Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Products

Abstract

INTRODUCTION The topic of this article is the valuation of a periodical minimum return embedded in life insurance contracts in several countries - for example, the United States and Norway. These guarantees are typically included in traditional life insurance products, such as endowment insurances, and are not necessarily connected to more modern products like variable life insurance. The guarantee is typically an annual percentage minimum return. It seems to be practice to set the level of the guarantee far below the actual interest rate at the time of issue. At contract initiation, the guarantee is (far) out-of-the-money and, accordingly, its market value is low. In practice, guarantees are therefore often disregarded when the economic condition of an insurance company is evaluated (see, e.g., Katt, 1996). However, insurers recently have experienced significantly lower rates of return than they did in the 1980s, and there are examples of annual guarantees expiring in-the-money. Clearly, such guarantees represent an obligation for the issuing company and should in principle be evaluated and occur at the company's balance sheet. Proper valuation methods for guarantees do not seem to be developed as no extra premium usually is charged for the guarantee. In this article, we develop a model for pricing such guarantees. Indeed, the very existence of interest rate guarantees reflects the volatile nature of interest rates. Accordingly, its valuation must be based on a framework incorporating random interest rates. The typical approach in the actuarial literature is to model the interest rate by a stochastic process and price life insurance contracts according to the traditional principle of equivalence, an approach widely accepted under deterministic interest rates. Our approach is different. In a companion article, Persson (1998) develops a model for pricing life insurance contracts under stochastic interest rates based on economic theory. The main result of this model is that single premiums of life insurance contracts may be deduced from market prices of bonds. Formally, the premiums still may be calculated as expected present values, but in the presence of stochastic interest rates a risk-adjusted probability measure must be applied, instead of the original probability measure. This result follows from the theory of arbitrage pricing in financial economics. Compared to the classical principle of equivalence, our approach includes a model of the financial market and restricts the insurer's investment possibilities to securities traded in this market. For simplicity, the security market in this model consists of bonds and a savings account only. By a participating policy we refer to a contract where the insured is entitled to a share of the surplus if the realized interest rate during the insurance period is above the assumed interest rate. This property is included in many real-life life insurance contracts. Instead of the absolute amount of benefit, it is natural to focus on the rate of return of the policy; that is, we study the amount of insurance available for one unit of account. In addition, the insured must pay a loading for the participating option. The market price of this loading is determined for a certain specification of a participating policy. By partitioning the insurance period, the periodical minimum guarantee described above may be considered as a sequence of participating policies. The market-based loading for a policy including a minimum guarantee is then determined. The article first describes the economic model. Then, we determine market-based loadings for participating policies and participating policies with minimum guarantees and compare the results for the two types of policies. ECONOMIC MODEL The financial market consists exclusively of certain bonds and a savings account. Life insurance-specific factors are introduced at the end of the section. …