ounMar 1 14396 bot 1.6050 1.4010 14,833dune 1.4444 tua ten ier — 0064 1.5304 1.4060 23,641Est vol 7,951; vol Wed 15,965; open int 36,510, —2,720.
Swiss Franc OME) 225,000 francs; if per franc
Mar ee Gost” 5325 5541 20,680dune 5885. an ae aa = 0052 8358 5585 32.622Est vol 14,447; vol Wed 34,342; open int 53,337, +4.514,Australien Dollar (CME)-100 100.0 900 ass $ fa" AS
4960 4960
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4942 4942 :Est vol 3.212; val Wed aon open int 42,329, +313.Mexican Peso (CME) Site 000 new Max. Bar $ per MPMar 10488 10445 10395 10403 = OT ae od
10298 J
ay 10198 > to “reo 09900 848dune 10155 10170 10085 “0108 - 00027 10170 09070 20,061Aug a 09908 — 00027 .09800 .09800 100Sept. oats — 00027 09880 09300 2.683Euro FX (CMI 10; $ per Euro
Mar S116 9120 8965 s — 0089 9009 8333 38.657dune 9121 9130 8960 9010 - 0082 9784 8356 59,081so ‘9071 8900 9013. = 0003 98M 8979 1,178Est vo! 33,027; val Wed 40,744; open Int 98,0683, 3000.
    
 
  
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Source: Reprinted by permission of Dow Jones, Inc., via copyright Clearance Center, Inc.© 2001 Dow Jones & Company, Inc. All Rights Reserved Worldwide.
Mexico interest rates were higher than in the United States. This corresponds to the r, > r situationand explains why the futures prices of these three currencies decrease with maturity.
Example 3.8 The futures price of the Japanese yen in Table 3.3 appears to be increasing at a rate ofabout 4.6% per annum with the maturity of the contract. The increase suggests that short-term interestrates were about 4.6% per annum higher in the United States than in Japan on March 15, 2001.
A Foreign Currency as an Asset Providing a Known Yield
Note that equation (3.13) is identical to equation (3.7) with g replaced by r,. This is not acoincidence. A foreign currency can be regarded as an investment asset paying a known yield.The yield is the risk-free rate of interest in the foreign currency.
To understand this, suppose that the one-year interest rate on British pounds is 5% per annum.(For simplicity, we assume that the interest rate is measured with annual compounding and interestis paid at the end of the year.) Consider a United States investor who buys | million pounds. Theinvestor knows that £50,000 of interest will be earned at the end of one year. The value of thisinterest in dollars depends the exchange rate. If the exchange rate in one year is 1.5000, the interestis worth $75,000; if it is 1.4000, the interest is worth $70,000; and so on. The value in dollars of the58 CHAPTER 3
 
interest earned is 5% of the value of the sterling investment. The 5% interest therefore represents aknown yield to the United States investor on the slerling investment.
3.12 | FUTURES ON COMMODITIES
 
 
 
We now move on to consider futures contracts on commodities. First we consider the impact ofstorage on the futures prices of commodities that are investment assets, such as gold and silver.°We assume that no income is earned on the assets.
Storage Costs
Equation (3.5) shows that in the absence of storage costs the forward price of a commodity that isan investment asset is given byFy = Sye’™ (3.14)
Storage costs can be regarded as negative income. If U is the present value of all the storage coststhat will be incurred during the life of a forward contract, it follows from equation (3.6) that
Fg = (Sy + U)e'™ (3.15)
Example 3.9 Consider a one-year futures contract on gold. Suppose that it costs $2 per ounce peryear to store gold, with the payment being made at the end of the year. Assume that the spot price is$450 and the risk-free rate is 7% per annum for all maturities. This corresponds to r= 0.07,So = 450, T = 1, and
U =2¢ °°! — 1.965
From equation (3.15) the futures price, Fo, is given byFy = (450 + 1.865)e"°7*! = $484.63
If Fy > 484.63, an arbitrageur can buy gold and short one-year gold futures contracts to lock in aprofit. If Fy < 484.63, an investor who already owns gold can improve the return by selling the goldand buying gold futures contracts.
If the storage costs incurred at any time are proportional to the price of the commodity, they canbe regarded as providing a negative yield. In this case, from equation (3.7),
Fo= Spel Or (3.16)
where uv denotes the storage costs per annum as a proportion of the spot price.
Consumption Commodities
For commodities that are consumption assets rather than investment assets, the arbitrage argu-ments used to determine futures prices need to be reviewed carefully. Suppose that instead of
® Recall that for an asset to be an investment asset it need not be held solely for investment purposes. What isrequired js that some individuals hold it for investment purposes and that these individuals be prepared to sell theirholdings and go long forward contracts, if the latter look more attractive. This explains why silver, although it hassignificant industrial uses, is an investment asset.Determination of Forward and Futures Prices 59
equation (3.15) we haveFy > (So + Ue (3.17)
To take advantage of this opportunity, an arbitrageur can implement the following strategy:
1. Borrow an amount Sy+U at the risk-free rate and use it to purchase one unit of thecommodity and to pay storage costs.
2. Short a forward contract on one unit of the commodity.
If we regard the futures contract as a forward contract, this strategy leads to a profit ofFy —(Sy + U)e™ at time 7. There is no problem in implementing the strategy for any commodity.However, as arbitrageurs do so, there will be a tendency for Sy to increase and Fy to decrease untilequation (3.17) is no longer true. We conclude that equation (3.17) cannot hold for any significantlength of time.Suppose next thatFy < (Sy + U)e™ (3.18)
In the case of investment assets such as gold and silver, we can argue that many investors hold thecommodity solely for investment. When they observe the inequality in equation (3.18), they willfind it profitable to:
1. Sell the commodity, save the storage costs, and invest the proceeds at the risk-free interestrate.
2, Take a long position in a forward contract.
The result is a riskless profit at maturity of (Sy + U)e’ — Fy relative to the position the investorswould have been in if they had held the commodity. It follows that equation (3.18) cannot hold forlong. Because neither equation (3.17) nor (3.18) can hold for long, we must have Fy = (Sp + Uje™.
For commodities that are not to any significant extent held for investment, this argument cannotbe used. Individuals and companies who keep such a commodity in inventory do so because of its
mption value—not because of its value as an investment. They are reluctant to sell thecommodity and buy forward contracts, because forward contracts cannot be consumed. There istherefore nothing to stop equation (3.18) from holding. All we can assert for a consumption
commodity is therefore
 
Fy < (Sp $+ U)e™ (3.19)If storage costs are expressed as a proportion u of the spot price, the equivalent result is
Fy < Se? (3.20)
Convenience Yields
We do not necessarily have equality in equations (3.19) and (3.20) because users of a consumptioncommodity may feel that ownership of the physical commodity provides benefits that are notobtained by holders of futures contracts. For example, an oil refiner is unlikely to regard a futurescontract on crude oil as equivalent to crude oil held in inventory. The crude oil in inventory can bean input to the refining process whereas a futures contract cannot be used for this purpose. Ingeneral, ownership of the physical asset enables a manufacturer to keep a production processrunning and perhaps profit from temporary local shortages. A futures contract does not do the60 CHAPTER 3
 
same. The benefits from holding the physical asset are sometimes referred to as the convenienceyield provided by the commodity. If the dollar amount of storage costs is known and has a presentvalue U, the convenience yield, y, is defined so that
Foe? = (Sy + U)e™
orFy = Set 3.21)
The convenience yield simply measures the extent to which the left-hand side is less than the right-hand side in equation (3.19) or (3.20). For investment assets the convenience yield must be zero;otherwise, there are arbitrage opportunities. Table 2.2 of Chapter 2 shows that the futures prices ofsome commodities such as Sugar—-World tended to decrease as the time to maturity of the contractincreased on March 15, 2001. This pattern suggests that the convenience yield, y, is greater thanr+u for these commodities.
The convenience yield reflects the market’s expectations concerning the future availability of thecommodity. The greater the possibility that shortages will occur, the higher the convenience yield.If users of the commodity have high inventories, there is very little chance of shortages in the nearfuture and the convenience yield tends to be low. On the other hand, low inventories tend to lead tohigh convenience yields.
3.13 COST OF CARRY
 
 
The relationship between futures prices and spot prices can be summarized in terms of the cost ofcarry. This measures the storage cost plus the interest that is paid to finance the asset less theincome earned on the asset. For a non-dividend-paying stock, the cost of carry is r, because thereare no storage costs and no income is earned; for a stock index, it is r—q, because income isearned at rate g on the asset. For a currency, it is r — rp; for a commodity with storage costs that area proportion u of the price, it is r+ u; and so on.
Define the cost of carry as c. For an investment asset, the futures price is
Fo = Se (3.22)For a consumption asset, it is
x elev0 = de
ons2New=
where y is the convenience yield.
3.14 DELIVERY OPTIONS
 
 
Whereas a forward contract normally specifies that delivery is to take place on a particular day, afutures contract often allows the party with the short position to choose to deliver at any timeduring a certain period. (Typically the party has to give a few days’ notice of its intention todeliver.) The choice introduces a complication into the determination of futures prices. ShouldDetermination of Forward and Futures Prices 61
the maturity of the futures contract be assumed to be the beginning, middle, or end of thedelivery period? Even though most futures contracts are closed out prior to maturity, it isimportant to know when delivery would have taken place in order to calculate the theoreticalfutures price.
If the futures price is an increasing function of the time to maturity, it can be seen from
equation (3,23) that ¢ > y, so that the benefits from holding the asset Gncluding convenience yieldequation (3.25) that ¢ y, SO that the oenefts irom hoiding the asset (inciuding convenience yleid
and net of storage costs) are less than the risk-free rate. It is usually optimal in such a case for theparty with the short position to deliver as early as possible, because the interest earned on the cashreceived outweighs the benefits of holding the asset. As a rule, futures prices in these circumstancesshould be calculated on the basis that delivery will take place at the beginning of the deliveryperiod. If futures prices are decreasing as time to maturity increases (c < y), the reverse is true. It isthen usually optimal for the party with the short position to deliver as late as possible, and futuresprices should, as a rule, be calculated on this assumption.
 
3.15 FUTURES PRICES AND THE EXPECTED FUTURE SPOT PRICE
A question that is often raised is whether the futures price of an asset is equal to its expected futurespot price. If you have to guess what the price of an asset will be in three months, is the futures pricean unbiased estimate? Chapter 2 presented the arguments of Keynes and Hicks. These authorscontend that speculators will not trade a futures contract unless their expected profit is positive. Bycontrast, hedgers are prepared to accept a negative expected profit because of the risk-reductionbenefits they get from a futures contract. If more speculators are long than short, the futures pricewill tend to be less than the expected future spot price. On average, speculators can then expect tomake a gain, because the futures price converges to the spot price at maturity of the contract.Similarly, if more speculators are short than long, the futures price will tend to be greater than theexpected future spot price.
Risk and Return
Another explanation of the relationship between futures prices and expected future spot prices canbe obtained by considering the relationship between risk and expected return in the economy. Ingeneral, the higher the risk of an investment, the higher the expected return demanded by aninvestor. Readers familiar with the capital asset pricing model will know that there are two types ofrisk in the economy: systematic and nonsystematic. Nonsystematic risk should not be important toan investor. It can be almost completely eliminated by holding a well-diversified portfolio. Aninvestor should not therefore require a higher expected return for bearing nonsystematic risk.Systematic risk, by contrast, cannot be diversified away. It arises from a correlation between returnsfrom the investment and returns from the stock market as a whole. An investor generally requires ahigher expected return than the risk-free interest rate for bearing positive amounts of systematicrisk, Also, an investor is prepared to accept a lower expected return than the risk-free interest ratewhen the systematic risk in an investment is negative.
The Risk in a Futures Position
Let us consider a speculator who takes a long futures position in the hope that the spot price of theasset will be above the futures price at maturity. We suppose that the speculator puts the present62 CHAPTER 3
value of the futures price into a risk-free investment while simultaneously taking a long futuresposition. We assume that the futures contract can be treated as a forward contract. The proceeds ofthe risk-free investment are used to buy the asset on the delivery date. The asset is thenimmediately sold for its market price. The cash flows to the speculator are:
Time 0: —Foe”Time 7: +S;
where S; is the price of the asset at time 7’.The present value of this investment is
—Fyew"? + E(Sp)e
where & is the discount rate appropriate for the investment (i.e., the expected return required byinvestors on the investment) and E denotes the expected value. Assuming that all investmentopportunities in securities markets have zero net present value,
—Foe" + E(S;)e*” =0orFy = E(Sp)e?*7 (3.24)
The value of k depends on the systematic risk of the investment. If S; is uncorrelated with the levelof the stock market, the investment has zero systematic risk. In this case k = r, and equation (3.24)shows that Fy = E(S,). If S; is positively correlated with the stock market as a whole, theinvestment has positive systematic risk. In this case k>r, and equation (3.24) shows thatFy < E(Sy). Finally, if S; is negatively correlated with the stock market, the investment hasnegative systematic risk. In this case k < r, and equation (3.24) shows that Fy > E(S;).
Empirical Evidence
If Fy = E(S,), the futures price will drift up or down only if the market changes its views about theexpected firevises its expectations about future spot prices upward as often as it does so downward. It followsthat when Fy = E(S;), the average profit from holding futures contracts over a long period of timeshould be zero. The Fy < E(Sr) situation corresponds to the positive systematic risk situation.Because the futures price and the spot price must be equal at maturity of the futures contract, afutures price should on average drift up, and a trader should over a long period of time makepositive profits from consistently holding long futures positions. Similarly, the Fy > E(S;) situa-tion implies that a trader should over a long period of time make positive profits from consistentlyholding short futures positions.
How do futures prices behave in practice? Some of the empirical work that has been carried outis listed at the end of this chapter. The results are mixed. Houthakker’s study looked at futuresprices for wheat, cotton, and corn from 1937 to 1957. It showed that significant profits could beearned by taking long futures positions. This suggests that an investment in one of these assets haspositive systematic risk and Fy < E(S7). Telser’s study contradicted the findings of Houthakker.His data covered the period from 1926 to 1950 for cotton and from 1927 to 1954 for wheat andgave rise to no significant profits for traders taking either long or short positions. To quote fromTelser: ‘The futures data offer no evidence to contradict the simple... hypothesis that the futuresprice is an unbiased estimate of the expected future spot price.” Gray’s study looked at corn futures
d of time, we can reas
re spot price, Over a long pe
       Determination of Forward and Futures Prices 63
prices during the period 1921 to 1959 and resulted in similar findings to those of Telser. Dusak’sstudy used data on corn, wheat, and soybeans from 1952 to 1967 and took a different approach. Itattempted to estimate the systematic risk of an investment in these commodities by calculating thecorrelation of movements in the commodity prices with movements in the S&P 500. The resultssuggest that there is no systematic risk and lend support to the Fy = E(S;) hypothesis. However,more recent work by Chany using ihe same commodities aud more advanced statistical iechuiquessupports the Fy < E(S;) hypothesis.
SUMMARY
 
For most purposes, the futures price of a contract with a certain delivery date can be considered tobe the same as the forward price for a contract with the same delivery date. It can be shown that intheory the two should be exactly the same when interest rates are perfectly predictable.
For the purposes of understanding futures (or forward) prices, it is convenient to divide futurescontracts into two categories: those in which the underlying asset is held for investment by asignificant number of investors and those in which the underlying asset is held primarily forconsumption purposes.
In the case of investment assets, we have considered three different situations:
1. The asset provides no income.2. The asset provides a known dollar income.3. The asset provides a known yield.
The results are summarized in Table 3.4. They enable futures prices to be obtained for contracts onstock indices, currencies, gold, and silver. Storage costs can be regarded as negative income.
In the case of consumption assets, it is not possible to obtain the futures price as a function ofthe spot price and other observable variables. Here the parameter known as the asset’s convenienceyield becomes important. It measures the extent to which users of the commodity feel thatownership of the physical asset provides benefits that are not obtained by the holders of thefutures contract. These benefits may include the ability to profit from temporary local shortages orthe ability to keep a production process running. We can obtain an upper bound for the futuresprice of consumption assets using arbitrage arguments, but we cannot nail down an equalityrelationship between futures and spot prices.
stinenier
Table 3.4 Summary of results for a contract with time to maturity T one t
cate Tana, cet mith arice S& when the cick-free interest pate far aan inv asset With price og When We TISK-Irce interest rate ior 4
T-year period is r
Forward/futures Value of long forward contractAsset price with delivery price K
 
Provides no income Soe? Sy — Kev?
Provides known incomewith present value / (Sp — De'™ So -1— Kev?
Provides known yield ¢ Spel OT Syewt? — Kew"?64 CHAPTER 3
The concept of cost of carry is sometimes useful. The cost of carry is the storage cost of theunderlying asset plus the cost of financing it minus the income received from it. In the case ofinvestment assets, the futures price is greater than the spot price by an amount reflecting the cost ofcarry. In the case of consumption assets, the futures price is greater than the spot price by anamount reflecting the cost of carry net of the convenience yield.
If we assume the capital asset pricing model is true, the relationship between the futures priceand the expected future spot price depends on whether the return on the asset is positively ornegatively correlated with the return on the stock market. Positive correlation will tend to lead toafutures price lower than the expected future spot price. Negative correlation will tend to lead toafutures price higher than the expected future spot price. Only when the correlation is zero will thetheoretical futures price be equal to the expected future spot price.
SUGGESTIONS FOR FURTHER READING
 
On Empirical Research Concerning Forward and Futures Prices
Cornell, B., and M. Reinganum, “Forward and Futures Prices: Evidence from Foreign ExchangeMarkets,” Journal of Finance, 36 (December 1981), 1035-45.
French, K., “A Comparison of Futures and Forward Prices,” Journal of Financial Economics, 12(November 1983), 311-42.
Park, H. Y., and A. H. Chen, “Differences between Futures and Forward Prices: A FurtherInvestigation of Marking to Market Effects,” Journal of Futures Markets, 5 (February 1985), 77-88.
Reondloman Rand © Carabini. “The Efficiency of the TrNMarcmMman, W., ana L. Laracini, “ine cinciency of & ir
Finance, 34 (September 1979), 895-914.
Viswanath, P. V., ‘Taxes and the Futures-Forward Price Difference in the 91-Day T-Bill Market,”Journal of Money Credit and Banking, 21, no. 2 (May 1989), 190-205.
On Empirical Research Concerning the Relationship Between Futures Prices andExpected Future Spot Prices
Chang, E. C., “Returns to Speculators and the Theory of Normal Backwardation,” Journal of Finance,40 (March 1985), 193-208.
Dusak, K., “Futures Trading and Investor Returns: An Investigation of Commodity Risk Premiums,”Journal of Political Economy, 81 (December 1973), 1387-1406.
Gray, R. W., “The Search for a Risk Premium,” Journal of Political Economy, 69 (June 1961), 250-60.
Houthakker, H. S., “Can Speculators Forecast Prices?” Review of Economics and Statistics, 39 (1957),143-51,
Telser, L. G., “Futures Trading and the Storage of Cotton and Wheat,” Journal of Political Economy,66 (June 1958), 233-55.
On the Theoretical Relationship Between Forward and Futures Prices
Cox, J. C., J. E. Ingersoll, and 8. A. Ross, “The Relation between Forward Prices and Futures Prices,”Journal of Financial Economics, 9 (December 1981), 321-46.Determination of Forward and Futures Prices 65
 
 
 
 
 
 
 
Jarrow, R. A., and G. S. Oldfield, “Forward Contracts and Futures Contracts,” Journal of FinancialEconomics, 9 (December 1981), 373-82.
Kane, E. J., “Market Incompleteness and Divergences between Forward and Futures Interest Rates,”Journal of Finance, 35 (May 1980), 221-34.
Richard, S., and M. Sundaresan, “A Continuous-Time Model of Forward and Futures Prices in aMultigood Economy,” Journal of Financial Economics, 9 (December 1981), 347-72.
3.
u
3.2.3.3.
34.
3.5.
3.6.
weco
3.9.
3.10.
3.12.
. An investor receives $1,100 in one yea:
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
. A bank quotes you an interest rate of 14% per annum with quarterly compounding. What is the
equivalent rate with (a) continuous compounding and (b) annual compounding?
Explain what happens when an investor shorts a certain share.
Suppose that you enter into a six-month forward contract on a non-dividend-paying stock whenthe stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% perannum. What is the forward price?
A stock index currently stands at 350. The risk-free interest rate is 8% per annum (withcontinuous compounding) and the dividend yield on the index is 4% per annum. What shouldthe futures price for a four-month contract be?
Explain carefully why the futures price of gold can be calculated from its spot price and otherobservable variables whereas the futures price of copper cannot.
Explain carefully the meaning of the terms convenience yield and cost of carry. What is therelationship between futures price, spot price, convenience yield, and cost of carry?
. Is the futures price of a stock index greater than or less than the expected future value of the
index? Explain your answer.An investor receives €1.190 in one. tipercentage return per annum with
a. Annual compounding
b. Semiannual compounding
c. Monthly compounding
d. Continuous compounding
What rate of interest with continuous compounding is equivalent to 15% per annum withmonthly compounding?
A deposit account pays 12% per annum with continuous compounding, but interest is actuallypaid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
. A one-year long forward contract on a non-dividend-paying stock is entered into when the stock
price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.a. What are the forward price and the initial value of the forward contract?b. Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. Whatare the forward price and the value of the forward contract?The risk-free rate of interest is 7% per annum with continuous compounding, and the dividendyield on a stock index is 3.2% per annum. The current value of the index is 150. What is the six-month futures price?66
3.13.
3.14.
3.15.
3.16.
3.17.
3.18.
3.19.
3.20.
3.21,
3.22.
3.23.
CHAPTER 3
Assume that the risk-free interest rate is 9% per annum with continuous compounding and thatthe dividend yield on a stock index varies throughout the year. In February, May, August, andNovember, dividends are paid at a rate of 5% per annum. In other months, dividends are paid ata rate of 2% per annum. Suppose that the value of the index on July 31, 2002, is 300. What is thefutures price for a contract deliverable on December 31, 2002?
Suppose that the risk-free interest rate is 10% per annum with continuous compounding and thatthe dividend yield on a stock index is 4% per annum. The index is standing at 400, and the futuresprice for a contract deliverable in four months is 405. What arbitrage opportunities does this create?
Estimate the difference between short-term interest rates in Mexico and the United States onMarch 15, 2001, from the information in Table 3.3.
The two-month interest rates in Switzerland and the United States are 3% and 8% per annum,respectively, with continuous compounding. The spot price of the Swiss franc is $0.6500. Thefutures price for a contract deliverable in two months is $0.6600. What arbitrage opportunitiesdoes this create?
The current price of silver is $9 per ounce. The storage costs are $0.24 per ounce per year payablequarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculatethe futures price of silver for delivery in nine months.
Suppose that F, and F, are two futures contracts on the same commodity with times to maturity,t, and t,, where ft, > t;. Prove thatFy< Fee)
where r is the interest rate (assumed constant) and there are no storage costs. For the purposes ofthis problem, assume that a futures contract is the same as a forward contract.
When a known future cash outflow in a foreign currency