Extreme Events – Specimen Question
A.4.2(c) – Answer/Hints
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Q. More generally, can you
identify two series where the linear combination of the series with the least
variance is also the one with the maximum kurtosis? Hint: try identifying two
series with very few terms in them as it simplifies the relevant mathematics.
Two simple series that have this property are:
Period

A

B

1

0.25

1

2

0.25

1

3

0.25

1

4

0.25

1

5

1

0




Mean

0

0

Standard deviation

0.56

1

(Excess) kurtosis

5

3

These series both have zero mean, and are chosen so that the
main contributor to kurtosis in series A is an observation which is zero in
Series B. We find that the linear combination with the maximum kurtosis is the
same as the linear combination with the minimum standard deviation, i.e. 100%
series A. This would remain true even if we multiplied B by anything greater
than approximately 0.6.
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