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) Determine the spot rate curve. (That is, determine s_0.5, s₁, and s_1.5 in yearly terms.) (Keep 4 decimal places, e.g. 0.1234)

     s_0.5:          s₁:            s_1.5 :

(b) 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:

(c) 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)

(d) 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)

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

Answer 1

(a) For Bond A: Coupon Rate = 0%, Price = $ 947.5572, Par Value = $ 1000, Tenure 1 year, Assumption: Coupon payment frequency and compounding frequency are both semi-annual

Let the 1 year spot rate s1 be 2y

Therefore, 947.5572 = 1000 / (1+y)^(2)

y = [(1000/947.5572)^(1/2) - 1] = 0.0273 or 2.73%

s1 = 2 x y = 2 x 2.73 = 0.0546 or 5.46 %

For Bond B: Coupon Rate = 7%, Price = $ 1014.898, Par Value = $1000, Tenure = 1 year,Assumption: Coupon payment frequency and compounding frequency are both semi-annual

Let the half-year spot rate s0.5 be 2z

Semi-Annual Coupon = 0.07 x 1000 x 0.5 = $ 35

Therefore, 1014.898 = 35 / (1+z) + 35 / (1.0273)^(2) + 1000 / (1.0273)^(2)

34.1763 = 35 / (1+z)

z = [(35/34.1763) - 1] = 0.0241 or 2.41 %

s0.5 = 2 x z = 2 x 2.41 = 4.82 %

For Bond C: Coupon Rate = 5%, Price = $ 981.4915, Face Value = $ 1000, Assumption: Coupon payment frequency and compounding frequency are both semi-annual

Semi-Annual Coupon = 0.05 x 1000 x 0.5 = $ 25

Let the 1.5 year spot rate s1.5 be 2m

Therefore, 981.4915 = 25 / (1.0241) + 25 / (1.0273)^(2) + 1025 / (1+m)^(3)

933.391 = 1025 / (1+m)^(3)

m = [(1025/933.391)^(1/3)-1] = 0.0317 or 3.17 %

s1.5 = 2 x 3.17 = 6.34 %

(b) Price of 0.5 year zero-coupon bond = Z0.5 = 1000 / (1+s0.5/2) = 1000 / (1.0241) = $ 976.467 ~ $ 976.47

Price of 1.5 year zero-coupon bond = Z.1.5 = 1000 / (1+s1.5/2) = 1000 / (1.0317)^(3) = $ 910.625 ~ $ 910.62

(c) f0.5,0.5 = 2 x [(1+s1/2)^(2) / (1+s0.5/2)-1] = [(1.0273)^(2)/(1.0241) - 1 ] x 2 = 0.06102 or 6.102 %

(d) f0.5,1.5 = [(1+s1.5/2)^(3) / (1+s0.5/2) - 1] = [(1.0317)^(3)/(1.0241) - 1] = 0.0723 or 7.23 %

(e) Price of 1.5 year bond 6 months later = 25 / (1+f0.5,0.5/2) + 1025 / (1+f5,1.5) = 25 / (1.03051) + 1025 / (1.0723) = $ 980.149 ~ $ 980.15