Martin Goldberg Head of Model Validation Risk Architecture Citigroup

Stress Testingfor Market RiskAdvanced Stress Testing TechniquesRisk TrainingNew York, Nov 9, 2006 Martin Goldberg Head of Model Validation Risk Architecture Citigroup The analysis and conclusions set forth are those of the authors. Citigroup is not responsible for any statement or conclusion herein, and opinions or theories presented herein do not necessarily reflect the position of the institution.

Outline • Types of Stress Testing • Monte Carlo with calibrated distribution • Named scenarios • Historical events • Hypothetical events • Stressed Distribution • Contagion and Concentration • Quantile selection • Use in setting Economic Capital • Combining Stress Tests, VaR, and Risk Manager Estimates • Communicating results

Monte Carlo - Calibrated univariate distribution • Naïve • Use the same historical timeseries as for VaR (if you use HVAR), or the same covariance matrix (if you use MC VaR) • Run many times • Result is a bad day in the current market - not actually a stress event • Somewhat better • Use a univariate distribution with “fat tails” • Examples are Johnson [1], Tukey g×h[2], or Levy[3] • Many more parameters - is it practical? • Run many times - sometimes the worst case for a portfolio is not in the tails of the distribution of any market factor, but at an “interior point”

“Fat Tails” occur frequently in financial markets • Extreme events in the market occur much more frequently than normal (Gaussian) distributions would predict. • Resultant distribution are “leptokurtic” or have “fat tails” Normal distribution Fat tail event

Calibrated multivariate distribution • Naïve • Use the same correlations as for VaR - equivalent to a Gaussian copula • A Gaussian Copula has zero tail dependence - a very extreme move in one asset is never simultaneous with a similarly big move in any other asset. • This ignores “tail dependence” - also called “contagion” • Somewhat better • Use a parametric copula with tail dependence • Examples are in Joe’s book[4] • Almost impossible to extend to high dimensionality • Use the observed historical copula (“empirical copula”) and extrapolate • Unclear how to extrapolate if you use HVaR

Joint market moves can lead to extreme $$$ losses

Example of Tail Dependence - Brent/ Kerosene

Shortcomings of using a calibrated MC • If the assumed functional form is too simple, there are no stress events generated, or only unrealistic ones • If you add parameters, calibration becomes impractical • The past may not be a good predictor of future stress events • Stationarity is unlikely for extreme moves • If this could be done analytically by even a roomful of Nobel Prize winners, using calibrated MC, then LTCM would still be a powerhouse

Named Scenarios - Historical • Historical means that we look at the impact of past events if they happened again today. However, it needs careful application: • Relative vs. absolute changes • Structural changes in financial environment (e.g. EUR convergence) • New markets • Typically, named historical scenarios might include • US Stock Crash of 1987 • Russian Default • The selection should be appropriate to your portfolio

Named Scenarios - Hypothetical • Hypothetical means that we propose scenarios that have not necessarily occurred in the past. • Needs careful construction. • Might help to involve economists, traders, etc. in constructing plausible but unlikely scenarios • Examples (my own ideas - I have no idea if anyone uses these) • US Congress can’t pass budget - US defaults • China invades Taiwan • “Mr Fusion” - free electricity • Be sure to include knock-on effects on all other markets • Historical correlations are irrelevant here

Stressed Distributions • Rather than specify the change in each asset for scenarios, this involves “tweaking” the regular VaR scenario generator • HVaR • Exaggerate the size of selected day’s (or all days’) changes • Scaling need not be uniform across assets • Do for as many sets of tweaks as desired • MC VaR • Exaggerate selected volatilites as desired • Change correlations, but must preserve non-negative definite matrix • Run VaR engine on this stressed dataset • Advantage: Easy to implement • Disadvantage: Difficult to create many plausible but novel stress scenarios with appropriate contagion

Contagion Stress • In extreme events, the concept of correlation is as misleading as trying to use volatilities for these sudden jumps and regime shifts • Historical relationships may not be relevant in stress events • Do not use historical covariance as a proxy for contagion estimates • Crashes/Skyrocketing in one asset might be contagious to other assets (“Tail dependence”) anti-contagious (“flight to quality”), or unexpectedly irrelevant (“circuit breakers”) • Assessing these changes in relationships is more an art than a science

Concentration Stress • A different but important kind of stress test is when liquidity changes either alone or simultaneously with price jumps • If you have a large exposure that suddenly becomes illiquid, there may be no meaningful price, but your capital is in effect frozen • A recent concentration stress example is the downfall of Amaranth. • They kept buying the same futures contract as the price went up, but the price was only going up due to their buying. What percent of the liquid float of any given asset does your firm hold? • If you are very long (short) one strategy, what happens if that one collapses?

How stressful of a stress do you use? • Although much of stress scenario construction is subjective, there are usually corporate guidelines on what they mean by a stress event. • Value at Risk is by definition a 99% worst ten-day event • Basel 2 looks at 99.9% worst one-year events • AA firms such as Citigroup and JPMorganChase nominally set economic capital as being large enough to withstand 99.97% worst one-year stress losses • The worst year in a thousand is very bad. Stress events of this magnitude might involve revolutions, global political upheavals, and such. No corporation has lasted 10,000 years, or even 1,000.

How stressful of a stress do you really use? • Often, what the guidelines really mean is that you look at the realized volatility in the recent past, assume it is a stationary lognormal or Gaussian pdf, and scale it up accordingly. • This is a difficult question to ask the Policy Committee, but usually someone can indicate what level of stress is appropriate • Ask “them” if they really mean: • Same stress as the crash of 1987 • Ten times worse • Private ownership of assets is outlawed • The various scenarios should be roughly of equal severity so each of them is a meaningful exercise

Setting Economic Capital • Economic capital is defined as the value of assets your firm needs to withstand a specified level of stress losses and still avoid bankruptcy. • It could be • Max(VaR*multiplier, worst stress test result) • Average(VaR*multiplier, average stress test result) • Etc. • Something even more clever • This means that the stress scenarios have to be comparable in magnitude so that it is not clear which scenario will dominate next time.

Combining Stress Tests, VaR, and Risk Manager Estimates • In some firms, the risk managers periodically give a subjective estimate of how risky their business is, in terms roughly comparable to a narrowly targeted stress test. • When scaling VaR multipliers, stress tests, and RM estimates to be aggregatable, part of the art form is to not punish desks for hedging against shocks. It may help to compare the impact on random portfolios of the desk’s asset classes rather than on the actual hedged desk holdings, to ensure fairness.

Communicating results • Part of the stressing process is to try to safeguard against desks gaming the specifics of the scenarios, so some secrecy may be desirable. • Part of the process is analyzing the results to point out weak areas, inadvertent side bets, and helping to decide where extra hedging or diversification might be warranted. This is where the process adds the most value. • Senior management, the desks, and the regulators all may have a keen interest in some or all of the results, presented in some digestible form - this too is an art.

Conclusions • Stress testing is not easy • Formulaic approaches are not optimal • The past may not be a good proxy for the future • Decide what level of stress to aim for • Use multiple scenarios and kinds of stress test, but make them comparable in stressfulness • Be creative in designing, and get economists and research involved in designing the tests • Know your audience

References • Johnson NL, Kotz S (1970), Distributions in Statistics: Continuous Univariate Distributions - 1, John Wiley & Sons, NY • http://fic.wharton.upenn.edu/fic/papers/02/0225.pdf and http://fic.wharton.upenn.edu/fic/papers/02/0226.pdf • http://en.wikipedia.org/wiki/Levy_distribution • H. Joe, “Multivariate Models and Dependence Concepts” Chapman&Hall, 1997

Martin Goldberg Head of Model Validation Risk Architecture Citigroup

Martin Goldberg Head of Model Validation Risk Architecture Citigroup

Presentation Transcript

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