مدل درونی از شرایط تسلیم شدن در بیمه عمر مرتبط با ارزش سهام

We propose a model for pricing a unit-linked life insurance policy embedding a surrender option. We consider both single and annual premium contracts. First we analyse a quite general contract, for which we obtain a backward recursive valuation formula based on the Cox et al. [Cox, J.C., Ross, S.A., Rubinstein, M., 1979. Option pricing: a simplified approach. J. Finan. Econ. 7, 229–263] binomial model. Then we concentrate upon a particular case, that is the famous model with exogenous minimum guarantees. In this case we extend our previous analysis in order to take into account the possibility that the guarantees at death or maturity and the surrender values are endogenously determined, and provide necessary and sufficient conditions for the premiums to be well defined.

The surrender option embedded in several types of life insurance contracts gives the policyholder the right to early terminate the contract and to receive a cash amount, called surrender value. To avoid adverse selection phenomenons, this option is usually granted only if the contract provides benefits both in case of death and in case of survival such as, e.g., endowment and whole-life insurance policies. The problem of fixing the surrender conditions when designing a new policy is very important, specially if the financial component of the policy is predominant. An over-simplified way to solve the problem could be that of fixing very low or even null surrender values. The (only) advantage of this solution is that the insurance company can completely ignore the surrender option and use consolidated actuarial techniques for pricing (and hedging) the contract. After all, the surrender is a unilateral decision of the policyholder that does not respect the initial terms of the contract and implies a loss of future gains expected by the insurance company. However, this solution may have a disastrous effect from a marketability point of view. First of all, it may spread discontent through those that originally did not consider the possibility of surrender but are forced to abandon their contracts for unexpected events and now feel swindled. Secondly, markets are populated also by investors that do not know exactly their time horizon, for which the surrender conditions may constitute one of the key-elements in the choice of an investment product such as a life insurance contract. For these and similar reasons, among which a very important one is also the market competition between insurers, the insurance company could decide to fix very competitive surrender conditions, but in this case it cannot afford to ignore the surrender option, that constitutes a component of the contract and must be suitably rewarded. Then an accurate valuation of this option, along with the other elements of the contract, is called for. In particular, given the surrender conditions, the valuation of such an option can be performed by following two different approaches: (i) According to the former, the surrender decision is treated just like death, i.e., it is considered an “exogenous” cause of termination of the contract. Actually, such decision can be driven by several “personal” reasons out of the control of the insurance company. For instance, the policyholder can fall into financial difficulties, or he(she) can become acquainted with a change in the health status of the insured, checked by the insurance company only at inception. The collection of sufficient statistics on surrenders (called also withdrawals) allows to estimate the expected surrender rates and to construct a multidecrement table with two possible causes of elimination: surrender and death. A very natural assumption combined with this approach is the stochastic independence between surrenders and the financial elements. (ii) The latter approach merges the contract into a contingent-claims framework, characterized by perfectly competitive and frictionless markets, populated by rational and non-satiated agents all sharing the same information. According to this approach, the surrender decision is not at all independent of the financial elements, since it is the consequence of a rational choice. Then, in this framework, the whole contract and, in particular, the surrender option, are typical American-style 1 contingent-claims, whose valuation is obtained by merging together the traditional actuarial techniques, based on pooling arguments, with the modern financial toolbox, based on the no-arbitrage principle. The above two approaches may appear to be completely incompatible. Nevertheless we are fully convinced that, although most policyholders are very likely induced to surrender their policies by “exogenous” reasons, the correct approach to follow in the valuation of the contract is the second one, because the fair value of a right, such as the surrender option, is independent of the behaviour of its owner. In other words, the policyholder has the right to act “optimally” when taking his(her) surrender decision, hence no “discounts” are admitted for anticipated non-optimal behaviours. In fact, even if such discounts were allowed, the insurance company could not subsequently forbid the policyholder to act optimally, and this could obviously turn out to be a serious threat to solvency. The literature concerning the valuation of the surrender option in a contingent-claims framework is not very abundant. The only papers of which we are aware are those by Albizzati and Geman, 1994, Grosen and Jørgensen, 1997, Grosen and Jørgensen, 2000, Jensen et al., 2001, Steffensen, 2002, Bacinello, 2003a, Bacinello, 2003b and Tanskanen and Lukkarinen, 2003, and Vannucci, 2003a and Vannucci, 2003b. Apart from the paper by Steffensen (2002), that acts in a very general framework, the above papers deal only with single-premium contracts (except Bacinello, 2003b) and traditional or participating life insurance (except Grosen and Jørgensen, 1997, Vannucci, 2003a and Vannucci, 2003b), and analyse a fixed-term life insurance policy without mortality risk, hence a purely-financial contract (except Bacinello, 2003a, Bacinello, 2003b, Vannucci, 2003a and Vannucci, 2003b). The introduction of mortality risk is a very delicate point, even under the usual assumption of stochastic independence between mortality and financial factors. In fact, unlike European-style 2 contracts, in the valuation of American contracts it is not possible to keep separate the financial elements from mortality, because the surrender decision involves continuous comparisons between the surrender value and the value of the contract, that obviously depends also on mortality factors. Hence there is a continuous interaction between mortality and financial factors. Another interaction arises when the analysis is shifted from single-premium to periodic-premium contracts, because the periodic premium depends on the value of the surrender option, that in turn depends on the premium amount, even if the guarantees and the surrender values are exogenously given. In this paper we propose a model for pricing, according to the second approach, a unit-linked life insurance contract embedding a surrender option. The methodology and the philosophy used are quite the same as in Bacinello, 2003a and Bacinello, 2003b, where they are applied to participating policies. The contract analysed is an endowment policy. We have chosen such type of policy because it is characterized by a high level of savings component and delivers benefits both in case of death and in case of survival at maturity. Moreover, our analysis can be straightforwardly applied also to whole-life insurance by simply taking, as maturity date, the date corresponding to the terminal age of the insured. 3 We consider both single-premium and annual-premium payments. In a first moment we analyse a quite general equity-linked contract, without specifying the way in which benefits and surrender values are linked to the reference portfolio. For this general contract we obtain a backward recursive valuation formula based on the Cox et al. (1979)binomial model. Then we concentrate upon a particular case of the general contract, that is the famous model with exogenous minimum guarantees pioneered by Brennan and Schwartz (1976) and Boyle and Schwartz (1977). In this particular case we extend our previous analysis in order to take into account the possibility that the minimum guarantees at death or maturity and the cash surrender values are endogenously determined, and provide necessary and sufficient conditions for the premiums to be well defined. Here the terms exogenous and endogenous are used with the same meaning given them by Bacinello and Ortu (1993). The paper is structured as follows. In Section 2 we introduce our valuation framework and analyse the single-premium contract, first of all in general terms and after by endogenizing the minimum guarantees and the surrender values. The same analysis is then extended, in Section 3, to periodic-premium contracts. The whole discussion is also accompanied by the presentation of some numerical results.

We have proposed a discrete time model for pricing a unit-linked life insurance policy, of the endowment type, embedding a surrender option. The valuation has been performed by merging the contract into a contingent-claims framework, characterized by perfectly competitive and frictionless markets, free of arbitrage opportunities. The contract considered is very general and allows, in particular, for minimum guarantees at death, maturity or surrender. Moreover, the (possible) guarantees can be exogenously given, i.e., independent of premiums, as well as endogenously determined. We have analysed both single and annual premium contracts. The numerical results have shown that annual premium contracts, although reproducing the same qualitative behaviour as single premium ones, can be very cheaper, specially when the endogenization of guarantees at death, maturity and surrender is taken into account. The discrete time model used in the paper leads to very complicated computations, especially for periodic-premium contracts. Then it could be interesting to see if this model can be generalized in continuous time. Another relevant aspect, which is probably the main driver of the surrender behaviour on life insurance markets, is the “stochasticity” of interest rates. As already said in Section 2.2, a great limit of our model is undoubtedly that of completely ignoring this aspect. Finally, the stochasticity of mortality trends should also be taken into account. Hence this paper is to be intended as a first step in the direction of endogenously modelling the surrender conditions/behaviours, and the developments of the model proposed with a more realistic financial market are very important topics of future research.