Scenario Optimization

1. Scenario Optimization

2. Contents • Introduction • Mean absolute deviation models • Regret models • Value at Risk in optimal portfolios Financial Optimization and Risk Management

3. Scenario optimization • Powerful models for risk management in both equities and fixed income assets (and other assets) • Tradeoff geared against risk when both measures are computed from scenario data • Scenarios can describe different types of risk (credit, liquidity, actuarial …) • Fixed income, equities and derivatives can be managed in the same framework Scenarios: future values rl of risky variables r (prices, exchange rates, etc.) with probabilities pl, l=1,…,N Financial Optimization and Risk Management

4. Mean absolute deviation models • Trades off the mean absolute deviation measure of risk against portfolio reward Financial Optimization and Risk Management

5. Mean absolute deviation models • The model is formulated as a linear program, large scale portfolios can be optimized using LP software • When returns are normally distributed the variance and mean absolute deviation are equivalent risk measures The model is formulated in the absolute positions • Notations: • Initial portfolio value, budget constraint • Future portfolio value • Mean of future portfolio value Financial Optimization and Risk Management

6. Mean absolute deviation models • Tradeoff between mean absolute deviation and expected portfolio value • How to solve this? Multidimensional integrals here. • No explicit functional form like in Markowitz problem. Only numerical solution is possible • Two possible approaches: • Specialized sampling optimization procedures • SCENARIO OPTIMIZATION with finite number of scenarios Financial Optimization and Risk Management

7. Scenario optimization for mean absolute deviation models • Finite number of scenarios: • No multidimensional integrals anymore. BUT, what about the objective function? It is still difficult to process directly. • Answer: let us reformulate it as a linear programming problem using auxilliary variables Financial Optimization and Risk Management

8. Scenario optimization for mean absolute deviation models • New functions: positive and negative deviations of portfolio from the mean • where • Similar to option payoffs Financial Optimization and Risk Management

9. Scenario optimization for mean absolute deviation models • Auxilliary variable for each scenario: • Deviation of portfolio from its mean for each scenario • Minimization of mean absolute deviation Financial Optimization and Risk Management

10. Scenario optimization for mean absolute deviation models • Maximization of portfolio value with constraints on risk: • Parameter w traces efficient frontier Financial Optimization and Risk Management

11. Scenario optimization for mean absolute deviation models • Different weights for upside potential and downside risk • Weights sum up to one Financial Optimization and Risk Management

12. Scenario optimization for mean absolute deviation models • Tracking models • Limits on maximum downside risk • Tracking index (or liabilities) Financial Optimization and Risk Management

13. Scenario optimization for regret models • Random target: index, competition, etc. • Regret function • Regret is positive when portfolio outperforms the target and negative otherwise • Our context for regret: portfolio value Financial Optimization and Risk Management

14. Scenario optimization for regret models • Decomposition of regret • Upside regret: measure of reward • Downside regret: measure of risk • Probability that regret does not exceed some threshold value: Financial Optimization and Risk Management

15. Scenario optimization for regret models • Expected downside regret against potfolio value • Scenario optimization model: Financial Optimization and Risk Management

16. Scenario optimization for regret models • e-regret models • Minimization of expected downside e-regret Financial Optimization and Risk Management

17. Scenario optimization for regret models • Portfolo optimization with e-regret constraints Financial Optimization and Risk Management

18. Value at Risk in portfolio optimization • Loss function • Probability that loss does not exceed some threshold • Probability of losses strictly greater than some threshold Financial Optimization and Risk Management

19. Value at Risk in portfolio optimization • Relation between different quantities Financial Optimization and Risk Management

20. Value at Risk in portfolio optimization • Distribution of returns of Long Term Capital Management Fund Financial Optimization and Risk Management

21. Value at Risk in portfolio optimization • Conditional Value at Risk Financial Optimization and Risk Management

22. Value at Risk: examples Sample VaR of Schlumberger, Morris and Commercial Metals portfolio, 95% probability, 1 trading day VaR, % blue - 500 trading days, red - 2000 trading days Fraction of portfolio 2 portfolio 1: (0.51283, 0, 0.48717), portfolio 2: (0, 0.67798, 0.32202) Financial Optimization and Risk Management

23. VaR and CVaR: comparison CVaR may give very misleading ideas about VaR VaR/CVaR fraction of portfolio 2 Financial Optimization and Risk Management

24. Value at Risk: examples Sample VaR of Ford/IBM portfolio VaR, % Gaivoronski & Pflug (1999) blue - 500 trading days, red - 2000 trading days Fraction of IBM stock portfolio 1: (0.51283, 0, 0.48717), portfolio 2: (0, 0.67798, 0.32202) Financial Optimization and Risk Management

25. Computational approach • Filter out or smooth irregular component • Use NLP software as building blocks • Matlab implementation with links to other software Financial Optimization and Risk Management

26. Smoothing (SVaR) Financial Optimization and Risk Management

27. Properties of the coefficients Financial Optimization and Risk Management

28. Why a special smoothing? • Avoid exponential growth of computational requirements with increase in the number of assets • In fact for SVaR it grows linearly Financial Optimization and Risk Management

29. Smoothed Value at Risk (SVaR) VaR fraction of portfolio 2 Financial Optimization and Risk Management

30. SVaR: larger smoothing parameter VaR fraction of portfolio 2 Financial Optimization and Risk Management

31. Mean-Variance/VaR/CVaR efficient frontiers return VaR 500 ten days observations Financial Optimization and Risk Management

32. Financial Optimization and Risk Management

33. Mean-Variance/VaR/CVaR efficient frontiers return CVaR Financial Optimization and Risk Management

34. Mean-Variance/VaR/CVaR efficient frontiers return StDev Financial Optimization and Risk Management

35. We developed capability to compute efficiently VaR-optimal portfolios Now what? - Serious experiments with portfolios of interest to institutional investor • 8 Morgan Stanley equity price indices for US, UK, Italy, Japan, Argentina, Brasil, Mexico, Russia • 8 J.P. Morgan bond indices for the same markets • time range: January 1, 1999 – May 15, 2002 • totally 829 daily price data • A nice set to test risk management ideas: 11 September 2001, Argentinian crisis July 2001, … • more than 80000 mean-VaR optimization problems solved Financial Optimization and Risk Management

36. Turbulent times … Financial Optimization and Risk Management

37. Turbulent times … Financial Optimization and Risk Management

38. In-sample experiments • Compute efficient frontiers from daily price data • 250 days time window • nonoverlapping 1 day observations • overlapping 60 days observations Financial Optimization and Risk Management

39. In-sample experiments: mean-VaR space Financial Optimization and Risk Management

40. In-sample experiments: mean-VaR space Financial Optimization and Risk Management