Help needed on a fixed income question

Question

Hi everyone, Im working on my CFA prep and bumped into a question. It's not tricky at all at first sight but now I have some doubt.. can anyone help me there? The question is as follows:Given the following spot and forward rates:Current 1 year spot rate (S1) is 5%1y1y is 7.63%2y1y is 12.18%3y1y is 15.5%The value of a 4 year 10% annually pay, 1000 USD par value bond is closest to:A 996B 1009C 1086The solution said answer is B, but I actually had two different answers using different methods.If I compute the spot rate of 4 year bond, which should be the YMT of the bond first, which is S4=[(1+S1)*(1+1y1y)*(1+2y1y)*(1+3y1y)]^(1/4)-1, S4 will be 10.134% and the PV of the bond will be near to 996.If I adopt the cash flow method and add up PV of coupons & FV of the bond, the answer will be 1009. But both methods seem correct to me. I don't know what went wrong??Thanks in advance for your thoughts!

cfachris · Answer

Hi, I assumed you had a typo in your original question for S1, that it should be 5.5% as I don't seem to get your exact answers with S1=5% (an answer of 1014 instead of 1009).Assuming that S1=5.5%, the correct answer is B, i.e. 1009. This is derived as: 1009 = 100/[(1.055)] + 100/[(1.055)*(1.0763)] + 100/[(1.055)*(1.0763)*(1.1218)] + 1100/[(1.055)*(1.0763)*(1.1218)*(1.155)]In your calculation (which yielded 996), you have assumed that the S4 spot rate is the same each year i.e. S1=S2=S3=S4, which is not true.S2 = (1.055*1.0763)^(0.5)-1 = 6.56%S3 = (1.055*1.0763*1.1218)^(1/3)-1 = 8.4%And you already have S1 and S4. So if you discount each cashflow with their respective period's spot rate, you'll get the same answer, i.e. 1009 = 100/(1.055) + 100/(1.0656^2) + 100/(1.084^3) + 1100/(1.1013^4)