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]

S₁ is the 1-year spot rate. This is equal to the yield of the 1-year zero coupon bond. The price of a zero coupon bond is the present value of its face value, discounted at the spot rate.

Therefore, $947.5572 = ($1,000 / (1 + S₁)¹)

S₁ = ($1,000 / $947.5572) - 1 = 0.0554

The price of a coupon bond is the present value of its cash flows. A coupon bond's cash flows are its coupon payments, and its face value receivable on maturity. To find the present value, each payment is discounted at the appropriate discount rate (6 months, 1 year, 1.5 years etc.)

The semiannual coupon payment = coupon rate * face value / 2 = 7% * $1,000 / 2 = $35

Therefore, price of 1-year bond = [semiannual coupon payment / (1 + S_0.5)^0.5] + [(semiannual coupon payment + face value) / (1 + S₁)¹]

1,014.8980 = [35 / (1 + S_0.5)^0.5] + [(35 + 1000) / (1 + 0.0554)¹]

[35 / (1 + S_0.5)^0.5] =  1,014.8980 - [(35 + 1000) / (1 + 0.0554)¹]

[35 / (1 + S_0.5)^0.5] = 34.1807

(1 + S_0.5)^0.5 = 35 / 34.1807

S_0.5 = (35 / 34.1807)^0.5 - 1

S_0.5 = 0.0485

Similarly, for the 1.5 year bond, semiannual coupon payment = coupon rate * face value / 2 = 5% * $1,000 / 2 = $25

Therefore, price of 1.5-year bond = [semiannual coupon payment / (1 + S_0.5)^0.5] + [semiannual coupon payment / (1 + S₁)¹] + [(semiannual coupon payment + face value) / (1 + S_1.5)^1.5]

981.4915 = [25 / (1 + 0.0485)^0.5] + [25 / (1 + 0.0554)¹] + [(25 + 1000) / (1 + S_1.5)^1.5]

[1025 / (1 + S_1.5)^1.5] = 933.389

(1 + S_1.5)^1.5 = 1025 / 933.389

S_1.5 = (1025 / 933.389)^1/1.5 - 1

S_1.5 = 0.0644

b]

The price of a zero coupon bond is the present value of its face value, discounted at the spot rate.

PZ_0.5 = $1,000 / (1 + 0.0485)^0.5 = $953.7310

PZ_1.5 = $1,000 / (1 + 0.0644)^1.5 = $939.4907

c]

The forward rate is a function of the spot rate.

(1 + S₁₎ = (1 + S_0.5) * (1 + f_0.5,1)

(1 + 0.0554) = (1 + 0.0485) * (1 + f_0.5,1)

(f_0.5,1) = 0.0065

d]

The forward rate is a function of the spot rate.

(1 + S_1.5) = (1 + S_0.5) * (1 + f_0.5,1.5)

(1 + 0.0644) = (1 + 0.0485) * (1 + f_0.5,1)

(f_0.5,1) = 0.0151