Some Remarks on T-copulas

Among the various distributions available, the Student-t distribution has gained popularity in these calculations for several reasons (as opposed to, say, α-stable distributions). The first is that for three or more degrees of freedom it possesses a finite variance, and so can be calibrated to the variance of observable data. The second is that t-variables are relatively easy and fast to generate for simulations.

However, one very desirable property that should be exhibited by any calculation of economic capital is the ability to capture concentrated risks. Put simply, asset movements -particularly large movements -should be correlated. Thus, it is necessary to generate correlated t-variables. A recent paper on this topic is [SL] -we refer the reader to this paper for the necessary background on t-copulas, and the references contained therein.

In this paper we examine three t-copulas in this context, in particular their properties regarding correlation and tail correlation.

Let X, Y ∼ N (0, 1) with correlation ρ(X, Y ) = ρ.

Typically, correlated Student-t distributions with n degrees of freedom, U and V , can be formed via the transformations:

where C is sampled from a chi-squared distribution with n degrees of freedom1

An alternative formulation is given by:

where C 1 and C 2 are independently sampled from a chi-squared distribution with n degrees of freedom. This formulation is suggested to be more desirable in [SL] as it gives rise to a product structure of the density function when ρ = 0.

However, we will show that this has a major impact on the correlation, and in particular the resulting bivariate distribution2 and tail correlation.

Another naïve method of constructing correlated t-variables, U and V , (assumed to have the same degrees of freedom) is the following: take uncorrelated t-variables, U and W , then put

(2.3) However, V will not have a t-distribution as the sum of two t-variables is not a t-variable. Note that for three degrees of freedom or more, the t-variable sums lie within the domain of attraction of the Normal distribution. However, since we are only performing one sum, the tail of the distribution is still a power law of order n. Despite this, the resulting distribution does posses some useful properties.

(2.1), (2.2), and (2.3) define the three t-copulas that we will examine. We refer to these t-copulas as being generated by the Same χ 2 , Independent χ 2 , and Correlated-t, respectively.

3 Independent χ 2

We will firstly examine the case of the Independent χ 2 t-variables. We now show that this construction has a major impact on the correlation as follows:

Assuming that A and B have the same distribution, we have

In fact, the amount by which the correlation is reduced, namely

can be determined explicitly. For the case where C 1 and C 2 have 3 degrees of freedom, this turns out to be 2/π ≈ 0.6366.

We now determine the amount by which the correlation is reduced by explicitly. We first begin with a calculation of the required moments -we could not find a convenient reference, and record it here for completeness: Lemma 1. The nth moment of the Inverse-Chi Distribution with ν degrees of freedom is given by:

If α = ν/2 and β = 2, then this is the chi-squared distribution with ν degrees of freedom.

We wish to make the transformation Y = 1/ √ X. Since this is a monotonic function, we use the transformation formula:

We now derive the formula for the n-th moment of Y . Firstly, note that

and we have:

Remark: Note that the moments for this distribution will not be defined when α -n/2 is a negative integer.

Consider now the case when α = ν/2 and β = 2:

as required.

Proposition 1. The factor by which the correlation is reduced by is given by

and for large ν we have

Proof: Let us first consider the case ν = 3:

Hence,

and

For general degrees of freedom we have

Remark: Compare (3.31) with the expression given in [SL], section 4.1, which is very similar, except that they (erroneously) give the square-root of this expression.

Using the properties of Beta functions and Stirling’s formula, we have that

as required.

To explicitly compute (3.31) for a given value of ν, we need to consider odd and even cases. We

We now provide empirical results for each of our three t-copulas. We simulated 1,000,000 observations using each of the three methods, and have provided below graph of the pdf’s of the distribution and the copulas. We have only considered here the case of the t-distribution with three degrees of freedom, and a base correlation of 0.9. (the graphs presented in this section are based on the first 5,000 observations) As can be clearly seen, the pdf’s of the Same χ 2 and Correlated-t are elliptical, but the pdf of the Independent χ 2 is quite splayed out.

We have also constru

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