This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CHAPTER 5
Determination of Forward and Futures Prices Problem 5.1.
Explain what happens when an investor shorts a certain share. The investor’s broker borrows the shares from another client’s account and sells them
in the usual way. To close out the position, the investor must purchase the shares. The
broker then replaces them in the account of the client from whom they were borrowed.
The party with the short position must remit to the broker dividends and other income
paid on the shares. The broker transfers these funds to the account of the client from
whom the shares were borrOWed. Occasionally the broker runs out of places from which to
borrow the shares. The investor is then short squeezed and has to close out the position
immediately. Problem 5.2.
What is the diﬁ'erence between the forward price and the value of a forward contract? The forward price of an asset today is the price at which you would agree to buy or
sell the asset at a future time. ri‘he value of a forward contract is zero when you ﬁrst enter
into it. As time passes the underlying asset price changes and the value of the contract
may become positive or negative. Problem 5.3. Suppose that you enter into a six—month forward contract on a nondividendpaying
stock when the stock price is $30 and the riskﬂee interest rate (with continuous com—
pounding) is 12% per annum. What is the forward price? The forward price is
30e012x05 = $31.86 Problem 5.4. A. stock index currently stands at 350. The riskfree interest rate is 8% per annum
(with continuous compounding) and the dividend yield on the index is 4% per annum.
What should the futures price for a fourmonth contract he? The futures price is
3506(0.OB—0.04)X03333 ____ $354.7 Problem 5.5.
Explain carefully why the futures price of gold can be calculated from its Spot price
and other observable variables whereas the futures price of copper cannot. 33 Gold is an investment asset. If the futures price is too high, investors Will ﬁnd it
proﬁtable to increase their holdings of gold and short futures contracts. If the futures
price is too low, they will ﬁnd it proﬁtable to decrease their holdings of gold and go long
in the futures market. Copper is a consumption asset. If the futures price is too high,
a strategy of buy copper and short futures works. However, because investors do not in general hold the asset, the strategy of sell cepper and buy futures is not available to them.
There is therefore an upper bound, but no lower bound, to the futures price. Problem 5.6.
Explain carefully the meaning of the terms convenience yield and cost of carry. What
is the relationship between futures price, spot price, convenience yield, and cost of carry? Convenience yield measures the extent to which there are beneﬁts obtained from
ownership of the physical asset that are not obtained by owners of long futures contracts. The cost of carry is the interest cost plus storage cost less the income earned. The futures
price, F0, and spot price, So, are related by F0 2 304%le where c is the cost of carry, 3; is the convenience yield, and T is the time to maturity of
the futures contract. Problem 5.7.
Explain why a foreign currency can be treated as an asset providing a known yield. A foreign currency provides a known interest rate, but the interest is received in
the foreign currency. The value in the domestic currency of the income provided by the foreign currency is therefore known as a percentage of the value of the foreign currency.
This means that the income has the properties of a known yield. Problem 5.8.
Is the futures price of a stock index greater than or less than the expected future value
of the index? Explain your answer. The futures price of a stock index is always iess than the expected future value of the
index. This follows from Section 5.14 and the fact that the index has positive systematic risk. For an alternative argument, let p. be the expected return required by investors or
the index so that E (ST) = Soe(“"‘J)T. Because ,u. > r and F0 = Soar”T, it follows thai
E(ST) > F0. Probiem 5.9. A oneyear long forward contract on a nondividend—paying stock is entered into whet
the stock price is $40 and the riskfree rate of interest is 10% per annum with continuou.
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 1'.
still 10%. What are the forward price and the value of the forward contract? 34 (a) The forward price, F0, is given by equation (5.1) as:
F0 = 40401x1 = 44.21 or $44.21. The initial value of the forward contract is zero.
(b) The delivery price K in the contract is $44.21. The value of the contract, f, after six
months is given by equation (5.5) as: f = 45 —— 44.21e—0'lxo5 = 2.95
i.e., it is $2.95. The forward price is: 45e01X05 = 47.31 or $47.31. Problem 5.10. The riskfree rate of interest is 7% per annum with continuous compounding, and the
dividend yield on a stock index is 3.2% per annum. The current value of the index is 150.
What is the sixmonth futures price? Using equation (5.3) the six month futures price is 1506(0.07——D.032) X0.5 = 152.88 or $152.88. ~ l‘ Problem 5.11. Assume that the riskfree interest rate is 9% per annum with continuous compounding
and that the dividend yield on a stock index varies throughoutthe year. In February, May,
August, and November, dividends are paid at a rate of 5% per annum. In other months,
dividends are paid at a rate of 2% per annum. Suppose that the value of the index On
July 31 is 1,300. What is the futures price for a contract deliverable on December 31 of
the same year? The futures contract lasts for five months. The dividend yield is 2% for three of the
months and 5% for two of the months. The average dividend yield is therefore 1
E(3x2+2x5)=3.2% The futures price is therefore 13008(0.09—0.D32)X0.4167 ___. 13 331.80 or $1331.80. 35 Problem 5.12. ing and that the dividend yield on a stock index is 4% per annum. The index is standing at 400, and the futures price for a contract deliverable in four months is 405. What arbitrage
opportunities does this create? The theoretical futures price is 4008(0.10—0.04) X4/12 = 408.08 The actual futures price is only 405. This shows that the index futures price is too low
relative to the index. The correct arbitrage strategy is 1. Buy futures contracts 2. Short the shares underlying the index. Problem 5.13. Estimate the diiference between short— term interest rates in Mexico and the United
States on January 8, 2007 from the information in Table 5.4. The settlement prices for the futures contracts are
Jan 0.91250
Mar 0.91025
The March 2007 price is about 0.25% below the January 2007 price. This suggests that
the short—term interest rate in the Mexico exceeded shortterm interest rates in the United
States by about 0.25% per two months or about 1.5% per year. Problem 5.14.
The twomonth interest rates in Switzerland and the United States are 2% and 5%
per annum, respectively, with continuous compounding. The spot price of the Swiss franc is $08000. The futures price for a contract deliverable in two months is 30.8100. What
arbitrage opportunities does this create? ’ The theoretical futures price is
0.8000€(0'05_0'02)X2/12 = 0.8040
The actual futures price is too high. This suggests that an arbitrageur should buy Swiss
francs and short Swiss francs futures. Problem 5.15. The spot price of silver is $9 per ounce. The storage costs are $0.24 per ounce per
year payable quarterly in advance. Assuming that interest rates are 10% per annum for
all maturities, calculate the futures price of silver for delivery in nine months. The present value of the storage costs for nine months are
0.06 + 0.06e—01m25 + 0.06e—0'10xo'5 = 0.176 or $0.176. The futures price is from equation (5.11) given by F0 where
F0 = (0.000 + 0.176)e°1><°75 = 9.89 La, it is $9.89 per ounce. Problem 5.16. Suppose that F1 and F2 are two futures contracts on the same commodity with times
to maturity, t1 and 132, where t2 > t1. Prove that F2 S F1 eT(t2t1) where r is the interest rate (assumed constant) and there are no storage costs. For the
purposes of this problem, assume that a futures contract is the same as a forward contract. If
F2 > Fleﬂh—t‘) an investor could make a riskless proﬁt by 1. Taking a long position in a futures contract which matures at time t1 2. Taking a short position in a futures contract which matures at time t;;
When the ﬁrst futures contract matures, the asset is purchased for F1 using funds borrowed
at rate 1'. It is then held until time t2 at which point it is exchanged for F2 under the
second contract. The costs of the funds borrowed and accumulated interest at time t2 is
Flam—‘1) A positive proﬁt of F2 _ F1 er(tg—t1) is then realized at time t2. This type of arbitrage opportunity cannot exist for long. Hence: F2 ..<.. F18r(t2_t1) Problem 5.17. When a known future cash outﬂow in a foreign currency is hedged by a company
using a forward contract, there is no foreign exchange risk. When it is hedged using
futures contracts, the markingtomarket process does leave the company exposed to some
risk. Explain the nature of this risk. In particular, consider whether the company is better
off using a futures contract or a. forward contract when a. The value of the foreign currency falls rapidly during the life of the contract b. The value of the foreign currency rises rapidly during the life of the contract 0. The value of the foreign currency ﬁrst rises and then falls back to its initial value
d. The value of the foreign currency ﬁrst falls and then rises back to its initial value Assume that the forward price equals the futures price. In total the gain or loss under a futures contract is equal to the gain or loss under the
corresponding forward contract. However the timing of the cash ﬂows is different. When
the time value of money is taken into account a futures contract may prove to be more
valuable or less valuable than a forward contract. Of course the company does not know in
advance which will work out better. The long forward contract provides a perfect hedge.
The long futures contract provides a slightly imperfect hedge. (a) In this case the forward contract would lead to a slightly better outcome. The company
will make a loss on its hedge. If the hedge is with a forward contract the whole of 37 the loss will be realized at the end. If it is with a futures contract the loss Will be
realized day by day throughou the contract. On a present value basis the former is
preferable. ‘ (b) in this case the futures contract would lead to a slightly better outcome. The Company
will make a gain on the hedge. If the hedge is with a forward contract the gain will be (d) In this case the forward contract would lead to a slightly better outcome. This is
because, in the case of the futures contract, the early cash ﬂows would be negative
and the later cash ﬂow would be positive. It is sometimes argued that a forward exchange rate is an unbiased predictor of future
exchange rates. Under what circumstances is this so? From the discussion in Section 5.14 of the text, the forward exchange rate is an
unbiased predictor of the future exchange rate when the exchange rate has no systematic risk. To have no systematic risk the exchange rate must be uncorrelated with the return
on the market. Problem 5.19.
Show that the growth rate in an index futures price equals the excess return of the index. over the riskfree rate. Assume that the riskfree interest rate and the dividend yield
are constant. F0 = SOe£T_q)T
F1 = 518(rq)(Tt1) where So and 31 are the spot price at times zero and t1, 1' is the riskfree rate, and q is
the dividend yield. These equations imply that Deﬁne the excess return of the index over the riskfree rate as :r. The total return is
r + :1: and the return realized in the form of capital gains is r + a: — q. It follows that
31 = Soe(°”+m"9)‘1 and the equation for F1 [F0 reduces to F1 ._=e Fe 37“] which is the required result. Problem 5.20. Show that equation (5.3) is true by considering an investment in the asset combined
with a short position in a futures contract. Assume that all income from the asset is
reinvested in the asset. Use an argument similar to that in footnotes 2 and 4 and explain
in detail What an arbitrageur would do if equation (5.3) did not hold. Suppose we buy N units of the asset and invest the income from the asset in the asset.
The income from the asset causes our holding in the asset to grow at a continuously com—
pounded rate q. By time T our holding has grown to N eqT units of the asset. Analogously
to footnotes 2 and 4 of Chapter 5, we therefore buy N units of the asset at time zero at a
cost of 80 per unit and enter into a forward contract to sell N e‘JT unit for F0 per unit at
time T. This generates the following cash ﬂows:
Time 0: *N80
Time T: N FoeqT
Because there is no uncertainty about these cash ﬂows, the present value of the time T inﬂow must equal the time zero outﬂow when we discount at the riskfree rate. rThis
means that N50 = (NFoeqT)c"'T or
so = soarW This is equation (5.3). If F0 > Soe(""q)T, an arbitrageur should borrow money at rate r and buy N units of
the asset. At the same time the arbitrageur should enter into a forward contract to sell
N eqT units of the asset at time T. As income is received, it is reinvested in the asset. At
time T the loan is repaid and the arbitrageur makes a proﬁt of N (Foe?T — Sne'T) at time
T. If F0 < Soe('"_9)T, an arbitrageur should short N units of the asset investing the
proceeds at rate 1‘. At the same time the arbitrageur should enter into a forward contract to
buy N e9T units of the asset at time T. When income is paid on the asset, the arbitrageur
owes money on the short position. The investor meets this obligation from the cash
proceeds of shorting further units. The result is that the number of units shorted grows at
rate q to N eqT. The cumulative short position is closed out at time T and the arbitrageur makes a proﬁt of N (SoeTT — Foe‘iT). Problem 5.21. Explain careﬁilly what is meant by the expected price of a commodity on a particular
future date. Suppose that the futures price of crude oil declines with the maturity of the
contract at the rate of 2% per year. Assume that speculators tend to be short crude oil
futures and hedgers tended to be long. What does the Keynes and Hicks argument imply
about the expected future price of oil? To understand the meaning of the expected future price of a commodity, suppose that
there are N different possible prices at a particular future time: P1, P2, . . ., PN. Deﬁne 39 qz as the (subjective) probability the price being P.; (with q1 — Q2 + . .. + qN = 1). The
expected future price is Different people may have different expected future prices for the commodity. The
expected future price in the market can be thought of as an average of the opinions of
diiferent market participants. Of course, in practice the actual price of the commodity at
the future time may pr0ve to be higher or lower than the expected price. Keynes and Hicks argue that speculators on average make money from commodity
futures trading and hedgers on average lose money from commodity futures trading. If
speculators tend to have short positions in crude oil futures, the Keynes and Hicks argu
ment implies that futures prices overstate expected future spot prices. If crude oil futures prices decline at 2% per year the Keynes and Hicks argument therefore implies an even
faster decline for the expected price of crude oil. Problem 5.22.
The Value Line Index is designed to reﬂect changes in the value of a portfolio of over 1,600 equally weighted stocks. Prior to March 9, 1988, the change in the index from one
day to the next was calculated as the geometric average of the changes in the prices of the understate the futures price? When the geometric average of the price relatives is used, the changes in the value of
the index do not correspond to changes in the value of a portfolio that is traded. Equation
(5.8) is therefore no longer correct. The changes in the value of the portfolio is monitored by
an index calculated from the arithmetic average of the prices of the stocks in the portfolio.
Since the geometric average of a set of numbers is always less than the arithmetic average,
equation (5.8) overstates the futures price. It is rumored that at one time (prior to 1988),
equation (5.8) did hold for the Value Line index. A major Wall Street ﬁrm was the ﬁrst
to recognize that this represented a trading opportunity. It made a ﬁnancial killing by
buying the stocks underlying the index and shorting the futures. Problem 5.23. A US. company is interested in using the futures contracts traded on the CME to
hedge its Australian dollar exposure. Deﬁne r as the interest rate (all maturities) on the
US. dollar and T'f as the interest rate (all maturities) on the Australian dollar. Assume
that r and T'f are constant and that the conipany uses a contract expiring at time T to
hedge an exposure at time t (T > t). a. Show that the optimal hedge ratio is ere—orr—t) b. Show that, when t is one day, the optimal hedge ratio is almost exactly So/Fo
where 80 is the current spot price of the currency and F0 is the current futures
price of the currency for the contract maturing at time T. 40 c. Show that the company can take account of the daily settlement of futures con
tracts for a hedge that lasts longer than one day by adjusting the hedge ratio so that it
always equals the spot price of the currency divided by the futures price of the currency. (a) The relationship between the futures price Ft and the spot price 3; at time t is
Ft 2 Stew—”“11””
Suppose that the hedge ratio is h. The price obtained with hedging is
MFG — Ft) + S,
where F0 is the initial futures price. This is
are + s. — asterWT”) if h = e(’"f'“"l(T‘t), this reduces to lab}; and a zero variance hedge is obtained. (b) When it is one day, it is approximately em “’")T = So/Fo. The appropriate hedge ratio
is therefore So/Fo. (c) When a futures contract is used for hedging, the price movements in each day should
in theory be hedged separately. This is because the daily settlement means that a
futures contract is closed out and rewritten at the end of each day. From (b) the
correct hedge ratio at any given time is, therefore, 3/ F where S is the spot price and
F is the futures price. Suppose there is an exposure to N units of the foreign currency
and M units of the foreign currency underlie one futures contract. With a hedge ratio
of 1 we should trade N /M contracts. With a hedge ratio of S/ F we should trade ﬂ
FM contracts. In other words we should calculate the number of contracts that should be
traded as the dollar value of our exposure divided by the dollar value of one futures
contract (This is not the same as the dollar value of our exposure divided by the dollar
value of the assets underlying one futures contract.) Since a. futures contract is settled
daily, we should in theory rebalance our hedge daily so that the outstanding number
of futures contracts is always (SN) / (FM). This is known as tailing the hedge. (See
Section 3.4 of the text.) 41 ...
View
Full Document
 Spring '08
 Matos

Click to edit the document details