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IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **

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IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION

Tobias Dillmann* and Jochen Ruß**

ABSTRACT

Insurance contracts often include so-called implicit or embedded options. There have been many attempts to calculate the value of such options and a fair price. In the present paper, we introduce a general model for the pricing of such options. The model allows for a wide range of analysis that was not possible with earlier models. First, our model allows for two sources of uncertainty, e.g. interest rates and some asset price. Thus, we are able to analyze interest-sensitive options in unit-linked contracts. Secondly, the model introduces the so-called concept of exercise probabilities. Whereas almost all previous work in this area assumed that policy holders exercise options in life insurance contracts if and only if the option is in the money, our model allows for more realistic cases, where some insured might base the decision to exercise an option on other than financial criteria. A third interesting point is that the model allows for a simultaneous analysis of several options within the same insurance contract.

We calculate the option price for two different options in deferred unit-inked annuities: A lump sum option and a combined option consisting of a lump sum option and an early retirement option. We perform extensive sensitivity analysis with respect to market scenarios and parameters describing the contract and analyze how the option price varies if we assume different patterns for the exercise probability.

KEYWORDS

Implicit option, embedded option, early retirement option, lump sum option, exercise probability, multivariate tree.

1 INTRODUCTION

A policy holder often has the right but not the duty to change some features of his life insurance contract during the term of the contract. Such a right is often referred to as an implicit or embedded option. It may have an influence on the cash flows of the policy, e.g. with respect to time, amount, or probability of occurrence.

Such options can create substantial additional benefits for the policy holder, which on the other hand poses a financial risk on the insurance company. Therefore, insurance companies need to quantify the value of such options in order to determine an appropriate premium and hedge themselves against the risks.

There have been several attempts to derive a fair value of such options, cf. e.g. [Ge 97], [Gr/Jo 97], [He/Kr 99], [Gr/Jo 00], [Di/Ru 01a], [Di/Ru 01b], and [Ba/Ha 02]. All these authors determine a financial option with a payoff pattern that is essentially equivalent to the difference between the cash flows of the insurance contract after exercising the option and the cash flows of the original insurance contract. One example is the lump sum option in deferred annuity contracts. This option is essentially equivalent to a put option on a coupon bond, where the strike of the put option equals the lump sum and the coupons correspond to the annuity payments. The authors then argue that since the implicit option is equivalent to the corresponding financial option, the market value of the financial option (adjusted for mortality where necessary) is an appropriate price for the implicit option. At first glance, this is very convincing, since this is the price that is needed to set up a hedge portfolio immunizing the risk resulting from the option.

However, Albizzati and Geman in [Al/Ge 94] pointed out that there is a significant difference between a “typical” financial option and a “typical” implicit option in an insurance contract: A financial option is usually

* Dr. Tobias Dillmann is consultant at the Institut für Finanz- und Aktuarwissenschaften. Address: Institut für Finanz- und Aktuarwissenschaften, Helmholtzstrasse 22, 89081 Ulm, Germany, phone: +49 731 50 31235, fax: +49 731 50 31239, email: t.dillmann@ifa-ulm.de

** Dr. Jochen Russ is managing director at the Institut für Finanz- und Aktuarwissenschaften and lecturer at the Uiversity of Ulm. Address: Institut für Finanz- und Aktuarwissenschaften, Helmholtzstr. 22, 89081 Ulm, Germany, phone: +49 731 50 31233, fax: +49 731 50 31239, email: j.russ@ifa-ulm.de

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bought for financial reasons only, e.g. speculation or hedging. Therefore, it will be exercised if and only if exercising is profitable, i.e. the option is in the money. In what follows, a policy holder exercising his option according to the same criteria will be referred to as a rational policy holder.1 If all policy holders were rational, then the price of an equivalent financial option would also be the fair price for an implicit option.

Whether or not a policy holder exercises an implicit option might however often depend on many other issues like his retirement planning, health, marital status, etc. Some policy holders may therefore find it preferable to exercise an option although the corresponding financial option is out of the money.

Technically speaking, the exercise probability, i.e. the probability of exercising an implicit option, needs to be taken into consideration. Albizzati and Geman proposed in [Al/Ge 94] a model how this can be done. In Section 2.1, we will elaborate on this model and include their concept of exercise probabilities in our previous general model described in [Di/Ru 01a] which assumed a rational policy holder.

In the present paper, we will further extend our previous model such that it will be possible to analyze options in unit-linked life insurance contracts that depend on both, interest rates and the asset price. Thus, we will need to model two underlying stochastic processes. This is done in Section 2.2. The pricing of non-European options is often done using so-called backward induction on tree structures. These techniques are well known for options that depend on one stochastic variable only. In Section 2.3, we construct a multivariate tree structure that enables us to use a backward induction algorithm for the pricing of non-European options depending on both, interest rates and an underlying asset. This tree can be used for the pricing of any kind of option, including implicit options in life insurance contracts.

In Section 3, we will apply our model to a unit-linked contract. First, we describe the insurance contract and the included options. We consider a single premium deferred unit-linked annuity. We start with a combination of two options giving the policy holder the following rights: At the end of the deferment period, he may choose a lump sum payment (the asset value) and terminate the contract instead of receiving the lifelong annuity payments. Alternatively, during the last couple of years, the policy holder can demand the immediate start of the annuity payments. This is referred to as an early retirement option. As a second and simpler case, we consider the lump sum option alone.

In Section 4, we will determine the price of the combined option for given market data and see how the value of the combined option compares to the sum of the prices of the individual options, still assuming the case of a rational policy holder. For the sake of simplicity, we then restrict ourselves to the simpler case where only the lump sum option is included in the contract. For this case, we analyze how different assumptions for the exercise probability influence the value of the option. Finally, Section 5 closes with a summary and an outlook for possible future research.

2 THE MODEL

2.1 Options in Life Insurance Contracts

In this section, we present a general model that describes implicit options in life insurance contracts and their impact on the cash flows of the corresponding life insurance contract. Furthermore, the model allows the calculation of the value of the corresponding option considering that the exercise probability may depend not only on financial circumstances.

By C, we denote the set of all possible insurance contracts. Any CC is fully described by the premiums, the benefits (at survival or death), and their corresponding probabilities. By Ct we denote the value of the contract C

at time t, i.e. the expected present value of future benefits minus future premiums.

An option is the right but not the duty to change the insurance contract CC into some other insurance contract C'∈C. By Tt, we denote the set of all possible exercise dates in [t, T] with T being the maturity of the

insurance contract. Furthermore, we denote the “payoff” of the option at time t by Ψt.

2

1 We use this notation for lack of better terms although policy holders who do not follow these criteria are not necessarily irrational.

2 In this paper, we assume that the option can be exercised at any exercise date, if the policy is still in force. There are options in the market which can only be exercised in case some trigger event happens (e.g. marriage).

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In case of a European option with the single exercise date t'≤T, we obviously have Ψt' =C't'−Ct'. Assuming a rational policy holder, the value of the option at time 't is Θt' =max[0;Ψt']=max[0;C't'−Ct'].

We use the standard risk neutral evaluation principle3 and assume independence between mortality and financial markets. Furthermore, we assume the insurance company to be risk neutral with respect to mortality.4 Then, for

't

t< , the value of the implicit option for a rational policy holder is given by (1)         − = Θ − + − t ds s r t t t x t t Q t E p C C e F t t | ] ' ; 0 max[ ' ) ( ' ' ' ,

where kpx is the probability of an insured person aged x years to survive the next k years.

In life insurance contracts, options can often be exercised at several dates, e.g. each policy anniversary date. Such options are called Bermuda options. A rational policy holder would seek an optimal exercise date. This corresponds to an optimal stopping time5

t

T

∈ '

τ , where Tt denotes the set of all stopping times on Tt. Hence, (1)

becomes (2)         − = Θ − + − ∈ t ds s r τ τ t x t Q t E p ;C C e F t t | ] ' 0 max[ sup ( ) τ τ τ T . 6

An option price determined according to (1) or (2) quantifies the cost to fully hedge the corresponding risk for the expected number of policy holders still alive at the exercise date of the option under the assumption that all policy holders act rational.7

In [Di/Ru 01a] and [Di/Ru 01b], we used this approach to calculate the value of the lump sum option in deferred non-linked annuity contracts and the flexible expiration option in endowment contracts, respectively. The values of these options were surprisingly high. If all policy holders acted rational, insurance companies could not offer these options for free.

Before we introduce the general concept of exercise probabilities, we will look at two special cases for European options: If we know that all policy holders will exercise the option, independent of any circumstances, the value of the option is given by

(

)

        − = Θ +t ds s r t t t x t t Q t E p C C e F t t | ' ' ) ( ' ' ' ,

which can of course become negative. On the other hand, if we know that no policy holder will ever exercise the option, the corresponding value is zero.

In terms of exercise probabilities, the first special case corresponds to a constant exercise probability of one, whereas the second special case means a constant exercise probability of zero. Furthermore, the case of a rational policy holder corresponds to an exercise probability of zero, if the option is out of the money, and one, if the option is in the money.

Such options can also be included into this model, cf. [Di 02]. Since this complicates the notation and is not needed in the examples and calculations below, we refrain from including this feature.

3 We assume a complete, continuous, and frictionless market with a probability space (Ω, Σ, P). Furthermore, F is a completed version of a filtration generated by a Wiener process WQ, where Q denotes the unique equivalent

martingale measure with respect to P. In our specification of the model in Section 2.2, we assume WQ to be a

two-dimensional Wiener process driving a process r for the short rate and S for the asset value. 4 Cf. e.g. [Aa/Pe 94]. For alternatives to this assumption, see footnote 7.

5 A non-negative random variable τ is called a stopping time with respect to a filtration

t F if {τ≤t}∈Ft for all ] ; 0 [ T

t∈ , cf. e.g. Section 8.1 in [Mu/Ru 97].

6 Cf. e.g. Section 8.1 in [Mu/Ru 97] or Section 8.E in [Du 96].

7 The approach used here assumes that the insurer uses the law of large number for mortality risks and sets up a hedge portfolio that hedges for the expected number of insured alive at the exercise date of the option. This means that we assume mortality risk to be diversifiable or the insurer to be risk neutral with respect to mortality (see above). See [Mo 98] for hedging strategies minimizing the insurer’s risk in a model without this assumption. Note however, that at outset (t=0) the strategies coincide.

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In general, the average policy holder will base his decision both on financial and non-financial circumstances. This means that, e.g. for the lump sum option in deferred annuities, many policy holders will take the lump sum when interest rates are high. However, when interest rates are low they will keep the pre-determined annuity and do not exercise the option. Nevertheless, since the decision is also based on other criteria like health8 or comfort9, there will always be some insured who take the lump sum, no matter how low interest rates are, and some who keep the annuity, no matter how high interest rates are.

If all policy holders based their decision only on personal circumstances like retirement planning, health, or marital status, and not on financial information like interest rates and the current unit value, then each person’s exercise probability would be independent of the inner value of the option. Particularly, these probabilities would not vary when market conditions change. The insurer could calculate with a constant average exercise probability.10

In case some policy holders include financial information in their decision, the average exercise probability will depend on financial circumstances. If we assume that all those insured who consider financial information do so in a profitable way, we can conclude that at every possible exercise date the average exercise probability is increasing in the inner value of the option.

We denote this exercise probability by π(Ψt') for any t'∈Tt. Thus, equations (1) and (2) become

(1’)         Ψ Ψ = Θ − + − t ds s r t t t x t t Q t E p e F t t | ) ( ' ) ( ' ' ' π and (2’)         Ψ Ψ = Θ − + − ∈ t ds s r t x t Q t E p e F t t | ) ( sup ) ( τ τ τ τ τ T π . Obviously, for    ≤ Ψ > Ψ = Ψ 0 for 0 0 for 1 ) ( t t t π , for all t,

(1) and (2) coincide with (1’) and (2’). For every other exercise pattern, (1’) and (2’) will lead to a lower value of the option than (1) and (2).

2.2 The Model for the Economy

Since we will consider an option in a unit-linked life insurance contract that depends on both, interest rates and some underlying unit11 (which we will refer to as “asset”), we use a two-dimensional stochastic model for the economy. For the short rater(t), we will use a Hull-White framework. The price of the unit at time t is denoted by S(t) and is modeled by a geometric Brownian motion like in the standard Black-Scholes framework.12 Thus, on the probability space (Ω, Σ, Q) we have

(3) dr(t)

(

(t) ar(t)

)

dt dW1 (t) Q r σ θ − + = and (4) dS(t) r(t)S(t)dt S(t) dW2 (t) Q S σ + = ,

for positive constants a, σ , and r σ , and for the two-dimensional Wiener process S ( 1 , 2 )

Q Q

Q W W

W = under the risk neutral measure Q.

8 If the policy holder expects to die within the next few years because of his bad health, he might prefer a lump sum payment over a lifelong annuity.

9 The policy holder may not want to manage his money but rather receive a lifelong annuity.

10 From an insurance company’s point of view, it is sufficient to know the average exercise probability of all insured, if the number of insured is large enough.

11 E.g. a mutual fund, a stock index, etc.

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If an option depends on only one of these two processes, e.g. an interest rate option or a stock option, there is extensive literature on how to price these options. If the option is non-European, the pricing is often done in a tree structure. This means that a discrete version of the process of the underlying is used and the pricing uses the so-called backward induction method.

Implicit options in life insurance contracts can be non-European options. Additionally, in case of an option that is included in a unit-linked insurance contract, the option depends on both, interest rates and the price of the underlying unit. Thus, for the calculations in Section 4, we use a discrete version of the above model.

There already exist separate tree structures for each of the two processes: A discrete version of the interest rate process (3) is given by the trinomial tree as proposed by Hull and White in [Hu/Wh 94]. For the asset process given by (4), the binomial tree according to Cox, Ross, and Rubinstein is a standard discrete time version (cf. [Co/Ro/Ru 79]). We will give a brief overview over these two models and then combine them to a multivariate tree structure that is a discrete version of our model, cf. (3) and (4).

Binomial tree structure for the asset price

Suppose our observation starts at time t=0and we consider time-steps of length ∆t>0. Furthermore, we let S denote the price of the asset at t=0. In each of the time intervals, the asset price can either move up or down, i.e., it can change to Su or Sd, with u>1 and d<1. The probability under the risk neutral measure for moving up will be denoted by p. Thus, the probability of a down movement is 1–p, cf. Figure 1.

Figure 1: Branching pattern of the binomial tree in one time period t

The transition probability p is determined using the concept of state prices.13 Therefore, the price of the asset at the beginning of the period can be written as a linear combination of the possible asset prices at the end of the period. Applying this to both the risky asset and a risk free asset, we get p=(ert d)(ud)−1 for given u

and d. This is the probability of an up movement in the risk-neutral world. Typically, u is chosen as ) exp( t u= σS ∆ . Furthermore, 1 − = d

u ensures that the binomial tree is recombining (i.e. an up movement followed by a down movement leads to the same asset price as a down movement followed by an up movement). In general, the node ( ki, ) corresponds to the time it with an asset price of k i k

d

Su with k=0,1,…,i. Figure 2 shows the structure of the binomial tree of the asset price.

Figure 2: Binomial tree of the asset price

13 Cf. e.g. Section 5.3 in [Pa 98] for details.

S Su Sd p 1–p S Su Su2 Su3 Su Sd Sd3 S Sd Sd2

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The valuation of an option on this asset is then done by the so-called method of backward induction, beginning at the maturity of the option. The value of the option at maturity T is known. For example, a put option on the asset is worth max

[

XST;0

]

, where ST is the asset price at time T and X is the strike price of the option. In a

risk-neutral world, the value of the option at some time 0<it<Tcan be computed as the expected value (using the risk-neutral probability p) of the option at time (i+ )1∆t discounted for a period ∆t at the risk-free rate r. If it is not a European-style option, it is necessary to check at each node whether early exercise is preferable to holding the option for a further time period ∆t. By working backwards through the tree like this, the value of the option at time zero is eventually obtained.

Cox, Ross, and Rubinstein also showed in [Co/Ro/Ru 79] that a special limiting case of their discrete model coincides with the continuous Black-Scholes model.

Trinomial tree structure for the interest rate

In [Hu/Wh 94], Hull and White propose the use of a trinomial tree for their model of the short rate. Its advantage over a binomial tree is that it offers one more degree of freedom that is helpful in including mean reversion properties.

Figure 3: Branching pattern of the trinomial tree in one time period ∆t

(a) (b) (c)

Usually, the tree branches as in Figure 3(a). If the short rate is small, the tree will branch off as in Figure 3(b). This reflects the mean reversion which implies a stronger upward drift whenever interest rates are low. In case the short rate is rather high, the branching pattern as shown in Figure 3(c) will be used. Therefore, the tree will have a structure as shown in Figure 4.

Figure 4: Trinomial tree of the short rate

Hull and White give a criterion for the step length ∆r of the short rate in the tree (depending on ∆t). They also give a criterion that specifies the point in time from when on the branching patterns given in Figure 3(b) and 3(c) will be used and the tree will stop growing vertically (cf. Figure 4). This not only reflects the mean reversion of the model but also has the advantage that the dimension of the tree is much smaller than that of a binomial tree.

) ( r u p ) ( r m p ) (r d p ) ( r u p ) ( r m p ) (r d p ) ( r u p ) ( r m p ) (r d p

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The transition probabilities in each node ( ji, ) to all its possible succeeding nodes not only depend on the position of the node within the tree, but also on the model parameters θ and σr. According to [Hu/Wh 94], they

are chosen such that both the expected change and the expected variance of the short rate for the next time step

t

∆ in the tree fit the corresponding values under the risk-neutral measure. As a third condition, the sum of all three transition probabilities has to be one. We will not go into the details of constructing and calibrating the tree but rather refer the reader to [Hu/Wh 94] for the formulae and other technical details. Note however, that

) , ( ) ( i j pr

µ for µ=u, µ=m, and µ=d is the probability of an up, a middle, and a down movement of the interest rate in node ( ji, ). This node corresponds to time it when the short rate has made a net of j up movements.14

2.3 The Multivariate Tree Structure

We will now combine the two tree structures described above to one multivariate tree. Thus, every node in the tree corresponds to a specific combination of the short rate and the asset price at some point in time. Therefore, each node is represented by a triple (i,j,k). The initial node is denoted as (0,0,0) with an initial asset price S and a starting interest rate r0 for the time interval [0,∆t], where ∆t is the length of times steps in the tree structure. All succeeding nodes (i,j,k) with i≥1 are characterized by the point in time, the corresponding interest rate, and the asset price:

• i specifies the point in time it

• j determines the interest rate for the interval [(i−1)∆t,it] • k indicates the asset price at time it.

The index j specifies the interest rate according to the trinomial tree described earlier, thus a positive (negative) index j corresponds to a node in the upper (lower) half of the tree, j=0 corresponds to the middle of the interest rate tree. The absolute value | j| specifies the number of up or down movements of the interest rate until time it.15 Since interest for any interval [(i1)t,it] corresponds to the interest rate at time (i− )1t and is earned at time it, the interest rate at node (i,j,k) of the combined tree corresponds to the interest rate at node

) , 1

(ij of the separate trinomial tree.

As in the separate binomial tree for the asset price, k indicates the number of up movements of the asset until time it, i.e. the asset price is Sukdik =Su2ki with k=0,1,...,i.16

Every node in the multivariate tree has six possible succeeding nodes. Because of the mean reversion of the trinomial tree for the short rate, the multivariate tree also has different branching pattern for very large and very small values of the interest rate. In the standard case, the succeeding nodes correspond to a combination of a movement of the asset (up or down) and a movement of the short rate (up, no change, or down), as illustrated in Figure 5(a). Similar to the trinomial tree of the short rate, the branching pattern when interest rates are very low (very high) is shown in Figure 5(b) (Figure 5(c)).

The transition probabilities depend on both the movement of the asset price and the interest rate as suggested by the double index of p, where u, m, and d correspond to an up, middle, and down movement, respectively (a middle movement is only possible for the short rate).

These probabilities are determined to be consistent with the separate trinomial tree of the short rate by Hull and White.

14 For given i, admissible values of j can be calculated as follows: If i*t denotes the point in time where the trinomial tree uses the non-standard branching pattern shown in Figure 3(b) and 3(c) for the first time, then

{

i i i i

}

j∈ −~,−~+1,K,0,K,~−1,~ with

[ ]

* ; min ~ i i i = and i

{

0,1,K

}

. 15 The interest rate tree is recombining.

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Figure 5: Branching pattern in the multivariate tree structure in one time period ∆t

Recalling the time lag of the interest rate explained above, we note that the interest rate for the interval ]

) 1 ( ,

[it i+ ∆t is allocated to the node with time index i+1 in the combined tree. Thus, for determining the transition probabilities from node (i,j,k) to its succeeding nodes, the relevant development of the interest rate is that of the previous ∆t period. At time t =it, this is known. Therefore, we can calculate conditional probabilities and get

) , , ( ) , 1 ( ) , , ( ( |) | ) ( i j p i j k p k j i p r Sr µ ν µ µν = − , with ()( 1, ) j i pr

µ as the transition probability for a µ-movement from node (i−1,j) in the separate trinomial tree of interest rate for µ∈{u,m,d}. Furthermore, ( |)(, , )

| i j k

p Sr

µ

ν is the conditioned probability of a ν -movement of the asset price in the interval [it,(i+1)∆t] with ν∈{ du, }, if the interest rate has made a µ-movement in the interval [(i−1)∆t,it].

These conditional transition probabilities are determined using the concept of state prices as in the separate binomial tree for the asset movement: To exclude arbitrage, there have to exist positive state prices ϕ and 1 ϕ , 2 that describe the price of the asset at time (i− )1∆t as a linear combination of the possible asset prices at time

t

i∆ .17 Since this also holds for the risk free rate, we have:

2 1 1=uϕ +dϕ 2 1 exp( ) ) exp( 1= rtϕ + rtϕ .

Since the conditional transition probability for an up movement can be determined from the state price ϕ by 1

1 ) exp(r t ϕ p= ∆ , we get d u d t j i r k j i pSr u − − ∆ =exp( (, ) ) ) , , ( * ) | ( |µ and d u t j i r u k j i pSr d ∆ − = exp( (, ) ) ) , , ( * ) | ( |µ .

Here, the index j corresponds to the index of the interest rate after a * µ-movement starting from index j, e.g., 1

* = j+

j in case of an up movement of the interest rate according to Figure 5a. The interest r( ji, ) is the value of the interest as in node ( ji, ) of the binomial tree. From these conditional probabilities, the probabilities

) , , ( ) , 1 ( ) , , ( (|) | ) ( i j p i j k p k j i p r Sr µ ν µ

µν = − that are needed to construct the multivariate tree can be calculated.

Then, option prices can be calculated in the multivariate tree with backward induction. The value of the option in node (i,j,k) is determined by discounting the expected value of the option in all six succeeding nodes. For the case of a European option, the value of the option Θ(i,j,k) in node (i,j,k) with it<T is determined as

17 Cf. e.g. Section 5.2 in [Pa 98] for a detailed description of this procedure and the corresponding mathematical background. (a) (b) (c) pdd pdu pud puu pmd pmu pdd pdu pud puu pmd pmu pdd pdu pud puu pmd pmu S S S r r r

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(

)

Θ + − + ∆ = Θ ∆ +∆ ) , , ( * * * *, ) (, , ) exp ( 1, , ) , 1 ( ) , , ( k j i N t i x tp ri j k t k j i p k j i k j i µν .

Here, N(i,j,k) is the set of all succeeding nodes (i+1,j*,k*)N(i,j,k) of node (i,j,k).

Further details about the construction procedure of this multivariate tree, the formulae for the transition probabilities, as well as the application of the option pricing formulae from Section 2.1 in this multivariate tree structure can be found in [Di 02].

3 THE INSURANCE CONTRACT

In this section, we first describe the considered insurance contract, a deferred unit-linked annuity. In Section 3.2, we look at a combined option, consisting of the right to receive a lump sum instead of the annuity (lump sum option) and the right to receive the annuity earlier than scheduled (early retirement option). We give formulae for the price of this combined option assuming a rational policy holder. In Section 3.3, we then derive the corresponding formulae including the concept of exercise probabilities, i.e. no longer assuming a rational policy holder. Here – in order to focus on the effect of the exercise probabilities – we consider the lump sum option only.

3.1 The Unit-Linked Annuity

We consider a single premium deferred unit-linked annuity.18 For the sake of simplicity, we ignore any costs. The insured, aged x, pays a single premium P at t=0. This is invested in some asset, e.g. a mutual fund. We assume the asset price to follow (4). At t=n, the end of the deferment period, the asset value PS(n) S(0) is transferred into a lifelong immediate annuity, paying an annual guaranteed amount A as long as the insured person lives. A is given by

k k n x kp i S n S P n A A − ∞ = + + ⋅ = =

(1 ) ) 0 ( ) ( ) ( 0 ,

where i denotes the interest rate credited to the contract.

In Germany, we currently have a guaranteed interest rate of 3.25% and a non-guaranteed surplus. Whenever the insurance company earns a return on the general assets (based on book values) that is higher than the guaranteed rate, at least 90% of the exceeding part has to be distributed to policy holders as surplus. In many products, a long term average value for the total return (i.e. guaranteed rate plus surplus) is assumed at outset and the annuity is calculated with this total return as i. Whenever the real surplus exceeds the assumed surplus, the annuity is increased and vice versa.

The surplus used to be very stable. Due to adverse market conditions, it has been reduced by many life insurance companies in 2001 and 2002. Nevertheless it is still less volatile than e.g. market rates. This low volatility is generated by accumulating hidden reserves in years where the return on the insurer’s assets (based on market values) exceeds i. These reserves are used to compensate for low returns in other years. This smoothing process is the main reason why many implicit options are quite valuable, since they allow the policy holder to switch in and out of a contract earning these rates, which are independent of market rates.

Note that in this setting, the amount of the annuity is not guaranteed at t=0, since it depends on the stochastic value S(n). However, we assume that the so-called conversion factor f(n)= A(n) S(n) is guaranteed.

3.2 A Combination of a Lump Sum Option and an Early Retirement Option for a Rational Policy Holder Now, we include a combined option into the insurance contract: Instead of taking the annuity at t=n as described in Section 3.1, the insured can optionally choose

a) at t=n to take his asset value PS(n) S(0) as a lump sum instead of the lifelong annuity (lump sum option)

b) or to take an immediate lifelong annual annuity of

18 Note that in spite of the name, this product is unit-linked only during the deferment period. In Germany and many other countries, products that are unit-linked during the annuity phase are rather rare and have no significant market share.

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k k t x kp i S t S P t A A − ∞ = + + ⋅ = =

(1 ) ) 0 ( ) ( ) ( 0

earlier than scheduled, i.e. at any time t∈{nm ,nm+1 ,...,n−1}, for some integer 1≤m<n (early retirement option). Note that for t<n, the conversion factor is f(t)= A(t)/A(n)< f(n).

In this setting, the insured can either once exercise option a), or b), or exercise none, but not exercise both. The set of all exercise dates after time t is given by Tt ={nm, nm+1, ..., n−1, n}∩[t;n].

To quantify the value of this option, we first need the present value of all future annuities at the time the annuity starts (t). This is given by

        =

∞ = − + − t t k ds s r t x t k Q p At e F E t A PV k t | ) ( ) ; ( ) ( .

When exercising the lump sum option at t=n, the insured “sells” his guaranteed annuity to receive the asset value. Thus, Cn =PV( nA; ) and C’ is a contract that pays the lump sum immediately. Hence,

) 0 ( ) ( ' P S n S

Cn= ⋅ and the payoff of option a) at t=n is

(5) ( ; ) ) 0 ( ) ( PV An S n S P n= ⋅ − Ψ .

The payoff of option b) at any time t∈{nm ,nm+1 ,...,n−1} is given by

t

t =PV AtC

Ψ ( ; ) ,

since the original contract C is sold in order to receive the immediate annuity A(t). Assuming some death benefit

Dk, payable at the end of year k during the deferment period if the insured dies within that year, the value Ct of

the original contract C is given by

        + ∫ =

− = − + + + − − + − + t n t k ds s r k k t x t x t k ds s r t x t n Q t E p PV Ane p q De F C k t n t | ) ; ( 1 () ) ( 1 ,

where qxdenotes the probability that an insured aged x dies within the next year. For the case of a rational policy

holder, the value of the combined option at time t is given by (6)         Ψ = Θ + − ∈ t ds s r t x t Q t E p e F t t | ] max[0; sup () τ τ τ τ T .

In Section 4, we will derive numerical results for this value in our model, using market data. 3.3 The Lump Sum Option considering Exercise Probabilities

We now include the concept of exercise probabilities as introduced in Section 2 into our example. This means that we no longer assume the policy holder to act rational. To keep the notation simple, we restrict ourselves to option a), i.e. the lump sum option only. Hence – using the payoff Ψ is as given in (5) and including the n

exercise probabilities from (1’) – the value of this option becomes (7)         Ψ Ψ = Θ +t ds s r n n t x t n Q t E p e F n t | ) ( () π ,

since t= is the only possible exercise date. n 4 RESULTS

4.1 Data

For our following analysis, we use market data from February 21, 2002. In particular, the term structure of interest rates is given by the prices of discount bonds B(0, t) displayed in Table 1.

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Table 1: Discount bond prices from February 21, 2002 (t in years)

t 1 2 3 4 5 6 7 8

B(0, t) 0.9640 0.9210 0.8762 0.8316 0.7877 0.7449 0.7037 0.6643

t 9 10 15 20 25 30 40 50

B(0, t) 0.6272 0.5923 0.4430 0.3315 0.2506 0.1912 0.1128 0.0660

The calibration of the Hull-White model was done using swap prices and forward volatilities from February 21, 2002. The resulting parameters are a=0.051 and σr =0.775%. For the volatility parameter of the asset price, we used the implied volatility VDAX of the German equity index DAX30 from February 21, 2002:

% 22 . 25 = S σ .

Furthermore, we assumed the insured person to be male, used the mortality table DAV94R of the German Society of Actuaries (DAV), and let the death benefits Dk in case of death during the deferment period be the

asset value at time k, i.e.Dk =PS(k) S(0). 4.2 Results for the Option from Section 3.2

We determine the value of the option described in Section 3.2 using the standard concept of backward induction in our multivariate tree.

We fix a single premium of 100,000 €, an exercise period of m=5 years, and a contract interest rate of i=5% to calculate the value of the option for different combinations of age x and deferment period n. The results are shown in Figure 6. The results for i=7% are given in Figure 7.19

Figure 6: Price of the combined option for i=5%

20 30 40 50 60 10 15 20 25 30 35 40 45 50 55 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 x n

As expected, the price of the option is decreasing in the initial age x, because of mortality: The option can only be exercised, if the insured person survives until the first possible exercise date. This probability is smaller for older persons. This effect can be seen in both Figures 6 and 7 and is of course stronger for longer deferment periods n. However, it can be seen that this effect is less pronounced when the guaranteed interest i is high as in Figure 7.

19 Note that i=5% is roughly the current average interest (guaranteed rate + surplus) in the German insurance market. The value i=7% used to be the average for many years before the recent market turbulences.

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The effect of the deferment period n on the option price is more complicated. On the one hand, the expected asset value at the end of the deferment period is higher for increasing n. Therefore, the value of the underlying and the strike price of the option increase accordingly, leading to an increase of the option price. On the other hand, the probability of surviving long enough to exercise the option decreases with increasing n, leading to a decrease of the option price. The first effect tends to be stronger for small values of n, whereas the second effect prevails for larger values of n. Thus, the option price is first increasing and then decreasing in n.

Figure 7: Price of the combined option for i=7%

20 30 40 50 60 10 15 20 25 30 35 40 45 50 55 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 x n

We will now perform sensitivity analysis with respect to the given market data. Again, we fix a single premium of 100,000 €, an exercise period of m=5 years, and a contract interest rate of i=7%. We furthermore let x=30 and

n=35. Then, we calculate the value of the option, assuming a parallel shift in the term structure of interest rates

of ∆r and a shift in the interest rate volatility of ∆ . The results are given in Figure 8. σr

As expected, the price of the combined option is increasing in the interest rate volatility σ . Since this is true for r

both separate options, the effect is also visible for the combined option. The reason is that an increased volatility leads to a greater variance of the value of the underlying of the option and thus to a higher price of the option. The sensitivity with respect to changes in the level of market interest rates is more complicated, because there are contrary effects on the separate options. For the early retirement option, an increase in the level of interest rates causes the option price to decrease. This is because both, the value of the underlying (the annuity as scheduled) and the strike price (the earlier annuity) are decreasing for higher levels of market rates. Since the earlier annuity includes more coupons with longer time to maturities, the reduction of the strike price is stronger, causing the price of the early retirement option to decrease. The price of the lump sum option however is increasing with the level of market interest rates. If rates are higher, then the value of any coupon bond decreases. Consequently, the underlying of the put option becomes cheaper. At the same time, the expected asset value at maturity, i.e. the expected strike price of the put option, increases for higher market rates because the drift of the asset corresponds to the market interest in the risk neutral world. Both effects lead to a higher price of the option.

Therefore, looking at the price of the combined option in Figure 8, we can see both an increase for very low and very high levels of market interest rates, which are caused by the effects of the separate options as just described.

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Figure 8: Price of the combined option for different market scenarios -3% -2% -1% 0% 1% 2% 3% -0.3% -0.1% 0.1% 0.3% 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000 ∆r ∆σr

We now look at the effect of the contract interest rate i in the option price: The early retirement option becomes quite valuable for high levels of i compared to the market interest. On the other hand, if market interest is high in comparison with the interest of the annuity, it is advantageous for the policy holder to exercise the lump sum option.

Figure 9: Comparison of combined option price with separate option prices

0 5,000 10,000 15,000 20,000 25,000 3% 4% 5% 6% 7% 8% 9% interest rate i early retirement option

lump sum option combined option price

sum of separate option prices

These effects can also be seen in Figure 9. Here, we calculate the separate values of the lump sum option and the early retirement option and compare these values with the value of the combined option for a single premium of 100,000 €, an exercise period m=5, x=30, n=35, and for different values of i.

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It can be seen that the price of the combined option is always slightly smaller than the sum of the prices of the separate options. The reason is that the separate options will be exercised in opposite situations depending on the level of market interest r and policy interest i. Thus, there exist only few situations in which a rational policy holder has to choose between the two options both being in the money.

4.3 Results for the Option from Section 3.3

In this Section, we calculate the value of the option described in Section 3.3, for several different patterns of the exercise probability π(Ψn). 1.    ≤ Ψ > Ψ = Ψ 0 for 0 0 for 1 ) ( n n n π 2. π(Ψn)= const 3.       Ψ Ψ > Ψ ≤ Ψ ≤ Ψ Ψ − Ψ Ψ − Ψ − + Ψ < Ψ = Ψ high high n high n low low high low n low high low low n low n p p p p p for for ) ( for ) ( π

The first pattern corresponds to a rational policy holder. The second pattern assumes that all policy holders act completely independent of financial circumstances, whereas the third pattern assumes that there are some policy holders acting rational and some who do not.

Table 2: Value of the lump sum option with exercise probability pattern 2 )

t'

π 0 0.2 0.4 0.6 0.8 1

Option value for %

7 =

i 0 € -1,539 € -3,079 € -4,618 € -6,158 € -7,697 €

Option value for %

5 =

i 0 € 1,632 € 3,264 € 4,895 € 6,527 € 8,159 €

For P=100,000 €, x=30, n=35, and i=7%, the value of the option in the case of rational policy holders is 2,948 €. The corresponding value for i=5% is 10,705 €. If we assume constant exercise probabilities, we get the option values displayed in Table 2.

From (1’), it is obvious, that for pattern 2, the values are linear in π(Ψn). Furthermore, the option values can become negative. This is no surprise, since in this model, the decision to exercise the option is independent of the question whether or not the option is in the money.

We now look at the more realistic exercise pattern 3. We calculate the option value for different values of

low

p and phigh . We furthermore fix Ψlow and Ψhigh as follows: We identify the node in the multivariate tree, for

which the inner value of the option is closest to zero. The short rate in this node is r*. We then let Ψlow be the

inner value at (r*+spread) and Ψhighthe inner value at (r*–spread). Varying these parameters, we get the results

shown in Table 3.

The value of the option is always decreasing in plow . This is as expected, since plow describes the percentage of

people who exercise the option, no matter how far out of the money it may be. The more people exercise the option when it is out of the money, the cheaper the option becomes. Since (1–phigh ) is the proportion of people

who never exercise the option, no matter how profitable exercising might be, it is obvious, that the value of the option is increasing in phigh .

The sensitivity with respect to the spread is rather small. This is important for practical use, since it should be rather simple to estimate the number of people who never exercise the option, as well as the number of people who always do. However, it is rather hard to estimate the exercise pattern of the remaining people. Our data

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indicate, that a good estimate of plow andphigh is sufficient to determine a good approximation for the option

value and that a rather rough estimate of the exact pattern might suffice.

Table 3: Value of the lump sum option with exercise probability pattern 3

low p 20% 20% 30 % 30 % 20% 20% 30 % 30 % high p 80% 80% 80% 80% 70% 70% 70% 70% spread 1% 3% 1% 3% 1% 3% 1% 3% i = 5% Option value 8,038 € 7,995 € 7,786 € 7,750 € 6,970 € 6,934 € 6,719 € 6,690 € low p 20% 20% 30 % 30 % 20% 20% 30 % 30 % high

p

80% 80% 80% 80% 70% 70% 70% 70% spread 1% 3% 1% 3% 1% 3% 1% 3% i = 7% Option value 221 € 178 € -841 € -878 € -71 € -107 € -1,135 € -1,164 € The option value for our exercise patterns 2 and 3 are significantly lower than for a rational policy holder. If we assume in pattern 3 for i=5% that 20% of the policy holders always exercise the option and 20% never exercise it, then the option value is reduced by about 25%. This means that an option that is sold to investors with that exercise pattern can be hedged 25% cheaper than a pure financial option. In the case i=7%, the option value is reduced by more than 90%.

On the other hand, a wrong estimation for plow and phigh bears significant risks. If an insurance company

estimates rather high values for plow and low values for phigh from historic data and then the exercise pattern

changes during the life of the policy, this might lead to a significant loss.

5 SUMMARY AND OUTLOOK

In the present paper, we have developed a general model for the pricing of implicit options in life insurance contracts. Previous work assumed a rational policy holder and hence applied standard principles of option pricing. Our paper is based on this approach but includes the concept of exercise probabilities. We implemented a special case of the model with stochastic processes for the short rate and some asset, thus creating a framework in which implicit options in unit-linked contracts can be considered.

We then derived prices for several options that are offered in the market for free and found that they are of significant value. Our results indicate that the price of such options depends heavily on the structure of the exercise probabilities and that ignoring these probabilities by assuming rational policy holders dramatically overestimates the price.

In a next step, it would be very interesting to analyze real data from life insurance companies in order to see how exercise frequencies correlate with market data. This could help identifying more realistic patterns for the exercise probabilities.

Previous papers concluded with the warning that implicit options can create a substantial risk for a life insurer, in particular when they are given away for free. Considering the results from the present paper, this message might be adapted as follows: Giving away implicit options for free is indeed a high risk, but the risk is not as high as previous work indicated. Assessing the “real” risk will require further investigation of exercise probabilities. REFERENCES

[Aa/Pe 94] Aase, Knut K. and Persson, Svein-Arne 1994: Pricing of Unit-Linked Insurance Policies.

Scandinavian Actuarial Journal, 1, 1994: 26–52.

[Al/Ge 94] Albizzati, Marie-Odlie and Geman, Hélyette 1994: Interest Rate Management and Valuation of the Surrender Option in Life Insurance Policies. Journal of Risk and Insurance, 61, 4: 616–637.

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[Ba/Ha 02] Ballotta, Laura and Haberman, Steven 2002: Valuation of Guaranteed Annuity Conversion Options.

Actuarial Research Paper No.141, Department of Actuarial Science & Statistics, City University London.

[Co/Ro/Ru 79] Cox, John C., Ross, Stephen A., and Rubinstein, Mark 1979: Option Pricing: A Simplified Approach. Journal of Financial Economics, 7: 229–263 .

[Di/Ru 01a] Dillmann, Tobias S. and Ruß, Jochen 2001: Implicit options in life insurance contracts: Part 1 - The case of lump sum options in deferred annuity contracts. Blätter der DGVM, XXV, 2: 211–223.

[Di/Ru 01b] Dillmann, Tobias S. and Ruß, Jochen 2001: Implicit Options in Life Insurance Contracts: Part 2 - The case of flexible expiration options in endowment contracts. Blätter der DGVM, XXV, 2: 225–235. [Di 02] Dillmann, Tobias S. 2002: Modelle zur Bewertung von Optionen in Lebensversicherungsverträgen.

IFA-Verlag, Ulm.

[Du 96] Duffie, Darrell 1996: Dynamic Asset Pricing Theory. Princeton University Press, Princeton, New Jersey. [Ge 97] Gerdes, Wolfgang 1997: Bewertung von Finanzoptionen in Lebensversicherungsprodukten. Der Aktuar,

3, 3: 117–124.

[Gr/Jo 97] Grosen, Anders and Jørgensen, Peter Løchte 1997: Valuation of Early Exercisable Interest Rate Guarantees. Journal of Risk and Insurance, 94, 3: 481–503.

[Gr/Jo 00] Grosen, Anders und Jørgensen, Peter Løchte 2000: Fair Valuation of Life Insurance Liabilities: The Impact of Interest Rate Guarantees, Surrender Options, and Bonus Policies. Insurance: Mathematics and

Economics, 26: 37–57.

[He/Kr 99] Herr, Hans-Otto and Kreer, Markus 1999: Zur Bewertung von Optionen und Garantien bei Lebensversicherungen. Blätter der DGVM, XXIV, 2: 179–193

[Hu 97] Hull, John C. 1997: Options, Futures, and other Derivatives. Third Edition, Prentice Hall International. [Hu/Wh 94] Hull, John C. and White, Alan 1994: Numerical Procedures for Implementing Term Structure

Models I: Single Factor Models. Journal of Derivatives, 2, 1: 7–16.

[Mo 98] Møller, Thomas 1998: Risk-minimizing hedging strategies for unit-linked life insurance contracts.

ASTIN Bulletin, 28, 1: 17–47.

[Mu/Ru 97] Musiela, Marek and Rutkowski, Marek 1997: Martingale Methods in Financial Modelling. Springer, New York.

[Pa 98] Panjier, Harry H. (editor) 1998: Financial Economics with Applications to Investments, Insurance and

References

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