correlation and dependence. It explains that correlation is designed to measure one partic-
ular type of dependence, namely linear dependence. The chapter explains that correlation
can be monitored in the same way as volatility. Either the exponentially weighted moving
average or GARCH (1,1) procedure can be used. It also explains the positive-semidefinite
condition for a consistent variance-covariance matrix.
Copulas are explained in this chapter because the Gaussian copula model of defaults is
necessary for an understanding of Basel II in Chapter 15. As preparation for the material
on copulas, the chapter discusses the properties of multivariate normal distributions and
explains factor models. The latter provide a convenient way of defining the correlation
matrix for a large number of variables. The capital asset pricing model which is likely to
be somewhat familiar to a lot of students provides an example of a one-factor model.
The notation used by mathematicians for copulas is too abstract for many students
of risk management. As will be evident from Chapter 11 and the slides that go with it, I
prefer to use a minimal amount of algebra. I go through the copula material fairly slowly.
(I find the copula material, once understood, is considered to be so straightforward that
students cannot understand why they had any difficulty with it in the first place!)
I start by asking students to suppose they know the marginal (unconditional) probabil-
ity distributions of two variables and want to define a correlation structure. If the variables
are normally distributed it is natural to assume that they are bivariate normal (although
this is by no means the only possibility). In other cases we can map the distributions on
a “percentile-to-percentile” basis to standard normal distributions and assume that these
are bivariate normal. Thus the correlation structure between the original distributions is
defined indirectly using other easier-to-deal-with distributions.
This procedure is illustrated in the text for two variables that have triangular dis-
tributions. (I choose triangular distributions because it is easy to calculate cumulative
probabilities for them.) The marginal distributions of the variables are shown in Figure
11.2. The transformation of the first variable to a standard normal distribution is shown in
Table 11.1 and that for the second variable is shown in Table 11.2. The joint distribution
of the variables when the correlation between the normal distributions is 0.5 is shown in
Table 11.3. The overall methodology is shown diagrammatically in Figure 11.3.
Figures 11.2 and 11.3 together with Tables 11.1, 11.2, and 11.3 provide an example of
the Gaussian copula model. Other copula models can be understood similarly. For example
we can transform two variables to Student t-distributions on a percentile-to-percentile basis
and assume the transformed variables have a bivariate Student t-distribution. As shown
in Figures 11.4 and 11.5 the Student t-copula has more tail dependence than the Gaussian
copula.
When many variables are involved and the Gaussian copula model is used, each vari-
able can be transformed into a normal distribution on a percentile-to-percentile basis. A
factor model can then be used to define the correlations between the normal distributions.
Gordy’s result has been moved to Chapter 15 in the fourth edition. The proof of
Vasicek’s result is now in a separate section from a statement of the result which I think
makes material flow better.
Problems 11.16 to 11.21 all make good hand-in assignment questions.
Chapter 12: Value at Risk and Expected Shortfall
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