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Setting Capital Requirements With Coherent Measures of Risk- Part 1
by Glenn G. Meyers

Insurers need capital to pay claims when premium revenues fall short. We actuaries have long sought a formula that determines this capital directly from the insurer's aggregate loss distribution. As Bob Butsic1 pointed out at our recent spring meeting, the derivation of such a formula is not an obvious process. We have to balance the cost of an insolvency with the cost of holding capital. Should we find such a formula, we could use it to quantify the effects of the cost of capital on a variety of pricing and reinsurance strategies.

The paper "Coherent Measures of Risk" by Artzner, et al.2 provides an axiomatic treatment of this problem. The CAS Committee on Dynamic Financial Analysis plans to add a chapter on this subject into the DFA Handbook. This is the first of two articles that summarize the main ideas in that paper.

Let X be a random variable representing an insurer's total loss. Let r (X) be a measure of risk that represents the assets that the insurer should have on hand to pay all losses for which it is liable. The insurer may account for a portion of its assets as a liability to cover what it expects to pay, but in some instances more money will be needed. The money set aside for this contingency is what we call capital. Let's now review some properties we want r (X) to have. Consider the following set of scenarios and the risk measure r (X) = Maximum(X) applied to the five loss scenarios.

Table 1

Scenario

X1

X2

X1+X2

2X1

X1+1

1

1

5

6

2

2

2

2

1

3

4

3

3

4

2

6

8

5

4

5

4

9

10

6

5

3

3

6

6

4

r (X)

5

5

9

10

6

Artzner, et al., begin by stating a set of axioms that define "coherent measures of risk":

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

You can see by inspection that the measure r (X) = Maximum(X) satisfies these axioms, and thus is a coherent measure of risk.

Let's discuss the meaning of these axioms. The subadditivity axiom captures the meaning of diversification. When two insurers merge, they do not need to increase their total assets. In fact, if the merger is effective, they can reduce their total assets. The monotonicity axiom means that if Insurer A always has losses that are less than Insurer B, it will need less total assets. The positive homogeneity axiom means that if an insurer buys a l percent quota share reinsurance contract on its entire book of business, it can reduce its assets by l percent.

An advantage to having a good axiomatic system to measure risk is that it frees us from any worry of making inconsistent decisions on managing risk.

Now if we use r (X) = Maximum(X) for most real insurance situations, we would find ourselves paying dearly for maintaining the necessary capital. Artzner, et al., provide us with a less conservative coherent measure of risk, called the Tail Value at Risk. This is calculated by the formula:

TVaRa(X) = Average of the Top (1 – a)% of Losses

 Table 2 gives = TVaRa(X) for a = 40% and 60% with the scenarios of Table 1.

Table 2

Scenario

X1

X2

X1+X2

2X1

X1+1

TVaR40%

4.0

4.0

7.0

8.0

5.0

TVaR60%

4.5

4.5

7.5

9.0

5.5

Now there are other measures of risk that we actuaries often use that are not coherent. One of these measures is the probability of ruin, a.k.a. the Value at Risk (VaR). Consider, for example, two insurers with the following loss scenarios.

Table 3

Scenario

Probability

Insurer A

Insurer B

Insurer A and B

1

0.9850

0

0

0

2

0.0075

100

0

100

3

0.0075

0

100

100

Let's suppose that we measure risk by setting r (X) equal to the 99th percentile of loss. For Insurers A and B,
r (XA) + r (XB) = 0, but r (XA + XB) = 100. This violates the subadditivity axiom, and shows that VaR is not a coherent measure of risk.

A second measure that is commonly used but is not coherent is given by setting r (X) equal to the expected value of X plus a constant, T, times the standard deviation of X.

Consider two insurers with the following loss scenarios.

Table 4

Scenario

Probability

Insurer A

Insurer B

1

0.5

0

65

2

0.5

100

115

If we set T = 2, we have r (XA) = 150 and r (XB) = 140. But since XA < XB for every scenario, this measure violates the monotonicity axiom and is not a coherent measure of risk.

So far, I have identified only Maximum(X) and TVaR(X) as coherent measures of risk. These measures are sensitive mainly to extreme events. You may want a coherent measure of risk that responds to the full range of losses. There are such measures. It turns out that there is a good way to describe all coherent measures of risk, and I will discuss this in the next article.

1 Robert Butsic, "Allocating the Cost of Capital," CAS Spring Meeting, May 19-22, 2002. www.casact.org/education/spring/2002/handouts/butsic1.ppt

2 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