Multivariate Extremes, Aggregation and Risk Estimation

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

2. Research Team • Höskuldur Ari Hauksson • Michel M. Dacorogna • Thomas Domenig • Ulrich A. Müller • Gennady Samorodnitsky • Work done while at Olsen & Associates

3. Overview • Multivariate extreme value theory • The empirical tails of extreme values for FX rates • Risk management with correlated extremes • Conclusion

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/x=a

5. Returns of Financial Assets • It is generally accepted that financial returns have Fréchet EVD with 2 < 1/x=a < 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. 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-1dimensional space, unlike the copula, which lives in d dimensions • In two dimensions the dependence function is one dimensional

7. Multivariate Extreme Value Theory (II) • A distribution is regularly varying, in n dimension, if there exists a constant a > 0 and a vector Qwith values in Sd-1, the unit sphere in Rd, such that the following limit exists for all x > 0 • where denotes vague convergence on Sd-1 and PQis the distribution of Q

8. Vague Convergence and Regular Variation • A sequence of probability measures (mn) is said to vaguely converge to a probability measure m if for all sets A such that we have • Regularly varying means that, asymptotically, the distribution in polarcoordinates can be represented by a product measure of the spectralmeasurePQ and a radial measure, which has a power decay

9. Low Frequency versusHigh 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 tailsof the distribution (99% quantile) • The question reduces to: How do the tails behave under aggregation?

10. Tail under Aggregation • Let X1 and X2 be two regularly varying random variables in Rd with tail index a and spectral measures . Define Y=X1+X2 . Assume that • Then Y is regularly varying and its spectral measure is a convex linear combination of

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. Elliptic Distributions • A random variable X is elliptic if there exists a constant vectorm and a positive definite matrixS such that the random variable Y=S-1/2(X-m) is spherically distributed • Spherically distributed i.e. invariant under rotation • The matrix S is a constant multiple of the covariance matrix and m is the mean

13. Elliptic Distributions (II) • The conditioned variable is also elliptic when Ws is defined as • In particular, X and have the same correlation matrix. Therefore the correlation as a function of s should be constant.

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. 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 unevenlyspaced in time: we use linear interpolation to obtain a regular time series • We study USD/DEM, USD/CHF, USD/JPY and GBP/USD

16. Spatial Dependence • We examine the spatial dependence of the tail structure with three different statistical analyses: • Conditional correlation • Symmetric/Antisymmetric exceedence probabilities • Spectral measure as a function of the angle

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. Empirical Results for theConditional 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 Ws. 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. Symmetric / Anti-symmetric Exceedence Probabilities • Let X and Y be two univariate random variables and let xq and yq 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. 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. Empirical Results for theSymmetric 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. The Spectral Measure • According to the theorems above the spectral measure capturescompletely the dependence structure of the EVD • We compute it by estimating the density of Q 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. Third Quadrant First Quadrant Empirical Results for theSpectral Density

24. Spectral Measure andLagged Returns • The measures are very similar for all frequencies of the returns, consistent with our theorem • A study of the spectral measure of a laggedtime 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. 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. Consequences for Risk Management • The tail index and the spectral measure can be estimated from thehigh frequency time series (Xi) • 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. 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. 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 variablekis the scaling exponent

29. Scaling Behavior of the VaR

30. Scaling Behavior of the ES

31. Scaling Exponent for VaR and ES

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 l such that a portfolio with l in one currency and 1-l in the other minimizes the risk

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. Portfolio Minimization with VaR

35. Portfolio Minimization with ES

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 smootherthan 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. 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. Reference Multivariate Extremes, Aggregation and Risk Estimation. Quantitative Finance, January 2001, vol. 1, page 79-95.