- Problem Set #3 will be due at the beginning of class on Thursday, September 14. Since I just made and posted some minor edits of Problem Set #3 this morning, be sure to re-download this assignment in case if you have previously downloaded the prior version.
Anyway, here are some (hopefully) helpful hints as you work on Problem Set #3 :
Problem 1. At the beginning of the life of the forward contract, the delivery price (K) is set equal to the forward price Ft, and the value of the contract, f = Ft – K = $0. This is why we always show the net “outlay” at the initiation of either a long or short forward contract as $0. However, as time passes, K stays the same (because it is a contract parameter), but the forward price subsequently changes as 1) the value of the underlying changes, and 2) the underlying pays dividends (if the underlying is a stock), interest (if the underlying is a bond), or a convenience yield (if the underlying is a commodity). Thus the net value of the forward contract may subsequently become either positive or negative. For this problem, you will want to use equation 6 in Hull’s “Determination of Forward and Futures Prices” chapter in order to find the value of the short position in the forward contract 3 months out.
Problem 2. This problem asks you to discuss “…whether the rate of interest on the gold loan is too high or too low in relation to the rate of interest on the cash loan”. Your discussion needs to include some type of analysis (based upon “no arbitrage” arguments) which unambiguously establishes whether the rate of interest on the gold loan is either too high or too low. Thus, the key to a successful discussion requires that you directly compare the cost of a cash loan with the cost of a gold loan. For example, if the corporate client borrows $1 million cash today, she will have to pay $1 million (1.11) = $1.11 million cash back in 1 year. Now suppose that the price per ounce for gold today is $1,333.33; thus 750 ounces of gold at today’s price is worth $1 million. Therefore, if the corporate client were to borrow 750 ounces of gold today, she would be required to pay back 750(1.02) = 765 ounces of gold in one year. Thus, she can replicate the value of the $1 million cash loan by borrowing 750 ounces of gold today and buying 765 ounces of gold in the forward market. The cost of borrowing $1 million in the gold rather than in the cash market depends upon what today’s price is for a gold forward contract.
Problem 3. Logically, the bank will price the product based upon the assumption that the (US) company will choose the delivery date that is least favorable to the bank. The company’s timing will depend upon whether it expects to receive foreign currency at a future date and will need to convert into US dollars (i.e., has a long position in the underlying), or whether it expects to have to pay in foreign currency and will need to convert US dollars into the foreign currency (i.e., has a short position in the underlying). Furthermore, it also matters whether the foreign interest rate is different (either higher or lower) than the domestic interest rate. During tomorrow’s class meeting, we will work through a simple foreign exchange numerical example which will unambiguously establish (based upon no arbitrage arguments) that whenever rforeign > rdomestic, then F(t, T) < S(t). Similarly, if rforeign < rdomestic, then F(t, T) > S(t).
Problem 4. The forward pricing examples that we discussed in class last Thursday implicitly assume that investors can borrow and lend at the same interest rates. Under this assumption, then there is only one price that is consistent with the no arbitrage principle. In this problem, borrowing and lending rates are different; specifically, the cost of borrowing exceeds the cost of lending. Under this assumption, then there exists a range of forward prices such that no arbitrage opportunity exists. The reason why is because the cost of the replicating portfolio for selling forward is more expensive than the cost of the replicating portfolio for buying forward, since selling forward is synthetically equivalent to a borrowing transaction, whereas buying forward is synthetically equivalent to a lending transaction.
Problem 5. In this problem, there will be two components to K2. The first component will be the forward price at time T1, based upon the spot exchange rate at that point in time. However, since we know at T1 that S1 > K1, the second component included in our to K2 calculation consists of the future value of the bank’s profit at T1. By asking the bank to roll the settlement of the contract forward by one period, the client company is simply taking out a loan in the amount of S1 – K1.