Assume that there are no transaction costs and that securities are infinitely divisible. If an 8 percent coupon paying bond (with semi-annual coupon payments) that has six months left to maturity trades at 97.54, and there is a zero-coupon bond with six months remaining to maturity that is correctly priced using a discount rate of 9 percent, is there an arbitrage opportunity? A) No, the coupon bond is correctly priced. B) Yes, the coupon bond price is too high. C) Yes, the coupon bond price is too low. D) The coupon bond is not correctly priced but no arbitrage trade can be set up using the zero-coupon bond. answer was C) Yes, the coupon bond price is too low. The coupon bond has a cash flow at maturity of 104, which discounted at 9% results in a bond price of 99.52. Therefore, the bond is underpriced. An arbitrage trade can be set up by short-selling 1.04 units of the zero-coupon bond at 99.52 and then using the proceeds to buy 1.02 units of the coupon bond. ----- I don’t understand how did they come to selling 1.04 & buying 1.02…units nos in the solution… Can someone help me understand this.
Here’s their thinking: The coupon bond is essentially a zero with par = 104 as there are no other coupon payments except the last one. The zero has par = 100. To hedge the coupon bond then you need to short 1.04 of the zero so that the proceeds of the long coupon bond cover the short. Now they suggest that you should use all the proceeds from the short sale to buy the coupon bond so that the arbitrage gain is 0.02 * coupon bond. Personally, I would take the extra money and buy stuff with it.
current coupon bond price < what the pv of its cash flows are… hence buy this coupon bond and short sell the zero.