48 Frequently Asked Questions In Quantitative Finance
What is a Coherent Risk Measure and
What are its Properties?
Short Answer
A risk measure is coherent if it satisfies certain simple,
mathematical properties. One of these properties,
which some popular measures donotpossess 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
Xconsists only of a far out-of-the-money put with one
day to expiry. PortfolioYconsists 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% con-
fidence 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,ρ(X)whereXis a set of
outcomes, should satisfy. These are as follows.
1.Sub-additivity:ρ(X+Y)≤ρ(X)+ρ(Y). This just says
that if you add two portfolios together the total risk
