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S H A U N S. W A N G


This paper presents a universal framework for pricing financial and insurance

risks. Examples are given for pricing contingent payoffs, where the underlying

asset or liability can be either traded or not traded. The paper also outlines an

application of the framework to prescribe capital allocations within insurance companies, and to determine fair values of insurance liabilities.


Currently there is a pressing need for a universal framework for the determination of the fair value of financial and insurance risks. In the insurance industry, this need is evident in the Society of Actuaries' "Symposium on Fair Value of Liabilities", and in the Casualty Actuarial Society's "Risk Premium Project" and "Task Force on Fair Valuing P/C Insurance Liabilities".

In the financial services industry, this pressing need is evidenced by the recent Basel Accords on regulatory risk management that require fair value, analogous to market prices, to be applied to all assets or liabilities, whether traded or not, on or off the balance sheet. In light of all these current events, this paper addresses a very timely subject.

The paper is comprised of three parts, summarized as follows:

Part One: The Framework introduces a new transform and correlation measure that extends CAPM to pricing all kinds of assets and liabilities, having any type of probability distribution, whether traded or underwritten, in finance or insurance. This transform is just as easily applied to contingent payoffs that are co-monotone with their underlying assets or liabilities.

In its simplest form, the new transform relies on a parameter called the "market price of risk", extending a familiar concept in C A P M to risks with non-normal distributions. The "market price of risk" can either be applied to, 1 SCOR, One Pierce Place, Itasca, IL 60143, E-mail: swang@scor.com ASTIN BULLETIN,Vol. 32, No. 2, 2002, pp. 213-234 214 SHAUNS. WANG or implied from, a distribution, in order to arrive at a "risk-adjusted price" for the underlying risk in question. The "market price of risk" increases continuously with duration, and is consistent at each horizon date between an underlying and its co-monotone contingent payoff.

When the return for an underlying asset has a normal distribution, the new transform replicates the C A P M price for that underlying asset, and recovers the Black-Scholes price for options on that underlying asset.

Part Two: Examples of Pricing Contingent Payoffs illustrates the application of the new framework to pricing call options on traded stocks, and to pricing weather derivatives.

Part Three: Capital Allocation & Fair Values of Liabilities illustrates the application of the new framework to insurance company capital allocations, and to the determination of fair values of insurance liabilities. In particular, it addresses a challenging issue concerning the long-term duration of liabilities.

Also, the framework is equally applicable to primary insurance business and excess-of-loss reinsurance when calculating fair values of liabilities.

–  –  –

Capital Asset Pricing Model CAPM is a set of predictions concerning equilibrium expected returns on assets.

Classic C A P M assumes that all investors have the same one-period horizon, and asset returns have multivariate normal distributions. For a fixed time horizon, let R~ and RM be the rate-of-return for asset i and the market portfolio M, respectively. Classic C A P M asserts that

–  –  –

In asset portfolio management, this is also called the Sharpe Ratio, after William Sharpe.


In terms of market price of risk, CAPM can be restated as follows:

–  –  –

where Pi, M is the linear correlation coefficient between Ri and R M. In other words, the market price of risk for asset i is directly proportional to the correlation coefficient between asset i and the market portfolio M.

CAPM provides powerful insight regarding the risk-return relationship, where only systematic risk deserves an extra risk premium in an efficient market.

However, CAPM and the concept of "market price of risk" were developed under the assumption of multivariate normal distributions for asset returns.

CAPM has serious limitations when applied to insurance pricing when loss distributions are not normally distributed. In the absence of an active market for insurance liabilities, the underwriting beta by line of business has been difficult to estimate.

Option Pricing Theory

Besides CAPM, another major financial pricing paradigm is modern option pricing theory, first developed by Fischer Black and Myron Scholes (1973).

Some actuarial researchers have noted that the payoff functions of a European call option and a stop-loss reinsurance contract are similar, and have proposed an "option-pricing" approach to pricing insurance risks. Unfortunately, the Black-Scholes formula only applies to lognormal distributions of market returns, whereas actuaries work with a large array of distributional forms.

Furthermore, there are subtle differences between option pricing and actuarial pricing (see Mildenhall, 2000). One way to better appreciate the difference between "financial asset pricing" and "insurance pricing", is to recognize the difference in types of data available for pricing.

Options pricing is performed in a world of Q-measure (using risk-adjusted probabilities), where the available data consists of observed market prices for related financial assets. On the other hand, actuarial pricing is conducted in a world of P-measure (using objective probabilities), where the available data consists of projected losses, whose amounts and likelihood need to be converted to a "fair value" price (see Panjer et al, 1998).

Because of this difference, the price of an option is determined from the minimal cost of setting up a hedging portfolio, whereas the price of insurance is based on the actuarial present value of costs, plus an additional risk premium for correlation risk, parameter uncertainty and cost of capital.

A Universal Pricing Method

–  –  –

where (1)is the standard normal cumulativedistribution. The key parameter2 is called the marketprice of risk, reflectingthe level of systematicrisk. The transform (i) is now better known as the Wang,transform among financial engineers and risk managers. The Wang transform was partly inspired by the work of severalprominentactuarial researchers,includingGaryVenter(1991,

1998) and Robert Butsic (1999).

For a givenasset X with cdf F(x), the Wang transformwillproducea "riskadjusted" cdf F*(x). The meanvalueunderF*(x), denotedby E*[X],willdefine a risk-adjusted "fair value" of X at time T, whichcan be further discountedto time zero, using the risk-freeinterest rate.

The Wang transformis fairlyeasyto numericallycompute.Many software packages have both (1) and (1) as built-in functions. In MicrosoftExcel, (1)(y)

-I can be evaluatedby NORMSDIST(y) and (1)-1(z)can be evaluatedby NORMSINV(z).

One fortunate property of the Wang transform is that normal and lognormal distributions are preserved:

• If F has a N o r m a l ( g, o 2) distribution, F* is also a normal distribution with I.t* = ~ - ~,o and o* = o.

• If F has a lognormal(g, o 2) distribution such that In(X) - N o r m a l (la, o2), F* is another lognormal distribution with g* = ~t- ~o and o* = o.

Stock prices are often modeled by lognormal distributions, which implies that stock returns are modeled by normal distributions. Equivalent results can be obtained by applying the Wang transform either to the stock price distribution, or, to the stock return distribution.

Consider an asset i on a one-period time horizon. Assume that the return R~ for asset i has a normal distribution with a standard deviation of oi. Applying

the Wang t r a n s f o r m to the distribution of Ri we get a risk-adjusted rate-ofreturn:

E * [/~.] : E[Ri]-2a i.

In a competitive market, the risk-adjusted return for all assets should be equal to the risk-free rate, r. Therefore we can infer that L = (E[R~] -r)loi, which is exactly the same as the market price of risk in classic C A P M. With ~ being the market price of risk for an asset, the Wang transform replicates the classic C A P M.

Unified Treatment of Assets & Liabilities A liability with loss variable X can be viewed as a negative asset with gain Y=

-X, and vice versa. Mathematically, if a liability has a market price of risk ~,


when treated as a negative asset, the market price of risk will be -2. That is, the market price of risk will have the same value but opposite signs, depending upon whether a risk vehicle is treated as an asset or liability. For a liability with loss variable X, the Wang transform in equation (1) has an equivalent representation.

S * (x) : * [ * - l (S(x)) + 2], (2)

where S(x) = 1 - F(x).

The following operations are equivalent:

1. Applying transform (1) with )~ to the cdf F(x) of a gain variable X,

2. Applying transform (1) with -)~ to the cdf F(y) of the loss variable Y= -X, and

3. Applying transform (2) with )~ to S(y) = 1 - F ( y ) of the loss variable Y= -X.

Their equivalence ensures that the same price is obtained for both sides of a risk transaction.

If a loss variable has a Normal(~t,a 2) distribution, the Wang transform (2) will produce another normal distribution with it* = ~ - )~a and ~* = ~. Thus, for a loss variable with a normal distribution, the Wang transform (2) recovers the traditional standard-deviation loading principle, with the parameter being the constant multiplier.

A New Measures of Correlation

According to CAPM, the market price of risk )~ should reflect the correlation of an asset with the overall market portfolio. When we generalize the concept of market price of risk to assets and liabilities with non-normal distributions, the Pearson linear correlation coefficient becomes an inadequate measure of correlation. Examples can be constructed such that a deterministic relationship has a Pearson correlation coefficient close to zero. Such an example was

provided in Wang (1998):

Consider the case where X ~ lognormal(0,1) and Y= (X) °. Despite this deterministic relationship, the linear correlation coefficient between X and Y approaches zero as ~ increases to infinity. That is, Px, r -~ 0 as ~y -~ oo.

This also implies that correlation should not be estimated by running linear regression, unless all of the variables have normal distributions.

Now we show a new way to extend the Pearson correlation coefficient to variables with non-normal distributions. For any pair of variables {X, Y} with

distributions Fx and Fr, we transform them into "standard normal variables":

–  –  –

Now, let us reconsider the case where X - lognormal(0,1) and Y= (X)% Consistent with this deterministic relationship, this new measure of correlation between X and Y is always 1. That is, p~ r = 1 for all ¢~ values.

Using this new measure of correlati3n we may extend classic CAPM as


~'i = Pi, M" 2M, where 2s and 2M are the respective market prices of risk in the Wang transform, without assuming normality.

Pricing of Contingent Payoffs

For an underlying risk X a n d a function h, we say that Y= h(X) is a derivative (or contingent payoff) of X, since the payoff of Y is a function of the outcome of X. If the function h is monotone, we say that Y is a co-monotone derivative of X. For example, a European call option is a co-monotone derivative of the underlying asset; in (re)insurance, an excess layer is a co-monotone derivative of the ground-up risk.

Theoretically, the underlying risk X and its co-monotone derivative Y should have the same market price of risk, ~., simply because they have the same correlation (as shown by using our new measure of correlation) with the market portfolio.

In pricing a contingent payoff Y= h(X), there are two ways of applying the Wang transform.

• Method I: Apply the Wang transform to the distribution Fx of the underlying risk X. Then derive a risk-adjusted distribution F y from F x using Y* = h(X*).

• Method II: First derive its own distribution Fr for Y=h(X). Then apply the Wang transform to Fr directly, using the same X as in Method I.

Mathematically it can be shown that these two methods are equivalent. This important result validates using the Wang transform for risk-neutral valuations of contingent payoffs.

Implied Z and the Effect of Duration For a traded asset, the market price of risk ~, can be estimated from observed market data. We shall now take a closer look at the implied market price of risk and how it varies with the time horizon under consideration.


–  –  –

where d Wi is a random variable drawn from a normal distribution with mean equal to zero and variance equal to dt. In equation (4), gi is the expected rate of return for the asset, and oi is the volatility of the asset return. Let Xi(0) be the current asset price at time zero. For any future time T, the prospective stock price Xi (7) as defined in equation (4) has a lognormal distribution (see

Hull, 1997, p. 229):

–  –  –

For any fixed future time T, a "no arbitrage" condition (or simply, the market value concept) implies that the risk-adjusted future asset price, when discounted by the risk-free rate, must equal the current market price. In this continuoustime model, the risk-free rate r needs to be compounded continuously.

As a result, we have an implied parameter value:

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