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DISTORTION RISK MEASURES AND ECONOMIC CAPITAL

By Werner Hürlimann, Switzerland

Abstract.

To provide incentive for active risk management, it is argued that a sound coherent distortion risk measure should preserve some higher degree stop-loss orders, at least the degree three convex order. Such risk measures are called free of tail risk or simply tail-free risk measures. It is shown that under some common axioms and other plausible conditions, a tail-free coherent distortion risk measure identifies necessarily with the Wang right-tail measure or the expected value measure. This main result is applied to derive an optimal economic capital formula.

Key words : coherent risk measure, distortion risk measure, tail-free risk measure, higher degree convex order, Bernoulli distribution, Pareto distribution, economic capital

  1. Introduction.

The axiomatic approach to risk measures is an important and very active subject, which applies to different topics of actuarial and financial interest like premium calculation and capital requirements. Besides the coherent risk measures by Arztner et al.(1997/99), one is interested in the distortion risk measures by Denneberg(1990/94), Wang(1995/96) and Wang et al.(1997). Under certain circumstances, distortion risk measures are coherent risk measures (e.g. Wang et al.(1997), Theorem 3). For this reason, they can be used to determine the capital requirements of a risky business, as suggested by several authors including Wirch and Hardy(1999), Goovaerts et al.(2002) and Wang(2002).

However, despite of being coherent, a lot of distortion risk measures, like conditional value-at-risk (identical to expected shortfall) or the very recent Wang transform risk measure, do not always provide incentive for risk management because they lack of giving a capital relief in some simple two scenarios situations of reduced risk (Examples 3.1 and 3.2). To prevent the existence of such pathological counterexamples, one is interested in distortion risk measures that preserve the higher degree stop-loss orders (e.g. Hürlimann(2000b), Yoshiba and Yamai(2001)). More precisely, it is known that a coherent distortion risk measure preserves the usual stochastic order and the usual stop-loss order (e.g. Hürlimann(1998a)).This is a desirable property because increased risk should be penalized with an increased risk measure. With equal means and variances, a stop-loss order relation between different random variables cannot exist. In this situation, increased risk can be modeled by the degree three stop-loss order or more specifically, by equal mean and variance, the degree three convex order. Thus, one is interested in distortion risk measures that preserve this higher degree convex orders. Such measures are called free of tail risk or simply tail-free distortion risk measures. The present contribution shows that under some plausible conditions, a non-trivial degree two tail-free coherent distortion risk measure must necessarily coincide with the Wang right-tail risk measure proposed by Wang(1998). A more detailed account of the paper follows.

Section 2 recalls the notions of coherent risk measure and distortion risk measure and identifies the distortion risk measures that induce coherent risk measures. Section 3 defines the notion of a higher degree tail-free risk measure by requiring the preservation property under a higher degree convex order. It is shown through counterexamples that the conditional value-at-risk or expected shortfall and the new Wang transform risk measure are not degree two tail-free coherent risk measures. In Proposition 4.1, we identify a necessary and sufficient condition for a coherent distortion risk measure to be a degree two tail-free risk measure on the subset of biatomic losses. Then, in Proposition 4.2, we show that the coherent proportional hazard or PH-distortion risk measure , is degree two tail-free if, and only if, one has or . In Section 5, it is shown that the special case , that is the Wang right-tail measure, remains finite and degree two tail-free on the subset of all positive affine transformed Pareto losses with finite means and variances. These simple results are the essential ingredients of our main Theorem 6.3. Suppose the set of losses contain all Bernoulli losses, all Pareto losses, and the positive affine transforms thereof. If the risk measure remains finite for losses with finite means and variances, and if it satisfies some plausible axioms, then the risk measure identifies necessarily with the Wang right-tail risk measure or the trivial expected value measure. This main result is applied to the determination of an optimal economic capital formula in Section 7. As a practical illustration, we recover the quite old “-rule” for normally distributed risks.

  1. Coherent distortion risk measures.

Let be a probability space such that is the space of outcomes or states of the world, A is the -algebra of events and P is the probability measure. For a measurable real-valued random variable X on this probability space, that is a map , the probability distribution of X is defined and denoted by .

In the present paper, the random variable X represents a financial loss such that for the real number is the realization of a loss and profit function with for a loss and for a profit. A set of financial losses is denoted by . A risk measure is a functional from the set of losses to the extended non-negative real numbers described by a map . A coherent risk measure is a risk measure, which satisfies the following desirable properties (e.g. Arztner et al.(1997/99)) :

(M)(monotonicity) If are ordered in stochastic dominance of first order, that is for all x, written , then

(P)(positive homogeneity) If is a positive constant and then

(S)(subadditivity) If then

(T)(translation invariance) If c is a constant and then

Definitions 2.1. A continuous increasing function such that and is called distortion function. The dual transform of a distortion function is called dual distortion function. For with probability distribution , the transform defines a distribution function, which is called distorted distribution function. The dual distortion function defines a transformed distribution function , which is called dual distorted distribution function.

Taking the mean value with respect to the distorted distribution of a loss with probability distribution , one obtains the distortion (risk) measure

.(2.1)

Similarly, the dual distorted distribution defines the dual distortion (risk) measure

(2.2)

One notes that the dual transform implies the following alternative dual representations of the distortion measures (2.1) and (2.2) in terms of the distorted survival function and the dual distorted survival function associated to the survival function :

(2.3)

(2.4)

Wang et al.(1997), Theorem 3, implies that the risk measures (2.3) and (2.4) are coherent risk measures provided that is a concave ( is a convex) function. This implies that (2.1) and (2.2) are coherent provided that is a convex ( is a concave) function. For completeness, let us also mention a further duality between losses and gains, the latter being defined as negative losses. With this result, it suffices to study risk measures of either losses or gains.

Lemma 2.1. Let be a loss random variable, a distortion function, and the dual distortion function. Then one has the relationships

.(2.5)

Proof. Using that and making the substitution one obtains

The second relation in (2.5) is shown similarly. 

  1. Tail-free distortion risk measures.

Besides monotonicity, that is preservation of stochastic dominance of first order, it is known that a distortion measure with concave distortion function preserves the stop-loss order or increasing convex order (e.g. Hürlimann(1998a)).This is a desirable property because increased risk should be penalized with an increased measure. With equal means and variances, a stop-loss order relation between different random variables cannot exist. In this situation, increased risk can be modeled by the degree three stop-loss order or equivalently, by equal mean and variance, the degree three convex order. Thus, one is interested in distortion measures, which preserve this higher degree convex orders. As suggested by Yoshiba and Yamai(2001), such measures should be called free of tail risk or simply tail-free distortion measures. Some more formal definitions and properties are required.

For any real random variable X with distribution , consider the higher order partial moments , called degree n stop-loss transforms. For n=0 the convention is made that coincides with the indicator function , hence is simply the survival function of X. For n=1 this is the usual stop-loss transform , written without upper index. It is not difficult to establish the recursion (e.g. Hürlimann(2000a), Theorem 2.1)

.(3.1)

It will be useful to consider the following variants of the higher degree stop-loss orders (see Kaas et al.(1994), Hürlimann(2000a) among others).

Definitions 3.1. For n=0,1,2,..., a random variable X precedes Y in degree n stop-loss transform order, written , if for all x one has . A random variable X precedes Y in degree n stop-loss order, written , if and the moment inequalities are satisfied. With equal moments for some , the relation is written . In particular, the one extreme case defines a general degree n stop-loss order and the other one defines the so-called (n+1)-convex order recently studied by Denuit et al.(1998). Note that the special case is identical with the usual stochastic order or stochastic dominance of first order, also denoted . For n=1 the stochastic order coincides with the usual stop-loss order or equivalently increasing convex order .

For fixed n, the above stop-loss order variants satisfy the following hierarchical relationship

.(3.2)

Moreover, the higher degree stop-loss orders build a hierarchical class of partial orders (Kaas et al.(1994), Theorem 2.2), that is one has

.(3.3)

Definition 3.2. A risk measure is called a degree n tail-free risk measure if it is preserved under the (n+1)-convex order, that is if satisfy then .

As mentioned above, it is known that a distortion measure with concave distortion function preserves for , and is thus a tail-free risk measure of degree zero and one. In the present paper, we are interested in specific concave distortion functions such that is a degree two tail-free risk measure. For motivation, it is very important to emphasize the practical relevance of tail-free distortion measures. Our field of application is here risk management.

Example 3.1 : Conditional value-at-risk versus Wang right-tail measure

Consider the coherent distortion measure (2.3) defined by the increasing concave distortion function , where is a small probability of loss, say . By definition, the measure associated to is denoted . It is known that this risk measure coincides with several other known risk measures like the conditional value-at-risk measure and the expected shortfall measure (e.g. Hürlimann(2001a)). In now standard notation, conditional value-at-risk at the confidence level , written , coincides with . For comparison, consider the distortion function . The coherent distortion measure (2.3), called Wang right-tail measure and denoted by , has been proposed by Wang(1998) as a measure of right-tail risk. For illustration, let now Y be a loss consisting of two scenarios with loss amounts 20$, 2100$ such that . Through active risk management, assume that the lower amount can be eliminated and that the higher loss amount can be reduced to 1700$. By equal mean and variance, this results in a loss X such that . Suppose a risk manager is weighing the cost of risk management against the benefit of capital relief. Then CVaR does not promote risk management because , which shows that there is a capital penalty instead of a capital relief for either removing or reducing the initial loss amounts. However, the Wang right-tail measure offers a capital relief because . Since Y is evidently a higher loss than X, the CVaR measure fails to recognize this feature. Even more, in this simple example X precedes Y in the degree three convex order. This shows through a meaningful counterexample that CVaR is not a degree two tail-free coherent risk measure.

Being aware of the fact that CVaR ignores useful information in a large part of the loss distribution, Wang(2002) has proposed a new coherent distortion measure, which should adjust more properly extreme low frequency and high severity losses. However, as the following counterexample shows, Wang’s most recent proposal does not generate a degree two tail-free coherent risk measure.

Example 3.2 : Wang transform measure versus Wang right-tail measure

Consider the distortion function , where is the standard normal distribution and is a small probability of loss, say . This interesting choice finds further motivation in Wang(2000/2001) and defines the Wang transform measure, where . Similarly to Example 3.1, consider a biatomic loss Y such that . Let X be a biatomic loss with the same mean and variance such that . Obviously Y is a higher loss than X, but the Wang measure does not provide incentive for risk management because . However, the Wang right-tail measure offers a capital relief because . Since X precedes Y in the degree three convex order, the Wang transform measure is not a degree two tail-free coherent risk measure.

  1. Tail-free distortion risk measures for biatomic losses.

To prevent counterexamples of the type presented in the Examples 3.1 and 3.2, we derive a condition on the distortion function, which guarantees that the measure defined in (2.3) is a degree two tail-free distortion measure when restricted to the subset of biatomic losses with equal mean and variance. For this one must show that the distortion measure is preserved under the degree three convex order.

Let denote the subset of all real-valued standard biatomic random variables with mean zero and variance one. Recall that an element is uniquely determined by its support with , and probabilities . It is convenient to identify X with its support and use the short-hand notation (e.g. Hürlimann(1998b), Theorem I.5.1).

Lemma 4.1. Let and belong to . Then one has if, and only if, one has .

Proof. If then one has necessarily , hence . Conversely, if the latter inequalities hold, one has and the difference in distributions has two sign changes in the order . The affirmation follows from Denuit et al.(1998), Theorem 4.3. 

An arbitrary biatomic loss with mean and standard deviation is a positive affine transform with . The distortion measure (2.3) satisfies the relationship with

.(4.1)

Applying Lemma 4.1 it follows that the distortion measure is tail-free of degree two for the subset of biatomic losses if, and only if, the function of one variable in (4.1) is monotone increasing.

Proposition 4.1. Let be a continuous and differentiable increasing concave distortion function. The coherent distortion risk measure is a degree two tail-free risk measure for the subset of biatomic losses if, and only if, the following condition holds :

.(4.2)

Proof. The function is monotone increasing if, and only if, one has . Making the change of variables this condition identifies with (4.2). 

Among the many concave distortion functions known in the literature, only a few turn out to generate degree two tail-free coherent risk measures for the subset of biatomic losses. For illustration, we present two attractive possibilities.

Example 4.1 : Proportional hazard or PH-distortion risk measure

The PH-transform , is due to Wang(1995) and has been justified on an axiomatic basis in Wang et al.(1997). If one has . Since the condition (4.2) is satisfied. Therefore, for , the PH-distortion measure is a degree two tail-free coherent risk measure for the subset of biatomic losses.

Example 4.2 : Lookback distortion risk measure

The lookback transform , is an interesting alternative to the PH-transform. For the lookback transform has been motivated in Hürlimann(1998c). We show that for the lookback distortion risk measure is a degree two tail-free coherent risk measure for the subset of biatomic losses. Denote by the left-hand side in condition (4.2). Using that , a calculation shows the inequality

.

It suffices to show that the curly bracket is non-negative. Setting this is equivalent to the inequality

.

Using the series expansion , one obtains that

which shows the desired inequality.

The following precise characterization will be crucial in Section 6.

Proposition 4.2. Let , be the PH-transform. Then the PH-distortion risk measure is degree two tail-free for the subset of biatomic losses if, and only if, one has or .

Proof. With the result of Example 4.1, it remains to prove that the PH-distortion measure is not degree two tail-free in case . Let be biatomic losses in such that . Using (4.1) we show that there exists such that

.(4.3)

With the substitution it suffices to show that there exists such that

.(4.4)

Clearly, for one has and . Since is continuous on there exists such that . 

Remarks 4.1.

The lookback distortion risk measure is degree two tail-free for biatomic losses if, and only if, one has (Hürlimann(2002), Proposition 5.2). Though the Wang transform in Example 3.2 is not degree two tail-free, numerical calculation suggests that the modified Wang transform