Question

For a given term structure, you are given the following information about two bonds that are re-
deemable at par and have face amount $100, and coupons payable semi-annually.
Bond 1: Coupon rate 4% per year, price $85.12
Bond 2: Coupon rate 10% per year, price $133.34
Find the yield-to-maturity for a 10-year zero coupon bond.

Answer 1

Let r1 be 0.5 year spot rate r2 be 1 year spot rate,....r20 be 10 year spot rate

All rates above are rates compounded semiannually

From Bond 1:
2/(1+r1/2)+2/(1+r2/2)^2......+102/(1+r20/2)^20=85.12 (1)

From Bond 2:
5/(1+r1/2)+5/(1+r2/2)^2......+105/(1+r20/2)^20=133.34 (5)

If we subtract 2.5 times (1) from (5), we get

(105-102*2.5)/(1+r20/2)^20=133.34-85.12*2.5
=>(1+r20/2)^20=(105-102*2.5)/(133.34-85.12*2.5)
=>r20=(((105-102*2.5)/(133.34-85.12*2.5))^(1/20)-1)*2
=>r20=6.4558%