Question

The table below lists maturities, coupons and prices for three bonds. All bonds have the same default risk and a face value of 100.

Bond	Maturity	Coupon	Price
A	3	6%	94
B	2	5%	98
C	2	3%	94.5

a) What is the yield to maturity of a two-year zero-coupon bond?

b) What is the price of a one-year zero-coupon bond with a face value of 100?

c) What is the implied one-year forward rate for the period between year 2 and year 3?

d) What is the price of a bond similar to Bond C but paying a 6% coupon instead?

Answer 1

Let S1, S2, S3 be the spot rates for year 1, 2 and 3 respectively. Let's define a = 1 / (1 + S1); b = 1 / (1 + S2)² and c = 1 / (1 + S3)³

Price of a bond A = C / (1 + S1) + C / (1 + S2)² + (C + FV) / (1 + S3)³

Hence, 94 = 6/(1 + S1) + 6 / (1 + S2)² + (6 + 100) / (1 + S3)³

Or, 94 = 6a + 6b + 106c -------------- (1)

Similarly based on price of bond B we get: 98 = 5a + 105b ------------------(2)

and Similarly based on price of bond C we get: 94.5 = 3a + 103b ------------(3)

3 x Equation (2) - 5 x equation (3):

(3 x 98 - 5 x 94.5) = (3 x 105 - 5 x 103)b Or, -178.5 = -200b

Hence, b = 0.8925

From equation (2): a = (98 - 105b) / 5 = (98 - 105 x 0.8925) / 5 = 0.8575

From equation (1): c = (94 - 6a - 6b) / 106 = 0.7877

a = 0.8575 = 1 / (1 + S1)

Hence, S1 = 1/0.8575 - 1 = 16.62%

b = 0.8925 = 1/(1 + S2)²

Hence, S2 = 5.85%

c = 0.7877 = 1/(1+S3)³

Hence, S3 = 8.28%

Part (a)

the yield to maturity of a two-year zero-coupon bond = S2 = 5.85%

Part (b)

the price of a one-year zero-coupon bond with a face value of 100 = 100 / (1 + S1) = 100 / (1 + 16.62%) = 85.75

Part (c)

the implied one-year forward rate for the period between year 2 and year 3 = (1 + S3)³ / (1 + S2)² - 1 = (1 + 8.28%)³ / (1 + 5.85%)² - 1 = 13.30%

Part (d)

the price of a bond similar to Bond C but paying a 6% coupon instead = 6a + 106b = 6 x 0.8575 + 106 x 0.8925 = 99.75