Scenario Optimization

Scenario Optimization

Financial Optimization and Risk ManagementProfessor Alexei A. Gaivoronski

Contents

Introduction Mean absolute deviation models Regret models Value at Risk in optimal portfolios


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


Mean absolute deviation models Trades off the mean absolute deviation measure of risk against portfolio

reward


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


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


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


Scenario optimization for mean absolute deviation models

New functions: positive and negative deviations of portfolio from the mean

where

Similar to option payoffs


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


Scenario optimization for mean absolute deviation models Maximization of portfolio value with constraints on risk:

Parameter traces efficient frontier


Scenario optimization for mean absolute deviation models Different weights for upside potential and downside risk

Weights sum up to one


Scenario optimization for mean absolute deviation models Tracking models

Limits on maximum downside risk

Tracking index (or liabilities)


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


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:


Scenario optimization for regret models Expected downside regret against potfolio value

Scenario optimization model:


Scenario optimization for regret models -regret models

Minimization of expected downside -regret


Scenario optimization for regret models Portfolo optimization with -regret constraints


Value at Risk in portfolio optimization Loss function

Probability that loss does not exceed some threshold

Probability of losses strictly greater than some threshold


Value at Risk in portfolio optimization Relation between different quantities


Value at Risk in portfolio optimization Distribution of returns of Long Term Capital Management Fund


Value at Risk in portfolio optimization Conditional Value at Risk


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


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


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


Computational approach

Filter out or smooth irregular component Use NLP software as building blocks Matlab implementation with links to other

software


Smoothing (SVaR)


Properties of the coefficients


Why a special smoothing?

Avoid exponential growth of computational requirements with increase in the number of assets

In fact for SVaR it grows linearly


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


VaR


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


VaR


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




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



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


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


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


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


In-sample experiments

Compute efficient frontiers from daily price data

250 days time window nonoverlapping 1 day observations overlapping 60 days observations


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


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

Date post:	06-Jan-2016
Category:	Documents
Upload:	kendra
View:	124 times
Download:	15 times

Scenario Optimization

Documents