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Setting Capital Requirements With Coherent Measures of Risk- Part 2

by Glenn G. Meyers

In the August edition of The Actuarial Review, I began a description of the paper, "Coherent Measures of Risk" by Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath.1 In this article, I will complete my description and provide a link between these measures and risk-adjusted probability measures.

Let's begin with a quick review of our definitions. Let X be a random variable representing insured losses. Let
r (X) be a measure of risk. r is a coherent measure of risk if it satisfies the following axioms:

1. SubadditivityFor all random losses X and Y,

r (X + Y) < r (X) + r (Y)

2. Monotonicity—For all random losses X and Y, if X < Y for all scenarios, then

r (X) < r (Y)

3. Positive Homogeneity—For all l > 0 and random losses X,

r (lX) = lr (X)

4. Translation Invariance—For all random losses X and constant loss amounts ,

r (X +a) = r (X) + a

Let X take its values over a finite set of scenarios. In the last article, we identified r (X) = Maximum(X) as a coherent measure of risk.

In most insurance situations, the maximum loss leads to capital requirements that are too conservative, that is to say, expensive. So other measures of risk may be appropriate. Artzner et al go on to prove that, "a measure of risk is coherent if and only if it can be expressed as the supremum of the expected losses taken over a class of probability measures on our finite set of scenarios." A simple example of a class of probability measures is one that assigns the probability of 1/n to each element of each subset of n scenarios. The supremum of the expected losses over this class of probability measures is the average of the n largest losses.

This example leads us to the Tail Value at Risk, TVaRa, which is equal to the average of the top 1 - a percent of the losses.

A problem with the Tail Value at Risk is that it reacts only to very large losses. So recently I began to look for other coherent measures of risk that respond to the full range of losses. Noting that Artzner's representation of coherent measures of risk allowed for risk-adjusted probabilities, I looked at a formula for transformed probabilities proposed by Shaun Wang:

W(x) = F(F-1(F(x))-l).

W(x) represents the cumulative distribution function for the transformed probability measure, F(x) is the cumulative distribution function for the objective probability measure, and F (x) is the cumulative distribution function for the standard normal distribution. l is a free parameter representing risk aversion. Wang has used this transform to establish links between traditional actuarial pricing methodologies and financial pricing methodologies such as the Black-Scholes option pricing formula and the Capital Asset Pricing Model.

It turns out that if you calculate expected values with the risk-adjusted probabilities generated by the Wang Transform, you get another coherent measure of risk. This is not an obvious statement. To calculate a measure of risk with the Wang Transform, you first arrange the possible values of X in increasing order, calculate the cumulative probabilities, and then calculate the transform using the above formula. It takes some effort to prove that this gives the same result as taking the supremum of expected values over a class of probability measures, as characterized by Artzner. In fact, by replacing F(x) with other cumulative distribution functions in the Wang Transform formula, I found examples where the resulting measure of risk is not subadditive.

When I discussed this with Wang, he referred me to a paper that he wrote jointly with Virginia Young and Harry Panjer2 that proposes a set of axioms that are satisfied if and only if a measure of risk, rg(X), can be represented as the expected value of a risk-adjusted probability measure. That is:

rg(X) = xi (g(F(xi)) - g(F(xi-1)))

where g is a nondecreasing function with g(0) = 0 and g(1) = 1. If, in addition, g is concave up, then rg(X) satisfies all of the axioms that define a coherent measure of risk. As an example, the Wang Transform uses

g(u)= F(F-1(u)-l).

If g(u) = Max(0, u-a)/(1-a), then rg(X) = TVaRa(X).

We say that two risks, X and Y, are comonotone if (Xi-Xj)(Yi -Yj) > 0 for all scenarios i and j. The Wang/Young/Panjer axioms replace the subadditivity axiom with an axiom that requires
r (X+Y) = r (X) + r (Y) for comonotone X and Y.

Table 1 provides a sample calculation of rg(X) for the Wang Transform with l = 2. I will leave it as an exercise to the reader to verify that TVaR85%(X) = 4.33 and
TVaR90%(X) = 4.50.

Table 1

xi

Pi

F(Xi)

W(xi)

W(xi)-W(xi-1)

1

0.50

0.50

0.0228

0.0228

2

0.20

0.70

0.0700

0.0473

3

0.15

0.85

0.1676

0.0976

4

0.10

0.95

0.3612

0.1936

5

0.05

1.00

1.0000

0.6388

E[X]=

2.00

 

rg(X)=

4.3784

Since any measure of risk written in the form of rg(X) above is coherent, we now have a good supply of coherent measures or risk which are comonotone additive. Are all coherent measures of risk comonotone additive? The answer is no. Table 2 gives an example of a coherent measure of risk that is not comonotone additive.

Table 2 consists of three scenarios. The measure of risk is a maximum of the expected values over two probability measures.

Table 2

Scenario

X

Y

X+Y

p1

p2

1

1.0

0.0

1.0

0.4

0.3

2

2.0

0.0

2.0

0.3

0.6

3

2.0

1.0

3.0

0.3

0.1

E1

1.6

0.3

1.9

 

 

E2

1.7

0.1

1.8

 

 

r

1.7

0.3

1.9

 

 

If we accept the proposition that an insurer's required assets should depend upon its distribution of losses, I believe that coherent measures of risk have desirable properties for the formula used to establish the required assets. Because of these fine papers, we have a very useful representation of these measures of risk.

1 Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath, "Coherent Measures of Risk," Math. Finance 9 (1999), no. 3, 203-228 www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf

2 Shaun S. Wang, Virginia R. Young, and Harry H. Panjer, "Axiomatic Characterization of Insurance Prices," Insurance Mathematics and Economics 21 (1997) 173-182.