multivariate extremes, aggregation and risk estimation by michel m. dacorogna risk measures &...
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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
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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
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Michel DacorognaCorrelated ExtremesOverview
Multivariate extreme value theory
The empirical tails of extreme values for FX rates
Risk management with correlated extremes
Conclusion
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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
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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
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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
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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
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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
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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?
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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.
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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
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Michel DacorognaCorrelated Extremes
Empirical Results for theSpectral Density
First Quadrant Third Quadrant
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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
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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”
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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
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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
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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
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Michel DacorognaCorrelated ExtremesScaling Behavior of the VaR
Eindhoven, 07.03.06
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Michel DacorognaCorrelated ExtremesScaling Behavior of the ES
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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
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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
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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
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Michel DacorognaCorrelated ExtremesPortfolio Minimization with VaR
Eindhoven, 07.03.06
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Michel DacorognaCorrelated ExtremesPortfolio Minimization with ES
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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
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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