The Mathematics of Financial Modelingand Investment Management

23-RiskManagement Page 747 Wednesday, February 4, 2004 1:13 PM

Risk Management 747

than in investment banking. Many asset management firms consider the

occurrence of losses due to operational risk to be irrelevant.

RISK MEASURES

Risk is embodied in a probability distribution of returns or of possible

losses. From a management point of view it is interesting to collapse this

probability distribution in a single number. The problem of measuring

risk with a single number has received much attention, even in contexts

other than finance.

Historically, the first measure of the risk contained in a distribution

is its variance, or the standard deviation, which is the square root of the

variance. The variance of a distribution gives an indication as to

whether the distribution is concentrated around some value or spread

over a large interval of values. If the standard deviation of a distribution

is high, it means that there is a high probability that the variable might

take values significantly different from its mean. A high standard devia-

tion, therefore, corresponds to a high risk. In the terminology of risk

management, standard deviation represents unexpected loss (UL).

Because risk is uncertainty (lack of information), the question of the

information conveyed by a probability distribution has led to the con-

cept of information and to Information Theory. In the case of finite

probabilities, information (I) in the sense of Information Theory is

defined as the average of the logarithms of probabilities (pi):

N

I = ∑ pilog pi

i = 1

Information reaches its maximum when the probability is concen-

trated in only one outcome, that is, pi = 1 for i = k, pi = 0 for i ≠ k. In

this case information is zero as the information of an outcome with

probability zero is conventionally set to zero. Information reaches its

minimum when all probabilities are equal, that is, when there is maxi-

mum uncertainty on the future outcome. In this case information is neg-

ative: I = –N log N. There is no lower bound to information.

Information with a minus sign is well known in statistical physics as

entropy, which is a measure of disorder: E = –I. The information associated

with an equi-probable binary scheme, that is, the information associated

with the choice between two equally probable possible outcome, is called