Question

Assume semi-annual compounding. We know that 6-month T-bill is trading at a yield of 2%; 12-month T-bill is trading at a yield of 2.5%; 3%-coupon 18-month T-note is trading at par ($100); 3.4%-coupon 2-year T-note is trading at par ($100). With the above information, compute the 2-year spot rate. Assume semi-annual compounding. Round your answer to 4 decimal places. For example if your answer is 3.205%, then please write down 0.0321.

Answer 1

Let r be the 18 month spot rate

Price of 18 month 3% coupon bond is 100

Hence,
1.5/(1+2%/2)+1.5/(1+2.5%/2)^2+(100+1.5)/(1+r/2)^3=100
=>r=(((100+1.5)/(100-(1.5/(1+2%/2)+1.5/(1+2.5%/2)^2)))^(1/3)-1)*2
=>r=3.010%

Let s be the 2 year spot rate

Price of 24 month 3.4% coupon bond is 100

Hence,
1.7/(1+2%/2)+1.7/(1+2.5%/2)^2+1.7/(1+3.010%/2)^3+(100+1.7)/(1+s/2)^4=100
=>s=(((100+1.7)/(100-(1.7/(1+2%/2)+1.7/(1+2.5%/2)^2+1.7/(1+3.010%/2)^3)))^(1/4)-1)*2
=>s=3.419023%

So 2 year spot rate is 0.0342