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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 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.

 


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