The Equilibrium Insurance Price and Underwriting Return in a Capital Market Setting

Abstract

This article develops a modified capital asset pricing model, CAPM, which integrates the markets for financial assets, real assets, and insurance. It is shown that the purchase of insurance should not be ignored in determining the equilibrium rate of return on assets, and that the equilibrium insurance premium should adjust for the sources of systematic risk associated with financial assets, real assets, and the insurance market. Consistent with some earlier studies, it was found that the insurer's equilibrium underwriting return could be negative, rather than positive, under certain conditions. The insurance pricing models based on financial theory differ widely in terms of parameter specifications, computation methods, and underlying assumptions. Models based on the Options Pricing Model (OPM) are based on total variability but are very sensitive to taxation (see Doherty and Garven, 1986). On the other hand, models based on the Capital Asset Pricing Model (CAPM) focus on risk and ignore other risk factors (see Fairley, 1979). Indeed, unsystematic risk and underwriting risk are also important in determining insurers' rate of return (see Cummins and Harrington, 1988; and Witt and Urrutia, 1983). It has also been suggested that the risk-free rate used to discount loss reserves may be excessive (see D'Arcy, 1988). Insurance pricing models based on financial theory that have gained some prominence as regulatory tools are discrete-time discounted cash flow, DCF, models (see Myers and Cohn, 1987; the National Council on Compensation insurance, NCCI, 1987; and Cummins, 1990). The main objective of this article is the derivation of equilibrium insurance prices and underwriting returns in a capital setting. The proposed models are based on the CAPM. Even though, the CAPM literature has generally assumed away the importance of risk and return effects of real assets and insurance contracts on an investor's optimal portfolio, several extensions of the standard CAPM have been developed that try to account for the missing assets and expand the concept of a market portfolio (see Mayers, 1972; Roll, 1977; Brito, 1978; Breeden, 1979; and Mayers and Smith, 1983). More recently, Ang and Lai (1987) and Turner (1987) have also developed insurance pricing formulas based on the CAPM. The Ang and Lai model is developed from the viewpoint of an insurance firm and does not explicitly incorporate real assets into the ratemaking formulas. The Turner model is derived from the household viewpoint and takes into account real assets, insurance shares, insurance policies and an individual's consumption. The model proposed in this article differs from the Ang and Lai model in that it is derived from the insured's point of view and explicitly recognizes both real assets and insurance contracts in the pricing formulas. The model presented in this article is similar to the Turner model. It differs from the Turner model in its assumption that insurance is provided by mutual rather than stock insurance companies. The present model was developed independently of the Turner model and has pedagogical value because it uses simpler notation and presents a more straightforward derivation of the pricing formula. The two-fold purpose of this article is to extend the traditional CAPM in order to recognize two important facts: first, most investors invest in real assets as well as financial assets; and, second, most investors also purchase insurance to protect themselves against real asset losses, and, therefore, they are also insureds. It is shown that the equilibrium insurance premium and the fair rate of return on an insurance contract should adjust for three sources of systematic risk: that associated with the markets for financial assets, real assets, and insurance. Even though the magnitude of several covariance terms are unknown, the equilibrium underwriting return for the insurer could be negative. If this were the case, such a result would be consistent with what previous studies have suggested and would imply a positive cost of capital for policyholder supplied funds. …