Question

Suppose you have the following three zero-coupon bond (ZCB) available: a 1-year ZCB that costs $97, a 2-year ZCB that costs $95, and a 3-year ZCB that costs $92. Assume that the par values are $100.

a. What must the price of a 3-year coupon bond with at 8% coupon rate?
b. How would you make an arbitrage profit if the coupon bond was trading at $100?
c. How much arbitrage profit would you make per $100 of the 3-year coupon bond trade?

Answer 1

Bond price = Par value / (1 + 1 year spot rate)1

$97 = $100 / (1 + 1 year spot rate)^1

(1 + 1 year spot rate)^1 = $100 / $97

(1 + 1 year spot rate) = 1.030928

1 year spot rate = 3.0928%

Bond price = Par value / (1 + 2 year spot rate)^2

$95 = $100 / (1 + 2 year spot rate)^2

(1 + 2 year spot rate)^2 = $100 / $95

(1 + 2 year spot rate)^2 = 1.052632

(1 + 2 year spot rate) = (1.052632)(1 / 2)

(1 + 2 year spot rate) = 1.025978

2 year spot rate = 2.5978%

Bond price = Par value / (1 + 3 year spot rate)^3

$92 = $100 / (1 + 3 year spot rate)^3

(1 + 3 year spot rate)^3 = $100 / $92

(1 + 3 year spot rate)^3 = 1.086957

(1 + 3 year spot rate) = (1.086957)(1 / 3)

(1 + 3 year spot rate) = 1.028184

3 year spot rate = 2.8184%

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

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

Coupon per period = $8

a) Bond price = Coupon / (1 + 1 year spot rate)^1 + Coupon / (1 + 2 year spot rate)^2 + (Coupon + Par value) / (1 + 3 year spot rate)^3

Bond price = $8 / (1 + 3.0928%)^1 + $8 / (1 + 2.5978%)^2 + ($8 + $100) / (1 + 2.8184%)^3

Bond price based on spot rates = $114.7199

b. Bond price based on spot rates is greater than traded bond price to exploit this arbitrage the following strategy must be implemented

The 3 year 8% coupon bond should be bought at $100.

Portfolio = -$100

1 year zero coupon bond with face value $8 must be sold

Portfolio = (Price of 1 year zero coupon bond / Face value) * Amount of Face value to be Sold

Portfolio = ($97 / $100) * $8

Portfolio = $7.76

2 year zero coupon bond with face value $8 must be sold

Portfolio = Price of 2 year zero coupon bond / Face value) * Amount of Face value to be Sold

Portfolio = ($95 / $100) * $8

Portfolio = $7.6

3 year zero coupon bond with face value $108 must be sold

Portfolio = Price of 3 year zero coupon bond / Face value) * Amount of Face value to be Sold

Portfolio = ($92 / $100) * $108

Portfolio = $99.36

Arbitrage profit = -$100 + $7.76 + $7.6 + $99.36

Arbitrage profit = $14.72

c) Arbitrage profit = Bond price based on spot rates - Traded Bond price

Arbitrage profit = $114.72 - $100

Arbitrage profit = $14.72

Arbitrage profit would you make per $100 = $14.72