Post Modern Portfolio Theory

1. Market Crashes and Modeling Volatile MarketsProf. Svetlozar (Zari) T.RachevChief-Scientist, FinAnalytica Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe Department of Statistics and Applied Probability University of California, Santa BarbaraThalesian seminar , London - December 2, 2009

2. Post Modern Portfolio Theory

3. MPT “translation” for Volatile Markets Normal (Gaussian) Distributions Correlation Sigmas Sharpe Ratios BS Option Pricing Markowitz Optimal Portfolios Fat-tailed Distributions Tail & Asymmetric Dependence Expected Tail Loss STARRPerformance Tempered-Stable Option Pricing Fat-tail ETL Optimal Portfolios Real World Old World

4. Models Map

5. Agenda • The Fat-tailed Framework • Univariate model (single asset) • Subordinated models • Stable model • Dependence • Risk and Performance measures • Applications • Option pricing - Some extension of the main fat-tailed model: Tempered Stable models • Modeling market crashes • Risk monitoring • Portfolio management and optimization

6. Fat-tail Modeling Framework

7. Phenomena of Primary Market Drivers - 1 • Univariate level • Time-varying volatility • Fat-tails • Asymmetry • Long-range dependence (intra-day) DJ Daily returns

8. Subordinator (g(W)) < 1 Subordinator (g(W)) < 1 g <0 Fat-tailed Fat-tailed “On the days when no new information is available, trading is slow and the price process evolves slowly. On days when new information violates old expectations, trading is brisk, and the price process evolves much faster”. Clark (1973) Emp.

9. Subordinator > 1

10. Stable Family • Rich history in probability theory • Kolmogorov and Levy (1930-1950), Feller (1960’s) • Long known to be useful model for heavy-tailed returns • Mandelbrot (1963) and Fama (1965)

11. Fat Tails Study: 17,000+ factors 85%, 95%, 97.5%, and 99%VaR tested

12. Fat Tails Study: Factors Breakdown

13. Alpha Tail Parameter:Varies Across Assets & Time • Important to: • Distinguish tail risk contributors and diversifiers • Changes in the market extreme risk after removing GARCH

14. Tail parameter Alpha for 41 indices after removing GARCH effect /May 15th 2009/ There is NO universal tail index!

15. Phenomena of Primary Market Drivers - 2 Zero tail dependence Gaussian copula • Tail Dependence

16. Dependence Models AsymmetricDependence

17. Dependence Models Bi-variate Normal Fat-tailed indices Gaussian Copula Fat-tailed indices Fat-tailed copula Modeling of Extreme Dependency in market crashes is critical for taking correct investment decisions Observed returns in Q3 1987 F is the multivariate cdf, C is the copula function and Fi are the one-dimensional cdf.

18. Risk & Performance Measures Symmetric risk penalty Downside risk penalty Downside risk penalty and upside reward

19. Why not Normal ETL?

20. Summary • Fat-tailed world is a complex one: • GARCH is not enough • Fat-tails are not enough • Copula choice is important • Fat-tails change across assets and across time • Beware of pseudo-fat-tailed models • Fat-tailed ETL as a risk measure is important

21. Application 1 – Option pricing Stable and Tempered Stable Distributions

22. Tempered Stable Models Introduction • The stable model does not allow for unique equivalent martingale measure • Take a stable model and make the very end of the tails lighter (still much heavier than the Gaussian) • All moments exist • No-arbitrage option pricing exists

23. Tempered Stable

24. Tempered Stable

25. Map of Tempered Stable Distributions Kim-Rachev (KR) Modified Tempered Stable (MTS) Normal Tempered Stable (NTS) Smoothly Truncated Stable (STS) Rapidly Decreasing Tempered Stable (RDTS) Classical Tempered Stable(CTS)

26. Incorporating GARCH Effect other tempered stable models

27. Is GARCH Enough? … No! • QQ plots between the empirical residual and innovation distributions for daily return /data for IBM/

28. Option Prices and GARCH Models SPX Call Prices (April 12, 2006) where N is the number of observation, is the n-th price determined by the simulation, and is the n-th observed price.

29. Model Universe • We studied the full spectrum of tractable (infinitely divisible) models • We see that Stable ARMA-GARCH is the best choice to model primary risk drivers • We propose a form of tempered stable (RDTS) for option pricing

30. The Option Pricing Models Universe

31. Application 2 - Modeling Market Crashes

32. Daily Returns: S&P 500 Index

33. Crash Probability: Black Monday On October 19 (Monday), 1987 the S&P 500 index dropped by 23%. Fitting the models to a data series of 2490 dailyobservations ending with October 16 (Friday), 1987 yields the following results:

34. Crash Probability: U.S. Financial Crisis On the September 29 (Monday),2008 the S&P 500 index dropped by 9%. Fitting the models to a data series of 2505 dailyobservations ending with the September 26 (Friday), 2008 yields the following results:

35. S&P Backtest

36. Application 3 – Risk Monitoring

37. Backtest Example • Long-short stock portfolio • 99% VaR backtest was run from 8/1/2007 to 5/15/2008 (206 days) • 250 rolling window used to fit the models • Models: • Historical method • Normal method • Constant Volatility • EWMA for Cov matrix • Asymmetric Stable with Copula • Constant Volatility • Volatility Clustering

38. Model Comparison • Quantitative - Number of exceedances • Average - must be on average 2 • Number of exceedances above 4 /95% CI is 0-4/ • Checked on portfolio and industry level • Qualitative • Visual check of VaR evolution vs returns Av. # of exceedances % Industries VaR rejected

39. Risk Backtest Fat-tailed VaR with constant volatility provides long-term equilibrium VaR Fat-tailed VaR with volatility clustering provides dynamic short-term view of the tail risk (VaR) Both are important! Returns Normal 99 Normal 99 EWMA Asym Stable Fat-tail Copula Asym Stable Fat-tail Copula Vol Clustering

40. Application 4 – Portfolio Management and Optimization

41. Portfolio Risk Budgeting The expression for marginal contribution to ETL is and the resulting risk decomposition: • Marginal Contribution to Risk Standard Approach: St Dev ETL:

42. Portfolio Optimization • Flexibility in problem types, a very general formulation is where the first ETL is of a tracking-error type, the second one measures absolute risk and l ≤ Aw ≤ u generalizes all possible linear weight constraints If future scenarios are generated, there are two choices: • Linearize the sample ETL function and solve as a LP • Solve as a convex problem

43. Summary • Modeling Fat-tailed world is a complex task BUT crucial for: • Option pricing • Explaining volatility smile • Identifying statistical arbitrage opportunities • Crash warning indicators • Helps identify changes in the market structure faster • Risk monitoring • Realistic understanding of risk and its evolution • Portfolio construction and optimization • Achieve higher risk-adjusted returns

44. Q&A… Thank you!