Raising Transition Probability Matrices to Non-Integer powers - a problem in Quantitative Finance
November 12, 2009
A few years back there was a big conference on Credit Risk organized by a big name conference organizer in Asia. Many top and high flying bankers and finance professionals participated in that conference. The lead roundtable was chaired by senior executive from a major rating agency. He was talking about how rating agencies were on top of all credit risk modelling and how his firm was using very sophisticated models to capture the default risks in fixed income securities and, lo and behold, the credit derivatives.
After a while, when the discussion turned to more practical matters such as capturing default probabilities and transition probabilities from historical data, one guy from the audience stood up and quite petulantly asked: "let's just forget these complex credit products and the instruments, let's just say we have a portfolio of corporate bonds and you have a one year transition probability matrix. How would you get the two year transition probability matrix from this?"
The speaker, this senior, quite high flying rating agency professional, got a bit irritated (one could see it on his face) and then answered: "that's simple, quite elementary. You simply multiply the first year rating transition matrix with itself and you get the two year rating transition matrix. If you want a three year transition matrix then simply multiply the one year transition matrix with itself and then again multiply that product with the one year matrix. All you are doing is raising the one year matrix to the power of two, three, etc. for finding out the two year, three year matrix."
The guy from the audience then asked: "...and what if you want to find the six month (half year) or a three month transition probability matrix from the one year matrix. How will you do it?"
A long silence! The chairman of the roundtable looked at him as if he was hit by a tornado. What happened later on is not important. But we learnt recently that the rating agency professional who saw a live ghost that day was eventually fired from his job and is now teaching finance in some business school.
Here's the problem: You have a portfolio of corporate bonds with four rating categories, A, B, C and D. The category D signifies default, i.e. that the bond has defaulted. This is a stylized, hypothetical example and all numbers are dummy numbers. A sample one year rating transition probability matrix can look like this:
It simply tells us the probability that a bond which is in a particular state (rating category) today, i.e. at time t = 0, will move to another state (rating category) at the next period, i.e. t = 1. Here, the next period signifies one year. Rating agencies estimate transition probability matrices for t = 1, 2, 3, ... which signify 1 yr, 2 yrs, 3yrs, etc.
Therefore, from the above matrix we can see that the probability that a bond which is presently (today) in rating category (state) A will be downgraded to the rating category C in one year is 8.5%. The probability that a bond which is currently in rating category C will default after one year is 0.4% and the probability that a bond which is in rating category B will be upgraded to the rating category A is 29%. The diagonals of the matrix gives us the probabilities that a bond which is in a particular rating category will remain in the same rating category after one year. And the last row of the matrix tells us that if a bond has already defaulted, i.e. currently in rating category D, it will remain there forever.
From the above one year rating transition probability matrix we can easily calculate the two year transition probability matrix. Simply multiply the one year matrix by itself.
The above matrix gives us the two year transition probabilities. From the above we can see that if a bond is currently in rating category A then there is a 55% probability that in two years it will be in the same rating category and 14% probability that it will be downgraded to rating category C.
The problem that the guy above was alluding to was this: from the above one year transition probability matrix you cannot directly deduce the half year transition probability matrix; this is because, while it is simply and mathematically trivial to raise a matrix to the power of 2, 3, 4, 5,...... and so on, i.e. integers, it is not possible to raise a matrix to powers of 0.5, 0.75, ...., i.e. non integer powers.
And this has got nothing to do with credit risk modelling or quantitative finance. It is purely a mathematical problem.
Getting 2 year transition probability matrix is easy. Raise the one year transition probability matrix to the power of 2. Same for 3 year probability matrix, simply raise the one year matrix to the power of 3. However, you cannot simply raise a one year transition probability matrix to the power of 0.5 (for six months) or 0.25 (for three months).
This is the real issue. Matrices, unlike numbers, cannot be simply raised to non-integer powers. Of course, you have a square root matrix, i.e. a matrix which in all ways behaves as if it is raised to the power of 0.5. You can have a matrix, L, which when multiplied by its transpose gives you the original matrix. That is the definition of a square root matrix. But then that is a different discussion.
Therefore, the challenge is how to estimate the six month or three month rating transition probability matrix from a one year rating transition probability matrix? The math for that is neither straightforward nor trivial.
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