Question

Q1 (Essential to cover) The following bonds are trading in the market: Bond Time-to-Maturity Face value Coupon rate $ 100 0% $ 100 10% $ 100 20% $ 100 0% AWN Price $ 95.24 $ 107.42 $ 140.51 $ 85.48 In addition to the bonds above, you also observe some other bond (bond E) trading in the market at $138. Bond E has a time-to-maturity of two years, a face value of $100 and pays a coupon rate of 25%. Show that there is an arbitrage opportunity and how to exploit it.

Answer 1

As a first step, we need the zero rates for year 1 and year 2. Let's call them z1 and z2 respectively.

Bond A is a zero coupon paying bond A.

Hence, price of bond A = Face value / (1 + z1)

Hence, 95.24 = 100 / (1 + z1)

Hence, z1 = 100 / 95.24 - 1 = 5.00%

Price of Bond B = PV of all coupons + PV of Face value = 107.42 = 10 / (1 + z1) + (100 + 10) / (1 + z2)² = 10 / (1 + 5%) + 110 / (1 + z2)² = 9.52 + 110 / (1 + z2)²

Hence, z2 = [110 / (107.42 - 9.52)]^1/2 - 1 = 6.00%

As per this zeroes, the no arbitrage price of bond E = Coupon / (1 + z1) + (Face value + coupon) / (1 + z2)² = 25 / (1 + 5%) + 125 / (1 + 6%)² = 135.06

However it's actual price = 138

Hence, there is an arbitrage opportunity.

Please see the year wise cash flows of the three bonds:

	Year 1	Year 2
Bond A	100
Bond B	10	110
Bond E	25	125

Let's say we need A number of Bond A and B number of Bond B to replicate the cash flows of Bond E

Hence, year 1 cash flows = 100 x A + 10 x B = 25

Year 2 cash flows = 110 x B = 125

Hence, B = 125 / 110 = 1.1364

And hence, A = (25 - 10B) / 100 = 0.1364

Hence, we can replicate the cash flows of Bond E by a portfolio comprising of 0.1364 number of bond A and 1.1364 number of Bond B. The price of this portfolio = 0.1364 x Price of A + 1.1364 x Price of Bond B = 0.1364 x 95.24 + 1.1364 x 107.42 = $ 135.06

Hence, arbitrage strategy should be:

• (Short) Sell Bond E
• Buy 0.1364 number of Bond A
• Buy 1.1364 number of Bond E

And the arbitrage profit = 138 - 135.06 = $ 2.94