Value-at-Risk Theory and Practice
The first advanced book on value-at-risk
Chapter 3, Page 151 Exercise 37
Consider random vector X ~ (, ) with
[3.181]
Let
[3.182]
where
[3.183]
Express Y as a linear polynomial of independent chi-squared and normal random variables.
Solution
Construct the Cholesky matrix z of :
[s1]
and a matrix u with rows equal to orthonormal eigenvectors of :
[s2]
Define the change of variables for X:
[s3]
and obtain
[s4]
[s5]
Multiplying [s4] out:
[s8]
"Complete the squares" for terms involving and to obtain
[s9]
This expresses Y as a linear polynomial of three independent random variables:
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(
,
)
with
:
for X:
and
to obtain
