Advances in Risk Management

STEPHEN JEWSON 163

much simpler estimate such as a correlation matrix based on independence.
A non-parametric approach to implementing such shrinkage for weather
data, based on the Quenouille–Tukey jackknife, has been described in Jew-
son (2005), while a parametric approach for implementing shrinkage for
financial data correlation matrices has been described in Ledoit and Wolf
(2003). As far as the author is aware there has been no attempt to compare
the two approaches.

8.7 DEALING WITH NON-NORMALITY

One of the most obvious assumptions in the BMVN method is that the
joint distribution of the weather indices underlying the weather derivative
portfolio is multivariate normal. The assumption of multivariate normality
consists of the assumption that the marginals are normal and the assump-
tion that the copula is a Gaussian copula. With respect to the first of these
assumptions, empirical tests have shown that normality is a good model
for the marginals for seasonal temperature contracts, but is often not satis-
factory for monthly temperature contracts. It is certainly not a good model
for counting indices (such as an index which counts the number of freezing
days) when the counts are small.
There is a simple, and standard, method for incorporating non-normal
marginals into multivariate simulations. This method was apparently first
described by Iman and Conover (1982), and has recently been popularized
by Wang (1998). The method works as follows:

1 Marginal distributions are fitted to each weather index.

2 Using the CDFs for these marginal distributions, and the inverse of

the CDF of the normal distribution, the historical weather indices are

transformed to come from a normal distribution.^3

3 The transformed values are simulated using the multivariate normal

distribution.

4 The simulated values are transformed back to the original marginal

distributions, using the reverse of step 2.

The simulated values produced by this method do not have the samelinear
correlations as the original historical values, but it is usually argued that
linear correlation is not a good measure of dependence anyway when con-
sidering non-normal distributions. Instead, the method yields simulated
values with the correctrankcorrelations.
This method is a simple example of the use of copulas. In this case,
the copula being used is the normal copula, since the historical values are
transformed to a multivariate normal for the simulation step. Alternatively,