# Risk Management in Turbulent Times

By Gilles Bénéplanc; Jean-Charles Rochet | Go to book overview

9
Risk Management in a
Normal World
Risk managers and financial modelers often assume that distributions of risks are jointly normal, even when normality is rejected by the data! There are two main reasons for this:
 • The return on any portfolio of risks that have a (jointly) normal distribution also has a normal distribution, whose mean and variance can be computed by simple formulas: linear for the mean, quadratic for the variance. • The distribution of any (scalar) normal random variable is completely determined by its mean μ and its variance σ2. In fact, it is equivalent to a simple transformation μ + σỹ of a normal random variable with mean 0 and variance 1 (such a random variable ỹ is called a standard normal).

In particular, the expected utility of a (scalar) normal random variable of mean μ and variance σ2 only depends1 on μ and σ:

This chapter explores the beauties of the Normal world: the mean–variance criterion can be used safely [Section (9.1)], portfolio choice is easy [Section (9.2)], the diversification principle works well [Section (9.3)] and portfolio efficiency can be measured by the Sharpe ratio [Section (9.4)]. More importantly, in a Normal world, risk premiums are easy to compute even when markets are incomplete. They are given by the Capital Asset Pricing Model (CAPM), which is presented in Section 9.5. Normality assumptions also imply that hedging decisions on futures markets obey simple rules [Section (9.6)] as well as the allocation of economic capital between several divisions of a bank [Section (9.7)]. Unfortunately, the real world is not Normal. This is why the chapter concludes by showing the dangers of persisting to view the world as Normal, in spite of contrary empirical evidence [Section (9.8)].

9.1 THE MEAN-VARIANCE CRITERION

When risks are (jointly) normal, the trade-off between risk and return on a portfolio of assets can entirely be captured by two numbers: the expectation (or mean)

of the gross return of the portfolio and its variance Indeed the expected utility of the investor equals2

-118-

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