Working Paper by C. Orozco-Garcia and H. Schmeiser/media/internet/content/dateien... ·...

Is Fair Pricing Possible? An Analysis of Participating Life Insurance Portfolios

Working Paper by C. Orozco-Garcia and H. Schmeiser

Universität Hamburg, 11/2015

November 2015Seite 2 Structure

1. Introduction

2. Motivation

3. Model Description

4. Numerical Illustration

5. Conclusion

November 2015Seite 3 1. Introduction

Risk-neutral valuation is the standard procedure in order to derive “fair” premiums inparticipating life insurance contracts, to analyze the solvency of an insurer, or toobtain combinations of equity- and debt capital that do not lead to wealth transfersbetween shareholders and debtholders (policyholders)

Cf. Briys & de Varenne (1997), Grosen & Jorgensen (2002), Ballotta et al. (2006),Ballotta (2009), Bacinello (2001), Tanskanen & Lukkarinen (2003), Schmeiser &Wagner (2014), and many more

The “fair” single upfront premium P0 for the savings part of the contract is given by

𝑃𝑃0 = 𝑃𝑃𝑃𝑃 𝐿𝐿𝑇𝑇 = 𝐸𝐸0ℚ 𝐿𝐿𝑇𝑇 = 𝐸𝐸0

ℚ min(𝑃𝑃𝑇𝑇 ,𝐴𝐴𝑇𝑇)


If fair pricing takes place, the shareholders will receive a risk-adequate return on theirinitial contributions (net present value NPV = 0)

𝐸𝐸𝐶𝐶0 = 𝑃𝑃𝑃𝑃 max(𝐴𝐴𝑇𝑇 − 𝑃𝑃𝑇𝑇 , 0) = 𝐸𝐸0ℚ max(𝐴𝐴𝑇𝑇 − 𝑃𝑃𝑇𝑇 , 0)

The solvency level of the insurer can be measured by the default put option

𝐷𝐷𝑇𝑇 = max(𝑃𝑃𝑇𝑇 − 𝐴𝐴𝑇𝑇, 0)

The policyholders receive a yearly interest rate rt on their initial contribution P0 inform of a cliquet-style option

𝑟𝑟𝑡𝑡 = max 𝑔𝑔,α(At/At−1 − 1)


Hence, we receive an infinite amount of fair equity / premium combinations betweenthe two stakeholder groups that lead to no wealth transfer (measured by the NPV)

EC0

P0

Fair premium without default

PV(DT)


1. Introduction

2. Motivation



5. Conclusion

November 2015Seite 7 2. Motivation

In insurance practice we typically have

… regular premium payments

… different contract starting points / maturity dates and

… the contracts are typically pooled in one legal entity

There is very little research so far that tries to capture these point (cf. Hansen &Miltersen (2002), Gerstner et al. (2008), Ibragimov et al. (2010), Doskeland & Nordahl(2008), Gollier (2008))

The following two research questions have – at least to our knowledge – not beenfocussed so far


Research question 1:

Is it possible to simultaneously charge fair premiums (NPV = 0) to all policyholders?

NPV in formal terms:

and

(Present value of the policyholder account (policyholder generation i) minus presentvalue of the first premium paid by policyholder i)


If only fair terms for the portfolio can be provided,

…. who pays more? Who pays less?

PV premiums

PV payoff

Net present value

T T+1 T+2 Portfolio

> 0 >= 0 <= 0 = 0

November 2015Seite 10

Premiums

Default

Risk level

T T+1 T+2

Low Medium High

2. Motivation

Research question 2:

Are all policyholders encountering the same default risk?

Is it possible to ensure the same value for the default put option (ratio) to allpolicyholders / generation of policyholders?


Summary:

Remark

FAIR PREMIUM FOR ALL

GENERATIONS

SAME DEFAULT RISK

LEVELFOR ALL

GENERATIONS

2. Motivation


1. Introduction

2. Motivation



5. Conclusion


Basic framework

Participating life insurance contract with cliquet-style interest rate guarantee

Surrender options and mortality risk are not taken into account

Regular premium payments

Fixed time to maturity for all contracts

Two asset classes: Risky assets and riskless rate of return

Risky assets are modelled through a GBM

Risk management measures that can be taken by the insurer:

Adjusting equity capital at discrete point in time (issuance date of each contract) andchoosing the asset allocation in t = 0

3. Model description


I. Basic model

II. Model variant with early default (accounting model)

+ 𝑃𝑃1 + 𝐸𝐸𝐶𝐶1

Not Default+(𝑃𝑃2 + 𝐸𝐸𝐶𝐶2)

Not Default+(𝑃𝑃3 + 𝐸𝐸𝐶𝐶3)

…

Not Default−𝐿𝐿1,𝜏𝜏1+𝑇𝑇



Default−𝐴𝐴𝜏𝜏3+𝑇𝑇Default

−𝐴𝐴𝜏𝜏2+𝑇𝑇Default−𝐴𝐴𝜏𝜏1+𝑇𝑇

…Default−𝐴𝐴𝜏𝜏3Default

−𝐴𝐴𝜏𝜏2

+ 𝑃𝑃1 + 𝐸𝐸𝐶𝐶1 +(𝑃𝑃2 + 𝐸𝐸𝐶𝐶2) +(𝑃𝑃3 + 𝐸𝐸𝐶𝐶3) …

Not Default−𝐿𝐿1,𝜏𝜏1+T



Default−𝐴𝐴𝜏𝜏3+𝑇𝑇Default

−𝐴𝐴𝜏𝜏2+𝑇𝑇Default−𝐴𝐴𝜏𝜏1+T

𝜏𝜏1= 0 𝜏𝜏2 𝜏𝜏3 … 𝜏𝜏1 + 𝑇𝑇 𝜏𝜏2 + 𝑇𝑇 𝜏𝜏3 + 𝑇𝑇

𝜏𝜏1= 0 𝜏𝜏2 𝜏𝜏3 … 𝜏𝜏1 + 𝑇𝑇 𝜏𝜏2 + 𝑇𝑇 𝜏𝜏3 + 𝑇𝑇

Pension scheme

Life insurer



Formal structure:

Policyholder account of the i-th policyholder generation

Evolution of the assets



Basic model (I):

Insurer pursued its activities as long as

holds at maturity.Thereby,

denotes the accumulated liabilities of the insurer. In case of a default, the cost ofinsolvency correspond to the difference between the accumulated liabilities and theavailable assets



Basic model (II):

Proportional insolvency costs for the i-th policyholder generation

Payoff when taken default risk into account



Basic model (III):

Fair valuation (risk neutral measure)

Dynamics of the rate of return of the assets

Present value of the contracts of the i-th policyholder generation

Values at issuance date τi



Basic model (IV):

Accumulated value of the portfolio

Value of the default put option for the contracts of the i-th policyholder genera-tion

NPV of the contracts



Basic model (V):

Fair pricing takes place if the net present value is zero (for both stakeholder groups)

Default put option ratio is defined as

Thereby,

denotes the value of the contract(s) without default

Accounting model: Please cf. working paper



1. Introduction

2. Motivation



5. Conclusion

November 2015Seite 22 4. Numerical Illustration


Basic model (I):

It is not always possible to achieve a fair price for the portfolio as a whole


Basic model (II):

Fair pricing and equal default ratios are possible (for the portfolio as a whole and foreach generation) only in the case without default


Basic model (III):

Fair pricing and equal default ratios are possible (for the portfolio as a whole and foreach generation) only in the case without default


Basic model (IV):

The portfolio can be fairly priced, but NPVs for different policyholder generationsand corresponding default ratios vary


Accounting model (I):

Again, we can derive fair pricing – for the portfolio as a whole and for eachgeneration – and equal default ratios in the case without any default risk


Accounting model (II):


Accounting model (III):

The portfolio can be fairly priced, but NPVs for different policyholder generationsand corresponding default ratio vary


Accounting model (IV):

For γ = 9.87 % we get the same results

For higher shares of risk assets, the required equity capital in the accounting model issubstantially lower compared to the basic model

In the accounting model later generations are better of

Default ratios develop differently, NPV discrepancies are smaller compared to thebasic model

In general, one combination of fair pricing and positive but equal default ratios exist(solvency requirements may not allow such combinations)

However, pension schemes are typically not based on the accounting modelframework


Accounting model (V):


1. Introduction

2. Motivation



5. Conclusion


In practice, we will face some wealth transfer within policyholder generations andbetween shareholders and policyholders

In addition, default ratios are typically different for each generation

Basic model: Fair pricing for the portfolio may be possible, but default ratios (if > 0)vary – NPVs for each generation differ too –.“Early” generations are better off

Accounting model: A combination of equity contributions and asset allocation mayexist, where fair pricing is possible and equal default ratios can be provided to eachpolicyholder generation

However, even for the accounting model, it is not possible to fix a desired or requiredsafety level for all policyholder generations and obtain fair pricing

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