See also: Credit risk, Insurance and finance risk analysis modeling introduction, Markov Chain models, Time series in ModelRisk
Markov chains are often used in finance to model the variation of corporations' credit ratings over time. Rating agencies like Standard & Poors and Moody's publish transition probability matrices that are based on how frequently a company that started with, say, a AA rating at some point in time, has dropped to a BBB rating after a year. Provided we have faith in their applicability to the future, we can use these tables to forecast what the credit rating of a company, or a portfolio of companies, might look like at some future time using matrix algebra.
Let's imagine that there are just three ratings: A; B and default, with the following probability transition matrix for one year:
We interpret this table as saying that a random A-rated company has an 81% probability of remaining A-rated, an 18% probability of dropping to a B-rating, and a 1% chance of defaulting on their loans. Each row must sum to 100%. Note the matrix assigns a 100% probability of remaining in default once one is there (called an absorption state). In reality, companies sometimes come out of default, but we keep this example simple to focus on a few features of Markov Chains.
Now let's imagine that a company starts with rating B and we want to determine the probability it has of being in each of the three states in 2 years. The possible transitions are:
Rating in 1 year (transition probability) |
Rating in 2 years (transition probability) |
Combined probability |
A (0.17) |
A (0.81) |
0.1377 |
A (0.17) |
B (0.18) |
0.0306 |
A (0.17) |
Default (0.01) |
0.0017 |
B (0.77) |
A (0.17) |
0.1309 |
B (0.77) |
B (0.77) |
0.5929 |
B (0.77) |
Default (0.06) |
0.0462 |
Default (0.06) |
A (0.0) |
0.0 |
Default (0.06) |
B (0.0) |
0.0 |
Default (0.06) |
Default (1.0) |
0.06 |
Transition scenarios and their probabilities for a two year period for a company with rating B
Thus, the probability that the currently B-rated company will have each rating after two years is:
P(A-rating) = 0.1377 + 0.1309 + 0.0 = 0.2686
P(B-rating) = 0.306 + 0.5929 + 0.0 = 0.6235
P(default) = 0.0017 + 0.0462 + 0.06 = 0.1079
The calculations get rather more tiresome when there are many possible states (not the three we have here), but fortunately we are simply performing a matrix multiplication. Excel's MMULT array function can do this for us quickly, as shown below: Cells D11:F11 give the above calculated probabilities and the table D10:F12 gives the probabilities for a company starting in any other state.
Example model Two_year_transition_matrix - calculating a two year transition matrix
Now let us imagine that we have a portfolio of 27 companies with an A-rating and 39 companies with a B-rating, and we would like to forecast what the portfolio might look like in two years. The figure below models the portfolio in three ways: using just Binomial distributions if that is all your simulation software has, using the Multinomial distribution, and using ModelRisk's VoseMarkovSample function.
Example model Joint_credit_ratings - modeling the joint distribution of credit ratings among a number of companies in a portfolio after two years
Methods 1 and 2 in this model have two limitations. The first is that the model can become very large if we want to forecast out many periods because we would have to repeat the MMULT calculation many times. The second, more important limitation, is that the model can only handle integer time steps. So, for example, our transition matrix may be for a year, but we might want to forecast out just 10 days (t = 10/365 = 0.027397). The VoseMarkovSample function removes these obstacles: if the transition matrix is positive definite, we can replace the value 2 in the function at Cells F19:F21 with any non-negative value as illustrated with the formula in Cells L19:L21.
We mentioned that in this model 'Default' is assumed to be an absorption state. This means that if a path exists from any other state (A-rating, B-rating) to the Default state then eventually all individuals will end up in Default. The model below shows the transition matrix for t = 1, 10, 50 and 200. You can see that for t = 50 years, there is about an 81% probability that an A-rated company will already have defaulted, and about 84% chance for a B-rated company. By 200 years, there is almost a 100% chance that any company will have already defaulted. The graph below this topic shows this affect as a time series model. If this Markov chain model is a reasonable reflection of reality one might wonder how it is that we have so many companies left. A crude but helpfully economic theory of business rating dynamics assumes that if a company loses its rating position within a business sector, a competitor will take its place (either a new company or an existing company changing its rating) so we have a stable population distribution of rated companies.
Example model Transition_matrix_large_t - transition matrices for large values of t showing the progressive dominance of the default absorption state
Example model Markov_tracking - Markov chain time series showing progressive dominance of Default state
Read on: Measures of risk - Value at Risk