# How exactly are correlated defaults used/analyzed?

I've read a lot about correlate defaults but I can't seem to understand how they're used practically in a portfolio theory setting. Suppose I have two (?) companies, X and Y, and historic default information for each. How does knowing the correlation of defaults between X and Y help me with constructing my portfolio/loss curves? And how does this change if I have a basket of securities instead? I can add more details if needed.

• what does mean "constructing my portfolio/loss curves" ? – MJ73550 Jul 20 '16 at 11:46

The correlation does not play any role for a linear portfolio, such as a CDS index, However, for a portfolio with nonlinear dependence on the loss of underlying entities, such as the case for a CDO or an $m$-th to default swap, the correlation plays a role. Here, certain techniques such as copula may be needed, depending on the complexity of the structure.

The way to model the correlation may depend on the portfolio you have.

For example you may model the default distribution for a small portfolio of loans using a mixed binomial model. In this case the state of the economy determine the probability of default, but in each scenario the individual defaults events share the same probability (homogeneity). Thus, we have correlation in default events through common dependence on a factor. This is obtained by making the mixture parameter stochastic: p (the probability of default) is randomized. Let $p ̃$ be the stochastic default probability, which we assume, has a distribution on [0, 1] which can be continuous (i.e., given by a density f) or discrete. Conditional on the value of $p ̃$ the number of defaults follows a binomial distribution with probability parameter $p ̃$.

For example, in the discrete case, with two possible scenarios, the probability that k out of N default is given by:

$P(D=k)=f(p_1)\binom Nk p_1^k (1-p_1 )^{N-k}+f(p_2)\binom Nk p_2^k(1-p_1)^{N-k}$

The variance of the number of defaults is:

$Var[D]=Np ̅(1-p ̅ )+N(N-1)Var[p ̃]$

The first term is the variance that would apply if the default probability were fixed (i.e. without correlation). The second term is additional variance. Variation in $p ̃$ is an important contributor to the variance in the number of defaults because it determines the correlation between defaults events. Let $X_i$ denote the default indicator of issuer i (equal to 1 if i defaults and 0 otherwise), we have:

$ρ(X_i ,X_j )=\frac{Var[p ̃ ]}{p ̅(1-p ̅)} , i≠j$

Note that, since the variance in $p ̃$ determines the correlation between default event indicators, we can model the correlation by properly chose the distribution of $p ̃$.

It can be proved that for homogeneous large portfolios of loans the distribution of the loss function is equal to the distribution of $p ̃$. Merton model has an economic meaning and fit whit in this framework so it is often used for this purpose. Assume all firms have the same correlation $ρ$ and same default probability $p ̅$. The "Fraction of defaults" cumulative distribution for a Large homogeneous loan portfolio may then be written as:

$P[p(M)≤θ] = N(\frac{1}{\sqrt{ρ}} (N^{-1} (θ) \sqrt{(1-ρ)}-N^{-1} (p ̅ )))$

This is also known known as one-factor Gaussian copula model and has many applications with CDO.