Tranche Declension.

800px-CDO_-_FCIC_and_IMF_Diagram

With the CDO (collateralized debt obligation) market picking up, it is important to build a stronger understanding of pricing and risk management models. The role of the Gaussian copula model, has well-known deficiencies and has been criticized, but it continues to be fundamental as a starter. Here, we draw attention to the applicability of Gaussian inequalities in analyzing tranche loss sensitivity to correlation parameters for the Gaussian copula model.

We work with an RN-valued Gaussian random variable X = (X1, … , XN), where each Xj is normalized to mean 0 and variance 1, and study the equity tranche loss

L[0,a] = ∑m=1Nlm1[xm≤cm] – {∑m=1Nlm1[xm≤cm] – a}

where l1 ,…, lN > 0, a > 0, and c1,…, cN ∈ R are parameters. We thus establish an identity between the sensitivity of E[L[0,a]] to the correlation rjk = E[XjXk] and the parameters cj and ck, from where subsequently we come to the inequality

∂E[L[0,a]]/∂rjk ≤ 0

Applying this inequality to a CDO containing N names whose default behavior is governed by the Gaussian variables Xj shows that an increase in name-to-name correlation decreases expected loss in an equity tranche. This is a generalization of the well-known result for Gaussian copulas with uniform correlation.

Consider a CDO consisting of N names, with τj denoting the (random) default time of the jth name. Let

Xj = φj-1(Fjj))

where Fj is the distribution function of τj (relative to the market pricing measure), assumed to be continuous and strictly increasing, and φj is the standard Gaussian distribution function. Then for any x ∈ R we have

P[Xj ≤ x] = P[τj ≤ Fj-1j(x))] = Fj(Fj-1j(x))) = φj(x)

which means that Xj has standard Gaussian distribution. The Gaussian copula model posits that the joint distribution of the Xj is Gaussian; thus,

X = (X1, …., Xn)

is an RN-valued Gaussian variable whose marginals are all standard Gaussian. The correlation

τj = E[XjXk]

reflects the default correlation between the names j and k. Now let

pj = E[τj ≤ T] = P[Xj ≤ cj]

be the probability that the jth name defaults within a time horizon T, which is held constant, and

cj = φj−1(Fj(T))

is the default threshold of the jth name.

In schematics, when we explore the essential phenomenon, the default of name j, which happens if the default time τis within the time horizon T, results in a loss of amount lj > 0 in the CDO portfolio. Thus, the total loss during the time period [0, T] is

L = ∑m=1Nlm1[xm≤cm]

This is where we are essentially working with a one-period CDO, and ignoring discounting from the random time of actual default. A tranche is simply a range of loss for the portfolio; it is specified by a closed interval [a, b] with 0 ≤ a ≤ b. If the loss x is less than a, then this tranche is unaffected, whereas if x ≥ b then the entire tranche value b − a is eaten up by loss; in between, if a ≤ x ≤ b, the loss to the tranche is x − a. Thus, the tranche loss function t[a, b] is given by

t[a, b](x) = 0 if x < a; = x – a, if x ∈ [a, b]; = b – a; if x > b

or compactly,

t[a, b](x) = (x – a)+ – (x – b)+

From this, it is clear that t[a, b](x) is continuous in (a, b, x), and we see that it is a non-decreasing function of x. Thus, the loss in an equity tranche [0, a] is given by

t[0,a](L) = L − (L − a)+

with a > 0.