wilmott.com

Hello all,I recently had this question come up in a exam - and was curious if my attempt at it was correct.The question was as follows:QuoteInterest Rate TheoryThe following information is available:- A 3% coupon bond with face value £100 and maturity 10 years costs £100.- The spot rate s5 is given by 4.5%, and the forward rate f5,10 amounts to 6%.What is the price of a 6% coupon bond with face value £100 and maturity 10 years (assume yearly compounding)?Where s5 is the 5 year spot rate and f5,10 is the forward rate between year 5 and 10.In my attempt of the question, I matched up the payoffs between a 6% coupon bond and two 3% coupon bonds - stating the only difference being the two 3% bonds paid out an additional £100 at the end of the 10 years. So the prices should be the same, less £100 discounted by 10 years.So my working was as follows:A colleage of mine came to a different answer - his reasoning was - Given the 3% bond is at par, we can use its 3% YTM as the YTM of the 6% bond... so:Is either of them correct? I'm not quite sure of the logic behind using the 3% YTM as the YTM of the 6% bond.Thanks,MdSalih

what would your colleague say about the price of a 6% coupon bond with face value £200 and maturity 10 years?

If you're replicating the 6% bond with two 3% bonds, then per your correct logic that the only difference is £100 discounted back 10 years, why not discount it using the 3% YTM? This results in the same answer as your colleague.stilyo: for the £200 face value, simply double the price of the £100 6% bond.N.

You are correct, assuming the spot rates are zero-coupon rates. Your colleague is not correct. His method only works if interest rates are constant throughout the ten years, which you know is not correct.Normally you would not get such a large difference between these methods, but the problem has implausible rates. The PV of the final payment of 103 is only 61.76. That means the first nine coupons, which total 27, have a PV of 38.24, meaning some discount rates are negative.

Aaron,I agree that there is some implausibility, which is why I ignored the possibility of negative interest rates. There seems to be a dichotomy here; you can't have the 3% trading at par if the implied ten-year curve is much greater than that yield.N.