Question

The 6-month and 1-year zero rates are both 5% per annum. For a bond that has a life of 18 months and pays a coupon of 4% per annum (with semiannual payments and one having just been made), the yield is 5.2% per annum. What is the bond’s price? What is the 18-month zero rate? All rates are quoted with semiannual compounding.

Answer 1

Value of BOnd = Coupon 1 / (1+r)^1 + Coupon 2 / (1+r)^2 + (Coupon 3 + Maturity Value) / (1+r)^3

r = 0.052 / 2 = 0.026

Coupon = 100 * 0.02 = 2

= 2 / (1+0.026)^1 + 2 / (1+0.026)^2 + 2 + 100 / (1+0.026)^3

= $98.29

Now we will equate the value of bond to calculate the rate for 18 months

Value of BOnd = Coupon 1 / (1+r1)^1 + Coupon 2 / (1+r1)^2 + (Coupon 3 + Maturity Value) / (1+r2)^3

r1 = 0.052 / 2 = 0.026

r2 = ?

Coupon = 100 * 0.02 = 2

98.29 = 2 / (1+0.026)^1 + 2 / (1+0.026)^2 + (2 + 100) / (1+r2)^3

98.29 = 3.84923756217 + 102 / (1+r2)^3

102 / (1+r2)^3 = 98.29 - 3.84923756217

102 / (1+r2)^3 = 94.4407624379

(1+r2)^3 = 102 / 94.4407624379

1+r2 = 1.08004210647^1/3

r2 = 1.02599890136 - 1

r2 = 2.599890169%

r2 = 2.599890169% *2 = 5.20%