1、Instructors Manual Chapter 14 Page 157CHAPTER 14FORWARD AND FUTURES PRICESObjectives To explain the economic role of futures markets To show what information can and cannot be inferred from forward and futures prices.Outline14.1 Distinctions Between Forward and Futures Contracts14.2 The Economic Fun
2、ction of Futures Markets14.3 The Role of Speculators14.4 Relation Between Commodity Spot and Futures Prices14.5 Extracting Information from Commodity Futures Prices14.6 Spot-Futures Price Parity for Gold14.7 Financial Futures14.8 The Implied Risk-Free Rate14.9 The Forward Price Is Not a Forecast of
3、the Spot Price14.10 Forward-Spot Parity with Cash Payouts14.11 Implied Dividends14.12 The Foreign-Exchange Parity Relation14.13 The Role of Expectations in Determining Exchange RatesSummary Futures contracts make it possible to separate the decision of whether to physically store a commodity from th
4、e decision to have financial exposure to its price changes. Speculators in futures markets improve the informational content of futures prices and make futures markets more liquid than they would otherwise be. The futures price of wheat cannot exceed the spot price by more than the cost of carry: Th
5、e forward-spot price parity relation for gold is that the forward price equals the spot price times the cost of carry:This relation is maintained by the force of arbitrage. One can infer the implied cost of carry and the implied storage costs from the observed spot and forward prices and the risk-fr
6、ee interest rate. The forward-spot parity relation for stocks is that the forward price equals the spot price times 1 plus the risk-free rate less the expected cash dividend. This relation can therefore be used to infer the implied dividend from the observed spot and forward prices and the risk-free
7、 interest rate. The forward-spot price parity relation for the dollar/yen exchange rate involves two interest rates: where F is the forward price of the yen, S is the current spot price, rY is the yen interest rate, and r$ is the dollar interest rate. If the forward dollar/yen exchange rate is an un
8、biased forecast of the future spot exchange rate, then one can infer that forecast either from the forward rate or from the dollar-denominated and yen-denominated risk-free interest rates.FSCrs()1FrD()1rY1$Instructors Manual Chapter 14 Page 158Solutions to Problems at End of ChapterForward Contracts
9、 and Forward-Spot Parity.1. Suppose that you are planning a trip to England. The trip is a year from now, and you have reserved a hotel room in London at a price of 50 per day. You do not have to pay for the room in advance. The exchange rate is currently $1.50 to the pound sterling.a. Explain sever
10、al possible ways that you could completely hedge the exchange rate risk in this situation.b. Suppose that r=.12 and r$=.08. Because S=$1.50, what must the forward price of the pound be?c. Show that if F is $0.10 higher than in your answer to part b, there would be an arbitrage opportunity. SOLUTION:
11、a. Ways to hedge the exchange rate risk:Pay for the room in advanceBuy the pounds you will need in the forward market.Invest the present value of the rental payments in a pound-denominated riskless asset.对冲外汇风险的几种方法：提前对这个房间付款；在期货市场购买英镑；将与现期价值的租金同等的英镑投资于无风险资产。b. F = S (1+r$)/(1+r) = $1.50 x 1.08/1.12
12、 = $1.4464 per pound1.4464 美元/英镑c. If F is $1.55 then arbitrage profits can be made by borrowing dollars, investing in pounds and selling them forward at the inflated forward price. After paying off principle and interest on the dollars borrowed, you would have pure arbitrage profits left over.如果 F
13、是 1.55 美元，那么套利可以通过买进美元以获取收益，将美元投资于英镑然后在通货膨胀的价格下卖出。在支付完所借到的美元的本金和利息后，你将得到剩下的纯套利收益。For example, Borrow $1.50,Convert it into 1 pound,Invest it in pound-denominated bonds to have 1.12 pounds a year from now,Instructors Manual Chapter 14 Page 159Sell 1.12 pounds forward at $1.55 per pound to have $1.736
14、 a year from now,例如，你可以借进 1.50 美元，使起转变为 1 英镑，将起投资于英镑，一年后你会得到 1.12 英镑的，将这 1.12 英镑以 1.55 美元卖出，你将得到 1.736 美元。After 1 year, pay off the principle and interest on the loan ($1.50x 1.08 = $1.62).This series of transactions leaves you with $.116 a year from now with no initial outlay of your money.在一年之后，将本
15、金和利息支付，也就是 1.50*1.08=1.62 美元，这一系列的交易在没有创始费用的状况下将给你留下 1.16 美元。Arbitrage Position Immediate Cash Flow Cash Flow 1 Year From NowBorrow $1.50借进 1.50 美元$1.50 -$1.62Buy pound-denominated bond购买英镑远期合约-$1.50 S1Sell 1.12 pounds forward at $1.55 per pound以 1.55 美元 /英镑的价格卖出 1.12 英镑0 $1.736-S1 Net Cash Flows净现金
16、流0 $1.736-$1.62 = $.116Forward-Spot Parity Relation with Known Cash Payouts2. Suppose that the Treasury yield curve is flat at an interest rate of 7% per year (compounded semiannually).a. What is the spot price of a 30-year Treasury bond with an 8% coupon rate assuming coupons are paid semiannually?
17、b. What is the forward price of the bond for delivery six months from now?Instructors Manual Chapter 14 Page 160c. Show that if the forward price is $1 lower than in your answer to part b, there should be an arbitrage opportunity.SOLUTION:a. The spot price of the 30-year Treasury is $1,124.724:n i P
18、V FV PMT Result60 3.5 ? 1000 40 PV =1124.724b. The forward price for delivery six months from now is $1,124.089:6 个月后交割的债券远期价格是 1,124.089 美元F = S(1+r) - C = $1,124.724 x 1.035 - 40 =$1,124.089d. If the forward price is only $1,123.089, then arbitrage profits can be made by selling the bond short and
19、 buying it forward at the low forward price. It can be described as follows:如果远期价格仅仅是 1,123.089 美元，那么套利获得收益可以这样进行：将近期的美元卖出，并在将来以较低的价格买进。可以通过如下来描述：Sell short a bond at $1,124.724; buy it forward at $1,123.089; invest the proceeds of the short sale to earn 3.5% for 6 monthsAfter 6 months, take deliver
20、y of the bond and cover your short sale将 $1,124.724 卖出，并在将来买进$1,123.089；6 个月后将从购买中获得 3. 5%的收益。Arbitrage Position Immediate Cash Flow Cash Flow 1 Year From NowSell short a 30-year T-bond卖出 30 年期国债$1,124.089 -(S1 + $40)Buy 6-month T-bills paying 3.5%买进 6 个月国债制服3.5%-$1,124.089 $1,163.432Buy a forward c
21、ontract for a 30-year T-bond0 S1-$1,123.089Instructors Manual Chapter 14 Page 161买进 30 年期的国债远期合同Net Cash Flows净现金流0 $1,163.432- ($1,123.089 + $40) = $.343Forward-Spot Parity Relation with Uncertain Dividends3. A stock has a spot price of $100; the riskless interest rate is 7% per year (compounded an
22、nually), and the expected dividend on the stock is $3, to be received a year from now. a. What should be the one-year futures price?b. If the futures price is $1 higher than your answer to part a, what might that imply about the expected dividend?SOLUTION:a. S = $100, r = .07, D = $3. F = S ( 1+r) -
23、 D = $104b. If F is $105, that might imply that D is really only $2.如果 F 是 105 美元，那么说明 D 真正数值是 2 美元。Storage Costs versus Dividend Yield4. Compare the forward-spot price-parity relation for gold to the one for stocks. Is it fair to say that stocks have a negative storage cost equal to the dividend yi
24、eld?SOLUTIONOne could definitely say that stocks have a negative storage cost equal to the dividend.确定的说应该是股票的储存成本是股息的负值。5. Suppose you are a distributor of canola seed and you observe the spot price of canola to be $7.45 per bushel while the futures price for delivery one month from today is $7.60.
25、 Assuming a $.10 per bushel carrying cost, what would you do to hedge your price uncertainty?SOLUTIONWe see that F S+C. If you short the futures contract, you can sell your seed at $7.60 per bushel.Instructors Manual Chapter 14 Page 162我们发现 FS+C.如果你缩短期货交易合同的时限，你将可以把你的种子以 7.6 美元/蒲式耳。6. Infer the spot
26、 price of an ounce of gold if you observe the price of one ounce of gold for forward delivery in three months is $435.00, the interest rate on a 91-day Treasury bill is 1% and the quarterly carrying cost as a percentage of the spot price is .2%. （相当于成本是现货价格的2%）SOLUTIONDeduce from the futures price p
27、arity condition for gold that F = S0 (1 + r + s) so that S0 = $429.84.推断黄金的期货价格应该是 F=S（1+r+s）因而 S=429.84 美元7.You are a dealer in kryptonite and are contemplating a trade in a forward contract. You observe that the current spot price per ounce of kryptonite is $180.00, the forward price for delivery
28、of one ounce of kryptonite in one year is $205.20, and annual carrying costs of the metal are 4% of the current spot price.a. Can you infer the annual return on a riskless zero-coupon security implied by the Law of One Price?b. Can you describe a trading strategy that would generate arbitrage profit
29、s for you if the annual return on the riskless security is only 5%? What would your arbitrage profit be, per ounce of kryptonite?SOLUTIONa. By no-arbitrage, we require that the riskless rate r satisfy:在没有套利的情况下，无风险零息债券的年利率是：F = S0 (1 + r + s)205.2 = 180 (1 +r +.04) = 187.2 + 180rr = 18/180 = .10 or
30、10%b. The implicit risk-free rate that you can earn by buying kryptonite, storing it, and selling it forward at $205.2 per ounce is 10%. If the riskless borrowing rate is five percent, you should borrow at that rate and invest in hedged kryptonite. If you buy an ounce of kryptonite for $180, you wil
31、l get $205.2 for it for sure a year from now. If you borrow the $180, you will have to pay principal and interest of $180 x 1.05 plus another .04 x Instructors Manual Chapter 14 Page 163$180 in storage costs. This totals $196.2, thus l eaving you with $9 in ar bitrage profits.你可以通过买 kryptonite 并储存后在
32、将来以 205.2 美元/盅司的价格卖出赚取 10%隐含的无风险利率。如果无风险借贷利率是 5%，你可以以那样的比例借进并投资于 kryptonite。如果你以$180/盅司买进，你将会在一年后得到$205.2。如果你借进的是 180 美元，你将归坏本金和利息$180 x 1.05 以及.04 x $180 的储藏成本。这总共是$196.2，因而你可以从套利中获得$9。8.Calculate the implicit cost of carrying an ounce of gold and the implied storage cost per ounce of gold if the c
33、urrent spot price of gold per ounce is $425.00, the forward price of an ounce of gold for delivery in 273 days is $460.00, the yield over 91 days on a zero-coupon Treasury bill is 2% and the term structure of interest rates is flat.SOLUTIONFirst, we solve it assuming a simple compounding method for
34、the risk free interest rate. Over 273 days, the Risk free rate is 2%*3=6%. Therefore we have, 首先，我们通过假设一个零风险利率的简单组合方法来解决这个问题。273 天，零息国债收益率是 2%*3=6%。因而我们有，F = S (1 + r + s )460 = 425 (1.06 + s)s = (460 - 450.5)/425 = 9.5/425 = .02235 for 273 days Thus the carrying costs are roughly 8.24% for 273 days
35、 or 10.98% per year.所以持有成本粗率的计算为 8.24%以 237 天计算或者是 10.98%/年。Second, we solve it assuming we need to compound the interest rates. The risk free rate over 273 days will be (1+2%)3-1=6.12%.plug in the above formulae we get s=.021145 for 273 days. Thus the carrying costs are roughly 8.23% for 273 days o
36、r 11.13% per year.其次，我们需要连续复利。零息无风险利率在 273 天应该是(1+2%) 3-1=6.12%。再上式中我们得出 s=021145 / 273 天。所以持有成本应该是8.23% / 273 天 or 11.13%/年。9The forward price for a share of stock to be delivered in 182 days is $410.00, whereas the current yield on a 91-day T-bill is 2%. If the term Instructors Manual Chapter 14 P
37、age 164structure of interest rates is fiat, what spot price for the stock is implied by the Law of One Price?SOLUTIONF = $410; r = .02 per quarter.（每季度）S = F/(1+r)2 = $394.0810You observe that the one-year forward price of a share of stock in Kramer, Inc., a New York tour-bus company and purveyor of
38、 fine clothing, is $45.00 while the spot price of a share is $41.00. If the riskless yield on a one-year zero-coupon government bond is 5%:a. What is the forward price implied by the Law of One Price?b. Can you devise a trading strategy to generate arbitrage profits? How much would you earn per shar
39、e?SOLUTIONa. The no-arbitrage value of the forward price is F = $43.05.股票的远期价格是 F = $43.05b. The observed forward price is excessive. Consider short-selling a forward contract and taking a long position in a portfolio consisting of one stock and the sale of a bond with face value of F. Future liabil
40、ities for this position are zero, while the current cash inflow is $1.86.观察到的远期价格是更高的。考虑短期卖出期货合同并采取面值为 F的股票和债券的组合。这样的决定将来的债务是 0，并且现金流是$1.86。11Infer the yield on a 273-day, zero-coupon Japanese government security if the spot price of a share of stock in Mifune and Associates is 4,750 yen whereas the
41、 forward price for delivery of a share in 273 days is 5,000 yen.SOLUTIONThe implied yield over the 273 day term is r = 5.26%.273 天的收益率是 5.26%。12On your first day of trading in Vietnamese forward contracts, you observe that the share price of Giap Industries is currently 54, 000 dong while the one-ye
42、ar forward price is 60, 000 dong. If the yield on a one-Instructors Manual Chapter 14 Page 165year riskless security is fifteen percent, are arbitrage profits possible in this market? If not, explain why not. If so, devise an appropriate trading strategy.SOLUTIONArbitrage profits would seem to be po
43、ssible, since the no-arbitrage forward price implied by these parameters isF = $62,100.The futures contract is underpriced, relative to this no-arbitrage value. Consider taking a long position in the forward contract and simultaneously selling a share of Giap stock and buying a riskless bond with a
44、face value equal to the observed forward price. The liabilities from these joint positions are zero, while the current cash inflow is $1826.09.套利获利是可能的，这些参数所推导出来的远期价格是 F = $62,100。期货交易合同是定价的，相对于没套利的价值而言。考虑签定长期合同，同时卖出部分 Giap 股票，买进面值与观察到的期货价格相当的无风险债券。这些组合中的负债为 0，现金流是$1826.09.13The share price of Schle
45、ifer and Associates, a financial consultancy in Moscow, is currently 10, 000 roubles whereas the forward price for delivery of a share in 182 days is 11,000 roubles. If the yield on a riskless zero-coupon security with term to maturity of 182 days is 15%, infer the expected dividend to be paid by Sc
46、hleifer and Associates over the next six months.SOLUTIONThe implied dividend is 500 roubles.在以后 6 个月中的预期股利是 500 卢布。14The spot rate of exchange of yen for Canadian dollars is currently 113 yen per dollar but the one-year forward rate is 110 yen per dollar. Determine the yield on a one-year zero-coupon Canadian government security if the corresponding yield on a Japanese government security is 2.21%.SOLUTIONThe implied Canadian rate over this term is approximately 5.00%.在这些条件下加拿大政府证券的收益率近似的等于 5.00%。Instructors Manual Chapter 14 Page 166
