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%

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