Valuing a Credit-Rating-Triggered Payment with a Transition Matrix

Summary

The document poses a valuation problem for a contingent payment tied to a bond’s credit-rating deterioration. It specifies a monthly rating transition matrix, a maturity horizon, continuous discounting, and a rule that pays when the bond first falls below its initial A rating. A transition to default ends the arrangement and caps total receipts. The stated goal is to find a fair monthly premium by calculating the present value of the expected payments.

These details provide the ingredients for a finite-state valuation: track rating probabilities over time, distinguish the first qualifying downgrade from later downgrades, and discount payments at their occurrence dates. The document itself does not carry out that calculation or explain how to structure the payment count after the first trigger. Its wording about paying each time a new low occurs may need clarification, since it can change the cash flows materially. No premium estimate or validation is supplied.

Key ideas

• The bond’s rating can be represented as a finite-state Markov process using the supplied transition probabilities.
• A payment depends on reaching a rating below the initial A rating, so trigger timing matters.
• Default is an absorbing state that terminates the arrangement and caps payments.
• A fair premium requires discounting expected cash flows over the contract horizon.
• The document gives the setup but does not resolve ambiguities in the payment rule or calculate a value.

Tags

# How do I calculate the present value of a credit default swap?


# How do I calculate the present value of a credit default swap?












I am paid 20 million every time a bond drops to a new low over a 120 month period. I need to know how to find the present value of such an arrangement if there is a continuously compound interest of 5 percent.

Additional information:

The bond starts at a A rating and is only paid the first time it reaches any rating lower than A. So A -> AA-> A-> B->CCC would pay 40 million.

I am given the following probability transition matrix (the probability of the rating on the left becoming the rating on the right):

```
   p[AAA][AAA] = 0.9725; p[AAA][AA] = 0.0275;
   p[AA][AAA] = 0.0020; p[AA][AA] = 0.9742; p[AA][A] = 0.0238;
   p[A][AA] = 0.0020; p[A][A] = 0.9825; p[A][BBB] = 0.0155;
   p[BBB][A] = 0.0073; p[BBB][BBB] = 0.9819; p[BBB][BB] = 0.0108;
   p[BB][BBB] = 0.0030; p[BB][BB] = 0.9783; p[BB][B] = 0.0187;
   p[B][BB] = 0.0010; p[B][B] = 0.9751; p[B][C] = 0.0239;
   p[C][B] = 0.0066; p[C][C] = 0.9852; p[C][D] = 0.0082;
   p[D][D] = 1.0;
```

When the bond reaches a rating of D, the bond defaults and cannot recover from that position thus the swap arrangement is closed and we receive the most money possible at $100 million. Otherwise the arrangement continues until the bond matures at 120 months.

I need to find a fair monthly premium, which I believe involves finding the present value of this arrangement.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.