simple time-varying copula estimation wolfgang aussenegg, vienna university of technology

23. Workshop der Austrian Working Group on Banking and Finance12.-13.12.2008, Technische Universität Wien

Simple Time-Varying Copula Estimation

Wolfgang Aussenegg, Vienna University of Technology

Christian Cech, University of Applied Sciences bfi Vienna

AWG 23, 12.12.2008, Vienna. Aussenegg and Cech, Simple Time-Varying Copula Estimation 22

• Most models that assess daily market risk assume a multivariate Gaussian distribution of the risk factor changes (e.g. returns on stocks, commodities, zero-coupon bonds).

• However, empirical evidence shows that

– daily (univariate) asset returns are not normally distributed but display „heavy tails“.

– the dependence structure (the copula) is non-Gaussian, as a higher probability of joint extreme co-movements is observed.

marginal distributions:non-Gaussian

C

Copula:non-Gaussian

Introduction

multivariate distribution:non-Gaussian


Introduction

• The present article examines the appropriateness of a time-varying– Gaussian copula– Student t copula

• To calibrate the copulas on a daily basis, the 250 most recent return observations of the Eurostoxx 50 and the Dow Jones Industrial 100 index returns are employed (“rolling window”). this corresponds to the (minimum) amount of trading days that have to be used according to the Basel regulations for market risk.

• Contributions of this article– There are „calm“ periods where the use of a Gaussian copula

seems justifiable and „stormy“ periods where this is not the case.– Using a “setup” that is currently used in market risk management

departments (i.e. a rolling window of 250 trading days as estimation basis), the use of a Student t copula is very promising.

– Periods with a high probability of joint extreme co-movements tend to coincide with time periods of

• high volatility in the markets• high correlation of the markets’ returns


Copula approaches

• Copulas are functions that combine (univariate) marginal distributions to a (multivariate) joint distribution.

• Sklar’s theorem: a n-dimensional joint distribution F(x) where x = (x1, x2, ..., xn) may be expressed in terms of the joint distribution’s copula C and its marginal distribution functions F1, F2, ..., Fn as

F(x) = C(F1(x1), F2(x2), ..., Fn(xn))

• The copula function C is by itself a multivariate distribution function with uniform marginal distributions on the interval U1 = [0, 1],

C: U1n → U1

• We examine the goodness-of-fit of two elliptical copulas with parameters– Gaussian copula:

• correlation matrix P– Student t copula:

• correlation matrix P• degrees of freedom (scalar parameter)

the lower , the higher is the probability of joint extreme co-movements

• The Gaussian copula is a special case of the Student t copula where → ∞.


Data

• 5,490 daily return observations of the Eurostoxx 50 and the Dow Jones Industrial 100 stock indices from January 2nd, 1987 to January 11th, 2008.

• Time periods with high volatility: e.g. 1987, 1997-2003

• Time periods with low volatility: e.g. 1993 to 1996, 2004 to 2006


Data

• Extreme returns tend to occur simultaneously: e.g. ‘Black Monday’ (October 19th, 1987) and the days thereafter, Asian crisis in 1997, Russian crisis in 1998, 9/11.


Empirical results:copula parameters

• We use a rolling window of 250 index returns (roughly one trading year) to estimate the copula parameters on a daily basis. Total number of trading days (n) = 5,240.

• The pseudo-log-likelihood method (Genest and Rivest 1993) is employed– Gaussian Copula: parameter G

– Student t Copula: parameters t and (upper bound: 100)– (G and t take similar values)

estimates for t and on a daily basis substantial variability



• Copula parameter t (“correlation”):– minimum value: 0.119 (in 1994)– maximum value: 0.667 (in 2003)

• Copula parameter (“probability of joint extreme co-movements”):– > 100 ( Gaussian copula seems appropriate):

07-10/1989, 09/1994-02/1996, 03-07/2005– < 5 ( Gaussian copula does not seem appropriate):

12/1987-02/1988, 08/1990-10/1992, 09/2001-03/2004, 07/2006-06/2007

• Likelihood ratio test (H0: Gaussian Copula, HA: Student t copula)Rejection of the null hypothesis of a Gaussian copula at the– 5% significance level: 66.4% of the trading days– 1% significance level: 50.5% of the trading days



• Examination of the relationship betweenmarket volatility and degrees of freedom .

• Model 1:

• Model 2:

the higher the volatility, the lower the copula-parameter

tDJIt

Eurostoxxtt 210

tDJIt

Eurostoxxtt 10

Model 1 Model 2

coefficient t-value p-value coefficient t-value p-value

0 39.84 39.365 0.000 39.87 39.389 0.000

1 -89.29 -11.401 0.000 -74.21 -26.018 0.000

2 -56.66 -6.322 0.000

F-value 340.82 0.000 676.94 0.000

adj. R2 11.5% 11.4%

n 5,240 5,240



• Examination of the relationship between“correlation” t and degrees of freedom

• Model 3:

the higher the copula-parameter t , the lower the copula-parameter

tttt ,10

Model 3

coefficient t-value p-value

0 30.94 28.088 0.000

1 -43.17 -15.109 0.000

F-value 228. 27 0.000

adj. R2 4.2%

n 5,240



• Summary:

– There exist “calm” and “stormy” time periods as far as the probability of joint extreme co-movements is concerned.

– The probability of joint extreme co-movements tends to be high when

• The markets’ volatility is high• The correlation between markets’ returns is high


Empirical results: hit test

• The appropriateness of both a Gaussian and a Student t copula is examined using a hit test (back test, Kupiec 1995).

• For every single trading day, both the calibrated Gaussian copula CG and the Student t copula Ct are used to find uG and ut such that

CG(uG, uG) = 0.01 and Ct(ut, ut) = 0.01

• Interpretation of uG and ut :– Consider the values obtained for the first trading on which the copulas are

calibrated, December 18th, 1987: uG = 0.0561 and ut = 0.0389

– Recall that the probability that the Dow Jones index returns or the Eurostoxx 50 returns are, individually, below their 0.0561-quantiles, respectively below their 0.0389-quantiles, amounts to 5.61% respectively 3.89%.

– Assuming that the Gaussian copula is the true copula, we expect that with a 1% probability both index returns will be jointly below their 5.61%-quantiles.

– Assuming that the Student t copula is the true copula, we expect that with a 1% probability both index returns will be jointly below their 3.89%-quantiles.



• In addition to uG and ut an “empirical u”, uE , that is based purely on the empirical copula of the 250 most recent pairs of observations was calibrated

The graph below displays the estimates for uG and ut and uE .

On the top there are three lines with cross-markers. They indicate whether the observed returns of both indices are jointly below their uG-, ut-, respectively uE –quantiles (computed as empirical quantiles from the 250 most recent return observations, using linear interpolation).



• The Kupiec test compares the number of exceptions (both index-returns below theiru-quantiles) to the number exceptions one would expect if the models were correctly specified, i.e. 5,240∙0.01 = 52.4 exceptions.

Number of exceptions, Kupiec-test test-statistic LR and p-value (rejecting of the null-hypothesis of a correctly specified model).

• Student t copula: Do not reject H0

• “empirical u”: reject at the 5% significance level• Gaussian copula: reject at the 1% significance level

Kupiec

# exceptions LR p-value

Gaussian copula 83 15.33 0.000

Student t copula 60 1.06 0.302

‚empirical‘ approach 68 4.29 0.038


Conclusion (1/2)

• This article examines the ability of Gaussian and Student t copulas to predict the probability of joint extreme co-movements.

• Using more than 5,000 daily return observations of the Eurostoxx 50 and Dow Jones 100 stock indices, time-varying Gaussian and Student t copulas are calibrated on a rolling window of the 250 most recent pairs of return observations.

• There are time-periods when the assumption of a Gaussian copula seems adequate while there are other “stormy” time periods when the hypothesis of a Gaussian copula is rejected in favor of a Student t copula.


Conclusion (2/2)

• The Gaussian copula is inadequate when– the market volatility is high– the index returns are highly correlated

• A hit test shows that the Gaussian copula underestimates the probability of joint strongly negative returns of both indices. The null hypothesis that the Gaussian copula accurately predicts the probability of joint strongly negative returns can be rejected at the 1% significance level, while for the Student t copula this hypothesis cannot be rejected.

• These findings are in line with the empirical literature that find that the Gaussian copula underestimates the probability of joint extreme co-movements.

• Hence, the Student t copula seems a promising first step to improve market risk models and it will be well understood by practitioners that already know how to interpret correlation matrices.

simple time-varying copula estimation wolfgang aussenegg, vienna university of technology

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