Answer:
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
