Non-Normal Distributions

In the previous example, we generated samples conditional on a set of parameters given by μ and V assuming that these default probabilities were distributed multivariate normal. Next, we use this basic setup again in the form of a Gaussian copula. But, for now, let's revisit the experiment once again and focus on the last step, which involves NORMDIST. Recall that this step produced a cumulative probability under the normal cumulative density that we used in our subsequent analysis on credit defaults. Let's change the object of interest to portfolio loss, specifically the probability that returns would be less than or equal to some critical loss value. The underlying assumption again is that returns are multivariate normal. What if, instead, we felt that they were distributed non-normally?

Consider, for example, the possibility that the returns have fat tails implying the likelihood of more extreme returns compared to the normal density. This is illustrated conceptually further on for the case in which the normal density underestimates the likelihood of losses in general and extreme losses in particular. In this case, the normal density would underestimate both VaR and ETL if the true distribution of returns followed some type of extreme value distribution (in this case, one that is skewed left with a fatter tail).

Suppose we felt that distributions of returns were more consistent with a heavy-tailed distribution such as the t-distribution? What implications would this assumption have on our loss measures? Intuitively, our answer would hinge on understanding that heavier tails make losses observed under the normal distribution more likely under the alternative heavy-tail distribution. Therefore, we'd expect that there would be fewer observed returns falling below some critical percentile in the left tail. For example, NORMINV(.05,0,1) = –1.645, which states that the fifth percentile of a standardized normal distribution is located 1.645 standard deviations below the mean. Suppose instead, that, in truth, returns were distributed under the t-distribution with six degrees of freedom. This suggests TINV(.05,6) = –1.943 as the fifth percentile, which is further into the tail than would otherwise be under the normal density. The important conclusion here is that with heavier tails, heretofore extreme events are no longer so extreme and that the assumption of normality will underestimate risks. This was the case as the world entered the credit crisis in 2008; risk managers, who were locked into VaR based on the normal density, began to experience losses at frequencies that were supposed to be virtually impossible. This prompted the turn of phrase Black Swans.