Tranche Declension.

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 R^N-valued Gaussian random variable X = (X₁, … , X_N), where each X_j is normalized to mean 0 and variance 1, and study the equity tranche loss

L_[0,a] = ∑_m=1^Nl_m1_[xm≤cm] – {∑_m=1^Nl_m1_[xm≤cm] – a}

where l₁ ,…, l_N > 0, a > 0, and c₁,…, c_N ∈ R are parameters. We thus establish an identity between the sensitivity of E[L_[0,a]] to the correlation r_jk = E[X_jX_k] and the parameters c_j and c_k, from where subsequently we come to the inequality

∂E[L_[0,a]]/∂r_jk ≤ 0

Applying this inequality to a CDO containing N names whose default behavior is governed by the Gaussian variables X_j 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 τ_j denoting the (random) default time of the j^th name. Let

X_j = φ_j^-1(F_j(τ_j))

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

P[X_j ≤ x] = P[τ_j ≤ F_j^-1(φ_j(x))] = F_j(F_j^-1(φ_j(x))) = φ_j(x)

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

X = (X₁, …., X_n)

is an R^N-valued Gaussian variable whose marginals are all standard Gaussian. The correlation

τ_j = E[X_jX_k]

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

p_j = E[τ_j ≤ T] = P[X_j ≤ c_j]

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

c_j = φ_j⁻¹(F_j(T))

is the default threshold of the j^th name.

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

L = ∑_m=1^Nl_m1_[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.

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