# Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz

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1 Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance Brennan Schwartz (976,979) Brennan Schwartz

6 π (p) = n p x E Q [ e r n max(s T,G T ) I 0 ] () where n p x = (- q x )(- 2 q x ) (- n q x ) is the probability that the insured is still alive at time t = n and E Q [ I t ] is the conditional expectation operator with respect to a equivalent martingale measure Q and a filtration I t.. Similarly, the single premium of the equity-linked term insurance contract with a guarantee maturing at time T represented by π (t) is n π (t) = t- q x E Q [ e r t max(s t,g t ) I 0 ] (2) where t- q x = (- q x )(- 2 q x ) (- t- q x ) tq x is the probability that the insured dies within the t-th period. Using () and (2), the market price at time 0 of the equity-linked endowment insurance policies represented by π (e), which put a equity-linked pure endowment contract and a equity-linked term contract together, can be written by π (e) = n p x E Q [e r n max(s T,G T ) I 0 ] + n t- q x E Q [e r t max( S t,g t ) I 0 ] (3) The pricing formula of (3) will be utilized to evaluate the ELEPAVG with a one-shot surrender option in the following sections. 3. Valuation for ELEPAVG with a surrender option ELEPAVG with a one-shot surrender option is considered in this section. The work is restricted to the case which a policyholder can exercise the surrender option only once. Suppose a policyholder is permitted to exercise the surrender option only at time t = k (0<k<n). The cash surrender value (exercise price) is set by W(S k ), which is always a function of S t. It is assumed that surrender proportion of fund value is not limited. Furthermore, the purpose for a

7 Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance policyholder s surrender decision is to maximize the market value of the life insurance contract. 3. Investment consideration at initiation of contract We suppose that the policyholder initially plan to surrender m* shares of the fund at time k. In another word, if the insured is alive at time k, he will surrender m * shares of the fund whatever the realized value of S k will be. According to his surrender plan, the market value at time 0 of such an endowment insurance contract may be written by π (e,u) k = m 0 { t q x E Q [e r t max( S t,g t ) I 0 ] + ξ ( - k t q x ) V 0 (W(S k )) n + (-ξ )( k+ t q x E Q [e r t t max(s t,g t ) I 0 ]+ n p x E Q [e r n t max(s T,G T ) I 0 ])} (4) where ξ = m * / m 0, called surrender ratio. According to (4), the policyholder will try to find the optimal amount of q to maximize the market value of the contract. However the insured can t determine the optimal value of q at the initial time. It is because he does not know what value of W(S k ) will become. In another word, the optimal value of q is contingent to different situations. The criterion by which the policyholder chooses the optimal surrender ratio is discussed in the next subsection. 3.2 The optimal surrender behavior Following previous subsection, we need to find the optimal surrender behavior for a rational policyholder. The optimal surrender behavior determines the surrender ratio. It is also assumed that policyholder s decision making is based on maximization of the market value of the insurance contract at time k. Thus

8 the problem to be solved is to find the optimal surrender ratio q*, which would maximize the market value of the contract at time k, subject to 0 ξ*. Similar to (3), if the policyholder is alive at time k, the market values at time k of the insurance contract, denoted by π (e,k), may be written by π (e,k) = m 0 {ξ W(S k ) + ( ξ ) ( n k+ t q x E Q [e r t max(s t,g t ) I k ] + n p x E Q [e r n max(s T,G T ) I k ] ) } = m 0 {ξ W(S k ) + ( ξ ) Θ(S k )} (5) where Θ(S k ) denote n k+ t q x E Q [e r t max(s t,g t ) I k ]+ n p x E Q [e r n max(s T,G T ) I k ], which is a function of S k. To maximize (5), the optimal surrender ratio ξ* which a policyholder will take would be determined by comparing W(S k ) with Θ(S k ). The optimal surrender ration ξ* depends on the realized value of S k. As W(S k ) >Θ(S k ), the optimal surrender ration ξ* will equal one. As W(S k ) <Θ(S k ), the optimal surrender ration ξ* will be equal to zero. A break-even point, S k, could be found such that W( S k ) is equal to Θ( S k ). The criterion of the surrender decision for a policyholder, which depends on S k, is summarized by following expression. as S k S k ξ* = (6) 0 as S k S k

9 Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance 3.3 Pricing the surrender option The fair single premium of ELEPAVG with a one-shot surrender option is calculated in this subsection. Since the insured s behavior follows the optimal investment strategy of (6), the break-even point, S k, will divide the distribution of S k into exercise and no exercise region. The payoff at time k depends on which situation occurs. There are four situations, which are determined by two factors, i.e. the realized value of S k and the survival status of insured in k-th period. The payoff per share in each situation is summarized in Table. Table : The payoffs in the four situations at time t = k The status of insured in k-th time interval Dead Alive S k > S k Situation : Max (S k, G k ) Situation 2: W(S k ) The value of S k S k < S k Situation 3: Max (S k, G k ) Situation 4: 0 According the payoff in each situation, the market value at time 0 of ELEPAVG with the surrender option represented by π (e,s) may be derived by π (e,s) k = m 0 { k + ( - t q x E Q [e r t max( S t,g t ) I 0 ] + t q x E Q [e r k max(s k, G k ) I 0 ] t q x )[E Q (e r k W*(S k ) I 0 )+ E Q (e r k Θ*(S k ) I 0 )] } (7) Where W*(S k ) = W(S k ), if S k > S k and W*(S k ) = 0, if S k S k ; Θ*(S k ) = Θ(S k ), if S k < S k and Θ*(S k ) = 0, if S k > S k. By making (7) minus (3), the fair market value of one-shot surrender option embedded in equity-linked insurance policies represented by π (s) may be derived by

10 π (s) = m 0 { k p x [E Q (e r k W*(S k ) I 0 )+ E Q (e r k Θ*(S k ) I 0 )] n - k+ t q x E Q [e r t max( S t,g t ) I 0 ] - n p x E Q [e r n max(s T,G T ) I 0 ]} (8) To make the expression more concise, (8) may be rewritten by π (s) = m 0 ( - ) (9) where and denote k p x [E Q (e r k W*(S k ) I 0 )+ E Q (e r k Θ*(S k ) I 0 )] and n k+ t- q x E Q [e r t max( S t,g t ) I 0 ] + n p x E Q [e r n max(s T,G T ) I 0 ] respectively. According to (9), it is not necessary that the value of π (s) is positive since it is not absolute that is larger than. Thus, whether the value of one-shot surrender option is positive depends on how to set the cash surrender value (exercise price function), W(S k ), and the guarantee function, G t. In another word, both the cash surrender option and the guarantees provided by insurer will affect insured s intention to pay the premium for such the additional surrender option. If the value of (9) is not positive, the surrender option is not valuable and the premium of ELEPAVG with the surrender option may be as same as that of ELEPAVG with no surrender option. 4. Implications on two-period case The two-period case is discussed in details in this section. Let the term of insurance contract divided into two subintervals. Three time points, i.e. time 0, time and time2, are considered now. The surrender option may be exercised only at time.to make the model simpler, it is assumed that the asset value guarantee is constant, i.e. G t = g and the cash surrender option, W(S ), is set by a linear function of S, i.e. (+ε)s, where ε is a constant. Now we also suppose that the policyholder initially have a preferred surrender

11 Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance plan. The surrender portion of reference fund shares is ξ. Even the results in previous section imply that the policyholder can t determine the optimal surrender portion at initiation of the contract. Given the surrender plan, the market value of the insurance contract denoted by π (2) may be written by π (2) = m 0 { d x [e -r g +C(S 0,,g)]+ ξ (- d x )V 0 ((+ε)s ) + (- ξ )(- d x )[e -r2 g +C(S 0,2,g)]} (0) where C(S 0,,g) denotes the European call option value with the current price of underlying asset, S 0, the time to maturity,, and the exercise price, g and V t (S τ ) denotes the market value of S at time t The premium charged by an issuer to provide both guarantee and surrender options, denoted by α, is given by α = π (2) m 0 S 0 = m 0 { d x [e -r g +C(S 0,,g)- S 0 ]+ ξ (- d x )V 0 ((+ε)s ) -(- d x )S 0 +(- ξ )(- d x )[e -r2 g +C(S 0,2,g)]} () Based on the Martingale pricing theory, i.e. E Q [e - r S ] = S 0, V 0 (S ) is equal to S 0. Thus (0) may be rewritten by π (2) = m 0 {S 0 + d x P(S 0,,g)+(- ξ )(- d x ) P(S 0,2,g)+ ξεs 0 } (2) where P(S 0,,g) denotes the European put option value with current price of underlying asset,s 0, time to maturity,, and the exercise price, g. Note that (2) with q = 0 is a special case of Brennan and Schwartz s general pricing formula. According (2), it is obvious that the market value of the contract is an increasing function of g and ε. As the results derived in the third section, the optimal surrender behavior as of

12 time depends on the realized value of S. The break-even point, S, must satisfy the following equations. S +P( S,, g) = (+ε) S (3) or ε = P( S,, g) / S (4) (3) or (4) implies that the value of ε can not allowed to be too small. If ε is smaller than P(S, t, g) / S for each S, such a surrender option become valueless. Therefore, the larger the guarantee g is, the more the fair premium that the insurer will charge should be, which. The more the guarantee offered by insurer is, the more the cash surrender value should be. Otherwise, the surrender option may not provide the additional value for a policyholder. For the extreme case, the exercise price function with ε = 0 will make the surrender option valueless. Thus, given a certain single premium for the equity-linked insurance policies, the relationship between guarantee value and cash surrender value should be specified. The implications of the result may be helpful for both insurance product designer and insurance buyer. 5. Conclusion The key feature of equity-linked life insurance policies is the uncertainty of the future insurance benefit. Our work has shown how to use the Martingale pricing theory and principle of equivalence to price the one-shot surrender option embedded in equity-linked life insurance contracts. The optimal surrender behavior for the policyholder is found. Then the single premium of ELEPAVG with the surrender option, which is charged fairly by issuers, has been derived. Further the pricing formula for the valuation of the surrender option is also found. The result of this research provides some implications in practice for both insurer and investor. This study could be extended to the cases with multiple surrender options or American-type surrender option.

14 Nielsen, J. and Sandmann, K. (995), Equity-Linked Life Insurance: A Model with Stochastic Interest Rates, Insurance: Mathematics and Economics, Vol. 6, Nonnenmacher, D. J. F. and Russ, J (997), Equity-Linked Life Insurance in Germany: Quantifying the Risk of Additional Policy Reserves, Actuarial Approach for Financial Risks, 7 th International Colloquium, Cairns, Australia, Vol. 2,

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