steiner 11 tail risk attribution

8/9/2019 Steiner 11 Tail Risk Attribution

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Research Note

Andreas Steiner Consulting GmbH

August 2011

Tail Risk Attribution

Introduction

Tail risk refers to the shape of the left tail of the distribution of investment returns. Returndistributions are traditionally described in terms of their first for moments: mean return,

volatility, skewness and kurtosis. Attribution is a descriptive approach used in portfolioanalysis to explain a certain magnitude as the sum of contributions from portfolioconstituents as well as contributions from constituent attributes. In this research note, wepropose a tail risk attribution methodology which allows to explain portfolio modified value-at-risk in terms of contributions from assets as well as mean, volatility, skewness andkurtosis. The approach is free of any residuals. The attribution scheme can be summarizedgraphically as follows

Asset 1 Return Contr Asset 1 + Vola Contr Asset 1 + Skew Contr Asset 1 + Kurt Contr Asset 1 = Total Contr Asset 1

+ + + +Asset 2 Return Contr Asset 2 + Vola Contr Asset 2 + Skew Contr Asset 2 + Kurt Contr Asset 2 = Total Contr Asset 2

+ + + +Asset 3 Return Contr Asset 3 + Vola Contr Asset 3 + Skew Contr Asset 3 + Kurt Contr Asset 3 = Total Contr Asset 3

+ + + +Asset i Return Contr Asset i + Vola Contr Asset i + Skew Contr Asset i + Kurt Contr Asset i = Total Contr Asset i

+ + + +Asset n Return Contr Asset n + Vola Contr Asset n + Skew Contr Asset n + Kurt Contr Asset n = Total Contr Asset n

= = = =Portfolio Total Contr Re tur n + Total Contr Volatility + Total Contr Skewness + Total Contr Kurtosis = Grand Total

Modeling and measuring the sources of risk are elementary steps in investment riskmanagement. Many methods in portfolio analysis are implicitly or explicitly based on theassumption of the normal distribution. Our tail risk attribution approach is a simpletechnique to better understand the sources of non-normal risk components. The strength ofour approach is analytical traceability and familiarity because it builds on concepts in riskmeasurement which are already well established.

Contributions to Modified Value-At-Risk

Many portfolio risk measures have been proposed. The most famous (partially due to a lot

of negative press) is Value-At-Risk. VaR is a specific point in the left tail of a distribution. A

portfolio with a more negative VaR figure is understood to be more risky.

A normal distribution is completely defined by its first two moments. Therefore, normal VaR

is defined by mean, volatility and the quantile function qc, which is the inverse distribution

function of the standard normal distribution at a certain probability which is usually referred

to as the confidence level c
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cqNVaR

Normal VaR was developed in the 90s. Given the non-normal features of many real-world

financial time series, accounting for non-normalities very early became an important topic.

Zangaris 1996 paper A VAR methodology for portfolios that include options (RiskMetrics)

proposed a Cornish & Fisher expansion to correct for skewness and fat tails in the returns,which then become popular under the name of Modified Value-At-Risk

2332 52

36

13

24

11

6

1SqqKqqSqqMVaR cccccc

S refers to skewness and K to excess kurtosis. Note that rearranging terms in the above

expression yields an additive decomposition into four risk attributes

KqqSqqSqqMVaR cccccc 3

24

152

36

11

6

1 3232

ContrExcessKurtSkewContrVolaContreturnContrRMVaR

The risk attributes are contributions to modified value-at-risk

KqqKurtContr

SqqSqSkewContr

qVolaContr

eturnContrR

cc

ccc

c

324

1

5236

11

6

1

3

232

Excess kurtosis and skewness both depend on volatility. This is due to the trivial fact that a

distribution without dispersion cannot be skewed, nor can it have fat tails. A non-trivial

consequence is that looking at skewness and kurtosis figures alone (which is a rather

common practice is investment risk analysis), is not sufficient for assessing the impact of

non-normal risk attributes on portfolio MVaR.

Equipped with the above formulas, we can decompose portfolio MVaR into additive

contributions from the distributional characteristics mean, dispersion, skewness and

kurtosis. This is a nice result, but what we are really interested in is to explain the overall

portfolio result from the distributional characteristics of its constituents and their exposures,

i.e. asset weights.

It has been shown that Normal portfolio VaR is a so-called linear-homogenous function in

asset weights. From this, it follows that the Euler theorem can be applied to calculate asset

contributions

n

i iP

NVaRContrNVaR1


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The contribution of asset i to the normal portfolio VaR is equal to the asset weight times its

marginal contribution

i

Pii

w

NVaRwNVaRContr

The normal VaR of a portfolio can be expressed as

cP qwwwNVaR

The marginal contribution to normal VaR from asset i is

c

ii

i

P qww

w

w

NVaR

2

From the above formulas, we see that the marginal contribution can be decomposed

additively into a contribution from return and volatility. Therefore, it is also possible to

additively decompose the NVaR contribution

iii weturnContrNVaRR

c

iii q

ww

wwntrNVaRVolaCo

2

Interestingly, although less known, Modified VaR is also linear-homogenous.

Boudt/Peterson/Croux(2007) have shown that the marginal contribution of asset i to

modified portfolio VaR is

i

PPcc

i

Pcc

i

Pc

PccPccPci

i

P

i

P

w

SSqq

w

Kqq

w

Sqww

SqqKqqSqww

w

w

NVaR

w

MVaR

5218

13

24

11

6

1

...5272

13

48

11

12

12

...

332

2332

The partial derivatives of skewness and kurtosis are rather complex expressions based on

the co-skewness and co-kurtosis matrices. For details, see Boudt/Peterson/Croux(2007).

As before on portfolio level, it is possible to isolate the contributions from skewness andkurtosis by regrouping terms...

i

PcciPcc

iii

w

KqqwwwKqq

ww

wwntrMVaRKurtCo 3

24

13

48

12 33

i

PPcc

i

Pci

PccPci

ii

w

SSqq

w

Sqwww

SqqSqww

wwntrMVaRSkewCo

52

18

11

6

1

...5272

11

12

12

32

232


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The sum of contributions from skewness and kurtosis can be called contribution from non-

normality and shows the aggregate impact of higher moments on portfolio risk.

Therefore, we have derived an additive decomposition of portfolio MVaR into asset

contributions, which can be further decomposed in an additive fashion into contributions

from the risk attributes mean, volatility, skewness and kurtosis.

Note that Boudt/Peterson/Croux(2007) also show that an additive decomposition exists for

Modified Conditional VaR. This means that it is also possible to apply our attribution

framework to Conditional VaR.

Example Analysis

In the remainder of this note, we would like to illustrate the proposed approach for a

portfolio consisting of five assets with the following distributional characteristics1

Asset1 Asset2 Asset3 Asset4 Asset5 Portfolio

Average Return 0.7370% 0.9995% 0.8272% 0.5749% 0.5100% 0.7158%

Volatility 1.4270% 1.8799% 1.8845% 1.3136% 1.0804% 1.1678%

Skewness -1.3645 -3.5366 -2.6689 -1.1950 -4.9489 -2.7376

(Excess) Kurtosis 3.1913 22.1904 15.5489 5.4446 34.4455 14.6723

Assets Weights 25.0000% 20.0000% 10.0000% 30.0000% 15.0000% 100.0000%

The full tail risk attribution for this portfolio at 97.5% confidence level looks like this

Contributions Portfolio Asset1 Asset2 Asset3 Asset4 Asset5

Return 0.7158% 0.1843% 0.1999% 0.0827% 0.1725% 0.0765%

Volatility -2.2889% -0.5461% -0.6266% -0.3327% -0.6226% -0.1609%

Norma l VaR -1.5731% -0.3619% -0.4267% -0.2500% -0.4502% -0.0844%

Skewness -0.2356% -0.1807% 0.1569% 0.0094% -0.1504% -0.0707%

Kurtosis -1.1775% 0.0121% -0.7179% -0.2565% -0.2765% 0.0614%

Non-Normal -1.4131% -0.1687% -0.5611% -0.2472% -0.4269% -0.0093%

Mod if ied VaR -2.9862% -0.5305% -0.9878% -0.4971% -0.8771% -0.0937%

In order to make the results more interpretable, we have generated various waterfall charts

from the above results.

1

All moments shown/analyzed in this paper are population figures. Excel does not have built-in population functions forskewness and kurtosis. All values were calculated with ourAdvanced Portfolio Analytics Excel add-in.
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-2.9862%

-2.2889%

-0.2356%

-1.1775%

0.7158%

-3.50%

-3.00%

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%

0.50%

1.00%

Return Volatility Skewness Kurtosis

Contributions to Modified VaR By Risk Attributes

We see that modified portfolio VaR was mainly driven by volatility and kurtosis effects. The

rather substantial impact of kurtosis indicates that non-normal risk characteristics play an

important role for this portfolio.

In terms of asset contributions, we see that asset 2 and asset 4 were the main risk

drivers

-2.9862%

-0.5305%

-0.9878%

-0.4971%

-0.8771%

-0.0937%

-3.50%

-3.00%

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%

Asset1 Asset2 Asset3 Asset4 Asset5 ModifiedVaR

Contributions to Modified VaR By Assets


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-2.2889%

-0.5461%

-0.6266%

-0.3327%

-0.6226%

-0.1609%

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%

Asset1 Asset2 Asset3 Asset4 Asset5 Vola Contr

Decomposition of Volatility Contribution By Asset

The relative importance of asset 2 and asset 4 prevails in their contributions to volatility. A

slightly different picture emerges when analyzing the contributions to kurtosis: asset 2

clearly dominates the kurtosis effect. Note that asset 1 and asset 5 contribute positively to

kurtosis, i.e. lowers tail risk.

-1.1775%

-0.7179%

-0.2565%

-0.2765%

0.0614%

0.0121%

-1.40%

-1.20%

-1.00%

-0.80%

-0.60%

-0.40%

-0.20%

0.00%

0.20%

Asset1 Asset2 Asset3 Asset4 Asset5 Kurt Contr

Decomposition of Kurtosis Contribution By Asset


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Summary

We have presented a residual-free and additive tail risk attribution scheme for modified

value-at-risk, which helps understanding the impact of non-normal tail risk characteristics

on total portfolio risk.

The main advantage of the approach is analytical traceability. Of course, the approach is

subject to all known disadvantages of modified value-at-risk and value-at-risk in general.

Some of these disadvantages can be addressed by using modified conditional VaR as the

relevant portfolio risk measure instead of modified VaR.

Due to perfect linearity of the decomposition, it would be possible to extend the approach

and analyze risk differences between a portfolio and a benchmark (active risk).

All calculations were performed with our Advanced Portfolio Analytics Library (see

http://www.andreassteiner.net/apalibfor more information).

Literature

Zangari, 1996: A VAR methodology for portfolios that include options, RiskMetrics Monitor

First Quarter, pages 412

Boudt/Peterson/Croux, 2007: Estimation and Decomposition of Downside Risk for

Portfolios with Non-Normal Returns, Working Paper, Facultiy of Economics and

Management, K.U. Leuven
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