Question

Betty and Bob must construct the ZCB yield curve for Freedonia. Freedonia has bonds of 6 months, 12 months, 18 months, and 24 months terms.

         A 6-month ZCB with maturity value of $100 is priced at $94.3396.

         A 1-year coupon bond with maturity value of $100 and a coupon rate of 8% per annum, payable semiannually is priced at $94.6112.

         An 18-month coupon bond with maturity value of $100 and a coupon rate of 19% per annum payable semiannually is priced at $105.44031.

         A 2-year coupon bond with maturity value of $100 and a coupon rate of 10% per annum payable semiannually is priced at $90.2871.

         a By viewing the coupon bond as the sum of the ZCBs find the per annum yield compounded semiannually for the 6 months, 1 year, 18 months and 2 years ZCB.

         b A special 2-year bond has maturity value of $100 and coupons of $4, $9, $5, and $7 in that order. Use the ZCB yield curve data to compute the price of the bond.

        

1. 6 month_______, 12 month______18month________24month________

The price of the special bond is___________

Answer 1

a). Let the per annum yields for the 6-month, 12-month, 18-month & 24-month ZCBs be z1, z2, z3 and z4.

z1 calculation:

Current bond price = sum of all discounted cash flows

94.3396 = 100/(1+z1/2)

z1 = [(100/94.3396) -1]*2 = 12.00%

z2 calculation:

Annual coupon = coupon rate*par value = 8%*100 = 8

94.6112 = 8/(1+z1/2) + (100+8)/(1+z2/2)^2

94.6112 = 8/(1+12%/2) + 108/(1+z2/2)^2

z2 = [(108/87.064)^(1/2) -1]*2 = 22.75%

z3 calculation:

Annual coupon = 19%*100 = 19

105.44031 = 19/(1+z1/2) + 19/(1+z2/2)^2 + (100+19)/(1+z3/2)^3

z3 = [(119/72.199)^(1/3) -1]*2 = 36.25%

z4 calculation:

Annual coupon = 10%*100 = 10

90.2871 = 10/(1+z1/2) + 10/(1+z2/2)^2 + 10/(1+z3/2)^3 + (100+10)/(1+z4/2)^4

z4 = [(110/66.7245)^(1/4) -1]*2 = 26.62%

b). Price of the special bond = 4/(1+z1/2) + 9/(1+z2/2)^2 + 5/(1+z3/2)^3 + (100+7)/(1+z4/2)^4

= 4/(1+6%) + 9/(1+11.38%)^2 + 5/(1+18.12%)^3 + 107/(1+13.31%)^4

= 78.97