Tail Dependence in the Bivariate t Distribution

Why Tail Dependence Matters

Two risks can have the same ordinary correlation yet behave differently in extreme conditions. For an insurer or financial institution, the key concern may be whether two loss variables that are both unusually large remain likely to be large together as the threshold becomes more severe. The bivariate Student t model is useful for studying this question because it allows asymptotic tail dependence: joint extreme outcomes need not become negligible relative to a single extreme outcome.

The t model has a degrees-of-freedom parameter that controls tail heaviness and a dependence parameter that controls the association between the two variables. Its tail-dependence coefficient combines both. A crucial implication is that even with a zero dependence parameter, a finite-degrees-of-freedom bivariate t model has positive upper and lower tail dependence.

Definition: Let \(X\) and \(Y\) have continuous marginal distribution functions \(F_X\) and \(F_Y\), and set \(U=F_X(X)\) and \(V=F_Y(Y)\). The upper tail-dependence coefficient is \(\lambda_U=\lim_{q\to1^-}P(V>q\mid U>q)\), when the limit exists. The lower tail-dependence coefficient is \(\lambda_L=\lim_{q\to0^+}P(V\le q\mid U\le q)\), when the limit exists. A positive coefficient means that the conditional probability of an equally extreme outcome in the other variable does not vanish as the threshold approaches the end of the distribution.

Using marginal probability levels makes these coefficients independent of the units and shapes of the individual margins, provided the margins are continuous. For example, replacing a loss variable by a strictly increasing transformation changes its numerical scale but not its percentile. This tutorial focuses on calculating the coefficients when the dependence is described by a bivariate Student t distribution.

The Bivariate Student t Formula

Let \((X,Y)\) have a bivariate Student t distribution with degrees of freedom \(\nu>0\) and dependence parameter \(\rho\), where \(-1<\rho<1\). Its two standardized marginal distributions are Student t with \(\nu\) degrees of freedom, and its scale matrix has diagonal entries 1 and off-diagonal entries \(\rho\). The parameter \(\rho\) describes the elliptical dependence structure. When \(\nu>2\), it is also the Pearson correlation; when \(\nu\le2\), the marginal variance is infinite or undefined, so Pearson correlation does not exist.

Theorem: For a nondegenerate bivariate Student t distribution with degrees of freedom \(\nu>0\) and dependence parameter \(-1<\rho<1\), the upper and lower tail-dependence coefficients are equal: \(\lambda_U=\lambda_L=2T_{\nu+1}\!\left(-\sqrt{\frac{(\nu+1)(1-\rho)}{1+\rho}}\right)\). Here \(T_{\nu+1}\) is the cumulative distribution function of a univariate Student t random variable with \(\nu+1\) degrees of freedom. Equivalently, by symmetry, \(\lambda_U=\lambda_L=2\overline T_{\nu+1}\!\left(\sqrt{\frac{(\nu+1)(1-\rho)}{1+\rho}}\right)\), where \(\overline T_{\nu+1}(z)=1-T_{\nu+1}(z)\).

The two forms of the formula are identical because the Student t distribution is symmetric about zero. The version using the upper-tail probability \(\overline T\) is often convenient for calculation: find the square-root argument, evaluate the one-sided t probability, and double it. Report a probability rounded to four decimal places unless a problem specifies a different precision.

Why the Formula Holds

A short conditional-distribution argument explains the formula’s structure. For the stated bivariate t model, the conditional distribution of \(Y\) given \(X=x\) is a Student t distribution with \(\nu+1\) degrees of freedom, location \(\rho x\), and scale \(\sqrt{(\nu+x^2)(1-\rho^2)/(\nu+1)}\). This conditional scale grows with the magnitude of \(x\). Thus, when \(X\) is extreme, uncertainty in \(Y\) grows too; that shared growth is the mechanism behind tail dependence.

For an upper threshold \(a\) tending to infinity, condition on \(X=as\), where \(s\ge1\). The conditional t statistic for \(Y>a\) tends to the threshold shown below. Also, for a univariate t variable with \(\nu\) degrees of freedom, the conditional distribution of \(X/a\) given \(X>a\) tends to a distribution on \(s\ge1\) with density \(\nu s^{-\nu-1}\). Combining these limits gives an integral expression for \(\lambda_U\).

\lambda_U = \int_1^\infty \nu s^{-\nu-1} \overline T_{\nu+1}\!\left[ \sqrt{\frac{\nu+1}{1-\rho^2}} \left(\frac{1}{s}-\rho\right) \right]\,ds = 2\overline T_{\nu+1}\!\left( \sqrt{\frac{(\nu+1)(1-\rho)}{1+\rho}} \right).

The first equality averages the limiting conditional probability of \(Y>a\) over the limiting distribution of \(X/a\) given \(X>a\). Evaluating that integral with the Student t density gives the second equality. The lower-tail result follows from the symmetry \((X,Y)\) and \((-X,-Y)\) of the centered bivariate t distribution. This derivation also shows why simply substituting \(\rho\) into a normal-model result would miss an important feature: the conditional scale here increases with \(|x|\).

Worked Calculations

Worked Example: Zero Dependence Parameter With Heavy Tails

A hypothetical model for two standardized loss indices uses a bivariate Student t distribution with \(\nu=1\) and \(\rho=0\). Calculate both tail-dependence coefficients and interpret the result.

Method. Apply the bivariate t formula. The degrees of freedom for the univariate t probability in the formula are \(\nu+1=2\). The square-root argument is \(\sqrt{2(1-0)/(1+0)}=\sqrt{2}=1.4142\).

For a \(t_2\) random variable, the upper-tail probability at \(x>0\) is \(\overline T_2(x)=\frac12(1-x/\sqrt{x^2+2})\). Substituting \(x=\sqrt2\) gives \(\overline T_2(\sqrt2)=\frac12(1-\sqrt2/\sqrt4)=\frac12(1-0.7071)=0.1464\), using unrounded values in the calculation.

Therefore \(\lambda_U=\lambda_L=2(0.1464)=0.2929\), rounded to four decimal places. The result is checked by noting that it lies between 0 and 1 and that the model is symmetric, so the upper and lower coefficients agree. Although \(\rho=0\), the coefficients are positive: zero dependence parameter does not imply tail independence in this finite-\(\nu\) t model. The common scale variation produces joint extreme outcomes.

Common mistake avoided: Setting the tail-dependence coefficient to zero just because \(\rho=0\). That conclusion applies to the Gaussian model with a nondegenerate correlation parameter, not to a finite-degrees-of-freedom bivariate t model.

Worked Example: Changing Degrees of Freedom

A risk analyst compares the hypothetical bivariate t models with \(\rho=0\) and degrees of freedom \(\nu=1\) and \(\nu=3\). Calculate the coefficient for each model and explain the effect of changing \(\nu\).

For \(\nu=1\), the argument and \(t\) degrees of freedom are \(\sqrt{2}\) and 2. As calculated from the \(t_2\) upper tail, \(\overline T_2(\sqrt2)=0.1464\), so \(\lambda_U=\lambda_L=2(0.1464)=0.2929\).

For \(\nu=3\), the argument is \(\sqrt{4(1-0)/(1+0)}=2\), and the relevant distribution is \(t_4\). Its upper-tail probability at 2 is \(\overline T_4(2)=0.0581\) (rounded to four decimals). Thus \(\lambda_U=\lambda_L=2(0.0581)=0.1161\), with the coefficient calculated using the unrounded tail probability.

The comparison is consistent with the model: at the same dependence parameter, the \(\nu=1\) model has heavier tails and a larger asymptotic tail-dependence coefficient than the \(\nu=3\) model. Both coefficients remain positive because both models have finite degrees of freedom. The computed probabilities are in the valid range, and upper and lower coefficients agree in each symmetric model.

Common mistake avoided: Comparing only the square-root arguments and forgetting that the t distribution also changes from \(t_2\) to \(t_4\). The argument and the degrees of freedom must both be updated when \(\nu\) changes.

Worked Example: Positive Dependence Parameter

A hypothetical pair of standardized portfolios is modeled with a bivariate Student t distribution having \(\nu=1\) and \(\rho=1/3\). Find the upper and lower tail-dependence coefficients, then compare them with the model having \(\nu=1,\rho=0\).

The formula uses a \(t_{\nu+1}=t_2\) distribution. Its square-root argument is \(\sqrt{2(1-1/3)/(1+1/3)}=\sqrt{2(2/3)/(4/3)}=\sqrt{1}=1\).

Using \(\overline T_2(x)=\frac12(1-x/\sqrt{x^2+2})\), we obtain \(\overline T_2(1)=\frac12(1-1/\sqrt3)=0.2113\). Therefore \(\lambda_U=\lambda_L=2(0.2113)=0.4226\), rounded to four decimal places. The coefficient is higher than the 0.2929 found for \(\rho=0\), as expected: with the degrees of freedom fixed, increasing \(\rho\) increases the formula’s tail coefficient. The two coefficients match because the model is symmetric.

Common mistake avoided: Using \(1-\rho\) in the numerator but omitting \(1+\rho\) from the denominator. The full ratio is required; changing it would produce the wrong t-tail threshold.

Interpreting the Parameters and Limits

For fixed \(\nu\), increasing \(\rho\) increases the coefficient: the square-root argument \(\sqrt{(\nu+1)(1-\rho)/(1+\rho)}\) decreases, and the upper-tail probability of a t distribution increases as its argument decreases. For fixed \(\rho\), reducing \(\nu\) generally makes the t tails heavier and increases tail dependence. As \(\nu\) tends to infinity, the t model approaches a Gaussian model; for \(-1<\rho<1\), its asymptotic tail-dependence coefficient tends to zero. At finite \(\nu\), the t model therefore captures a feature that a Gaussian dependence model with the same nondegenerate correlation does not.

These coefficients describe a limiting conditional probability, not the probability of joint exceedance at a particular practical threshold. For finite thresholds, the conditional probability can differ materially from its limit. Also, equal upper and lower coefficients do not mean every pair of risks has equal finite-threshold behavior; the equality follows from the centered, symmetric bivariate t model specified here.

Key takeaway: For a bivariate Student t model, calculate the square-root threshold \(\sqrt{(\nu+1)(1-\rho)/(1+\rho)}\), evaluate the upper tail of a \(t_{\nu+1}\) distribution, and double it. The result is both \(\lambda_U\) and \(\lambda_L\). Finite degrees of freedom imply positive asymptotic tail dependence even when the dependence parameter is zero.

Check Your Understanding

Use the bivariate Student t formula in each question. Give probabilities to four decimal places where needed.

A model has \(\nu=1\) and \(\rho=0\). State the upper and lower coefficients and explain why setting them to zero is incorrect. (Answer: \(\lambda_U=\lambda_L=0.2929\); finite degrees of freedom give positive t tail dependence even though \(\rho=0\).)
A model has \(\nu=1\) and \(\rho=1/3\). Calculate the square-root argument and both coefficients. (Answer: argument \(1\); \(\lambda_U=\lambda_L=0.4226\).)
A model has \(\nu=1\) and \(\rho=-1/3\). Calculate the square-root argument and both coefficients. (Answer: argument \(2\); \(\lambda_U=\lambda_L=0.1835\).)
Two models have \(\rho=0\), with \(\nu=1\) in the first and \(\nu=3\) in the second. Which has greater asymptotic tail dependence, and what are the coefficients? (Answer: the \(\nu=1\) model; coefficients are 0.2929 and 0.1161, respectively, for both upper and lower tails.)
A bivariate t model has \(\nu=1\), and an analyst reports that its dependence parameter \(\rho=0\) is the Pearson correlation. Identify the issue and state the correct interpretation. (Answer: at \(\nu=1\), marginal variance is undefined, so Pearson correlation does not exist; \(\rho=0\) is the model’s dependence parameter.)

What You'll Learn