Rating Transition Matrix

FRM
#1

Consider the following one period transition matrix:

Initial Period State Next Period State

A B Default

A 95% 5% 0%

B 10% 80% 10%

Default 0% 0% 100%

If a company is originally in State A, What is the probability that company will have defaulted strictly before the fourth transition period from now ?

A) 0.875%

B) 0.5%

C) 1.375%

D) 1.875%

Can some one explain this problem in step by step procedure ?

#2

The correct answer is C) 1.375%.

Ignoring (for the moment) the 0% transitions, the possible three-transition future states are:

  • AAA (transition to state A, then transition to state A, then transition to state A)
  • AAB
  • AAD
  • ABA
  • ABB
  • ABD
  • ADA
  • ADB
  • ADD
  • BAA
  • BAB
  • BAD
  • BBA
  • BBB
  • BBD
  • BDA
  • BDB
  • BDD
  • DAA
  • DAB
  • DAD
  • DBA
  • DBB
  • DBD
  • DDA
  • DDB
  • DDD

However, AD has a probability of 0%, as do DA and DB, so the (non-zero probability) future states are:

  • AAA
  • AAB
  • ABA
  • ABB
  • ABD
  • BAA
  • BAB
  • BBA
  • BBB
  • BBD
  • BDD

Those in which the company defaults are:

  • ABD
  • BBD
  • BDD

The probability of

  • ABD is 95% × 5% × 10% = 0.475%
  • BBD is 5% × 80% × 10% = 0.400%
  • BDD is 5% × 10% × 100% = 0.500%

Thus, the probability of default is 0.475% + 0.400% + 0.500% = 1.375%.

#3

Kudos magician

#4

My pleasure.

(I went a bit overboard in the explanation, but I wanted to make sure that I covered everything. I hope that I understood correctly what you meant by “before the fourth transition period”.)

#5

thks for going overboard. that was required for concept clarity

#6

You’re welcome.

#7

Initial Period State Next Period State

A B Default

A 85% 10% 5%

B 10% 80% 10%

If a company is originally in State B, What is the probability that company will default over a given 2 year period?

A) 10%

B) 18%

C) 18.5%

D) 20%

Can some one explain this problem in step by step procedure and how does it differ from the one above? When do we have to make AAA Vs AA in terms of options?

Also what is meant, practicaly by the below:

The easiest way to determine the answer is to make this a square matrix including default in the intitial. Then self multiplying the matrix to get the two year transition matrix??? dont know what is meant by this.

Best regards,

Joe

#8

In 2 years, starting at B, you can go:

  • BAA, 10% × 85% = 8.5%
  • BAB, 10% × 10% = 1%
  • BADefault, 10% × 5% = 0.5%
  • BBA, 80% × 10% = 8%
  • BBB, 80% × 80% = 64%
  • BBDefault, 80% × 10% = 8%
  • BDefault, 10%

The bond defaults 0.5% + 8% + 10% = 18.5% of the time.

The second part says that if you include a third row where the bottom row has Default on the left – and 0%, 0%, 100% as the matrix entries, reading left to right – then you can multiply that matrix by itself (a standard technique in linear algebra) to get a new 3×3 matrix for two years instead of only one year.

#9

Thank you very much indeed.

#10

My pleasure.