Tail Dependence in Factor Copulas

Summary

Despite the lack of a closed-form copula, the tail dependence of a factor copula can be derived analytically using extreme value theory (EVT), exploiting the simple linear factor structure. When the common factor and idiosyncratic shocks have regularly varying (fat) tails with a common tail index, the factor copula generates non-zero tail dependence; asymmetry of the common factor (skew) makes upper and lower tail dependence differ — capturing crashes being more correlated than booms. Proposition 2 gives the explicit coefficients for the skew $t$-$t$ factor copula.

Overview

Tail dependence measures the probability that two variables are jointly extreme. For $X_i,X_j$ with marginals $G_i,G_j$:

$${\tau }_{ij}^L\equiv \underset{q\to 0}{\lim }\frac{\mathrm{Pr}[X_i\le G_i^{-1}(q),X_j\le G_j^{-1}(q)]}{q},{\tau }_{ij}^U\equiv \underset{q\to 1}{\lim }\frac{\mathrm{Pr}[X_i>G_i^{-1}(q),X_j>G_j^{-1}(q)]}{1-q}$$

Lower (upper) tail dependence is the probability both variables lie below (above) their $q$-quantile as $q\to 0$ ($q\to 1$), scaled by the probability that one does. The Normal copula has ${\tau }^L={\tau }^U=0$. The key insight: under regularly varying tails, the probability of the sum $\beta Z+\epsilon$ exceeding a diverging threshold equals the sum of the component exceedance probabilities (EVT), and joint exceedance of two sums is driven entirely by the shared component $Z$.

Main Content

Regularly varying tails (tail index)

A distribution $F$ has regularly varying tails with tail index $\alpha >0$ if its tail probabilities decay like a power law: $\mathrm{Pr}[\cdot >s]\approx A{s}^{-\alpha }$ as $s\to \infty$, where $A>0$ is a constant. Equivalently the density satisfies $f(s)\approx \alpha A{s}^{-\alpha -1}$, so $A={\lim }_{s\to \infty }f(s)/(\alpha s^{-\alpha -1})$. Fat-tailed distributions (Student’s $t$, skew $t$) are regularly varying; the tail index equals the degrees of freedom $\nu$. The Normal has thinner-than-power-law tails and is not regularly varying (giving zero tail dependence).

Proposition 1 — Tail dependence for a single-factor copula

Consider the factor copula generated by $X_i={\beta }_{i}Z+{\epsilon }_i$. Suppose $F_z$ and $F_{\epsilon }$ have regularly varying tails with a common tail index $\alpha >0$:

$$\mathrm{Pr}[Z>s]=A_z^U{s}^{-\alpha },\mathrm{Pr}[{\epsilon }_i>s]=A_{\epsilon }^U{s}^{-\alpha }\text{as }s\to \infty$$ $$\mathrm{Pr}[Z<-s]=A_z^L{s}^{-\alpha },\mathrm{Pr}[{\epsilon }_i<-s]=A_{\epsilon }^L{s}^{-\alpha }\text{as }s\to \infty$$

where $A_z^L,A_z^U,A_{\epsilon }^L,A_{\epsilon }^U>0$ are positive constants. Then:

(a) if ${\beta }_j\ge {\beta }_i>0$ (same sign, positive):

$${\tau }_{ij}^L=\frac{{\beta }_i^{\alpha }{A}_z^L}{{\beta }_i^{\alpha }{A}_z^L+A_{\epsilon }^L},{\tau }_{ij}^U=\frac{{\beta }_i^{\alpha }{A}_z^U}{{\beta }_i^{\alpha }{A}_z^U+A_{\epsilon }^U}$$

(b) if ${\beta }_j\le {\beta }_i<0$ (same sign, negative):

$${\tau }_{ij}^L=\frac{|{\beta }_i{|}^{\alpha }{A}_z^U}{|{\beta }_i{|}^{\alpha }{A}_z^U+A_{\epsilon }^L},{\tau }_{ij}^U=\frac{|{\beta }_i{|}^{\alpha }{A}_z^L}{|{\beta }_i{|}^{\alpha }{A}_z^L+A_{\epsilon }^U}$$

(c) if ${\beta }_i{\beta }_j=0$, or (d) if ${\beta }_i{\beta }_j<0$ (opposite signs), then both tail dependence coefficients are zero.

Interpretation: when loadings share sign and factor/idiosyncratic tail indices match, the copula has both upper and lower tail dependence. If $Z$ or $\epsilon$ is asymmetric (so $A^U\ne A^L$), then ${\tau }^U\ne {\tau }^L$ — the model captures differing probabilities of joint crashes vs joint booms. Note tail dependence depends only on the smaller loading $\min \{|{\beta }_i|,|{\beta }_j|\}$.

Boundary case (unequal tail indices)

If the tail index of $Z$ is strictly greater than that of $\epsilon$ and ${\beta }_i{\beta }_j>0$, then tail dependence equals one (the common factor dominates the joint tail). If the tail index of $Z$ is strictly less than that of $\epsilon$, then tail dependence is zero (idiosyncratic tails dominate).

Proposition 2 — Tail dependence for a skew $t$-$t$ factor copula

Consider the factor copula with $F_z=\text{Skew }t(\nu ,\lambda )$ (Hansen 1994) and $F_{\epsilon }=t(\nu )$ (standardized Student’s $t$). Then the tail indices of $Z$ and ${\epsilon }_i$ both equal $\nu$ (so $\alpha =\nu$), and the constants from Proposition 1 are:

$$A_z^L=\frac{bc}{\nu }{\left(\frac{b^2}{(\nu -2)(1-\lambda {)}^2}\right)}^{-(\nu +1)/2},A_z^U=\frac{bc}{\nu }{\left(\frac{b^2}{(\nu -2)(1+\lambda {)}^2}\right)}^{-(\nu +1)/2}$$ $$A_{\epsilon }^L=A_{\epsilon }^U=\frac{c}{\nu }{\left(\frac{1}{\nu -2}\right)}^{-(\nu +1)/2}$$

where

$$a=4\lambda c\frac{\nu -2}{\nu -1},b=\sqrt{1+3{\lambda }^2-a^2},c=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)\sqrt{\pi (\nu -2)}}$$

Substituting these constants into Proposition 1 gives the tail dependence coefficients for this copula. Because the idiosyncratic shock is symmetric ($A_{\epsilon }^L=A_{\epsilon }^U$) but the factor is skewed ($A_z^L\ne A_z^U$ when $\lambda \ne 0$), upper and lower tail dependence differ; with $\lambda <0$, lower tail dependence exceeds upper. When $\lambda =0$ the skew $t$ reduces to standardized $t$ ($a=0$, $b=1$), giving the symmetric $A_{\epsilon }^U=A_{\epsilon }^L$ form.

Proposition 3 — Tail dependence for a multi-factor copula

Consider the $K$-factor copula $X_i={\sum }_{k=1}^K{\beta }_{ik}{Z}_k+{\epsilon }_i$. Assume $F_{\epsilon },F_{z_1},\dots ,F_{z_K}$ have regularly varying tails with common tail index $\alpha >0$ and upper/lower coefficients $A_{\epsilon }^U,A_1^U,\dots ,A_K^U$ and $A_{\epsilon }^L,A_1^L,\dots ,A_K^L$. Then if ${\beta }_{ik}\ge 0$ for all $i,k$:

$${\tau }_{ij}^L=\frac{{\sum }_{k=1}^{K}1\{{\beta }_{ik}{\beta }_{jk}>0\}{A}_k^L{\beta }_{ik}^{\alpha }{\delta }_{L,ijk}^{\alpha }}{A_{\epsilon }^L+{\sum }_{k=1}^K{A}_k^L{\beta }_{ik}^{\alpha }},{\tau }_{ij}^U=\frac{{\sum }_{k=1}^{K}1\{{\beta }_{ik}{\beta }_{jk}>0\}{A}_k^U{\beta }_{ik}^{\alpha }{\delta }_{U,ijk}^{\alpha }}{A_{\epsilon }^U+{\sum }_{k=1}^K{A}_k^U{\beta }_{ik}^{\alpha }}$$

where the adjustment factors $\delta$ account for the (generally distinct) thresholds when loadings differ:

$${\delta }_{L,ijk}^{-1}\equiv \{\begin{array}{ll}\max \{1,{\gamma }_{L,ij}{\beta }_{ik}/{\beta }_{jk}\},& \text{if }{\beta }_{ik}{\beta }_{jk}>0\\ 1,& \text{if }{\beta }_{ik}{\beta }_{jk}=0\end{array},{\delta }_{U,ijk}^{-1}\equiv \{\begin{array}{ll}\max \{1,{\gamma }_{U,ij}{\beta }_{ik}/{\beta }_{jk}\},& \text{if }{\beta }_{ik}{\beta }_{jk}>0\\ 1,& \text{if }{\beta }_{ik}{\beta }_{jk}=0\end{array}$$ $${\gamma }_{L,ij}\equiv {\left(\frac{A_{\epsilon }^L+{\sum }_{k=1}^K{A}_k^L{\beta }_{jk}^{\alpha }}{A_{\epsilon }^L+{\sum }_{k=1}^K{A}_k^L{\beta }_{ik}^{\alpha }}\right)}^{1/\alpha },{\gamma }_{U,ij}\equiv {\left(\frac{A_{\epsilon }^U+{\sum }_{k=1}^K{A}_k^U{\beta }_{jk}^{\alpha }}{A_{\epsilon }^U+{\sum }_{k=1}^K{A}_k^U{\beta }_{ik}^{\alpha }}\right)}^{1/\alpha }$$

Only factors loading on both variables (${\beta }_{ik}{\beta }_{jk}>0$) contribute to joint tail dependence. Unlike the one-factor case, a general ranking of thresholds cannot be obtained, so the $\delta$ terms have no simple closed form (computed via the $\gamma$ expressions). This proposition is used to compute the implied inter- and intra-industry tail dependence in the empirical application.

Proof sketch (Appendix A — EVT mechanics)

The driving EVT fact (Hyung & de Vries 2007): for regularly varying tails, the exceedance probability of a sum equals the sum of component exceedance probabilities as the threshold diverges, $\mathrm{Pr}[\beta Z+\epsilon >s]\approx s^{-\alpha }(A_z^U{\beta }^{\alpha }+A_{\epsilon }^U)$, since $\mathrm{Pr}[\beta Z>s]\approx A_z^U(s/\beta {)}^{-\alpha }$. For the joint numerator, $\mathrm{Pr}[\beta Z+{\epsilon }_1>s,\beta Z+{\epsilon }_2>s]\approx \mathrm{Pr}[\beta Z>s,\beta Z>s]=s^{-\alpha }{A}_z^U{\beta }^{\alpha }$ — only the shared component $Z$ can drive both into the tail simultaneously (the independent ${\epsilon }_i$ jointly-exceeding is higher order $o(s^{-\alpha })$). The ratio gives ${\tau }^U$. When loadings differ, thresholds $s_1,s_2$ are matched so $G_1(s_1)=G_2(s_2)$, diverging at the same rate; the proof tracks which of $s_i/{\beta }_i$ is larger. Proposition 2 differentiates the regularly-varying density of Hansen’s (1994) skew $t$ to extract $A_z^U,A_z^L$ (verified via Mathematica).

Examples

Quantile dependence and the flexibility of skew $t$-$t$

Setup: Quantile dependence ${\tau }_q$ (the empirical proxy that converges to tail dependence as $q\to 0$ or $q\to 1$):

$${\tau }_q\equiv \{\begin{array}{ll}\frac{1}{q}\mathrm{Pr}[U_1\le q,U_2\le q],& q\in (0,0.5]\\ \frac{1}{1-q}\mathrm{Pr}[U_1>q,U_2>q],& q\in (0.5,1)\end{array},U_i\equiv G_i(X_i)\sim \text{Unif}(0,1)$$

Result (Figure 2): Symmetric copulas (Normal, $t$-$t$) have ${\tau }_q$ symmetric about $q=0.5$; fat-tailed factors raise ${\tau }_q$ near both tails. The skew $t(4)$-$t(4)$ factor copula generates weak upper but strong lower quantile dependence. Interpretation: Near a tail, an extreme observation in one variable is more likely to have come from the fat-tailed common factor $Z$ than from the thin-tailed idiosyncratic ${\epsilon }_i$, making an extreme in the other variable more likely. Negative skew concentrates this in the lower tail — exactly the “crashes more correlated than booms” feature wanted for equity returns.

Connections

• Factor Copula Construction — the latent structure from which these coefficients are derived.
• Multi-Factor and Block Dependence Structures — Proposition 3 supplies the implied tail dependence for the block model.
• Factor Copula Application - S&P 100 and Systemic Risk — uses Propositions 2 & 3 to report estimated tail-dependence coefficients.

See Also

• Factor Copulas - Overview
• Econometrics