The economic and financial developments over the last 15 years have brought to life insurers both new opportunities and new challenges. On one hand, the uncertainty of pay-as-you-go retirement schemes generated by longer life expectancies and lower birth rates has entailed a shift of savings in the French economy toward life insurance policies. Nonmortality-related contracts account today for more than 80 percent of French life insurance contracts. They represent more than half of the national savings and amounted to Fr 250 billion in 1993, climbing from Fr 49 billion in 1983.
On the other hand, volatile interest rates, disintermediation, and competition from banks and financial institutions offering similar types of products have forced life insurers to promise and guarantee higher rates of return on the savings components of life insurance and annuity contracts and, hence, to assume investment risks associated with higher book yields. Moreover, if the guaranteed return is not high enough compared to other forms of investment, mainly in the case of a rise in interest rates, policyholders may decide on early termination of their existing policies and choose a higher yield alternative offered in the capital markets (e.g., money market funds). It is the valuation of this surrender option in the context of stochastic interest rates that this article addresses, after an overview of interest rate risk management for life insurers.
Asay, Bouyaoucos, and Marciano (1989) have offered an option-adjusted-spread approach to estimating the financial value of outstanding policies. Their methodology is not given explicitly, however, because it is the proprietary approach used by Goldman Sachs for the valuation of callable bonds or mortgage-backed securities to take into account the prepayment option. Moreover, as pointed out by Babbel and Zenios (1992), the spread is difficult to estimate and strongly depends on the volatility assumption in the model. Lastly, whether one looks at a single contract or at a pool of contracts, the property of path-dependency must be taken into account since several lapses cannot be observed on the same policy. This article looks at the problem the other way around and directly calculates the value of the surrender option embedded in life insurance policies.
This surrender option, which is indeed an exchange option, cannot be priced by the formula provided in the seminal paper by Margrabe (1978), who assumed deterministic interest rates (as did Black and Scholes, 1973). Our problem, by definition, is set in the framework of stochastic interest rates. Moreover, Margrabe's exchange option was a European option; the surrender option has an exercise date that is an optimal stopping time. We partly solve the latter difficulty by considering a pool of homogeneous life insurance policies and using the fundamental averaging effect of the insurance mechanism. This leads to an evaluation of the option by arbitrage under the risk-neutral probability as the expectation of random cash flows occurring at well-defined dates and discounted with stochastic interest rates. We use under a generalized form the forward neutral probability measure introduced by Geman (1989) and Jamshidian (1989), which proved very powerful in pricing interest rate derivative instruments such as floating-rate notes and interest rate swaps (see El Karoui and Geman, 1991, 1993).
The next section presents some elements of life insurer asset-liability management. Then, we provide a closed-form expression of the surrender option value in the case of a single-premium policy when its dollar amount is invested in a fixed-term zero-coupon bond of the asset portfolio, and the dynamics of the term structure of interest rates are assumed to be driven by a one-factor model with a deterministic term structure of volatilities. The option price is computed under different interest rate volatilities, using both the closed-form expression and Monte Carlo simulations. …