Extreme Value Theory

Standard estimators of the dependence between assets are the correlation coefficient or the Spearman’s rank correlation for instance. However, as stressed by [Embrechts et al. ], these kind of dependence measures suffer from many deficiencies. Moreoever, their values are mostly controlled by relatively small moves of the asset prices around their mean. To cure this problem, it has been proposed to use the correlation coefficients conditioned on large movements of the assets. But [Boyer et al.] have emphasized that this approach suffers also from a severe systematic bias leading to spurious strategies: the conditional correlation in general evolves with time even when the true non-conditional correlation remains constant. In fact, [Malevergne and Sornette] have shown that any approach based on conditional dependence measures implies a spurious change of the intrinsic value of the dependence, measured for instance by copulas. Recall that the copula of several random variables is the (unique) function which completely embodies the dependence between these variables, irrespective of their marginal behavior (see [Nelsen] for a mathematical description of the notion of copula).

In view of these limitations of the standard statistical tools, it is natural to turn to extreme value theory. In the univariate case, extreme value theory is very useful and provides many tools for investigating the extreme tails of distributions of assets returns. These new developments rest on the existence of a few fundamental results on extremes, such as the Gnedenko-Pickands-Balkema-de Haan theorem which gives a general expression for the distribution of exceedence over a large threshold. In this framework, the study of large and extreme co-movements requires the multivariate extreme values theory, which unfortunately does not provide strong results. Indeed, in constrast with the univariate case, the class of limiting extreme-value distributions is too broad and cannot be used to constrain accurately the distribution of large co-movements.

In the spirit of the mean-variance portfolio or of utility theory which establish an investment decision on a unique risk measure, we use the coefficient of tail dependence, which, to our knowledge, was first introduced in the financial context by [Embrechts et al.]. The coefficient of tail dependence between assets X_i and X_j is a very natural and easy to understand measure of extreme co-movements. It is defined as the probability that the asset X_i incurs a large loss (or gain) assuming that the asset X_j also undergoes a large loss (or gain) at the same probability level, in the limit where this probability level explores the extreme tails of the distribution of returns of the two assets. Mathematically speaking, the coefficient of lower tail dependence between the two assets X_i and X_j , denoted by λ−_ij is defined by

λ−_ij = lim_u→0 Pr{X_i<F_i⁻¹(u)|X_j < F_j⁻¹(u)} —– (1)

where F_i⁻¹(u) and F_j⁻¹(u) represent the quantiles of assets X_i and X_j at level u. Similarly the coefficient of the upper tail dependence is

λ+_ij = lim_u→1 Pr{X_i > F_i⁻¹(u)|X_j > F_j⁻¹(u)} —– (2)

λ−_ij and λ+_ij are of concern to investors with long (respectively short) positions. We refer to [Coles et al.] and references therein for a survey of the properties of the coefficient of tail dependence. Let us stress that the use of quantiles in the definition of λ−_ij and λ+_ij makes them independent of the marginal distribution of the asset returns: as a consequence, the tail dependence parameters are intrinsic dependence measures. The obvious gain is an “orthogonal” decomposition of the risks into (1) individual risks carried by the marginal distribution of each asset and (2) their collective risk described by their dependence structure or copula.

Being a probability, the coefficient of tail dependence varies between 0 and 1. A large value of λ−_ij means that large losses occur almost surely together. Then, large risks can not be diversified away and the assets crash together. This investor and portfolio manager nightmare is further amplified in real life situations by the limited liquidity of markets. When λ−_ij vanishes, these assets are said to be asymptotically independent, but this term hides the subtlety that the assets can still present a non-zero dependence in their tails. For instance, two normally distributed assets can be shown to have a vanishing coefficient of tail dependence. Nevertheless, unless their correlation coefficient is identically zero, these assets are never independent. Thus, asymptotic independence must be understood as the weakest dependence which can be quantified by the coefficient of tail dependence.

For practical implementations, a direct application of the definitions (1) and (2) fails to provide reasonable estimations due to the double curse of dimensionality and undersampling of extreme values, so that a fully non-parametric approach is not reliable. It turns out to be possible to circumvent this fundamental difficulty by considering the general class of factor models, which are among the most widespread and versatile models in finance. They come in two classes: multiplicative and additive factor models respectively. The multiplicative factor models are generally used to model asset fluctuations due to an underlying stochastic volatility for a survey of the properties of these models). The additive factor models are made to relate asset fluctuations to market fluctuations, as in the Capital Asset Pricing Model (CAPM) and its generalizations, or to any set of common factors as in Arbitrage Pricing Theory. The coefficient of tail dependence is known in close form for both classes of factor models, which allows for an efficient empirical estimation.

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