Advances in Risk Management

STEPHEN JEWSON 161

be tempted to fit a distribution to the historical payoffs of the portfolio, and

use the fitted distribution to extrapolate the risk to higher levels (we have

occasionally called this extended burn analysis). However, such fitting and

extrapolation is more guesswork than science since we have no a priori idea

which distribution to fit. The payoff distribution is seldom close to normal,

even for very large portfolios, because of the non-linearity of many of the

contracts in a typical portfolio. The second shortcoming of burn analysis is

that estimates of the greeks of the portfolio, and other important diagnos-

tics such as the regression coefficients, or “betas”, of the portfolio, cannot be

calculated in an accurate way.

These shortcomings motivate the use of simulations, as we now describe.

8.4.2 Basic use of the multivariate normal

Both the shortcomings described above can be overcome by using simula-

tions, and the simplest example of using simulations is one particular use

of the multivariate normal distribution, as follows:^2

1–3 As for burn analysis.

4 A multivariate normal distribution is fitted to the detrended historical

settlement indices. The covariance matrix of the multivariate normal is

estimated using the empirical covariance matrix of these indices.

5 10,000 years of simulated settlement indices are created from this fitted

distribution.

6 The simulated settlement indices are converted into payoffs for each

contract and each simulated year.

7 The simulated payoffs are aggregated over the portfolio, giving a port-

folio simulated payoff for each simulated year.

8 The 10,000 portfolio simulated payoffs thus obtained are taken as

an empirical estimate for the distribution of possible payoffs of the

portfolio.

Riskcanthenbeestimatedatlevelsashighas1in10,000years, orevenhigher

levels if more years of simulations are used. Of course, these estimates are

only as good as the data and assumptions on which they are based, and these

assumptions become more dubious the further one progresses into the tail,

but at least this method gives ussomeway for estimating the probabilities

of extreme losses.

In the following sections we now discuss various extensions and alterna-

tives to the use of the multivariate normal distribution as described above.