# Finance 350: Problem Set 6 Alternative Solutions

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1 Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas This section contains the formulas you will need for this homework set: 1. Present Value (PV) Formula (a.k.a. Zero Coupon Bond Formula): V 0 = V N (1 + R/m) mt (1) where V N is the dollar amount to be received N periods in the future. m is the number of compounding periods per annum, T is the number of years until the money is received and R is the nominal interest rate (a.k.a. APR). Note that this formula is equivalent to: V 0 = V N (1 + i) N where i = R/m and N = mt. I will use this notation interchangeably throughout. 2. Future Value (FV) Formula: V N = V 0 (1 + i) N (2) where V 0 is the dollar amount received today. Note that the future value formula is just an algebraic manipulation of the present value formula. 1

2 3. Present Value of an Annuity Formula: A 0 = a (1 + i) + a (1 + i) a 2 (1 + i) + a N 1 (1 + i) N = 1 (1 + i) N a i where a is the amount of the annuity payment and i and N are as defined above. 4. Future Value of an Annuity Formula: A N = a (1 + i)n 1 i where a is the amount of the annuity payment and i and N are as defined above. 5. Index Forward Price Formula: (3) (4) F = S 0 e (r f d)t (5) where S 0 is the price of the underlying security at time 0, r f is the risk-free rate, d is the dividend yield and T is the time to maturity. 6. Interest Rate Futures Price: F = 100 InterestRate (6) where F is the futures price. The interest rate is typically LIBOR. II. Problems 1. 1.a The farmer is going to be selling the wheat in the future so he is concerned that the future spot price will be low, which would result in low revenues from the sale of his wheat. He can hedge this risk by selling futures on wheat. 1 1 An easy way to remember whether one should buy or sell futures contracts to hedge price risk is to simply consider what action will be taken in the future. Since the farmer needs to sell his wheat in the future, he should sell futures contracts. 2

3 This short position in the futures contract obligates him to sell the wheat in the future at the delivery price (i.e. the future price at the inception date of the contract). Assuming his expectation is correct, the farmer is going to harvest 60,000 bushels in September. Since each contract is for 5,000 bushels, she will need to sell 60,000/5,000 = 12 contracts. And, since the harvest will take place in September, she should sell September contracts. 1.b Table (1) below considers three scenarios for the future spot price of wheat. The gain from the futures position, assuming a cash settlement, is computed using the formula: (F S T ) Quantity. The quantity in parentheses is the payoff to the seller of a futures contract. We multiply by the quantity since the payoff is per unit, which in this case is a bushel. For example, the \$9,600 gain from the futures position when the spot price is \$3.25 is computed as (\$3.41 \$3.25) 60, 000. The revenue from the sale at the spot price is simply the total number of bushels times the price per bushel in the market, or 60, 000 \$3.25 = \$195, 000. It is important point to recognize that, regardless of the future spot price, the total revenue from the farmer s position is \$204,600, as shown on the last row of the table. That is, the farmer is perfectly hedged. Table 1: Future Spot Price of Wheat \$3.25 \$ 3.35 \$3.45 Revenue from Sale at Spot Price \$195,000 \$201,000 \$207,000 Gain/Loss from Futures Position \$9,600 \$3,600 -\$2,400 Total \$204,600 \$204,600 \$204,600 1.c The strategy here is to compare the money he receives in September under his two options: (1) earn a salary and lease the farmland or (2) farm the wheat himself. Begin by computing the future value of all money the farmer will receive under the first scenario. He receives a salary of \$3,500 per month for nine months (January to September), with each payment at the end of the month. Therefore, the first 3

4 payment he will receive is in one month. We should recognize this exercise as computing the future value of an annuity, where the compounding is monthly (m = 12) and the number of periods (N) is nine, implying i = 0.048/12 = Thus, F V = \$3, 500 ( ) = \$32, (see equation (4)) This is the value of the salary in September, which is in addition to the \$5,000 he receives for leasing the land. Finally, we must account for the opportunity cost of using the \$150,000 to pay for seed and equipment. Assume he invested this money by placing it into a bank account earning 4.8% p.a. The future value of this money (equation (2)) is: \$150, 000 ( ) 9 = \$155, Summing these three future values yields: \$32, \$5, \$155, = \$192, If he decides to farm the wheat himself, he is going to have 60,000 bushels harvested in September. The per bushel price he needs to get in order to be indifferent between the salaried position and farming is found by equating the total revenue from the salaried position (\$192,495.94) to the total revenue he will receive from the harvest (60, 000 P ), where P is the price per bushel. Thus, \$192, = 60, 000 P. Solving for P yields \$3.208 per bushel. If he receives more than this price, he should farm himself. Anything less than this price and the farmer should choose to take the salaried position. 1.d We need to compare the revenue generated from selling in December, less storage costs, with the revenue generated from selling in September, while accounting for the time value of money. We determined in part 1.b that the farmer was guaranteed \$204,600 if he entered into a September futures position. 4

5 Consider the revenue generated from a future sale of the wheat in December, assuming the farmer takes a hedging position in futures and pays the storage costs. Locked in at \$3.515 per bushel under the terms of the futures contracts, the farmer is guaranteed to receive: 60, 000 \$3.51 = \$210, 900 in December. Discounting this figure back to September yields \$210, 900/( ) 3 = \$208, We must deduct from this figure the present value (as of September) of the storage costs, which may be computed as the present value of a 3-period annuity: 1 ( ) 3 \$1, 200 = \$3, (7) The net revenue from selling in December is \$208, \$3, = \$204, Thus, storing the wheat and selling it forward in December results in greater net revenue for the farmer (a difference of \$ \$204, = \$217.92). 1.e If the farmer can store an additional 40,000 bushels of wheat at no additional cost (beyond the \$1,200 per month), there is an opportunity to increase net revenues. The farmer should buy 40,000 bushels of wheat in September, store the wheat for three months and then sell it in December. He can execute the September purchase and December sale using futures contracts to lock in his costs and revenues. To buy 40,000 bushels of wheat in September, the farmer must buy 40,000/5,000 = 8 September contracts. Under the future price specified in the contracts, the farmer must pay 5, 000 \$3.41 = \$17, 050 per contract, or 8 \$17, 050 = \$136, 400 in total. To sell these additional bushels in December, the farmer must sell 8 December contracts. Under the future price specified in the contracts, the farmer will receive 5, 000 \$3.515 = \$17, 575 per contract, or 8 \$17, 575 = \$140, 600 in total. Discounting this amount back to September yields \$140, 600/( ) 3 = \$138, We now add this figure to the farmer s initial cost to get a net gain, in September dollars, equal to \$138, \$136,400 = \$2, (Note: we could compute the net profits at any point in time by simply discounting this figure appropriately). 5

6 1.f The problem here is that we ve introduced another source of uncertainty in addition to the price uncertainty, namely, crop uncertainty. The hedging strategy devised above will not perfectly hedge the farmer s revenue uncertainty (price * quantity), only the price uncertainty. We want to devise a strategy to hedge the farmer s revenue uncertainty. Consider the two possible revenue outcomes of the farmer. In the bad harvest, the farmer brings to market 50,000 bushels and sells them at the spot price of \$3.60/bushel for revenue of \$180,000. In the good harvest, the farmer brings to market 70,000 bushels and sells them at the spot price of \$3.20/bushel for revenue of \$224,000. If the farmer takes a position in September futures, we know that his per bushel payoff will be (S t F ). Since we know the futures price and (are assuming) we know the future spot price we can compute this quantity under both scenarios. In the good harvest year, the futures payoff is \$3.20 \$3.41 = \$ In the bad harvest year, the futures payoff is \$3.60 \$3.41 = \$0.19. The aggregate payoff in the bad harvest is: 50, 000 \$ X (0.19), (8) where X is the number of bushels. This equation says that the farmer will receive 50, 000 \$3.60 from the sale of his wheat in a bad year, plus some amount from his futures position. In a good harvest, the farmer s revenue is: 70, 000 \$ X ( 0.21). (9) For hedging purposes, the farmer wants these two quantities to be equal. That is, regardless of whether there is a good harvest or a bad harvest, the farmer wants exactly the same revenue. Setting equation (8) and (9) equal and solving for X yields X = 110, 000. This tells the farmer how many bushels he must buy in the future. With 5,000 bushels per contract, the farmer needs to buy 110,000/5,000 = 22 contracts. Table (2) below shows that this position does indeed completely hedge revenue risk. Revenue from the harvest is simply price times quantity. For example, in the good harvest, \$ , 000 = \$224, 000. Revenue from the futures position is just the difference between the delivery price and the spot price times the quantity. For example, in the good harvest, (\$3.20 \$3.41) 110, 000 = \$23,

7 Table 2: State Harvest Futures Total Revenue Revenue Revenue Good Harvest \$224,000 \$-23,100 \$200,900 Bad Harvest \$180,000 \$20,900 \$200, Consider the forward pricing relation in equation (5). Solving for d yields: d = r f 1 T ln(f/s 0) Substituting for the right hand side variables d = ln(971.67/966.67) = produces an implied dividend yield of 4.436%. 3. The futures contract is trading in the market for \$ true price (according to the forward price formula) is: F = S 0 e (r f d)t = \$966.67e ( )0.25 = \$ However, the Thus, the futures contract is trading at a discount, implying we should buy the contract in the market and sell the underlying asset (the index). The cash we receive from the sale of the asset can be invested in a T-bill earning the risk-free rate. The arbitrage table in Table 3 outlines the strategy. As Table 3 illustrates, by shorting the index, investing the proceeds in the a T-bill and buying the future, we ensure ourself a risk-free profit of \$1.04 per unit of index. Since each contract is for 250 times the index, the total profit per contract is 250 \$1.04 = \$260. Alternatively, we could have realized the profit in the future, as illustrated in Table 4. This two strategies are, in fact, monetarily equivalent since the future profits in Table 4, when discounted back 3 months, are equal to the profits found in table 3 (1.04 = 1.06e ). 7

8 Table 3: Position T = 0 (Today) T = 0.25 Buy Future 0 S T - \$ Sell e dt Units of the Index \$ S T Invest in T-Bill -\$971.67e = \$ \$ Net Position \$ Table 4: Position T = 0 (Today) T = 0.25 Buy Future 0 S T - \$ Sell e dt Units of the Index \$ S T Invest in T-Bill -\$ \$957.05e = \$ Net Position 0 \$ Since the manager owns the fund, he would like to hedge this position by selling futures on the index. Thus, if the value of the index falls, his long position in the fund will fall, but will be exactly offset by his profits on his short position in the futures. The question is, how many contracts must he sell? By selling forward the index, the manager ensures a return of ( )/800 = 16.67%. Thus, the value of his fund is ( ) \$10 million = \$ million at year end. Each futures contract on the S&P500 has a notional value of 250 times the index or = \$233, Thus, the manager must sell \$11, 670, 000/\$233, = 50 contracts. To illustrate the hedge, consider the scenario where the index level falls to in December. The value of the fund declines to /800) 10 million = \$10.42 million. However, the short futures position gains ( ) = \$1.25 million. This follows from being short 50 contracts and each contract being worth 250 times the level of the index. Thus, the value of the manager s aggregate position is \$10.42 million + \$1.25 million = \$11.67 million. No consider a rise in the level of the S&P500 to 1,000. The value of the fund will be (1, 000/800) 10 million = \$12.5 million. However, the futures 8

9 position will lose ( , 000) = \$0.83 million. Thus, the value of the manager s aggregate position is \$12.5 million \$0.83 million = \$11.67 million. Remark: Note that this hedging strategy is equivalent to the manager liquidating his position in the fund and investing the proceeds at the risk-free rate. Although this rate is not explicitly given in the problem, it is implied by the futures contract (see equation (5)). As discussed in class and the lecture notes, the manager may not want to liquidate his position either because of transaction costs or a desire to maintain his position in the index s assets. 5. Our company is going to be receiving Swiss Francs in the future, which we will then want to exchange (i.e. sell) for \$US. Thus, we want to sell SFR/\$ futures. By doing so, we ensure ourselves of receiving: 8 million SFR 0.8 SFR/\$ = \$10 million. Table 5 illustrates the hedge. Table 5: Future Exchange Rate Position 0.4 SFR/\$ 1.6 SFR/\$ Revenue from Sale 20 5 Forward Contract Net Position \$10 million \$10 million Regardless of the future exchange rate, our \$US income is the same. 6. Borrowing at LIBOR (i.e. a floating rate loan) exposes the company to interest rate risk: if the LIBOR rises, so do the payments on the loan. Thus, we want to enter into another contract that offsets this risk, a contract that pays us when the interest rate increases. In a eurodollar futures contract, if the interest rate (LIBOR) rises, the futures price will fall. This can be seen in equation (6). Since the Eurodollars 9

10 futures contract is currently trading at 92, equation (6) implies that the interest rate is at 8%. To lock in this rate, we must sell two March Eurodollar futures contracts. Underlying each contract is a notional 90-day, \$1 million loan to be entered into upon expiration of the futures contract (sometime in the middle of March). To determine if the hedge works, first consider what the total interest payment would be under a fixed 8% loan for one quarter: (0.08/4) \$2 million = \$40, 000. Now suppose that LIBOR rises to 10% prior to entering the loan. The interest on the loan will be (0.10/4) \$2 million = \$50, 000, an increase of \$10,000. However, our position in the futures contracts will yield: (2/4) 1, 000, 000 (.92.90) = \$10, 000, exactly offsetting the increased interest payment. 2 Alternatively, should LIBOR fall to 6%, the interest payment on our loan would fall to (0.06/4) \$2 million = \$30, 000, an decrease of \$10,000. However, our position in the futures contracts will lose: (2/4) 1, 000, 000 (.92.94) = \$10, 000, exactly offsetting the decreased interest payment. In both cases, the interest payment is fixed at \$40, a The futures price should be, according to equation (5), The contract is underpriced. F = e ( )(1/12) = b Since the market price for the contract is too low, we should buy the contract and sell the replicating portfolio. Consider the arbitrage table in Table 6. Alternatively, as in problem 3, we could have taken a position to obtain the arbitrage profits in the future. 7.c The intuition was laid out above: we buy the underpriced security (the futures contract) and short the replicating portfolio (the underlying asset and a T-bill). The actual mechanics can be described in more detail as follows: 2 The gain from the futures position comes from having sold 2 contracts, each for a loan in the amount of \$1 million for a length of 0.25 years. 10

11 Table 6: Position T = 0 (Today) T = 1/12 Buy Future 0 S T Sell e dt Units of the Index S T Invest in T-Bill 701e 0.05 (1/12) = Net Position At time 0 (today) buy the futures contract, which costs nothing, short a fraction (exp( dt )) of the index and invest part of the proceeds from the short sale in a T-bill. We need not short the entire index because of the dividend yield since the owner will receive dividends that they can reinvest in the index that make up for owning slightly less than a full unit. We could have placed the entire proceeds from the short sale in the T-bill, but this would have postponed our profits until the future (one month from now). Instead, we withhold In one month, we must buy back a full unit of the index, which just offsets the unit of index we get from owning the futures contract. The T-bill matures and pays off 701, just offsetting the 701 we owed from the futures contract. 7.d Intuitively, a fall in the dividend yield works in our favor since we are short the asset in our arbitrage position. Thus, we have to pay out less in the form of dividends to the holder of the index. The new arbitrage table is now seen in Table 7 Table 7: Position T = 0 (Today) T = 1/12 Buy Future 0 S T Sell e dt Units of the Index S T e (1/12) Invest in T-Bill 701e 0.05 (1/12) = Net Position 0.25 (1 e (1/12) S T In addition to the 0.25 we make today, we also receive some (small) 11

12 fraction of the future value of the index. 8. The Japanese investors are going to be receiving \$US 5 million in two months, which they will then want exchange (i.e. sell) for Yen. Thus, the hedge SEI wants to establish involves selling futures contracts. The question is how many? The current futures price is \$/Yen, implying the exposed Yen is million Yen. Since each contract is for 12.5 million Yen, SEI wants to buy /12.5 = 45 contracts. That is, they are contracting today to buy Yen in the future (equivalently they are selling \$US). To illustrate the hedge, Table 8 considers two scenarios: (1) the exchange rate decreases to \$/Yen and (2) the exchange rate increases to \$/Yen. Table 8: Future Exchange Rate Position \$/Yen \$/Yen \$US Yen \$US Yen Capital Raised Buy Forward Contracts Net Position Note that the net position in Yen is the same (except for rounding error as a result of our inability to buy fractional contracts). In the Capital Raised column the \$US amount raised is independent of the future exchange rate. However, the future Yen amount is exchange rate dependent and changes accordingly (5/0.007 = , 5/0.010 = 500). To compute the value of the long position in the forward contracts, consider the \$US payoff to a short position in one contract: (X T F T ) or ( )= Since there are 12.5 million Yen per contract, this amounts to million = million Yen per contract. We are short 45 contracts implying a total payoff of = \$US1.06 million. To translate this figure into Yen, simply divide by the prevailing exchange rate, 0.007, to get million Yen. 12

13 9. 9.a Recall the interest rate futures price from equation (6). F = 100 InterestRate Given the prices, it is straightforward to back out the implied LIBOR rates by subtracting the price from 100. The rates are reproduced in Figure 1 and plotted against maturity in Figure 2. Figure 1: Expiration Date LIBOR Future Prices Implied LIBOR (=100-Price) Mar Jun Sep Dec Mar Jun Sep Dec Mar Jun Sep Dec b The company needs to borrow \$12 million in September, which it will repay in December. Each contract is equivalent to \$1 million loan, implying the company will need to sell twelve contracts. The intuition is that the company is undertaking a loan that will become more expensive as LIBOR rises. Therefore, they need an instrument that will payoff when LIBOR rises to offset the additional interest expense on the loan. Since the price of the futures contract decreases as LIBOR rises, they should sell the contract(s). 13

14 Figure 2: Futures Implied LIBOR Mar-97 May-97 Jul-97 Sep-97 Nov-97 Jan-98 Mar-98 May-98 Jul-98 Maturity Date Sep-98 Nov-98 Jan-99 Mar-99 May-99 Jul-99 Sep-99 Nov-99 9.c Table 9 shows illustrates the working of the hedge when the interest rate increases or decreases by one percentage point. All dollar values are in millions. Table 9: Future (LIBOR) Interest Rate Position 4.78% 5.78% 6.78% Interest Expense on Loan \$ \$ \$ Short 12 Futures Contracts -\$ \$0 \$ Total Expense \$ \$ \$ Note that regardless of the future interest rate, the total expense is the same. Consider the \$ in the Interest Expense on Loan row. The company has taken out a four month loan of \$12 million with an interest rate equal to LIBOR+150bp (bp = basis points). Thus, the effective periodic 14

15 interest rate is ( )/4 = Multiplying the principal amount of the loan (\$12 million) by this rate yields the interest expense of \$ million. Similar computations yield the interest expenses under the other two interest rate scenarios. 9.d This is the opposite of the situation of above since now they are in a position to lend instead of borrow. Thus, they should buy 3 contracts. 9.e The December Eurodollar contracts are at \$94.04, which implies a LIBOR of 5.96%. Table 10 shows the net payoffs to different LIBOR rates from a position in Eurodollar futures contracts and lending. Table 10: Future (LIBOR) Interest Rate Position 4.96% 5.96% 6.96% Interest Earned on Loan \$7,500 \$0 -\$7,500 Long 3 Futures Contracts \$29,700 \$37,200 \$44,700 Total Gain \$37,200 \$37,200 \$37,200 These solutions are produced by Michael R. Roberts. Thanks go to Jen Rother for her excellent assistance, and to an anonymous TA. Any remaining errors are mine. 15

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