1. Multivariate Extremes, Aggregation and Risk Estimation By Michel M. Dacorogna Risk Measures & Risk Management for High Frequency Data Workshop Eindhoven, 6 - 8 March 2006

2. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 2 Research Team Höskuldur Ari Hauksson Michel M. Dacorogna Thomas Domenig Ulrich A. Müller Gennady Samorodnitsky Work done while at Olsen & Associates

3. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 3 Overview Multivariate extreme value theory The empirical tails of extreme values for FX rates Risk management with correlated extremes Conclusion

4. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 4 Univariate Extreme Value Theory The celebrated Fisher-Tippett Theorem states that, if the Extreme Value Distribution (EVD) exists then it is either a Fréchet or a Weibull or a Gumbel distribution The generalized extreme value distribution is determined by a single parameter 1/ 

5. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 5 Returns of Financial Assets It is generally accepted that financial returns have Fréchet EVD with 2  4 These distributions have heavy tails and not all the moments exist The n-th moment only exists if n < a Generally, the second moment and thus the standard deviation exists for financial returns

6. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 6 Multivariate Extreme Value Theory A multivariate EVD is completely determined by the univariate marginal EVD and a dependence function describing the dependence between the variables This dependence function lives in a d-1 dimensional space, unlike the copula, which lives in d dimensions In two dimensions the dependence function is one dimensional

7. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 7 Multivariate Extreme Value Theory (II) A distribution is regularly varying, in n dimension, if there exists a constant a > 0 and a vector  with values in S d-1, the unit sphere in R d, such that the following limit exists for all x > 0 where denotes vague convergence on S d-1 and P  is the distribution of 

8. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 8 Vague Convergence and Regular Variation A sequence of probability measures (  n ) is said to vaguely converge to a probability measure  if for all sets A such that we have Regularly varying means that, asymptotically, the distribution in polar coordinates can be represented by a product measure of the spectral measure P Q and a radial measure, which has a power decay

9. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 9 Low Frequency versus High Frequency Risks Risk management is not interested in one minute logarithmic price changes but rather in daily, weekly or monthly returns We need to find the relationship between the risk estimated on short time horizon return and that based on long time horizon return Modern risk management is mainly interested in the tails of the distribution (99% quantile) The question reduces to: How do the tails behave under aggregation?

10. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 10 Tail under Aggregation Let X 1 and X 2 be two regularly varying random variables in R d with tail index a and spectral measures. Define Y=X 1 +X 2. Assume that Then Y is regularly varying and its spectral measure is a convex linear combination of

11. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 11 A Model of Multivariate Distributions Elliptic distributions are a popular choice for modelling financial assets They are closed under linear combinations and marginal distributions (useful for portfolio) We want to find out which from the elliptic distributions or the regularly varying distributions capture the actual dependence structure in the tails

12. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 12 Elliptic Distributions A random variable X is elliptic if there exists a constant vector  and a positive definite matrix  such that the random variable Y=  -1/2 (X-  ) is spherically distributed Spherically distributed i.e. invariant under rotation The matrix  is a constant multiple of the covariance matrix and  is the mean

13. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 13 Elliptic Distributions (II) The conditioned variable is also elliptic when  s is defined as In particular, X and have the same correlation matrix. Therefore the correlation as a function of s should be constant.

14. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 14 Exploring the Empirical Tails We consider 10 minutes to biweekly returns of the foreign exchange rates The returns are defined as We study 12 years from January 1st, 1987 to December 31st, 1998

15. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 15 Empirical Setting We have more than 630,000 data points for 10 minutes and 210,000 for 30 minutes The time series from the market are unevenly spaced in time: we use linear interpolation to obtain a regular time series We study USD/DEM, USD/CHF, USD/JPY and GBP/USD

16. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 16 Spatial Dependence We examine the spatial dependence of the tail structure with three different statistical analyses: 1.Conditional correlation 2.Symmetric/Antisymmetric exceedence probabilities 3.Spectral measure as a function of the angle

17. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 17 Conditional Correlation First, we fit an elliptical distribution to the entire data set. We then examine the correlation of the data outside an ellipse We find that, in all the cases, the correlation increases as we get further into the tails Financial assets are more strongly dependent when the market is in an excited state Thus, we have established that the spatial dependence in the tails is not well captured by elliptical distributions fitted to the full distribution

18. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 18 Empirical Results for the Conditional Correlation This figure shows the correlation of the data lying outside an ellipse. The quantile indicates the fraction of data points lying inside the ellipse, the complement of W s. The data used is 10 minute (solid), 30 minute (dotted), 2 hour (short-dashed) and daily (long-dashed) returns. All currencies are quoted against the US Dollar.

19. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 19 Symmetric / Anti-symmetric Exceedence Probabilities Let X and Y be two univariate random variables and let x q and y q denote the q-th quantile of X and Y respectively The positive symmetric exceedence probabilities are the following limit The anti-symmetric exceedence probabilities are defined in a similar way for (-X,Y) and (X,-Y). The negative symmetric exceedence probabilities are defined in a similar way for (-X,-Y)

20. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 20 Symmetric / Anti-symmetric Exceedence Probabilities (II) If these limits (symmetric and antisymmetric) are all zero we say that X and Y are asymptotically independent Normal and Student-t are asymptotically independent Our empirical study shows limits that are clearly greater than 0 for the positive/negative symmetric exceedence probabilities: there is dependence in the tails of these processes

21. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 21 Empirical Results for the Symmetric Exceendence Probabilities Symmetric exceedence probabilities as a function of the quantile. The data used is 10 minute (solid), 30 minute (dotted), 2 hour (short-dashed) and daily (long-dashed) returns. All currencies are quoted against the US Dollar.

22. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 22 The Spectral Measure According to the theorems above the spectral measure captures completely the dependence structure of the EVD We compute it by estimating the density of  conditional on R (radius) being in the 99% quantile The empirical studies show that probability mass is more concentrated in the first and third quadrant, consistent with the symmetric exceedence probabilities

23. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 23 Empirical Results for the Spectral Density First Quadrant Third Quadrant

24. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 24 Spectral Measure and Lagged Returns The measures are very similar for all frequencies of the returns, consistent with our theorem A study of the spectral measure of a lagged time series versus a non- lagged time series shows that the two variables are independent in the tails This indicates that the GARCH effect is not present in the extremes. It is a phenomenon concentrating in the middle of the distribution

25. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 25 Elliptic Distributions and Financial Returns The spatial dependence in the tails is not well captured by elliptical distributions Optimal portfolios computed using elliptical distributions are sub- optimal in case of extreme movements in the market It confirms an old saying among traders: “Diversification works the worst when one needs it the most”

26. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 26 Consequences for Risk Management The tail index and the spectral measure can be estimated from the high frequency time series (X i ) The scale and location of the tail need, however, to be estimated from the low frequency data for risk management An alternative is to scale them up from those of the high frequency time series

27. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 27 Risk Measures Value-at-Risk (VaR) is the most popular risk measure in risk management VaR is not always subadditive and an alternative measure has been proposed: the Expected Shortfall (ES) We examine how these two measures scale under aggregation

28. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 28 Scaling of Risk Measures We compute the VaR and the ES at the 99% quantile as function of the return frequencies We fit straight lines to these points on a double logarithmic scale The variable k is the scaling exponent

29. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 29 Scaling Behavior of the VaR

30. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 30 Scaling Behavior of the ES

31. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 31 Scaling Exponent for VaR and ES DEMJPYGBPCHFMean VaR0.47 0.490.460.47 ES0.450.440.460.440.45

32. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 32 Minimizing the Risk of a Portfolio The scaling exponent is different than 0.5 for Brownian motion We investigate the Allocation of the capital between two foreign currencies to minimize the risk for an US investor Risk is here defined as the VaR and the ES respectively We find the parameter such that a portfolio with in one currency and 1- in the other minimizes the risk

33. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 33 Minimizing the Risk of a Portfolio (II) Both the VaR and ES are computed for a two week horizon of the allocation parameter The risk measures are computed with hourly, daily and biweekly data Daily and hourly curves are similar in shapes and lie at the same level Biweekly data are too few to be reliable

34. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 34 Portfolio Minimization with VaR

35. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 35 Portfolio Minimization with ES

36. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 36 Minimizing the Risk of a Portfolio (III) The general level of risk is correctly estimated by the hourly data The curves for hourly and daily data for ES are smoother than those for VaR Doing a risk minimization using VaR as a measure is dangerous as VaR is not capable of detecting concentration of risk

37. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 37 Conclusions Regularly varying rather than elliptical distributions are suited for capturing dependence structure in the tails HF Data considerably increase quality of estimates of extreme events and can be used to analyze dependence between various risks From the HF estimates it is possible to scale up the risk on longer time horizons Optimal portfolio against extreme risk should be analyzed with HF data using expected shortfall as risk measure

38. Eindhoven, 07.03.06 © Converium Michel Dacorogna Correlated Extremes Page 38 Reference Multivariate Extremes, Aggregation and Risk Estimation. Quantitative Finance, January 2001, vol. 1, page 79-95.