Academic journal article Journal of Risk and Insurance

Fair Valuation of a Guaranteed Life Insurance Participating Contract Embedding a Surrender Option

Academic journal article Journal of Risk and Insurance

Fair Valuation of a Guaranteed Life Insurance Participating Contract Embedding a Surrender Option

Article excerpt

ABSTRACT

In this article we deal with the problem of pricing a guaranteed life insurance participating policy, sold in the Italian market, which embeds a surrender option. This feature is an American-style put option that enables the policyholder to sell back the contract to the insurer at the cash surrender value. Employing a recursive binomial formula patterned after the Cox, Ross, and Rubinstein (1979) discrete option pricing model we compute, first of all, the total price of the contract, which also includes a compensation for the participation feature ("participation option," henceforth). Then this price is split into the value of three components: the basic contract, the participation option, and the surrender option. The numerical implementation of the model allows us to catch some comparative statics properties and to tackle the problem of suitably fixing the contractual parameters in order to obtain the premium computed by insurance companies according to standard actuarial practice.

INTRODUCTION

Life insurance contracts and pension plans are often very complex contingent claims that embed several financial options, both of European and of American style. According to Smith (1982) and Walden (1985), they can even be seen as option packages.

A typical example of European (put) option is implied by the maturity guarantees present in most types of equity-linked life insurance products. An accurate analysis of such guarantees is undoubtedly one of the main challenges in the life insurance business, more and more forced to broaden the range of its own products in order to force the increasing competition with financial markets. Large contributions in this direction have been made by financial and actuarial researchers since the pioneering work by Brennan and Schwartz (1976, 1979a, 1979b) and Boyle and Schwartz (1977). (1)

Another example of European (call) option is implied by the participation mechanism that characterizes policies with profits. Such a mechanism applies when "dividends" are credited to the policy reserve, thus producing an increase in the insurer's liabilities (benefits). This special feature has been studied, for instance, by Briys and de Varenne (1997), Grosen and Jorgensen (2000, 2002), Miltersen and Persson (2000), and Bacinello (2001), and in Europe is often referred to as "bonus."

In particular, Bacinello (2001) analyses a life insurance endowment policy with a minimum interest rate guaranteed in which both the benefit and the periodical premiums are annually adjusted according to the performance of a special investment portfolio. This contract is actually offered in the Italian market. Under the Black and Scholes (1973) and Merton (1973) framework, Bacinello (2001) expresses, first of all, the fair price of such a policy in terms of one-year call options, and then derives a very simple closed-form relation that characterizes fair contracts. However, as a concluding remark, Bacinello (2001) points out that an important issue connected with participating policies, which has not been dealt with in the article, is constituted by the presence of a surrender option. A surrender option is an American-style put option that entitles its owner (the policyholder) to sell back the contract to the issuer (the insurer) at the cash surrender value. Fair valuation of such an option, as well as an accurate assessment of the cash surrender values, are clearly crucial topics in the management of a life insurance company, both on the solvency and on the competitiveness sides.

The aim of the present article is just to fill this gap. We consider the single-premium version of the contract analyzed by Bacinello (2001) and define, first of all, a rule for computing the cash surrender values, which introduces an additional contractual parameter in the model. Then, by modeling the assets according to Cox, Ross, and Rubinstein (1979), we obtain a recursive algorithm for computing the fair price of the whole contract. …

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