Alternative Risk Measures for Alternative Investments
A. Chabaane
Y. Malevergne
BNP Paribas ACA Consulting
ISFA Actuarial School Lyon
JP. Laurent
F. Turpin
ISFA Actuarial School Lyon
BNP Paribas
BNP Paribas
email email ::
[email protected] [email protected]
http://laurent.jeanpaul.free.fr/
Evry April 2004 1
Outline Optimizing under VaR constraints z
Estimation techniques
z
VaR analytics and efficient portfolios comparison
Optimizing under alternative risk constraints z
Expected Shortfall, Downside Risk measure,…
z
Risk measures analytics and efficient portfolios comparison
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Data set 16 individual Hedge Funds Fund Style AXA Rosenberg Equity Market Neutral Discovery MasterFund Ltd Equity Market Neutral Aetos Corp Event Driven Bennett Restructuring Event Driven Calamos Convertible Convertible Arbitrage Sage Capital Convertible Arbitrage Genesis Emerging Markets Emerging Markets RXR Secured Note Fixed Income Arbitrage Arrowsmith Fund Funds of Funds Blue Rock Capital Funds of Funds Dean Witter Cornerstone Global Macro GAMut Investments Global Macro Aquila International Long Short Equity Bay Capital Management Long Short Equity Blenheim Investments LP Managed Futures Red Oak Commodity Managed Futures
Mean 5,61% 6,24% 12,52% 16,02% 10,72% 9,81% 10,54% 12,29% 26,91% 8,65% 13,95% 24,73% 9,86% 10,12% 16,51% 19,80%
Std Skewness Kurtosis 8,01% 0,82 13,65 14,91% -0,27 0,25 8,13% -1,69 7,78 7,48% -0,74 7,37 8,09% 0,71 2,59 2,45% -3,19 3,00 20,03% -3,34 6,40 6,45% 2,33 4,84 27,08% 14,51 60,70 3,47% 1,66 7,51 23,19% 7,42 9,17 14,43% 3,38 4,61 16,88% -1,22 2,32 19,31% 1,94 0,70 29,59% 3,07 10,25 29,08% 1,94 3,52 Hedge funds summary statistics
Granger VaR 3,72% 6,78% 2,73% 1,79% 3,14% 0,60% 8,44% 1,84% 6,67% 0,76% 7,55% 4,45% 7,99% 7,31% 11,80% 11,33%
ES 5,59% 8,98% 5,17% 3,67% 4,24% 1,05% 13,15% 2,84% 12,84% 1,40% 8,78% 6,27% 10,98% 9,68% 17,47% 16,00%
Correl / underlying index -28,36% 3,27% 34,05% 64,15% 32,75% 52,30% 88,06% 1,14%
31,62% 57,58% 72,07% 27,85% 22,77% 21,60%
Data structure z z
monthly data 139 observations
Non Gaussian features (confirmed by Jarque Bera statistics) Wide range of correlation with the CSFB tremont indexes Evry April 2004
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Data set (2) Rank correlation
Skewness Kurtosis Std Semi-variance Granger VaR ES
Skewness 100% 38% 40% 32% 23% 25%
Kurtosis 38% 100% 15% 15% -3% 6%
Std 40% 15% 100% 99% 93% 95%
Semi-variance Granger VaR 32% 23% 15% -3% 99% 93% 100% 95% 95% 100% 98% 96%
ES 25% 6% 95% 98% 96% 100%
Risk measured with respect to kurtosis and VaR are almost unrelated Std, semi-variance, VaR and ES are almost perfect substitutes for the risk rankings of hedge funds
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Data set (3) Correlations
Wide range of correlations z
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Some of them negative
5
Data set (4) Betas with respect to the S&P 500 index
12 funds have a significant positive exposure to market risk, but usually with small betas.
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Factor analysis Results of a Principal Component Analysis with the correlation matrix z z
8 factors explain 90% of variance 13 factors explain 99% of variance Î Î
z
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high potential of diversification some assets are not in the optimal portfolios but may be good substitutes
Factor-loadings lead to a portfolio which is high correlated with the S&P 500 (60%)
7
Value at Risk estimation techniques Empirical quantile z
Quantile of the empirical distribution
“L-estimator” (Granger & Silvapulle (2001)) z
Weighted average of empirical quantiles
Kernel smoothing: (Gourieroux, Laurent & Scaillet (2000) ) z
Quantile of a kernel based estimated distribution
Gaussian VaR z
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Computed under the assumption of a Gaussian distribution
8
VaR estimators analysis (1)
We denote by (a’r)1:n ≤…≤ (a’r)n:n the rank statistics of the portfolio allocation a VaR estimators depend only on the rank statistics VaR estimators are differentiable and positively homogeneous of degree one (with respect to the rank statistics) Thus, we can decompose VaR using Euler ’s equality :
∂VaR(a' R) VaR(a' R) = ∑ (a' r ) i:n i =1 ∂ ( a ' r ) i:n n
see J-P. Laurent [2003]
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VaR estimators analysis (2) Weights associated with the rank statistics for the different VaR estimators Partial Partial derivatives derivatives zoom zoom on on the the left left skew skew 0,2 0,2 00 00
22
44
66
88
10 10
12 12
14 14
16 16
18 18
20 20
-0,2 -0,2 -0,4 -0,4 -0,6 -0,6 -0,8 -0,8 -1 -1 -1,2 -1,2
Granger Granger VaR VaR
Gaussian Gaussian VaR VaR
Empirical Empirical VaR VaR
GLS GLS VaR VaR
Empirical VaR is concentrated on a single point Granger VaR is distributed around empirical VaR GLS VaR : smoother weighting scheme Gaussian VaR involves an even smoother pattern
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Mean VaR optimization A non-standard optimization program z
VaR is not a convex function with respect to allocation
z
VaR is not differentiable
z
Local minima are often encountered
Genetic algorithms z
(see Barès & al [2002])
z
Time consuming: slow convergence
z
1 week per efficient frontier
Approximating algorithm Larsen & al [2001] z
Based on Expected Shortfall optimization program
z
We get a sub-optimal solution
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Mean VaR efficient frontier 1,6%
1,5%
Expected return / Empirical VaR 1,4%
1,3%
1,2%
1,1%
1,0%
0,9% -0,2%
0,0%
0,2%
0,4%
0,6%
0,8%
1,0%
1,2%
1,4%
1,6%
1,8%
Emp ir ical V aR M ean / S&M VaR (GA)
M ean / Empirical VaR (GA)
M ean / empirical VaR (Larsen)
M ean / Variance
M ean / Kernel VaR (GA)
VaR efficient frontiers are close Far from the mean-Gaussian VaR efficient frontier Larsen & al. approximating algorithm performs poorly Evry April 2004
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Mean VaR efficient portfolios (1) 11
Efficient Efficient portfolios portfolios according according to to em empirical pirical VaR VaR (GA) (GA)
11
0.9 0.9
0.9 0.9
0.8 0.8
0.8 0.8
0.7 0.7
0.7 0.7
0.6 0.6
0.6 0.6
0.5 0.5
0.5 0.5
0.4 0.4
0.4 0.4
0.3 0.3
0.3 0.3
0.2 0.2
0.2 0.2
0.1 0.1
0.1 0.1
00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return 11
Efficient Efficient portfolios portfolios according according to to Kernel Kernel VaR VaR (GA) (GA)
00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return 11
0.9 0.9
0,9 0,9
0.8 0.8
0,8 0,8
0.7 0.7
0,7 0,7
0.6 0.6
0,6 0,6
0.5 0.5
Efficient Efficient portfolios portfolios according according to to Gaussian Gaussian VaR VaR
0,5 0,5
0.4 0.4
0,4 0,4
0.3 0.3
0,3 0,3
0.2 0.2
0,2 0,2
0.1 0.1 00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78%
Return
Evry April 2004
Efficient Efficient portfolios portfolios according according to to Granger Granger VaR VaR (GA) (GA)
AXA AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy Strategy LP LP Bennett Bennett Restructuring Restructuring Fund Fund LP LP Genesis Genesis Emerging Emerging Markets Markets Fund Fund Ltd Ltd Blue Blue Rock Rock Capital Capital Fund Fund LP LP Aquila International Fund Ltd Aquila International Fund Ltd Red Red Oak Oak Commodity Commodity Advisors Advisors Inc Inc
0,1 0,1 00 0,88% 0,88% 0,96% 0,96% 1,03% 1,03% 1,11% 1,11% 1,19% 1,19% 1,26% 1,26% 1,34% 1,34% 1,42% 1,42% 1,49% 1,49% 1,57% 1,57% 1,65% 1,65% 1,72% 1,72% 1,80% 1,80% Return Return
Discovery Discovery MasterFund MasterFund Ltd Ltd Calamos Calamos Convertible Convertible Hedge Hedge Fund Fund LP LP RXR RXR Secured Secured Participating Participating Note Note Dean Witter Cornerstone Fund IV Dean Witter Cornerstone Fund IV LP LP Bay Bay Capital Capital Management Management
Aetos Aetos Corporation Corporation Sage Sage Capital Capital Limited Limited Partnership Partnership Arrowsmith Arrowsmith Fund Fund Ltd Ltd GAMut Investments Inc GAMut Investments Inc Blenheim Blenheim Investments Investments LP LP (Composite) (Composite)
13
Mean VaR optimal portfolios (2) Optimal allocations with respect to the expected mean z
Empirical VaR leads to portfolio allocations that change quickly with the return objectives
z
GLS VaR leads to smoother changes in the efficient allocations
z
Gaussian VaR implies even smoother allocation
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Optimal allocations Almost the same assets whatever the VaR estimator
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2,5% 2,5%
Efficient Efficient frontiers frontiers in in aa Mean-Em Mean-Empirical pirical VaR VaR diagram diagram Ar Arrowsmit rowsmithhFund FundLt Ltdd
2,0% 2,0%
GAMut GAMut Invest Investment mentssInc Inc
11,5% ,5% Red RedOak Oak Commodit CommodityyAdvisor AdvisorssInc Inc Bennet Bennettt Rest Restrruct uctur uring ingFund FundLP LP
11,0% ,0%
Aet Aetos osCor Corporat poration ion RXR RXRSecur Secured edPart Participat icipating ingNot Notee Calamos CalamosConver Converttible ibleHedge HedgeFund Fund LP LP Sage Bay SageCapit Capital alLimit Limited edPart Partnership nership BayCapit Capital alManagement Management Blue BlueRock Rock Capit Capital alFund FundLP LP
0,5% 0,5%
-2% -2%
0,0% 0,0% 0% 0%
AXA AXARosenber RosenberggMar Market ket Neut Neutral ral St Strrat ategy egy LP LP
2% 2%
Mean Mean -- Granger Granger VaR VaR
4% 4%
6% 6%
Mean Mean -- Empirical Empirical VaR VaR
Blenheim BlenheimInvest Investment mentssLP LP Dean DeanWit Wittter er Cor Cornerst nerstone oneFund FundIV IV ((Composit Composite) e) LP LP Genesis GenesisEmer Emerging gingMar Market ketssFund Fund Lt Ltdd Aquila AquilaInt Internat ernational ionalFund FundLt Ltdd
Discovery Discovery Mast MasterFund erFundLt Ltdd
8% 8%
110% 0%
Mean Mean -- GLS GLS VaR VaR
112% 2%
114% 4%
Mean Mean -- Gaussian Gaussian VaR VaR
Since the rankings with respect to the four risk measures are quite similar, the same hedge funds are close to the different efficient frontiers VaR is not sub-additive but…we find a surprisingly strong diversification effect Malevergne & Sornette [2004], Geman & Kharoubi [2003] find less diversification…but work with hedge funds indexes Evry April 2004
16
Diversification Analysis of the diversification effect using : Participation ratio =
1 n
∑ ai2
Participation Participation ratio ratio
i =1
77
66 55
44
33
22 11
00 0,8% 0,8%
1,0% 1,0% Granger Granger VaR VaR
1,2% 1,2%
1,4% 1,4% Expected Expected return return
Empirical Empirical VaR VaR
GLS GLS VaR VaR
1,6% 1,6%
1,8% 1,8%
2,0% 2,0%
Gaussian Gaussian VaR VaR
Gaussian VaR leads to less diversified efficient portfolios Against « common knowledge » : non subadditivity of VaR implies risk concentration increases Evry April 2004
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Analysis including S&P 500 Analysis including S&P 500… 2,5% 2,5%
Efficient Efficient frontiers frontiers in in aa Mean-Em Mean-Empirical pirical VaR VaR diagram diagram Ar Arrowsmit rowsmithhFund FundLt Ltdd
2,0% 2,0%
GAMut GAMut Invest Investment mentssInc Inc
11,5% ,5% Red RedOak Oak Commodit CommodityyAdvisor AdvisorssInc Inc Bennet Bennettt Rest Restrruct uctur uring ingFund FundLP LP
11,0% ,0%
S&P S&P 500 500
Aet Aetos osCor Corporat poration ion RXR RXRSecur Secured edPart Participat icipating ingNot Notee Calamos CalamosConver Converttible ibleHedge HedgeFund Fund LP LP Sage Bay SageCapit Capital alLimit Limited edPart Partnership nership BayCapit Capital alManagement Management Blue BlueRock Rock Capit Capital alFund FundLP LP
0,5% 0,5%
-2% -2%
0,0% 0,0% 0% 0%
AXA AXARosenber RosenberggMar Market ket Neut Neutral ral St Strrat ategy egy LP LP
2% 2%
Mean Mean -- Granger Granger VaR VaR
4% 4%
6% 6%
Mean Mean -- Empirical Empirical VaR VaR
Blenheim BlenheimInvest Investment mentssLP LP Dean DeanWit Wittter er Cor Cornerst nerstone oneFund FundIV IV ((Composit Composite) e) LP LP Genesis GenesisEmer Emerging gingMar Market ketssFund Fund Lt Ltdd Aquila AquilaInt Internat ernational ionalFund FundLt Ltdd
Discovery Discovery Mast MasterFund erFundLt Ltdd
8% 8%
110% 0%
Mean Mean -- GLS GLS VaR VaR
112% 2%
114% 4%
Mean Mean -- Gaussian Gaussian VaR VaR
…no change in the efficient frontiers
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Alternative Risk Measures
19
Alternative risk measures Recent works about risk measures properties z
Artzner & al [1999], Tasche [2002], Acerbi [2002], Föllmer & Schied [2002]
z
Widens the risk measure choice range
Some choice criteria z
Coherence properties
z
Numerical tractability
Properties of optimal portfolios analysis z
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Comparison of different optimal portfolios
20
Expected shortfall Definition: mean of “losses “ beyond the Value at Risk Properties z
Coherent measure of risk
z
Spectral representation
Î
optimal portfolio may be very sensitive to extreme events if α is very low
Algorithm z
Linear optimization algorithms (see Rockafellar & Uryasev [2000]) Î
z
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may be based on the simplex optimization program
Quick computations
21
Downside risk Definitions z
Let x1, x2, …xn be the values of a portfolio (historical or simulated)
z
The downside risk is defined as follows
Properties z
2
See Fischer [2001]
No spectral representation Î
z
] −x
Coherent measure of risk Î
z
[
1 n + SV ( X ) = ∑ ( x − xi ) n i =1
fails to be comonotonic additive
Could be a good candidate to take into account the investors positive return preference
Algorithms z
Athayde’s recursive algorithm ( [2001]) Î
z
Konno et al ( [2002]) Î
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Derived from the mean - variance optimization
Use of auxiliary variables 22
Contribution of rank statistics Decomposition of the risk measures as for the VaR case Partial derivatives zoom on the left skew
0,05 0 0
5
10
15
20
-0,05 -0,1 -0,15 -0,2 -0,25 -0,3 Granger VaR
DSR
ES
STDV
VaR and ES weights are concentrated on extreme rank statistics Variance and Downside risk weights exhibit a smoother weighting scheme Evry April 2004
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Efficient frontiers: the Variance point of view 1.8% 1.8%
Efficient Efficient frontiers frontiers in in an an expected expected return return -- standard standard deviation deviation diagram diagram
1.7% 1.7%
1.6% 1.6%
1.5% 1.5%
1.4% 1.4%
1.3% 1.3%
1.2% 1.2%
1.1% 1.1%
1.0% 1.0%
0.9% 0.9%
0.8% 0.8% 0.0% 0.0%
0.0% 0.0%
0.0% 0.0%
M Mean ean // S&M S&M VaR VaR (GA) (GA)
0.0% 0.0%
0.0% 0.0%
0.1% 0.1%
S Sttaanda ndard rd de devvia iattio io nn
M Mean ean // Gaussian Gaussian VaR VaR
0.1% 0.1%
M Mean ean // ES ES (Uryasev) (Uryasev)
0.1% 0.1%
0.1% 0.1%
M Mean ean // DSR DSR
Variance and downside risk are very close Contrasts created by the opposition of z
Small events based measure: variance and downside risk
z
Large events based measure: VaR and Expected Shortfall
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The VaR point of view 1,8% 1,8%
Efficient frontiers in an expected return - Granger VaR diagram
1,7% 1,7%
1,6% 1,6%
1,5% 1,5%
1,4% 1,4%
1,3% 1,3%
1,2% 1,2%
1,1% 1,1%
1,0% 1,0%
0,9% 0,9% 0,0% 0,0%
0,2% 0,2%
0,4% 0,4%
M M ean ean // Granger Granger VaR VaR
0,6% 0,6%
0,8% 0,8%
1,0% 1,0%
M M ean ean // ES ES (Uryasev) (Uryasev)
1,2% 1,2%
1,4% 1,4%
M M ean ean // Standard Standard deviation deviation
1,6% 1,6%
1,8% 1,8%
2,0% 2,0%
M M ean ean // DSR DSR
VaR efficient frontier is far from the others (even from Expected Shortfall) z
VaR estimation involve a few rank statistics than the other risk measures
No differences between downside risk, Variance, Expected shortfall in the VaR view Evry April 2004
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Optimal portfolios 11
Efficient Efficient portfolios portfolios according according to to standard standard deviation deviation
11
0.9 0.9
0.9 0.9
0.8 0.8
0.8 0.8
0.7 0.7
0.7 0.7
0.6 0.6
0.6 0.6
0.5 0.5
0.5 0.5
0.4 0.4
0.4 0.4
0.3 0.3
0.3 0.3
0.2 0.2
0.2 0.2
0.1 0.1
0.1 0.1
00 0.88% 0.88% 0.96% 0.96% 1.03% 1.03% 1.11% 1.11% 1.19% 1.19% 1.26% 1.26% 1.34% 1.34% 1.42% 1.42% 1.49% 1.49% 1.57% 1.57% 1.65% 1.65% 1.72% 1.72% 1.80% 1.80% Return Return
11
Efficient Efficient portfolios portfolios according according to to sem semi-variance i-variance
00 0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return
11
0.9 0.9
0.9 0.9
0.8 0.8
0.8 0.8
0.7 0.7
0.7 0.7
0.6 0.6
0.6 0.6
0.5 0.5
0.5 0.5
0.4 0.4
0.4 0.4
0.3 0.3
0.3 0.3
0.2 0.2
0.2 0.2
Efficient Efficient portfolio portfolio according according to to ES ES (Uryasev) (Uryasev)
0.1 0.1
0.1 0.1
00
00 0.86% 1.32% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return
Evry April 2004
Efficient Efficient portfolios portfolios according according to to Granger Granger VaR VaR (GA) (GA)
AXA AXA Rosenberg Rosenberg Market Market Neutral Neutral Strategy Strategy LP LP Bennett Bennett Restructuring Restructuring Fund Fund LP LP Genesis Genesis Emerging Emerging Markets Markets Fund Fund Ltd Ltd Blue Blue Rock Rock Capital Capital Fund Fund LP LP Aquila Aquila International International Fund Fund Ltd Ltd Red Red Oak Oak Commodity Commodity Advisors Advisors Inc Inc
0.86% 0.86% 0.94% 0.94% 1.01% 1.01% 1.09% 1.09% 1.17% 1.17% 1.24% 1.24% 1.32% 1.32% 1.40% 1.40% 1.47% 1.47% 1.55% 1.55% 1.63% 1.63% 1.70% 1.70% 1.78% 1.78% Return Return
Discovery Discovery MasterFund MasterFund Ltd Ltd Calamos Calamos Convertible Convertible Hedge Hedge Fund Fund LP LP RXR RXR Secured Secured Participating Participating Note Note Dean Dean Witter Witter Cornerstone Cornerstone Fund Fund IV IV LP LP Bay Bay Capital Capital Management Management
Aetos Aetos Corporation Corporation Sage Sage Capital Capital Limited Limited Partnership Partnership Arrowsmith Arrowsmith Fund Fund Ltd Ltd GAMut GAMut Investments Investments Inc Inc Blenheim Blenheim Investments Investments LP LP (Composite) (Composite)
26
Optimal portfolios Almost the same assets whatever the risk measure
Some assets are not in the optimal portfolios but may be good substitutes As for the VaR, risk measures with smoother weights leads to more stable efficient portfolios. Evry April 2004
27
Diversification Analysis of the diversification effect Participation Participation ratio ratio 88
77
66
55
44
33
22
11
00 0,8% 0,8%
1,0% 1,0% Granger Granger VaR VaR
1,2% 1,2%
1,4% 1,4% Expected Expected return return ES semi ES (Uryassev) (Uryassev) semi variance variance
1,6% 1,6%
1,8% 1,8%
2,0% 2,0%
Gaussian Gaussian VaR VaR
Expected Shortfall leads to greater diversification than other risk measures Gaussian VaR leads to less diversified efficient portfolios
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Rank correlation analysis between risk levels and optimal portfolio weights
Optimal ptf sc semi-v. Optimal ptf sc VaR Optimal ptf sc ES Semi-variance Granger VaR ES
Optimal ptf sc semi-v. Optimal ptf sc VaR Optimal ptf sc ES 100% 38% 40% 38% 100% 60% 40% 60% 100% 37% 39% 15% 53% 43% 28% 38% 35% 16% Rank correlation
Semi-variance Granger VaR 37% 53% 39% 43% 15% 28% 100% 95% 95% 100% 98% 96%
ES 38% 35% 16% 98% 96% 100%
No direct relation
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Conclusion The same assets appear in the efficient portfolios, but allocations are different z z
The way VaR is computed is quite important Expected shortfall leads to greater diversification
No direct relation between individual amount of risk and weight in optimal portfolios: Large individual risk Small individual risk z
⇒ ⇒
low weight in optimal portfolios large weight in optimal portfolios
Importance of the dependence between risks in the tails
The risk decomposition (can be compared to spectral representation) allows to understand the structure of optimal portfolios Open question: z
Evry April 2004
Relation between risk measures and investors’ preferences
30
