Extreme Events – Specimen Question
A.4.2(b) – Answer/Hints
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Q. Given a linear combination
of Index A and B what is the maximum possible kurtosis of a linear combination
of A and B?
The linear combination of the form 
with the
highest kurtosis can be found by finding the 
(
)
which maximises the kurtosis of this combination. As kurtosis is scale
invariant, this will also provide the maximum possible kurtosis. As A
has a much higher kurtosis than B (4.62 vs -0.84) we expect that the
resulting linear combination will be more biased towards A than was the one in
(a) that maximised standard deviation and hence characterised the first
principal component.
As in (a) we can find the solution using Microsoft Excel,
e.g. using the Solver Add-in. It has = -0.088,
i.e., as expected, close to A. The kurtosis of the resulting combination is 4.71.
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