The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees

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1 The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow and Mark Willder Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, U.K. Abstract In this paper we consider how an insurer should invest in order to hedge the maturity guarantees inherent in participating policies. Many papers have considered the case where the guarantee is increased each year according to the performance of an exogenously given reference portfolio subject to some guaranteed rate. However, in this paper we will consider the more realistic case whereby the reference portfolio is replaced by the insurer s own investments which are controlled completely at the discretion of the insurer s management. Hence in our case any change in the insurer s investment strategy leads to a change in the underlying value process of the participating contract. We use a binomial tree model to show how this risk can be hedged, and hence calculate the fair value of the contract at outset. Key words: Participating Policy; Maturity Guarantees; Management Discretion; Hedging; Fair Values; Binomial Tree Model JEL classification: G13, G22 Subj. class: IM30; IE50; IB10 1 Introduction Many insurance companies sell participating policies in which the policyholders share in the profits of the insurer. The amount paid out from these policies is subject to some form of minimum guarantee on at least one of maturity, death, or surrender. These minimum guarantees are increased by the addition Corresponding author. address: m.willder@hw.ac.uk (Mark Willder). Preprint submitted to Insurance: Mathematics and Economics 19 May 2006

2 of bonuses throughout the term of the contract. In this paper we will focus on maturity guarantees and will assume that all policies reach maturity. The majority of the literature has considered the case where bonuses are set according to the performance of some exogenously given reference portfolio, see the references in Section 2. The insurer is unable to change the reference portfolio or change the way it is invested. We will call this the traditional case. The insurer is assumed to announce how the bonus distribution mechanism works in advance. The mechanism will describe how the policyholder will participate in the performance of the reference fund and will normally include a guaranteed minimum rate of bonus. Authors have then shown how derivative pricing techniques can be used to determine a fair value for such contracts. However, in practice the insurer will base its bonuses on the performance of its own investments. The insurer usually has considerable discretion in the choice of investments, although regulations, solvency, and policyholder expectations, may place limits on their discretion. For example, in the U.K. a falling stock market has resulted in insurers switching from high equity strategies to portfolios with little or even no equity exposure. Any change in the underlying assets will change the value of the guarantees the insurer has offered. Therefore the insurer has the power to manipulate the investments in order to reduce the risk to itself. In the traditional case the insurer can set up a hedge portfolio, separate from the reference portfolio, so that they have completely matched the liabilities. Alternatively the traditional insurer can choose not to match, but it will then run the risk of making a loss if the guarantee bites. Either way the policyholder is entitled to the same benefit based on the performance of the reference fund and the bonuses declared as a function of this performance. The practical case that we consider is very different. The bonuses declared and the final payout are determined by the performance of all the insurer s assets and hence the insurer s decision whether to match or not will directly effect the policyholder. In this paper we will show how the insurer should use its investment discretion to remove the risk from the policy. We will then be able to derive the fair price that the policyholder should pay given that they know that the insurer will alter the investments to remove the risk to the insurer. This is the first paper to consider an insurer which uses its full discretion in setting the investment policy, but also calculates the policholder s bonuses according to the performance of its total assets. Hence we want to focus on the key issue of finding assets to match the guarantees given by these bonuses. We will use a simple model, but one that is rich enough to demonstrate some important results regarding the assets that the insurer should choose. We 2

3 will assume that the available assets follow a binomial model and will ignore mortality and expenses. The remainder of the paper is organised as follows. In Section 2 we review the literature on calculating the fair value of participating policies. Then in Section 3 we describe the participating policy we will consider in this paper. We show how the guaranteed maturity benefit is related to the insurer s actual investment portfolio. In Section 4 we describe a binomial tree model for the performance of bonds and equity shares. Then in Section 5 we use this model to derive the perfect hedging strategy so that the value of the insurer s investments at maturity are exactly equal to the maturity guarantee. In Section 6 we discuss superhedging strategies such that at maturity the insurer has more funds than are required to pay the maturity guarantee. A numerical example is given in Section 7. Finally in Section 8 we give our conclusions. 2 Background The payout from many insurance contracts depends in some way upon the actual investment return earned on the assets subject to some minimum guarantee. We will consider policies that participate in some way in the profits of the insurer. Hence these policies are called participating policies or with-profits policies. The guarantee may take many forms, for example there may be a guaranteed payout on death or early surrender. However, in this paper we will consider only maturity guarantees. At the beginning of the policy a minimum guaranteed amount will be set. This initial guarantee is supplemented by some form of bonus which can be varied to reflect the performance of the underlying investments. Once declared, bonuses become part of the guarantee. One of the key problems to solve for policies with guarantees is the calculation of their fair value, that is the value of the policy which is consistent with the market value of traded assets. Brennan and Schwartz (1976) and Brennan and Schwartz (1979) were the first to apply derivative pricing techniques to the fair value of policies with guarantees, although they did not consider policies with bonuses. Fair values of participating policies with maturity guarantees were then considered by Persson and Aase (1997), Miltersen and Persson (1999), Miltersen and Persson (2003), and Bacinello (2001). The fair value of participating poli- 3

4 cies with surrender guarantees has also been considered by Grosen and Jorgensen (2000), Miltersen and Hansen (2002), Miltersen and Persson (2000), Bacinello (2003), and Tanskanen and Lukkarinen (2003). Bauer et al. (2005) have extended this work by considering how accounting rules effect the bonus distribution mechanism. A reduced fair value has been calculated to take account of the possibility of insurer insolvency by Briys and de Varenne (1997), Grosen and Jorgensen (2002), Jorgensen (2001), and Ballotta (2005). In addition Guillén et al. (2006) have considered participating policies with a maturity payout determined by a return smoothing mechanism. The papers described above all contain the assumption that the policyholder s assets are held in a reference portfolio of shares (or, in the case of Persson and Aase (1997), cash) and that the insurer cannot influence the performance of this portfolio. The insurer instead must hedge the risk using a separate account. However as Brennan and Schwartz (1979) pointed out, in practice the underlying asset is the insurer s own investment portfolio and the insurer s management have considerable discretion in setting the investment policy, and hence can reduce the value of the policyholder s guarantee by reducing the riskiness of the portfolio. Wilkie (1987) and Willder (2004) show for any time prior to maturity how to match the maturity guarantee including all bonuses declared to that date. In these papers the insurer is invested in a mixture of shares and put options or the equivalent hedge portfolio. However, in these papers, in contrast to our own, the hedging strategy does not ensure that bonuses can continue to be declared in the future. Hibbert and Turnbull (2003) describe how to calculate a fair value of a withprofits policy where the management have limited discretion in choosing the assets. They assume that the management follows a fixed rule such that they will reduce the equity exposure of the reference portfolio if the value of the guarantees rises too close towards the value of the assets. However, in contrast to our paper, they do not find the optimum investment strategy for the insurer, and they do not base their bonus strategy on the performance of the insurer s assets. We will assume that the insurer s management uses its discretion to invest so as to minimise the risk to the insurer. Brennan and Schwartz (1979) stated that for a policy with no bonuses the guarantees could be made worthless by investing in the risk-free asset from outset. In this paper we show that no such static investment strategy exists for policies where a series of bonuses are declared. However, we will show how the insurer can hedge the risks within their investment portfolio. We then calculate the fair value that the policyholder 4

5 should pay for this contract given the insurer s investment discretion. 3 Insurance Contract We will consider a single premium participating insurance policy with a maturity guarantee. This guarantee will be increased each year depending on the performance of the insurer s assets. We wish to focus on the value of this maturity guarantee and so we assume that all policies reach maturity for simplicity. We also ignore expenses. The insurance company sets up an account for the policyholder with initial account balance X(0) = 1. The policyholder is guaranteed to receive the account balance at maturity. We note that, as discussed for example by Guillén et al. (2006), the account balance is not the same as the fair value of the contract before maturity. One of the aims of this paper is to find a fair value for this contract given the mechanism for crediting investment returns to the account balance described below. The account increases at an annual return of r X (t) in year t. In a similar way to Ballotta (2005), this annual return is given as the maximum of a guaranteed rate g and the return of the insurance company s portfolio multiplied by a participation rate δ. Let V (t) denote the value of the portfolio of the insurance company at time t (this differs from Ballotta (2005) where V (t) was a given external reference portfolio). X(t) is then given by X(t) = X(t 1)e r X(t) with r X (t) = max {g, δr V (t)} (1) where r V (t) is the return on the portfolio of the insurance company during the period (t 1, t], that is r V (t) = log V (t) V (t 1), the guaranteed rate g is a real number and δ (0, 1) is the participation rate. g and δ are assumed to be given at time 0 and are independent of t. In practice the guaranteed rate g used by insurers has been non-negative so that the policyholder s account cannot fall, but in theory g can take any value. A negative value of g ensures that the policyholder s loss in any given year is limited. After rearranging (1) we obtain ( ) δ V (t + 1) X(t + 1) = X(t) max eg, V (t). (2) 5

6 We assume that the insurance company has the right to change the portfolio V at any time t. By allowing the insurer to exercise its discretion over the choice of assets we are considering the situation of insurers in practice. This contrasts with the exogenously given reference portfolio assumed in the papers described in Section 2. The insurance company wants to hedge this contract by setting up a portfolio with final value V (T ) = X(T ). The problem arising here is that any attempt to hedge the contract would result in a change of the portfolio V and therefore the underlying price process of the contract X would change. This again is in contrast with the papers described in Section 2 which set up a separate hedging portfolio with the price process of the contract X still based on the reference portfolio. If the interest rate r is assumed to be constant, the insurance company can invest V (0) into a zero-coupon bond with maturity T, as suggested by Brennan and Schwartz (1979), and would receive V (0)e T r at time T. In this case the portfolio of the insurance company would consist of a number of bonds with maturity T and the account of the policyholder at time T would be X(T ) = X(0) exp{t max[g, δr]}. This payoff at time T can be achieved by investing V (0) = V (T )e T r = X(T )e T r = X(0) exp{t max[g r, δr r]} into the T -period zero-coupon bond. Hence, we have found a static hedging strategy for the contract using the T -year zero-coupon bond. In practice the interest rate r is not constant over the lifetime of an insurance contract. However, it is not possible to match the liabilities of the contract using only the T -period zero-coupon bond if r is not constant. The T -year zerocoupon bond still has a guaranteed return when we consider the complete term of the policy. However, we must declare bonuses each year according to (2), but the return on the zero-coupon bond in any given year is not guaranteed. Bond yields can fall in some years pushing up the value of the insurer s portfolio and hence increasing the bonuses, while in other years bond yields could rise to such an extent that the value of the insurer s portfolio increases by less than g and hence the bonuses increase at the guaranteed rate g. Hence, we cannot know at time zero what the final value of the policyholder s account will be, and so we cannot match with only the T -year zero-coupon bond. This is in contrast with Brennan and Schwartz (1979) who could match with the T -year zero-coupon bond even if r was stochastic because no bonuses were declared on their policies. For the remainder of this paper we will consider the more interesting case 6

7 where the interest rate r is not constant. In Section 4 we will describe a stochastic financial market model. In Section 5 we then show how the insurer should invest given this stochastic model. 4 Financial Market Model We assume that trading takes place at discrete times t = 0, 1,..., T N and that the return r X (t) of the policyholder s account X is calculated at times t = 1,..., T. The insurance company can invest into a zero-coupon bond with maturity t + 1, and a zero-coupon bond with maturity t + 2. As trading takes place only at discrete times the zero-coupon bond with maturity t + 1 is risk-free. However, the zero-coupon bond with maturity t + 2 is a risky asset as its price will not be known prior to time t + 1. The insurance company can also invest into another risky asset. This risky asset represents any kind of financial instrument that has non-interest rate risk. For simplicity we will refer to this asset as equities, but it could in fact be any asset with non-interest rate risk including portfolios or options. At the beginning of the period (t, t + 1] the insurer invests a fraction a(t) of their assets into the bond with maturity t+2, and a fraction b(t) into equities, so that the remaining fraction 1 a(t) b(t) is invested into the bond with maturity t + 1. We will consider a binomial tree model for both the interest rate and the price of shares. Let (Ω, F, P) be a probability space and F = (F t ) t=0,...,t be a filtration on (Ω, F, P). The interest rate during the period (t 1, t] is denoted by r(t). We assume that the rate of interest r follows a binomial tree model. That is r(1) is a constant, or F 0 -measurable, and r(t + 1) = Z(t)r(t), t = 1,..., T 1 with independent and identically distributed random variables Z(t) with u Z(t) = d P[Z(t) = u] = p P[Z(t) = d] = 1 p, t = 1,..., T 1 (3) with p (0, 1) and 0 < d < u. We assume that r(t + 1) is known at the beginning of the period (t, t + 1], i.e. Z(t) and r(t + 1) are F t -measurable which means that r(t) is a predictable stochastic process. 7

8 To set up the portfolio V and to hedge the liability from the contract specified in (2) the insurance company can invest into several assets. As mentioned above, at any time t T 2 there exist zero-coupon bonds with maturities t + 1 and t + 2. P (s, t) denotes the price at time s of a zero-coupon bond with maturity t and we assume that P (s, t) is F s -measurable and P (t, t) = 1 for all t. To avoid arbitrage opportunities the following relationships must hold P (t 1, t) = e r(t) and e ur(t) < e r(t) P (t 1, t + 1) < e dr(t) (4) since P (t, t + 1) = e ur(t) if Z(t) = u and P (t, t + 1) = e dr(t) if Z(t) = d. The no-arbitrage condition (4) implies that there exists a measure Q on (Ω, F) such that P (t 1, t + 1) = E Q [ e r(t) r(t+1) F t 1 ] = e r(t) [ q(r(t))e ur(t) + [1 q(r(t))]e dr(t)] (5) with Q[Z(t) = u] = q(r(t)) and Q[Z(t) = d] = 1 q(r(t)). We assume from now on that the measure Q is fixed and the price of the two-year bond at time t is therefore a deterministic function of r(t + 1). A particular example for Q is provided in Section 7. In addition to the two zero-coupon bonds there exists another risky asset which we have called equities and its value is also modelled by a binomial tree. Let S(t) denote the value of an equity index at time t with all income reinvested. The value at time t + 1 is then given by S(t + 1) = e r(t+1) Y (t + 1)S(t) t = 0,..., T 1, (6) S(0) (0, ) is a constant and Y (t) is a sequence of independent, identically distributed and F t -measurable random variables on (Ω, F, P). The distribution of Y (t) is given by u S with P[Y (t) = u S ] = p S Y (t) = (7) d S with P[Y (t) = d S ] = 1 p S where 0 < p S < 1 and d S < 1 < u S for all t = 1,... T. This choice of d S and u S together with (6) guarantees that the proposed model is arbitrage-free in every period. Note that r is predictable and therefore there are only two possible states for S(t + 1) given F t. 8

9 5 Perfect Hedging We first consider the case where the issuer of the contract, the insurance company, wants to hedge the liabilities perfectly. This means they want to invest in a way such that at maturity the value of their portfolio is equal to the liabilities, i.e. V (T ) = X(T ). Let us first consider the last period (T 1, T ]. Since r(t ) is known at time T 1 we obtain with the same argument as in the case of a constant interest rate that the value of the portfolio at time T 1 can be chosen to equal V (T 1) = X(T 1) exp{max[g r(t ), δr(t ) r(t )]} (8) and this amount is invested into a one-period zero-coupon bond or the bank account to achieve V (T ) = X(T ) at maturity. In the following we are interested in the ratio Π(t, r(t + 1)) = X(t)/V (t). We do not know the value of X(t) or V (t) but we can calculate their ratio. We then work backwards calculating the ratio at times T 1, T 2,..., 1, 0. However we know that X(0) = 1 so that we can then work forwards to calculate the value of X(t) and V (t) at each time. Π(t, r(t+1)) = X(t)/V (t) depends on the interest rate r(t+1) that is payable during the period (t, t + 1]. We therefore rewrite equation (8) as Π(T 1, r(t )) = X(T 1) V (T 1) = exp{ max[g r(t ), δr(t ) r(t )]}. (9) To see that this is the only way to achieve V (T ) = X(T ) given F T 1 we now assume that the insurance company invests into another asset, which might be a bond with a time to maturity larger than one period, or the risky asset S. Since these opportunities are equivalent at time T 1 we assume that the insurance company wants to invest into S. We then obtain V (T ) from [ ] V (T ) = V (T 1) (1 b(t 1))e r(t ) S(T ) + b(t 1) S(T 1) (10) where b(t 1) is the fraction of V (T 1) that is invested into the risky asset during (T 1, T ]. The policyholder s account at maturity X(T ) is then [ ] δ X(T ) = X(T 1) max eg, (1 b(t 1))e r(t ) S(T ) + b(t 1) S(T 1). (11) 9

10 With the notation f T (y, b) = (1 b)e r(t ) + by max { e g, [(1 b)e r(t ) + by] δ} (12) for y {d S e r(t ), u S e r(t ) } and b R and since V (T ) = X(T ) we have Π(T 1, r(t )) = ( ) X(T 1) S(T ) V (T 1) = f T, b(t 1) S(T 1) which must hold for both possible outcomes of S(T )/S(T 1). Therefore, the following equality has to be fulfilled by b(t 1) f T (e r(t ) u S, b(t 1)) = f T (e r(t ) d S, b(t 1)). (13) Lemma 1 b(t 1) = 0 is the only solution for (13). Proof: (1 b)e r(t ) + bx is strictly increasing in b for x = e r(t ) u S since u S > 1 and therefore f T (e r(t ) u S, b) is also strictly increasing in b as δ < 1. Similarly we can show that f T (e r(t ) d S, b) is strictly decreasing in b. Therefore there exists only one solution of (13). As the interest rate over the last period is a known constant we know from Section 3 that b(t 1) = 0 is a solution. Hence b(t 1) = 0 is the only solution. Hence we have shown that the only possible hedge portfolio at time T 1 requires the insurer to invest entirely into the 1-year bond. At any time t T 2 we consider a self-financing portfolio V (t) consisting of the two zero-coupon bonds with maturity t + 1 and t + 2. Recall that a(t) is the fraction of money that is invested into the two-period bond. 1 a(t) is then the fraction of money that is invested into the one-period bond at time t, as at this stage we do not want to invest into the risky asset S. We will consider the possibility of investing in equities later. The value V (t+1) of this portfolio at time t + 1 given the value V (t) at time t is therefore V (t)(1 a(t)) V (t + 1) = P (t + 1, t + 1) + V (t)a(t) P (t + 1, t + 2) P (t, t + 1) P (t, t + 2) [ ] 1 a(t) = V (t) P (t, t + 1) + a(t) P (t + 1, t + 2). (14) P (t, t + 2) The policyholder s account at time t + 1, X(t + 1), is given by (2). We already know the value X(T )/V (T ) which is one by definition. We also know the ratio Π(T 1, r(t )), given in (9), at time T 1 from the arguments above. We are now interested in the ratio Π(T 2, r(t 1)) at time T 2. From (9) we obtain 10

11 1 V (T 1) = X(T 1). (15) Π(T 1, r(t )) Inserting (2) and (14) for t = T 2 into (15) we obtain V (T 2) = [ 1 a(t 2) P (T 2, T 1) 1 Π(T 1, r(t )) ] a(t 2) + P (T 1, T ) P (T 2, T ) ( ) δ V (T 1) X(T 2) max eg, V (T 2). (16) Given F T 2 the value of r(t 1) is known and there are only two possible scenarios for r(t ) which are dr(t 1) and ur(t 1). Since Π(T 1, r(t )), P (T 1, T ), and V (T 1)/V (T 2) are deterministic functions of r(t ), (16) must be fulfilled for both outcomes of r(t ). With Π(t, r(t + 1)) = X(t)/V (t) and the notation f(t, z, a) = Π(t + 1, zr(t + 1)) [ (1 a)e r(t+1) + { max e g, [ (1 a)e r(t+1) + a P (t,t+2) e zr(t+1)] ] } δ (17) a P (t,t+2) e zr(t+1) for t = 0,..., T 1, z {d, u} and a R we choose a(t 2) such that the equation f(t 2, d, a) = f(t 2, u, a) (18) is fulfilled for a = a(t 2). If this solution exists we obtain from (16) X(T 2) Π(T 2, r(t 1)) = V (T 2) = f(t 2, d, a(t 2)) = f(t 2, u, a(t 2)). Note that Π(T 2, r(t 1)) depends on P (T 2, T ) but since, by equation (5), P (T 2, T ) can be expressed as a function of r(t 1) this dependency is already incorporated. With a backward induction argument we obtain for any time t = 0,..., T 2 that a(t) is the solution of and f(t, d, a) = f(t, u, a) (19) Π(t, r(t + 1)) = X(t) V (t) = f(t, d, a(t)) = f(t, u, a(t)). (20) 11

12 It remains to show that a unique solution of (19) exists. Lemma 2 Let d and u be positive numbers with d < u and let t be any positive integer with t T 2. Assume that the no-arbitrage condition (4) holds. Then the solution of (19) exists and is unique. Proof: Fix t {0,..., T 2} define the return on V during (t, t + 1] by R V (t + 1, x, a) = (1 a)e r(t+1) a + P (t, t + 2) e xr(t+1) ( ) e xr(t+1) = a P (t, t + 2) er(t+1) + e r(t+1). (21) From (4) it follows for x = d that R V (t, d, a) is a linear increasing continuous function of a. For we have For we obtain a > ( e r(t+1) e g) ( e r(t+1) ) 1 e dr(t+1) P (t, t + 2) f(t, d, a) = Π(t + 1, dr(t + 1))R V (t, d, a) 1 δ. a ( e r(t+1) e g) ( e r(t+1) ) 1 e dr(t+1) P (t, t + 2) f(t, d, a) = Π(t + 1, dr(t + 1))e g R V (t, d, a). Therefore f(t, d, a) is a strictly increasing continuous function of a that takes values in (, ). With a similar argument we obtain for x = u that f(t, u, a) is a strictly decreasing continuous function of a that takes values in (, ). Therefore, there exists a unique a R with f(t, d, a) = f(t, u, a) which proves the assertion. We remark that the solution of (19) does not need to be in (0, 1). Negative values for a(t) or 1 a(t) are possible and correspond to short positions in the two-period zero-coupon bond or the one-period zero-coupon bond respectively. From Lemma 1 and Lemma 2 we know that there is a single investment strategy that solves (13) and (19) for t = 0,..., T 1 using only the one-year and two-year zero-coupon bonds. We will denote this strategy by a 0 (t). The superscript is chosen to indicate that the insurance company does not invest into the risky asset S. From (20) we have the corresponding processes Π 0 (t). 12

13 Furthermore from (20) we can construct a pair of processes (V 0 (t), X 0 (t)) such that X 0 (0) = 1 and V 0 (T ) = X 0 (T ). So we have shown that a pair of processes (V 0 (t), X 0 (t)) have the required properties, but we have not shown that this pair is the only possible combination of processes with these properties. The insurance company might want to invest something into equities at some point in time. We have already shown in Lemma 1 that given F T 1 there is only one way to achieve V (T ) = X(T ), which is to invest everything into a one-period zero-coupon bond. Let b(t) denote the fraction of money that is invested into S(t) at time t. This means that the amount invested into the one-period bond will now be 1 a(t) b(t). We consider again a self-financing portfolio. With the notation R V (t, y, z, a, b) = [1 a b]e r(t+1) + bye r(t+1) + a e zr(t+1) P (t, t + 2) (22) we obtain for the value of the portfolio V (t + 1) at time t + 1 V (t + 1) V (t) [1 a(t) b(t)] S(t + 1) P (t + 1, t + 2) = + b(t) + a(t) P (t, t + 1) S(t) P (t, t + 2) = R V (t, Y (t + 1), Z(t + 1), a(t), b(t)) (23) where Y and Z are defined in (7) and (3) respectively. We now define the function h h(t, y, z, a, b) = Π0 (t + 1, zr(t + 1))R V (t, y, z, a, b) max {e g, R V (t, y, z, a, b) δ }. (24) Hence, with similar arguments as above, for any time t = 0,..., T 2 we obtain with a backward induction argument that a(t) and b(t) must solve the equations h(t, d S, d, a(t), b(t)) = h(t, d S, u, a(t), b(t)) (25) = h(t, u S, d, a(t), b(t)) (26) = h(t, u S, u, a(t), b(t)) (27) in order to achieve V (T ) = X(T ). Lemma 3 The only solution of the equations (25) to (27) is (a(t), b(t)) = (a 0 (t), 0) for t = 1,..., T. Proof: Similarly to the proof of Lemma 2 we find that R V (t, d S, d, a, b) is a strictly increasing continuous function in a and a strictly decreasing continuous 13

14 function in b. The same holds for h(t, d S, d, a, b). Similar results hold for the other combinations of y and z as summarised below. h(t, d S, d, a(t), b(t)) increasing in a decreasing in b h(t, d S, u, a(t), b(t)) decreasing in a decreasing in b h(t, u S, d, a(t), b(t)) increasing in a increasing in b h(t, u S, u, a(t), b(t)) decreasing in a increasing in b Therefore, there exists at most one solution to equations (25) to (27). Since we know already that (a(t), b(t)) = (a 0 (t), 0) is a solution, it must be the only one. We summarise our results in the following theorem. Theorem 1 Assume the model given in Section 4, and assume that V is given by (23) and X is given by (2). Then V (T ) = X(T ) if and only if (1) b(t) = 0 for all t = 0,..., T 1, (2) a(t 1) = a 0 (T 1) = 0 and (3) a(t) = a 0 (t) is the solution of (19) for all t = 0,..., T 2. In this case we obtain V (t) = 1 Π 0 (t, r(t + 1)) X(t) where Π 0 is given by (20) for a(t) = a 0 (t). Hence we have shown how to construct the hedging strategy that will perfectly match the guarantee at maturity. Unlike previous papers it is the performance of this hedge portfolio which determines the bonuses. We have shown that the only risky asset in the hedge portfolio is the two-year bond. The necessary value of the insurer s assets at time T 1, V (T 1), is a function of r(t ). Therefore at time T 2, V (T 1) can be seen as the value of an interest rate derivative that can be hedged by investing into bonds. Since the particular interest rate model considered here is a one-factor model, we can choose any asset whose price process is adapted to the filtration generated by Z(t). We have chosen the two-year bond. The only risk that the insurer has is coming from unknown future interest rates, and therefore the insurer would not invest into any other financial risks. We note that the financial model we consider is a quadronomial model since at the end of each period there are four possible states determined by movements 14

15 in interest rates and equity prices. The model is incomplete as there are only three securities; the two bonds and equities. Hence, we cannot hedge every derivative whose price depends on the performance of these three assets, see for example Section 6.5 of Dothan (1990). However, by excluding equities from the insurer s investments we return to a binomial model of the interest rate risk, which is complete given the two bonds. An alternative complete model is achieved by assuming a constant interest rate and a binomial model for equities, and as we have seen in Section 3 this leads to a trivial hedge in bonds. Although our results are in keeping with the discussion of completeness in books such as Dothan (1990), our results do not follow directly. Dothan (1990) considers the traditional case where the hedge portfolio is separate from the reference portfolio. The central idea of our paper is that we require that the hedge portfolio and the reference portfolio be the same. An important consequence is that in the traditional case the insurer can hedge a payoff based on the price of equities using only equities and a bond earning a constant risk-free rate in the hedge portfolio, but the hedge portfolio is different to the reference portfolio and so is not a solution to the problem in this paper even though the market is complete. Also, we have seen in Theorem 1 that the insurer can perfectly hedge its liabilities at any time t by investing in the risk-free asset and the two-year bond. This is a consequence of the complete interest-rate market. If we change the model for the bond price processes such that the interest rate market becomes incomplete a perfect hedge might not be possible. The construction of the hedge portfolio allows us to find the fair value of the contract V (0) given the initial value of the policyholder s account X(0). We have shown that there is a unique hedge portfolio given by a 0 (t) which ensures that there is no risk to the insurer. The insurer will not be prepared to sell the contract for less than V (0) because they will be unable to set up the riskless hedge, and so will be left with risk for which they are not compensated. If the policyholder pays more than V (0) the insurer has an arbitrage opportunity, because the insurer can invest according to the riskless strategy a 0 (t) so that they are guaranteed that V (T ) > X(T ). The fair value may be greater or less than the initial value of the policyholder s account depending on the generosity of g and δ in the bonus distribution mechanism given by equation (2). We can consider the amount V (0) X(0) as the initial charge that the insurer must make for giving the guarantees. 15

16 6 Superhedging In the last section the policyholder paid a premium V (0) and in return the insurer guaranteed to pay exactly the value of the guarantee X(T ) at maturity. However, the hedge strategy required investment in one-year and two-year bonds, but did not include equities. From now on we will denote the premium and final maturity payout by V 0 (0) and X 0 (T ) respectively whenever the insurer follows the strategy a 0 (t) throughout the policy as given by Theorem 1. Similarly we will define Π 0 (t, r(t+1)) as the ratio required at time t to ensure that the insurer can follow the riskless strategy a 0 (t) from time t onwards to obtain X(T ) = V (T ). We now consider the case where the policyholder wants some exposure to equities, but still wants to enjoy bonuses as given by equation (2) on the policyholder s initial account X(0) = 1. The insurer will only accept this contract if the policyholder is willing to pay more than V 0 (0). However, there is a danger here that the insurer will use its discretion to follow the investment strategy a 0 (t) so that the policyholder will loose the value of the extra premium V (0) V 0 (0) and the insurer will make arbitrage profits. So we will now assume that the insurer returns the full value of the assets V (T ) at maturity by declaring a terminal bonus equal to V (T ) X(T ). Given this modified contract we want to find the range of investment options open to the insurer such that V (T ) X(T ) for all possible scenarios. That is, we want to show how the insurer should use its discretion over the choice of investments in order to eliminate risk to the insurer. Consider the insurer at some integer time t + 1 < T during the lifetime of the contract. From Section 5 the insurer is assured that V (T ) will be at least as large as X(T ) if it follows the riskless strategy a(τ) = a 0 (τ), b(τ) = 0 for each τ t + 1 and the current ratio of the policyholder s account to the assets X(t + 1)/V (t + 1) is less than Π 0 (t + 1, r(t + 2)). Any ratio greater than Π 0 (t + 1, r(t + 2)) excludes the possibility of a riskless strategy for the remaining lifetime of the policy. Therefore at time t the insurer should invest so that in all possibles cases it has ratio X(t+1)/V (t+1) of at most Π 0 (t+1, r(t+2)) at time t+1. Similarly at time T 1 the insurer should invest so that X(T ) V (T ) in all possible cases. From equations (2) and (23) we have that 16

17 b(t) f T (y,b(t-1)) a(t) b(t-1) Fig. 1. The left hand side shows a contour plot of min y {ds,u S }, z {d,u} {h(t, y, z, a(t), b(t))}. The right hand side shows f T (y, b) as a function of b(t 1). The dotted line is the graph of f T ( e r(t ) d S, b(t 1) ) and the dashed line is the graph of f T ( e r(t ) u S, b(t 1) ). The minimum of the two functions is shown as the solid line. The parameters for this example are: g = 0.025, δ = 0.9, r(t ) = 0.03, u S = 1.3 and d S = 1/1.3. X(t) V (t) = X(t + 1) V (t + 1) R V (t, y, z, a, b) max {e g, R V (t, y, z, a, b) δ }. (28) We require at time t + 1 that X(t + 1)/V (t + 1) Π 0 (t + 1, zr(t + 1)) for each possible scenario of the interest rate and equity share price at t + 1. Hence, with the definition of h in (24) we have to choose a(t) and b(t) such that X(t) V (t) min {h(t, y, z, a(t), b(t))}. (29) y {d S,u S }, z {d,u} The left hand side of Figure 1 shows a typical contour plot of the function min y,z h(t, y, z, a(t), b(t)) for different values of a(t) and b(t). From the arguments in Section 5 it follows that min y,z h(t, y, z, a(t), b(t)) has a maximum at (a(t), b(t)) = (a 0 (t), 0). Therefore, whenever the insurance company decides to invest something, short or long, into equities they need more money than for the safe strategy a 0 (t). Hence for a given value of the rate X(t)/V (t) we can find a set of values for a(t) and b(t) given by the appropriate contour line X(t)/V (t) = min y,z h(t, y, z, a(t), b(t)) which ensures that the insurer can adopt a riskless strategy at time t + 1 regardless of the performance of the assets over the year (t, t + 1]. Further, any point within the contour represents a strategy where X(t)/V (t) < min y,z h(t, y, z, a(t), b(t)) which ensures that the insurer will have 17

18 more than enough assets to invest in the riskless portfolio and so will allow some form of risky strategy at time t + 1 also. Similarly at time T 1, to ensure V (T ) X(T ) we must choose b(t 1) so that X(T 1)/V (T 1) min y {ds,u S } {f T (y, b(t 1))}, where f T (y, b) is defined by equation (12). Recall that b(t 1) is the fraction of V (T 1) that is invested into risky assets during (T 1, T ]. The right hand side of Figure 1 shows the graph of f T (e r(t ) d S, b(t 1)) and f T (e r(t ) u S, b(t 1)) in a typical situation, with the minimum of the two functions shown as the solid line. So in this section we have shown how the insurer can determine a range of strategies which eliminate the risk to the insurer. The insurer s management have the discretion to choose which of the strategies they prefer. We leave for future research the consideration of which available strategy is optimal for the policyholder. If at time t the insurer invests in a strategy on a particular contour line so that X(t)/V (t) = min y,z h(t, y, z, a(t), b(t)) there is always the possibility that the worst scenario occurs so that the insurer is then forced to invest in the riskless strategy from time t + 1 onwards and therefore the policyholder will receive no final bonus so that V (T ) = X(T ). However, by choosing a strategy within the contour each year the insurer may be able to invest some proportion in equities throughout the contract. At maturity the policyholder will receive the value of V (T ), which must be at least as great as the guarantee X(T ). However, note that it will be very unlikely that either of these values will equal V 0 (T ) or X 0 (T ) obtained in the riskless case. The policyholder hopes that by investing a proportion in equities that they will benefit from high annual returns, and hence high bonuses, so that V (T ) > V 0 (T ) and X(T ) > X 0 (T ). However the policyholder runs the risk that the investments perform poorly such that V (T ) < V 0 (T ) and X(T ) < X 0 (T ). However, in either case the insurer will have chosen its investments so that V (T ) X(T ) and hence the insurer is free of risk. We have shown how to superhedge the policy, but what about its fair value? The terminal bonus at maturity ensures that the policyholder receives the full value of the assets V (T ). The hedging strategy ensures that the insurer will not need to make an additional payment to cover the cost of the guarantee X(T ). Hence the fair value of the contract at time t is simply the current asset value V (t). 18

19 Table 1 Interest rate r(t + 1). 7 Numerical Example In this section we give a brief numerical example to show how the methodology of Section 5 can be used to perfectly hedge the participating policy. We consider a policy with term T of four years. The bonus declaration mechanism is given by equation (2) with guaranteed growth rate g = and participation rate δ = 0.9. We model r(t), the interest rate during the period (t 1, t], with the binomial model given in equation (3). In practice the initial value for the interest rate would be known and we assume in this case that r(1) = The interest rate develops through each step of the binomial model with u = 1.5 and d = 1/u = 2/3. The interest rates derived from this model are given in Table 1. Note that the binomial tree is recombining. In order to derive the price of the two year bond we will assume that the yield curve is flat so that P (t 1, t + 1) = e 2r(t). This assumption satisfies the no arbitrage requirement given by (4). The measure Q appearing in equation (5) is then given by q(r(t)) = e r(t) e dr(t). (30) e ur(t) e dr(t) We now work backwards from time T = 4 using the results of Section 5 to obtain the investment into the one-period zero-coupon bond 1 a 0 (t) given on the left hand side of Table 2, and the value of the ratio Π 0 (t) = X 0 (t)/v 0 (t) given on the right hand side of Table 2. Note again that these two tables are recombining. Also note that on the table on the right hand side of Table 2 19

20 1 a 0 (t) X 0 (t)/v 0 (t) Table 2 The left hand side shows the fraction invested into the one-period zero-coupon bond 1 a 0 (t). The right hand side shows the relative price X 0 (t)/v 0 (t). the tree does not bifurcate in the final year because the insurer invests all its assets in the one-year zero-coupon bond which has known return. We see on the right hand side of Table 2 that the final ratio is 1 as required. The initial ratio of 0.99 means that the fair value is a little above the initial balance of the policyholder s account X 0 (0) = 1, and hence the policyholder should be required to pay extra for the guarantees. During the life of the policy the ratio can be either above or below 1 indicating that there are times when the assets are both greater than and less than the policyholder s account. On the left hand side of Table 2 we see an interesting feature at time 2. There are two cases where the insurer invests entirely into the two-period zero-coupon bond. This will occur in the penultimate year whenever we know for certain whether the guarantee will bite or not in the final two years. Hence the two-period zero-coupon bond provides a guaranteed return and we have a known final guarantee. For example, with r(2) = , we know that the guarantee must bite in the final two years because the binomial tree ensures that the return each year on the two-year zero-coupon bond cannot rise above the guarantee of 2.5%. Once we have calculated the ratio Π 0 (t) and the investment strategy a 0 (t) for each time period we can then work forwards from the initial condition X 0 (0) = 1 to calculate the values of V 0 (t) and X 0 (t) which are shown in Table 3. Note for these two tables that the path dependent nature of the payoff means that the binomial tree does not recombine. Again note that for each table the tree does not bifurcate in the final year because we have invested all the assets in the one-year zero-coupon bond which has known return. 20

21 V 0 (t) X 0 (t) Table 3 The left hand side shows the value of the portfolio V 0 (t). The right hand side shows the value of the policyholder s account X 0 (t). The left hand side of Table 3 shows that in the early years falling interest rates lead to higher returns on the portfolio because the value of the two-year zerocoupon bonds rises. However, higher reinvestment rates mean that scenarios with high interest rates ultimately lead to higher asset values at maturity. The right hand side of Table 3 follows a similar pattern to the left hand side. However, when returns are low the guaranteed growth rate ensures that the policyholder s account rises more quickly than the assets. Conversely when returns are high the participation rate means that the policyholder s account rises more slowly than the assets. 8 Conclusion Traditionally the bonuses and the final payout on a participating policy have been modelled as depending on an exogenously given reference portfolio. In this paper we have instead assumed that bonuses and payouts are related to the total investments of the insurer which can be changed at the discretion of the managment at any time. Our approach is more practical than the traditional approach. The insurer s accounts will show the value of its own assets and the return earned on these over the year. The policyholder will expect to receive a bonus related to the return published in the accounts. Hence the bonuses and payouts given to the policyholders will depend on the insurer s choice of investments. We have shown how the insurer should invest in order to perfectly match these liabilities and hence have shown how to 21

22 calculate a fair value for this kind of contract. Firstly in Section 3 we considered the case where interest rates were constant and showed that the insurer could match its liabilities by investing in a zerocoupon bond with the same term as the policy, as was seen by Brennan and Schwartz (1979). However, in contrast to the unit-linked case considered by Brennan and Schwartz (1979), we showed that it was risky to invest into this bond if the interest rate was stochastic. Then in Section 5 we considered a more realistic investment model with stochastic interest rates and found that the insurer could ensure that their assets were exactly equal to the policyholder s account at maturity by investing in a mixture of one-year and two-year bonds. Alternative hedges would also be possible if the two-year bond was replaced in our model by another asset driven only by interest rate risk. Finally we considered the addition of further risky assets such as equities to the insurer s portfolio. We showed that a superhedging strategy can be created so that the assets are always at least as great as the policyholder s account at maturity. This approach allows the insurer to invest in equities without taking any risk itself, but does offer the potential for higher policyholder returns. The superhedging strategy requires a larger sum to be invested at outset than under the perfectly matched case, and so in compensation the policyholder receives a final bonus so that they receive the full value of the assets. We have shown how to calculate a range of portfolios which will superhedge the risk. We showed that for each year the insurer can take any investment strategy it wishes as long as it will have at least enough assets at the year end to follow the perfectly hedged strategy if necessary. Therefore the perfectly hedged portfolio plays a central role in placing a limit on the investment choices the management can take. Acknowledgements We would like to thank Andrew Cairns, David Forfar, Angus Macdonald, David Wilkie and an anonymous referee for their useful comments and suggestions. References Bacinello, A. R., Fair pricing of life insurance participating policies with a minimum interest rate guaranteed. ASTIN Bulletin 31, Bacinello, A. R., Pricing guaranteed life insurance participating policies 22

23 with annual premiums and surrender option. North American Actuarial Journal 7, Ballotta, L., A Lévy process-based framework for the fair valuation of participating life insurance contracts. Insurance: Mathematics and Economics 37 (2), Bauer, D., Kiesel, R., Kling, A., Ruß, J., Risk-neutral valuation of participating life insurance contracts, working paper, University of Ulm. Brennan, M. J., Schwartz, E. S., The pricing of equity-linked life insurance policies with an asset value guarantee. Journal of Financial Economics 3, Brennan, M. J., Schwartz, E. S., Pricing and Investment Strategies for Guaranteed Equity-Linked Life Insurance. Monograph No. 7. The S. S. Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania. Briys, E., de Varenne, F., On the risk of life insurance liabilities: Debunking some common pitfalls. Journal of Risk and Insurance 64, Dothan, M. U., Prices in Financial Markets. Oxford University Press. Grosen, A., Jorgensen, P. L., Fair valuation of life insurance liabilities: The impact of interest rate guarantees, surrender options, and bonus policies. Insurance: Mathematics & Economics 26, Grosen, A., Jorgensen, P. L., Life insurance liabilities at market value: An analysis of insolvency risk, bonus policy, and regulatory intervention rules in a barrier option framework. Journal of Risk and Insurance 69, Guillén, M., Jorgensen, P. L., Nielsen, J. P., Return smoothing mechanisms in life and pension insurance: Path-dependent contingent claims. Insurance: Mathematics & Economics 38, Hibbert, A. J., Turnbull, C. J., Measuring and managing the economic risks and costs of with-profits business. British Actuarial Journal 9, Jorgensen, P. L., Life insurance contracts with embedded options. Working paper 96, University of Aarhus, Denmark, Miltersen, K. R., Hansen, M., Minimum rate of return guarantees: The Danish case. Scandinavian Actuarial Journal, Miltersen, K. R., Persson, S.-A., Pricing rate of return guarantees in a Heath-Jarrow-Morton framework. Insurance: Mathematics & Economics 25, Miltersen, K. R., Persson, S.-A., A note on interest rate guarantees and bonus: The Norwegian case. AFIR Conference, Miltersen, K. R., Persson, S.-A., Guaranteed investment contracts: Distributed and undistributed excess return. Scandinavian Actuarial Journal, Persson, S.-A., Aase, K. K., Valuation of the minimum guaranteed return embedded in life insurance products. Journal of Risk and Insurance 64, Tanskanen, A. J., Lukkarinen, J., Fair valuation of path-dependent par- 23

24 ticipating life insurance contracts. Insurance: Mathematics & Economics 33, Wilkie, A. D., An option pricing approach to bonus policy. Journal of the Institute of Actuaries 114, Willder, M., An option pricing approach to charging for maturity guarantees given under unitised with-profits policies. Ph.D. thesis, Heriot-Watt University. 24