Question

Assume the spot rates for one-, two-, and three-year zero coupon bonds are 2%, 3%, and 4%. (a)Calculate P(1), P(2), and P(3).(b)Calculate the price of a three-year 8% coupon bond, with interest paid annually.(c)Calculate ?1,1, ?1,2, and ?2,1(d)Calculate F(1,1), F(1,2), and F(2,1).(e)If ?1,3= 5%, calculate P(4).

Answer 1

a)

Assuming face value = $100

P(1) = Face value / (1 + 1 year spot rate)

P(1) = $100 / (1 + 2%)

P(1) = $98.04

P(2) = Face value / (1 + 2 year spot rate)²

P(2) = $100 / (1 + 3%)²

P(2) = $94.26

P(3) = Face value / (1 + 3 year spot rate)³

P(3) = $100 / (1 + 4%)³

P(3) = $88.90

b)

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (8% / 1) * $100

Coupon per period = $8

Bond Price = Coupon / (1 + 1 year spot rate)¹ + Coupon / (1 + 2 year spot rate)² + Coupon / (1 + 3 year spot rate)³ + Face value / (1 + 3 year spot rate)³

Bond Price = $8 / (1 + 2%)¹ + $8 / (1 + 3%)² + $8 / (1 + 4%)³ + $100 / (1 + 4%)³

Bond Price = $111.40

c)

Let us compute F(1,1)

(1 + 2 year spot rate)² = (1 + 1 year spot rate) * (1 + forward rate F(1,1))

(1 + 3%)² = (1 + 2%) * (1 + forward rate F(1,1))

(1 + forward rate F(1,1)) = (1 + 3%)² / (1 + 2%)

(1 + forward rate F(1,1)) = 1.040098

Forward rate F(1,1) = 4.0098%

Let us compute F(1,2)

(1 + 3 year spot rate)³ = (1 + 2 year spot rate)² * (1 + forward rate F(1,2))

(1 + 4%)³ = (1 + 2%)² * (1 + forward rate F(1,2))

(1 + forward rate F(1,2)) = (1 + 4%)³ / (1 + 3%)²

(1 + forward rate F(1,2)) = 1.060292

Forward rate F(1,2) = 6.0292%

Let us compute F(2,1)

(1 + 3 year spot rate)³ = (1 + 1 year spot rate) * (1 + forward rate F(2,1))²

(1 + 4%)³ = (1 + 2%) * (1 + forward rate F(2,1))²

(1 + forward rate F(2,1))² = (1 + 4%)³ / (1 + 2%)

(1 + forward rate F(2,1))² = 1.102808

(1 + forward rate F(2,1)) = 1.050146

Forward rate F(2,1) = 5.0146%

d)

(1 + 4 year zero rate)⁴ = (1 + 3 year zero rate)³ * (1 + forward rate F(1,3))

(1 + 4 year zero rate)⁴ = (1 + 4%)³ * (1 + 5%)

(1 + 4 year zero rate)⁴ = 1.181107

(1 + 4 year zero rate) = 1.042491

4 year zero rate = 4.2491%

Assuming face value = $100

P(4) = Face value / (1 + 4 year spot rate)⁴

P(4) = $100 / (1 + 4.2491%)

P(4) = $84.67