CT8, April 2016, Question 9: Why can't one use transition probability matrix?

[zusammen]

Hi,

I'm doing Question 9, April 2016, CT8, which asks students to calculate the probability that the company is in default at time 2. Why can't I construct a transition probability matrix, then multiply it by itself to arrive at the result? And when is a transition probability matrix appropriate for calculating this kind of question?

 

[Mark Willder]

Hi

Yes, you could construct a matrix and then multiply it by itself as we have just two time steps. It would give exactly the same result. However, most of the entries in the matrix aren't required to answer this question, so it's simpler to follow the alternative approach given in the solution.

The matrix would look like:

0.88 0.10 0.02
0.05 0.65 0.30
0.00 0.00 1.00

and all that is required to answer the question is to multiply the first row by the third column.

Best wishes

Mark

 

[zusammen]

Hi

Yes, you could construct a matrix and then multiply it by itself as we have just two time steps. It would give exactly the same result. However, most of the entries in the matrix aren't required to answer this question, so it's simpler to follow the alternative approach given in the solution.

The matrix would look like:

0.88 0.10 0.02
0.05 0.65 0.30
0.00 0.00 1.00

and all that is required to answer the question is to multiply the first row by the third column.

Best wishes

Mark

Hi Mark,

Thanks for the clarification. I have some other questions. I actually constructed the matrix like this

-0.12 0.1 0.02
0.05 -0.35 0.3
0 0 0

following the generator matrix of the JLT model (section 5, chapter 19 of the CMP).

How can your matrix is an identity one but the JLT generator matrix have the sum of each row equals to zero? Are they different?

 

[Mark Willder]

Hi Mark,

Thanks for the clarification. I have some other questions. I actually constructed the matrix like this

-0.12 0.1 0.02
0.05 -0.35 0.3
0 0 0

following the generator matrix of the JLT model (section 5, chapter 19 of the CMP).

How can your matrix is an identity one but the JLT generator matrix have the sum of each row equals to zero? Are they different?

Hi

Chapter 19 describes two different matrices for the JLT model.

First of all it shows the transition matrix with the transition probabilities. This is the matrix that I have used. The top left hand corner is the probability of starting in the first state and still being in the first state one time step later. So in this case 1 - 0.1 - 0.02 = 0.88.

Secondly it shows the generator matrix with the transition intensities. This is the matrix that you have tried to use. To use this matrix you need to take its exponential, which can be done by considering the exponential as an infinite series (so you can't just square it as I think you tried to do). In this matrix the top left hand corner is the sum of the other entries in the first row multiplied by minus one, ie -0.1 -0.02 = -0.12 as you suggest.

Unfortunately you can't really use your approach in this question as you have been given transition probabilities (similar to p and q in mortality) rather than intensities (similar to forces of mortality). The generator matrix will only work in the exam if there is a simple relationship between the matrix and its square to enable simplification of the exponential expansion.

I hope this helps to clarify how the different matrices are used.

Best wishes

Mark

 

[Bill SD]

Thanks Mark. A related Q to your post: CM2 Paper B Apr 2021 Q3(ii) asks "Construct, for each year, the 1-year transition rate matrix".
Having recently taken CS2, when i see the word 'rate', the question implies a generator matrix with transition intensities (and so rows sum to 0) rather than probabilities (where rows would sum to 1). But from the answer it is clear that examiner simply wanted a transition matrix of probabilities. Does 'transition rate' usually refer to a probability rather than a a transition intensity?

 

[Alvin Kissoon]

In CM2 this could refer to either intensity or probability, but the question will indicate which one they are looking for. The hint in this case is "using your answer from part (i)" , which was the expected number of assets moving between credit states; therefore using that information we could determine the probabilities.