You are given the following information: at t = 0, the price of a 10?year zero coupon bond with FV = $10,000 is $7,000; the price of a 3?year zero coupon bond with FV = $5,000 is $4,300; f3,12 = 6%. A bank is offering the following product: for every $Z that you give the bank at t = 10, the bank will give you back $1 at t = 15, or for every $Z that you borrow from the bank at t = 10, you will have to pay back $1 at t = 15. (In other words, money gets multiplied once by (1/Z) going from t = 10 to t = 15.)
a) What should Z be if there is no arbitrage?
b) Suppose the bank is offering the product described above, but instead of Z being equal to your answer from part a, it is equal to 0.95 times your answer to part a. Describe how you would construct an arbitrage strategy. Assume that part of your strategy involves buying or selling $1 worth of the 10? Year zero coupon bond and also either buying or selling $1 worth of the 3? year zero coupon bond. Construct the arbitrage in such a way that your profit is realized at t = 15, and the net cash flows at all other points in time are 0. You must specify for f 3, 12 = 6% whether you are borrowing or lending at that rate, and how much. You must also specify how much you are borrowing or lending using the bank’s product. Make sure to also specify for each bond if you are buying or shorting $1 worth of the bond. Finally, make sure to specify for each bond, how many units you are buying or selling.
c) What is the magnitude of your arbitrage profit at t = 15?
d) Repeat question b, but this time for 1 unit of the 10? Year bond bought or sold at t = 0.
e) What is the magnitude of the arbitrage profit at t = 15 from part d?