Ambiguies Maximum Drawdown
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. Return distributions are traditionally described in terms of their first for moments: mean return, volatility, skewness and kurtosis. Attribution is a descriptive approach used in portfolio analysis to explain a certain magnitude as the sum of contributions from portfolio constituents as well as contributions from constituent attributes. In this research note, we propose a tail risk attribution methodology which allows to explain portfolio modified valueat-risk in terms of contributions from assets as well as mean, volatility, skewness and kurtosis. The approach is free of any residuals. The attribution scheme can be summarized graphically as follows… Asset 1
Return Contr Asset 1
+
+ Asset 2
Return Contr Asset 2
Return Contr Asset 3
+
Return Contr Asset i
+
Return Contr Asset n
+
Total Contr Return
+
Skew Contr Asset 2
Vola Contr Asset 3
Vola Contr Asset i
+
Vola Contr Asset n
+
Skew Contr Asset 3
+
Skew Contr Asset i
Total Contr Volatility
+
Kurt Contr Asset 2
Skew Contr Asset n
+
Kurt Contr Asset 3
Total Contr Skew ness
=
Total Contr Asset 2
=
Total Contr Asset 3
=
Total Contr Asset i
=
Total Contr Asset n
=
Grand Total
+ +
Kurt Contr Asset i
+ +
Kurt Contr Asset n
= +
Total Contr Asset 1
+
+ +
=
+
+
= +
Kurt Contr Asset 1
+
+
= Portfolio
Vola Contr Asset 2
+
+
+
+ Asset n
Skew Contr Asset 1
+
+ Asset i
+
+
+ Asset 3
Vola Contr Asset 1
= +
Total Contr Kurtosis
Modeling and measuring the sources of risk are elementary steps in investment risk management. Many methods in portfolio analysis are implicitly or explicitly based on the assumption of the normal distribution. Our tail risk attribution approach is a simple technique to better understand the sources of non-normal risk components. The strength of our approach is analytical traceability and familiarity because it builds on concepts in risk measurement 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…
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
1/7
NVaR qc
Normal VaR was developed in the 90’s. Given the non-normal features of many real-world financial time series, accounting for non-normalities very early became an important topic. Zangari’s 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…
1 1 3 1 MVaR qc qc2 1 S qc 3 qc K 2 qc3 5 qc S 2 6 24 36
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…
1 1 1 3 MVaR qc qc2 1 S 2 qc3 5 qc S 2 qc 3 qc K 36 6 24
MVaR ReturnContr VolaContr SkewContr ExcessKurtContr
The risk attributes are contributions to modified value-at-risk ReturnContr VolaContr qc
1 1 SkewContr qc2 1 S 2 qc3 5 qc S 2 36 6
1 3 KurtContr qc 3 qc K 24
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
NVaRP NVaRContri i 1
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
2/7
The contribution of asset i to the normal portfolio VaR is equal to the asset weight times its marginal contribution… NVaRContri wi
NVaRP wi
The normal VaR of a portfolio can be expressed as… NVaRP w w w qc
The marginal contribution to normal VaR from asset i is… NVaRP 2 wi i qc wi w w
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… NVaRReturnContri wi i
NVaRVolaCo ntri wi
2 wi qc w w
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… MVaRP NVaRP ... wi wi
2 wi 1 1 1 qc2 1 S P qc3 3 qc K P 2 qc3 5 qc S P2 ... 48 72 w w 12
1 S 1 K P 1 S w w qc2 1 P qc3 3 qc 2 qc3 5 qc S P P wi 24 wi 18 wi 6
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 and kurtosis by regrouping terms... MVaRKurtCo ntri wi
MVaRSkewContri wi
1 2 wi 1 K P qc3 3 qc K P wi w w qc3 3 qc wi w w 48 24
2 wi 1 1 qc2 1 S P 2 qc3 5 qc S P2 ... 72 w w 12
1 S 1 S wi w w qc2 1 P 2 qc3 5 qc S P P wi 18 wi 6
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
3/7
The sum of contributions from skewness and kurtosis can be called “contribution from nonnormality” 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
25.0000%
20.0000%
10.0000%
30.0000%
15.0000%
100.0000%
Assets Weights
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%
Normal 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%
Modified 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 for skewness and kurtosis. All values were calculated with our Advanced Portfolio Analytics Excel add-in.
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
4/7
Contributions to Modified VaR By Risk Attributes 1.00% 0.50%
0.7158%
0.00% -0.50% -2.2889% -1.00% -1.50%
-2.9862% -0.2356%
-2.00% -1.1775%
-2.50% -3.00% -3.50% Return
Volatility
Skewness
Kurtosis
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…
Contributions to Modified VaR By Assets 0.00% -0.5305% -0.50% -1.00%
-0.9878%
-1.50%
-2.9862% -0.4971%
-2.00% -2.50%
-0.8771% -0.0937%
-3.00%
-3.50% Asset1
Asset2
Asset3
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
Asset4
Asset5
Modified VaR
5/7
Decomposition of Volatility Contribution By Asset 0.00% -0.5461% -0.50% -0.6266% -1.00% -2.2889% -0.3327% -1.50% -0.6226%
-2.00% -0.1609%
-2.50% Asset1
Asset2
Asset3
Asset4
Asset5
Vola Contr
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.
Decomposition of Kurtosis Contribution By Asset 0.20% 0.0121% 0.00% -0.20% -0.7179%
-0.40%
-1.1775%
-0.60% -0.80%
-0.2565%
-1.00%
-0.2765% -1.20%
0.0614%
-1.40% Asset1
Asset2
Asset3
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
Asset4
Asset5
Kurt Contr
6/7
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/apalib for more information).
Literature Zangari, 1996: “A VAR methodology for portfolios that include options”, RiskMetrics Monitor First Quarter, pages 4–12 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
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
7/7
Tail Risk Attribution Introduction Tail risk refers to the shape of the left tail of the distribution of investment returns. Return distributions are traditionally described in terms of their first for moments: mean return, volatility, skewness and kurtosis. Attribution is a descriptive approach used in portfolio analysis to explain a certain magnitude as the sum of contributions from portfolio constituents as well as contributions from constituent attributes. In this research note, we propose a tail risk attribution methodology which allows to explain portfolio modified valueat-risk in terms of contributions from assets as well as mean, volatility, skewness and kurtosis. The approach is free of any residuals. The attribution scheme can be summarized graphically as follows… Asset 1
Return Contr Asset 1
+
+ Asset 2
Return Contr Asset 2
Return Contr Asset 3
+
Return Contr Asset i
+
Return Contr Asset n
+
Total Contr Return
+
Skew Contr Asset 2
Vola Contr Asset 3
Vola Contr Asset i
+
Vola Contr Asset n
+
Skew Contr Asset 3
+
Skew Contr Asset i
Total Contr Volatility
+
Kurt Contr Asset 2
Skew Contr Asset n
+
Kurt Contr Asset 3
Total Contr Skew ness
=
Total Contr Asset 2
=
Total Contr Asset 3
=
Total Contr Asset i
=
Total Contr Asset n
=
Grand Total
+ +
Kurt Contr Asset i
+ +
Kurt Contr Asset n
= +
Total Contr Asset 1
+
+ +
=
+
+
= +
Kurt Contr Asset 1
+
+
= Portfolio
Vola Contr Asset 2
+
+
+
+ Asset n
Skew Contr Asset 1
+
+ Asset i
+
+
+ Asset 3
Vola Contr Asset 1
= +
Total Contr Kurtosis
Modeling and measuring the sources of risk are elementary steps in investment risk management. Many methods in portfolio analysis are implicitly or explicitly based on the assumption of the normal distribution. Our tail risk attribution approach is a simple technique to better understand the sources of non-normal risk components. The strength of our approach is analytical traceability and familiarity because it builds on concepts in risk measurement 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…
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
1/7
NVaR qc
Normal VaR was developed in the 90’s. Given the non-normal features of many real-world financial time series, accounting for non-normalities very early became an important topic. Zangari’s 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…
1 1 3 1 MVaR qc qc2 1 S qc 3 qc K 2 qc3 5 qc S 2 6 24 36
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…
1 1 1 3 MVaR qc qc2 1 S 2 qc3 5 qc S 2 qc 3 qc K 36 6 24
MVaR ReturnContr VolaContr SkewContr ExcessKurtContr
The risk attributes are contributions to modified value-at-risk ReturnContr VolaContr qc
1 1 SkewContr qc2 1 S 2 qc3 5 qc S 2 36 6
1 3 KurtContr qc 3 qc K 24
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
NVaRP NVaRContri i 1
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
2/7
The contribution of asset i to the normal portfolio VaR is equal to the asset weight times its marginal contribution… NVaRContri wi
NVaRP wi
The normal VaR of a portfolio can be expressed as… NVaRP w w w qc
The marginal contribution to normal VaR from asset i is… NVaRP 2 wi i qc wi w w
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… NVaRReturnContri wi i
NVaRVolaCo ntri wi
2 wi qc w w
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… MVaRP NVaRP ... wi wi
2 wi 1 1 1 qc2 1 S P qc3 3 qc K P 2 qc3 5 qc S P2 ... 48 72 w w 12
1 S 1 K P 1 S w w qc2 1 P qc3 3 qc 2 qc3 5 qc S P P wi 24 wi 18 wi 6
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 and kurtosis by regrouping terms... MVaRKurtCo ntri wi
MVaRSkewContri wi
1 2 wi 1 K P qc3 3 qc K P wi w w qc3 3 qc wi w w 48 24
2 wi 1 1 qc2 1 S P 2 qc3 5 qc S P2 ... 72 w w 12
1 S 1 S wi w w qc2 1 P 2 qc3 5 qc S P P wi 18 wi 6
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
3/7
The sum of contributions from skewness and kurtosis can be called “contribution from nonnormality” 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
25.0000%
20.0000%
10.0000%
30.0000%
15.0000%
100.0000%
Assets Weights
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%
Normal 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%
Modified 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 for skewness and kurtosis. All values were calculated with our Advanced Portfolio Analytics Excel add-in.
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
4/7
Contributions to Modified VaR By Risk Attributes 1.00% 0.50%
0.7158%
0.00% -0.50% -2.2889% -1.00% -1.50%
-2.9862% -0.2356%
-2.00% -1.1775%
-2.50% -3.00% -3.50% Return
Volatility
Skewness
Kurtosis
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…
Contributions to Modified VaR By Assets 0.00% -0.5305% -0.50% -1.00%
-0.9878%
-1.50%
-2.9862% -0.4971%
-2.00% -2.50%
-0.8771% -0.0937%
-3.00%
-3.50% Asset1
Asset2
Asset3
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
Asset4
Asset5
Modified VaR
5/7
Decomposition of Volatility Contribution By Asset 0.00% -0.5461% -0.50% -0.6266% -1.00% -2.2889% -0.3327% -1.50% -0.6226%
-2.00% -0.1609%
-2.50% Asset1
Asset2
Asset3
Asset4
Asset5
Vola Contr
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.
Decomposition of Kurtosis Contribution By Asset 0.20% 0.0121% 0.00% -0.20% -0.7179%
-0.40%
-1.1775%
-0.60% -0.80%
-0.2565%
-1.00%
-0.2765% -1.20%
0.0614%
-1.40% Asset1
Asset2
Asset3
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
Asset4
Asset5
Kurt Contr
6/7
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/apalib for more information).
Literature Zangari, 1996: “A VAR methodology for portfolios that include options”, RiskMetrics Monitor First Quarter, pages 4–12 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
© 2011, Andreas Steiner Consulting GmbH. All rights reserved.
7/7
