Question

Consider the following three bonds:

Bond	Coupon Rate	Maturity (years)	Price
A	0%	1.0	$947.5572
B	7%	1.0	$1,014.8980
C	5%	1.5	$981.4915

Assume that coupons are paid every 6 months and the face values of all the bonds are $1,000.

(a) Suppose that the 0.5- and 1.5-year zero-coupon bonds are available. Determine their respective prices. (Keep 2 decimal places, e.g. xxx.12)

     P_Z0.5:               P_Z1.5:

(b) Determine the forward rate f _0.5,1 (in yearly term) on a 6-month Treasury bill 6 months from now. (Keep 4 decimal places, e.g. 0.1234)

(c) Determine the forward rate f_0.5,1.5 (in yearly term) on a 12-month Treasury bill 6 months from now. (Keep 4 decimal places, e.g. 0.1234)

(d) Price the 1.5-year coupon bond 6 months from now. (Keep 2 decimal places, e.g. xxx.12)?

Answer 1

Answer 1) The price of a zero coupon Bond will calculated by discounting of maturity value of bond by effective interest rate ,

i.e P = M / (1+r)ⁿ

First we have to calculate Interest rate for the respective period as,

from Bond A, 947.55 = 1000/(1+S1)^1

=> S1 =0.05535

From Bond B , coupon is 7% , and matures in 1 year ,0.5 years we get 0.07*1000/2 = $35

so, 1014.898 = 35/(1+S0.5)^0.5 + 1035/(1+S1)

=> 1014.898 = 35/(1+S0.5)^0.5 + 1035/(1+0.05535)

=> S0.5 = 0.04851

From Bond C , Coupon is 5% , matures in 1.5 year ,0.5 years we get 0.05*1000/2 = $25

So, 981.4915 = 25/(1+0.04851)^0.5 + 25/(1+0.05535) + 1025/(1+S1.5)^1.5

=> S1.5 = 0.064407.

Price of zero coupon bond : P0.5 = 1000/(1+0.04851)^0.5 = $ 976.59

Price of zero coupon bond : P1.5 = 1000/(1+0.064407)^1.5 = $ 910.62

Answer 2) Forward rate f 0.5,1 .

(1+S1) = (1+f(0.5, 1))(1+S0.5)^0.5

(1+0.05535) = (1+f(0.5, 1))(1+0.04851)^0.5

=> f 0.5,1 = 0.3065.

Answer 3) Forward rate f 1,1.5

(1+S1.5) = (1+f(1,1.5))(1+S0.5)^0.5

(1+0.064407) = (1+f(1,1.5))(1+0.04851)^0.5

=>f 1,1.5 = 0.03949