Academic journal article Journal of Risk and Insurance

A Class of Distortion Operators for Pricing Financial and Insurance Risks

Academic journal article Journal of Risk and Insurance

A Class of Distortion Operators for Pricing Financial and Insurance Risks

Article excerpt


This article introduces a class of distortion operators, [g.sub.[alpha]] (u) = [phi][[[phi].sup.-1](u) + [alpha]], where [phi] is the standard normal cumulative distribution. For any loss (or asset) variable X with a probability distribution [S.sub.x](x) = 1-[F.sub.x](x), [g.sub.[alpha]][[S.sub.x](x)] defines a distorted probability distribution whose mean value yields a risk-adjusted premium (or an asset price). The distortion operator [g.sub.[alpha]] can be applied to both assets and liabilities, with opposite signs in the parameter [alpha]. Based on CAPM, the author establishes that the parameter [alpha] should correspond to the systematic risk of X. For a normal ([micro],[[sigma].sup.2]) distribution, the distorted distribution is also normal with [micro]' = [micro] + [alpha][sigma] and [sigma]' = [sigma]. For a lognormal distribution, the distorted distribution is also lognormal. By applying the distortion operator to stock price distributions, the author recovers the risk-neutral valuation for options and in particular the Black-Scholes formula.


This study discusses the price of risk for both insurance and financial risks. The price of an insurance risk is also called risk-adjusted premium, excluding expenses. Numerous and diverse theories exist on the price of risk in the literatures of economics, finance, and actuarial science. The objective of this study is to take a unified approach and integrate economic, financial, and actuarial pricing theories.

There are two competing economic theories for the price of risk. The expected utility theory has dominated the financial and insurance economics for the past half century. Its influence in actuarial risk theory is evident (see Borch, 1961; Buhlmann, 1980; and Goovaerts et. al., 1984). Over the past decade, a dual theory of risk has been developed in the economic literature by Yaari (1987) and others. Based on Venter's (1991) observation on insurance layer prices, Wang (1995, 1996) proposed calculating insurance premium by transforming the decumulative distribution function, which turned out to coincide with Yaari's economic theory of risk.

The first major financial pricing theory is the capital asset pricing model (CAPM). Built on Harry Markowitz's portfolio theory, CAPM was developed by William Sharpe, John Lintner, Jan Mossin, and others. CAPM is a set of predictions concerning equilibrium expected returns on assets. It has greatly affected our perception of risk and our ways of thinking when making investment decisions. However, CAPM has serious drawbacks when applied to insurance pricing. The CAPM assumption that asset returns are normally distributed is no longer valid for insurance if loss distributions are skewed. Another difficulty with insurance CAPM is the estimation errors associated with the underwriting beta (see Cummins and Harrington, 1985).

Another centerpiece of the financial pricing paradigm is option-pricing theory. Over the past two decades, the financial field has witnessed tremendous growth of activities using options and other derivatives. The wide acceptance of the Black-Scholes formula contributed to this financial revolution. Some researchers noted the resemblance between an option and a stop-loss reinsurance cover, which called for an analogous approach to pricing insurance risks. Unfortunately, the Black-Scholes formula applies only to lognormal distributions, while actuaries work with a large array of distribution forms. Furthermore, there are significant differences between option pricing and actuarial pricing. Mildenhall (1999) provides an excellent discussion of the differences between these two approaches. Option-pricing methodology defines prices as the minimal cost of setting up a hedging portfolio, while actuarial pricing is based on the actuarial present value of costs and the law of large numbers. Using financial jargon, op tion pricing is done in a world of Q-measure, whereas actuarial pricing is done in a world of P-measure. …

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