Cost of Capital and Surrender Options for Guaranteed Return Life Insurance Contracts

Until recently traditional valuation methods in the life insurance industry were based on deterministic projections of risk factors. As long as life insurers were primarily underwriting diversifiable risks, the deterministic approach although not sufficiently appropriate could provide good approximations. The delay in the recent realization by the insurers for the need for more efficient valuation standards was mainly due to the opacity of traditional accounting systems.

Cost of Capital and Surrender Options

for Guaranteed Return Life Insurance Contracts

Alexis Bailly February 27, 2005

Master of Advanced Studies in Finance ETH - University of Zürich

Abstract

The opacity of traditional accounting systems for insurance companies is well known This wasconfirmed recently by unexpected repercussions of stock market and interest rates movements on thefinancial strength of many insurance companies To improve transparency, new valuation standardsare initiated by regulators or by professional bodies such as actuaries or accountants Whether thepurpose is pricing or risk management, the new standards are all based on a market consistentframework where assets and liabilities are valued at market value

Traditionally the pricing and the risk capital assessment are treated separately In this thesis

we build a unifying valuation framework where these two components can not be dissociated Toreflect the incompleteness of insurance markets and the limited access to equity capital, we introducethe notion of cost of capital We analyse the impact of the cost of capital on the valuation of lifeinsurance contracts with guarantees

In the last part of this thesis we focus on surrender options that represent today one of the mostobscure risk for life insurers We present models where the pricing can be precisely performed, but

we also discuss certain aspects of policyholder’s behaviour

1

I would like to thank the supervisors of my master’s thesis, Prof Damir Filipovi´c for his valuablecomments and for his availability, Prof Paul Embrechts for useful discussions and Jon Bardola whosupported me with the necessary time of work and also encouraged me in the selection of this practicaltopic

2 Pricing and capital requirement for life insurance policies 5

2.1 The structure of the contract 5

2.1.1 The model 5

2.2 The case of a ’true’ guarantee 6

2.3 The case of a conditional guarantee 8

3 Cost of Capital 11 3.1 Incomplete markets and cost of capital 11

3.2 Pricing under cost of capital 13

3.2.1 The case of a ’true’ guarantee 13

3.2.2 The case of a conditional guarantee 15

4 Multi-period extension 17 4.1 General framework 17

4.2 Simplified framework 18

4.2.1 The two-period case 19

5 Surrender options 21 5.1 Surrender options in the case of conditional guarantee 21

5.1.1 Finite difference approach 23

5.2 Surrender triggered by interest rates movements 25

3

1 Introduction

Until recently traditional valuation methods in the life insurance industry were based on deterministicprojections of risk factors As long as life insurers were primarily underwriting diversifiable risks, thedeterministic approach although not sufficiently appropriate could provide good approximations Thedelay in the recent realization by the insurers for the need for more efficient valuation standards wasmainly due to the opacity of traditional accounting systems

The statutory accounting system is a summary of the cash flows and reserve movements duringthe year This accounting system could be suitable for industrial companies but not for life insurerswhere cash flows from one individual contract would be spread over long term periods Consideringthe usual cash flow pattern of insurance contracts described by an initial loss followed by a sequence

of small profits, the statutory accounting is unable to give any indication on the profitability of anew policy To overcome this deficiency new standards such as US GAAP have been introduced toallow a smoothing of expenses and revenues over the term of the contract Unfortunately these lateststandards will be even more opaque and will not help an active management due to the attenuation

by smoothing of the volatility risk

During the 90’s, actuaries introduced the Embedded Value standards Basically, the EmbeddedValue is the discounted value of future statutory profits, where all economic and operational as-sumptions are projected in a deterministic way Embedded Value as a nominal value may not bevery significant but when comparing with the previous standards, the change in Embedded Valuecould give an important insight into the sources of value creation (for instance investments, mortality,lapses ) The main drawback remains the inability to value the costs of options and guarantees.The use of deterministic methods, allowed life insurance companies to ignore the sources of volatil-ity, and in particular the occurrence of extreme events In the last 10 years, insurance companiesstarted to record extreme losses resulting from unanticipated changes in interest rates, stock markets

or longevity

The pressure from the investment community is becoming more and more intense The modernstandards currently in development are either called Fair Value, Market Consistent Value or StochasticEmbedded Value; the objectives are the same, valuing assets and liabilities at purchase value by taking

in account all sources of variability Traded assets and liabilities will be valued at market value andthe non traded ones will be valued at the price of the replicating portfolio The main challengefor insurance companies is the implementation of stochastic models in order to value the costs ofembedded options and guarantees

In this paper, our focus will be on market consistent valuation at the level of an individual policy.Traditionally the issue of the pricing of an insurance contract was completely separated from theissue of risk capital assessment In this thesis, we propose a unifying valuation framework, where thetwo issues are treated simultaneously

The thesis is organised as follows In Section 2 we describe the structure of the contract and

we introduce the general framework for pricing participating policies with guaranteed return InSection 3 we first introduce a definition for the cost of capital by considering the incompleteness ofthe insurance market Then we analyse the impact of the cost of capital on the valuation of insurancecontracts In Section 4 we extend our framework to a multiperiod situation, we discuss some practicalissues and analyse a simplified two-period case In Section 5, we focus on the pricing of surrenderoptions and discuss the issue of the policyholder behaviour We end with some conclusions

2 Pricing and capital requirement for life insurance cies

In this section we develop a framework to analyse a single premium participating insurance contractwith minimum guarantee Under this contract the policyholder pays a single premium P0at time t=0and the insurer is obliged to pay at maturity t=T a specified amount depending on the performance

of a reference fund

The participating contract consists of two elements - a fixed guaranteed payment and a call optionwhich gives the policyholder a fraction of the surplus We define the surplus as the policyholder’sfund value less the guaranteed amount

Such contracts provide generally a guaranteed benefit in case of death, but in our analysis we willignore mortality and will only focus on financial risks

Moreover, the default possibility of an insurance company is never zero and in case of such an eventthe company will not be able to fulfil its obligations towards the policyholder Therefore the default

of the insurance company must be considered in a market consistent framework In practice thisdefault is often ignored by the insurance companies concerning their retail business It is difficult forthe policyholder to estimate the impact of the solvability of the insurer on the price of their contract.However in an insurer-reinsurer relationship, this risk is not ignored and the financial soundness ofthe reinsurer will have an important impact on the price of a treaty We will treat both situations

- true guarantee under the assumption of a non defaultable company and - conditional guaranteeunder the assumption of a defaultable company

2.1.1 The model

We consider an economy with two traded assets - a risk free bank account with price process B and

a risky asset with price process S

We assume that the financial market is frictionless and can be represented by a probability space(Ω, F, {Ft} , P ) {Ft}t ∈[0,T ] is the filtration generated by a one-dimentional Brownian motion, W,which represents the financial uncertainty in the economy

We assume that there exists a constant risk free interest rate r such that the dynamic of the bankaccount is given by

dBt= rBtdt, B0= 1 (1)The dynamic of the risky asset is described by the following stochastic differential equation,

dSt= µStdt + σStdWt (2)The insurance contract offers a guaranteed minimum return g We assume that a fraction λ ∈ (0, 1]

of the surplus is given to the policyholder at maturity, λ is defined at the beginning of the contract

At time t=0 the total premium received by the insurance company is P0 In addition, we assumethat the shareholders of the company have to inject a certain level of target capital TC0in order toguarantee the solvency of the company

We define the ruin probability ψ as the probability that at maturity t=T, the value of the assets

of the company AT is lower than the value of the liabilities LT,

ψ (u) = P [AT< LT | Initial shareholder capital = u]

We distinguish the physical probability P used here to express the ruin probability from the riskneutral probability Q that we will use later in the valuation of the prices of insurance contracts

5

The management of the company fixes a target α ∈ (0, 1) for the maximum value of the ruinprobability The target capital TC0 is the minimum capital satisfying this constraint on the ruinprobability and is given by

T C0= inf {u : ψ (u) ≤ α} Moreover, we assume that immediately after receiving the premium, the company decides thefollowing investment strategy:

- F0= S0 is invested in the risky asset,

- P0− F0= P0− S0 is invested in the riskless asset,

- TC0 is invested in the riskless asset

It is appropriate to assume that the target capital is entirely invested in the riskless asset given thatits main purpose is a role of buffer against adverse movements in the financial market Concerningthe split of the initial premium between risky and riskless asset, we assume that it is a pure decision

of the management of the company Our choice here is justified by the fact that we will consider

F0= S0 to be the reference fund for the attribution of the policyholder’s benefits

In this section we assume that the insurance company will fulfil in any situation its obligationstowards the policyholder This is equivalent to assume that the shareholder will inject at maturityadditional capital in the company if required For instance this could be observed in a situation where

a holding company decides to inject additional capital in an affiliate in order to avoid reputationalissues We can also assume that the regulator is monitoring closely the solvency of the company and

by constraining the investment strategy prevents the company from becoming insolvent

For an initial fund of the policyholder L0= S0,the payoff at maturity to the policyholder can bedescribed as

The price of a contract having such a payoff is calculated in the following way:

The initial investment strategy is as follows:

- F0= S0 is invested in the risky asset

- V0− F0= V0− S0 is invested in the riskless asset

- TC0 is invested in the riskless asset

The total value of the assets of the company at date t=0 is given by A0 = V0+ T C0

At maturity T, the value of the assets of the company is defined in the following way,

AT = (T C0+ (V0− S0)) erT+ ST (7)The benefits to be paid to the policyholder are given by

7

Therefore the initial target capital is given by

T C0= S0egT − β e−rT− (V0− S0) (12)

In this section we assumed that at maturity the shareholder will honour its obligations in anysituation, therefore the target capital has no importance from the policyholder’s point of view Thecontract is clearly defined; the only information important for the policyholder is the price Here,the target capital is only a requirement from the regulator and there is no additional cost associated.The price of the contract is independent of the level of the target capital and therefore when thepricing is performed there is no necessity to consider simultaneously the question of the minimumcapital requirement

The non consideration of the default risk of the insurer within the pricing of a contract fairlyrepresents the common practice in the market This practice results from the information asymmetryexisting between policyholders and shareholders

Many insurance companies are set in the form of a limited liability company For these companies,

in case of insolvency there is no obligation for the shareholders to inject additional capital

By considering the default of the company, the payoff to the policyholder becomes

LT = S0e

gT + λM ax 0, ST− S0egT if ST ≥ β

AT= (T C0+ (V0− S0)) erT+ ST if ST < β , (13)where β defines the threshold value of the stock price indicating insolvency,

{ST < β} ⇔ AT< LgT = S0egT

We can also write the payoff in the following way,

LT = S0egT + λM ax 0, ST− S0egT · 1{S T ≥β}

+ (T C0+ (V0− S0)) erT+ ST · 1{S T <β }, (14)or

LT = S0egT · 1{S T ≥β}+ λ(ST− S0egT) · 1{ST≥Max{S 0 e gT ,β}}

+ (T C0+ (V0− S0)) erT+ ST · 1{S T <β } (15)

0.60.8

1.21.41.61.8

Payoff at Maturity as a function of ST considering the default of the insurer

Comparing to the true guarantee case, the contract includes an additional put option in favour

of the company Ignoring default implies an overpricing of the contract and results therefore in anadditional value creation for the shareholders

The price of such contracts are calculated in the following way

which leads to

V0· erT = S0egT · Q [ST ≥ β] − λS0egT· Q ST ≥ Max S0egT, β

+λEQ ST· 1{ST≥Max{S0e gT ,β}} +Q [ST< β] · (T C0+ (V0− S0)) erT+EQ ST· 1{S T <β } ,

or

V0 = S0e(g−r)T· N (d2) − λS0e(g−r)T · N (h2) + λS0· N (h1)

+ (T C0+ (V0− S0)) · N (−d2) + S0· N (−d1) , (17)with

9

The following example shows the variation of the price of the contract and the target capital with

a change in the level of solvency α

0.5 0.6 0.7 0.8 0.9 1− α

0.20.40.60.81

We note also that the price of the contract is increasing with the level of the target capital Inorder to charge the policyholder with a maximum price, the company could be tempted to increaseindefinitely the level of the target capital But as we will see in the next section, capital has a cost,and the company can only have a limited resort to shareholders’ capital

3 Cost of Capital

Insurance companies are required to maintain a minimum level of capital in order to satisfy thesolvency level In the previous section this capital was represented by the target capital TC0 Arational client will usually prefer to buy his contract from the more secure company A simple wayfor an insurance company to reduce its ruin probability is to increase the target capital to relativelyhigh levels In practice we note that the access to equity capital has a cost This cost is represented

by an extra return requirement by the shareholders on their invested assets Part of this extra returncovers the risk that in case of default shareholders may not receive the full value of their investment.Covering the default risk is equivalent to going back to the situation where pricing was performedunder the assumption of a "true" guarantee In which case the value of the limited liability putoption was not deducted from the price of the contract The practice shows that in addition to theassumption of a "true" guarantee, the shareholders will still require an extra compensation Thisextra compensation is justified not only by capital costs resulting from market imperfections such

as frictional costs, double taxation or agency costs but mainly from the fact that the policyholderagrees to pay more than the cost of the contract for the shareholder We will summarize all theseextra compensations to the shareholder under the term of "cost of capital" And we consider thecost of capital to be entirely explained by the risk aversion of the policyholder who can not access

or replicate the payoff of the contract outside the offer of the company In such incomplete marketsthere is usually no uniqueness of prices satisfying no-arbitrage conditions The final prices will bechosen by the insurance company according to the level of the competition in the market For a newcontract, the insurer will fix the price at the highest level, and afterwards the price will be adjustedwith the inflow of new competitors

Let us consider the following model to illustrate the incomplete market situation

Let us assume that the insurance company wants to sell a contract with a term of one year (T=1)

We assume that this contract covers entirely a contingent claim C faced by the policyholder.The preferences of the policyholders and the shareholders at time T are described by exponentialutilities with different absolute risk aversions αP and αS respectively

Let uP be the utility function of the policyholder and uSthe utility function of the shareholder

uP(x) = 1 − e−αP x

uS(x) = 1 − e−αS x

And let WP and WSbe the initial wealth of the policyholder and shareholder respectively

We will define the price of the contract by the indifference price

We assume that if the economic agents are not buying or selling the contract, then they will investtheir initial capital in the riskless asset and will wait to face the contingent claim at time t=T.The indifference price pS for the shareholder is defined in the following way:

uS(WS(1 + r)) = E [uS((WS+ pS) (1 + r) − C)] (22)this gives,

pS= 1

αS(1 + r)ln E e

αSC

(23)

pS is the minimal price at which the shareholder will accept to sell the contract

The indifference price of the policyholder is defined as follows:

E [uP(WP(1 + r) − C)] = E [uP((WP− pP) (1 + r) + C − C)] = uP((WP− pP) (1 + r)) (24)The first term can also be written as

The second term in (24) is given by

uP((WP − pP) (1 + r)) = 1 − e−αP ((WP−p P )(1+r))

.Plugging in the equation (24) gives

be defined This will be the purpose of the next sections

The following example gives an illustration of the relation between the indifference price and thelevel of risk aversion

Indifference prices

A transaction between the policyholder and the company can occur only if pS ≤ pP.In general,

a sufficient condition for the occurrence of a transaction will be a sufficiently large price differential

pP− pS to cover the default risk on the shareholder’s assets but also any frictional costs

In this chapter we consider the same models as defined earlier, and we will look at the additionalimpact of the cost of capital The company fixes at the beginning of the contract the maximumlevel of the insolvency risk α that they can tolerate The target capital corresponds to the minimumcapital injected initially in the company such that the ruin probability at maturity is lower than thelevel α

We assume that the cost of the capital is defined by an additional return γ on the target capital

TC0 injected by the shareholders This additional return is in excess of the invested returns and isassumed to take in account the risk aversion of the policyholders For instance if TC0 is invested inthe riskless asset returning r then the return allocated to the shareholder will be γ + r

In such a situation the company can not increase indefinitely the level of the target capital withoutaltering the performance of their business An inappropriately high level of the target capital maydiscourage certain policyholders from buying their contract from the company, because they mayconsider the cost of the capital to be too high

3.2.1 The case of a ’true’ guarantee

The payoff to the policyholder is defined in the same way as in the previous sections:

LT= M ax S0egT, λ(ST− S0egT) + S0egT (25)

We assume that in addition to the fair price of the contract, the policyholder is required to payinitially an amount B to cover the cost of the shareholders’ capital Instead of applying such anupfront charge, an other possibility could be to charge at maturity a portion of the surplus of thepolicyholder

The required value by the shareholder at time T is given by T C0e(r+γ)T

13

If we assume that B is invested in the riskless asset, we should have the following relation

This additional premium B will also have an impact on the target capital

At maturity the value of the assets is given by

AT = (T C0+ (V0− S0) + B) erT+ ST (29)The target capital TC0 has been defined such that the following condition is satisfied:

P [AT < LT | Initial shareholder capital = T C0]6 α

The insolvency event is defined as follows:

{AT< LT} = (T C0+ (V0− S0) + B) erT + ST < S0egT + λM ax 0, St− S0egT

When the company is in an insolvency situation, there is no surplus distributed to the policyholder,therefore

{AT < LT} ⇔ AT< LgT = S0egTor

{AT < LT} = (T C0+ (V0− S0) + B) erT + ST < S0egT ,which implies

T C0= S0egT − β e−rT− (V0− S0) − T C0 eγT− 1 (31)Finally,

T C0= e−γT S0egT − β e−rT− (V0− S0) (32)The target capital is a decreasing function of the cost of capital γ A shareholder expecting ahigher excess return γ has to inject a lower amount of capital in the company because the targetcapital is partly financed by the policyholder

Comparing to the case studied in the section 2.2, we notice that the target capital becomes nowrelevant for the policyholder’s decision through the cost of capital B that the policyholder has to bear

In addition, for a similar value of the solvency level α, the required shareholder’s capital injectionhas reduced by an amount corresponding to the cost of capital B