Alternative Risk Measures for Alternative Investments A. Chabaane BNP Paribas ACA Consulting Y. Malevergne ISFA Actuarial School Lyon JP. Laurent ISFA Actuarial School Lyon BNP Paribas F. Turpin BNP Paribas email : francoise.turpin@bnpparibas.com http://laurent.jeanpaul.free.fr/
Outline! Optimizing under VaR constraints " Estimation techniques " VaR analytics and efficient portfolios comparison! Optimizing under alternative risk constraints " Expected Shortfall, Downside Risk measure, " Risk measures analytics and efficient portfolios comparison 2
Main result Decomposition of risk measures " VaR " Expected Shortfall " Downside Risk A way to understand optimal portfolios structure 3
Data set! 6 hedge funds! Data structure " monthly data " 39 observations! High skewness and kurtosis! Low (or negative) correlations " diversification potentiality 4
Value at Risk estimation techniques! Empirical quantile! Weighted average of quantiles: L-estimator (Granger & Silvapulle (2))! Kernel smoothing: (Gourieroux, Laurent & Scaillet (2) )! Gaussian VaR : computed under Gaussian assumption 5
Contribution of rank statistics ()! We denote by (a r) :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 : see J-P. Laurent [23] n VaR( a' R) VaR ( a' R) = ( a' r) : a r i= ( ' ) i: n i n 6
Contribution of rank statistics (2)! Weights associated with the rank statistics for the different VaR estimators,2 Partial derivatives zoom on the left skew 2 4 6 8 2 4 6 8 2 -,2 -,4 -,6 -,8 - -,2 Granger VaR Gaussian VaR Empirical VaR GLS 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 7
Mean VaR optimization! A non-standard optimization program " VaR is not a convex function " VaR is not differentiable in the general case (with respect to allocation) " Local minima are often encountered! Genetic algorithms (see P.-A. Barès, R.Gibson and S. Gyger [22]) " Derived from the natural selection process " Time consuming: slow convergence! Approximating algorithm Larsen, Mausser & Uryasev " Based on Expected Shortfall (see below) optimization program " We get a sub-optimal solution 8
Mean VaR efficient frontier,6%,5%,4% Expected return / Empirical VaR,3%,2%,%,%,9% -,2%,%,2%,4%,6%,8%,%,2%,4%,6%,8% Empirical VaR Mean / S&M VaR (GA) Mean / Empirical VaR (GA) Mean / Kernel VaR (GA) Mean / empirical VaR (Larsen) Mean / Variance! VaR efficient frontiers are close! Far from the mean-gaussian VaR efficient frontier! Larsen & al. approximating algorithm performs poorly 9
Mean VaR efficient portfolios () Efficient portfolios according to empirical VaR (GA).9.8.7.6.5.4.3.2..86%.94%.%.9%.7%.24%.32%.4%.47%.55%.63%.7%.78% Return Efficient portfolios according to Kernel VaR (GA).9.8.7.6.5.4.3.2..86%.94%.%.9%.7%.24%.32%.4%.47%.55%.63%.7%.78% Return Efficient portfolios according to Granger VaR (GA).9.8.7.6.5.4.3.2..86%.94%.%.9%.7%.24%.32%.4%.47%.55%.63%.7%.78% Return Efficient portfolios according to Gaussian VaR,9,8,7,6,5,4,3,2,,88%,96%,3%,%,9%,26%,34%,42%,49%,57%,65%,72%,8% Return AXA Rosenberg Market Neutral Strategy LP Discovery MasterFund Ltd Aetos Corporation Bennett Restructuring Fund LP Calamos Convertible Hedge Fund LP Sage Capital Limited Partnership Genesis Emerging Markets Fund Ltd RXR Secured Participating Note Arrowsmith Fund Ltd Blue Rock Capital Fund LP Dean Witter Cornerstone Fund IV LP GAMut Investments Inc Aquila International Fund Ltd Bay Capital Management Blenheim Investments LP (Composite) Red Oak Commodity Advisors Inc
Mean VaR optimal portfolios (2)! Interpretations of the previous graphs " Empirical VaR leads to portfolio allocations that change quickly with the return objectives since it is based on a single rank statistic " As expected (according to the decomposition) GLS VaR leads to smooth changes in the efficient allocations 2,5% " VaR is not sub-additive but we find a surprisingly strong diversification effect " Taking into account Hedge Funds indexes leads to different result (see Y. Malevergne and D. Sornette [22], H. Geman and C. Kharoubi [23]) 2,%,5%,%,5% Efficient frontiers in a Mean-Empirical VaR diagram GAMut Investments Inc Bennett Restructuring Fund LP Aetos Corporation RXR Secured Participating Calamos Convertible Note Hedge Fund Sage Capital Limited Partnership LP Bay Capital Management Blue Rock Capital Fund LP Arrowsmith Fund Ltd Discovery MasterFund Ltd AXA Rosenberg Market Neutral Strategy LP Blenheim Investments LP Dean Witter Cornerstone Fund (Composite) IV LP Genesis Emerging Markets Fund Lt d Aquila International Fund Ltd Red Oak Commodity Advisors Inc,% -2% % 2% 4% 6% 8% % 2% 4% Mean - Granger VaR Mean - Empirical VaR Mean - GLS VaR Mean - Gaussian VaR
Comparison of efficient portfolios! Comparison of efficient portfolios under VaR constraints for a given.2% level of expected return Optimal portfolios for % level of return Emprical Granger VaR GLS VaR VaR AXA Rosenberg Market Neutral Strategy LP 3,9% 2,2%,% Discovery MasterFund Ltd,3% 2,%,2% Aetos Corporation 4,5% 2,4%,6% Bennett Restructuring Fund LP 2,% 26,% 25,4% Calamos Convertible Hedge Fund LP,%,%,% Sage Capital Limited Partnership 28,6% 3,4% 29,8% Genesis Emerging Markets Fund Ltd,2%,%,% RXR Secured Participating Note 2,% 8,8% 6,5% Arrow smith Fund Ltd 2,% 2,2%,2% Blue Rock Capital Fund LP 23,6% 7,6% 25,% Dean Witter Cornerstone Fund IV LP,%,%,% GAMut Investments Inc 9,9% 6,9% 9,% Aquila International Fund Ltd,%,%,% Bay Capital Management 2,6%,6%,% Blenheim Investments LP (Composite),3%,%,% Red Oak Commodity Advisors Inc,9%,%,% Aetos Corporation RXR Secured Participating Note Blue Rock Capital Fund LP -,35 -,25 -,5 -,5,5,5,25 2
Behavior of left tails! Comparison of efficient portfolios under VaR constraints for a given.2% level of expected return Optimal portfolios for % level of return Emprical Granger VaR GLS VaR VaR AXA Rosenberg Market Neutral Strategy LP 3,9% 2,2%,% Discovery MasterFund Ltd,3% 2,%,2% Aetos Corporation 4,5% 2,4%,6% Bennett Restructuring Fund LP 2,% 26,% 25,4% Calamos Convertible Hedge Fund LP,%,%,% Sage Capital Limited Partnership 28,6% 3,4% 29,8% Genesis Emerging Markets Fund Ltd,2%,%,% RXR Secured Participating Note 2,% 8,8% 6,5% Arrow smith Fund Ltd 2,% 2,2%,2% Blue Rock Capital Fund LP 23,6% 7,6% 25,% Dean Witter Cornerstone Fund IV LP,%,%,% GAMut Investments Inc 9,9% 6,9% 9,% Aquila International Fund Ltd,%,%,% Bay Capital Management 2,6%,6%,% Blenheim Investments LP (Composite),3%,%,% Red Oak Commodity Advisors Inc,9%,%,% Aetos Corporation RXR Secured Participating Note Blue Rock Capital Fund LP -,5 -,3 -, -,9 -,7 -,5 -,3 -,! Analysis of 3 asset distributions and resulting allocations " Aetos C. has a fat tail : penalized by Granger VaR and even more by GLS VaR " RXR S. has a thin tail : favored by Granger VaR and GLS VaR " Blue R. has a thin extreme tail which quickly thickens : penalized by Granger VaR 3
Alternative Risk Measures Alternative Risk Measures 4
Alternative risk measures! Recent works about risk measures properties (P. Artzner, F.Delbaen, J-M. Eber and D. Heath [999], D. Tasche [22], C. Acerbi [22], H. Föllmer and A. Schied [22]) " widens the risk measure choice range! Some choice criteria " coherence properties " numerical tractability! Properties of optimal portfolios analysis " comparison of different optimal portfolios 5
Expected shortfall! Definition: mean of losses beyond the Value at Risk! Properties " Coherent measure of risk " Spectral representation! Algorithm # optimal portfolio may be very sensitive to extreme events if α is very low " Linear optimization algorithms (see R.T Rockafellar & S. Uryasev [2]) # may be based on the simplex optimization program " Quick computation 6
! Definitions " Let x, x 2, x n be the values of a portfolio (historical or simulated) " The downside risk is defined as follows! Properties " Coherent measure of risk (see T. Fischer [2]) " No spectral representation # fails to be comonotonic additive " Could be a good candidate to take into account the investors positive return preference! Algorithm n SV ( X ) = n i= " Derived from the variance optimization # the Athayde s recursive algorithm [( ) ] + 2 x x x i Downside risk 7
! Decomposition of the risk measures as for the VaR case Contribution of rank statistics,5 Partial derivatives zoom on the left skew 5 5 2 -,5 -, -,5 -,2 -,25 -,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 8
Optimal portfolios Efficient portfolios according to standard deviation Efficient portfolios according to Granger VaR (GA).9.9.8.8.7.7.6.6.5.5.4.4.3.3.2..88%.96%.3%.%.9%.26%.34% Re turn.42%.49%.57%.65%.72%.8% Efficient portfolios according to semi-variance.9.2..86%.94%.%.9%.7%.24%.32%.4%.47%.55%.63%.7%.78% Re turn Efficient portfolio according to ES (Uryasev).9.8.7.6.5.4.3.2..86%.94%.%.9%.7%.24%.32% Return.4%.47%.55%.63%.7%.78%.86%.94%.%.9%.7%.24%.32%.4%.47%.55%.63%.7%.78% Return AXA Rosenberg Market Neutral Strategy LP Discovery MasterFund Ltd Aetos Corporation Bennett Restructuring Fund LP Calamos Convertible Hedge Fund LP Sage Capital Limited Partnership Genesis Emerging Markets Fund Ltd RXR Secured Participating Note Arrowsmith Fund Ltd Blue Rock Capital Fund LP Dean Witter Cornerstone Fund IV LP GAMut Investments Inc Aquila International Fund Ltd Bay Capital Management Blenheim Investments LP (Composite) Red Oak Commodity Advisors Inc.8.7.6.5.4.3.2. 9
Comparison of efficient portfolios! Comparison of efficient portfolios under risk measure constraints for a given % level of expected return Optimal portfolios for % level of return ES VaR DSR AXA Rosenberg Market Neutral Strategy LP 2,9% 2,2%,5% Discovery MasterFund Ltd,4% 2,%,8% Aetos Corporation,% 2,4% 2,3% Bennett Restructuring Fund LP 3,% 26,% 8,% Calamos Convertible Hedge Fund LP,%,%,% Sage Capital Limited Partnership 2,7% 3,4% 3,% Genesis Emerging Markets Fund Ltd,%,%,% RXR Secured Participating Note,5% 8,8% 9,5% Arrow smith Fund Ltd 2,9% 2,2%,8% Blue Rock Capital Fund LP 22,2% 7,6% 24,6% Dean Witter Cornerstone Fund IV LP,%,%,% GAMut Investments Inc 4,8% 6,9%,9% Aquila International Fund Ltd,%,%,% Bay Capital Management,6%,6%,4% Blenheim Investments LP (Composite),%,%,% Red Oak Commodity Advisors Inc,%,%,% Arrowsmith Fund AXA Rosenberg Market Neutral Strategy LP Blue Rock Capital Fund LP -,35 -,25 -,5 -,5,5,5,25 2
Behavior of left tails! Comparison of efficient portfolios under risk measure constraints for a given % level of expected return Optimal portfolios for % level of return ES VaR DSR AXA Rosenberg Market Neutral Strategy LP 2,9% 2,2%,5% Discovery MasterFund Ltd,4% 2,%,8% Aetos Corporation,% 2,4% 2,3% Bennett Restructuring Fund LP 3,% 26,% 8,% Calamos Convertible Hedge Fund LP,%,%,% Sage Capital Limited Partnership 2,7% 3,4% 3,% Genesis Emerging Markets Fund Ltd,%,%,% RXR Secured Participating Note,5% 8,8% 9,5% Arrow smith Fund Ltd 2,9% 2,2%,8% Blue Rock Capital Fund LP 22,2% 7,6% 24,6% Dean Witter Cornerstone Fund IV LP,%,%,% GAMut Investments Inc 4,8% 6,9%,9% Aquila International Fund Ltd,%,%,% Bay Capital Management,6%,6%,4% Blenheim Investments LP (Composite),%,%,% Red Oak Commodity Advisors Inc,%,%,% Arrowsmith Fund Ltd Blue Rock Capital Fund LP AXA Rosenberg Market Neutral Strategy LP -,35 -,3 -,25 -,2 -,5 -, -,5! Analysis of 3 asset distributions and resulting allocations " Axa R. has very few points in the extreme tail : favored by Expected Shortfall " Arrowsmith. has an extreme low return : penalized by Downside Risk " Blue R. has a thin extreme tail which quickly thickens : penalized by VaR 2