Question

The current market price of a two-year 25% coupon bond, paying annual coupons with $100 face value is trading at $121.97. The current market price of a one year zero coupon bond with $100 face value is $89.28.  

1. What must the price of a two-year zero coupon bond with a $100 face value be in order to avoid arbitrage? (6 points)

(Hint: Let P be the value of a two year zero coupon bond with $100 face value if needed.)

2. Suppose the price of the two year zero coupon bond with a face value of $100 is $76. Is there an arbitrage opportunity? If yes, describe in details the transactions you will put together in order to capture the profits upfront.  (9 points)
3. Practically, what are the constraints you might face when executing the trade you proposed in b.  (5 points)

Answer 1

Ans a) Yield rate of year 1 and year 2 will be find using the given zero bond and coupon bond.

For zero coupon bond yield is given as below

FV = 100, PV = -89.28, PMT = 0, N = 1, CPT I/Y

Thus I/Y = 12%

Similarly we will use this 12% to find the rate for 2 year yield using two year coupon bond

Bond price = coupon/( 1 + yield rate in year 1) + (face value + coupon)/(1 + yield rate in year 2)^2

121.97 = 25/1.12 + 125/(1 + YTM in year 2)^2

YTM in year 2 = 12%

thus the price of 2 year zero coupon bond = 100/(1.12)^2

= $79.72

Ans b) Yes there is arbitrage opportunity.

I will sell the current two year bond and will get $79.72 and buy the $76 bond in that case will make a profit of (79.72 - 76) = $3.72

Ans c) There is interest rate which is the main constraint that will be impact in this trade because due to change in interest rate the deal which is looking good now may be start looking as a loss, not only that there is risk of finding the counter party also risk of liquidity.

There is cost associated with this transaction and if this transaction cost is more than the risk free gain then there is no point in entering in this position.