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Chapter 1 An Overview of Derivative Contracts 1.1. Because the forward price has declined between July and September, the short (party B) is profiting, and the long (party A) is experiencing paper losses. Thus, party A is the only party that has an incentive to default, in order to avoid its loss. Party A has contracted to buy at a price of $38, which is high, given that recent contracts have delivery prices of only $35. 1.2. OTC (over-the-counter) derivatives are custom made contracts. They do not trade on an exchange. They are created when two parties (one of whom is typically a derivatives dealer or market maker) agree on the terms of the contract. OTC derivatives positions are usually held until the delivery date. They are not marked to market; therefore, there is more default risk. Collateral is negotiated. Exchange traded derivatives trade on an exchange, such as a futures exchange or an options exchange. It is easier to offset a position when the contract trades on an exchange. Thus, many more exchange traded derivatives do no end up with actual delivery of the underlying asset. They are standardized contracts. They are marked to market daily. This prevents a large buildup in value (positive for one party, and equal but negative for the other party). The exchange clearinghouse stands behind every trade. There is less risk of loss due to your counterparty defaulting. 1.3. A futures contract is like a portfolio of one-day forward contracts. Each day, the one-day forward contract is settled up, with one party realizing a gain, and the other party realizing an equal loss. Then, a new one-day forward contract is entered into, with a delivery price equal to that day’s settlement price. Consider a futures contract with 3 months until delivery. You buy it at the close of trading on Monday, when the futures price is 25. The settlement prices on the next 3 days are: Day Settlement Price Tuesday 24 Wednesday 26 Thursday 27 The futures contract is marked to market daily, so your profits are as follows: Day Profit or Loss on one long futures contract Tuesday -1 Wednesday +2 Thursday +1 Now consider a time series of one-day forward contracts. On Monday, you go long one contract with a delivery price of 25. On Tuesday, the price is 24, so you lose $1. Also on Tuesday, you go long a forward contract for delivery on Wednesday, with a forward price of 24. On Wednesday, the price is 26, so your profit is $2. Also on Wednesday, you go long a forward contract for delivery on Thursday with a forward price of 26. On Thursday, the price is 27, so your profit is $1, and you also go long a forward contract for delivery on Friday with a forward price of 27. 1 The above example illustrates how a futures contract is equivalent to a time series portfolio of one-day forward contracts. A swap is equivalent to a portfolio of forward contracts. Each of them is originated today. Each of them has a different delivery day. Each of them has typically has the same delivery price. See figure 1.3. 1.4. Futures contracts are marked to market daily. Thus, the profits are realized sooner with a long futures position (if futures prices rise), than with a long forward position. With a long forward position, the profit is not realized until the delivery day. 1.5. The notional principal for the derivatives market does not reflect the values of the contracts. It reflects the base quantity underlying the contract. Consider an interest rate swap, with a notional principal of $40 million. The $40 million only reflects the base on which the payments are determined. A typical quarterly payment might be on the order of about $100,000. This $100,000 payment is determined if, say, the difference between the fixed interest rate and the floating interest rate is 1% per year, which is 0.25% per quarter. Then 0.0025 X $40,000,000 = $100,000. The value of a typical swap is zero when it is originated, if it properly priced. The value is zero regardless of the notional principal. The value as time passes and as interest rates change, but the value will always be a small fraction of the notional principal. The value of a futures contract is zero after it is marked to market. Thus, the notional amount of the asset underlying a futures contract in no way reflects the value of the contract. 1.6. (a) is the best answer. A student could argue that (d) is correct, too, as an arbitrageur buys a cheap asset and sells an equivalent more expensive asset today. Then, at some time in the future, the arbitrageur will offset those two trades. Thus an arbitrageur buys low today and sells high today. Then, she will sell high and buy low in the future. 1.7. (d) is the best answer. Futures have less default risk because both parties to a futures contract actually deal with the clearinghouse. Thus, neither has to consider the creditworthiness of the other. In addition, default risk of a futures contract is reduced because futures are marked to market each day. In contrast, a forward contract can experience a large buildup in value (positive for one party, and negative for the other). 2 Chapter 2 Risk and Risk Management 2.1. Financial price risk is the risk created by changes in prices. If the prices of inputs or outputs, exchange rates, interest rates, or stock prices create uncertainty for an individual or business, then financial price risk exists. 2.2. Why firms should manage risk is discussed in section 2.3 of this chapter. The reasons are to reduce the costs of financial distress to increase the likelihood that the firm will invest in all future profitable opportunities because it is less costly for firms to hedge than for most individuals to hedge because firms have access to superior information, so that they may properly hedge at the right times because many owners of the firm may not be well diversified because hedging may increase debt capacity because hedging may decrease taxes paid by the firm because risk averse managers may want to hedge to lower interest rates the firm must pay on debt to give the firm flexibility 2.3. Hedging reduces taxes when the tax schedule is convex (see figure 2.5). Thus, if the existing tax code has provisions that makes the tax schedule convex for most firms, then changing it to a flat tax will remove that convexity, and reduce the incentives to hedge. If the current tax code is concave for most firms, then changing it to a flat tax should not affect the hedging decision. 2.4. Gold is an input (a raw material) for this firm. Thus, if the price of gold rises, its costs will rise, and this will reduce profits, all else equal. However, if the firm can pass on the higher cost of gold to consumers, and they will still buy the same number of rings, then the firm will not care about rising gold prices. If demand for its rings is independent of the price of gold, then the firm will not need to hedge. It will merely pass on the higher cost of the rings to consumers, and its profits will remain unchanged. However, if the demand for rings is price elastic, then elastic demand: % quantity demanded >1 % price This means that if prices rise by some percentage amount, the quantity demanded will decline by a greater percentage amount. Thus, the firm is exposed to the risk of rising gold prices. Its costs are rising. If the firm does not raise the price of its final product (same demand), then its profits will be reduced because its costs are higher. If the firm does raise the price of its rings, demand will decline, and profits will also decline. In this case, the firm should hedge by buying gold in the forward market in or in the futures market. 2.5. Define P as the price. Then the expected price is: E(P) = (0.22)(40) + (0.58)(50) + (0.12)(60) + (0.08)(70) = 50.6 3 (1) (2) (3) (4) (5) (col. (1) * col. (2)) (col. (2) – E(P))2 (col. (4) * col. (2)) Squared Deviation from Prob * (sqrd devn Probability Outcome Outcome * Probability the expected price from the expected price) 0.22 40 8.8 112.36 24.7192 0.58 50 29 0.36 0.2088 0.12 60 7.2 88.36 10.6032 0.08 70 5.6 376.36 30.1088 Sum = expected price = 50.6 Sum = variance = 65.64 The standard deviation is the square root of the variance, and equals $8.10 2.6. New discoveries No new discoveries Probability 0.3 0.7 EBIDT 8,000,000 70,000,000 depreciation 12,000,000 12,000,000 interest 5,000,000 5,000,000 taxable income -9,000,000 53,000,000 taxes 0 21,200,000 net income -9,000,000 31,800,000 Cash flow 3,000,000 43,800,000 Expected net income = (0.3)(-9,000,000) + (0.7)(31,800,000) = $19,560,000. Cash flow is net income plus depreciation. Expected cash flow = (0.3)(3,000,000) + (0.7)(43,800,000) = $31,560,000. Variance of net income = (0.3)(-9,000,000-19,560,000)2 + (0.7)(31,800,000-19,560,000)2 = 349,574,400,000,000 The standard deviation of net income is $18,696,908.84 If the firm hedges, then: New discoveries No new discoveries Probability 0.3 0.7 EBIDT 43,000,000 55,000,000 depreciation 12,000,000 12,000,000 interest 5,000,000 5,000,000 taxable income 26,000,000 38,000,000 taxes 10,400,000 15,200,000 net income 15,600,000 22,800,000 Cash flow 27,600,000 34,800,000 Expected net income = (0.3)(15,600,000) + (0.7)(22,800,000) = $20,640,000. Cash flow is net income plus depreciation. Expected cash flow = (0.3)(27,600,000) + (0.7)(34,800,000) = $32,640,000. 4 Variance of net income = (0.3)(15,600,000-20,640,000)2 + (0.7)(22,800,000-20,640,000)2 = 10,886,400,000,000 The standard deviation of net income is $ 3,299,454.50 The example illustrates how hedging reduces the volatility of a firm’s net income and cash flow. 2.7. If the random variable is symmetric, such as with a normal probability distribution, it is irrelevant whether we use the variance (standard deviation) or the semi-variance. But the criticism is valid when the random variable’s probability distribution is not symmetric (i.e., skewed). Note that it seems as if negative, bad, events occur much more frequently than they “should”. This would provide a rationale for using the semivariance. 2.8. a. The firm can examine historical cash flows to determine their sensitivity to prices, interest rates, exchange rates, etc. To determine cash flow sensitivity to prices, the firm should run univariate and multivariate regressions. The good point to this approach is that it is simple to estimate these models. There are several drawbacks to this approach. First, it focuses on what has happened in the past. There is no guarantee that its exposure will remain constant in the future. Second, the firm must be sure that the cash flows it estimates using past data do not include the impact of any risk management activities it may have previously undertaken. The firm must be sure that it is dealing with actual cash flows, and not accounting variables that are created with a great deal of discretionary decisions. Third, historical cash flows may fluctuate because of factors other than prices. But because these factors were prominent in periods during which one or more of the prices changed, the regression models will conclude that cash flows are dependent on those prices. b. The firm can regress the percentage changes in the firm’s stock price (rates of return) against changes in prices of raw materials, exchange rates, interest rates, etc. This measures how investors regard the relationship between equity value and changes in these prices. Again, these models are simple to estimate, and because market returns are used, distortions caused by accounting conventions are not relevant. However, this approach depends on investors’ abilities to identify the firm’s risk exposures accurately, and to impound them into security prices. c. The firm can estimate its own model of how its cash flows are determined, and simulate the results of the model using Monte Carlo simulation techniques. The Monte Carlo approach is good because it is forward looking, and because it disciplines a firm into analyzing the nature of its business risks. The impact of changing prices on all of the firm’s activities can be estimated using Monte Carlo simulation. But, the Monte Carlo approach suffers from the drawback that the individual setting up the model will often impose his biases, consciously or unconsciously, into the model. This could ensure that some particular results that reflect these biases could be obtained. 2.9. (a) (Hedging may actually lower a firm’s cost of capital) 2.10. (c) 2.11. (c) 2.12. (e) 5 Chapter 3 Introduction to Forward Contracts 3.1. There are D = 153 days between June 2 and November 2 (the forward period). The problem does not state how many days are defined to be in a year (B). Equation 3.1 determines the loss; a loss is realized because the FRA was sold, and interest rates rose. The spot 5-month Eurodollar rate (6.04%) on June 2 determines the loss for the firm. Assuming B = 360, the firm must pay $36,463.97: P*[ r ( t 1,t 2 ) fr ( 0 ,t 1,t 2 )]*( D/ B ) 1[ r ( t 1,t 2 )*( D/ B )] $80million[ 0.0604 0.0593](153 / 360)] $36,463.97 = 1 [(0.0604)(153 / 360)] 3.2. The firm can either a) go to its original counterparty and buy itself out (if it is losing money), or negotiate a payment to be received (if it is currently making a profit), or b) enter into a new forward contract with three months until delivery. If the old contract was to buy the good, the new offsetting contract will be to sell the good; if the old contract obligated the firm to deliver the good, the new offsetting contract must obligate the firm to buy the good. 3.3. The contract is now a liability for Goldfinger. The original contract was to buy at $340. Three months later, a contract that would offset Goldfinger’s obligation would require them to sell at $331/oz. Thus, the forward price has declined from what it was originally, 3.4. a. It should sell the 4 X 10 FRA, which obligates BilboBank to lend $40 million for a six month period beginning on November 6. The original 5 X 11 FRA obligated Bilbobank to borrow $40 million for six months beginning on November 6. b. BilboBank will have to make a payment; it is losing money as of July 6. BilboBank bought a forward rate of 6.25%, and the relevant forward rate on July 6 is 6.06%. The buyer of a 5 X 11 FRA profits if interest rates rise. Here, interest rates fell. c. BilboBank will have to pay the present value of the difference in forward rates. Roughly, this will be the present value of the expected payment to be made four months hence. Given the change in the quoted forward rate, the expected payment is $40million(0.0625 0.0606)(1 / 2) $36,882.46 [1 (0.0606)(1 / 2)] 6 But this is the expected payment to be made four months hence. So this amount must be discounted back to the present time, using the spot rate for discounting: $36,882.46 $36,159.27 [1 ( 0.06 )( 4 / 12 )] 3.5. The party that was long the contract made a profit of [S(T) - F(0,T)] * N = $[32.20- 31]/bbl * 24,000 bbls = $28,800. 3.6. You buy the FRA at the asked yield of 8.18%. The spot rate on the delivery date is 9.02%. Assuming that there are 60 days in the contract period, your profit will be received three months hence, and the amount received will be GBP60million( 0.0902 0.0818)( 60 / 365) GBP81,638.82 [1 ( 0.0902)( 60 / 365)] 3.7. a. On the settlement day, the firm receives £5 million, and pays ¥840 million (£5 million times ¥168/£ = ¥840 million). b. The firm realizes a profit because it is long the forward contract to buy GBP at a price of ¥168/£. The actual spot price is higher than the originally agreed upon forward price. S(T) > F(0,T). The profit is [S(T) - F(0,T)]N = [¥180/£ - ¥168/£](£5 million) = ¥60 million. 3.8. $0.79 SFR1 The agreement to exchange $0.79 for one SFR, one year hence, has no value if it is an equilibrium forward price. If a zero coupon bond that will pay off its $1 face value one year hence is today worth $0.939, then it follows that a zero coupon bond that will pay off its $0.79 face value one year hence is today worth 0.939 * 0.79 = $0.74181. Using today’s exchange rate of $0.75/SFR, we can now compute the SFR-denominated value of $0.74181. It is $0.74181 * SFR/$0.75 = SFR0.98908. It might be added that given these spot prices and forward prices, we can also compute the U.S. and Swiss one-year interest rates. For the U.S., given PV = 0.939, FV = 1, and n = 1 year, we can compute the one-year interest rate to be 6.496%. For Switzerland, given PV = 0.98908, FV = 1, and n = 1 year, the one year interest rate is computed to be 1.104%. In chapter 5, the following formula for forward exchange pricing is presented: (1 rUS ) F S (1 rSwiss ) 7 where F and S are in terms of $/SFR. Given the spot exchange rate and the two countries’ one- year interest rates, the forward price of a SFR for delivery one year hence can be computed: . (106496) F 0.75 $0.79 / SFR . (101104) 3.9. a. FFr10 million X $0.1429/FFr = $1,429,000. Figure 3.5 shows that the forward price for French francs is $0.1429/FFr, for delivery 3-months forward. b. The table does not provide the forward price of SFr, denominated in terms of FFr. But since we know that FFr10 million = $1,429,000 (for delivery 3-months hence), we can now convert this dollar amount to SFR as follows: $1,429,000 X SFR1.6642/$ = SFr2,378,141.8. Figure 3.7 shows that the SFr- denominated forward price of a dollar is 1.6642. 3.10. (d) is definitely false. (c ) could be considered by many to be a false statement. 3.11. (b) is true 3.12. (d) 3.13. (d) 3.14. (a) 8 Chapter 4 Using Forward Contracts to Manage Risk 4.1. A gold mining firm faces the risk that gold prices will decline, if demand for its output is price inelastic. Revenues (= price * amount of gold sold) will then decline if prices fall. The firm should sell gold forward to protect itself against a decline in the price of its output, gold. Firm A, which will see demand for its product be totally unaffected by a price decline, will have to sell more gold forward to manage its risk exposure. Its dollar-revenue will decline in direct proportion to the price decline. However, firm B will observe a small increase in the demand for its output, so a price decline will not affect total revenues as much as it will firm A. Firm B will have to sell less gold forward to hedge its risk exposure. 4.2. Grains are an input for the cereal manufacturer. Thus, the cereal manufacturer faces the risk that grain prices will increase, and it should buy wheat and corn forward to protect itself. But grains are an output for the farmer. The farmer will sell wheat and corn forward to protect itself from falling prices. Price elasticity is not a factor for the farmer because he can sell only the amount of wheat and corn he has planted, at the market price. 4.3. Most of the assets of banks, thrifts and other financial institutions have floating rates. Any loans they make with rates that reset periodically represent floating rate assets. Issuing floating rate debt serves to hedge their exposure to interest rate risk. If interest rates rise, they will receive more interest revenue from their assets, and they will also have to pay out added interest expense on the floating rate debt. This illustrates the concept of matching the characteristics of a financial institution’s assets and liabilities. 4.4. The bank is exposed to the risk that interest rates will decline. More specifically, it fears that the three-month forward rate that exists nine months hence will be lower than expected. If the 3-month rate is lower than expected, nine months hence, then the bank’s profit will be lower than expected. If the 3-month rate declines by a large amount, the bank may find itself paying more for its borrowed funds than it will receive on any new 3-month loan it makes. To hedge this risk exposure, the bank should sell a 9 X 12 FRA. This will lock in a lending rate for three months, beginning nine months hence. If interest rates do decline (below the contractually agreed upon forward rate), then the bank will earn a profit on the FRA at the same time that it loses money in the spot market. The balance sheet approach can also be used to answer the question. The bank’s asset has a shorter duration (nine months) than the duration of its liability (twelve months). Thus, if interest rates decline, its liability will rise in value by a greater amount than its asset. Thus, the value of the bank’s common stock must decline if interest rates decline. 4.5. The bank is exposed to the risk that interest rates will rise. More specifically, it fears that the three-month forward rate that exists nine months hence will be higher than expected. To hedge this risk exposure, the bank should buy a 9 X 12 FRA. This will lock in a borrowing rate from time t=9 until time t=12. If interest rates do rise (above the contractually agreed upon forward rate), then it will earn a profit on the FRA at the same time that it loses money in the spot market (because its borrowing rate is higher nine months hence). 9 Alternatively, consider that the bank’s asset has a greater duration than that of its liability. Thus, if interest rates rise, the value of the asset will decline by a greater amount than will its liability. Thus, a rise in interest rates will lead to a decline in equity value. 4.6. It faces the risk that interest rates will rise during the next four months. It should buy a 4 X 6 FRA to hedge itself. If interest rates do rise, it will have added interest expense on the money it borrows, but it will also realize a profit on the long FRA position. 4.7. a. The firm should buy a 6 X 15 FRA. This will lock in a borrowing rate beginning six months hence for a nine-month forward period. If interest rates rise, then it will face a higher interest rate on its commercial paper, but it will profit on the FRA. b. Typically, interest on the commercial paper that the firm will issue will be paid 15 months hence, based on the spot nine-month rate that exists six months hence. Thus, interest on the commercial paper if r(6,15) = 0.05 will be $25,000,000 * 0.05 * 9/12 = $937,500. The profit or loss on the FRA is computed using equation 3.1. Keeping in mind that the interest expense on the commercial paper will be paid at fifteen months hence, and the profit/loss on the FRA will be realized six months hence, here is the results of the hedge: r(6,15) interest on the loss (+) or profit (-) Total interest commercial paper on the FRA expense 0.05 937500 180722.9 1118223 0.055 1031250 90036.01 1121286 0.06 1125000 0 1125000 0.065 1218750 -89392.1 1129358 0.07 1312500 -178147 1134353 Suppose instead that we find the future value of the profit/loss on the FRA. Basically, this means that the FRA cash flow is not discounted. E.g., if r(6,15) is 5%, the future value of the loss on the FRA is $25,000,000 * (0.06-0.05) * (9/12) = $187,500. Here are the results of the hedge if all cash flows are computed on a future value (15 months hence) basis: 0.05 937500 187500 1125000 0.055 1031250 93750 1125000 0.06 1125000 0 1125000 0.065 1218750 -93750 1125000 0.07 1312500 -187500 1125000 This example illustrates how the timing of the cash flows on the spot position and the hedging instrument (the FRA) can affect the results of the hedge. 4.8. The assets (the floating rate loans) of the financial institution have a short duration. Rates will be reset frequently. Its liabilities are longer term. If interest rates decline, then the value of its assets will not change very much, but the value of its liabilities will increase. This means that the value of its common stock must decline. FRA’s should be sold. Alternatively, think in terms of its revenues and expenses. If interest rates decline, interest income on its portfolio of floating rate loans will decline, while interest expense will remain unchanged. Again, we conclude that the financial institution faces the risk of declining interest rates, and FRA’s should be sold to hedge this risk. 10 4.9. a. The Japanese firm faces the risk that 8-month French interest rates will decline during the next four months. If this happens, then it will earn less FFR-denominated interest when it actually deposits the FFR45 million. The firm also faces the risk that the yen value of FFR will be lower one year hence; i.e., that the ¥/FFR exchange rate will decline. It should sell FFR forward at the forward ¥/FFR exchange rate for delivery one year hence. b. The Japanese firm will want to sell a 4 X 12 FFR-denominated FRA. The profit on the FRA, if B =360 and D = 242 is: FFR45million( 0.09 0.078)( 242 / 360) FFR344,914.96 [1 ( 0.078)( 242 / 360)] 4.10. The Japanese firm faces the risk that the ¥/$ exchange rate will rise, and it should therefore buy U.S. dollars forward to hedge this risk. If the ¥/$ exchange rate rises, then it will cost more yen to acquire the dollars it needs to pay $-denominated interest. But it will profit on the forward contract on dollars that it buys. 4.11. a. The fund is exposed to the risk that the $/£ exchange rate will decline. The fund has GBP denominated assets, but no GBP-denominated liabilities. b. GBP should be sold forward to hedge. c. Initially: buy $16 million in British stocks. Thus £9,523,810 worth of British stocks can be purchased: $16 million * £ = £9,523,810 $1.68 If £9,523,810 are sold forward then the dollar loss of $957,143 is computed as follows: Sell the British stocks six months later for £9 million Exchange the £9 million for $ at the spot exchange rate: £9 million * $1.70 = $15,300,000 £ Loss on the forward contract = £9,523,810 * ($1.673/£ - $1.70/£) = -$257,143 The dollar loss is $15,300,000 - $257,143 - $16,000,000 = -$957,143 Dollar rate of return is $957,143/$16,000,000 = -5.982% during the six month period. This can be annualized in either of two ways: simple: -0.05982 * 2 = -0.1196 = -11.96% compounded: (1+r(0,0.5))2 - 1 = (1 - 0.05982)2 - 1 = (0.94018)2 - 1 = 0.88394 - 1 = -0.11606 = -11.606% Had the fund not hedged, then the dollar loss of $700,000 is computed as follows: Sell the British stocks six months later for £9 million Exchange the £9 million for $ at the spot exchange rate: £9 million * $1.70 = $15,300,000 £ The dollar loss is $15,300,000 - $16,000,000 = -$700,000 Dollar rate of return is ($15,300,000 - $16,000,000)/$16,000,000 = -4.375% during the six month period. This can be annualized in either of two ways: simple: -0.04375 * 2 = -0.0875 = -8.76% compounded: (1+r(0,0.5))2 - 1 = (1 - 0.04375)2 - 1 = (0.95625)2 - 1 = 11 -0.08559 = -8.559% 4.12. The Japanese firm faces the risk that the yen-denominated price of apples will rise. This may happen either if the dollar-denominated price of apples rises, or if the JPY/$ exchange rate increases. The apples imported are a cost for the Japanese firm; it wants to minimize its costs, cet. par. To hedge its price risks, it should buy forward contracts on apples, priced in terms of yen. Alternatively, it can buy dollar denominated forward contracts on apples, and also buy dollars in the forward market. It should buy these forwards with delivery dates that correspond to the dates on which it believes it will be purchasing apples. 4.13. The insurance company faces the risk that interest rates will decline. If interest rates decline, it will have to reinvest coupons at lower rates, and the total compounded rate of return that it ultimately earns will decline. Thus, a strip of FRA’s should be sold. There are 55 months until maturity. Coupons will be received 1 month, 7 months, 13, 19, 25, 31, 37, 43, and 49 months hence (the last coupon, received at maturity, will not be reinvested). The strip should consist of 1 X 55, 7 X 55, 13 X 55, …., 43 X 55, and 49 X 55 FRA’s . The principal amount of each FRA should be (0.09/2)*($45 million) = $2,025,000 4.14. The portfolio manager faces the risk that the DEM/$ exchange rate will decline. If the DEM price of the dollar declines, then he will receive fewer DEM to invest when he converts the dollars to DEM, one month from today. In addition, he faces the risk that the German stock market will increase between today and when the $60 million is received. To manage his currency risk exposure, he should buy DEM (i.e., sell dollars) in the forward market. He should agree to deliver $60 million and receive the appropriate amount of DEM at the forward DEM/$ exchange rate. 4.15. If one-year LIBOR rises by 1 basis point, the firm will experience a cash flow shortfall of $4500. This is derived from the regression equation. The firm should buy a 12 X 24 FRA to hedge against the risk that one year hence, one year LIBOR will be higher than the current contracted forward rate. Choosing the notional principal of the FRA is challenging. One response is for the firm to estimate the expected rise in one-year LIBOR. If it expects one-year LIBOR to rise by 100 bp, then its cash flow will be reduced by $450,000. Thus, it will want to profit by $450,000. The firm should solve for the notional principal amount that will provide it with a $450,000 profit on the FRA to offset the cash flow decline: $ X (0.01)( 360 / 360 ) $450 ,000 [1 (0.09 )( 360 / 360 )] In this formula we arbitrarily chose the expected increase in one-year LIBOR from 8% (contracted forward rate) to 9% (spot one-year rate one year hence). Solving for X, we compute that the notional principal should be $49,050,000. 4.16. a. It would buy forward exchange contracts as a hedge against a planned future purchase of foreign securities. Suppose it expects a cash inflow next week that it plans to invest. Then, during the next week, it faces the risks that 1) the prices of foreign securities will rise, denominated in the foreign currencies, and 2) that the dollar price of the foreign currencies will 12 rise. Thus, to hedge the foreign currency risk, it would buy forward contracts committing it to buy the foreign currencies at a fixed price. It would sell forward exchange contracts to hedge against a decline in the dollar value of the foreign currencies for the securities it already owns. If, for example, it owns some German stocks, it will sell DEM forward to hedge against the possibility that a decline in the $/DEM exchange rate will occur, causing the dollar denominated value of its assets to decline. b. Foreign exchange forwards hedge only the currency risk. There is no protection against the possibility that the home-country, DEM-denominated value of the securities will decline. Furthermore, hedging locks in a future price. There is a (roughly 50%) probability that the $/FX rate will rise, and the fund will not enjoy the dollar denominated profits that would be created by that rise, if it has hedged. 4.17. It owns British bonds, which are an asset. a. It fears that British interest rates will rise, which will cause the value of its bonds, denominated in £, to decline. To hedge, it can buy a £-denominated FRA. If British interest rates then rise, the profit on the FRA can offset the loss on the value of the bonds. b. Assets Liabilities £ (* €/£ = €) € owners equity € If €/£ falls, €-denominate assets will fall in value, causing losses for the stockholders. The fund can sell £ in the forward market (and be paid € for those £), locking in the forward €/£ rate. 4.18. a. The bank fears that the € interest rate will rise during the next month. The bank has committed to lend at a fixed 5% rate. It must lock in a borrowing (funding) rate. b. Buy a 1 X 13 FRA. This locks in a 12 month borrowing rate, beginning one month hence. If rates do rise, then the bank will lose in the spot market, but make a profit on the FRA. c. The bank wants it to be less than 5%. Then it will be borrowing at less than 5%, and lending at 5%. d. It fears that the ¥/€ exchange rate will fall. It will receive € 13 months hence. If ¥/€ falls, it will receive less ¥ for those €10.5 million. e. It can sell € forward. Then if ¥/€ falls, it will profit in the forward market. Selling € forward locks in the ¥ it will receive. Sell €10.5 million forward. 4.19. d 4.20. a 13 Chapter 5 Determining Forward Prices and Futures Prices 5.1. If an arbitrageur can use only x% of the proceeds from the short sale, then the cost of carry formula becomes an inequality. First ask: What if F>S+CC-CR? In this case, there is no change in the cost of carry proof. An individual can borrow, buy the (underpriced) spot good, and sell the (overpriced) forward contract, realizing an arbitrage profit. The spot good is not sold short. Thus, the upper bound (highest possible forward price) equals S+CC-CR. However, the lower bound is affected, because upon selling short the spot good, the arbitrageur gets to lend only x% of S to earn interest. Assume that the carry rate of return is unaffected by the fact that she gets to use only a fraction of the proceeds. Define CC* = xS0h(0,T). We then have: What if F<S+CC*-CR? Then F-S-CC*+CR<0 -F+S+CC*-CR>0 Today buy forward at F0 0 sell short the spot good + S0 lend - S0; x% of S0 is lent to earn h(0,T)% interest; (1-x)% of S0 can be considered to be lent at zero interest. _____________________________ 0 At delivery buy the good -F0 receive principal & interest +xS0(1+h(0,T)) + (1-x)S0 = S0(1+xh(0,T)) pay any carry return -CR ______________________________ -F0 + S0 + xS0h(0,T) - CR > 0 or -F0 + S0(1+xh(0,T)) - CR > 0 Therefore, the lower bound for the forward price when only x% of the proceeds of the short sale can be used by the arbitrageur equals S0 + CC* -CR, where the carry costs (the income that can be earned from the proceeds of the short sale) equal xS0h(0,T). Overall, the cost of carry pricing inequality is: S+CC-CR > F > S+CC*-CR 5.2. a. Cash and carry arbitrage determines the upper bound: Today buy 1 share @ $63/share = -$63 borrow @ 7% +$63 14 sell forward @ F0 ______________ CF0 = 0 Eight months hence repay loan with interest -$63 * (1 + (0.07)(8)/(12)) = -$65.94 deliver the stock +F0 __________________ F0 < $65.94 Reverse cash and carry arbitrage determines the lower bound: Today sell 1 share @ $62.50/share = +$62.50 lend @ 6% -$62.50 buy forward @ F0 ______________ CF0 = 0 Eight months hence get repaid with interest +$62.50 * (1 + (0.06)(8)/(12)) = + $65 buy the stock -F0 __________________ F0 > $65 Conclusion $65 < F0 < $65.94 The following screens use the FinancialCAD function aaCDF to verify these results. Note that in order to conform to the solution, we used aaAccrual_days to search for an expiration date that was 240 days hence, and used a 360 day year; in this way the delivery date was 0.66667 years (8 months) hence. The upper bound is 65.94: aaCDF spot price per unit of underlying commodity 63 rate - simple interest 0.07 value (settlement) date 5-Feb-2001 expiry date 3-Oct-2001 actual/360 accrual method 2 storage cost 0 convenience value 0 fair value statistic 2 fair value 65.94 15 aaAccrual_days effective date 5-Feb-2001 terminating date 3-Oct-2001 accrual method 2 actual/360 number of business days from an effective date to a terminating date 240 The lower bound is $65: aaCDF spot price per unit of underlying commodity 62.5 rate - simple interest 0.06 value (settlement) date 5-Feb-2001 expiry date 3-Oct-2001 actual/360 accrual method 2 storage cost 0 convenience value 0 fair value statistic 2 fair value 65 b. Today (CF = 0): borrow +62.75 buy the stock -62.75 sell forward One, four and seven months hence: receive dividend +0.40 lend -0.40 Eight months hence Pay off loan -62.75(1 + (0.06)(8)/(12)) = -65.26 receive FV(Dividends) +0.40(1 + (0.06)(7/12)) + 0.40(1 + (0.06)(4/12)) + 0.40(1 + (0.06)(1/12) = +1.224 sell the stock +F0 ______________________________________ F0 = 64.036 The FinancialCAD function aaEqty_fwd solves this problem as follows: 16 aaEqty_fwd value (settlement) date 5-Feb-2001 expiry date 5-Oct-2001 30/360 accrual method 4 rate - simple interest 0.06 cash price of the underlying equity index 62.75 fair value of forward or futures statistic 2 dividend payment table t_14 t_14 dividend payment table all the dividend dates from the value date to the expiry date dividend amount 5-Mar-2001 0.4 5-Jun-2001 0.4 5-Sep-2001 0.4 fair value of forward or futures 64.036 c. If F = 63, then the implied repo rate is F0 CR S 0 63 1224 62.75 . 0.02349 2.349% S0 62.75 This is a holding period return. It can be annualized using simple interest (0.02349*12/8 = 3.5235%) or using compounded interest ((1.02349)1.5 - 1 = 3.544%). The FinancialCAD function aaCDF_repo can be used to compute the same answer. However, the function does not allow you to input the dividends. To finesse this shortcoming, put the future value of dividends (1.224) into the cell for “convenience value”. Then, aaCDF-repo will get the solution that the annualized (using simple interest) implied repo rate is 3.5235%: aaCDF_repo spot price per unit of underlying commodity 62.75 17 futures price 63 value (settlement) date 5-Feb-2001 expiry date 3-Oct-2001 actual/360 accrual method 2 storage cost 0 convenience value 1.224 annual repo rate incl. convenience value statistic 1 annual repo rate incl. convenience value 0.03523506 If F = 63, the forward price is too low (the theoretical forward price is 64.036). Therefore, an arbitrageur would buy the cheap forward contract and sell the underlying stock. Equivalently, if the arbitrageur’s lending rate is greater than the repo rate, reverse cash and carry arbitrage is possible. Here, the repo rate is 3.5235%, and the lending rate is 6%, so the arbitrageur should essentially borrow in the forward market (sell the stock and buy forward is tantamount to borrowing), and lend in the cash market at 6%. d. The upper boundary is determined by cash and carry arbitrage. Borrow $62.75 and sell forward. The theoretical upper forward price boundary is 62.75 * (1 + 0.06*8/12) = $65.26. But the lower forward price boundary is determined by reverse cash and carry arbitrage. Today sell the stock +62.75 lend -62.75 ($43.925 is lent at 6%; $18.825 is lent at 0%) buy forward at F0 __________________________________________ CF0 = 0 Eight months hence buy the stock -F0 get repaid $18.825 + $43.925 * (1 + 0.06*8/12) = $64.507 ________________________________________ F0 = 64.507 Thus, 64.507 < F0 < 65.26. 5.3. a. There is no carry return on gold. Therefore, F = S + CC = S + h(0,T)S h(0,T) = (0.10)(5)/12 = 0.041667 Therefore, F = 300(1.041667) = 312.5 FinancialCAD solves the problem as follows: 18 AaCDF spot price per unit of underlying commodity 300 rate - simple interest 0.1 value (settlement) date 5-Feb-2001 expiry date 5-Jul-2001 actual/360 accrual method 2 storage cost 0 convenience value 0 fair value Statistic 2 fair value 312.5 aaAccrual_days effective date 5-Feb-2001 terminating date 5-Jul-2001 actual/360 accrual method 2 number of business days from an effective date to a terminating date 150 b. The actual forward price ($310) is below the theoretical forward price ($312.50). Therefore buy the cheap forward and sell gold in the spot market. This is reverse cash and carry arbitrage. Today: buy forward zero cash flow sell 100 oz. of gold +$30,000 lend proceeds -$30,000 __________________ total cash flow 0 At delivery, five months hence: buy 100 oz. of gold @ F0 -$31,000 get repaid principal and interest +$30,000 * (1+ (0.1)(5/12)) = +$31,250 ________________________________ total cash flow $ 250 = arbitrage profit c. The implied repo rate over the five-month period is 19 F0 S 0 310 300 0.0333 3.33 % S0 300 If this is annualized using simple interest then the annual implied repo rate is .0333 * 12/5 = 8% aaCDF_repo spot price per unit of underlying commodity 300 futures price 310 value (settlement) date 5-Feb-2001 expiry date 5-Jul-2001 actual/360 accrual method 2 storage cost 0 convenience value 0 annual repo rate incl. convenience value statistic 1 annual repo rate incl. convenience value 0.08 5.4. a. rb = borrowing rate = 0.10. Therefore hb = 0.041667 over 5 months. rl = lending rate = 0.09. Therefore hl = 0.0375 over 5 months. The reverse cash and carry trades establish the lower bound for the forward price. This involves selling the spot good short, lending the proceeds, and buying the cheap forward contract. If the forward price is below this lower bound, then perform a reverse cash and carry arbitrage. Therefore, F > S (1+hl) F > 300(1.0375) F > 311.25 The cash and carry arbitrage requires that you borrow, buy the spot good, and sell the forward contract. This determines the upper bound for the forward price. If F is above this upper bound, then perform the cash and carry trades. Therefore, F < S (1+hb) F < 300 (1.041667) F < 312.5 The no-arbitrage pricing bounds are 311.25 < F < 312.5. If the highest possible forward price, 312.5, was surpassed, then arbitrage opportunities would exist. In other words, if the forward price rises too high (above F0(max)), then you should borrow, buy spot gold, and sell the overpriced forward contract: Today 20 borrow $30000 at 10%/yr, which is h(0,5mo.) = 4.1667%. buy 100 oz of gold at S0 = $300/oz. sell 1 forward contract at F0, which lies above F0(max) These acts result in a zero cash flow today. 5 months hence deliver your 100 oz of gold, and receive F0 repay principal and loan of S0(1+h(0,5mo.)) = $30,000(1.041667) = $31,250. For there to be no arbitrage profit, the cash flow in 5 months cannot be positive. Thus, F0(max) < 312.5/oz. If the forward price fell below the lowest possible price, F0(min), then arbitrage would be possible by selling spot gold short, lending the proceeds for 5 months, and buying a cheap forward contract. Today buy one forward contract at F0, which is below F0(min) sell short 100 oz of gold at $300/oz., providing you with $30,000 lend $30,000 at 9%/year, which is h(0, 5mo) = 3.75%. These trades result in a zero cash flow today. 5 months hence buy 100 oz. of gold at F0 get repaid $31,125. ($30,000(1.0375)=$31,125). For there to be no arbitrage profit, the cash flow in 5 months cannot be positive. Thus, F0(min)> 311.25. b. In part (a), we established that S(1+hl) < F < S(1+hb) The left hand side, the lower bound, is set by the reverse cash and carry trades, in which the spot good is sold, at the bid price. The right hand side is set by the cash and carry arbitrage, in which the spot good is bought. It will be bought at the asked price. Therefore, Sbid(1+hl) < F < Sask(1+hb) 300(1.0375) < F < 301(1.041667) 311.25 < F < 313.54 FinancialCAD solves for the lower bound as follows: aaCDF spot price per unit of underlying commodity 300 rate - simple interest 0.09 21 value (settlement) date 24-Sep-2001 expiry date 21-Feb-2002 actual/360 accrual method 2 storage cost 0 convenience value 0 fair value statistic 2 fair value 311.25 aaAccrual_days effective date 24-Sep-2001 terminating date 21-Feb-2002 actual/360 accrual method 2 number of business days from an effective date to a terminating date 150 FinancialCAD solves for the upper bound as follows: aaCDF spot price per unit of underlying commodity 301 rate - simple interest 0.1 value (settlement) date 24-Sep-2001 expiry date 21-Feb-2002 actual/360 accrual method 2 storage cost 0 convenience value 0 fair value statistic 2 fair value 313.5416667 c. If transactions costs to trade forward are introduced, the old lower bound of $311.25/oz must be even lower so that a profit can be realized after transactions costs. The old upper bound of $313.54 must be even higher in order to arbitrage. $50 on a contract covering 100 oz of gold equals $0.50/oz. Therefore the lower bound is 0.50 lower, and the upper bound is now 0.50 higher. 22 Sbid(1+hl) - TC < F < Sask(1+hb) + TC 311.25 - 0.50 < F < 313.54 + 0.50 310.75 < F < 314.04 d. Finally, if you get to lend only 10% of the proceeds of the short sale, your carry costs for the reverse cash and carry trades are only 10% of the amount computed above. In other words, before, you could lend $300 to earn 9% per year. You earned $11.25 per ounce of gold in 5 months. Now, you can lend only $30 (10% of $300) to earn 9% per year. You will earn only $1.125 in 5 months. Only the lower bound (reverse cash and carry) is affected, because the spot good is sold short when reverse cash and carry arbitrage is performed. The new lower bound is Sbid + interest on the short sale - TC 300 + 1.125 - 0.5 = 300.625 Therefore, the no-arbitrage pricing bounds are 300.625 < F < 314.04. These represent the prices that preclude arbitrage for you, given that the spot price of gold is $300/oz. Others may have better borrowing and lending rates, and yet others may get to use more than 10% of the short sale proceeds to earn interest. For them, the arbitrage band will be narrower. 5.5. A risk premium for a forward contract is the difference between the forward price and the expected price for the underlying good at delivery. Thus, if the forward price is 400, but "the market" expects that at delivery, the spot price of the good will be 398, the risk premium is 2. A risk premium will exist if the conditions of normal backwardation or contango exist. Suppose that hedgers are predominantly short forward contracts. This means that speculators must be long forward on balance. But speculators will not go long forward unless they are compensated with an additional expected return. Thus, the forward price must then be below the price that speculators expect will exist at delivery. The forward price is expected to rise over the life of the contract. This is normal backwardation. If, however, hedgers are net long forward contracts, then speculators must be short forward on balance. Speculators will not go short forward unless they expect to make profits; that is, they must expect that the forward price will fall. Equivalently, the current forward price must be above the spot price that they expect will exist at the delivery date. This is contango, and it too postulates a risk premium for forward contracts. 5.6. The repo rate is the interest rate that exists for borrowers who use securities that they own as collateral. Essentially, a prospective borrower sells her securities for a price, and agrees to repurchase the securities at a later date (most frequently, tomorrow) at a slightly higher price. If the borrower defaults, then she loses ownership of the securities. The implied repo rate is also virtually a riskless rate of return. It is earned by buying the spot good and selling a forward contract or futures contract. Going short a forward contract virtually locks in a selling price for the spot good. The implied repo rate is a lending rate; you will have lent money by buying the spot good (a cash outflow). Arbitrageurs will trade when their repo rate is below the implied repo rate. The implied reverse repo rate is a nearly riskless borrowing rate. It is realized by selling the spot good and going long forward. The long forward position essentially locks in a buying 23 price. Arbitrage is possible when traders can lend in the cash market at rates above the implied reverse repo rate that exists with forward or futures contracts. 5.7. More distant forward prices are lower because oil likely has a convenience return. Users of crude oil need the oil for production purposes, and they are unwilling to sell spot oil for $12.68, and lock in the purchase price of oil in the future by going long forward, even though it would be cheaper to buy the oil in the forward market. If the expected spot price in January 2000 is $12, and the current forward price for delivery in January 2000 is $10.43, then the forward price is expected to rise. This is normal backwardation. 5.8. S = 125 r = 0.05 h(0,4) = 1.6667% FV(divs) = $1.50(1+(0.05)(2/12)) = $1.5125 F = S(1+h)-FV(divs) = 125(1.01667)-1.5125 = 125.5708 aaEqty_fwd value (settlement) date 24-Sep-2001 expiry date 24-Jan-2002 30/360 accrual method 4 rate - simple interest 0.05 cash price of the underlying equity index 125 fair value of forward or futures statistic 2 dividend payment table t_14 t_14 dividend payment table all the dividend dates from the value date to the expiry date dividend amount 24-Nov-2001 1.5 fair value of forward or futures 125.5708333 b. The actual forward price of $125.40 is too low. Therefore, to arbitrage, sell the stock short, lend the proceeds, and buy forward. Selling the stock short and buying it forward represents synthetic borrowing at an implied repo rate that is less than your lending rate of 5%/annum. The unannualized implied repo rate over a four-month period is F0 CR S 0 125 .40 1.5125 125 0.0153 1.53 % S0 125 24 Annualized using simple interest, this is 0.0153 * (12/4) = 4.59%. FinCAD gets the same solution. Think of the carry return (the dividend plus interest on the dividend) as a convenience value for holding the cash asset (the stock). Note that to annualize the return, and get the same answer of 4.59%, we must compute the number of days that will get us 0.33333333 year, or 120 days in a 360 day-year. aaCDF_repo spot price per unit of underlying commodity 125 futures price 125.4 value (settlement) date 24-Sep-2001 expiry date 22-Jan-2002 actual/360 accrual method 2 storage cost 0 convenience value 1.5125 annual repo rate incl. convenience value statistic 1 annual repo rate incl. convenience value 0.0459 aaAccrual_days effective date 24-Sep-2001 terminating date 22-Jan-2002 actual/360 accrual method 2 number of business days from an effective date to a terminating date 120 5.9. The foreign exchange forward pricing equation is: S [1 hd ( 0, T )] 86.25[1 ( 0.01)(98 / 365)] F 85.33597 [1 h f ( 0, T )] 1 ( 0.05)(98 / 365) aaFXfwd FX spot - domestic / foreign 86.25 rate - domestic – annual 0.01 rate - foreign – annual 0.05 value (settlement) date 14-Mar-97 forward delivery or repurchase date 20-Jun-97 accrual method - domestic rate 1 actual/ 365 (fixed) accrual method - foreign rate 1 actual/ 365 (fixed) 25 Statistic 2 fair value of forward (domestic / foreign) fair value of forward (domestic/ 85.3359692 foreign) b. The actual forward price is too low. Sell C$ in the spot market, and buy C$ in the forward market. On March 14: Borrow one C$ at 5% +C$1 Sell one C$, and receive ¥86.25 - C$1 + ¥86.25 Lend ¥86.25 at 1% in Japan - ¥86.25 Sell ¥86.25(1+(0.01)(98/365)) = ¥86.481575 forward (& hence, buy C$ forward) @ F0 = ¥85.25/C$ ________________ Total CF on March 14: 0 0 On June 20: Receive interest and principal on the ¥86.25 ¥86.25(1+(0.01)(98/365)) = +¥86.481575 Deliver the ¥86.481575 to satisfy the terms of the forward contract, at the originally agreed upon forward price +C$1.0144466 - ¥86.481575 Repay the C$ loan with interest - C$1(1+(0.05(98/365)) = - C$1.0134247 __________________________________ Total CF on June 20: +C$0.0010219 The unannualized implied rep rate is: F [1 h f ( 0, T )] S 85.25[1 ( 0.05)(98 / 365)] 86.25 0.0016748 S 86.25 Annualized, the implied repo rate is then (0.0016748)*(365/98) = 0.0062378 = 0.62378%. FinancialCAD solves the problem as follows: aaFXfwd_repo_d FX spot - domestic / foreign 86.25 FX forward price - domestic / foreign 85.25 rate - foreign - annual 0.05 value (settlement) date 14-Mar-97 forward delivery or repurchase date 20-Jun-97 accrual method - domestic rate 1 actual/ 365 (fixed) accrual method - foreign rate 1 actual/ 365 (fixed) domestic repo rate 0.0062378 An arbitrageur will borrow at the implied repo rate of 0.62378%, and lend in the cash market at 1%. You can borrow at the implied repo rate by selling the C$ and buying it forward. 26 5.10. By investing in Great Britain, you will have £4 million (1.058)2 = £4,477,456. Instead, convert £4 million into FFR at the spot exchange rate of £0.10/FFR, and you will have FFR40 million. You know that two years hence, you will have FFR40 million (1.08)2 = FFR46,656,000. So today, sell FFR46,656,000 forward at the forward price of £0.098/FFR, and you know that you will have £4,572,288 two years hence. Thus, investing in France, and selling FFR forward, is preferred. 5.11. a. (1+h(0,6)) = (1+h(0,4))(1+h(4,6)) 1.063 = (1.041)(1+h(4,6)) h(4,6) = 2.11335% Annualized, r(4,6) = 0.021135 (12/2) = 12.68%. FinancialCAD’s solution is: aaFRAi value (settlement) date 7-Feb-97 effective date 7-Jun-97 terminating date 7-Aug-97 FRA contract rate 0.035 notional principal amount 1000000 accrual method 4 30/ 360 discount factor curve t_43_1 interpolation method 1 linear statistic 2 implied forward rate t_43_1 discount factor curve grid date discount factor 7-Feb-97 1 7-Jun-97 0.96061479 7-Aug-97 0.94073377 implied forward rate 0.12680116 b. ____________________ 0 4 6 <-----borrow-------------> <--lend--------> @t=0 buy 4 month securities (lend) $132,000/1.041 = -$126,801.15 sell 6 month securities +$126,801.15 _________________________ CF0 = 0 27 @t=4 the 4 month securities mature +$132,000 @t=6 the 6 month securities mature: -$126,801.15(1.063) = -$134,789.62 The unannualized forward borrowing rate from t=4 to t-6 is then: $134,789.62 $132,000 fh(4,6) = 0.021133 21133% . $132,000 Annualized, this is 2.1133% * (12/2) = 12.68% 5.12. The general formula is t2 t1 t2 - t1 (1 + r(0,t2) ) = (1 + r(0,t1) ) (1 + fr(t1,t2) ) Here we have t1 = 10, t2 = 14, r(0, t1) = 10%, and r(0, t2) = 11% (1.11)14 = (1.1)10 (1 + fr (t0, t4))14-10 (1.11)14/(1.1)10 = 1.6618 = [1 + fr (10, 14)]4 1.1354 = 1 + fr (10, 14) fr(10, 14) = 13.54% So the investor can borrow at time 10 for a period of four years at a rate of 13.54%. This can be locked in at time zero by lending at the 10 year pure discount rate and borrowing for 14 periods. ___________________________________ 0 10 14 0 10 14 = borrow borrow lend Time 0 Borrow $38,554 for 14 years at 11%/yr + $38,554 Lend $38,554 for 10 years at 10%/yr - $38,554 $0 Time 10 28 Get repaid on the loan $38,554 (1.1)10 = +$100,000 Time 14 Repay borrowed funds -$38,554 (1.11)14 = -$166,185 Given FV = $166,185, PV = -$100,000, and n = 4, solve for the interest rate, and we find that r = 13.54% 5.13. a. F = 120 * (1.026)10 / (1.066)10 = ¥81.86/$ b. $10,000,000(1.066)10 = $18,948,378 Or, convert $10,000,000 into ¥1,200,000,000 at the spot exchange rate. Then ¥1,200,000,000(1.026)10 = ¥1,551,153,774. If this amount is sold forward at the forward exchange rate of ¥81.86/$, then $18,948,861 will be received (This differs slightly from the $ figure above due to a rounding error). 5.14. This problem requires that the student find the current spot nine-month interest rate (r(0,9)) and spot four-month interest rate (r(0,4)). Then use this equation to solve for fr(4,9): (1+r(0,9))0.75 = (1+r(0,4))0.333 (1+fr(4,9))0.416667 5.15. This problem requires that the student find the current spot nine-year interest rate (r(0,9)) and the spot four-year interest rate (r(0,4)). Then use this equation to solve for fr(4,9): (1+r(0,9))9 = (1+r(0,4))4 (1+fr(4,9))5 5.16. This problem requires that the student find the current spot one-year US interest rate (rUS), the spot one-year Canadian interest rate (rCan), and spot exchange rate (S, expressed as the number of $US per Canadian dollar). Then use this equation to solve for F: F = S (1+rUS) / (1+rCan) 5.17. a. Section 5.2.2 focused on lenders. This question concerns borrowers. An Indonesian borrower could borrow 2300 rupiah today, and repay 2760 rupiah one year hence. In contrast, lets assume that the spot exchange rate is 2300 rupiah per dollar. If the Indonesian government was successful, then a borrower would assume that one year hence, the spot exchange rate will bee 2415 rupiah per dollar (that is 5% more than the original spot exchange rate). So an Indonesian borrower could borrow $1 today and convert it to 2300 rupiah at today’s spot exchange rate. One year hence, the Indonesian borrower will need $1.10 to repay its dollar debt. But that $1.10 will be worth $1.10 X 2415 rupiah/$ = 2656.50 rupiah. Thus, as long as the Indonesian government is successful at managing the depreciation of the rupiah, Indonesian borrowers would be better off borrowing dollars. b. The Indonesian borrower was betting on only a 5% increase in the price of a US dollar. Instead, the price of a dollar rose by 152%. $1.10 X 5800 rupiah/$ = 6380 rupiah would be needed to get the $1.10 needed to repay its dollar debt. 29 c. The Indonesian borrower could have borrowed dollars at 10% interest, and bought dollars (sold rupiah) in the forward market. This would have locked in the forward exchange rate, and eliminated its exchange rate risk. However, given the spot exchange rate and the prevailing spot US and Indonesian interest rates, the forward price of a dollar would likely have been 2300 (1.20)/(1.10) = 2509.09 rupiah per dollar. At this forward price, the Indonesian borrower would have been indifferent between borrowing rupiah or borrowing dollars. 5.18. a. F = 48(1.06) = 50.88 b. Three months later, there are only nine months remaining until delivery. Thus r(0,3/4) is the appropriate interest rate to use. The new theoretical forward price is F 42(1 0.07 3 ) 44.205 4 The value of the original contract is then 44 .205 50 .88 6.675 value 6.342 [1 0.07 (3 / 4)] 1.0525 The original contract is an asset for the short, because the forward price has declined. The short originally agreed to sell the underlying asset for 50.88. It is a liability (value = -6.342) for the long. 5.19. You are losing. It is a liability for you. You have committed to sell at 1.02. Now, new contracts are being initiated to have you sell at 1.04. The value of your position is $0.02 / € € 250,000 5000 $4938.27 1 0.05 1.0125 4 You have the incentive to default. 5.20. In terms of ¥/€, we have 1 h¥ F S 1 0.008 13 12 113.42 1.008667 ¥108.25/€ 1 h 113.42 13 € 1 0.0525 1.056875 12 The FinCAD function aaFXfwd computes the same result: aaFXfwd FX spot - domestic / foreign 113.42 rate - domestic 0.008 rate - foreign 0.0525 value (settlement) date 30-May-2002 forward delivery or repurchase date 30-Jun-2003 30 30/360 accrual method - domestic rate 4 accrual method - foreign rate 430/360 statistic 2fair value of forward (domestic /foreign) fair value of forward (domestic /foreign) 108.2464561 5.21. a. (1+h(0,12)) = [1+h(0,9)][1+fh(9,12)] 1 (0.079 9 12 ) 1 (0.08 3 ) 12 1.05925 1.02 1.080435% FinancialCAD cannot (to our knowledge) directly solve this problem. However, aaFRAi can indirectly solve it, by using the two spot rates (h(0,12) = 8.0435% and h(0,9) = 7.9%) to compute the implied forward rate of 8%. This is shown as follows: aaConvertR_DFcrv value (settlement) date 1-Jan-2002 rate curve t_48_4 simple interest rate basis rate quotation basis 7 accrual method of rate 430/360 t_48_4 rate curve maturity date yield to maturity 1-Oct-2002 0.079 1-Jan-2003 0.080435 discount factor curve - aaConvertR_DFcrv grid date discount factor 1-Jan-2002 1 1-Oct-2002 0.944064196 1-Jan-2003 0.925553134 aaFRAi value (settlement) date 1-Jan-2002 effective date 1-Oct-2002 terminating date 1-Jan-2003 FRA contract rate 0.035 notional principal amount 1000000 31 30/360 accrual method 4 discount factor curve t_43_1 linear interpolation method 1 statistic 2implied forward rate t_43_1 discount factor curve grid date discount factor 1-Jan-2002 1 1-Oct-2002 0.944064196 1-Jan-2003 0.925553134 implied forward rate 0.08 b. Nine months hence, on October 1. c. The new spot 3-month rate of 8.6% on October 1 is greater than the contract rate of 8%. Hence, the firm loses 50MM 0.086 0.08 92 365 75616.44 $74,012 1 0.086 92365 d. 1.02168 5.22. The theoretical futures price is F = 295(1.06)0.41667 = 302.25. Alternatively, it is F = 295 X (1 + (0.06)(5/12)) = 302.375. FinCAD’s solution depends on the number of days during the five month period. Thus, if there are 153 days in the five month period, and there are 365 days in a year, then F = 295 X (1 + (0.06)(153/365)) = 302.4194521. This is the answer provided by FinCAD’s function aaCDF: aaCDF spot price per unit of underlying commodity 295 rate - simple interest 0.06 value (settlement) date 30-May-2002 expiry date 30-Oct-2002 actual/365 (fixed) accrual method 1 storage cost 0 32 convenience value 0 fair value statistic 2 fair value 302.4194521 With a storage cost of 0.1, the result provided by FinCAD is higher; the fair value is 302.5194521. To explain why the theoretical futures is higher when there is a storage cost, consider the choice between buying spot gold or a gold futures contract. The fact that futures contracts require no initial outlay lets you invest the money until the futures delivery date. This advantage to the futures contract makes the futures price higher than the spot price. Now we are considering yet another advantage for buying futures; i.e., the purchase of a futures contract lets you avoid the storage costs that must be paid if the gold is purchased today. Thus, the futures price of gold is even higher. Basically, the futures price should be set so that the marginal investor is indifferent between buying gold in the spot market or in the futures market. The higher futures price reflects the advantages of buying futures. 5.23. On May 29, 2002, we found that S = ¥124.43/$. The six-month “domestic” interest rate (in Japan) was 0.8%, and the six-month foreign interest rate (U.S.) was 1.893%. The actual forward exchange rate was ¥123.17/$. FinCAD computed the theoretical forward rate to be ¥123.74/$ as follows: aaFXfwd FX spot - domestic / foreign 124.43 rate - domestic 0.008 rate - foreign 0.01893 value (settlement) date 30-May-2002 forward delivery or repurchase date 30-Nov-2002 actual/360 accrual method - domestic rate 2 accrual method - foreign rate 2actual/360 statistic 2fair value of forward (domestic /foreign) fair value of forward (domestic /foreign) 123.7415398 5.24. The domestic (U.S.) six-month interest rate is computed to be 6.9455% as follows: aaFXfwd_repo_d FX spot - domestic / foreign 0.6495 FX forward price - domestic / foreign 0.6624 rate - foreign - annual 0.03 value (settlement) date 30-May-2002 forward delivery or repurchase date 30-Nov-2002 33 actual/360 accrual method - domestic rate 2 accrual method - foreign rate 2actual/360 domestic repo rate 0.069455166 The instructor might consider having students look up the existing S, F, and rSwitz. Then, students could compare the domestic repo rate to existing spot six-month interest rates in the U.S. 5.25. The unannualized interest rate for the four-month holding period is (4/12) X 5% = 1.6667%. Given the information, the only implied forward rate that can be computed is fr(4,12). It can be found using (1+h(0,12)) = (1+h(0,4))(1+fh(4,12)) = 1.06 = 1.016667(1+fh(4,12)). Solving, we compute that fh(4,12) = 4.262295%. Since the length of this forward period is 8 months, we can annualize that forward rate to be 4.262295% X (12/8) = 6.3934%. Here is the solution using aaFRAi: aaFRAi value (settlement) date 30-May-2002 effective date 30-Sep-2002 terminating date 30-May-2003 FRA contract rate 0.035 notional principal amount 1000000 30/360 accrual method 4 discount factor curve t_43_1 linear interpolation method 1 statistic 2implied forward rate t_43_1 discount factor curve grid date discount factor 30-May-2002 1 30-Sep-2002 0.983606554 30-May-2003 0.943396226 implied forward rate 0.063934421 34 Chapter 6 Introduction to Futures 6.1. a. $0.6308/gal. X 42,000 gal. = $26,493.60. b. You lost $0.0173/gal. X 42,000 gal. = $726.60. c. Your profit would have been $(0.6260 – 0.3950)/gal. X 42,000 gal. = $9702. A particularly astute student would have noted that the August contract hit a new high on the day shown, so that the profit would have been $(0.6308 – 0.3950)/gal. X 42,000 gal. = $9903.60. d. The basis is the cash price less the futures price. Basis = 0.635 – 0.6308 = 0.0042. e. Except for an upward ripple in March – April 2000, it is inverted. f. Gasoline has a convenience yield. Those with gasoline need it for their business. They require the physical good during the next several months, and therefore will not engage in reverse cash and carry arbitrage, which requires selling the spot good (either selling it out of inventory or selling it short). 6.2. a. You lose (279-250) X 100 ounces = $2900, on August 27. b. You lose $2900 on April 3. You must borrow that amount at an annual interest rate of 8%, for 146 days. Your total loss is $2900 X (1+(0.08)(146)/365) = $2992.80. c. Your mark to market profit is (420-279)(100 ounces) = $14,100 on April 3. You get to lend $14,100 for 146 days at an annual interest rate of 8%. By August 27, you will have 14,100 + 14,100(0.08)(146/365) = $14,551.20. Then, on August 27, your mark to market loss is (250 – 420) X 100 ounces = $17,000. Thus, your total loss is $17,000 - $14,551.20 = $2448.80. d. Scenario (c ) is the most attractive. At least you have earned some interest on your initial mark to market profit. 6.3. Forward Contracts Futures Contracts a. All elements are negotiated: what is to be a. Standardized contracts. Only the price is delivered, where it is to be delivered, how negotiated much and when it its to be delivered b. Generally illiquid, because they are custom- b. Liquid. They trade on futures exchanges made contracts c. Usually result in delivery c. Rarely delivered. Almost always offset prior to the delivery date. Some cannot be delivered because they are cash settled. d. Default risk is likely to be higher than for d. Extremely low probability that the futures. The exceptions occur when a large, clearinghouse, which becomes a party to every very well capitalized institution is the trade, will default. All members of the counterparty. clearinghouse are obligated to support it, should it ever fall into financial distress. e. Up front money, and collateral, can be e. Initial margin is required, as specified by the negotiated exchange. FCM often makes margin even higher. f. One time cash flow at delivery. Profit or loss f. Marked to market daily. equals the difference between the original forward price that was negotiated, and the spot price on the delivery date 35 6.4. Initial margin is $23,438. Maintenance margin is $18,750, which represents a loss of $4688. For S&P500 futures, one point equals $250. A loss of $4688 would occur if the futures price fell by 4688/250 = 18.75 points. 1335.50 – 18.75 = 1316.75 will trigger a margin call for a long. 6.5. You lost 45 ticks. One tick is worth $10. Thus, you lost $450. 6.6. You went long at 114-27. Each tick is $31.25. Each point (32 ticks) is $1000. To receive a margin call, you must lost $700, which is 22.4 ticks. Rounded up, you must lose 23 ticks, which would occur if the futures price fell to 114-04. 6.7. Basis = cash price - futures price. 1326.65-1367.90 = -41.25. 6.8. You went long at 0.8596, which is $0.008596/¥. You must take delivery of ¥12,500,000. You must pay ¥12,500,000 X $0.0085/¥ = $106,250. This is your invoice amount. Your mark to market loss equals 0.8596 – 0.8500 = 96 ticks X $12.50 = $1200, which was paid out as daily resettlement cash flows between “today” and the delivery day of the futures contract. Effectively, you paid $106,250 + $1200 = $107,450 for the ¥12,500,000, which is a price of $0.008596, which was the original futures price on the day you bought the contract. 6.9. a. Date Settlement Initial Mark to Equity Maint. Final Final Price Cash market Margin Cash Equity Balance cash flow Call Balance 11/5 30.68 $3000 +$230 $3230 $3000 $3230 11/6 31.02 3000 +340 3570 3000 3570 11/7 30.74 3000 -280 3290 3000 3290 11/8 30 3000 -740 2550 3000 2550 11/9 29.64 3000 -360 2190 810 3810 3000 11/12 29.19 3810 -450 2550 3810 2550 11/13 29.84 3810 +650 3200 3810 3200 11/14 29.98 3810 +140 3340 3810 3340 11/15 29.45 3810 -530 2810 3810 2810 11/16 28.86 3810 -590 2220 3810 2220 11/19 28.44 3810 -420 1800 1200 5010 3000 11/20 28.94 5010 +370* 3370 * This cash flow is based on the 28.81 price at which this trade was offset. Upon offsetting the contract, $3370 is transferred to your cash account. Since you moved $5010 out of your cash account, your loss is $5010 - $3370 = $1640. This is verified by just observing the difference between the initial and final futures prices: (30.45 – 28.81) X 1000 barrels = $1640. b. If you had instead used a bank letter of credit to satisfy the initial margin requirement , your variation margin cash flows would be those under the column labeled “mark to market cash flow” in the above table: +$230 on Nov. 5, +$340 on Nov. 6, ….. +$370 on Nov. 20. All mark to market cash flows must be paid/received in cash when using a bank letter of credit or Treasury 36 Bills to satisfy the initial margin requirement. The sum of the daily cash flows in the column equals -$1640. 6.10. Let’s say that you went long at a futures price of F0. You have not bought anything. You have only agreed to buy something in the future, and a mutually agreed upon price. The value of the contract at initiation is zero, since others were willing to enter into positions at that same futures price. But having the right and obligation to buy something in the future at F0 does have value just before it is marked to market. If F1> F0, then it has positive value. If F1 < F0, then it has negative value. Marking to market forces the two parties to settle up each day. The party with a positive value realizes that cash flow at the end of the day. She is paid by the party with a contract having a negative value. After marking to market, the value of the contract is again zero. 6.11. A futures contract is marked to market daily. The amount at risk, should a default occur (by the loser), only equals the profit realized on that day. In contrast, a forward contract can build up a large amount of value for the winner, and an equally large liability (negative value) for the loser. This gives a greater incentive for the loser on a forward contract to default, if possible. Also, the clearinghouse and all of its members stand behind every futures contract. Thus, a futures trader is not concerned that the counterparty to his trade will default. He is concerned only about the creditworthiness of the clearinghouse and its members. 6.12. The corporation fears that interest rates will rise. If interest rates rise, then bond prices fall, and the corporation will receive less dollars for the bonds it intends to sell. Equivalently, if interest rates rise, the firm will have to pay a higher interest rate on the bonds. The corporation should do a short hedge by selling long term interest rate futures, like Treasury Bond futures. Then, if interest rates do rise, it will lose in the spot market, but it will profit in the futures market. 6.13. The meaning of this statement is illustrated by problems 8 and 9 in this chapter. Suppose that the underlying asset is one unit of the good. If the initial futures price is F 0, and the delivery day futures price is F1, then the variation margin cash flows will have equaled F1 – F0. F1 is the amount actually paid at delivery (the invoice amount). The difference between F1 and F0 is resettled daily. 6.14. If you look at Figure 6.5, you see that 3-month Euribor and Euroswiss futures trade. In Figure 6.5, June Euribor futures are trading at 95.60, which reflects an interest rate of 4.40%. June Euroswiss futures are trading at 96.81, reflecting an interest rate of 3.19%. Problem 6.14 asks what you would do if the Euroswiss interest rate exceeded the Euribor rate, and you wanted to speculate that it will revert to “normal”. You would buy Euroswiss futures (expecting the futures price to rise and Euroswiss interest rates to decline), and also sell Euribor futures (and profit if the futures price fell and Euribor interest rates rose). 6.15. June futures prices are as follows: U.S.: 94.79, UK (top of middle column): 94.60, Japan (SGX): 99.66, EU: 95.60, Canada: 95.03. This reflects the following interest rates (where the interest rate equals 100 minus the futures price): U.S.: 5.21%, UK: 5.40%, Japan: 0.34%, EU: 4.40%, Canada: 4.97%. Thus, interest rates are highest in the UK, and lowest in Japan. 6.16. The underlying asset of the contract is C$100,000. Thus, one tick is $0.0001/C$ X C$100,000 = $10.. 37 6.17. We can use Figure 6.5 to get a sample answer to this question. Currency Contract size Nearby price change Mark to Market Cash Flow for a Long Position Japanese yen 12,500,000 +.(00)0063 +$787.50 Deutschemark 125,000 -.0046 -$575 Canadian $ 100,000 -.0040 -$400 British pound 62,500 -.0082 -$512.50 Swiss franc 125,000 -.0065 -$812.50 Australian $ 100,000 -.0005 -$50 Mexican peso 500,000 -.00072 -$360 Euro 125,000 -.0090 -$1125 6.18. If traders are offsetting their positions, then open interest will decline. If traders are establishing new positions, then open interest will increase as the number of new contracts in existence will rise. For example, suppose that open interest is just one contract; Lisa is long and Gabriel is short. Then, on the next day, Lisa buys another contract; volume is one and open interest increases to two contracts. If Gabriel buys (offsets) a contract from Jack, then volume is one and open interest remains at one contract. If Gabriel buys a contract from Lisa, then both offset their positions, and volume is one and open interest drops to zero. 6.19. Livestock has a convenience yield. Those with the cattle may not wish to sell their inventory, since they are in business, and don’t want to alienate their customers. 6.20. Figure 6.4 provides some information for a sample answer. Suppose that in the July 29, 1999 WSJ, we read that the spot price of crude was $20.75/bbl. We see in Figure 6.4 that the February 2000 futures price is $19.86/bbl. Thus, the unannualized impled repo is (19.86- 20.75)/20.75 = -0.04289. If the delivery date is February 1, 2000, then there are 188 days between July 28 and February 1. The annualized implied repo rate is –0.04289 X 365/188 = - 8.327%. aaCDF_repo spot price per unit of underlying commodity 20.75 futures price 19.86 value (settlement) date 28-Jul-1999 expiry date 1-Feb-2000 actual/365 (fixed) accrual method 1 storage cost 0 convenience value 0 annual repo rate incl. convenience value statistic 1 annual repo rate incl. convenience value -0.08327352 38 Chapter 7 Risk Management With Futures Contracts An interesting project is to have students collect historical price data of a good, and historical futures prices for a related futures contract. Then, the students can regress changes in the spot price on changes in the futures price, to obtain a hedge ratio. The spot good does not have to be exactly the same as the good underlying the futures contract, so that some basis risk is created. Finally, have the students monitor the effectiveness of the hedge, using subsequent (out of sample) price data. You might even arrange the situation so that the students must roll the hedge forward. 7.1. He will lose if prices fall; i.e., if interest rates rise. Using futures, he should be a short hedger. If he sells Treasury bill futures and if interest rates do rise, he will suffer losses in the cash market but realize profits on his short futures position. If he uses FRAs to hedge, he would buy the FRAs. When buying an FRA, profits are realized when interest rates rise. 7.2. If the dollar price of the € falls (i.e., the $/€ exchange rate declines), then the US corporation will receive fewer dollars for its euros. It should therefore sell euro futures. I.e., it should be a short hedger. 7.3. A short hedger owns the spot good and is short futures. Ideally, while the hedge is in place, the spot price will rise and the futures price will decline; this is highly unlikely to happen since spot and futures price changes should be correlated. Nonetheless, the short hedger therefore hopes that the basis will strengthen (increase). 7.4. Alternative methods of hedging include transacting in the spot market (just sell the asset if you fear a decline in its value, or buy it today in anticipation of a price increase), using options, forward contracts, or swaps. Transaction costs are important when deciding which to use. These include commissions, paying the bid-ask spread, and any price pressure from the trades. The speed at which the trades can take place can be important. Finally, the hedger must consider relative mispricing of the hedging instrument. Always buy undervalued assets and contracts, and sell overvalued assets and contracts. 7.5. This is illustrated in the text, using monthly data, in Table 7.2. 7.6. You will enter into a long hedge. You fear that tin prices will rise. This is an anticipatory cross hedge. a. The copper regression has a higher R2. Thus, copper price changes are more highly correlated with tin prices, and copper futures would likely produce the more effective hedge. Other factors to be considered are liquidity, and relative mispricing of the futures. Note that tin and silver may have a convenience yield, causing an inverted term structure of futures prices; in this case buying a contract with a futures price less than the current spot price may be desirable. b. There are 25,000 pounds of copper underlying a futures contract. Therefore, buy (1.92)(100,000)/25,000 = 7.68 contracts. 39 7.7. a. You will buy crude oil futures. Without the tail, buy (1.109)(20,000)/1000 = 22.18 contracts. With the tail, buy the present value of 22.18 contracts, or 22.18/(1+(0.16)(5)/12) = 20.79 contracts. b. With the tail, you are always long the present value of 22.18: t # of contracts long Profit 0 20.79 1 21.06 (21.06)(1000)(20.04-19.93) = $2316.60 3 21.60 (21.60)(1000)(20.95-20.04) = $19,656.00 4 21.89 (21.89)(1000)(21.09-20.95) = $3064.60 5 22.18 (22.18)(1000)(22.35-21.09) = $27,946.80 You pay $23.82 X 20,000 = $476,400 for the oil. The sum of your futures profits is $52,984. The net effective purchase price is $476,400 - $52,984 = $423,416 for 20,000 barrels of oil, or $21.17/bbl. c. The original basis is $21.40 – 19.93 = $1.47. The basis at delivery is $23.82 - $22.35 = $1.47. In contrast, if ST = 24.50, then the final net effective purchase price is (24.50)(20,000) – 52,984 = $437,016 for 20,000 barrels of oil, or $21.85/bbl. d. In part (b), the basis was constant. In part (c ), the basis widened from 1.47 to 2.15, which worked to the disadvantage of this long hedger. 7.8. The current price of one note is $1000/(1.1)2 = $826.45. The firm fears that interest rates will rise (prices of debt instruments will fall). Therefore, sell Eurodollar futures contracts to hedge. If Eurodollar interest rates rise by 10 bp, the futures profit will be $250/contract. If the YTM of the notes rises 7 bp, the new price will decline to $1000/(1.1007) 2 = $825.40. This is $1.05/note less than originally anticipated. On 10,000 notes, the firm will lose $10,500. Dollar equivalency states that S = nF. This means that $10,500 = 250n. Then, n = 42 Eurodollar futures contracts should be sold. 7.9. a. The theoretical futures price is F = S (1+h) = 300(1.12) = $336/oz. b. Without a tail, you will sell 100 futures contracts: number of units of the spot number of futures contracts to trade in order tohave a risk h * X position to be hedged number of units underlying minimizing hedge one futures contract = 1.0 X (10,000/100) = 100 If you tailed your hedge, you would sell 100/1.12 = 89.2857 contracts. c. You have sold 100 futures. The initial F = 336, and one month later, F = 392. Thus, you lost $56/oz. X 100 oz. / contract X 100 contracts = $560,000. You must borrow this amount for eleven months, so at delivery, you owe $560,000 X (1+(0.12)(11)/12) = $621,600. You then sell your 10,000 oz. of gold for $392/oz. X 10,000 = $3,920,000. Your effective selling price is 40 therefore $3,920,000 - $621,600 = $3,298,400. This selling price of $329.84 is less than the original futures price of $336/oz because the hedge was not tailed. You lost $61,600 in interest expense (= $6.16 per ounce). d. If a tail was used, then only 89.2857 contracts would have been sold at time zero. One month later, you would be short 100/(1+(0.12)(11)/12) = 90.09 contracts. You will lose (392-336 = ) $56/oz X 100 oz. / contract X 90.09 contracts = $504,504 one month after the hedge origination date. Borrow this amount for eleven months. At delivery, you will owe $504,504 X 1.11 = $600,000. Your effective selling price is now $3,920,000 - $600,000 = $3,320,000, or $332/oz. 7.10. You should sell a gold futures contract. It’s a normal market, because the futures price is above the spot price. The basis (S – F) equals –10. You profit by $1300 if the futures price declines to 297. But after marking to market, the value of the gold futures contract is zero. Cash, or spot market Futures market Today: Today: You own 100 oz. of gold. The spot You sell one gold futures contract at price is $300. You fear that the price of the futures price of 310 gold will decline Two weeks hence: Two weeks hence: The spot price of gold declines to 285 Buy (offset) one gold futures contract at the futures price of 297 Loss: 100(300-285) = -$15,000 Profit: (31000 – 29700) = +$13,000 Because the basis weakened from –10 to –12, the short hedger did not have a perfect hedge. The hedger, however, should have initially considered it unlikely for the basis to weaken for gold, unless interest rates rose sharply during the two week period. But the example illustrates that unless the hedge lifting date coincides with the futures’ delivery date, the hedger must be prepared to deal with basis risk. 7.11. If a mutual fund receives an unexpected cash inflow, it has the problem of deciding what to do with the money. If it wants exposure to stocks, it might go long stock index futures as a substitute for buying individual equities. It may be unsure exactly which equities it wants to invest in. It may regard the desired exposure to equities as being short term. It may believe that the cash investment is short term, and it wants the liquidity to get out of “the market” should many investors want their money back in the near future. Stock index futures allow the fund to liquidate when investors make withdrawals, rather than deciding exactly which stocks should be sold. Selling individual equities will be more costly in terms of commissions and bid-ask spreads, and they may upset a carefully constructed portfolio. Sometimes, stock index futures will be priced below their theoretical values. If so, then going long futures will be advantageous relative to buying the underlying stocks. Stock index futures require margin that is only a fraction of the size of funds required to actually buy the stocks. Thus, cash reserves can be maintained while simulating full investment. Stock index futures can typically be traded more easily and quickly than individual stocks, and at lower costs. Finally, they allow funds to make asset reallocation decisions by switching exposures among stocks, bonds and cash (Tbills) quickly and at low cost. 41 7.12. Short hedgers typically own the underlying asset, fear that its price will decline, and therefore sell futures. Thus they are +S and –F. S-F equals the basis, so short hedgers are effectively long the basis. Long hedgers anticipate buying the asset in the future, fear a price rise, and therefore buy futures. Thus, they are -S and +F. –(S-F) is effectively short the basis. 7.13. The basic cost of carry pricing model is F = S(1+h), or F-S = Sh, which is the negative of the basis. Sh is the cost of carrying the spot good; i.e., it’s the interest expense of tying your money up with inventory. Thus basis equals the net cost of carry. 42 7.14. DS DF May 10 5.2 June -4 -13.0 July -18 -14.5 SUMMARY OUTPUT August 6 2.9 Sept. -17 -25.4 Regression Statistics Oct. 20 26.1 Multiple R 0.949114 Nov. 15 7.3 R Square 0.900817 Dec. -6 -12.4 Adjusted R 0.894982 Square Jan. 8 11.5 Standard Error 3.871599 Feb. -2 -3.1 Observations 19 March -24 -34.5 April -8 -3.5 ANOVA May -14 -21.5 df SS MS F Significance F June 5 1.2 Regression 1 2314.34 2314.34 154.3997 5.89E-10 July 1 0.1 Residual 17 254.8178 14.98928 August -2 -3.7 Total 18 2569.158 Sept. -4 -9.6 Oct. 13 17.9 Coefficie Standard t Stat P-value Lower Upper nts Error 95% 95% Nov. 6 5.5 Intercept 1.752543 0.911461 1.922785 0.071425 -0.17047 3.67556 mean -0.789473684 -3.3421053 X Variable 1 0.760603 0.061212 12.42577 5.89E-10 0.631458 0.889749 variance 142.7309942 222.24813 Std. devn 11.94700775 14.907989 The mean of column B is computed by typing =average(b2:b20) The variance of column B is computed by typing =var(b2:b20); this is the sample variance The standard deviation of column B is computed by typing =stdev(b2:b20); this is the sample standard deviation 43 Chapter 8 Stock Index Futures 8.1. a. With no dividends, F = S(1+h) = 600(1+0.05(61/365)) = 605.014. b. First, we must compute the number of shares of each stock in one unit ($600 worth) of the D&M2 Index, and/or the number of shares in 250 units (250 X 600 = $150,000) of the index. Market Value Fraction $ in 250 units of the index Dubo $400 million 0.7273 $109,090.91 Miller $150 million 0.2727 $ 40,909.09 Thus, given their respective stock prices: Shares in 250 units of Index Shares in one unit of the index Dubo 109,090.91/40 = 2727.27 2727.27/250 = 10.9091 Miller 40,909.09/30 = 1363.64 1363.64/250 = 5.4546 Now, we can compute the theoretical futures price: F = S + carry costs – carry returns F = 600(1+0.05(61/365)) – 10.9091(0.50)(1+(0.05)(44/365)) – 5.4546(0.20)(1+(0.05)(10/365) = 605.014 – 5.487 – 1.0924 = 598.43 c. Lower bound: Established by reverse cash and carry trades: sell stock, lend proceeds, borrow to pay the dividends. Buy cheap futures Upper bound: Established by cash and carry trades: Borrow to buy stock. Lend the dividends. Sell overpriced futures if the upper bound is violated. 600(1+0.06(61/365)) - 10.9091(0.50)(1+(0.07)(44/365)) – 5.4546(0.20)(1+(0.07)(10/365) < F < 600(1+0.07(61/365)) - 10.9091(0.50)(1+(0.06)(44/365)) – 5.4546(0.20)(1+(0.06)(10/365) = 606.0164 – 5.5006 – 1.0930 < F < 607.0192 – 5.4940 – 1.0927 599.4228 < F < 600.4325 d. Transaction costs are six cents/share times the total number of shares of stock needed to replicate 250 units of the index. 2727.27 shares of Dubo 1363.64 shares of Miller 4090.91 total shares times $0.06/share = $245.45 + 15.00 futures commission $260.45 Now, divide $260.45, which is the total commission on 250 units of the index, by 250, in order to get a per unit commission of $1.04. Lower the lower bound, and raise the upper bound by this amount: 599.42 – 1.04 < F < 600.43 + 1.04 598.38 < F < 601.47 e. The actual F of 625 violates the upper bound of 601.47. Perform cash and carry arbitrage: Borrow, buy the stock, and sell futures. November 15 44 Buy 2727.27 shares of Dubo -109,090.80 Buy 1363.64 shares of Miller - 40,909.20 Borrow $150,000 @ 7% +150,000 Sell one futures contract @ F = 625 0 Total Cash Flow 0 December 1 Receive dividends (2727.27)(0.50) +1363.635 Lend @ 6% -1363.635 Total Cash Flow 0 January 5 Receive dividends (1363.64)(0.20) +272.728 Lend @ 6% -272.728 Total Cash Flow 0 January 15 Sell shares +(250)(ST) Offset futures position (250)(625) – (250)(FT) = 156,250 – (250)(ST) Repay original loan - 150,000 X (1 + (0.07)(61)/365)) = - 151,754.79 Get repaid for lending dividends + 1363.635 X (1 + (0.06)(45)/365) + 272.728 X (1 + (0.06)(10)/365) = + 1646.90 Pay transaction costs - 260.45 Total Cash Flow +$5881.66 Note that the upper bound is 601.47 and the futures price is 625. Then the arbitrage profit is (625 – 601.47) X 250 = $5882.50. Except for rounding errors, this should equal the total cash flow arbitrage profit computed on January 15. 8.2. We apologize for the ambiguous wording for this problem. The carry costs (hS) on a $100 million portfolio equals $100 million X (0.05)(84)/365 = $1,150,684.93. This means that the FV of dividends (dividends plus interest on the dividends) equals $1,420,000 - $1,150,684.93 = $269,315.07. This means that the unannualized dividend yield on the S&P (assuming that your portfolio is identical to the S&P500) is 269,315.07/100,000,000 = 0.00269 = 0.269%. Thus, the theoretical futures price is 1333.58(1 + (0.05)(84)/365) – 0.00269(1333.58) = 1345.33. The futures price is too high, and you should not quasi-arbitrage. Here is a better problem (one that creates the opportunity to quasi-arbitrage by selling your stock portfolio, lending the proceeds, and buying the cheap futures. Let the problem read: “At the end of 84 days, your estimate of the future value of dividends is $269,315” (this eliminates the ambiguous wording. Also, let the actual futures price be 1340 (which will permit the quasi- arbitrage). Quasi-arbitrage requires selling the stock portfolio, lending the proceeds, and buying the cheap futures contracts. You should buy $100 million/(1333.58)(250) = 299.945 futures contracts. After 84 days, you will have $100 million X (1+(0.05)(84)/365) + 299.945 X 250 X (FT – 1340) = $101,150,685 + 74,986.25FT - $100,481,575. This will exceed what you will have if you just held on to the stock itself. If you held on to the stock, you would have $100 million X (1 45 + (ST – 1333.58)/1333.58) + FV (Dividends). Note also that ST = FT due to convergence. (ST – 1333.58)/1333.58 is just the rate of return on the spot index over the next 84 days. Consider four possible scenarios for the two alternative investments: ST = FT Long Futures & Long Treasuries Long Stock 1300 $98,151,235 $97,751,281 1333.58 $100,669,273 $100,269,315 1340 $101,150,685 $100,750,726 1400 $105,649,860 $105,249,894 Selling your portfolio, lending the proceeds, and buying the cheap futures dominates holding on to the stock. 8.3. The total market value of the Belch Index is (49 X $5 X 1 million) + (1 X $100 X 1 million) = $345 million. $345 million/divisor = 690 implies that the divisor is 500,000. Thus, there are two shares of each of the low-priced stocks, and two shares of the high priced stock, in one unit of the Belch: (49 X 2 shares X $5) + (1 X 2 shares X $100) = 690. The weight of each low priced stock is 10/690 = 0.01449, and the weight of the high priced stock is 200/690 = 0.28986. F = S + CC – CR S = 690 CC = hS = (0.10)(63/365)(690) = 11.9096 The carry return, CR, equals the future value of the dividends = 49 stocks X $0.25 X 2 shares X (1+(0.10)(33/365))+(1 stock X $1 X 2 shares X (1+(0.10)(13/365)) = 26.7286 The theoretical futures price is 690 + 11.9096 – 26.7286 = 675.18. 8.4. a. F = S + CC – CR S = 110 CC = 110 X 0.07 X 45/365 = 0.9493 The index is value weighted. Thus, it consists of 60% Gomuco and 40% of Bookuco. 0.6 X 110 = 66, so there are 2.2 shares of Gomuco in one unit of the index. 0.4 X 110 = 44, so there are 1.1 shares of Bookuco in the index. The future value of dividends = $0.50/share X 2.2 shares X (1 + (0.07)(30/365)) = 1.1063. The theoretical futures price is 110 + 0.9493 – 1.1063 = 109.8430. b. The lower bound is established by reverse cash and carry arbitrage, where you buy cheap futures, sell the stock, lend the proceeds, and borrow to pay the dividends. The upper bound is established by cash and carry arbitrage, where you borrow to buy the stock, lend the dividends, and sell overpriced futures. 110 + (110 X 0.06 X 45/365) – (0.50 X 2.2 X (1 + (0.08)(30/365))) < F < 46 110 + (110 X 0.08 X 45/365) – (0.50 X 2.2 X (1 + (0.06)(30/365))) 109.706 < F < 109.980 8.5. Because each stock has the same market value, and there are 500 stocks in the index, then there is 1000/500 = $2 of each stock in one unit of the index. Each stock sells for $10/share. Therefore 0.20 share of each stock is in the index. 0.20 shares X $10/share X 500 stocks = 1000 = index value. F = 1000 (1 + (0.07)(32/365)) – {250 stocks X 0.2 X $0.50 X (1 + (0.07)(17/365))} - 250 stocks X 0.2 X $0.60 = 1006.1370 – 25.0815 - 29.9197 = 951.1358 1 (0.07 )(14 / 365 ) Note that the dividends that are paid on June 30 are actually discounted back to June 16. 8.6. First, lets check that the futures price is indeed theoretically correct. F = S (1+h) = 1350 (1 + (0.05)(0.25)) = 1367. Portfolio A consists of $256 million in stocks with a beta of 1.0375, partially financed with $96 million in borrowed funds. A stock market decline from 1350 to 1215 is a 10% decline. This means that the stocks (with a beta of 1.0375) will likely decline by 10.375%, from $256 million to $229.44 million. But you must also repay your loan with interest, and this equals $96 million (1 + (0.05)(0.25)) = $97.2 million. Thus, the residual value of portfolio A is $229.44 – 97.2 = $132.24 million. Portfolio B consists of $160 million in stocks, and a long position in 293.59 futures. The stocks will decline by 10.375% to $143.4 million. The futures position will lose 293.59 X 250 X (1367 – 1215) = $11,156,420. Thus the value of portfolio B will be $143.4 – 11.15642 = $132.24 million. 8.7. a. port = (150/160)0.94 + (10/160)0 = 0.88125 b. You should sell stock index futures. Equation (8.12) computes the number of stock index futures to trade to change the beta of a portfolio from p to d: Total Portfolio Value Number of futures contracts to trade to change a portfolio d p Currently Held Number of dollars underlying beta from p to d the futures contract $160,000,000 (250) X (1300) 138.46 contracts = (0.6 0.88125) X Alternatively, see section 8.3.2.1. c. Sell $160,000,000 (250) X (1300) 433.85 contracts (0.0 0.88125) X 47 d. The stock market fell from 1297 to 1100, which is a decline of 15.19%. The beta of the stock (only) portfolio is 0.94, so they should fall in value by (0.94)(15.19%) = 14.28%, from $150 million to $128,583,655. If dividends are zero, and the initial futures price was correct, then the unannualized interest rate during the life of the futures contract is h = (F-S)/S = (1300-1297)/1297 = 0.002313 = 0.2313%. Then, the Treasury Bills will increase in value to $10 million X 1.002313 = $10,023,130. You profited on the futures contracts. The profit equals (250) X (433.85 contracts) X (1300 – 1100) = $21,692,500 The total value of the portfolio is $128,583,655 + $10,023,130 + $21,692,500 = $160,299,285. Portfolio B increases in value to $160,000,000 X 1.002313 = $160,370,080. 8.8. It is appropriate to do a long hedge, and buy futures. Other information you should want includes: 1) what index is used in the regression that produced these betas. 2) What R2 exists for each market model regression. The appropriate index is the one that produces the highest R2, all else equal. 3) What is the liquidity for the different stock index futures contracts that you can use to hedge. The total value of the portfolio to be purchased is $1 million. The portfolio beta is (0.1)(0.4) + (0.2)(1.0) + (0.3)(1.2) + (0.4)(1.8) = 1.32. Buy $1,000,000 (250)(1253.1) 4.21contracts 1.32 X If the interest rate was 50%/year, you would want to tail the hedge, and buy the present value of 4.21 contracts: 4.21 3.52 contracts (0.5)(144 ) 1 365 8.9. a. Synthetic long stock is created by going long stock index futures and buying Treasury Bills. Buying Treasury Bills is a lending transaction, thus the relevant interest rate is 4.5%. You would create a synthetic long stock position if stock futures prices are too low. A synthetic T-bill position is created by buying stock and selling stock index futures. Cash and carry arbitrage states that F = S + Sh Long stock index futures = long stock + borrowing Rearrange this to conclude that Lending = long stock + short stock index futures b. Assume no transaction costs. The theoretical, no-arbitrage pricing bounds are 1300 X (1 + (0.045/2)) < F < 1300 X (1 + (0.05/2)) = 1329.25 < F < 1332.5. The actual stock index futures price of 1315 is too low. Therefore, you should sell your stock, invest the proceeds in T-bills, and buy cheap futures. Put another way, sell your stock and buy synthetic stock (long stock index futures and lending). This assumes that your portfolio is similar to the composition of the SP 500. 48 c. Had you maintained your original position, your stocks would have risen in value by 13.33%, to $11,333,333. The T-bills would have risen in value by 2.5%, to $10,250,000. Thus, the total value of your portfolio would have risen to $21,583,333. Suppose instead that you had sold your stocks for $10 million, and invested the proceeds in T-bills. You would now have $20 million invested in 6-month T-bills. You would also buy ($10,000,000/(250 X 1300)) = 30.769 futures at a futures price of 1315. Six months later, your T bills would be worth $20 million X 1.025 = $20,500,000. You would also have realized a profit on the futures contracts equal to 30.769 X 250 X (1500 – 1315) = $1,423,066. Thus, the total value of your portfolio is $20,500,000 + $1,423,066 = $21,923,066. This is better than your original portfolio. 8.10. a. Your $250,000 can be converted to (¥108/$) X $250,000 = ¥27 million. This means that ¥13.5 million will be invested in each stock. The number of shares of Hitachi that must be purchased is ¥13,500,000/(¥1389/share) = 9719.22 shares. The number of shares of Fujitsu that must be purchased is ¥13,500,000/(¥3340/share) = 4041.916 shares. b. The rate of return on Hitachi was (1500-1389)/1389 = 7.99%. The rate of return on Fujitsu was (4000-3340)/3340 = 19.76%. Because equal amounts were invested in each stock, the rate of return on the portfolio is equally weighted. The rate of return in yen is (0.5)(7.99%) + (0.5)(19.76%) = 13.875%. c. (1 + rhc) = (1 + rs ) (1 + rfx ). It is important to realize that rfx is the percentage change in the value of the foreign currency. Here, the foreign currency is the yen. The problem states that the dollar rose in value from ¥108/$ to ¥118/$. But this means that the yen fell in value from $0.009259/¥ to $0.008475/¥. This is a –8.472% decline in the value of the yen. Thus, (1 + rhc) = (1 + rs ) (1 + rfx) = 1.13875 X 0.91528 = 1.042275, and rhc = 4.2275%, denominated in dollars. Your dollar profit is $250,000 X 0.042275 = $10,568.75 d. Each point on a Nikkei Dow futures contract is worth $5. Here, the Nikkei Dow rose by 1000 points, which is $5000 per contract. Since you bought two contracts, your profit would have been $10,000, which is less than the $10,568.75 computed in part c. There is no foreign exchange price risk with futures contracts on foreign stock indexes. 8.11. a. F3-month = S(1+h(0,1)) – FV (Divs during the next month) F6-month = S(1+h(0,4)) – FV (Divs during the next 4 months) If the yield curve steepens, h(0,4) will increase more than h(0,1). So, the 6-month futures price will likely increase more than the 3-month futures price, and the spread between the two futures prices will likely increase. b. We don’t know what FV (Divs during the next month) and FV (Divs during the next 4 months) are. And of course there are just random fluctuations in futures prices that are unrelated to the theoretical futures pricing model. 49 Chapter 9 Treasury Bond and Treasury Note Futures 9.1. The asked price in Figure 9.1 is 124:30, or $1249.375 for a $1000 face value bond. The previous coupon was paid 16 days ago, on November 15, 2000. Accrued interest is (16/184) X ($120/2) = $5.2174 The total price paid is then $1249.375 + $5.2174 = $1254.5924. The next coupon will be paid 168 days hence, on May 15, 2001. Thus d = 168/184 = 0.913 of a half year. There are N = 8 subsequent coupon payments of $60 each after May 15, 2001. Thus, the semi-annual YTM is found by trial and error in the following equation: 1254.5924 = [60 + 60(PVIFA, h%, 8) + 1000(PVIF, h%, 8)][PVIF, h%, 0.913] By trial and error, h = 2.799%. Doubling this, the annual YTM is 5.598%. Using excel, input the following into a cell: =YIELD("12/01/2000","5/15/05",0.12,124.9375,100,2,1), and the answer is 5.5972%. The YTM shown in Figure 9.1 is 5.59%. The FinCAD solution is: aaBond_y settlement date 1-Dec-2000 maturity date 15-May-2005 dated date - date from which accrued interest date of first coupon after dated date date of last coupon prior to maturity date coupon 0.12 principal at maturity 100 semi-annual date frequency 2 accrual method 7actual/actual (bond basis) (eom) price 124.9375 yield is simple if in last period, compounded otherwise rate basis in last coupon period 1 number of days bond is ex-dividend 0 yield to maturity statistic 1 yield to maturity 0.055972179 9.2. A bond’s flat price does not include accrued interest. Its gross price includes accrued interest. 50 9.3. The trader hopes that short term interest rates will fall. She has lent money forward, from September 1999 to December 1999, by doing this calendar spread. She has agreed to buy T- bonds in September, and sell them in December; effectively, she has lent money to the government for a three month forward period, at a locked-in rate. Just like any money lender who has locked in a lending rate, she hopes that interest rates subsequently fall. Given the prices, the trader lost 1-09 on the September contract, and profited by 1-11 on the December contract. Her net profit is two ticks. At $31.25/tick, her profit is $62.50. 9.4. The 6.25% of May 2030 is the worst possible bond for the short to deliver. It has the most negative value (-3.1345) using equation 9.4b. It also has the lowest, most negative, implied repo rate (-29.34%). The best bond to deliver is the one that costs the least, and gets the short the highest invoice price. The worst bond is the one that costs the most, and gets the short the lowest invoice price. 9.5. The conversion factor for the 9 ¼% of February 2016 T-bond is 1.3185 (see Figure 9.4). Then, the invoice amount is $102,062.50 X 1.3185 plus accrued interest. Accrued interest is ($9,250/2)(114/184) = $2,865.49. Thus, the total invoice amount is $137,434.90. The short may decide to deliver these bonds if he was planning on selling them anyway, or if it could be part of a portfolio adjustment, or for tax reasons. 9.6. Figure 9.4 shows that the four conversion factors are 1.1364, 1.1355, 1.1349, and 1.1340 for December 2000, March 2001, June 2001, and September 2001, respectively. Delivery Time to Rounded Time Formula to Compute Date Maturity To Maturity Conversion Factor Dec 00 22 yrs, 2 ½ mos y=22, m=0 9-3a: .035625(PVIFA,3%,44)+1(PVIF,3%,44) = 1.1364 Mar 01 21 yrs, 11½ mos y=21, m=9 9-3c: .035625+.035625(PVIFA,3%,43)+1(PVIF,3%,43) -.01781 =1.1355 1.014889157 Jun 01 21 yrs, 8 ½ mos y=21, m=6 9-3b: .035625(PVIFA,3%,43)+1(PVIF,3%,43) = 1.1349 Sep 01 21 yrs, 5 ½ mos y=21, m=3 9-3d: .035625+.035625(PVIFA,3%,42)+1(PVIF,3%,42)-.01781 = 1.1340 1.014889157 9.7. a. First the conversion factor for the CTD T-bond must be computed, or at least read from Figure 9.4. The conversion factor is 1.3272. AI(0) on April 20, 2001 is the AI from 11/15/00 until 4/20/01: (9.125/2)(156/181) = 3.9323. S = 135.46875 R = 5%/year AI(T) = (9.125/2) X (1 + (0.05)(36/365)) + (9.125/2)(36/184) = 5.4563 Then, using equation (9.7), we compute that 1 (0.05)(61) F (135.46875 3.9323)1 5.4563 101.8005 1.3272 365 It was not asked, but FinCAD computes the theoretical futures price to be 101.7844: 51 aaBondFwd settlement date 20-Apr-2001 forward delivery or repurchase date 20-Jun-2001 repo (financing) rate 0.05 simple interest rate basis frequency of the repo rate 7 accrual method - repo rate 3actual/365 (actual) bond price (per 100) 135.46875 maturity date 15-May-2018 dated date - date from which accrued interest date of first coupon after dated date date of last coupon prior to maturity date coupon 0.09125 principal 100 semi-annual date frequency 2 accrual method 17actual/actual (bond basis) holiday list t_26_4 end of month - no adjustment business day convention (see Glossary) 1 number of days bond is ex-dividend 0 clean forward price statistic 4 t_26_4 holiday list holiday date 25-Dec-2005 clean forward price 135.0882657 101.7844075= clean forward price/1.3272 b. The party who is short T-bond futures has the delivery options. See Section 9.2.7 for a description of the options. The actual futures will be below the theoretical futures price that we just computed, because of these delivery options. Indeed from question 9.7.c, we see that the actual futures price is 100.5, which is less than the theoretical futures price we just computed c. 52 [(1.3272)(100.5) 5.4563 135.46875 3.9323 365 [135.46875 3.9323] 61 100 2.41% It is unlikely that the 9 1/8% of May 2018 bond is the cheapest to deliver. 9.8. You believe that T-bond prices will fall; therefore you believe that long term riskless interest rates will rise. You sold at 101-28. The contract settled at 102-03. Thus you lost seven ticks. At $31.25/tick, you have a mark to market cash outflow of $218.75. You will receive a margin call when you have lost $1500. At $31.25/tick, this is 48 ticks, or 1-16 points. Thus, you will receive a margin call at 103-12. 9.9. First, you must compute the conversion factor for the CTD bond. Use the procedure described in Section 9.1.6. Use equation 9.3b: CF = 0.040625 (PVIFA, 3%, 57) + 1(PVIF, 3%, 57) = 1.2885 S =95.3125 AI(0) = AI from 8/15 to 11/9 = (86/184)(8.125/2) = 1.8988 AI(T) = AI on 12/20 = (127/184)(8.125/2) = 2.8040 1 (0.0775)(4 1) F (95.3125 1.8988)1 2.8040 73.9259 1.2885 365 There are no interim coupon payments between today and the target delivery date. Therefore the implied repo rate (IRR) is: (CF )(F ) AIT ( S AI 0) 365 IRR T S AI 0 (1.2885)(92.53125) 2.8040 (95.3125 1.8988) 365 41 227.29% 95.3125 1.8988 9.10. If the CTD T-bond is defined to be the one that maximizes the raw basis (equation 9.4c: [(CF)(F) – S]), then bond [(CF)(F) – S] 6% of Feb 2026 (0.7782)(110.40625) – 88.59375 = -2.676 8% of Nov 2021 (1)(110.40625) – 112 = -1.59375 9% of Nov 2018 (1.1019)(110.40625) – 122.75 = -1.0934 Thus, the 9% of November 2018 is the CTD by this method. To use the maximum IRR method, a target delivery date must be assumed. Assume it is March 20, 1997. Then, T = 66. For the 6% of February 2026 bond, AI(0) = (59/181)(6/2) = 0.9779. AI(T) = (125/181)(6/2) = 2.0718. 53 For the 8% of November 2021 bond, AI(0) = (59/181)(8/2) = 1.3039. AI(T) = (125/181)(8/2) = 2.7624. For the 9% of November 2018 bond, AI(0) = (59/181)(9/2) = 1.4669. AI(T) = (125/181)(9/2) = 3.1077. (CF )(F ) AIT ( S AI 0) 365 IRR T S AI 0 For the 6% of Feb 2026 bond : (0.7782)(110.40425) 2.0718 (88.59375 0.9779) 365 88.59375 0.9779 66 9.78% For the 8% of Nov. 2021bond : (1)(110.40425) 2.7624 (112 1.3039) 365 112 1.3039 66 0.67% For the 9% of Nov. 2018 bond : (1.1019)(110.40425) 3.1077 (122.75 1.4669) 365 66 2.43% 122.75 1.4669 Thus, the maximum IRR is the 9% of November 2018 bond, and this method also concludes it is the CTD. 9.11. T-bond futures prices are determined by the cost of carry formula. If the term structure of interest rates is rising, as it normally is, then the carry costs (short term interest rates) will be less than the carry return. Futures prices for contracts close to delivery will then be less than those for contracts with more distant delivery dates. This situation is similar to the case when dividends (the carry return) on the stocks in a stock index exceed carry costs, in which case stock index futures prices will be below the spot index. If, however, you argue that the term structure is upward sloping due to investor expectations (i.e., the unbiased expectations theory determines the term structure), then you can infer that futures prices are also determined by investor expectations. In this case, the rising yield curve means that investors expect interest rates to rise in the future, and it is investor expectations that cause an inverted term structure of T-bond futures prices. Arbitrage is the most powerful force in financial markets. No matter what investor expectations are, it must be true that the cost of carry pricing model will hold. 9.12. In the corrected version of this problem, we would have cited a conversion factor of 1.1630 in part b, and an actual futures price of 72 in part c. a. The firm fears that interest rates will rise; i.e., that prices will fall. It should sell T- bond futures to hedge. Then, if interest rates do rise, it will lose in the spot market because it will have to pay a higher interest rate. But, it will also profit in the futures market. b. The time to maturity on June 1, 1989 is 27 years, 5 ½ months. Rounded down, the modified time to maturity is 27 years and 3 months. Equation (9.3d) is used to compute the conversion factor: 0.0375 0.0375 (PVIFA,3%, 54) 1(PVIF,3%, 54) CF 0.01875 1.1630 (1.03) 0.5 54 The conversion factor in the text (0.9447) is incorrect. It was actually computed when the benchmark yield for the T-bond futures contract was 8%, rather than the 6% that it is today. The yield to maturity of the bond is 9%. It can be computed using Excel or FinCAD. Using Excel, enter =YIELD("4/10/1989","11/15/2016",0.075,84.78125,100,2,1). c. The following data are needed to compute the theoretical futures price: CF = 1.1630 S = 84.78125 AI(0) = AI from 11/15 to 4/10: (146/181)(7.5/2) = 3.0249 R = 0.08 T = 71 days until delivery AI(T) = the coupon to be paid on 5/15 + the interest on the coupon from 5/15 until 6/20 + The AI from 5/15 until 6/20 = 7.5/2 + (7.5/2)(0.08)(36/365) + (7.5/2)(36/184) = 4.5133 1 (0.08)(71) F (84.78125 3.0249)1 4.5133 72.7939 1.1630 365 If the conversion factor given in the text (0.9447) is used, the theoretical futures price is 89.6150. Almost always, the actual futures price will be less than the theoretical futures price because the trader who is short futures has valuable delivery options. 88-06 = 88.1875 is less than 89.6150. In the corrected version of this problem, we would have cited a conversion factor of 1.1630 in part b, and an actual futures price of 72 in part c. 72 is less than 72.7939 because of the shorts’ delivery options. d. Assume that the firm is planning to issue 50,000 bonds, each with a face value of $1000. The market value is also $1000, because they are being issued at par. If the annual interest rate was to rise from 10.25% to 11.25%, then the firm would receive only $921.07 for each bond: 51.25(PVIFA, 5.625%, 40) + 1000(PVIF, 5.625%, 40) = $921.07 For 50,000 bonds, the total loss is $78.93 X 50,000 = $3,946,500. (Alternatively, find the present value of the added interest that the firm would have to pay if interest rates rose to 11.25%, and the bonds were issued at par. You should come close to the figure of $3,946,500.) Using the dollar equivalency approach, first find the profit realized by being short one futures contract if interest rates rise by 100 basis points. Assume that the yield curve shifts in a parallel fashion, and that the same bond remains cheapest to deliver. The original YTM on the bond was computed in part (b) to be 9%/year. Compute the new gross price (market price plus accrued interest) of this bond if it rose to 10%: [37.50 + 37.50(PVIFA, 5%, 56) +1000(PVIF, 5%, 56)] [PVIF, 5%, 0.1989] = 796.01 To compute the theoretical futures price, the only other variable that changes (ever so slightly) is the carry return. The interest earned on the coupon rises from 8% to 9%: 7.5/2 + (7.5/2)(0.09)(36/365) + (7.5/2)(36/184) = 4.5170. 55 Thus: 1 (0.09)(71) F (79.601)1 4.5170 65.7589 1.1630 365 (Using the conversion factor in the text, 0.9447, the theoretical futures price is 80.9539) So, if interest rates rise by 100 basis points, the theoretical futures price declines from 72.7939 to 65.7589, which will be a profit of (72.7939 – 65.7589) X $1000/point = $7035 per contract. This assumes that the value of the delivery options remains constant. The number of futures contracts to sell is found by dollar equivalency: h = VS/VF = $3,946,500/7035 = 561 contracts e. The results of the regression model predict that if the YTM on the CTD bond rises by 100 basis points (leading to a profit of $7035 per contract sold), the YTM on the spot bond being hedged would rise by only 91 basis points. If the annual interest rate was to rise by 91 basis points, from 10.25% to 11.16%, then the firm would receive only 51.25(PVIFA, 5.58%, 40) + 1000(PVIF, 5.58%, 40) = $927.75 for each $1000 face value bond. On 50,000 bonds, the firm would lose $72.25 X 50,000 = $3,612,500. The firm would then want to sell 3,612,500/7035 = 513.5 T-bond futures to hedge its planned issuance of bonds. 9.13. Here, the pension fund manager fears that interest rates will fall (prices will rise) between April 10 and April 30. Therefore a long position in T-bond futures is needed to hedge. Assume that the same bond that is the CTD in Problem 9.12 is the CTD here. Assume that yields on the two bonds will move in a one-for-one fashion; i.e., a regression of yields on the 12% of August 15, 2008-13 bond on yields for the CTD bond would produce a slope coefficient of 1.0 Assume that the spot bond being hedged will be called early, in 2008. Given the quoted price of 124, the YTM of the spot bond being hedged is 9.3%. This can be computed using excel (=YIELD("4/10/1989","8/15/2008",0.12,124,100,2,1)), or FinCAD: aaBond_y settlement date 10-Apr-1989 maturity date 15-Aug-2008 dated date - date from which accrued 56 date of first coupon after dated date date of last coupon prior to maturity date coupon 0.12 principal at maturity 100 semi-annual date frequency 2 accrual method 17actual/actual (bond basis) price 124 yield is always compounded rate basis in last coupon period 2 number of days bond is ex-dividend 0 yield to maturity statistic 1 yield to maturity 0.093003929 (If you assume the bond won’t be called early, the YTM is 9.46%). If the YTM on this bond fell by 100 basis points, to 8.3%, its price would rise to $1404.30: aaBond_p settlement date 10-Apr-1989 maturity date 15-Aug-2008 dated date - date from which accrued date of first coupon after dated date date of last coupon prior to maturity date coupon 0.12 principal at maturity 100 semi-annual date frequency 2 accrual method 17actual/actual (bond basis) yield to maturity 0.0803 yield is always compounded rate basis in last coupon period 2 number of days bond is ex-dividend 0 fair value + accrued interest statistic 3 fair value + accrued interest 140.429625 This is an increase in price of $164.30 per $1000 face value. This implies that if interest rates fall by 100 basis points, the pension fund manager will have to pay $9.858 million more for the $60 million face value bonds. In problem 9.13, we found that a 100 basis point change would lead to a profit in the futures contract of $7035 per contract. Thus, dollar equivalency states that the manager should go long $9,858,000/$7035 = 1401 futures contracts to hedge. 57 Note that the solution here assumes that a 100 basis point increase in yields produces the same profit in the futures market as a 100 basis point decrease in yields. This is, in fact, an incorrect assumption. Because of the convexity of the price-yield curve, a decrease in yields in fact will lead to a larger change in the bond price than what will occur of yields increase by the same amount. 9.14. The current YTM of the RJR note is 7.85%. If its YTM fell by 100 basis points, the new YTM would be 6.85%, and its quoted price (not including accrued interest) would be 107.58 per $100 par value. This is $4.08 higher than its current quoted price of 103.50. You can use FinCAD to compute the new 107.58 price: aaBond_p settlement date 15-Jul-1999 maturity date 15-Apr-2004 dated date - date from which accrued date of first coupon after dated date date of last coupon prior to maturity date coupon 0.0875 principal at maturity 100 semi-annual date frequency 2 accrual method 17actual/actual (bond basis) yield to maturity 0.0685 yield is always compounded rate basis in last coupon period 2 number of days bond is ex-dividend 0 fair value (clean price) statistic 1 fair value (clean price) 107.5778338 If the price rises by $4.08 per $100 face value, the insurance company will have to pay 189,250.57 X 4.08 = $772,142 to purchase the notes (see section 9.3.3). To hedge, go long $772,142/$4788.30 = 161.26 T-note futures contracts. 9.15. a. He faces the risk that interest rates will rise, which will create a decline in value of his portfolio. To hedge, he should sell T-bond futures. b. Equation (9.13) provides the basis point value (BPV) for a bond. Here, BPV = dP = [8.513/(1+(.107/2))] X $10,000,000 X 0.0001 = $8080.68. Assume that the duration of the CTD is the same as the duration of the futures contract. Then, the BPV for the CTD is [9.17/(1+(.1096/2))] X $79,500 X 0.0001 = $69.114. There is no specified holding period for the hedge. It may be for a day, or it may be longer. Thus, in the BPV for the futures contract, given by equation (9.15), h = 0. Then, BPV = 58 F = (1/1.244) X 69.114 = $55.56. The BPV of the target is 0. Thus, use equation (9.16) to solve for the number of futures contracts to sell: $8080.68/$55.56 = 145.45. c. If the spot YTM rises from 10.70% to 10.80%, then the loss in value is estimated using equation A9.7: P = -(duration/(1+r)) X price X r Thus, the change in value = -(8.513/1.107) X $10 million X 0.001 = $76,901.54. The decline in value of $100,000 of the CTD bond is –(9.17/1.1096)X$79,500X0.001 = $657.01 The profit for being short one futures contract is then 657.01/1.244 = $533. The portfolio manager was short 145.45 contracts. Thus, his profit is 145.45 X $533 = $77,522.75. This was a nearly perfect hedge. The loss in portfolio value ($76,901.54) is almost perfectly matched by the futures profit ($77,522.75). 9.16. a. The duration of the 30-year bond can be computed using equation A9.2. Or, you can use FinCAD to find that the duration is 8.495. aaBond_p settlement date 19-Jun-2002 maturity date 19-Jun-2032 dated date - date from which accrued date of first coupon after dated date date of last coupon prior to maturity date coupon 0.12 principal at maturity 100 semi-annual date frequency 2 accrual method 17actual/actual (bond basis) yield to maturity 0.1213 yield is always compounded rate basis in last coupon period 2 number of days bond is ex-dividend 0 duration statistic 4 duration 8.495333461 In Section 9.4.1, the change in the value of a position if interest rates change by one basis point is defined to be dV = -(duration/(1+r)) X value X dr = dur X V X dr/(1+r) = dur X V X %(1+r). Thus to immunize against a change in interest rates, BOGUS wants the change in the value of its assets to equal the change in the value of its liabilities. Then, owners’ equity will not change if interest rates change. In other words, let A = assets and L = liabilities. Set 59 durA VA %(1+rA) = durL VL %(1+rL). The duration of liabilities is 1.0. We are assuming that %(1+rA) = %(1+rL). Thus durA = durL VL / VA = (1)(10)/12 = 0.8333. The duration of the 30-year bond is 8.495. The duration of the 91-day T-bills is 0.25. Define x as the number of dollars to be invested in the 91-day T-bills. Then solve for x in the following equation: (x/12)(0.25) + ((12-x)/12)(8.495) = 0.8333 x = $11.15 million invested in 91-day T-bills 12-x = $0.85 million invested in 30-year bonds. b. Section 9.4.1 explains how to change the duration of a portfolio using futures contracts. Here, we wish to change the duration of its assets from 8.495 to 1. Assume that the YTM of the CTD bond is 12.13%, and that its duration equals that of the futures contract. Then, the BPV of the CTD bond ($100,000 face value) is 9.75/(1+(0.1213/2)) X 122,406.25 X 0.0001 = 112.52. The time that the hedge will be in effect is unknown. Thus, in equation 9.15, assume that h = 0. Then, the BPV of the futures contract is 112.52/1.3513 = 83.27. The BPV of the BOGUS’ investment in the 30-year bond is 8.495/(1+(0.1213/2)) X $12,000,000 X 0.0001 = $9611.09. One-year securities are yielding 8%. Then the target BPV is 1.0/(1+0.08/2)) X $10,000,000 X 0.0001 = 961.54. Thus, to change the duration from 8.495 to 1.0, use equation 9.16: NF = (961.54 – 9611.09)/83.27 = -103.87. Sell 104 T-bond futures contracts. 9.17. a. He should be concerned about rising prices. He does not want to pay more money for the strips. b. He should buy T-bond futures to hedge. If prices do rise, he will then profit in the futures market. c. The price of the August 2020 ci is $32 per $100 par value. Its YTM is 5.87% according to Figure 9.1. This is computed using PV = -32, FV = 100, N = 39.4 half-years. Then h = 2.934%. Double this and the annualized YTM = 5.87%. (However, if PV = -32, FV = 100, and N = 19.7 years, we can compute YTM = 5.95%.). If interest rates decline by 10 basis points (5 basis points on a half year basis) then h = 2.884%, and the price of a $100 par value bond will rise to 32.62, which is a 1.9375% increase in price. Thus the intended original $50 million investment will cost an additional 0.019375 X $50 million = $968,750 The price of the CTD bond is 136:05, or $136,156.25 per $100,000 face value. Its YTM is 5.80% (h = 2.9%). Use FinCad to compute the change if value of the CTD bond if the YTM falls from 5.8% (FinCAD’s price is 136.2021) to 5.7% (FinCAD’s price is 137.5664). Thus the rise in value of the CTD bond, given a 10 basis point decline in its YTM, is 1.3643 per $100 face value. Use equation 9.12 to estimate the change in the futures price, given a change in the CTD. The 3-month interest rate is 6.2%. Thus h(0, 0.25 year) = 0.0155. The CF for the CTD bond is 1.3328 (figure 9.4). Then, F = (1.0155/1.3328) X 1.3643 = 1.0395. Since one point equals 60 $1000, 1.0395 points equals $1039.50. Dollar equivalency requires that $968,750/$1039.50 = 931.94 contracts be purchased. 9.18. a. He should be concerned that yields will rise, in which case the value of his portfolio will decline. b. He should sell T-bond futures to hedge. Rising yields will then result in profits in the futures market. c. If the YTM of the CTD bond rises by one basis point, the YTM of the manager’s portfolio will likely rise by 1.035 basis points to 7.0135% then equation 9.13 predicts the change in the value of the portfolio: dP = 11.3/(1+(0.07/2)) X 100,000,000 X 0.0001035 = $113,000. We must either estimate the YTM of the CTD bond, or estimate the duration of the CTD bond. We don’t know its time to maturity, but it must be between 15 years and 30 years. Here is a table that presents the possible times to maturity, and the associated YTM and duration (given that the semiannual coupon payment is $43.75 for a $1000 face value bond, and its price of $1120): Time to maturity YTM Duration 30 years 0.07717 11.84894 29 years 0.07709 11.74080 28 years 0.07699 11.62450 27 years 0.07689 11.49931 26 years 0.07678 11.36459 25 years 0.07665 11.21963 24 years 0.07651 11.06352 23 years 0.07635 10.89543 22 years 0.07618 10.71441 21 years 0.07598 10.51943 20 years 0.07577 10.30931 19 years 0.07552 10.08294 18 years 0.07524 9.83888 17 years 0.07492 9.57582 16 years 0.07456 9.29210 15 years 0.07414 8.98613 For simplicity, we will consider only the two extreme cases, if the CTD bond has 30 years to maturity or 15 years to maturity. We don’t know the length of time that the hedge will be in effect. Thus, we can ignore the fact that the short term interest rate is 6.1%. If it has 30 years to maturity, then by equation 9.13, its BPV is (11.84894/(1+(0.07717/2))) X $112,000 X 0.0001 = $127.77. The BPV of the futures contract is then 127.77/1.0709 = 119.32. The number of T-bond futures to sell is $113,000/$119.32 = 947.05. If it has 15 years to maturity, then its BPV is (8.98613/(1+(0.07414/2)) X $112,000 X 0.0001 = $97.05. The BPV of the futures contract is $97.05/1.0709 = $90.62. The number of futures contracts to sell is $113,000/$90.62 = 1246.94. 61 d. This is a cross hedge. The estimated regression equation that leads us to predict that if the YTM of the CTD bond rises by one basis point, the YTM of the bond portfolio will rise by 1.035 basis points may prove to be incorrect. The value of the delivery options might change, so that we cannot predict the change in the futures price. Some of the portfolio’s bond ratings could change, creating unexpected changes in the value of the portfolio. Yield spreads between high rated corporates and treasuries could change. 9.19. Once a T-bond has less than 15 years to maturity, as of the first day of the delivery month, it is no longer deliverable into that particular T-bond futures contract. Thus, as of March 1, 2001, the February 15, 2016 T-bond has less than 15 years to maturity. Thus, it cannot be delivered into the March 2001 futures contract. 62 Chapter 10 Treasury Bill and Eurodollar Futures 10.1 a. h(0,265) = (10,000-9259)/9259 = 0.08 = 8% b. 0.08 X (365/265) = 0.1102 ==11.02% c. (1.08)365/265 – 1 = (1.08)1.3774 – 1 = 0.1118 = 11.18% 10000 9259 360 d. dy 0.1007 10 .07 % 10000 265 e. (c ) exceeds (b) because the interest earned over the 265 day holding period also earns interest during the remaining 0.3774 of a year in (c ). When it is compounded, interest also earns interest. Simple interest does not earn interest. (b) exceeds (d) for two reasons. First, 10,000 is in the denominator of the discount yield. Normally, the original purchase price (9259) is in the denominator of an unannualized rate of return. Second, the discount yield multiplies the first expression by 360/# days to maturity. It should be multiplied by 365/# days to maturity. Thus, the discount yield will always be less than the rate of return computed using simple interest. 10.2. a. For $1 million face value, the spot prices of the spot T-bill are: (0.0458)(90) P $1 million 1 $988,550 360 (0.0483)(181) P $1 million 1 $975,715.83 360 b. From equation 5.12: P(0,181) = P(0,90) [FP(90,181)] 975,715.83 = (988,550) [FP(90,181)] [FP(90,181)] = 0.98701718 per $1 of face value, or $987,017.18 per $1 million face value. c. Given the IMM index value of 95.13, the futures discount yield is 4.87%. Then, the value of the 91 day T-bills underlying the contract is (0.0487 )(91) P $1 million 1 $987 ,689 .72 360 This is slightly above the theoretical value computed in part (b) 10.3. Equation 10.14 is T 90 T 90 1 (r(0, T 90)) 360 1 (r(0, T)) 360 1 (fr(T, T 90)) 360 Multiply both sides by 360: fr(T , T 90) 360 (r (0, T 90))(T 90) [360 (r (0, T ))(T )][1 ] (1) 4 Since I = 100(1-fr(T,T+90)), fr(T,T+90 = (100-I)/100 = 1 – 0.01I. 63 Also note that the lat term in equation (1), 1+fr(T,T+90)/4, equals 1+0.25fr(T,T+90). Substituting these into equation (1), we get 360 + (r(0,T+90))(T+90) = [360+(r(0,T))(T)][1.25 – 0.0025I] (2) Rearranging, [360 (r (0, T 90 ))(T 90 )] 1.25 – 0.0025I = (3) 360 r (0, T )(T ) Equation (3) then becomes [360 (r (0, T 90 ))(T 90 )] 0.0025I = 1.25 - (4) 360 r (0, T )(T ) Multiply and divide 1.25 by [360+r(0,T)T]: 450 1.25 (r (0, T )(T ) [360 (r (0, T ()))( T 90 ) 0.0025I = (5) 360 r (0, T )(T ) Finally, divide both sides by 0.0025 to get equation 10.15: 360 500 (T )( r (0, T )) 400 (T 90 )[ r (0, T 90 )] I= 360 (T )[ r (0, T )] 10.4. If the firm borrows $1,000,000 from the bank for 100 days at an annual rate of 6.5%, then it will owe the bank $1,000,000 X [1+(0.065)(110/360)] = $1,019,861 at the end of 110 days. If the firm borrows $1,000,000 from the bank for 20 days at an annual rate of 6.25%, then it will owe the bank $1,000,000 X [1+(0.0625)(20/360)] = $1,003,472 at the end of 20 days. Suppose that spot LIBOR is 6.68%, 20 days hence. This means that the final settlement futures price is 93.32, and that a profit of 38 ticks was realized on the futures contract. (38)($25/tick) = $950. Thus the firm will have to borrow $1,003,472 - $950 = $1,002,522 at a rate of 6.68% + 0.50% = 7.18%. Then, the firm will owe the bank $1,002,522 X (1+(0.0718)(90/360)) = $1,020,517 at the end of 100 days. This is inferior to just borrowing for 110 days at the original spot interest rate. Suppose instead that spot LIBOR is 5.92%, 20 days hence. This means that the final settlement futures price is 94.08, and that a loss of 38 ticks was realized on the futures contract. (38)($25/tick) = $950. Thus the firm will have to borrow $1,003,472 + $950 = $1,004,422 at a rate of 5.92% + 0.50% = 6.42%. Then, the firm will owe the bank 64 $1,004,422 X (1+(0.0642)(90/360)) = $1,020,543 at the end of 100 days. This is also inferior to just borrowing for 110 days at the original spot interest rate. 10.5. Equation 10.15 determines the theoretical futures price. For the December contract, we have T = 24 and T+90 = 114, r(0,24) = 0.07875 and r(0,114) = 0.08. Then 360 500 (24 )( 0.07875 )) 400 (114 )[ 0.08 ] I= 92 .01 360 (24 )[ 0.07875 ] For March, we have T = 115, T+90 = 205, r(0,115) = 0.08, and r(0,205) = 0.079375. Then 360 500 (115 )( 0.08 )) 400 (205 )[ 0.079375 ] I= 92 .34 360 (115 )[ 0.08 ] For June, we have T = 206, T+90 = 296, r(0,206) = 0.079375, and r(0,296) = 0.079375. Then 360 500 (206 )( 0.079375 )) 400 (296 )[ 0.079375 ] I= 92 .41 360 (206 )[ 0.079375 ] For September, we have T = 297, T+90 = 387, r(0,297) = 0.079375, and r(0,387) = 0.07875. Then 360 500 (297 )( 0.079375 )) 400 (387 )[ 0.07875 ] I= 92 .80 360 (297 )[ 0.079375 ] 10.6. Equation 5.11 can be used to compute the theoretical forward rate. Assume a 360 day year. 1+h(0,t2) = (1+h(0,t1))(1+fh(t1,t2)) 1+(0.055)(97/360) = (1+(0.05)(7/360)) X (1+x(90/360)) 1.014819444 = 1.000972222 (1+0.25x) Solving for x, we find x = 0.0553. Hence, the theoretical futures price is 94.47. You would want to buy the cheap futures at 94.34, borrow for 97 days, and lend for seven days. Buying futures commits you to lending from day 7 to day 97, at a rate of 5.66%. Borrowing for 97 days and lending for 7 days creates a forward borrowing situation, using spot instruments, at a rate of 5.53%. You can quickly approximate the size of your arbitrage profit to equal (94.47-94.34) X $25/tick = $325. At time zero, lend the present value of $1,000,000 = 1,000,000/(1+0.05(7/360)) = $999,028.72 for seven days. Also borrow $999,028.72 for 97 days. Buy the futures contract. The net cash flow at time zero is zero. Seven days hence, you will receive $1,000,000 from the original loan you made. You also will profit in the futures market by 25(5.66-x), where x equals 90-day LIBOR on day 7. You will lend 65 this total amount ($1,000,000 + 25(5.66-x)) for 90 days at the spot annual rate of x%. 90 days later, you will receive [$1,000,000 + 25(5.66-x)] X (1+x/400). Thus, if x = 5%, you have profited in the futures market by 66 ticks ($1650), and seven days later you will be able to lend $1,000,000 + 25(5.66-5) = $1,001,650 at the spot annual rate of 5%. On day 97, you receive $1,001,650 X 1.0125 = $1,014,170.625. But recall that, at time zero, you also borrowed $999,028.72 for 97 days at 5.5%, so that you owe $1,013,833.77 on that loan. Your arbitrage profit is $336.86, realized on day 97. If x = 6%, you have lost 34 ticks in the futures market, and seven days later you will be able to lend $1,000,000 - 25(5.66-6) = $999,150 at an annual rate of 6%. On day 97, you will receive $999,150 X 1.015 = $1,014,137.25. You owe $1,013,833.77 on the original 97-day loan. Your arbitrage profit is $303.48, realized on day 97. 10.7. The theoretical futures price is computed as follows: 1+h(0,t2) = (1+h(0,t1))(1+fh(t1,t2)) 1+(0.055)(187/360) = (1+(0.05)(97/360)) X (1+x(90/360)) 1.028569444 = 1.013472222 (1+0.25x) Solving for x, we find x = 0.0596. Hence, the theoretical futures price is 94.04. You would want to sell the overpriced futures at 94.34, borrow for 97 days, and lend for 187 days. Selling futures commits you to borrowing from day 97 to day 187, at a rate of 5.66%. Borrowing for 97 days and lending for 187 days creates a forward lending situation, using spot instruments, at a rate of 5.96%. Your arbitrage profit will be about (94.34-94.04) X $25/tick = $750. 10.8. a. The firm fears that interest rates will rise (prices will fall). If that happens, it will receive less money for its $20 million in commercial paper; equivalently, its interest expense will be greater. To hedge, it should sell Eurodollar futures. b. If 30-day CP rates rise by 10 basis points, the firm will receive less money when it issues its CP. At the current rate of 7%, it will receive $19,884,010. This is computed as follows. The formula for the add-on yield (aoy) is F P 360 aoy P 30 Thus 20 ,000 ,000 P 360 0.07 P 30 P = $19,882,363. Thus, the firm will receive $1647 less. With the regression model, if the 30-day CP rate rises by 10 bp, Eurodollar futures aoy will likely rise by 8.333 bp. Then, the profit on the sale of one Eurodollar futures contract will be 8.333 ticks X $25/tick = $208.33. Using dollar equivalency, VS = h F = 1647 = h(208.33). Therefore, sell h = 7.9056 Eurodollar futures contracts. 10.9. Note that the firm’s initial interest payment is $350,000 (not $550,000). 66 a. Today, November 19, spot LIBOR is 6%, and the nearby (December) futures aoy is 5.78%. Thus, the firm might expect spot LOBOR on 2/19, 5/19, and 8/19 to be 22 basis points below the futures LIBOR that currently exist. Then, it might expect to pay 78 bp above the current futures LIBOR’s that exist in the term structure of futures prices: Feb. 19: pay 5.75% + 78 bp = 6.53% May 19: pay 5.93% + 78 bp = 6.71% August 19: pay 6.14% + 78 bp = 6.92% Then, its total interest expense is expected to be: $20,000,000 X [(0.07/4) + (0.0653/4) + (0.0671/4) + (0.0692/4)] = 0.0679 X $20,000,000 = $1,358,000. (Other answers might be acceptable. In particular, one common answer is that the firm expects to pay exactly 100 bp above the futures LIBORs. Then, the expected interest expense is $20,000,000 X [(0.07/4) + (0.0675/4) + (0.0693/4) + (0.0714/4)] = 0.06955 X $20,000,000 = $1,391,000). b. With a strip hedge: Today, sell 20 March, 20 June, and 20 September futures contracts. With a rolling hedge, sell 60 March futures today at 94.25. On Feb. 19, offset the 60 March contracts and sell 40 June contracts at 94.00. On May 19, offset the 40 June contracts and sell 20 Sept. contracts at 92.62. On August 19, offset the 20 Sept. contracts. Final results: With no hedge, the firm’s interest expense would be $20,000,000 X [(0.07/4) + (0.06/4) + (0.0706/4) + (0.0794/4)] = 0.07 X $20,000,000 = $1,400,000. This is slightly more than what it originally expected ($1,358,000) With the strip hedge, the firm would profit by (94.25 – 94.02 = 23 ticks X $25/tick = ) $575 X 20 contracts = $11,500 on the 20 March contracts. It would profit by (94.07 – 92.90 = 117 ticks X $25/tick = ) $2925 X 20 contracts = $58,500 on the 20 June contracts. It would profit by (93.86 – 91.92 = 194 ticks X $25/tick = ) $4850 X 20 contracts = $97,000 on the 20 Sept. contracts. Thus, its total interest expense with the strip hedge is $1,400,000 - $11,500 - $58,500 - $97,000 = $1,233,000. This is $125,000 less than it expected. With the rolling hedge, the firm would profit by (94.25 – 94.02 = 23 ticks X $25/tick = ) $575 X 60 contracts = $34,500 on the 60 March contracts. It would profit by (94.00 – 92.90 = 110 ticks X $25/tick = ) $2750 X 40 contracts = $110,000 on the 40 June contracts. It would profit by (92.62 – 91.92 = 70 ticks X $25/tick = ) $1750 X 20 contracts = $35,000 on the 20 Sept. contracts. Thus, its total interest expense with the strip hedge is $1,400,000 - $34,500 - $110,000 - $35,000 = $1,220,500. This is $135,500 less than it expected. The rolling hedge performed the best. 10.10. a. The firm fears that interest rates will rise (prices will fall) between today and December 17th. Thus it will sell 25 December futures to hedge. The loan maturity will be for 90 days (Dec. 17 to March 17), and the corporate treasurer is using Eurodollar futures to hedge. 67 b. If it did not hedge, it would have to pay 8.81% on its loan. Equation 10.11 is used to compute the interest on the loan, paid at maturity: $25,000,000 X 0.0881 X (90/360) = $550,625. The futures price on December 17 will be 92.19. The firm originally sold 25 December contracts at 91.98. Thus, it lost 21 ticks per contract, for a total loss of 21 ticks X $25/tick X 25 contracts = $13,125. After hedging, the firm’s total interest expense is $550,625 + $13,125 = $563,750. The annualized interest rate on the hedged position is (563,750/25,000,000) X (360/90) = 9.02%. This is 100 basis points above the November 26 futures interest rate of 8.02% (100 – 91.98 = 8.02%). c. If it uses a strip hedge, the firm should sell 25 contracts each of the December, March, June, and September contracts (100 contracts in all). Alternatively, it can use a rolling hedge, and sell 100 December contracts. With the latter, there is greater basis risk, since the firm is using 90 day Eurodollar futures to hedge a one-year loan. d. The answer is the same answer as (c ). Here, however, a strip hedge will remove all of the basis risk. If the distant contracts are sufficiently liquid (for Eurodollar futures, they typically are very liquid), the strip hedge is recommended. A rolling hedge is much riskier, and should only be used if the nearby contract is more overpriced than the others, and/or it has considerably greater liquidity than the more distant contracts. 10.11. a. The hedger initially sells 200 September contracts at 94.555. She offsets them on September 13 at 94.350. This is a profit of 20.5 ticks/contract X $25/tick X 200 contracts = $102,500. On September 13, the hedger also sells 150 December contracts at 94.320. On December 13, she offsets them at 94.150. This is a profit of 17 ticks/contract X $25/tick X 150 contracts = $63,750. On December 13, the hedger also sells 100 March contracts at 94.22. On March 13, she offsets them at 94.00. This is a profit of 22 ticks/contract X $25/tick X 100 contracts = $55,000. On March 13, the hedger also sells 50 June contracts at 94.17. On June 19, she offsets them at 94.10. This is a profit of 7 ticks/contract X $25/tick X 50 contracts = $8750. b., c. Quarter Firm’s Quarterly Gain/loss Total Effective Borrowing Interest on Futures Interest Borrowing Rate Expense Positions Expense Rate Sep99-Dec99 5.65% $706,250 $25,625 $680,625 5.445% Dec99-Mar00 5.85% $731,250 $21,250 From Sep/Dec $25,625 $684,375 5.475% Mar00-Jun00 6% $750,000 $27,500 From Dec/Mar $21,250 From Sep/Dec $25,625 $675,625 5.405% Jun00-Sep00 5.9% $737,500 $ 8,750 From Mar/Jun $27,500 From Dec/Mar $21,250 From Sep/Dec $25,625 $654,375 5.235% 68 d. The average effective loan rate is (5.445 + 5.475 + 5.405 + 5.235)/4 = 5.39%. If effectiveness is defined in terms of the lowest possible borrowing rate, then this hedge in problem 10.11 is more effective than the hedge in Table 10.4. If hedging effectiveness is measured in terms of the original interest rates implied by the futures curve (5.445%, 5.81%, 5.815%, and 6.07%), then while this hedge did better, it also deviated a great deal from those original rates. e. This rolling hedge did better because interest rates fell during the period, to levels less than those implied by the original futures curve. Thus, the hedger realized considerable profits from the hedge. Had interest rates risen a great deal initially, the hedger would have performed poorly. Note that an alternative response is NOT to spread the earlier futures profits over all subsequent quarters. This is more accurate in terms of measuring the exact timing of cash flows, but less accurate in terms of attributing the results of the hedge to the intent of the hedge. In this case, the table would look like this: Quarter Firm’s Quarterly Gain/loss Total Effective Borrowing Interest on Futures Interest Borrowing Rate Expense Positions Expense Rate Sep99-Dec99 5.65% $706,250 $102,500 $603,750 4.83% Dec99-Mar00 5.85% $731,250 $63,750 $667,500 5.34% Mar00-Jun00 6% $750,000 $55,000 $695,000 5.56% Jun00-Sep00 5.9% $737,500 $ 8,750 $728,750 5.83% 69

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posted: | 6/30/2011 |

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