Overview
The Maximum Likelihood default model seeks to compute the maximum likelihood value of the default correlation given a dataset of historical defaults.
The model presented here assumes the latent variable model.
Default Correlations
The probability of the {% i^{th} %} asset defaulting given a realization of the latent factor {% Z %} is given by
{% p_i(Z) = Prob(A_i \leq \Phi^{-1}(p_i)|Z) %}
{% p_i(Z) = Prob(w_iZ + \sqrt{1-w_i^2} \epsilon_i \leq \Phi^{-1}(p_i)|Z) %}
{% = \Phi[\frac{\Phi^{-1}(p_i) - w_iZ}{\sqrt{1-w_i^2}}] %}
where {% \Phi %} is the
cumulative normal distribution.
Computations
The computations here follow those presented in Loffler
- Single Sector - this model assumes that all the loans have the same probability of default and a single default correlation.
- Multiple Sectors - this model assumes a finite set of loan categories, or sectors, each with a different probability of default. (i.e. the probability of default is the same for all loans in a sector, but difference across sectors)
Inference
Once the correlations have been measured, as above, it is often necessary to build a confidence interval around the measured value in order to understand how firm the estimate is. (see for example statistical inference)
The simplest way to do this is to utilize resampling methods to build the interval.