Modeling Default Correlations

Overview


One of the critical aspects of managing a credit portfolio quantitatively is coming up with a measure for the correlations between defaults in the portfolio. Theoretically, if the portfolio is large enough, a zero correlation implies near zero risk, hence the correlation becomes a critical number in default modeling.

Default Correlations


The correlation between random variables {% X %} and {% Y %} is defined to be
{% Corr(X,Y) = \frac{Cov(X,Y)}{\sigma_x \sigma_y} %}
(see moments)



For an asset, default can be modeled as a Bernoulli Distributed Variable. For two Bernoulli distributed variables, the correlation between them is given by
{% cov(X,Y) = E(X,Y) - E(X)E(Y) = prob_{x,y} - prob_x \times prob_y %}
where {% prob_x %} is the probability of default of asset {% X %} and {% prob_{x,y} %} is the probability of joint default between {% X %} and {% Y %}. Then the correlation between {% X %} and {% Y %} , is given as
{% \rho _{xy} = \frac{prob_{xy} - prob_x \times prob_y }{\sqrt{prob_x(1-prob_x)prob_y(1-prob_y)}} %}

Measuring Joint Probability of Default


Measuring a default correlation is fairly hard. To simplify calculations, we often assume that the correlation is the same between any two loans. In addition to this assumption, we will assume also that the probability of default is the same for every loan. This is just a simiplifying assumption for the purpose of calculating the correlation. Once we have a correlation number, we will use then revert to using the modeled default probabilities.

We can start by calculating an average annula default rate, by averaging the yearly default rates of our portfolio:
{% \hat{prob} = \frac{1}{T} \sum_i^T \frac{D_t}{N_t} %}
where {% D_t %} is then number of defaults in year t, and {% N_t %} is the number of loans in that year.

Then, the number of different pairs of defaulted assets among D defaults is given by
{% \dbinom D2 = \frac{D_t(D_t -1)}{2} %}
The maximum number of possible default pairs out of N loans is
{% \dbinom N2 = \frac{N_t(N_t -1)}{2} %}
The joint default rate in year t is then
{% \frac{D_t(D_t -1)}{N_t(N_t -1)} %}
so the average joint default rate can be computed as the average of these values.

Topics


  • Latent Variables - a simplifying model for measuring and calculating correlations between defaults.
  • Portfolio Diversification - the effects of pooling a set of loans together on the distribution of losses due to default.

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