A Class of Distortion Operators for Pricing Financial and Insurance Risks

This article introduces a class of distortion operators, gα (u) = Φ Φ [ ( ) ] − + 1 u α , where Φ is the standard normal cumulative distribution. For any loss (or asset) variable X with a probability distribution SX(x) = 1– FX(x), gα [SX(x)] defines a distorted probability distribution whose mean value yields a risk-adjusted premium (or an asset price). The distortion operator gα can be applied to both assets and liabilities, with opposite signs in the parameter α . Based on CAPM, the author establishes that the parameter α should correspond to the systematic risk of X. For a normal ( μ σ , 2 ) distribution, the distorted distribution is also normal with ′ = + ′ = μ μ ασ σ σ and . 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. INTRODUCTION 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; Bühlmann, 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). Shaun Wang is with SCOR Reinsurance Company, Itasca, Illinois. The author gratefully thanks Phelim Boyle, Stephen Mildenhall, Harry Panjer, Gary Venter, Julia Wirch, and Virginia Young for helpful comments. 16 THE JOURNAL OF RISK AND INSURANCE 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, option pricing is done in a world of Q-measure, whereas actuarial pricing is done in a world of P-measure. In an age in which financial and insurance risks are becoming more integrated, it is highly desirable to have a unified pricing theory. Many researchers, including Smith (1986), Cummins (1990, 1991), Embrechts (1996), and others, have expressed this viewpoint. Considerable efforts have been made by actuaries and financial economists to connect financial and insurance pricing theories (see D’Arcy and Doherty, 1988, and Gerber and Shiu, 1994). Although researchers are still trying to put together various pieces of pricing theory puzzles, an overall picture has not yet emerged. In the actuarial literature on the price of risk, the proportional hazards (PH) transform is gaining the attention of actuaries. The PH-transform, as a special member of the general class of Wang (1996), exhibits many desirable properties, especially in pricing insurance layers. However, the PH-transform fails to reproduce the BlackScholes formula for lognormal risks. Moreover, the PH-transform cannot be applied simultaneously to assets and liabilities. This article proposes a new distortion operator in the general class of Wang (1996). Unlike the PH-transform, this new distortion operator is equally applicable to assets and losses. For stop-loss reinsurance covers, this distortion operator resembles a riskneutral valuation of financial options. This distortion operator connects four different approaches: (i) the traditional actuarial standard deviation loading principle, (ii) Yaari’s economic theory of risk, (iii) CAPM, and (iv) option-pricing theory. The flow of this article is as follows: The “Distortion Operator and Insurance Pricing” section introduces the concept of a distortion operator within the context of insurance layer pricing. The “Choquet Pricing of Assets and Losses” section discusses the pricing of assets and losses using distortion operators. The next section introA CLASS OF DISTORTION OPERATORS FOR PRICING FINANCIAL AND INSURANCE RISKS 17 duces a new distortion operator and discusses its properties. In “The Implied α From Asset Prices” section, the author derives the implied distortion parameter from asset prices. In “The Parameter α and Systematic Risk” section, based on the capital asset pricing model, the author shows that the distortion parameter should correspond to the systematic risk. In the “Recovery of the Black-Scholes Formula” section, by applying the new distortion operator to stock price distributions, the author recovers a risk-neutral valuation of options, in particular the Black-Scholes formula. The next section discusses the fundamental difference between a distortion operator and a transformed distribution. The “Measure of Downside Risk and Tail Thickness” section discusses some related measures of downside risk and tail thickness. The following section discusses some practical issues in pricing insurance, and the final section gives two examples of pricing insurance using the new distortion operator. DISTORTION OPERATOR AND INSURANCE PRICING Let X be a non-negative loss random variable with cumulative distribution function FX(x) = P X x ≤ ( ) . The decumulative distribution function, denoted by SX(x) = 1 – FX(x), has a special role in calculating insurance premiums based on the fact that E X S y dy X [ ] ( ) = ∞ ∫0 . An insurance layer X(a, a+m] is defined by a payoff function X a a m X a X a a X a m m a m X ( , ] , , , + = ≤ < − ≤ < + + ≤      0, when , when , when 0 where a is the attachment point (also called deductible or retention) and m is the limit. The decumulative distribution function for the layer X(a, a + m] is related to that of the underlying risk X by the following equation: S y S a y y m m y X a a m X ( , ]( ) ( ) , . + = + ≤ < ≤    , when , when 0 0 The expected loss for the layer X(a,a+m] can be calculated by E X a a m S y dy S x dx X a a m X a a m [ ( , ]] ( ) ( ) . ( , ] + = = + ∞ + ∫ ∫ 0 For a very small layer X(a, a + ε ], the net premium (expected loss) is S a X ( ) ⋅ ε . This explains why SX is also called the “layer net premium density.” Lee (1988) gives a detailed account of SX in relation to expected layer loss cost. Venter (1991) showed that, for any given risk, market prices by layer always imply a transformed distribution. Inspired by Venter’s insightful observation, Wang (1996) suggested calculating premium by directly transforming the decumulative distribution function: 18 THE JOURNAL OF RISK AND INSURANCE Hg X X g S x dx [ ] [ ( )] . = ∞ ∫0 The function g: [0, 1]→ [0, 1] is an increasing function with g(0) = 0 and g(1) = 1. This study refers to g as a distortion operator. A distortion operator transforms a probability distribution SX to a new distribution g[SX]. The mean value under the distorted distribution, Hg[X], represents risk-adjusted premium, excluding acquisition or internal expenses. It is obvious that Hg X a a m X a a m X a a m g S y dy g S x dx [ ( , ]] [ ( )] [ ( )] . ( , ] + = = + + ∞ ∫ ∫0 For layer X(a,a+m] the risk-adjusted premium is the same whether (i) the layer X(a, a + m] is treated as a stand-alone risk and g is applied to its decumulative distribution function S y X a a m ( , ]( ) + or (ii) the ground-up loss distribution is transformed to g[SX(x)], from which we calculate the expected loss to the layer. As demonstrated in Wang (1996), the desirable distortion operator for pricing insurance layers should meet the following criteria: • 0 < g(u) < 1, g(0) = 0, and g(1) = 1. These conditions ensure that (i) for each value of x, g[SX(x)] defines a valid probability and (ii) non-zero probability events will still have (non-)zero probability after applying the distortion operator g. • g(u) is an increasing function (where it exists, ′ ≥ g u ( ) 0 ). This is to ensure that (i) the distorted probability g[SX(x)] defines another distribution and (ii) the riskadjusted layer premium decreases as the layer increases for fixed limit. • g(u) is concave (where it exists, ′′ ≥ g u ( ) 0 ). This is to ensure that (i) the risk load is non-negative for every risk or layer and (ii) the relative risk loading increases as the attachment point (retention) increases for a fixed limit. • ′ = +∞ g ( ) . 0 This is needed to ensure unbounded relative loading at extremely high layers. Unbounded relative loading at high reinsurance layers seems to be supported by observed market reinsurance premiums (see Venter, 1991). Butsic (1999) also showed that the loss beta is unlimited at very high layers. Wang (1996) considered a number of elementary one-parameter functions and concluded that only the power function g(u) = ur, ( 0 1 < ≤ r ) satisfied all these requirements. The power function corresponds to the PH-transform in Wang (1995). Wang, Young, and Panjer (1997) give a characterization of the PH-transform by an axiom regarding evaluation of compound Bernoulli risks. Although the PH-transform has some unique and desirable characteristics, researchers and practitioners have expressed some concerns, enumerated as follows: 1. The PH-transform of a lognormal distribution is no longer a lognormal distribution. To some this is a bit of a disappointment since it does not yield an analogy to the Black-Scholes formula for pricing financial options. (1) A CLASS OF DISTORTION OPERATORS FOR PRICING FINANCIAL AND INSURANCE RISKS 19 2. The PH-transform has a very simple functional form. However, this simplicity also comes with a limitation in terms of flexibility in its shape. To some insurance market price observers, the PH-transform sometimes yields a relative loading that increases too fast at high layers. 3. The PH-transform cannot be applied simultaneously to both assets and liabilities, as explained in the next section. CHOQUET PRICING OF ASSETS AND LOSSES With a broader perspective, we allow a loss variable X to be negative to include assets, and allow an asset variable A to be negative to include losses. For a consistent valuation, an asset A can be viewed as a negative loss X = –A and vice versa. For any variable X with decumulative distribution function SX(x), ( −∞ < < ∞ x ), the Choquet integral with respect to distortion operator g is defined by Hg X X X g S x dx g S x dx [ ] ( ) ( ) . = [ ] − { } + [ ] ∞