On valuation of equity-linked life insurance products and credibility models

Abstract

Equity-linked life insurance products offer the policyholder both an insurance guarantee and a chance to engage in potential gains from a reference risk asset, say a stock or an equity index. Whether the insured dies before the contract maturity, a benefit payment to a designated beneficiary is due. Determining such a product's value is a crucial task for insurance companies. Since the payout may occur upon the insured's death, the concept of the time-until-death random variable is introduced. It is a non-negative random variable modeling the remaining lifetime of the insured. The valuation problem can be reduced to a particular form where the time-until-death distribution is exponential. The payment amount depends on the then-current value of the policyholder's account. The calculation follows a cliquet-style (or ratchet) structure. It incorporates the greater value of a guaranteed minimum return and a proportion of the return from the linked equity into the accumulation.

After presenting some preliminary results, we first focus on the case where a geometric Brownian motion models the dynamic of the reference asset. A drifted Brownian motion then describes the log-asset price. A specific portion of the return rate increment of the linked equity in a year is credited into accumulation. Moreover, another pattern for accumulation is discussed, where a part of the highest return rate increment ever reached during a year is taken into calculation. Closed-form valuation formulas for both cases are presented. Second, we examine the case where the modeling of the reference asset is extended to follow a jump-diffusion model. Two maturity scenarios are addressed. The valuation formula corresponding to constant maturity is firstly presented. On the other hand, motivated by the erlangization technique, an erlangized version of the valuation problem is proposed. This part is demonstrated by assuming a generic Erlang distributed random variable to model the inter-period time, thereby approximating the deterministic counterpart. The erlangized valuation problem is well explained and explicitly solved.

The following part addresses a problem related to the credibility theory, whose role in various research areas of actuarial science is significant. Among others, the hypothetical mean and process variance are two quantities that convey crucial information to insurance companies when determining insurance premiums. Enlightened by the prestigious mean-variance premium principle, we propose a credibility approach to estimate the linear combination of hypothetical mean and process variance (i.e., the individual mean-variance premium principle) under the quadratic loss function. Our proposed estimator consists of several parts, in the linear form of observations and their quadratic terms, as well as some quantities representing population information. Meanwhile, a spin-off result is compared with the classical simple Bühlmann model and the q-credibility model. Moreover, the non-parametric estimators of structural quantities are also provided for ease of its practical usage, yielding the empirical credibility estimator.