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Scenario Optimization
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Contents
Introduction Mean absolute deviation models Regret models Value at Risk in optimal portfolios
Financial Optimization and Risk ManagementProfessor 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 rl of risky variables r (prices, exchange rates, etc.) with probabilities pl, l=1,…,N
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Mean absolute deviation models Trades off the mean absolute deviation measure of risk against portfolio
reward
Financial Optimization and Risk ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor 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 ManagementProfessor Alexei A. Gaivoronski
Scenario optimization for regret models Expected downside regret against potfolio value
Scenario optimization model:
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Scenario optimization for regret models -regret models
Minimization of expected downside -regret
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Scenario optimization for regret models Portfolo optimization with -regret constraints
Financial Optimization and Risk ManagementProfessor 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 ManagementProfessor Alexei A. Gaivoronski
Value at Risk in portfolio optimization Relation between different quantities
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Value at Risk in portfolio optimization Distribution of returns of Long Term Capital Management Fund
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Value at Risk in portfolio optimization Conditional Value at Risk
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Value at Risk: examples
0 0.2 0.4 0.6 0.8 11.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5VaR, %
blue - 500 trading days, red - 2000 trading days
Sample VaR of Schlumberger, Morris and Commercial Metals portfolio, 95% probability, 1 trading day
portfolio 1: (0.51283, 0, 0.48717), portfolio 2: (0, 0.67798, 0.32202)Fraction of portfolio 2
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
VaR and CVaR: comparison
0 0.2 0.4 0.6 0.8 16
6.5
7
7.5
8
8.5
9
9.5
10CVaR may give very misleading ideas about VaR
VaR/CVaR
fraction of portfolio 2
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Value at Risk: examples
Gaivoronski & Pflug (1999)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.55
1.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2VaR, %
Fraction of IBM stockblue - 500 trading days, red - 2000 trading days
portfolio 1: (0.51283, 0, 0.48717), portfolio 2: (0, 0.67798, 0.32202)
Sample VaR of Ford/IBM portfolio
Financial Optimization and Risk ManagementProfessor 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 ManagementProfessor Alexei A. Gaivoronski
Smoothing (SVaR)
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Properties of the coefficients
Financial Optimization and Risk ManagementProfessor 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 ManagementProfessor Alexei A. Gaivoronski
Smoothed Value at Risk (SVaR)
0 0.2 0.4 0.6 0.8 11.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
fraction of portfolio 2
VaR
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
SVaR: larger smoothing parameter
0 0.2 0.4 0.6 0.8 11.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
fraction of portfolio 2
VaR
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Mean-Variance/VaR/CVaR efficient frontiers
4.5 5 5.5 6 6.5 7 7.5 8 8.50.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
VaR
return
500 ten days observations
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Mean-Variance/VaR/CVaR efficient frontiers
6 7 8 9 10 11 120.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9return
CVaR
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Mean-Variance/VaR/CVaR efficient frontiers
3 3.5 4 4.5 5 5.5 6 6.50.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9return
StDev
Financial Optimization and Risk ManagementProfessor 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, 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
We developed capability to compute efficiently VaR-optimal portfolios
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Turbulent times …Morgan Stanley Equity Indices* USA MSDUUS Index USD
0.0000
200.0000
400.0000
600.0000
800.0000
1000.0000
1200.0000
1400.0000
1600.0000
1 56 111
166
221
276
331
386
441
496
551
606
661
716
771
826
881
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
Turbulent times …J.P. Morgan Bond Indices (Developed Markets and EMBI+ for
Emergin Markets) Argentina JPEMAR Index USD
0.0000
50.0000
100.0000
150.0000
200.0000
250.00001 55 10
916
321
727
132
537
9
433
487
541
595
649
703
757
811
865
Financial Optimization and Risk ManagementProfessor 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 ManagementProfessor Alexei A. Gaivoronski
In-sample experiments: mean-VaR space
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
1
2
3
4
5
6
7
8
9
VaR (%)
Ret
urn
(%)
Stdev-optimalCVaR-optimalVaR-optimal
Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski
In-sample experiments: mean-VaR space
0 1 2 3 4 5 60
1
2
3
4
5
6
7
8
9
10
CVaR (%)
Ret
urn
(%)
day 125, 12-Oct-2000
Stdev-optimalCVaR-optimalVaR-optimal