# What is a Coherent Risk Measure and What are its Properties?

Table of Contents:

Short answer

A risk measure is coherent if it satisfies certain simple, mathematical properties. One of these properties, which some popular measures do not possess is sub-additivity, that adding together two risky portfolios cannot increase the measure of risk.

Example

Artzner et al. (1997) give a simple example of traditional VaR which violates this, and illustrates perfectly the problems of measures that are not coherent. Portfolio X consists only of a far out-of-the-money put with one day to expiry. Portfolio Y consists only of a far out-of-the-money call with one day to expiry. Let us suppose that each option has a probability of 4% of ending up in the money. For each option individually, at the 95% confidence level the one-day traditional VaR is effectively zero. Now put the two portfolios together and there is a 92% chance of not losing anything, 100% less two lots of 4%. So at the 95% confidence level there will be a significant VaR. Putting the two portfolios together has in this example increased the risk. 'A merger does not create extra risk' (Artzner et al. 1997).

Long answer

A common criticism of traditional VaR has been that it does not satisfy all of certain commonsense criteria. Artzner et al. (1997) defined the following set of sensible criteria that a measure of risk, p(X)where X is a set of outcomes, should satisfy. These are as follows:

1. Sub-additivity: p(X + Y) < p(X) + p(Y). This just says that if you add two portfolios together the total risk can't get any worse than adding the two risks separately. Indeed,

there may be cancellation effects or economies of scale that will make the risk better.

2. Monotonicity: If X < Y for each scenario then p(X) > p(Y). If one portfolio has better values than another under all scenarios then its risk will be better.

3. Positive homogeneity: For all X > 0, p(XX) = Xp(X). Double your portfolio then you double your risk.

4. Translation invariance: For all constant c,

p(X + c) = p(X) — c. Think of just adding cash to a portfolio; this would come off your risk.

A risk measure that satisfies all of these is called coherent. The traditional, simple VaR measure is not coherent since it does not satisfy the sub-additivity condition. Sub-additivity is an obvious requirement for a risk measure, otherwise there would be no risk benefit to adding uncorrelated new trades into a book. If you have two portfolios X and Y then this benefit can be defined as

Sub-additivity says that this can only be non-negative.

Lack of sub-additivity in a risk measure and that can be exploited can lead to a form of regulatory arbitrage. All a bank has to do is create subsidiary firms, in a reverse form of the above example, to save regulatory capital.

With a coherent measure of risk, specifically because of its sub-additivity, one can simply add together risks of individual portfolios to get a conservative estimate of the total risk.

## Coherent measures

A popular measure that is coherent is Expected Shortfall. This is calculated as the average of all the P&Ls making up the tail percentile of interest. Suppose we are working with the 5% percentile, rather than quoting this number (this would be traditional VaR) instead calculate the average of all the P&Ls in this 5% tail.

## Attribution

Having calculated a coherent measure of risk, one often wants to attribute this to smaller units. For example, a desk has calculated its risk and wants to see how much each trader is responsible for. Similarly, one may want to break down the risk into contributions from each of the greeks in a derivatives portfolio. How much risk is associated with direction of the market, and how much is associated with volatility exposure, for example.

References and Further Reading

Acerbi, C & Tasche, D On the coherence of expected shortfall. www-m1.mathematik.tu-muenchen.de/m4/Papers/Tasche/shortfall.pdf

Artzner, P, Delbaen, F, Eber, J-M & Heath, D 1997 Thinking coherently. Risk magazine 10 (11) 68-72