Overview
Maximum Likelihood
Setup
The model assumes that there are a set of loans, each of which belongs to one of a discrete set of loan classes (or loan grades). That is, the loans are placed into a set of buckets such that the default characteristics of loans within a given bucket is assume to be similar.
That is, it is assume that the probability of default of each loan within a bucket is the same, which is denoted as {% p_k %} for the probability of default in the kth bucket. Also, the correlation of default of one loan with another loan is determined solely by the buckets that each loan belongs to.
The model is built using the latent variable approach. That means that each bucket is assigned a weight, {% w_k %} which represents the weight assigned to the latent variable {% Z %}.
That is to say, each asset i in the the kth bucket is driven by a latent variable
{% A_{ik} = w_kZ + \sqrt{1 - w_k^2} \epsilon_{ik} %}
Constructing the Scorecard
The Likelihood of the number of defaults in the kth bucket at time t, where {% N_{kt} %}
is the number of loans in the kth bucket at time interval t, and {% D_{kt} %}
is the number of defaults in the kth bucket at time interval t, is given by
{% %}
{% L_{kt} = \int_{\infty}^{\infty} \binom {N_{kt}}{D_{kt}} p_k(Z)^{D_{kt}} (1-p_k(Z))^{N_{kt}-D_{kt}} d \Phi(Z)%}
then
{% L_k = \Pi_{t=1}^T L_{kt} %}
and likewise, the Likelihood at time t is
{% L_{t} = \int_{\infty}^{\infty} \Pi_{k=1}^K \binom {N_{kt}}{D_{kt}} p_k(Z)^{D_{kt}} (1-p_k(Z))^{N_{kt}-D_{kt}} d \Phi(Z)%}
The total Likelihood is given by
{% L = \Pi_{t=1}^T \int_{\infty}^{\infty} \Pi_{k=1}^K \binom {N_{kt}}{D_{kt}} p_k(Z)^{D_{kt}} (1-p_k(Z))^{N_{kt}-D_{kt}} d \Phi(Z)%}
As always, the log Likelihood is given by
{% loglikelihood = ln L %}
Implementation
Maxium Likelihood - the maximum likelihood library implements the calculation of the log likelihood.