Question

Consider the following US government (risk-free) bonds:

Bond A: 2-year note issued one year ago with a coupon rate of 5%

Bond B: 3-year note issued two years ago with a coupon rate of 5%

The price of the first bond is 100 and the price of the second bond is 101. For simplicity, assume that investors do not face margin requirements or interest payments to short-sell assets.

a. Assume there are no transaction costs. Establish an arbitrage trade to profit from the pricing of these bonds. What would be the profit in USD per pair of bonds traded?

b. Assume that transaction costs are 1% of the face value of the bond per transaction. Selling and buying a bond are two separate transactions. What is the net gain/loss of implementing the strategy from the previous question?

c. The US government issues a one-year bond that makes semi-annual coupons with one year maturity and $100 face value. Under the absence of arbitrage assumption, what would be the price of this bond if:

i. Bond A is correctly priced and Bond B is incorrectly priced?

ii. Bond B is correctly priced and Bond A is incorrectly priced?

Answer 1

a. Assume there are no transaction costs. Establish an arbitrage trade to profit from the pricing of these bonds. What would be the profit in USD per pair of bonds traded?

Both the bonds have one year to go and have identical cash flows of one coupon + Face value at the end of one year. But they have different prices. Hence, there is an arbitrage opportunity that can be exploited by short selling the expensive bond and buying the cheaper bond.

Hence, the arbitrage trade should be:

1. (Short) Sell 1 no. of Bond B
2. Buy 1 no. of bond A

At t = 0, the profit = $ 101 received by short selling 1 no. of Bond - $ 100 paid to buy 1 no. of bond A = $ 1

Hence, the profit in USD per pair of bonds traded = $ 1

b. Assume that transaction costs are 1% of the face value of the bond per transaction. Selling and buying a bond are two separate transactions. What is the net gain/loss of implementing the strategy from the previous question?

Net gain / loss = $ 101 received by short selling 1 no. of Bond - 1% x Face value of Bond B as transaction cost - $ 100 paid to buy 1 no. of bond A - 1% x Face value of Bond B as transaction cost = 101 - 1% x 100 - 100 - 1% x 100 = - $ 1 (i.e. net loss of $ 1)

c. The US government issues a one-year bond that makes semi-annual coupons with one year maturity and $100 face value. Under the absence of arbitrage assumption, what would be the price of this bond if:

i. Bond A is correctly priced and Bond B is incorrectly priced?

The coupon on the bonds has not been mentioned. Assuming that coupon is 5%, the price of the bond = price of the correctly priced bond = price of bond A = $ 100

Hence, price of the US government bond = $ 100

ii. Bond B is correctly priced and Bond A is incorrectly priced?

The coupon on the bonds has not been mentioned. Assuming that coupon is 5%, the price of the bond = price of the correctly priced bond = price of bond B= $ 101