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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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  • $\begingroup$ you did not precise the time period over which your proability should be considered. I give an example : p[AAA][AAA]=0.9725 means that if the bond is AAA, then it has 97.25% to stay AAA the next day ? the next month ? $\endgroup$ – MJ73550 Apr 12 '16 at 7:29
  • $\begingroup$ As MJ73550 mentions, a time period is critical. For the transition matrix, the maturity of the contract, and the bond (if different). 2nd point - who owns the bond and collects the 5% coupon - you? $\endgroup$ – Todd Page Apr 12 '16 at 14:30
  • $\begingroup$ Apologies, I'm awful at making sure I include all the information. You own the underlying asset and the bond has a chance of changing ratings a total of 120 times, every month for 10 years. You are collecting the 5% on the bond (semiannual coupons), but I don't believe that is relevant to finding the premium of the swap. But I could be wrong. $\endgroup$ – CcS Apr 12 '16 at 19:46
  • $\begingroup$ Just curious - why would the coupon not be relevant to the premium? $\endgroup$ – Todd Page Apr 12 '16 at 21:00
  • $\begingroup$ You're paying for essentially an insurance plan that operates outside the bond, but is dependent on the bond. We want to know what I should be paying monthly for the swap arrangement. I could be wrong and these things can somehow correlate, but I don't see it. $\endgroup$ – CcS Apr 14 '16 at 23:08

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