study material chap 4 question 3.18 exam style question

sorry ! question is from chapter 3 of study material


The credit-worthiness of debt issued by companies is assessed at the end of each year by
a credit rating agency. The ratings are A (the most credit-worthy), B and D (debt
defaulted). Historic evidence supports the view that the credit rating of a debt can be
modelled as a Markov chain with one-year transition matrix:

3x3 matrics

|0.92 0.05 0.03 |
|0.05 0.85 0.1 |
|0 0 1 |

(i) Determine the probability that a company currently rated A will never be rated B
in the future.

i could not got under stand the expression mentioned below

0.03 + 0.92 * 0.03 + (0.92)2 * 0.03 + (0.92)3 * 0.03 +....

please clarify

 

first of all see the matrix...Its transition matrix from AA,AB,AD in first row,BA,BB,BD in 2nd row and DA,DB,DD in third row.
I hope this is clear.
Next out of these transition states we have to find out prob that A can never go to B i.e A either goes to D directly(.03) or goes like AAD(.92*.03),AAAD(.92*.92*.03)...and so on.Hence we add these values and approximate it with geometric progression.

 

Can't the A rating remain at A only?
It's not specified that it has to go to D.
It's just that it should never be rated B

 

thanks its clear

thanks once again

 

what i believe as its having 3 states if its being barred to reach b , then it would move to B or D and GP series would follow.