Question

11. (a) Calculate the price of a five year fixed interest security redeemable at 105 % with 8% annual coupons given the following information: Yi = Y2 = $2,1 = F3,1 = $2.3 = 9% 8% 6% 5% 0%. (b) The prices of zero coupon bonds for various terms are as follows: 6 years = £70% 12 years =£50% 18 years = £20%. Calculate Y12 and F12,6-

Answer 1

(a) We first need to find the spot rates using the series of forward rates

The 1-year, 2-year, 3-year spot rates are denoted by f(0,1), f(0,2), f(0,3) and so on

f(0,1) = 9%

f(0,2) = 8%

(1+f(0,3))^3 = ((1+f(0,2))^2) * (1+f(2,1))

(1+f(0,3))^3 = 1.08^2 * 1.06

f(0,3) = 0.0733 = 7.33%

(1+f(0,4))^4 = (1+f(0,3))^3 * (1+f(3,1))

(1+f(0,4))^4 = 1.0733^3 * 1.05

f(0,4) = 0.06743 = 6.743%

(1+f(0,5))^5 = (1+f(0,2))^2 * (1+f(2,3)^3

(1+f(0,5))^5 = 1.08^2 * (1.0)^3

f(0,5) = 0.03126 = 3.126%

We now need to discount the cash-flows of 5-year security using the spot rates calculated above

Let the face value be 100 and the current price be P

Cash-flows in year 5 = 105%*100 + 8 = 113

$P = \frac{8}{1.09} + \frac{8}{1.08^{2}} + \frac{8}{1.0733^{3}} + \frac{8}{1.06743^{4}} + \frac{113}{1.03126^{5}}$

P = 123.711

(b) Par value of 12-year zero coupon bond = 100

Then the current price of 12-year zero coupon bond = 50%*100 = 50

Yield on the 12-year zero coupon bond = (F/PV)^(1/n) - 1

Yield on the 12-year zero coupon bond = (100/50)^(1/12) - 1 = 0.05946 = 5.946%

Y(12) = 5.946%

Par value of 18 year zero coupon bond = 100

Then the current price of 18-year zero coupon bond = 20%*100 = 20

Yield on the 18-year zero coupon bond = (F/PV)^(1/n) - 1

Yield on the 18-year zero coupon bond = (100/20)^(1/18) - 1 = 0.09353 = 9.353%

(1+Y(12))^12 * (1+F(12,6))^6 = (1+f(0,18)^18

(1.05946)^12 *(1+F(12,6))^6 = (1.09353)^18

F(12,6) = 0.1650 = 16.50%