StressTesting Better Portfolio Mgmt Steven P Greiner Ph

Agenda • Why do Stress-Testing? Governance, that’s why!! • Extreme-Event Stress-Testing • Going Non-Linear: Markov-Chain MC • Conclusions 2

Governance – Ethics – Survey Results + We are painfully aware of the public opinion towards the financial sector in the wake of continued financial crisis PRESENTATION FROM FACTSET RESEARCH SYSTEMS 3

Extreme Event Stress-Testing Practical Example 4

Some Stress-Testing Methodologies EXTREME EVENT 1) Begins with a risk model, you need some way of estimating correlations (covariance) across assets 2) Obtain the covariance (or factor returns) from some historical “stressed” market environment or your own innovation 3) Use this covariance to compute risks &/or these factor returns to compute returns on today’s portfolio 5 All data and charts sourced from Fact. Set Research Systems Inc.

You run a risk report and see the Va. R increase over the last several weeks and you think. . . Risk = <w*E*C*Et*wt> + <w*V(ε)*wt> Is this risk level change caused by trades (w), exposure changes (E), or market volatility (systemic risk) itself (C)? 6

Observations + 1 1/17 + 2 1/24 + 3 1/31 + 4 2/7 + 5 2/14 + 6 2/21 + 7 2/26 7

Recipe to Interpret Effects • Select several sequential weekly time periods • Compute 95% Va. R using all the combinations of actual portfolios, frozen portfolios (i. e. exposures) & covariance on those dates • Choose 7 weeks: one obtains a 7 X 7 matrix of exposure changes on one axis & covariance changes on the other 8 All data and charts sourced from Fact. Set Research Systems Inc.

Recipe to Interpret Effects • When exposures are fixed & covariance evolves, one observes impact of changing correlations • Covariance follows VIX • Allows observation of volatility impact 9 All data and charts sourced from Fact. Set Research Systems Inc.

Recipe to Interpret Effects • When covariance is frozen & exposures change, one observes pricing impact • prices detached from VIX • Implies exposure change causes increase in risk 10 All data and charts sourced from Fact. Set Research Systems Inc.

Recipe to Interpret Effects • Move further out to 99% Value-at-Risk • Even stronger affect out in the tail • Exposures dominating 11 All data and charts sourced from Fact. Set Research Systems Inc.

Recipe to Interpret Effects • Monitor difference between 99% and 95% Va. R • Observe tail widening over time • Though VIX muted. . ? ? • Exposures increasing risk though volatility is stable 12 All data and charts sourced from Fact. Set Research Systems Inc.

Conclusions. . . What’s Happening is. . . • Current 95% Va. R is increasing mildly => • Covariance isn’t resulting in the increased risk => • VIX volatility signals are subdued => • Rising tail risks are due to exposures changes (spreading of difference between 99% & 95% Va. R) => Implies increasing probability of event risk Q for PM’s: WOULD YOU DO ANYTHING? 13 All data and charts sourced from Fact. Set Research Systems Inc.

Markov Chain-MC Stress-Testing Practical Example 14

Correlations of “Stresses” with S&P 100 Drawback? Correlations tie directly to linear stress-testing 15

Some Stress-Testing Methodologies MARKOV-CHAIN MONTE-CARLO 1) Begins with a risk model, you need some way of estimating correlations across assets. Use when your subject to data starvation for tail estimates 2) Generate synthesized data that matches joint probability distribution between the stress & all risk model factors. . . simultaneously. . . to populate the tail 3) Calculate the “beta(s)” between stress & risk model factors: Factor = beta 1*stress + beta 2*stress 2 + others 4) For a given stress (i. e. -30%), compute a value of F given the applied stress & compute return estimate 16 All data and charts sourced from Fact. Set Research Systems Inc.

Markov Chain Monte-Carlo (MCMC) • Generates sequence of random variables from an “unknown” multi-variate probability density while incorporating the correlations from each variable with every other • Sequential values tend to be auto-correlated, so delete early trials • Optimize the search width parameter to achieve ~25% acceptance ratio • Especially useful for re-populating “tail” density • However, it requires “trial” density? ? ? 17

Use “Normal Projection” to create easy trial density 18 Multivariate Weibull Distributions for Asset Returns: I Yannick Malevergne & Didier Sornette; Finance Letters, 2004 2(6), 16 -32

Consider Bi-Modal Multi-Variate MCMC Example Empirical Pairs Plots (500 x 5) MCMC Replicates (2500 x 5) QA: Run Kolmogorov-Smirnov 2 -sample test that measures whether “x” and “y” are drawn from same distribution 19

Empirical Scatter Plot Close Up MCMC Reproduction 20

EURUSD joint with Risk Model Factors 21

MCMC EURUSD Forex Kolmogorov. Smirnov p-value is typically order of ~65% 22

MCMC JPYUSD Forex 23

MCMC Wheat Futures 24

MCMC Results allow for Non-Linear ST 25

Cooliolusions! Stress-Testing is good “Governance” • Should be part of the investment process and requires cooperation between RM & PM • Use it to complement traditional risk measures and to deploy your own insights • Shouldn’t solely be based on naive inputs alone. Let your inner “Michelangelo” out, and be creative with it Fact. Set offers complete system. . 26

…more examples 27