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### Extreme Events – Specimen Question A.4.2(a) – Answer/Hints

Q. Identify the series corresponding to the principal components of Indices A and B

There are several different ways of identifying the series corresponding to the principal components of A and B.

The first point to note is that these series are only computable up to a mean drift term and up to a scalar multiple, although it is conventional to arrange for the series to have zero mean and for them to be scaled in magnitude according to some suitable convention.

Thus, it is easiest to work with the following adjusted data, which involve application of constant adjustments to the original data so that they now have zero means:

 Period Adjusted logged return A (, say) Adjusted logged return B (, say) 1 -0.2856959 0.0511585 2 0.0043199 -0.1719851 3 0.2200434 -0.1231949 4 0.0793551 -0.0280909 5 -0.0107785 -0.0952445 6 0.0083080 0.0013678 7 -0.0026977 -0.0777577 8 0.0132707 0.0299835 9 0.0737789 0.1254705 10 -0.1245130 0.0172569 11 0.0102960 -0.1085632 12 0.0112886 0.1130148 13 -0.0354204 0.1670349 14 0.0309356 0.0949475 15 -0.0406315 0.0821033 16 -0.0087522 -0.0627037 17 -0.0692503 -0.0788417 18 0.0662952 -0.0209003 19 0.0386876 0.0549608 20 0.0211603 0.0299835

The first principal component can be found by finding the  () which maximises the standard deviation of . The weights to give to A and B in constructing the principal component series are then  and  respectively. Subsequent principal components can be found by Gram-Schmidt orthogonalisation. This is possible to do in Microsoft Excel, e.g. using the Solver Add-in, but is rather convoluted. With this data the  (in radians) corresponding to the first principal component is -0.7136              and that for the second (i.e. in this case, other) principal component is 0.8572

Simpler is to use a standard statistics package that identifies the principal component weights directly from the underlying data or to use the corresponding web service function available via the Nematrian website, i.e. MnPrincipalComponentsWeights which using this data returns an array:

 Principal Component Multiplier to apply to Multiplier to apply to 1 0.756 -0.655 2 0.655 0.756

The Nematrian website also provides a web service function which calculates the principal component series directly, i.e. applies these weights to the underlying (adjusted) data series. This is MnPrincipalComponents.