steiner 11 tail risk attribution
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8/9/2019 Steiner 11 Tail Risk Attribution
1/7Electronic copy available at: http://ssrn.com/abstract=1908148
2011, Andreas Steiner Consulting GmbH. All rights reserved. 1 / 7
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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8/9/2019 Steiner 11 Tail Risk Attribution
2/7Electronic copy available at: http://ssrn.com/abstract=1908148
2011, Andreas Steiner Consulting GmbH. All rights reserved. 2 / 7
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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2011, Andreas Steiner Consulting GmbH. All rights reserved. 7 / 7
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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