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 = (X_{1}, … , 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}^{N}l_{m}1_{[xm≤cm]} – {∑_{m=1}^{N}l_{m}1_{[xm≤cm]} – a}

where l_{1} ,…, l_{N} > 0, a > 0, and c_{1},…, 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_{j}X_{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 = (X_{1}, …., X_{n})

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

τ_{j} = E[X_{j}X_{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}^{N}l_{m}1_{[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.