Credit Transition Matrix - Markov Chain
Overview
The credit transition matrix is an approach to modeling the credit dynamics of loans that are assigned discrete credit ratings (i.e. there are a finite
set of credit scores assigned to each loan).
Ratings are assigned to a loan based on its perceived risk of default. Ratings can change over time as the creditworthiness of a
borrower changes.
The transition matrix approach models the probability of a loan trainsitioning from one rating to another, and from any
rating to default. It makes the Markov assumption ,
(see
markov chains)
that is, the probability of a loan either defaulting or transitioning to any rating is dependent only on the
current rating of the loan. In particular, it does not depend on what the ratings of the loan were in the past.
Matrix Representation
The state of any loan at a particular time is represented as a column vector
{%
\begin{bmatrix}
rating1 \\
rating2 \\
rating3 \\
defaulted \\
\end{bmatrix}
%}
As an example, a loan would have the following representation if it were non-defaulted and having a rating of
rating2.
{%
\begin{bmatrix}
0 \\
1 \\
0 \\
0 \\
\end{bmatrix}
%}
Transition Probabilities as a Matrix
The probabilties of tranisitioning from one rating to another over a single period is arranged as a matrix.
{%
M = \begin{bmatrix}
p_{1,1} & p_{1,2} & p_{1,3} & p_{1,d} \\
p_{2,1} & p_{2,2} & p_{2,3} & p_{2,d} \\
p_{3,1} & p_{3,2} & p_{3,3} & p_{3,d} \\
0 & 0 & 0 & 1
\end{bmatrix}
%}
Here {% p_{1,j} %} is the probability of transitioning from rating i to rating j. d represents the defaulted state.
The probabilty matrix of trasitioning from one rating to another over n periods is given by {% M^n %}
Single Loan Calculation
When a loan's current rating is encoded as a vector {% \vec{v} %}, the probability of the loan being in any of the given ratings
(or defaulted) is then given by the
matrix product
{% probabilities = M^n \vec{v} %}
Examples
Smoothing Calculated Probabilities
Once a set of probabilities have been calculated for each risk grade, once may find that there appears to be noise in the curve for each risk grade when
plotted. Assuming that the probabilities should follow an exponential by grade, one may wish to smooth the probabilities by running an exponential regression,
and then using the resulting curve as the assumed probability curve.
(see
fitting an exponential)
