1. Portfolio Optimization with Conditional Value-at-Risk and Chance Constraints David L. Olson University of Nebraska Desheng Wu University of Toronto; University of Reykjavik

3. Risk & Business Taking risk is fundamental to doing business – Insurance Lloyd’s of London – Hedging Risk exchange swaps Derivatives/options Catastrophe equity puts (cat-e-puts) – ERM seeks to rationally manage these risks Be a Risk Shaper

4. Financial Risk Management Evaluate chance of loss – PLAN Hubbard [2009]: identification, assessment, prioritization of risks followed by coordinated and economical application of resources to minimize, monitor, and control the probability and/or impact of unfortunate events – WATCH, DO SOMETHING

5. Our Paper PLAN – Markowitz [1952] risk = variance Control by diversifying Take advantage of correlation to get build-in hedging – Generate portfolios on efficient frontier Chance constrained programming Value-at-risk Conditional value-at-risk

6. Value-at-Risk One of most widely used models in financial risk management (Gordon [2009]) Maximum expected loss over given time horizon at given confidence level – Typically how much would you expect to lose 99% of the time over the next day (typical trading horizon) Implication – will do worse (1-0.99) proportion of the time

7. VaR = 0.64 expect to exceed 99% of time in 1 year Here loss = 10 – 0.64 = 9.36 Finland 2010

8. Use Basel Capital Accord – Banks encouraged to use internal models to measure VaR – Use to ensure capital adequacy (liquidity) – Compute daily at 99 th percentile Can use others – Minimum price shock equivalent to 10 trading days (holding period) – Historical observation period ≥1 year – Capital charge ≥ 3 x average daily VaR of last 60 business days Finland 2010

9. VaR Calculation Approaches Historical simulation – Good – data available – Bad – past may not represent future – Bad – lots of data if many instruments (correlated) Variance-covariance – Assume distribution, use theoretical to calculate – Bad – assumes normal, stable correlation Monte Carlo simulation – Good – flexible (can use any distribution in theory) – Bad – depends on model calibration Finland 2010

10. Limits At 99% level, will exceed 3-4 times per year Distributions have fat tails Only considers probability of loss – not magnitude Conditional Value-At-Risk – Weighted average between VaR & losses exceeding VaR – Aim to reduce probability a portfolio will incur large losses Finland 2010

11. Optimization Maximize f(X) Subject to: Ax ≤ b x ≥ 0 Finland 2010

12. Minimize Variance Markowitz extreme Min Var [Y] Subject to: Pr{Ax ≤ b} ≥ α ∑ x = limit= to avoid null solution x ≥ 0 Finland 2010

13. Chance Constrained Model Maximize the expected value of a probabilistic function Maximize E[Y] (where Y = f(X)) Subject to: ∑ x = limit Pr{Ax ≤ b} ≥ α x ≥ 0 Finland 2010

14. Maximize Probability Max Pr{Y ≥ target} Subject to: ∑ x = limit Pr{Ax ≤ b} ≥ α x ≥ 0 Finland 2010

15. Minimize VaR Min Loss Subject to: ∑ x = limit -Loss = initial value - z 1-α √[var-covar] + E[return] where z 1-α is in the lower tail, α= 0.99 x ≥ 0 Equivalent to the worst you could experience at the given level Finland 2010

16. Demonstration Data 5 stock indexes – Morgan Stanley World Index (MSCI) – New York Stock Exchange Composite Index (NYSE) – Standard & Poors 500 (S&P) – Shenzhen Composite (China) – Eurostoxx 50 (Euro)

17. Data Daily – 1992 through June 2009 (4,292 observations) MSCINYSES&PChinaEuro Mean 0.000180.000270.0002520.0007830.0003 Covariance(MSCI)9.69E-059.91E-050.00017.21E-060.000101 Covariance(NYSE)0.0001350.0001377.89E-079.13E-05 Covariance(S&P)0.000147-2.2E-068.9E-05 Covariance)China)0.0005433.32E-06 Covariance(Euro)0.000204

18. Correlation China uncorrelated Eurostoxx low correlation with first 3 MSCINYSES&PChina Eurostoxx MSCI1 NYSE0.8675711 S&P0.8412610.9753291 China0.0314340.002916-0.007941 Eurostoxx0.7226920.5516720.514560.0099751

19. Distributions Used Crystal Ball software – Chi-squared, Kolmogorov-Smirnov, Anderson- Darling for goodness of fit Results stable across methods Student-t best fit – Logistic 2 nd, Normal & Lognormal 3 rd or 4 th – IMPLICATION: Fat tails exist Symmetric

20. Impact of Distribution on VaR Fat tails matter

21. Models Maximize expected return s.t. budget ≤ 1000 Minimize Variance s.t. investment = 1000 Maximize probability{return>specified level} for levels [1000, 950, 900, and 800]. Maximize expected return s.t. probability{return ≥ specified level} ≥ α for α [0.9, 0.8, 0.7, and 0.6]. Minimize Value at risk for an α = 0.99 Minimize CVaR constrained to attain given return

22. Optimization Solutions Excel SOLVER – Maximize return linear – Others nonlinear Generalized Reduced Gradient Some instability in solutions across runs

23. Simulated Solutions to evaluate Monte Carlo Simulation – Crystal Ball – 10,000 runs of one year each (long-term view) Correlation: Daily (short-term) – Crystal Ball allows use of correlation matrix Correlation: Annual data (245 days) – Couldn’t reasonably enter that many within software – Used Cholesky decomposition

24. Optimization Solutions Objective MSCINYSES&PChinaEuro Max E[return]0001000.00 Min Variance123.4876.6000 Max Pr{E[Ret]>1000}0671.60166.6161.8 Max Pr{E[Ret]>950}0877.3091.531.2 Max Pr{E[Ret]>900}0938.3061.70 Max Pr{E[Ret]>800}0964.4035.60 CC {Pr>.9[Ret>800]}0547.30208.3244.4 CC{Pr>.8[Ret>800]}000571.5428.5 CC{Pr>.8[Ret>900]}0408.60255.8335.6 CC{Pr>.7[Ret>900]}000885.7114.3 Min VaR at 0.99 level113.1886.9000

25. Solution Expected Performances Objective ReturnVariancePr{>1000}Pr{>950}Pr{>900}Pr{>800} Max E[return]1275.9*5464220.64290.66720.69080.7353 Min Variance1067.428818*0.65160.75010.83020.9320 Max Pr{E[Ret]>1000}1106.6461310.6865*0.76140.82430.9130 Max Pr{E[Ret]>950}1088.9342000.68130.7680*0.83840.9305 Max Pr{E[Ret]>900}1082.3313960.67550.76660.8400*0.9340 Max Pr{E[Ret]>800}1076.9298900.66860.76290.83880.9350* CC {Pr>.9[Ret>800]}1116.5557360.68560.75430.81320.9000 CC{Pr>.8[Ret>800]}1194.42073160.66230.70040.73620.8000 CC{Pr>.8[Ret>900]}1127.8691520.68300.74540.80000.8841 CC{Pr>.7[Ret>900]}1254.24370350.64700.67400.70000.7487 Min VaR at 0.99 level1067.6288196519750583049321

26. Simulation – Max Return

27. Trials10,000 Mean1,217.16 Median996.42 Standard Deviation883.36 Variance780,316.87 Skewness2.51 Kurtosis16.69 Minimum0.00 Maximum12,984.16

28. Simulation – Min Variance

29. Trials10,000 Mean1,054.31 Median1,034.20 Standard Deviation208.39 Variance43,424.47 Skewness0.5960 Kurtosis3.72 Minimum354.32 Maximum2,207.00

30. Comparison ModelModel Return Model Variance Model VaR CVaR Sim return Sim Variance Sim VaR Max return1275.95464221649 1217.2780317849 Min variance1067.428818374 1063 1054.343424353 Max Prob{Ret>1000}1106.646372452 1102 1082.447101320 Max Prob{Ret>950}1088.934200392 1085 1070.040820327 Max Prob{Ret>900}1082.331396379 1078 1065.340397377 Max Prob{Ret>800}1076.929890373 1073 1061.340629332 Max Ret st Pr{Ret>800}>0.91116.555736498 1111 1088.854031325 Max Ret st Pr{Ret>800}>0.81194.4207316604 1184 1146.1250461532 Max Ret st Pr{Ret>900}>0.81127.869152557 1122 1096.467862383 Max Ret st Pr{Ret>900}>0.71254.24370351446 1194.8579255762 Min VaR at the 0.99 level1076.129724226 1072 1061.341039343

31. CVaR Models ratio f(α)/(1-α) RatioMSCINYSES&PChinaEuro 1.000100000 1.0800931.768.30 1.10127.9230.6292.1148.0201.4 1.130414.9286.7298.40 1.1500598.7401.30 1.1800456.0544.00 1.2000360.9639.10 1.2200265.8734.20 1.222055.7132.6735.476.3 1.22250.344.7133.6737.284.2

32. Model Results Return constraintReturnVarianceMinMaxVaR CVaR 1.01056.7339064332094309 1052 1.081066.5327655162434290 1062 1.101078.7425705284281298 1074 1.131104.6917844247685361 1097 1.151122.314458239010025443 1113 1.181141.52252173145367501 1130 1.201156.33057462486071571 1143 1.221171.14002891836774635 1156 1.2221171.34009231846742702 1156 1.22251171.64027731816752634 1157

33. Correlation Makes a Difference Daily Models t-distribution

34. Correlation impact on Variance Daily Models t-distribution 3 outliers – China mixed with others

35. Correlation impact on Value-at-Risk Daily Models t-distribution Directly proportional to Variance

36. Conclusions Can use a variety of models to plan portfolio Expect results to be jittery – Near-optimal may turn out better – Sensitive to distribution assumed Trade-off – risk & return