23-RiskManagement Page 748 Wednesday, February 4, 2004 1:13 PM
748 The Mathematics of Financial Modeling and Investment Managementbit. As information is additive, it represents the number of bits necessary to
characterize a choice.
This definition of information can be extended to a continuous
probability distribution. However, in the continuous case, information
looses its meaning.^5 For this and for other reasons, information cannot
be used effectively as a measure of risk.^6
When JP Morgan released its RiskMetrics model in 1994, it proposed
a measure of risk called Value at Risk (VaR).^7 Defined as a confidence
interval, VaR is the maximum loss that can be incurred with a given prob-
ability. Suppose we choose a confidence level of 95%. We say that a port-
folio has a given VaR, say $1 million, if there is a 95% probability that
losses above $1 million will not be incurred. This does not mean that the
given portfolio cannot lose more than $1 million, it means only that
losses above $1 million will happen with a probability of 5%. If we trans-
late probabilities into relative frequencies, this means, in turn, that losses
above $1 million will happen approximately 5 times every 100. If we
measure VaR daily this means 5 days out of 100 days.
As a measure of risk, VaR has many drawbacks. It does not specify
the amount of losses exceeding VaR. Different distributions might have
the same VaR but totally different distributions of extreme values. For
instance, in the above example of a VaR of $1 million at 95%, 5 times
every 100 a portfolio might lose just above $1 million or a much larger
amount. Perhaps the most serious drawback of VaR is the fact that it is
not subadditive. The VaR of aggregated portfolios might be larger than
the sum of individual VaRs. This is unreasonable as one expects risk to
decrease in aggregate due to diversification and anticorrelations. Despite
these drawbacks, and despite the fact that confidence intervals are ulti-
mately a rather complex probabilistic concept, VaR has become
extremely popular as a risk measure.
In 1998 Artzner, Delbaen, Eber, and Heath^8 published an important
paper where they defined the conditions for risk measures to be coher-(^5) This fact is well known in statistical physics where the entropy associated with a
continuous scheme is somewhat arbitrary.
(^6) The pioneering work of Arnold Zellner has started a new strain of econometric lit-
erature based on Information Theory. See Arnold Zellner, “Bayesian Method of Mo-
ments (BMOM) Analysis of Mean and Regression Models,” in J.C. Lee, W.D.
Johnson, and A. Zellner (eds.), Prediction and Modeling Honoring Seymour Geisser
(New York: Springer, 1994), pp. 61–74.
(^7) Note that RiskMetrics and VaR are not related. The concept of VaR can be applied
to any probability distribution of return and not only to RiskMetrics.
(^8) Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath, “Coherent
Measures of Risk,” Mathematical Finance 9 (1999), pp. 203–228.