Multivariate Extremes, Aggregation andRisk Estimation

By Michel M. Dacorogna

Risk Measures & Risk Management for High Frequency Data Workshop

Eindhoven, 6 - 8 March 2006

Eindhoven, 07.03.06

Page 2

© Converium

Michel DacorognaCorrelated ExtremesResearch Team

Höskuldur Ari Hauksson Michel M. Dacorogna Thomas Domenig Ulrich A. Müller Gennady Samorodnitsky

Work done while at Olsen & Associates

Eindhoven, 07.03.06

Page 3

© Converium

Michel DacorognaCorrelated ExtremesOverview

Multivariate extreme value theory

The empirical tails of extreme values for FX rates

Risk management with correlated extremes

Conclusion

Eindhoven, 07.03.06

Page 4

© Converium

Michel DacorognaCorrelated ExtremesUnivariate 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/

1/exp( (1 ) , 0( )

exp( exp( )), 0

x ifG x

x if

Eindhoven, 07.03.06

Page 5

© Converium

Michel DacorognaCorrelated ExtremesReturns 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 <

Generally, the second moment and thus the standard deviation exists for financial returns

Eindhoven, 07.03.06

Page 6

© Converium

Michel DacorognaCorrelated ExtremesMultivariate 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

Eindhoven, 07.03.06

Page 7

© Converium

Michel DacorognaCorrelated Extremes

Multivariate Extreme Value Theory (II)

A distribution is regularly varying, in n dimension, if there exists a constant > 0 and a vector with 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 P is the distribution of

v

, / .lim (.)v

t

P X tx X Xx P

P X t

v v

Eindhoven, 07.03.06

Page 8

© Converium

Michel DacorognaCorrelated Extremes

( ) ( )n A A

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 and a radial measure, which has a power decay

0A

Eindhoven, 07.03.06

Page 9

© Converium

Michel DacorognaCorrelated Extremes

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 tails of the distribution (99% quantile)

The question reduces to: How do the tails behave under aggregation?

Eindhoven, 07.03.06

Page 10

© Converium

Michel DacorognaCorrelated Extremes

1 2and .P P

1 2andP P

Tail under Aggregation

Let X1 and X2 be two regularly varying random variables in Rd with

tail index and spectral measures . Define Y=X1+X2 . Assume that

Then Y is regularly varying and its spectral measure is a convex linear combination of

1 2lim 0.rP X r X r

Eindhoven, 07.03.06

Page 11

© Converium

Michel DacorognaCorrelated ExtremesA 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

Eindhoven, 07.03.06

Page 12

© Converium

Michel DacorognaCorrelated ExtremesElliptic 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

Eindhoven, 07.03.06

Page 13

© Converium

Michel DacorognaCorrelated ExtremesElliptic 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.

sX

sX

1( , ) { ( ) ( ) }, 0s x x x s s

sX

Eindhoven, 07.03.06

Page 14

© Converium

Michel DacorognaCorrelated ExtremesExploring 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

ln( ) ln( )t t tt

x xr

t

Eindhoven, 07.03.06

Page 15

© Converium

Michel DacorognaCorrelated ExtremesEmpirical 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

Eindhoven, 07.03.06

Page 16

© Converium

Michel DacorognaCorrelated ExtremesSpatial 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

Eindhoven, 07.03.06

Page 17

© Converium

Michel DacorognaCorrelated ExtremesConditional 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

Eindhoven, 07.03.06

Page 18

© Converium

Michel DacorognaCorrelated Extremes

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 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.

Eindhoven, 07.03.06

Page 19

© Converium

Michel DacorognaCorrelated Extremes

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)

1lim q qq

Y y X x

Eindhoven, 07.03.06

Page 20

© Converium

Michel DacorognaCorrelated Extremes

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

Eindhoven, 07.03.06

Page 21

© Converium

Michel DacorognaCorrelated Extremes

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.

Eindhoven, 07.03.06

Page 22

© Converium

Michel DacorognaCorrelated ExtremesThe 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

Eindhoven, 07.03.06

Page 23

© Converium

Michel DacorognaCorrelated Extremes

Empirical Results for theSpectral Density

First Quadrant Third Quadrant

Eindhoven, 07.03.06

Page 24

© Converium

Michel DacorognaCorrelated Extremes

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 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

Eindhoven, 07.03.06

Page 25

© Converium

Michel DacorognaCorrelated Extremes

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”

Eindhoven, 07.03.06

Page 26

© Converium

Michel DacorognaCorrelated ExtremesConsequences for Risk Management

The tail index and the spectral measure can be estimated from the high 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

Eindhoven, 07.03.06

Page 27

© Converium

Michel DacorognaCorrelated ExtremesRisk 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

( ) ( )ES X E X P X

1( ) ( )VaR X P X P X

Eindhoven, 07.03.06

Page 28

© Converium

Michel DacorognaCorrelated ExtremesScaling 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 variableis the scaling exponent

( ) (1)( )VaR t VaR t

Eindhoven, 07.03.06

Page 29

© Converium

Michel DacorognaCorrelated ExtremesScaling Behavior of the VaR

Eindhoven, 07.03.06

Page 30

© Converium

Michel DacorognaCorrelated ExtremesScaling Behavior of the ES

Eindhoven, 07.03.06

Page 31

© Converium

Michel DacorognaCorrelated ExtremesScaling Exponent for VaR and ES

DEM JPY GBP CHF Mean

VaR 0.47 0.47 0.49 0.46 0.47

ES 0.45 0.44 0.46 0.44 0.45

Eindhoven, 07.03.06

Page 32

© Converium

Michel DacorognaCorrelated ExtremesMinimizing 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

Eindhoven, 07.03.06

Page 33

© Converium

Michel DacorognaCorrelated ExtremesMinimizing 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

Eindhoven, 07.03.06

Page 34

© Converium

Michel DacorognaCorrelated ExtremesPortfolio Minimization with VaR

Eindhoven, 07.03.06

Page 35

© Converium

Michel DacorognaCorrelated ExtremesPortfolio Minimization with ES

Eindhoven, 07.03.06

Page 36

© Converium

Michel DacorognaCorrelated ExtremesMinimizing 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

Eindhoven, 07.03.06

Page 37

© Converium

Michel DacorognaCorrelated ExtremesConclusions

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

Eindhoven, 07.03.06

Page 38

© Converium

Michel DacorognaCorrelated ExtremesReference

Multivariate Extremes, Aggregation and Risk Estimation. QuantitativeFinance, January 2001, vol. 1, page 79-95.