Scenario Optimization

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Contents Introduction Mean absolute deviation models Regret models Value at Risk in optimal portfolios

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 r l of risky variables r (prices, exchange rates, etc.) with probabilities p l, l=1,…,N

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Mean absolute deviation models Trades off the mean absolute deviation measure of risk against portfolio reward

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski Scenario optimization for mean absolute deviation models Maximization of portfolio value with constraints on risk: Parameter  traces efficient frontier

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski Scenario optimization for mean absolute deviation models Tracking models  Limits on maximum downside risk  Tracking index (or liabilities)

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski Scenario optimization for regret models Expected downside regret against potfolio value Scenario optimization model:

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Scenario optimization for regret models  -regret models Minimization of expected downside  -regret

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Scenario optimization for regret models Portfolo optimization with  -regret constraints

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski Value at Risk in portfolio optimization Relation between different quantities

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Value at Risk in portfolio optimization Distribution of returns of Long Term Capital Management Fund

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Value at Risk in portfolio optimization Conditional Value at Risk

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Value at Risk: examples VaR, % blue trading days, red trading days Sample VaR of Schlumberger, Morris and Commercial Metals portfolio, 95% probability, 1 trading day portfolio 1: ( , 0, ), portfolio 2: (0, , ) Fraction of portfolio 2

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

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Value at Risk: examples Gaivoronski & Pflug (1999) VaR, % Fraction of IBM stock blue trading days, red trading days portfolio 1: ( , 0, ), portfolio 2: (0, , ) Sample VaR of Ford/IBM portfolio

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski Smoothing (SVaR)

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Properties of the coefficients

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski Smoothed Value at Risk (SVaR) fraction of portfolio 2 VaR

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski SVaR: larger smoothing parameter fraction of portfolio 2 VaR

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Mean-Variance/VaR/CVaR efficient frontiers VaR return 500 ten days observations

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Mean-Variance/VaR/CVaR efficient frontiers return CVaR

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Mean-Variance/VaR/CVaR efficient frontiers return StDev

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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, totally 829 daily price data - A nice set to test risk management ideas: 11 September 2001, Argentinian crisis July 2001, … - more than mean-VaR optimization problems solved We developed capability to compute efficiently VaR-optimal portfolios

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Turbulent times …

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Turbulent times …

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski 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 Professor Alexei A. Gaivoronski In-sample experiments: mean-VaR space

Financial Optimization and Risk Management Professor Alexei A. Gaivoronski In-sample experiments: mean-VaR space