Portfolio Optimization with Conditional Value-at-Risk and Chance Constraints

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 = 0.64expect to exceed 99% of time in 1 yearHere 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 99th 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 VarianceMarkowitz 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 - z1-α √[var-covar] + E[return] where z1-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. DataDaily – 1992 through June 2009 (4,292 observations)

18. CorrelationChina uncorrelatedEurostoxx low correlation with first 3

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 2nd, Normal & Lognormal 3rd or 4th • IMPLICATION: • Fat tails exist • Symmetric

20. Impact of Distribution on VaRFat tails matter

21. Models • Maximize expected returns.t. budget ≤ 1000 • MinimizeVariances.t. investment = 1000 • Maximize probability{return>specified level} for levels [1000, 950, 900, and 800]. • Maximize expected returns.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

25. Solution Expected Performances

26. Simulation – Max Return

27. Simulation – Max Return • Trials 10,000 • Mean 1,217.16 • Median 996.42 • Standard Deviation 883.36 • Variance 780,316.87 • Skewness 2.51 • Kurtosis 16.69 • Minimum 0.00 • Maximum 12,984.16

28. Simulation – Min Variance

29. Simulation – Min Variance • Trials 10,000 • Mean 1,054.31 • Median 1,034.20 • Standard Deviation 208.39 • Variance 43,424.47 • Skewness 0.5960 • Kurtosis 3.72 • Minimum 354.32 • Maximum 2,207.00

30. Comparison

31. CVaR Modelsratio f(α)/(1-α)

32. Model Results

33. Correlation Makes a DifferenceDaily Models t-distribution

34. Correlation impact on VarianceDaily Models t-distribution3 outliers – China mixed with others

35. Correlation impact on Value-at-RiskDaily Models t-distributionDirectly 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