies (Wiener process)
V
σ : Credit risk volatility of reinsurance companies
V
φ : Instantaneous interest rate elasticity of assets of reinsurance companies
Among them, credit risk is perpendicular to interest rate risk, where credit risk
represents all risks other than interest rate risk.
2.2 Interest rate dynamic process
The study honors the interest rate dynamic process developed by Duan and Yu
(2005) and Lee and Yu (2007) by assuming that instantaneous interest rate change
process is in line with square root process developed by Cox et al (1985). This type of
configuration will avoid getting Vasicek interest rates to be less than zero.
Instantaneous change process of interest rate under real-world probability measure (P)
is as follows:
( )t t t t
dr k m r dt v r dZ= − + (2)
Where:
k : Mean reversion strength value
m : Long term interest rate average
v : Interest rate volatility
t
Z : Wiener process unrelated to ,V t
W
Since interest rate value is an important factor affecting the value of the assets of
reinsurance companies, in the following we will combine interest rate change process
and asset change process together by substituting Equation (2) into Equation (1). The
asset change process then becomes as follows:
,
( )t V V V t V t t V V t
dV km kr dt v r dZ dWµ φ φ φ σ= + − + + (3)
The standard practice to make evaluation easier is to convert the original
real-world probability measure (P) to risk neutral probability measure (Q). The
8
interest rate change process under risk neutral probability measure (Q) is expressed as
follows:
* * *( )t t t t
dr k m r dt v r dZ= − + (4)
Where:
*
*
*
r
r
r t
t t
k k
kmm
k
rdZ dZ dt
v
λ
λ
λ
= +
=+
= +
Among them, r
λ is market price of interest rate risk, which is set to constant by
Cox et al (1985). *
tZ is the Wiener process under risk neutral probability measure.
The asset change process under risk neutral probability measure (Q) is expressed
as follows:
* *
,t t t V t t t V t V tdV rV dt v r V dZ V dWφ σ= + + (5)
Among them, *
,V tW is the Wiener process unrelated to *
tZ under risk neutral
probability measure (Q).
2.3 Liability dynamic process
With regard to reinsurance company liabilities, they come from not only
catastrophe reinsurance contracts, but also from other types of insurance coverage
liabilities. Any types of liability that have nothing to do with catastrophe reinsurance
contracts will represent future claims of present value, which belong to interest rate
sensitive liabilities, therefore, based on Lee and Yu (2007), the liability change
process of reinsurance companies under real world probability measure (P) is
expressed as follows:
,t L t L t t L t L t
dL L dt L dr L dWµ φ σ= + + (6)
Where:
t
L : Value of liability of reinsurance companies in period t
L
µ : Expected change in liability value
L
φ : Interest rate elasticity of liabilities of reinsurance companies
L
σ Liability value volatility
,L t
W : Wiener process
Such a continuous diffusion process reflects the effect of interest rate change and
the risk of small daily variations. Since small daily variation ,L t
W is part of the
homogeneous change in capital market, the study assumes that it is a no risk premium.
The result from the interest rate change process under risk neutral probability measure
(Q) in Equation (4) is carried into Equation (6). Substituting t
Z with *
tZ to obtain
interest rate change process under risk neutral probability measure (Q), which can be
9
expressed as follows:
* *
,t t t L t t t L t L tdL r L dt v r L dZ L dWφ σ= + + (7)
2.4 Catastrophe losses processes
The study follows the traditional insurance established in literature (see Bowers,
et al (1986)) that uses compound Poisson process to express change in total
catastrophe losses. That is, using a series of loss jumps to describe the process of
catastrophe losses. The catastrophe loss change process in catastrophe reinsurance
contract underwriting standard in period t can be expressed in the follow change
process:
( )
1
N t
t t
j
C c=
= ∑
(8)
To estimate the impact of basis risk on catastrophe reinsurance contract, we
additionally listed the change process of composite index of catastrophe losses, which
is expressed as follows:
( )
, ,
1
N t
index t index t
j
C c=
= ∑ (9)
Where, { }0
( )t
N t≥
represents the number of changes in catastrophes occurred in
each period. The study assumes that such a change process is driven by the Poisson
process with frequency λ . t
c represents the actual amount of loss in jth
catastrophic
loss event underwritten by catastrophe reinsurance contract in a specific period; ,index t
c
represents the amount of composite index of losses. In terms of the amount of actual
losses (t
c ) and the amount of composite index of losses (,index t
c ), the number of
catastrophic occurrence for each time is 1,2,..., ( )j N t= , the study assumes that
catastrophic events are independent of each other, identical to each other, and obey the
variable nature of lognormal distribution, and that the amount of actual losses (t
c ) and
the amount of composite index of losses (,index t
c ) are unrelated to the number of
catastrophic events occurred. Logarithmic mean and variance are represented by the
symbol c
µ (index
µ ) and c
σ (index
σ ), respectively. In addition, assume that each time the
number of catastrophe occurrences is at 1,2,..., ( )j N t= , the correlation between the
logarithm of the amount of actual losses (t
c ) and the amount of composite index of
losses (,index t
c ) is equal to c
ρ .
For the purpose of calculation, the study followed the practice of Merton (1976)
by assuming that the overall economy will only be affected by regional catastrophic
events, so the change process number { }0
( )t
N t≥
and catastrophe loss amounts t
c and
,index tc
in a catastrophic event are risk-free premiums. Therefore, the change process of
loss in Equation (8) and (9) will not alter the original distribution characteristics as a
result of converting real world probability measure (P) to risk neutral probability
10
measure (Q) (Lee and Yu, 2007).
3. Valuation of Catastrophe Reinsurance Contracts
Once the change processes of assets, liabilities, and losses under risk neutral
probability measure (Q) are known, the payoff functions in a wide range of
circumstances under risk neutral measure (Q) can be discounted to measure the value
of reinsurance contract. The study first had to list out basic payoff functions of those
reinsurance companies that did not issue insurance event linked securities. Then, the
study had to extend the analysis to include the valuation of reinsurance contracts that
are affected when different insurance linked products (CAT bonds and hybrid Cat
bonds) are issued.
3.1 Non-issuance situations
The study first had to analyze the profit function of reinsurance contracts in
situations when reinsurance companies have not issued any insurance linked securities.
A non-issuance scenario is viewed as the most basic situation and is easily understood,
and its profit functions can be extended to a variety of complex situations, such as
reinsurance companies issuing CAT bonds and hybrid Cat bonds.
3.1.1 No default risk
The type of reinsurance contract used in this study is excess of loss policy. When
the total amount of loss insured exceeds the minimum contract trigger value A of the
original insurance company due to catastrophe, the reinsurance company must then
pay compensation to the original insurance company as stipulated in the contract, with
upper trigger loss limit M. When taking into account no risk of contract breach by
reinsurance companies, the payoff function of the catastrophe reinsurance contract is
expressed as follows:
,
,
0 ,
T
T T T
M A C M
P C A M C A
otherwise
− >
= − > >
(10)
T
P : Payoff function of reinsurance contract at maturity.
T
C : Aggregate catastrophe losses insured by the insurance company at maturity.
M : Cap level
A : Attachment point
3.1.2 Default risk
When reinsurance companies cannot manage to fulfill claims obligations
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stipulated in the provisions of catastrophe reinsurance contracts, there is the risk of
default. The payoff function of reinsurance contracts under default risk at time of
contract maturity is expressed in Equation (11). Where, the asset and liability values
of reinsurance companies are represented by T
V andT
L , respectively, upon maturity of
reinsurance contract. We learned, through default risk considerations, that profit
function of this kind of catastrophe reinsurance contracts is based on end-period
financial position ratio analysis of reinsurance companies. This study assumes that
reinsurance contract liabilities underwritten by reinsurance companies have the same
priority claims as other types of insurance liabilities. When reinsurance companies
default because they cannot fulfill claims obligations, these liabilities will be shared
equally among the remaining asset values of the reinsurance companies on a pro rata
basis. As the profit function shows, default risk will reduce the expected cash inflows
of insurance companies and further contribute to the decline in the value of
reinsurance contracts.
,
,
,
( ) ,
( ) ,
0 ,
T T T
T T T T T
T
d T T T T
T
T T
T T T T
T T
M A if C M and V L M A
C A if M C A and V L C A
M A VP if C M and V L M A
L M A
C A Vif M C A and V L C A
L C A
othe
− ≥ ≥ + −
− ≥ ≥ ≥ + −
−= ≥ < + −
+ −
−≥ ≥ < + −
+ −
rwise
(11)
3.2 Issuance of catastrophe bonds
Reinsurance Companies can issue catastrophe bonds to increase their own
underwriting capacity to provide more insurance programs or increase the scope of
protection. These types of CAT bond-based contingent payments are used to pay
liabilities claims. Therefore, in this type of situations, the size of the reinsurance
contract value not only determines the financial positioning of reinsurance companies,
but also determines the correlation between the size of the CAT bond-based
contingent payments and the composite index of catastrophe losses.
In this section, we will consider the situation of reinsurance companies providing
reinsurance contract protections while issuing CAT bonds. In research for our thesis
paper, it is assumed that the CAT bonds are zero-coupon bonds that do not require to
pay any coupon when bond contract period is still in effect. From CAT bond investor
point of view, the payoff function of CAT bonds upon maturity is expressed as
follows:
*
, *
,
,
CAT CAT
CAT T
CAT CAT
F if C KP
rp F if C K
≤=
× > (12)
12
Where:
CAT
F : Denomination of CAT bond
*
C : Could be T
C or,Index T
C , depending on what is stated in the CAT bond
contract.
rp : Ratio between catastrophe loss trigger level and the required
repayment of principal when the level is triggered, with 0 1rp≤ <
When the cumulative catastrophe losses reach the catastrophe bond stipulated
trigger level (CAT
K ), reinsurance companies can obtain capital injections (CAT
δ ) from
CAT bonds as expressed below:
*
,( )CAT CAT CAT TC F Pδ = − (13)
Payoff function Re,dCAT T
P of the reinsurance contract upon maturity is expressed
as follows:
CATRe,
, ( )
, ( )
( )( ), ( )
( )( )
T T CAT T
T T T CAT T T
T CAT
d T T T CAT T
T
T T CAT
M A if C M and V L M A
C A if M C A and V L C A
M A VP if C M and V L M A
L M A
C A V
L
δ
δ
δδ
δ
− > + > + −
− > > + > + −
− += > + < + −
+ −
− +, ( )
0 ,
T T CAT T T
T T
if M C A and V L C AC A
otherwise
δ
> > + < + −+ −
(14)
3.2.1 Basis risk
When compensation claims that catastrophe linked products are based on are
actual catastrophe losses (T
C ), that is, when *( ) ( )
TC Cδ δ= , then from the perspective of
reinsurance companies, basis risk does not exist. However, if the claims are based on
composite index of catastrophe losses (,index T
C ) rather than actual catastrophe losses,
that is, when *
,( ) ( )Index T
C Cδ δ= , then basis risk does exist. When basis risk exists, the
size of reinsurance company's actual underwriting losses and that of composite index
of catastrophe losses will not be in agreement. At such a time, the capital that CAT
bonds should be injected with will not be the same as the actual capital injection. Such
a situation will increase the default risk of reinsurance companies, thereby reducing
their debt repayment ability. Therefore, basis risk is also a risk factor affecting
reinsurance contract value.
3.3 Issuance of hybrid CAT bonds
The study actually combined CAT bonds and CatEPut options into hybrid CAT
bonds. Reinsurance companies can use the issuance of hybrid CAT bonds to solve the
problem of insufficient capital at the time when CAT bonds are issued. However,
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reinsurance company issued CAT bonds still cannot completely transfer the
underwritten catastrophe risk to the capital market (Cummins, Lewis, and Phillips,
2002; Nell and Richer, 2004). The study proposed instead that reinsurance company
issue hybrid CAT bonds to solve the problem of insufficient capital injections. Based
on product design, hybrid CAT bonds possess more post-loss financing characteristics
than CAT bonds, providing reinsurance companies with more capital injections and
underwriting capabilities.
Next is the description of payoff function during the issuing of hybrid CAT
bonds. Hybrid CAT bonds consist of two products, CAT bonds and CatEPut options,
and therefore have payoff function (Hybrid
δ ) that is sum total of payoff functions of
CAT bonds (CAT
δ ) and CatEPut (CatEPuts
δ ). The payoff function of these products is
expressed as follows:
CAT bond portion of the payoff function:
From the point of view of CAT bond investors, the payoff function of matured
CAT bonds is as follows:
*
, *
,
,
CAT Hybrid
CAT T
CAT Hybrid
F if C KP
rp F if C K
≤=
× >
(15)
Where:
Hybrid
K : Trigger level of hybrid CAT bonds
Capital injections (CAT
δ ) obtained by reinsurance companies is as follows:
*
,( )CAT CAT CAT TC F Pδ = −
CatEPut option portion of the payoff function:
This article assumes that the market is efficient. That is, equity value of
reinsurance companies in the market will fully reflect the actual equity value of
reinsurance companies upon maturity. At such time, the equity value of reinsurance
companies in the market is expressed as follows:
Equity value (T
E ) = Asset value (T
V ) - Liability value (non-CAT + CAT
reinsurance contracts;T
L )
Under the assumption of market efficiency, the equity value of reinsurance
companies in the market can reflect the true equity value of reinsurance companies, so
catastrophic events will increase the value of liability that further reduces equity value.
Therefore, downside risk protection of equity value is critically important.
Reinsurance companies also use CatEPuts to increase post catastrophe capital
injections to protect downside risk of equity value, thereby transferring catastrophe
risk to capital market. The payoff function of CatEPuts upon maturity is expressed as
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follows:
*,
0 ,
CatEPuts T CatEPuts T T Hybrid
CatEPuts
E E if E E and C K
if otherwiseδ
− > >=
(16)
Where:
CatEPuts
δ : Capital injection of CatEPut option in the hybrid CAT bond
CatEPuts
E : Strike price of CatEPut option in the hybrid CAT bond
Hybrid
K : Trigger level of hybrid Cat bonds
Payoff function of matured hybrid CAT bonds is expressed as follows:
Hybrid CAT CatEPuts
δ δ δ= + (17)
Where:
Hybrid
δ : Payoff function of hybrid CAT bonds
CAT
δ : Payoff function of CAT bond in the hybrid CAT bond
CatEPuts
δ : Payoff function of CatEPut option in the hybrid CAT bond
Payoff function of reinsurance contracts upon maturity is expressed as follows:
Re,
, ( )
, ( )
( )( ), ( )
T T Hybrid T
T T T Hybrid T T
T Hybrid
dHybrid T T T Hybrid T
T
M A if C M and V L M A
C A if M C A and V L C A
M A VP if C M and V L M
D M A
δ
δ
δδ
− > + > + −
− > > + > + −
− += > + < + −
+ −
( )( ), ( )
0 ,
T T Hybrid
T T Hybrid T T
T T
A
C A Vif M C A and V L C A
D C A
otherwise
δδ
− + > > + < + −
+ −
(18)
Therefore, the value of insurance contracts is not only related to initial capital
position of reinsurance companies, but is also related to the size of capital injections
of hybrid CAT bond
3.4 Rate on line (ROL)
Based on the payoff function of reinsurance contracts under various
circumstances , as well as on the dynamic processes of assets, liabilities, and interest
rates described in the previous section, ROL (fair pricing premium rate) is expressed
as follows:
0
0
1r
Sr dSQ
TROL E e POM A
− ∫= × × −
(19)
ROL is the proportion of each dollar of loss recoverable by the reinsurance
contract. 0
QE represents expected value of product issue date under risk neutral
probability measure (Q). , Re, Re,, , ,
T T d T dCAT T dHybrid TPO P P P P= represents payoff function of
reinsurance contact at maturity under various circumstances discussed above. The
15
structure analyzed in the study has now been explained in detail. The study does not
expect to find any closed form solution under these complex circumstances. Thus, in
the next section we will use numerical analysis to estimate the value of reinsurance
contracts.
4. Numerical Analysis
This section will use Monte Carlo simulation method to estimate ROL for
reinsurance contracts under various circumstances. The simulations are run based on a
weekly basis with 20,000 paths. Table 1 shows a list of basic parameter values that
can be regarded as baseline values for Monte Carlo simulation. First, simulation
results obtained under non-issuance situation are served as baseline values, which are
then compared with cases where various insurance linked products are issued. This
will obtain the level of impact of various factors associated with reinsurance
companies, such as initial capital position, catastrophe frequency, standard deviation,
CAT bonds, and hybrid CAT bonds issuance, on ROL of reinsurance contracts when
these factors change.
[Insert Table 1]
The study considered setting the initial financial position ratio for the reinsurance
companies to 1.1, 1.2, 1.3, 1.4, and 1.5. The initial liability (0L ) is set at 100. The
period underwritten by the reinsurance contract is one year, and the minimum trigger
level and the maximum upper limit level is set to 20 and 100, respectively. Next, the
study introduced the product contract portion of the insurance event-linked products.
In the CAT bond portion, the effective maturity period of CAT bond contracts is set to
one year. When cumulative catastrophe losses exceed CAT bond trigger level (CAT
K ),
the portion of principal needed to be repaid to CAT bondholders, ( rp ), is set to be 0.5.
The CAT bond issuing denomination is set to 10. That is, the ratio of issuance
denomination to initial liability of reinsurance companies is 0.1. In addition, CAT
bond contract trigger level (CAT
K ) is set to 80,100, or 120, respectively. Next, the
trigger level (CatEPuts
K ) of CatEPut part is set to 80,100, or 120, respectively. The
intrinsic value (CatEPuts
E ) is set to 10, which is a European style multiple
trigger-condition option. The hybrid CAT bonds are created from CAT bonds and
CatEPut options, with effective maturity period for the hybrid CAT bond set to one
year and trigger level (Hybrid
K ) set to 80, 100, or 120, respectively. The part on CAT
bonds and CatEPut options is the same as those discussed previously. The above is the
part on products setting.
Next is the discussion on other parameter settings. The correlation (VD
ρ ) between
dynamics process of assets and liabilities of reinsurance companies is set to 0.2, initial
16
interest rate (0
r ) is set to 5%, long-term average interest rate ( m ) is also set to 5%,
strength of interest rate mean reversion ( k ) is set to 0.2 and interest rate volatility ( v )
is set to 10 %, and the market price of interest rate risk (r
λ ) is set to be -0.01. The
values of these parameters are all set to the same as those used in literature. To reflect
the number of times catastrophes occur each year, the frequency of occurrence of
catastrophic events ( λ ) is set to annual occurrence of 0.5, 1, and 2 times, respectively.
The study also assumes that the parameters for actual catastrophe losses and
composite catastrophe losses are the same. The logarithmic average of actual
catastrophe losses and composite catastrophe losses ( ,C index
µ µ ) is set to ( 2, 2 ), and that
the value of logarithmic standard deviation ( ,C index
σ σ ) is set to 0.5, 1, and 2,
respectively. Attention of the study will also be focused on the relationship between
actual catastrophe losses and composite catastrophe losses.
4.1 Default risk
The study first describes the value of the contract by considering default and
no-default risks in various cases of catastrophe frequencies and standard deviation of
losses, then analyzed various cases of issuance of insurance linked products. In cases
of no-default risk, the soundness of the initial financial position will not affect the
value of the contract. In cases of default risk, the size of financial position ratio will
affect the value of the reinsurance contract. The difference in the value of contract
between the two cases is called default risk premium. Next, we will list the simulation
results in Table 2.
[Inset Table 2]
The result is consistent with the expected results. Compared to reinsurance
contracts with default risk factors taken into consideration, the value of reinsurance
contracts with no-default risk considerations is larger. The higher the initial capital
ratio of reinsurance companies is, the lower the risk of default; the value of
reinsurance contract value will also be higher. In addition, the observed default risk
premium will increase with the increase in standard deviation of catastrophe
frequency. For example, when the initial capital ratio of reinsurance was 1.3 and
standard deviation of loss was 2, and when frequency of catastrophe occurrence
increased from 0.5 to 1 and 2, the default risk premium went up from 156.5 basis
points to 312.9 and 623.9 basis points in value. Default risk premium increases as
reinsurance companies' degree of financial leverage, frequency of catastrophe
occurrences, and catastrophe loss volatility increase. Since the consideration of
catastrophic risk will lead to the increase of default risk premium, catastrophe risk is
an important factor in reinsurance contract evaluation. For example, in the situation
when standard deviation of catastrophe frequency is 2, and when the initial financial
17
ratio of reinsurance companies decreased from 1.5 to 1.1, the default risk premium
rose from 329 basis points to 963.3 basis points in value, a total increase of 634.3
basis points. In that case, when standard deviation of catastrophe frequency dropped
to 1, the amount of increase in default risk premium fell to 75.4 basis points in value.
In addition, when catastrophe risk is low (low catastrophe frequency and low
catastrophe loss standard deviation), the default risk premium becomes insignificant.
For example, when standard deviation of catastrophe frequency is 0.5, the range of
default risk premium is from 0 to 1.3 basis points in value.
4.2 Issuance of catastrophe bonds
Table 3 shows how reinsurance companies use the issuing of CAT bonds to
facilitate the transfer of catastrophic risk, reduce default risk of reinsurance companies,
and raise the value of reinsurance contracts. The study set the ratio of CAT bond
denomination to total liability is set to be 0.1, and set this type of CAT bonds to no
basis risk. The trigger level (CAT
K ) was set to 80,100, and 120, respectively. Result of
the simulation is shown in Table 3.
[Insert Table 3]
Next, we compared the reinsurance contract values listed in Table 2 and 3. We
observe that no matter what the capital position, trigger level, catastrophe frequency,
and standard deviation are at the beginning, reinsurance companies can use the
issuance of CAT bonds to reduce default risk premium and increase the value of the
reinsurance contracts. Here we should notice that default risk premium and value of
reinsurance contract would reduce and increase, respectively, following the decrease
in trigger level. That is because when the value of trigger level is higher, the
probability of total amount of catastrophe losses reaching trigger level is lower, which
also reduces the ability of reinsurance companies to pay catastrophe claims. In cases
of high catastrophe frequency, high standard deviation, and low initial capital position,
the impact of CAT bonds on the raising of reinsurance contract value is even more
significant. For example, in the situation with standard deviation of catastrophe
frequency equals to 2 and initial capital position equals to 1.1, when the trigger levels
are 80 and 120, respectively, the issuance of CAT bonds can add 61.7 and 40.2 basis
points, respectively, to the reinsurance contract value (or subtract from default risk
premium).
4.2.1 Basis risk
When the size of the capital injected by CAT bonds is based on composite index
rather than actual underwriting losses, basis risk exists in CAT bonds. Table 4 shows
how basis risk affects reinsurance contract value when reinsurance companies issue
18
CAT bonds. Table 5 shows the corresponding default risk premium under the same
condition.
[Insert Table 4]
[Inset Table 5]
Table 4 shows that when correlation between catastrophe losses underwritten by
reinsurance companies and composite index of losses is low, the level of basis risk
faced by reinsurance companies is high. When correlation coefficient is equal to 1,
basis risk does not exist. This means that reinsurance contract value is the same as
those listed in Table 3 under the same condition. The difference between reinsurance
contract value with other different correlations and those with correlation of 1 can be
viewed as a basis risk premium. Under correlation of 0.3, 0.5, or 0.8, we can see that
basis risk lowers reinsurance contract value and raises default risk premium. The
degree of decline in value increases with the increase in basis risk, catastrophe
frequency, and standard deviation. For example, when catastrophe frequency is equal
to 2, the standard deviation of both actual losses and composite index of losses of
reinsurance companies equal to 2, trigger level equals to 80, and initial capital
position equals to 1.1. With correlation dropping from 0.8 to 0.5, the reinsurance
contract decreased (default risk premium will rise) by 12.6 basis points in value. As
correlation continued to drop from 0.5 to 0.3, reinsurance contract continued to
decrease (default risk premium will continue to rise) by 9.3 basis points in value. It
should be noted that basis risk premium decreases with the increase in trigger level
and initial financial position, but increases with the increase in catastrophe frequency
and standard deviation of losses. Although basis risk premium is significant, but
compared to the issuance of CAT bonds, its effect on increasing the value of
reinsurance contract is less significant.
[Insert Figure 1]
Figure 1 shows, under the condition when catastrophe frequency is 2, standard
deviation of both actual and composite index is 2, trigger level is 80, and initial
capital position is 1.1, how the value of insurance contract and basis risk (loss
correlation coefficient) are related to catastrophe liability structure ratio (CAT bond
issuing denominations and total liabilities of reinsurance companies). Figure 1 shows
that reinsurance contract value increases with the increase in proportion of catastrophe
liability structure, which also explains how a higher proportion of CAT bond liability
structure improves the ability of reinsurance companies paying catastrophe claims,
and thus increases the value of reinsurance contracts. Figure 1 also shows that when
CAT bonds are issued with very large denominations, the effect of basis risk becomes
much intense, with increasing basis risk premium following the proportional increase
in catastrophic liability structure. When the proportion of CAT liability structure is
19
lowered, basis risk premium with varying correlation is almost non-existent.
4.3 Issuance of hybrid CAT bonds
Next, we explore the situation of reinsurance companies issuing hybrid CAT
bonds. Although issuance of CAT bonds can transfer catastrophe risks faced by the
reinsurance companies to the capital market, it is not enough to handle all the
catastrophe risks when standard deviation of catastrophe frequency is high. The study
proposes that reinsurance companies issue hybrid CAT bonds to further solve the
problem with insufficient capital created by catastrophe risks.
[Insert Table 6]
Table 6 shows the increase in underwriting ability of reinsurance companies
themselves after they had issued hybrid CAT bonds, which further increase the value
of catastrophe reinsurance contracts. The hybrid CAT bonds proposed in this study is
a combination of CAT bonds and CatEPuts that also assumes that there is no need to
further consider the default risk of this types of insurance linked products. The study
assumes that catastrophe loss trigger level referenced by the CAT bonds and CatEPuts
in hybrid CAT bond products is the same. The trigger level settings for hybrid CAT
bond contract (Hybrid
K ) are 80,100, and 120, respectively. For the CAT bond part, the
ratio of CAT bond denomination to initial liability of reinsurance companies is set to
0.1. For the CatEPuts part, the option is set to a common European style multiple
trigger-condition option, with exercise equity value of 10. The resulting reinsurance
contract value after the simulation is shown in Table 6. Next, we compared the
reinsurance contract value in Table 2 and 6. We observed that, regardless of the
situation with initial capital position, trigger level, catastrophe frequency, and
standard deviation, reinsurance companies could reduce the risk of default and
increase the value of reinsurance contracts by issuing hybrid CAT bonds. It should be
noted that default risk premium and reinsurance contract value will decrease and
increase, respectively, following the drop in trigger level. This is also because that the
higher the trigger level, the lower the probability that the total amount of catastrophe
losses will reach the trigger level. This reduces the ability of reinsurance companies to
pay catastrophe claims. Under conditions of high catastrophe frequency, high standard
deviation of losses, and low initial capital position, the impact of hybrid CAT bond on
the increase of reinsurance contract value is even more significant. For example, when
standard deviation of catastrophe frequency equals to 2, initial capital position equals
to 1.1, and trigger levels equal to 80 and 120, respectively, the issuance of hybrid CAT
bonds can add 185.5 and 120.8 basis points to reinsurance contract value (or subtract
from default risk premium), respectively.
20
4.3.1 Basis risk
When the amount of capital injection from hybrid CAT bonds is based on
composite catastrophe index rather than actual losses, then basis risk exists. Table 7
shows how basis risks affect the value of reinsurance contracts in the situation of
reinsurance companies issuing hybrid CAT bonds. Table 8 shows the corresponding
default risk premium under the same situation.
[Insert Table 7]
[Insert Table 8]
Table 7 shows that when the correlation between catastrophe losses underwritten
by reinsurance companies and composite index of losses is low, reinsurance
companies face higher basis risk. When the correlation coefficient is 1, basis risk does
not exist, and the reinsurance contract values are the same as those shown in Table 6
under the same condition. The standard deviation between reinsurance contract value
with other different correlations and those with correlation of 1 is viewed as the basis
risk premium. Under correlation of 0.3, 0.5, or 0.8, we observe that basis risks
reduced reinsurance contract value and increased default risk premium. The level of
decline in value increased following the increase in basis risk, catastrophe frequency,
and standard deviation of losses. For example, with catastrophe frequency equals to 2,
standard deviations of actual loss and composite index of reinsurance companies are
both equal to 2, trigger level equals 80, and initial capital position equals 1.1, when
correlation dropped from 0.8 to 0.5, the reinsurance contract subtracted (default risk
premium added) 37.6 basis points in value. As correlation continued to decline from
0.5 to 0.3, the reinsurance contract continued to loss (or default risk premium gained)
27.9 basis points in value. It should be noted that basis risk premium reduces with the
increase in trigger level and initial financial position, but increases with the increase
in standard deviation of catastrophe frequency. Although basis risk premium is
significant, compared to issuance of hybrid CAT bond, the impact on increasing the
value of reinsurance contracts is less significant.
Next, we made comparisons to see whether hybrid CAT bonds are still better at
raising the value of reinsurance contracts when basis risk is considered. First,
comparison between reinsurance contract values in Table 4 and 7 showed that hybrid
CAT bonds could raise the value of reinsurance contracts even more than CAT bonds.
The difference between the value of hybrid CAT bond reinsurance contracts and that
of CAT bond reinsurance contracts under other different correlations can be attributed
to the effect of hybrid CAT bonds. We were able to observe the result of basis risk in
reducing the effect of hybrid CAT bonds. The degree of decline in effect increases as
basis risk increases. For example, when catastrophe frequency was 2, standard
deviations of actual and composite index of losses of reinsurance companies were
21
both 2, trigger level was 80, initial capital position ratio was 1.1, and when correlation
dropped from 0.8 to 0.5, the value of hybrid CAT effect drops by 25 basis points.
When correlation continues to drop from 0.5 to 0.3, the value of hybrid CAT bonds
effect will also continue to drop by 18.6 basis points. We also observed that the
impact of hybrid CAT bonds reduces with the increase in trigger level and initial
financial position, but increases with the increase in standard deviation of catastrophe
frequency. Based on the above analysis, we learn that despite the consideration of
basis risk, hybrid CAT bonds are more able to enhance the value of reinsurance
contracts than CAT bonds.
4.4 Issuance comparison in terms of percentage increase in ROL
Table 9 shows the comparison of percentage increase in the value of reinsurance
contracts because of issuance of CAT bonds and hybrid CAT bonds under various
catastrophe frequency and standard deviation conditions. First, in defining the rate of
increase in issuance value, the numerator is defined as the increase in value of
reinsurance contracts because of issuance of CAT bonds or hybrid CAT bonds, the
denominator is defined as no default risk reinsurance contract value under specific
catastrophic conditions, and is expressed as follows:
[Insert Table 9]
Rate of increase in issuance value = increase in issuance value /no default risk ROL
For example, when standard deviation of catastrophe frequency is 2 and value of
reinsurance contract with no default risk is 0.29939, the issuance value of CAT bonds
with trigger level of 80 will increase by 61.7 basis points and the rate of increase with
CAT bond issuing is 0.00617/0.29939 = 2.06%. Table 9 shows that when standard
deviation of catastrophe frequency is 0.5, regardless of what security is issued, there is
no real effect on raising the value of reinsurance contracts. In other catastrophic
circumstances, regardless of CAT bonds or hybrid CAT bonds, when standard
deviation of catastrophe frequency becomes greater, the rise in the rate of increase of
reinsurance contract value also becomes greater. It means that when catastrophic
impact is more serious, the catastrophe-linked securities issued by reinsurance
companies can more rapidly raise the value of reinsurance contracts. In other words,
the effect of issuing these securities to increase the value of reinsurance contracts will
lead to the rise of rate of increase with growing severity of catastrophic events. For
example, when standard deviation of catastrophe frequency is 2, trigger level is 80,
and initial capital structure is 1.1, and when the rate of increase in catastrophe
frequency rises from 0.5 to 2, then the rate of increase in issuance value of CAT bonds
will rise from 1.96% to 2.06% and the rate of increase in the value of hybrid CAT
bonds will rise from 5.89% to 6.2%. When initial capital position are high, the rate of
22
increase in the issuance value will drop. For example, when the standard deviation of
catastrophe frequency is 2 and trigger level is 80, and when initial capital ratio
increases from 1.1 to 1.5, the rate of increase in issuance value of CAT bonds will
drop from 2.06% to 2.02% and the rate of increase in the issuance value of hybrid
CAT bonds will drop from 6.20% to 5.81%. When trigger levels become higher, the
rate of increase in the issuance value will decline. For example, when the standard
deviation of catastrophe frequency is 2 and opening capital ratio is 1.1, the rate of
increase in issuance value of CAT bonds will decline from 2.06% to 1.34% and the
rate of increase in issuance value of hybrid CAT bonds will decline from 6.20% to
4.03%.
Hybrid CAT bonds are made up of CAT bonds and CatEPuts, thus the effect of
hybrid CAT bonds on raising the value of reinsurance contracts is stronger than CAT
bonds. For example, when standard deviation of catastrophe frequency is 2, trigger
level is 80, and initial capital ratio is 1.1, the rate of increase in issuance value of
hybrid CAT bonds (6.20%) is larger than that of CAT bonds (2.06%). The difference
in the rate of increase in issuance value of both bonds is the CatEPut effect. The above
analysis shows that the issuance of hybrid CAT bonds is better in raising the rate of
increase in the value of reinsurance contracts than CAT bonds.
5. Conclusion
The hybrid CAT bonds proposed in this study can provide an effective solution to
address problems arising from insufficient valuation of catastrophe reinsurance
contracts faced by reinsurance companies and insufficient hedging made by these
reinsurance companies during catastrophic events. The study developed a reinsurance
contract valuation model designed to measure default risk, basis risk, catastrophe risk,
and interest rate risk that can also monitor as to how reinsurance companies raise the
value of reinsurance contracts by way of issuance of CAT bonds or hybrid CAT bonds.
In addition, the study adopted the concept of hybrid products proposed by Pauline and
Henri (2009) to create a type of hybrid CAT bonds composed of CAT bonds and
CatEPuts. The results showed that default risk premium has a significant value in
most cases involving catastrophic risk, and should not be ignored when evaluating
reinsurance contracts. The results also showed that the issuance of CAT bond and
hybrid CAT bond products could reduce default risk premium and increase the value
of the reinsurance contracts. Under condition of high catastrophe frequency, high
catastrophe losses, low capital position, and low trigger levels, the issuance of CAT
bonds and hybrid CAT bonds have a significant effect on raising the value of
reinsurance contracts. Basis risk will increase default risk premium and reduce the
23
value of reinsurance contract. The effect of basis risk will decrease with the increase
in reinsurance companies' initial capital structure ratios and trigger levels, but will
increase with the increase in catastrophe frequency and standard deviation of losses.
However, after considering all the factors, regardless of whether the issuance is CAT
bonds or hybrid CAT bonds, the impact on reinsurance contract is greater than that
from basis risk. In other words, even with potential basis risk, CAT bond or hybrid
CAT bond issuance are still able to reduce the default risk of reinsurance companies
and thus increase the value of reinsurance contracts.
The study assumes a completely efficient stock market truly reflects the equity
value of reinsurance companies. We found that, similar to designs of CAT bond
products, hybrid CAT bond combination product design is more able at reducing
default risk premiums than CAT bond single product design, thereby increasing the
value of reinsurance contracts even further. The reason is that combination type
products could attract additional capital injections as opposed to single type products.
This type of insurance-linked product design not only can increase the issuance
volume in risk capital market (Pauline and Henri, 2009), but the issuance of hybrid
CAT bond products can mitigate catastrophe risk faced by reinsurance companies
despite the existence of basis risk, as the study found.
24
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Vasicek, O.A., 1977. An equilibrium characterization of term structure. Journal of Financial Economics
5, 177-188.
Vaugirard, V.E., 2003. Pricing catastrophe bonds by an arbitrage approach. Quaterly Review of
Economics and Finance 43 (1), 119-132.
27
Table 1:
Parameters definitions and base values
Asset parameters Values
V
Vµ
Vφ
Vσ
,V tW
Reinsurer’s assets
Drift due to credit risk
Interest rate elasticity of asset
Volatility of credit risk
Wiener process for credit shock
0 0 1.1, 1.3, 1.5V L =
Irrelevant
-7, 0
5%
Interest rate parameters
r
κ
m
ν
rλ
,r tW
Initial instantaneous interest rate
Magnitude of mean-reverting force
Long-run mean of interest rate
Volatility of interest rate
Market price of interest rate risk
Wiener process for interest rate shock
5%
0.2
5%
10%
-0.01
Liability parameters
L
Lµ
Lφ
Lσ
,L tW
Reinsurer’s liabilities
Drift due to credit risk
Interest rate elasticity of liability
Volatility of credit risk
Wiener process for credit risk
100 不相關
-7, 0
0.05
Catastrophe loss parameters
λ
Cµ
,Index Cµ
Cσ
,Index Cσ
Cρ
( )N t
Catastrophe intensity
Mean of the logarithm of CAT losses for the insurer
Mean of the logarithm of CAT losses for the composite loss index
Standard deviation of the logarithm of CAT losses for the insurer
Standard deviation of the logarithm of CAT losses for the composite loss index
Correlation coefficient of the logarithm of CAT losses of the reinsurer and the
composite index
Poisson process for the occurrence of catastrophes
0.5,1,2
2
2
0.5 , 1 , 1.5
0.5 , 1, 1.5
0.3, 0.5, 0.8, 1
Other parameters
M
A
CATK
HybridK
CatEPutsE
rp
Cap level of loss paid by a reinsurance contract
Attachment level of a reinsurance contract
Trigger level in the CAT bond
Trigger level in the Hybrid CAT bond
Exercise price in the Hybrid CAT bond
The ratio of principal needed to be paid if debt forgiveness is triggered
100
20
80, 100, 120
80, 100, 120
10
0.5
28
Table 2: Values of reinsurance contracts (ROL) without issuing
( λ 、 Cσ ) Default-free ROL Default-risky ROL Default-risky premium
1.1 1.3 1.5 1.1 1.3 1.5
(0.5,0.5) 0.00298 0.00284 0.00297 0.00298 0.00013 0.00001 0.00000
(0.5,1) 0.02081 0.01709 0.01910 0.02016 0.00372 0.00171 0.00065
(0.5,2) 0.07886 0.05431 0.06320 0.07072 0.02455 0.01565 0.00814
(1,0.5) 0.01151 0.01079 0.01139 0.01150 0.00072 0.00012 0.00001
(1,1) 0.04870 0.03936 0.04418 0.04690 0.00934 0.00452 0.00180
(1,2) 0.15552 0.10666 0.12423 0.13932 0.04886 0.03129 0.01620
(2,0.5) 0.04865 0.04422 0.04771 0.04850 0.00443 0.00094 0.00015
(2,1) 0.12512 0.09885 0.11203 0.11983 0.02627 0.01309 0.00529
(2,2) 0.29939 0.20306 0.23700 0.26649 0.09633 0.06239 0.03290
This table presents the ROLs for alternative sets of catastrophe intensities (λ) and catastrophe loss volatilities ( Cσ ) for default
free and default risky reinsurance contracts. The difference of their values are the default risk premiums.
V/L represents the initial asset–liability structure or capital position of the reinsurers.
All estimates are computed using 20,000 simulation runs.
29
Table 3: Values of reinsurance contracts (ROL) with CAT bonds but no basis risk
( , )C
λ σ Contract value (ROL) Default risk premium
1.1 1.3 1.5 1.1 1.3 1.5
80CAT
K =
(0.5,0.5) 0.00284 0.00297 0.00298 0.00013 0.00001 0.00000
(0.5,1) 0.01723 0.01924 0.02030 0.00358 0.00157 0.00052
(0.5,2) 0.05586 0.06475 0.07223 0.02300 0.01411 0.00663
(1,0.5) 0.01079 0.01139 0.01150 0.00072 0.00012 0.00001
(1,1) 0.03971 0.04454 0.04724 0.00899 0.00416 0.00146
(1,2) 0.10976 0.12733 0.14233 0.04577 0.02819 0.01319
(2,0.5) 0.04425 0.04774 0.04852 0.00440 0.00092 0.00013
(2,1) 0.09992 0.11309 0.12084 0.02520 0.01202 0.00428
(2,2) 0.20924 0.24318 0.27253 0.09016 0.05621 0.02686
100CAT
K =
(0.5,0.5) 0.00284 0.00297 0.00298 0.00013 0.00001 0.00000
(0.5,1) 0.01718 0.01918 0.02024 0.00364 0.00163 0.00057
(0.5,2) 0.05562 0.06451 0.07202 0.02324 0.01435 0.00684
(1,0.5) 0.01079 0.01139 0.01150 0.00073 0.00012 0.00001
(1,1) 0.03957 0.04439 0.04711 0.00913 0.00431 0.00159
(1,2) 0.10931 0.12688 0.14194 0.04621 0.02864 0.01358
(2,0.5) 0.04423 0.04771 0.04850 0.00442 0.00094 0.00015
(2,1) 0.09943 0.11261 0.12042 0.02569 0.01251 0.00470
(2,2) 0.20843 0.24238 0.27182 0.09097 0.05702 0.02757
120CAT
K =
(0.5,0.5) 0.00284 0.00297 0.00298 0.00013 0.00001 0.00000
(0.5,1) 0.01713 0.01914 0.02019 0.00369 0.00168 0.00062
(0.5,2) 0.05526 0.06416 0.07167 0.02360 0.01470 0.00719
(1,0.5) 0.01078 0.01139 0.01150 0.00073 0.00012 0.00001
(1,1) 0.03943 0.04425 0.04697 0.00928 0.00445 0.00173
(1,2) 0.10867 0.12624 0.14131 0.04685 0.02928 0.01421
(2,0.5) 0.04422 0.04771 0.04850 0.00443 0.00094 0.00015
(2,1) 0.09902 0.11220 0.12001 0.02610 0.01292 0.00511
(2,2) 0.20708 0.24103 0.27050 0.09231 0.05836 0.02890
This table presents ROLs with CAT bond issuance, but no basis risk. ROLs are calculated and reported for alternative sets of
trigger values ( CATK ),
catastrophe intensities (λ), and catastrophe loss volatilities ( Cσ ). The default risk premium represents the difference in the
values of reinsurance
contracts with CAT bond issuance and risk free reinsurance contracts.
V/L represents the initial asset–liability structure or capital position of the reinsurers.
All estimates are computed using 20,000 simulation runs.
30
Table 4: Values of reinsurance contracts (ROL) with CAT bonds and basis risk
Trigger( CATK ) 80 100 120
Cρ 0.3 0.5 0.8 0.3 0.5 0.8 0.3 0.5 0.8
( , , )C indexλ σ σ / 1.1V L =
(0.5,0.5,0.5) 0.00284 0.00284 0.002844 0.00284 0.00284 0.00284 0.00284 0.00284 0.00284
(0.5,1,1) 0.01712 0.01716 0.01721 0.01710 0.01712 0.01716 0.01710 0.01710 0.01712
(0.5,2,2) 0.05491 0.05516 0.05555 0.05476 0.05499 0.05535 0.05457 0.05477 0.05503
(1,0.5,0.5) 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079
(1,1,1) 0.03946 0.03955 0.03968 0.03939 0.03945 0.03954 0.03937 0.03938 0.03941
(1,2,2) 0.10798 0.10850 0.10925 0.10767 0.10813 0.10881 0.10732 0.10770 0.10826
(2,0.5,0.5) 0.04424 0.04425 0.04425 0.04422 0.04423 0.04423 0.04422 0.04422 0.04422
(2,1,1) 0.09924 0.09954 0.09984 0.09901 0.09919 0.09938 0.09888 0.09892 0.09900
(2, 2,2) 0.20618 0.20711 0.20837 0.20552 0.20638 0.20756 0.20464 0.20534 0.20633
/ 1.3V L =
(0.5,0.5,0.5) 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297
(0.5,1,1) 0.01913 0.01917 0.01922 0.01911 0.01913 0.01917 0.01911 0.01911 0.01913
(0.5,2,2) 0.06380 0.06406 0.06445 0.06366 0.06389 0.06425 0.06347 0.06366 0.06393
(1,0.5,0.5) 0.01139 0.01139 0.01139 0.01139 0.01139 0.01139 0.01139 0.01139 0.01139
(1,1,1) 0.04429 0.04437 0.04450 0.04422 0.04428 0.04436 0.04419 0.04421 0.04424
(1,2,2) 0.12555 0.12607 0.12682 0.12524 0.12570 0.12638 0.12489 0.12527 0.12583
(2,0.5,0.5) 0.04773 0.04774 0.04774 0.04771 0.04771 0.04771 0.04771 0.04771 0.04771
(2,1,1) 0.11242 0.11272 0.11301 0.11219 0.11237 0.11256 0.11206 0.11210 0.11218
(2, 2,2) 0.24013 0.24106 0.24231 0.23947 0.24033 0.24150 0.23858 0.23928 0.24027
/ 1.5V L =
(0.5,0.5,0.5) 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298
(0.5,1,1) 0.02019 0.02023 0.02028 0.02017 0.02019 0.02023 0.02017 0.02017 0.02019
(0.5,2,2) 0.07131 0.07156 0.07194 0.07117 0.07140 0.07176 0.07099 0.07118 0.07144
(1,0.5,0.5) 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150
(1,1,1) 0.04700 0.04709 0.04721 0.04694 0.04700 0.04708 0.04692 0.04693 0.04696
(1,2,2) 0.14061 0.14112 0.14185 0.14032 0.14077 0.14144 0.13997 0.14035 0.14090
(2,0.5,0.5) 0.04851 0.04852 0.04852 0.04850 0.04850 0.04850 0.04850 0.04850 0.04850
(2,1,1) 0.12023 0.12050 0.12078 0.12001 0.12018 0.12037 0.11987 0.11991 0.11999
(2, 2,2) 0.26959 0.27049 0.27172 0.26895 0.26980 0.27096 0.26807 0.26877 0.26975
This table presents ROLs with CAT bond issuance and the payoffs to CAT bonds are linked to a catastrophe loss index. ROLs are
calculated and reported for alternative sets of trigger values ( CATK ), catastrophe intensities (λ), catastrophe loss volatilities
( Cσ , ,C indexσ ) and coefficients of correlation between the reinsurer’s catastrophe loss and the composite loss index ( Cρ ).
V/L represents the initial asset–liability structure or capital position of the reinsurers.
All estimates are computed using 20,000 simulation runs.
31
Table 5: Default risk premium of reinsurance contracts with CAT bond and basis risk
Trigger( CATK ) 80 100 120
Cρ 0.3 0.5 0.8 0.3 0.5 0.8 0.3 0.5 0.8
( , , )C indexλ σ σ / 1.1V L =
(0.5,0.5,0.5) 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013
(0.5,1,1) 0.00370 0.00366 0.00360 0.00371 0.00369 0.00365 0.00372 0.00371 0.00370
(0.5,2,2) 0.02395 0.02370 0.02331 0.02410 0.02387 0.02351 0.02428 0.02409 0.02383
(1,0.5,0.5) 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072
(1,1,1) 0.00924 0.00916 0.00903 0.00931 0.00925 0.00917 0.00934 0.00932 0.00929
(1,2,2) 0.04755 0.04702 0.04627 0.04785 0.04740 0.04671 0.04821 0.04783 0.04726
(2,0.5,0.5) 0.00441 0.00441 0.00440 0.00443 0.00443 0.00443 0.00443 0.00443 0.00443
(2,1,1) 0.02587 0.02558 0.02528 0.02610 0.02593 0.02574 0.02624 0.02620 0.02612
(2, 2,2) 0.09321 0.09228 0.09103 0.09387 0.09301 0.09184 0.09476 0.09406 0.09307
/ 1.3V L =
(0.5,0.5,0.5) 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001
(0.5,1,1) 0.00169 0.00165 0.00159 0.00170 0.00168 0.00164 0.00170 0.00170 0.00169
(0.5,2,2) 0.01506 0.01480 0.01441 0.01520 0.01497 0.01461 0.01539 0.01519 0.01493
(1,0.5,0.5) 0.00012 0.00012 0.00012 0.00012 0.00012 0.00012 0.00012 0.00012 0.00012
(1,1,1) 0.00442 0.00433 0.00420 0.00448 0.00442 0.00434 0.00451 0.00449 0.00447
(1,2,2) 0.02998 0.02945 0.02870 0.03028 0.02983 0.02914 0.03064 0.03026 0.02969
(2,0.5,0.5) 0.00092 0.00092 0.00092 0.00094 0.00094 0.00094 0.00094 0.00094 0.00094
(2,1,1) 0.01270 0.01240 0.01210 0.01292 0.01275 0.01256 0.01306 0.01302 0.01294
(2, 2,2) 0.05926 0.05833 0.05708 0.05993 0.05906 0.05789 0.06081 0.06011 0.05912
/ 1.5V L =
(0.5,0.5,0.5) 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
(0.5,1,1) 0.00063 0.00059 0.00054 0.00064 0.00063 0.00058 0.00065 0.00064 0.00063
(0.5,2,2) 0.00755 0.00730 0.00691 0.00768 0.00746 0.00710 0.00787 0.00768 0.00742
(1,0.5,0.5) 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001
(1,1,1) 0.00170 0.00161 0.00149 0.00176 0.00170 0.00162 0.00179 0.00177 0.00175
(1,2,2) 0.01491 0.01440 0.01367 0.01520 0.01475 0.01408 0.01555 0.01517 0.01462
(2,0.5,0.5) 0.00014 0.00014 0.00013 0.00015 0.00015 0.00015 0.00015 0.00015 0.00015
(2,1,1) 0.00489 0.00462 0.00434 0.00511 0.00494 0.00475 0.00525 0.00521 0.00513
(2, 2,2) 0.02981 0.02890 0.02767 0.03044 0.02959 0.02843 0.03132 0.03063 0.02965
This table presents the default risk premium with CAT bond issuance and the payoffs to CAT bonds are linked to a catastrophe
loss index. The default risk premiums are calculated and reported for alternative sets of trigger values ( CATK ), catastrophe
intensities (λ), catastrophe loss volatilities ( Cσ , ,C indexσ ) and coefficients of correlation between the reinsurer’s catastrophe loss
and the composite loss index ( Cρ ).
V/L represents the initial asset–liability structure or capital position of the reinsurers.
All estimates are computed using 20,000 simulation runs.
32
Figure 1: ROLs for alternative sets of loss correlation ( Cρ ) and reinsurer’s debt
structure ( CAT L ) while fixing ( , , ) (2,2,2)i indexλ σ σ = , ( , ) (0,0)V Lφ φ = .
33
Table 6: Values of reinsurance contracts (ROL) with Hybrid CAT bonds but no basis risk
( , )Cλ σ Contract value (ROL) Default risk premium
1.1 1.3 1.5 1.1 1.3 1.5
80Hybrid
K =
(0.5,0.5) 0.00284 0.00297 0.00298 0.00013 0.00001 0.00000
(0.5,1) 0.01751 0.01952 0.02053 0.00331 0.00130 0.00029
(0.5,2) 0.05895 0.06784 0.07506 0.01991 0.01101 0.00380
(1,0.5) 0.01079 0.01140 0.01150 0.00072 0.00011 0.00001
(1,1) 0.04042 0.04524 0.04783 0.00828 0.00346 0.00087
(1,2) 0.11597 0.13353 0.14800 0.03955 0.02199 0.00752
(2,0.5) 0.04429 0.04778 0.04855 0.00436 0.00087 0.00010
(2,1) 0.10206 0.11523 0.12261 0.02306 0.00989 0.00251
(2,2) 0.22161 0.25555 0.28390 0.07778 0.04385 0.01550
100Hybrid
K = (0.5,0.5) 0.00284 0.00297 0.00298 0.00013 0.00001 0.00000
(0.5,1) 0.01733 0.01934 0.02038 0.00349 0.00148 0.00043
(0.5,2) 0.05823 0.06712 0.07449 0.02063 0.01174 0.00437
(1,0.5) 0.01079 0.01139 0.01150 0.00073 0.00012 0.00001
(1,1) 0.03998 0.04480 0.04750 0.00873 0.00390 0.00121
(1,2) 0.11463 0.13220 0.14699 0.04089 0.02332 0.00854
(2,0.5) 0.04423 0.04772 0.04850 0.00442 0.00093 0.00015
(2,1) 0.10060 0.11378 0.12152 0.02452 0.01134 0.00360
(2,2) 0.21918 0.25313 0.28201 0.08021 0.04627 0.01738
120Hybrid
K =
(0.5,0.5) 0.00284 0.00297 0.00298 0.00013 0.00001 0.00000
(0.5,1) 0.01718 0.01919 0.02024 0.00364 0.00163 0.00057
(0.5,2) 0.05716 0.06605 0.07346 0.02170 0.01281 0.00540
(1,0.5) 0.01078 0.01139 0.01150 0.00073 0.00012 0.00001
(1,1) 0.03954 0.04437 0.04707 0.00916 0.00434 0.00163
(1,2) 0.11272 0.13029 0.14515 0.04281 0.02523 0.01037
(2,0.5) 0.04422 0.04771 0.04850 0.00443 0.00094 0.00015
(2,1) 0.09936 0.11254 0.12034 0.02576 0.01258 0.00478
(2,2) 0.21514 0.24909 0.27815 0.08425 0.05030 0.02125
This table presents ROLs with Hybrid CAT bond issuance, but no basis risk. ROLs are calculated and reported for alternative sets
of trigger values ( HybridK ), catastrophe intensities (λ), and catastrophe loss volatilities ( Cσ ). The default risk premium
represents the difference in the values of reinsurance contracts with CAT bond issuance and risk free reinsurance contracts.
V/L represents the initial asset–liability structure or capital position of the reinsurers.
All estimates are computed using 20,000 simulation runs.
34
Table 7: Values of reinsurance contracts (ROL) with Hybrid CAT bonds and basis risk
Trigger( HybridK ) 80 100 120
Cρ 0.3 0.5 0.8 0.3 0.5 0.8 0.3 0.5 0.8
( , , )C indexλ σ σ / 1.1V L =
(0.5,0.5,0.5) 0.00284 0.00284 0.00284 0.00284 0.00284 0.00284 0.00284 0.00284 0.00284
(0.5,1,1) 0.01716 0.01728 0.01744 0.01711 0.01716 0.01729 0.01710 0.01711 0.01715
(0.5,2,2) 0.05610 0.05686 0.05804 0.05567 0.05636 0.05744 0.05511 0.05568 0.05648
(1,0.5,0.5) 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079 0.01079
(1,1,1) 0.03965 0.03992 0.04030 0.03945 0.03963 0.03989 0.03937 0.03942 0.03950
(1,2,2) 0.11063 0.11221 0.11445 0.10971 0.11108 0.11312 0.10865 0.10979 0.11149
(2,0.5,0.5) 0.04427 0.04429 0.04429 0.04423 0.04423 0.04423 0.04422 0.04422 0.04422
(2,1,1) 0.10004 0.10092 0.10181 0.09935 0.09987 0.10044 0.09894 0.09906 0.09931
(2, 2,2) 0.21245 0.21524 0.21900 0.21046 0.21305 0.21657 0.20781 0.20991 0.21288
/ 1.3V L =
(0.5,0.5,0.5) 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297 0.00297
(0.5,1,1) 0.01917 0.01929 0.01945 0.01912 0.01917 0.01930 0.01911 0.01912 0.01916
(0.5,2,2) 0.06499 0.06576 0.06694 0.06456 0.06525 0.06634 0.06400 0.06458 0.06538
(1,0.5,0.5) 0.01140 0.01140 0.01140 0.01139 0.01139 0.01139 0.01139 0.01139 0.01139
(1,1,1) 0.04448 0.04474 0.04513 0.04428 0.04445 0.04471 0.04420 0.04425 0.04433
(1,2,2) 0.12819 0.12978 0.13201 0.12728 0.12865 0.13069 0.12622 0.12736 0.12906
(2,0.5,0.5) 0.04776 0.04778 0.04778 0.04772 0.04772 0.04772 0.04771 0.04771 0.04771
(2,1,1) 0.11321 0.11410 0.11499 0.11253 0.11305 0.11362 0.11211 0.11224 0.11249
(2, 2,2) 0.24640 0.24918 0.25294 0.24441 0.24700 0.25051 0.24176 0.24386 0.24683
/ 1.5V L =
(0.5,0.5,0.5) 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298 0.00298
(0.5,1,1) 0.02022 0.02033 0.02048 0.02018 0.02023 0.02035 0.02017 0.02018 0.02022
(0.5,2,2) 0.07241 0.07313 0.07422 0.07203 0.07269 0.07373 0.07149 0.07204 0.07280
(1,0.5,0.5) 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150 0.01150
(1,1,1) 0.04717 0.04740 0.04775 0.04699 0.04716 0.04742 0.04692 0.04696 0.04704
(1,2,2) 0.14306 0.14454 0.14662 0.14225 0.14358 0.14553 0.14124 0.14235 0.14396
(2,0.5,0.5) 0.04853 0.04855 0.04855 0.04850 0.04850 0.04850 0.04850 0.04850 0.04850
(2,1,1) 0.12091 0.12167 0.12243 0.12032 0.12082 0.12137 0.11992 0.12005 0.12029
(2, 2,2) 0.27542 0.27804 0.28157 0.27364 0.27614 0.27952 0.27108 0.27310 0.27596
This table presents ROLs with Hybrid CAT bond issuance and the payoffs to CAT bonds are linked to a catastrophe loss index.
ROLs are calculated and reported for alternative sets of trigger values ( HybridK ), catastrophe intensities (λ), catastrophe loss
volatilities ( Cσ , ,C indexσ ) and coefficients of correlation between the reinsurer’s catastrophe loss and the composite loss index
( Cρ ).
V/L represents the initial asset–liability structure or capital position of the reinsurers.
All estimates are computed using 20,000 simulation runs.
35
Table 8: Default risk premium of reinsurance contracts with Hybrid CAT bond and basis risk
Trigger( HybridK ) 80 100 120
Cρ 0.3 0.5 0.8 0.3 0.5 0.8 0.3 0.5 0.8
( , , )C indexλ σ σ / 1.1V L =
(0.5,0.5,0.5) 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013 0.00013
(0.5,1,1) 0.00366 0.00354 0.00337 0.00370 0.00365 0.00353 0.00371 0.00371 0.00366
(0.5,2,2) 0.02276 0.02200 0.02082 0.02319 0.02250 0.02142 0.02375 0.02318 0.02238
(1,0.5,0.5) 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072 0.00072
(1,1,1) 0.00905 0.00879 0.00840 0.00925 0.00907 0.00882 0.00933 0.00928 0.00920
(1,2,2) 0.04490 0.04331 0.04108 0.04581 0.04445 0.04240 0.04688 0.04574 0.04403
(2,0.5,0.5) 0.00438 0.00436 0.00436 0.00442 0.00442 0.00442 0.00443 0.00443 0.00443
(2,1,1) 0.02508 0.02420 0.02331 0.02577 0.02525 0.02468 0.02618 0.02606 0.02581
(2, 2,2) 0.08694 0.08416 0.08039 0.08893 0.08634 0.08283 0.09158 0.08948 0.08651
/ 1.3V L =
(0.5,0.5,0.5) 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001
(0.5,1,1) 0.00165 0.00153 0.00136 0.00169 0.00164 0.00152 0.00170 0.00170 0.00165
(0.5,2,2) 0.01387 0.01310 0.01192 0.01430 0.01361 0.01252 0.01486 0.01428 0.01348
(1,0.5,0.5) 0.00012 0.00012 0.00011 0.00012 0.00012 0.00012 0.00012 0.00012 0.00012
(1,1,1) 0.00422 0.00396 0.00357 0.00442 0.00425 0.00399 0.00450 0.00446 0.00437
(1,2,2) 0.02733 0.02575 0.02351 0.02824 0.02688 0.02483 0.02931 0.02817 0.02646
(2,0.5,0.5) 0.00089 0.00087 0.00087 0.00094 0.00093 0.00093 0.00094 0.00094 0.00094
(2,1,1) 0.01191 0.01102 0.01013 0.01259 0.01207 0.01150 0.01300 0.01288 0.01263
(2, 2,2) 0.05300 0.05021 0.04645 0.05499 0.05240 0.04888 0.05764 0.05554 0.05257
/ 1.5V L =
(0.5,0.5,0.5) 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
(0.5,1,1) 0.00060 0.00049 0.00033 0.00063 0.00058 0.00046 0.00064 0.00064 0.00059
(0.5,2,2) 0.00645 0.00573 0.00464 0.00683 0.00617 0.00513 0.00737 0.00682 0.00605
(1,0.5,0.5) 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001
(1,1,1) 0.00153 0.00130 0.00096 0.00171 0.00154 0.00129 0.00178 0.00174 0.00166
(1,2,2) 0.01247 0.01098 0.00890 0.01327 0.01194 0.00999 0.01429 0.01317 0.01156
(2,0.5,0.5) 0.00012 0.00010 0.00010 0.00015 0.00015 0.00015 0.00015 0.00015 0.00015
(2,1,1) 0.00421 0.00345 0.00269 0.00480 0.00430 0.00375 0.00519 0.00507 0.00483
(2, 2,2) 0.02397 0.02135 0.01782 0.02575 0.02326 0.01987 0.02831 0.02629 0.02343
This table presents the default risk premium with Hybrid CAT bond issuance and the payoffs to Hybrid CAT bonds are linked to a
catastrophe loss index. The default risk premiums are calculated and reported for alternative sets of trigger values ( HybridK ),
catastrophe intensities (λ), catastrophe loss volatilities ( Cσ , ,C indexσ ) and coefficients of correlation between the reinsurer’s
catastrophe loss and the composite loss index ( Cρ ).
V/L represents the initial asset–liability structure or capital position of the reinsurers.
All estimates are computed using 20,000 simulation runs.
36
Table 9:
Comparison between the CAT bond and the Hybrid CAT bond issuing on the rate of increase in
issuance value without basis risk
( , )Cλ σ With CAT Bond With Hybrid CAT Bond
1.1 1.3 1.5 1.1 1.3 1.5
*80K =
(0.5,0.5) 0.00% -0.01% 0.00% 0.00% -0.01% 0.00%
(0.5,1) 0.67% 0.67% 0.64% 1.98% 1.98% 1.75%
(0.5,2) 1.96% 1.96% 1.92% 5.89% 5.88% 5.50%
(1,0.5) 0.01% 0.02% 0.02% 0.05% 0.06% 0.04%
(1,1) 0.73% 0.73% 0.70% 2.18% 2.18% 1.91%
(1,2) 1.99% 1.99% 1.94% 5.98% 5.98% 5.58%
(2,0.5) 0.05% 0.05% 0.04% 0.15% 0.15% 0.11%
(2,1) 0.85% 0.85% 0.81% 2.56% 2.56% 2.22%
(2,2) 2.06% 2.06% 2.02% 6.20% 6.19% 5.81%
*100K =
(0.5,0.5) -0.01% -0.01% 0.00% -0.01% -0.01% 0.00%
(0.5,1) 0.39% 0.39% 0.37% 1.11% 1.12% 1.05%
(0.5,2) 1.66% 1.66% 1.65% 4.97% 4.97% 4.78%
(1,0.5) -0.02% -0.01% 0.00% -0.02% -0.01% 0.00%
(1,1) 0.43% 0.43% 0.43% 1.27% 1.27% 1.23%
(1,2) 1.70% 1.70% 1.69% 5.13% 5.13% 4.93%
(2,0.5) 0.01% 0.00% 0.01% 0.02% 0.02% 0.02%
(2,1) 0.46% 0.47% 0.47% 1.40% 1.40% 1.35%
(2,2) 1.79% 1.79% 1.78% 5.38% 5.39% 5.18%
*120K =
(0.5,0.5) -0.01% -0.01% 0.00% -0.01% -0.01% 0.00%
(0.5,1) 0.16% 0.17% 0.14% 0.41% 0.41% 0.37%
(0.5,2) 1.21% 1.21% 1.20% 3.61% 3.61% 3.48%
(1,0.5) -0.03% -0.01% 0.00% -0.03% -0.01% 0.00%
(1,1) 0.14% 0.14% 0.14% 0.37% 0.38% 0.36%
(1,2) 1.29% 1.29% 1.28% 3.89% 3.89% 3.75%
(2,0.5) 0.01% 0.00% 0.00% 0.01% 0.00% 0.00%
(2,1) 0.13% 0.14% 0.15% 0.41% 0.41% 0.41%
(2,2) 1.34% 1.35% 1.34% 4.03% 4.04% 3.89%