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Extreme Events – Specimen Question A.2.2(b) – Answer/Hints

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Q. Using a first order autoregressive model, de-smooth the observed returns for Index B to derive a return series that you think may provide a better measure of the underlying behaviour of the relevant asset category.

 

The approach for de-smoothing (or ‘de-correlating’) such returns suggested in the book Extreme Events involves assuming that there is some underlying ‘true’ return series, , and that the observed series, , derives from it via a first order autoregressive model, . We will assume that the autoregressive model actually applies to the logged returns, which are given in ExtremeEventsQuestionsAndAnswers2_1a.

 

If we assume that the corresponding  are independent, identically distributed (normal) random variables with standard deviation  (and mean 0 given that we have standardised the data) then the (expected) variance of the series , will be  and the (expected) covariance of the series  with the series  will be .

 

One way of proceeding would be to estimate  as the solution to the equation:

 

 

More precisely, since the above does not differentiate between  and  we would probably choose the  closest to zero (and, ideally, we would expect it to be positive and smaller than 0.5, which also requires  to be positive, as a value of  outside this range would be implausible).

 

In this instance:

 

Statistic (using logged returns)

Value

(Sample*) Variance of  (=)

0.008575828

(Sample*) Covariance of  with  (=)

0.002896449

Ratio (=)

0.337746

Estimated

0.279955

 

* Ideally, the Variance and Covariance should both include the same small sample size adjustment, i.e. both be “sample” or both be “population” estimates. The Microsoft Excel functions, VAR, VARP and COVAR, are somewhat confusing in this respect, since COVAR is calculated using a multiplier , and is therefore properly a “population” statistic and consistent with VARP, whilst VAR is calculated using a multiplier of , and so is a “sample” statistic. The Nematrian website’s web functions are clearer as it provides both a MnCovariance function and a MnPopulationCovariance function.

 

Ignoring small sample size adjustments, estimates of  can be then be found using:

 

 

However, if we do this in practice we find that there are second order effects that mean that estimating  as above still leaves some residual autocorrelation:

 

Period

 (logged returns of B)

 (first pass)

1

0.0487902

0.0487902

2

-0.1743534

-0.2611119

3

-0.1255632

-0.0728617

4

-0.0304592

-0.0139731

5

-0.0976128

-0.1301320

6

-0.0010005

0.0492060

7

-0.0801260

-0.1304104

8

0.0276152

0.0890558

9

0.1231022

0.1363395

10

0.0148886

-0.0323317

11

-0.1109316

-0.1414913

12

0.1106465

0.2086780

13

0.1646666

0.1475549

14

0.0925792

0.0712046

15

0.0797350

0.0830516

16

-0.0650720

-0.1226626

17

-0.0812101

-0.0650933

18

-0.0232686

-0.0070071

19

0.0525925

0.0757649

20

0.0276152

0.0088945

 

 

 

Variance (=)

0.0085758

0.0138041

Covariance (=)

0.0028964

0.0007054

Ratio (=)

0.3377457

0.0511003

Estimated

0.2799546

 

 

Better, therefore, is to use a root search algorithm in which we explicitly search for the  (ideally between 0 and 0.5) for which  has zero autocorrelation. This can be done using the Nematrian MnDesmooth_AR1 or MnDesmooth_AR1_rho functions (the former returns the desmoothed series, the latter returns the value of  for which  has zero autocorrelation). Given the form of the problem given here, these provide the following de-smoothed series (with rho equal to 0.31094682):

 

Period

 (logged returns of B)

 (de-smoothed)

1

0.0487902

0.0487902

2

-0.1743534

-0.2750507

3

-0.1255632

-0.05810445

4

-0.0304592

-0.01798381

5

-0.0976128

-0.13354672

6

-0.0010005

0.05881321

7

-0.0801260

-0.14282465

8

0.0276152

0.10452905

9

0.1231022

0.13148365

10

0.0148886

-0.03772687

11

-0.1109316

-0.14396646

12

0.1106465

0.22554488

13

0.1646666

0.13719425

14

0.0925792

0.07244591

15

0.0797350

0.08302433

16

-0.0650720

-0.13190296

17

-0.0812101

-0.05833410

18

-0.0232686

-0.00744471

19

0.0525925

0.07968530

20

0.0276152

0.00411769

 

 

 

Variance (=)

0.0085758

0.01487681

Covariance (=)

0.0028964

0

Ratio (=)

0.3377457

0

 

Note:

 

(a)          In general de-smoothing increases the variance of the return series being analysed. Here it has gone from 0.0926 to 0.1219.

 

(b)          Whilst the problem would usually be stated as shown, it perhaps makes more sense not to make the arbitrary implicit assumption that the smoothing is around a mean of zero, but around some mean (that is, for example, estimated from the data), i.e. as if the model was .

 

(c)           We might not necessarily want to give equal weight to each observation. This is possible using the Nematrian MnWeightedDesmooth_AR1 and MnWeightedDesmooth_AR1_rho functions.

 


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