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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

Data Daily – 1992 through June 2009 (4,292 observations) MSCINYSES&PChinaEuro Mean Covariance(MSCI)9.69E E E Covariance(NYSE) E E-05 Covariance(S&P) E-068.9E-05 Covariance)China) E-06 Covariance(Euro)

Correlation China uncorrelated Eurostoxx low correlation with first 3 MSCINYSES&PChina Eurostoxx MSCI1 NYSE S&P China Eurostoxx

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

Impact of Distribution on VaR Fat tails matter

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

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

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

Optimization Solutions Objective MSCINYSES&PChinaEuro Max E[return] Min Variance Max Pr{E[Ret]>1000} Max Pr{E[Ret]>950} Max Pr{E[Ret]>900} Max Pr{E[Ret]>800} CC {Pr>.9[Ret>800]} CC{Pr>.8[Ret>800]} CC{Pr>.8[Ret>900]} CC{Pr>.7[Ret>900]} Min VaR at 0.99 level

Solution Expected Performances Objective ReturnVariancePr{>1000}Pr{>950}Pr{>900}Pr{>800} Max E[return]1275.9* Min Variance * Max Pr{E[Ret]>1000} * Max Pr{E[Ret]>950} * Max Pr{E[Ret]>900} * Max Pr{E[Ret]>800} * CC {Pr>.9[Ret>800]} CC{Pr>.8[Ret>800]} CC{Pr>.8[Ret>900]} CC{Pr>.7[Ret>900]} Min VaR at 0.99 level

Simulation – Max Return

Trials10,000 Mean1, Median Standard Deviation Variance780, Skewness2.51 Kurtosis16.69 Minimum0.00 Maximum12,984.16

Simulation – Min Variance

Trials10,000 Mean1, Median1, Standard Deviation Variance43, Skewness Kurtosis3.72 Minimum Maximum2,207.00

Comparison ModelModel Return Model Variance Model VaR CVaR Sim return Sim Variance Sim VaR Max return Min variance Max Prob{Ret>1000} Max Prob{Ret>950} Max Prob{Ret>900} Max Prob{Ret>800} Max Ret st Pr{Ret>800}> Max Ret st Pr{Ret>800}> Max Ret st Pr{Ret>900}> Max Ret st Pr{Ret>900}> Min VaR at the 0.99 level

CVaR Models ratio f(α)/(1-α) RatioMSCINYSES&PChinaEuro

Model Results Return constraintReturnVarianceMinMaxVaR CVaR

Correlation Makes a Difference Daily Models t-distribution

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

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

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