Extreme Events – Specimen Question
A.4.2(a) – Answer/Hints
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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 GramSchmidt orthogonalisation.
This is possible to do in Microsoft Excel, e.g. using the Solver Addin, 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.
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