1.7.6 Example: Australian Equities (Quadratic Transformation)
For a third approach to calculating value-at-risk for our Australian equities portfolio, assume that 
is conditionally joint-normal with conditional mean vector 1|0μ = 
and covariance matrix 1|0Σ obtained previously. Our original portfolio mapping 
defines 
as a quadratic polynomial of a conditionally joint-normal random vector . As we will discuss in Section 3.13, any real-valued quadratic polynomial of a joint-normal random vector can be expressed as a linear polynomial of independent normal and chi-squared random variables. In this case, the expression takes the form
[1.50]
where X1 and X2 are independent chi-squared random variables, each with 1 degree of freedom and respective non-centrality parameters 674.2 and 14,195.5 This is not an approximation. The representation is exact.
There are various ways to extract a quantile of portfolio loss from a representation such as [1.50]. Two approaches that we shall discuss in Section 3.17 are:
- approximate the desired quantile using the Cornish-Fisher (1937) expansion,
- invert the characteristic function of using numerical integration,
Applying the first approach to our Australian equities portfolio yields an approximate 1-day 95% GBPvalue-at-risk of GBP 5854.