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Assignment 9

1. Do Chapter 15 Problem 15 in BKM [Hint: Treat di§erent time periods as different states] P489

2. Do Chapter 15 Problem 18 in BKM P490

3. Do Chapter 22 Problem 2 in BKM P771
4. Do Chapter 22 Problem 4 in BKM P771

5. Do Chapter 22 Problem 13 in BKM [Assume the investor is on long side of the futures contract] P772

6. Inflation-indexed and nominal perpetuities (consol bonds), whose coupon payments next period are each $100 are selling for $3,333 and $2,000 respectively.

a) What are the yields of the two bonds?

b) Assume that the two bonds have the same expected returns, so that there is no risk premium for holding the nominal perpetuity. Assume that these expected returns are constant over time and also assume that the expected ináation rate is constant over time. What is the expected ináation rate?

!”# P A R T V I Options, Futures, and Other Derivatives

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The open interest on the contract is the number of contracts outstanding. (Long and
short positions are not counted separately, meaning that open interest can be defined either
as the number of long or short contracts outstanding. The clearinghouse’s position nets out
to zero, and so is not counted in the computation of open interest.) When contracts begin
trading, open interest is zero. As time passes, open interest increases as progressively more
contracts are entered.

There are many apocryphal stories about futures traders who wake up to discover a
small mountain of wheat or corn on their front lawn. But the truth is that futures contracts
rarely result in actual delivery of the underlying asset. Traders establish long or short
positions in contracts that will benefit from a rise or fall in the futures price and almost
always close out, or reverse, those positions before the contract expires. The fraction of
contracts that result in actual delivery is estimated to range from less than 1% to 3%,
depending on the commodity and activity in the contract. In the unusual case of actual
deliveries of commodities, they occur via regular channels of supply, most often ware-
house receipts.

You can see the typical pattern of open interest in Figure 22.1. In the gold contract,
for example, the July contract is approaching maturity, and open interest is small; most
contracts have been reversed already. The greatest open interest is in the August contract.
For other contracts, for example, crude oil, for which the nearest maturity date isn’t until
August, open interest is highest in the nearest contract.

The Margin Account and Marking to Market
The total profit or loss realized by the long trader who buys a contract at time 0 and closes,
or reverses, it at time t is just the change in the futures price over the period, Ft ! F0. Sym-
metrically, the short trader earns F0 ! Ft.

The process by which profits or losses accrue to traders is called marking to market. At
initial execution of a trade, each trader establishes a margin account. The margin is a secu-
rity account consisting of cash or near-cash securities, such as Treasury bills, that ensures
the trader is able to satisfy the obligations of the futures contract. Because both parties to a
contract are exposed to losses, both must post margin. To illustrate, return to the first corn
contract listed in Figure 22.1. If the initial required margin on corn, for example, is 10%,
then each trader must put up $1,720 per contract. This is 10% of the value of the contract,
$3.44 per bushel ” 5,000 bushels per contract.

Because the initial margin may be satisfied by posting interest-earning securities, the
requirement does not impose a significant opportunity cost of funds on the trader. The ini-
tial margin is usually set between 5% and 15% of the total value of the contract. Contracts
written on assets with more volatile prices require higher margins.

On any day that futures contracts trade, futures prices may rise or may fall. Instead of
waiting until the maturity date for traders to realize all gains and losses, the clearinghouse
requires all positions to recognize profits as they accrue daily. If the futures price of corn
rises from 344 to 346 cents per bushel, the clearinghouse credits the margin account of the
long position for 5,000 bushels times 2 cents per bushel, or $100 per contract. Conversely,
the clearinghouse takes this amount from the margin account of the short position for each
contract held.

This daily settling is called marking to market. It means the maturity date of the con-
tract does not govern realization of profit or loss. Instead, as futures prices change, the pro-
ceeds accrue to the trader’s margin account immediately. We will provide a more detailed
example of this process shortly.

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Assume the !-day maturity futures price for silver is currently $”#.$# per ounce. Suppose
that over the next ! days, the futures price evolves as follows:

The daily mark-to-market settlements for each contract held by the long position will be
as follows:

The profit on Day $ is the increase in the futures price from the previous day, or ($”#.”# !
$”#.$#) per ounce. Because each silver contract on the Commodity Exchange (CMX) calls
for purchase and delivery of !,### ounces, the total profit per contract is !,### times $.$#,
or $!##. On Day %, when the futures price falls, the long position’s margin account will be
debited by $%!#. By Day !, the sum of all daily proceeds is $!!#. This is exactly equal to
!,### times the difference between the final futures price of $”#.”$ and the original futures
price of $”#.$#. Because the final futures price equals the spot price on that date, the sum of
all the daily proceeds (per ounce of silver held long) also equals PT ! F#.

Example 22.2 Marking to Market

Day Profit (Loss) per Ounce ! !,””” Ounces/Contract = Daily Proceeds

$ “#.”# ! “#.$# = &&&#.$# $!##
” “#.”! ! “#.”# = & &#.#! &”!#
% “#.$’ ! “#.”! = !#.#( &!%!#
) “#.$’ ! “#.$’ = & &# & && &#
! “#.”$ ! “#.$’ = & &#.#% & $!#
& & Sum = $!!#

Day Futures Price

# (today) $”#.$#&
$ &”#.”#
” &”#.”!
% &”#.$’
) &”#.$’
! (maturity) &”#.”$

Cash versus Actual Delivery
Most futures contracts call for delivery of an actual commodity such as a particular grade
of wheat or a specified amount of foreign currency if the contract is not reversed before
maturity. For agricultural commodities, where quality of the delivered good may vary, the
exchange sets quality standards as part of the futures contract. In some cases, contracts
may be settled with higher- or lower-grade commodities. In these cases, a premium or dis-
count is applied to the delivered commodity to adjust for the quality difference.

Some futures contracts call for cash settlement. An example is a stock index futures
contract where the underlying asset is an index such as the Standard & Poor’s 500. Deliv-
ery of every stock in the index clearly would be impractical. Hence the contract calls for
“delivery” of a cash amount equal to the value that the index attains on the maturity date of
the contract. The sum of all the daily settlements from marking to market results in the long
position realizing total profits or losses of ST ! F0, where ST is the value of the stock index
on the maturity date T and F0 is the original futures price. Cash settlement closely mimics
actual delivery, except the cash value of the asset rather than the asset itself is delivered.

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More concretely, the widely traded E-mini S&P 500 index contract calls for delivery of
$50 times the value of the index.2 Suppose that at maturity, the S&P 500 is at a level of
2,000. Instead of delivering shares of all 500 stocks included in the index, cash settlement
would require the short trader to deliver $50 ! 2,000, or $100,000. This provides the trader
exactly the same profit as would result from directly purchasing 50 units of the index for
$100,000 and then delivering it for $50 times the original futures price.

Regulations
Futures markets are regulated by the federal Commodities Futures Trading Commission.
The CFTC sets capital requirements for member firms of the futures exchanges, authorizes
trading in new contracts, and oversees maintenance of daily trading records.

The futures exchange may set limits on the amount by which futures prices may change
from one day to the next. For example, if the price limit on silver contracts were set at $1
and silver futures close today at $22.10 per ounce, then trades in silver tomorrow may
vary only between $21.10 and $23.10 per ounce. The exchanges may increase or reduce
price limits in response to perceived changes in the price volatility of the underlying asset.
Price limits are often eliminated as contracts approach maturity, usually in the last month
of trading.

Price limits traditionally are viewed as a means to limit violent price fluctuations. This
reasoning seems dubious. Suppose an international monetary crisis overnight drives up the
spot price of silver to $30. No one would sell silver futures at prices for future delivery as
low as $22.10. Instead, the futures price would rise each day by the $1 limit, although the
quoted price would represent only an unfilled bid order—no contracts would trade at the
low quoted price. After several days of limit moves of $1 per day, the futures price would
finally reach its equilibrium level, and trading would occur again. This process means no
one could unload a position until the price reached its equilibrium level. We conclude that
price limits offer no real protection against fluctuations in equilibrium prices.

Taxation
Because of the mark-to-market procedure, investors do not have control over the tax year
in which they realize gains or losses. Instead, price changes are realized gradually, with
each daily settlement. Therefore, taxes are paid at year-end on cumulated profits or losses,
regardless of whether the position has been closed out. As a general rule, 60% of futures
gains or losses are treated as long term, and 40% are treated as short term.

2The original S&P 500 contract had a multiplier of $500, later reduced to $250. However, the all-electronically
traded version of the contract, usually referred to as the E-mini, has a multiplier of $50. The great majority of
futures trading on the S&P 500 is now conducted on the E-mini rather than the original “big” contract.

22.3 Futures Markets Strategies
Hedging and Speculation
Hedging and speculating are two polar uses of futures markets. A speculator uses a futures
contract to profit from movements in futures prices, a hedger to protect against price
movement.

If speculators believe prices will increase, they will take a long position for expected
profits. Conversely, they exploit expected price declines by taking a short position.

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Suppose you believe that crude oil prices are going to increase and therefore decide to pur-
chase crude oil futures. Each contract calls for delivery of !,””” barrels of oil, so for every $!
increase in the futures price of crude, the long position gains $!,””” and the short position
loses that amount.

Conversely, suppose you think that prices are heading lower and therefore sell a con-
tract. If crude oil prices do in fact fall, then you will gain $!,””” per contract for every $! that
prices decline.

If the futures price for delivery in February is $#$ and crude oil is selling at the contract
maturity date for $#%, the long side will profit by $!,””” per contract purchased. The short
side will lose an identical amount on each contract sold. On the other hand, if oil has fallen to
$#!, the long side will lose, and the short side will gain, $!,””” per contract.

Example 22.3 Speculating with Oil Futures

Suppose the initial margin requirement for the oil contract is !”%. At a current futures price
of $#$, and contract size of !,””” barrels, this would require margin of .!” ! #$ ! !,””” =
$#,$””. A $! increase in oil prices represents an increase of !.&$%, and results in a $!,”””
gain on the contract for the long position. This is a percentage gain of !&.$% on the $#,$””
posted as margin, precisely !” times the percentage increase in the oil price. The !”-to-!
ratio of percentage changes reflects the leverage inherent in the futures position, because the
contract was established with an initial margin of one-tenth the value of the underlying asset.

Example 22.4 Futures and Leverage

Consider an oil distributor planning to sell !””,””” barrels of oil in February that wishes to
hedge against a possible decline in oil prices. Because each contract calls for delivery of
!,””” barrels, it would sell !”” contracts that mature in February. Any decrease in prices
would then generate a profit on the contracts that would offset the lower sales revenue from
the oil.

To illustrate, suppose that the only three possible prices for oil in February are $#!, $#$,
and $#% per barrel. The revenue from the oil sale will be !””,””” times the price per barrel.

Example 22.5 Hedging with Oil Futures

Why does a speculator buy a futures contract? Why not buy the underlying asset
directly? One reason lies in transaction costs, which are far smaller in futures markets.

Another important reason is the leverage that futures trading provides. Recall that
futures contracts require traders to post margin considerably less than the value of the asset
underlying the contract. Therefore, they allow speculators to achieve much greater leverage
than is available from direct trading in a commodity.

Hedgers, by contrast, use futures to insulate themselves against price movements. A
firm planning to sell oil, for example, might anticipate a period of market volatility and
wish to protect its revenue against price fluctuations. To hedge the total revenue derived
from the sale, the firm enters a short position in oil futures. As the following example illus-
trates, this locks in its total proceeds (i.e., revenue from the sale of the oil plus proceeds
from its futures position).

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Some speculators try to profit from movements in the basis. Rather than betting on
the direction of the futures or spot prices per se, they bet on the changes in the difference
between the two. A long spot–short futures position will profit when the basis narrows.

Consider an investor holding !”” ounces of gold, who is short one gold-futures contract.
Suppose that gold today sells for $!,#$! an ounce, and the futures price for June delivery
is $!,#$% an ounce. Therefore, the basis is currently $&. Tomorrow, the spot price might
increase to $!,#$&, while the futures price increases to $!,#$$, so the basis narrows to $’.

The investor’s gains and losses are as follows:

Gain on holdings of gold (per ounce): $!,#$& ! $!,#$! = $’
Loss on gold futures position (per ounce): $!,#$$ ! $!,#$% = $#

The net gain is the decrease in the basis, or $! per ounce.

Example 22.6 Speculating on the Basis

Consider an investor who holds a September maturity contract long and a June contract
short. If the September futures price increases by & cents while the June futures price
increases by ‘ cents, the net gain will be & cents ! ‘ cents, or ! cent. Like basis strategies,
spread positions aim to exploit movements in relative price structures rather than profit from
movements in the general level of prices.

Example 22.7 Speculating on the Spread

A related strategy is a calendar spread position, where the investor takes a long posi-
tion in a futures contract of one maturity and a short position in a contract on the same
commodity, but with a different maturity.4 Profits accrue if the difference in futures prices
between the two contracts changes in the hoped-for direction, that is, if the futures price on
the contract held long increases by more (or decreases by less) than the futures price on the
contract held short.

4Yet another strategy is an intercommodity spread, in which the investor buys a contract on one commodity and
sells a contract on a different commodity.

22.4 Futures Prices
The Spot-Futures Parity Theorem
We have seen that a futures contract can be used to hedge changes in the value of the
underlying asset. If the hedge is perfect, meaning that the asset-plus-futures portfolio has
no risk, then the hedged position must provide a rate of return equal to the rate on other
risk-free investments. Otherwise, there will be arbitrage opportunities that investors will
exploit until prices are brought back into line. This insight can be used to derive the theo-
retical relationship between a futures price and the price of its underlying asset.

Suppose for simplicity that a stock market index such as the S&P 500 currently is
at 1,000 and an investor who holds $1,000 in an indexed mutual fund wishes to tempo-
rarily hedge her exposure to market risk. Assume that the indexed portfolio pays year-
end dividends of $20. Finally, assume that the futures price for year-end delivery of the

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contract is 1,010.5 Let’s examine the end-of-year proceeds for various values of the stock
index if the investor hedges her portfolio by entering the short side of the futures contract.

5Actually, the E-mini futures contract on the S&P 500 calls for delivery of $50 times the value of the index,
so that each contract would be settled for $50 times the index. With the index at the assumed value of 1,000,
each contract would hedge about $50 ! 1,000 = $50,000 worth of stock. Of course, institutional investors would
consider a stock portfolio of this size to be quite small. We will simplify by assuming that you can buy a contract
for one unit rather than 50 units of the index.

Final value of stock portfolio, ST $ ! “#$ $ ! “”$ $%,$%$ $%,$&$ $%,$’$ $%,$#$
Payoff from short futures position
(equals F$ ” FT = $%,$%$ ” ST)

! ! ! !($ ! ! ! !)$ ! ! ! ! !$ ! ! “)$ ! ! “($ ! ! “*$

Dividend income ! !++ ! !)$ ! ! !++ !)$ ! !++ ! )$ ! !++ ! !)$ ! ++! ! !)$ ! ++! ! !)$
Total $%,$&$ $%,$&$ $%,$&$ $%,$&$ $%,$&$ $%,$&$

The payoff from the short futures position equals the difference between the original
futures price, $1,010, and the year-end stock price. This is because of convergence: The
futures price at contract maturity will equal the stock price at that time.

Notice that the overall position is perfectly hedged. Any increase in the value of the
indexed stock portfolio is offset by an equal decrease in the payoff of the short futures posi-
tion, resulting in a final value independent of the stock price. The $1,030 total payoff is
the sum of the current futures price, F0 = $1,010, and the $20 dividend. It is as though the
investor arranged to sell the stock at year-end for the current futures price, thereby elimi-
nating price risk and locking in total proceeds equal to the futures price plus dividends paid
before the sale.

What rate of return is earned on this riskless position? The stock investment requires an
initial outlay of $1,000, whereas the futures position is established without an initial cash
outflow. Therefore, the $1,000 portfolio grows to a year-end value of $1,030, providing a
rate of return of 3%. More generally, a total investment of S0, the current stock price, grows
to a final value of F0 + D, where D is the dividend payout on the portfolio. The rate of
return is therefore

Rate#of#return#on#hedged#stock#portfolio = ( F 0 + D) ” S 0 __________
S 0

This return is essentially riskless. We observe F0 at the beginning of the period when
we enter the futures contract. While dividend payouts are not perfectly riskless, they are
highly predictable over short periods, especially for diversified portfolios. Any uncertainty
is extremely small compared to the uncertainty in stock prices.

Presumably, 3% must be the rate of return available on other riskless investments. If not,
then investors would face two competing risk-free strategies with different rates of return,
a situation that could not last. Therefore, we conclude that

( F 0 + D) ” S 0 __________
S 0

= r f

Rearranging, we find that the futures price must be
F 0 = S 0 (1 + r f ) ” D = S 0 (1 + r f ” d ) (22.1)
where d is the dividend yield on the indexed stock portfolio, defined as D/S0. This result
is called the spot-futures parity theorem. It gives the normal or theoretically correct
relationship between spot and futures prices. Any deviation from parity would give rise to
risk-free arbitrage opportunities.

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on the arbitrage opportunity. Ultimately prices would change until the year-end cash flow
is reduced to zero, at which point F0 would equal S0(1 + rf ) ! D.

The parity relationship also is called the cost-of-carry relationship because it asserts
that the futures price is determined by the relative costs of buying a stock with deferred
delivery in the futures market versus buying it in the spot market with immediate delivery
and “carrying” it in inventory. If you buy stock now, you tie up your funds and incur a
time-value-of-money cost of rf per period. On the other hand, you receive dividend pay-
ments with a current yield of d. The net carrying cost advantage of deferring delivery of
the stock is therefore rf ! d per period. This advantage must be offset by a differential
between the futures price and the spot price. The price differential just offsets the cost-of-
carry advantage when F0 = S0(1 + rf ! d).

The parity relationship is easily generalized to multiperiod applications. We simply rec-
ognize that the difference between the futures and spot price will be larger as the maturity
of the contract is longer. This reflects the longer period to which we apply the net cost of
carry. For contract maturity of T periods, the parity relationship is
F 0 = S 0 (1 + r f ! d ) T (22.2)
Notice that when the dividend yield is less than the risk-free rate, Equation 22.2 implies that
futures prices will exceed spot prices, and by greater amounts for longer times to contract
maturity. But when d > rf , as is the case today, the income yield on the stock exceeds the for-
gone (risk-free) interest that could be earned on the money invested; in this event, the futures
price will be less than the current stock price, again by greater amounts for longer maturities.
You can confirm that this is so by examining the S&P 500 contract listings in Figure 22.1.

Although dividends of individual securities may fluctuate unpredictably, the annual-
ized dividend yield of a broad-based index such as the S&P 500 is fairly stable, recently
in the neighborhood of a bit more than 2% per year. The yield is seasonal, however, with
regular peaks and troughs, so the dividend yield for the relevant months must be the one
used. Figure 22.5 illustrates the yield pattern for the S&P 500. Some months, such as
January or April, have consistently low yields, while others, such as May, have consis-
tently high ones.6

We have described parity in terms of stocks and stock index futures, but it should be
clear that the logic applies as well to any financial futures contract. For gold futures, for
example, we would simply set the dividend yield to zero. For bond contracts, we would let
the coupon income on the bond play the role of dividend payments. In both cases, the par-
ity relationship would be essentially the same as Equation 22.2.

The arbitrage strategy described above should convince you that these parity relation-
ships are more than just theoretical results. Any violations of the parity relationship give
rise to arbitrage opportunities that can provide large profits to traders. We will see in
Chapter 23 that index arbitrage in the stock market is a tool to exploit violations of the
parity relationship for stock index futures contracts.

Spreads
Just as we can predict the relationship between spot and futures prices, there are similar
ways to determine the proper relationships among futures prices for contracts of different
maturity dates. Equation 22.2 shows that the futures price is in part determined by time
to maturity. If the risk-free rate is greater than the dividend yield (i.e., rf > d), then the

6The high value for the dividend yield in 2009 reflects the financial crisis. You learned in your corporate finance
class that firms are reluctant to reduce dividends. When the economy entered the crisis and stock prices fell dra-
matically, dividend payouts did not fall as precipitously. Therefore, the ratio of dividends to stock prices increased.

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level below the expected spot price and will rise over the life of the contract until the matu-
rity date, at which point FT = PT.

Although this theory recognizes the important role of risk premiums in futures markets,
it is based on total variability rather than on systematic risk. (This is not surprising, as
Keynes wrote almost 40 years before the development of modern portfolio theory.) The
modern view refines the measure of risk used to determine appropriate risk premiums.

Contango
The polar hypothesis to backwardation holds that the natural hedgers are the purchasers of
a commodity, rather than the suppliers. In the case of wheat, for example, we would view
grain processors as willing to pay a premium to lock in the price that they must pay for
wheat. These processors hedge by taking a long position in the futures market; therefore,
they are called long hedgers, whereas farmers are short hedgers. Because long hedgers
will agree to pay high futures prices to shed risk, and because speculators must be paid a
premium to enter the short position, the contango theory holds that F0 must exceed E(PT).

It is clear that any commodity will have both natural long hedgers and short hedgers.
The compromise traditional view, called the “net hedging hypothesis,” is that F0 will be
less than E(PT) when short hedgers outnumber long hedgers and vice versa. The strong
side of the market will be the side (short or long) that has more natural hedgers. The strong
side must pay a premium to induce speculators to enter into enough contracts to balance
the “natural” supply of long and short hedgers.

Modern Portfolio Theory
The three traditional hypotheses all envision a mass of speculators willing to enter either
side of the futures market if they are sufficiently compensated for the risk they incur. Mod-
ern portfolio theory fine-tunes this approach by refining the notion of risk used in the
determination of risk premiums. Simply put, if commodity prices pose positive systematic
risk, futures prices must be lower than expected spot prices.

To illustrate this approach, consider once again a stock paying no dividends. If E(PT)
denotes the expected time-T stock price and k denotes the required rate of return on the
stock, then the price of the stock today must equal the present value of its expected future
payoff as follows:

P 0 =
E( P T ) ______

(1 + k) T
(22.4)

We also know from the spot-futures parity relationship that

P 0 =
F 0 _______

(1 + r f ) T
(22.5)

Therefore, the right-hand sides of Equations 22.4 and 22.5 must be equal. Equating these
terms allows us to solve for F0!in terms of the expected spot price:

F 0 = E( P T ) ( 1 + r f _____ 1 + k )
T

(22.6)

You can see immediately from Equation 22.6 that F0 will be less than the expectation of PT
whenever k is greater than rf , which will be the case for any positive-beta asset. This means
that the long side of the contract will make an expected profit [F0 will be lower than E(PT)]
when the commodity exhibits positive systematic risk (k is greater than rf ).

Why should this be? A long futures position will provide a profit (or loss) of PT ” F0.
If the ultimate value of PT entails positive systematic risk, so will the profit to the long

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forward contract
futures price
long position
short position
single-stock futures
clearinghouse

KEY TERMSopen interest
marking to market
maintenance margin
convergence property
cash settlement
basis

basis risk
calendar spread
spot-futures parity theorem
cost-of-carry relationship

Spot-futures parity:! F 0 ( T ) = S 0 !(1 + r ” d ) T
Futures spread parity:! F 0 ( T 2 ) = F 0 ( T 1 )!(1 + r ” d ) ( T 2 ” T 1 )

Futures vs. expected spot prices: F 0 = E( P T ) ( 1 + r f _____ 1 + k )
T

KEY EQUATIONS

1. Why is there no futures market in cement?
2. Why might individuals purchase futures contracts rather than the underlying asset?
3. What is the difference in cash flow between short-selling an asset and entering a short futures

position?
4. Are the following statements true or false? Why?

a. All else equal, the futures price on a stock index with a high dividend yield should be higher
than the futures price on an index with a low dividend yield.

b. All else equal, the futures price on a high-beta stock should be higher than the futures price
on a low-beta stock.

c. The beta of a short position in the S&P 500 futures contract is negative.
5. What is the difference between the futures price and the value of the futures contract?
6. Evaluate the criticism that futures markets siphon off capital from more productive uses.
7. a. Turn to the Mini-S&P 500 contract in Figure 22.1. If the margin requirement is 10% of the

futures price times the contract multiplier of $50, how much must you deposit with your
broker to trade the September maturity contract?

b. If the September futures price were to increase to 2,090, what percentage return would you earn
on your net investment if you entered the long side of the contract at the price shown in the figure?

c. If the September futures price falls by 1%, what is your percentage return?
8. a. A single-stock futures contract on a non-dividend-paying stock with current price $150 has

a maturity of 1 year. If the T-bill rate is 3%, what should the futures price be?
b. What should the futures price be if the maturity of the contract is 3 years?
c. What if the interest rate is 6% and the maturity of the contract is 3 years?

9. Determine how a portfolio manager might use financial futures to hedge risk in each of the fol-
lowing circumstances:
a. You own a large position in a relatively illiquid bond that you want to sell.
b. You have a large gain on one of your Treasuries and want to sell it, but you would like to

defer the gain until the next tax year.
c. You will receive your annual bonus next month that you hope to invest in long-term corpo-

rate bonds. You believe that bonds today are selling at quite attractive yields, and you are
concerned that bond prices will rise over the next few weeks.

10. Suppose the value of the S&P 500 stock index is currently 2,000.
a. If the 1-year T-bill rate is 3% and the expected dividend yield on the S&P 500 is 2%, what

should the 1-year maturity futures price be?
b. What if the T-bill rate is less than the dividend yield, for example, 1%?

11. Consider a stock that pays no dividends on which a futures contract, a call option, and a put
option trade. The maturity date for all three contracts is T, the exercise price of both the put and
the call is X, and the futures price is F. Show that if X = F, then the call price equals the put
price. Use parity conditions to guide your demonstration.

12. It is now January. The current interest rate is 2%. The June futures price for gold is $1,500,
whereas the December futures price is $1,510. Is there an arbitrage opportunity here? If so, how
would you exploit it?

PROBLEM SETS

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!!” P A R T V I Options, Futures, and Other Derivatives

bod77178_ch22_747-774.indd 772 04/08/17 06:01 PM

13. OneChicago has just introduced a single-stock futures contract on Brandex stock, a company
that currently pays no dividends. Each contract calls for delivery of 1,000 shares of stock in
1 year. The T-bill rate is 6% per year.
a. If Brandex stock now sells at $120 per share, what should the futures price be?
b. If the Brandex price drops by 3%, what will be the change in the futures price and the change

in the investor’s margin account?
c. If the margin on the contract is $12,000, what is the percentage return on the investor’s position?

14. The multiplier for a futures contract on a stock market index is $50. The maturity of the contract is
1 year, the current level of the index is 1,800, and the risk-free interest rate is .5% per month. The
dividend yield on the index is .2% per month. Suppose that after 1 month, the stock index is at 1,820.
a. Find the cash flow from the mark-to-market proceeds on the contract. Assume that the parity

condition always holds exactly.
b. Find the holding-period return if the initial margin on the contract is $5,000.

15. You are a corporate treasurer who will purchase $1 million of bonds for the sinking fund in
3 months. You believe rates will soon fall, and you would like to repurchase the company’s
sinking fund bonds (which currently are selling below par) in advance of requirements. Unfor-
tunately, you must obtain approval from the board of directors for such a purchase, and this can
take up to 2 months. What action can you take in the futures market to hedge any adverse move-
ments in bond yields and prices until you can actually buy the bonds? Will you be long or short?
Why? A qualitative answer is fine.

16. The S&P portfolio pays a dividend yield of 1% annually. Its current value is 2,000. The T-bill
rate is 4%. Suppose the S&P futures price for delivery in 1 year is 2,050. Construct an arbitrage
strategy to exploit the mispricing and show that your profits 1 year hence will equal the mispric-
ing in the futures market.

17. The Excel Application box in the chapter (available in Connect; link to Chapter 22 material)
shows how to use the spot-futures parity relationship to find a “term structure of futures prices,”
that is, futures prices for various maturity dates.
a. Suppose that today is January 1, 2016. Assume the interest rate is 3% per year and a stock

index currently at 2,000 pays a dividend yield of 2.0%. Find the futures price for contract
maturity dates of: (i) February 14, 2016, (ii) May 21, 2016, and (iii) November 18, 2016.

b. What happens to the term structure of futures prices if the dividend yield is higher than the
risk-free rate? For example, what if the dividend yield is 4%?

18. a. How should the parity condition (Equation 22.2) for stocks be modified for futures contracts
on Treasury bonds? What should play the role of the dividend yield in that equation?

b. In an environment with an upward-sloping yield curve, should T-bond futures prices on
more-distant contracts be higher or lower than those on near-term contracts?

c. Confirm your intuition by examining Figure 22.1.
19. Consider this arbitrage strategy to derive the parity relationship for spreads: (1) enter a long

futures position with maturity date T1 and futures price F(T1); (2) enter a short position with
maturity T2 and futures price F(T2); (3) at T1, when the first contract expires, buy the asset and
borrow F(T1) dollars at rate rf ; (4) pay back the loan with interest at time T2.
a. What are the total cash flows to this strategy at times 0, T1, and T2?
b. Why must profits at time T2 be zero if no arbitrage opportunities are present?
c. What must the relationship between F(T1) and F(T2) be for the profits at T2 to be equal to

zero? This relationship is the parity relationship for spreads.

e X c e l
Please visit us at
www.mhhe.com/Bodie11e

Spot price for commodity $!”#
Futures price for commodity expiring in ! year $!”$
Interest rate for ! year %%%%%%%&%

1. Joan Tam, CFA, believes she has identified an arbitrage opportunity for a commodity as indicated
by the following information:

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772 PART VI Options, Futures, and Other Derivatives

bod77178_ch22_747-774.indd 772 04/08/17 06:01 PM

13. OneChicago has just introduced a single-stock futures contract on Brandex stock, a company

that currently pays no dividends. Each contract calls for delivery of 1,000 shares of stock in

1 year. The T-bill rate is 6% per year.

a. If Brandex stock now sells at $120 per shar e, what should the futures price be?

b. If the Brandex price drops by 3%, what will be the change in the futures price and the change

in the investor’s margin account?

c. If the margin on the contract is $12,000, what is the percentage return on the investor’s position?

14. The multiplier for a futures contract on a stock market index is $50. The maturity of the contract is

1 year, the current level of the index is 1,800, and the risk-free interest rate is .5% per month. The

dividend yield on the index is .2% per month. Suppose that after 1 month, the stock index is at 1,820.

a. Find the cash flow from the mark-to-market proceeds on the contract. Assume that the parity

condition always holds exactly.

b. Find the holding-period return if the initial margin on the contract is $5,000.

15. You are a corporate treasurer who will purchase $1 million of bonds for the sinking fund in

3 months. You believe rates will soon fall, and you would like to repurchase the company’s

sinking fund bonds (which currently are selling below par) in advance of requirements. Unfor-

tunately, you must obtain approval from the board of directors for such a purchase, and this can

take up to 2 months. What action can you take in the futures market to hedge any adverse move-

ments in bond yields and prices until you can actually buy the bonds? Will you be long or short?

Why? A qualitative answer is fine.

16. The S&P portfolio pays a dividend yield of 1% annually. Its current value is 2,000. The T-bill

rate is 4%. Suppose the S&P futures price for delivery in 1 year is 2,050. Construct an arbitrage

strategy to exploit the mispricing and show that your profits 1 year hence will equal the mispric-

ing in the futures market.

17. The Excel Application box in the chapter (available in Connect; link to Chapter 22 material)

shows how to use the spot-futures parity relationship to find a “term structure of futures prices,”

that is, futures prices for various maturity dates.

a. Suppose that today is January 1, 2016. Assume the interest rate is 3% per year and a stock

index currently at 2,000 pays a dividend yield of 2.0%. Find the futures price for contract

maturity dates of: (i) February 14, 2016, (ii) May 21, 2016, and (iii) N ovember 18, 2016.

b. What happens to the term structure of futures prices if the dividend yield is higher than the

risk-free rate? For example, what if the dividend yield is 4%?

18. a. How should the parity condition (Equation 22.2) for stocks be modified for futures contracts

on Treasury bonds? What should pla y the role of the dividend yield in t hat equation?

b. In an environment with an upward-sloping yield curve, should T-bond futures prices on

more-distant contracts be higher or lower than those on near-term contracts?

c. Confirm your intuition by examining Figure 22.1.

19. Consider this arbitrage strategy to derive the parity relationship for spreads: (1) enter a long

futures position with maturity date T

1

and futures price F(T

1

); (2) enter a short position with

maturity T

2

and futures price F(T

2

); (3) at T

1

, when the first contract expires, buy the asset and

borrow F(T

1

) dollars at rate r

f

; (4) pay back the loan with interest at time T

2

.

a. What are the total cash flows to this strategy at times 0, T

1

, and T

2

?

b. Why must profits at time T

2

be zero if no arbitrage opportunities are present?

c. What must the relationship between F(T

1

) and F(T

2

) be for the profits at T

2

to be equal to

zero? This relationship is the parity relationship for spreads.

e

X

cel

Please visit us at

www.mhhe.com/Bodie11e

Spot price for commodity $120

Futures price for commodity expiring in 1 year$125

Interest rate for 1 year        8%

1. Joan Tam, CFA, believes she has identified an arbitrage opportunity for a commodity as indicated

by the following information:

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C H A P T E R !! Futures Markets “”#

bod77178_ch22_747-774.indd 773 04/08/17 06:01 PM

a. Describe the transactions necessary to take advantage of this specific arbitrage opportunity.
b. Calculate the arbitrage profit.

2. Michelle Industries issued a Swiss franc–denominated 5-year discount note for SFr200 million.
The proceeds were converted to U.S. dollars to purchase capital equipment in the United States.
The company wants to hedge this currency exposure and is considering the following alternatives:
! At-the-money Swiss franc call options.
! Swiss franc forwards.
! Swiss franc futures.
a. Contrast the essential characteristics of each of these three derivative instruments.
b. Evaluate the suitability of each in relation to Michelle’s hedging objective, including both

advantages and disadvantages.
3. Identify the fundamental distinction between a futures contract and an option contract, and briefly

explain the difference in the manner that futures and options modify portfolio risk.
4. Maria VanHusen, CFA, suggests that using forward contracts on fixed-income securities can be

used to protect the value of the Star Hospital Pension Plan’s bond portfolio against the possibility
of rising interest rates. VanHusen prepares the following example to illustrate how such protec-
tion would work:
! A 10-year bond with a face value of $1,000 is issued today at par value. The bond pays an

annual coupon.
! An investor intends to buy this bond today and sell it in 6 months.
! The 6-month risk-free interest rate today is 5% (annualized).
! A 6-month forward contract on this bond is available, with a forward price of $1,024.70.
! In 6 months, the price of the bond, including accrued interest, is forecast to fall to $978.40

as a result of a rise in interest rates.
a. Should the investor buy or sell the forward contract to protect the value of the bond against

rising interest rates during the holding period?
b. Calculate the value of the forward contract for the investor at the maturity of the forward con-

tract if VanHusen’s bond-price forecast turns out to be accurate.
c. Calculate the change in value of the combined portfolio (the underlying bond and the appro-

priate forward contract position) 6 months after contract initiation.
5. Sandra Kapple asks Maria VanHusen about using futures contracts to protect the value of the Star

Hospital Pension Plan’s bond portfolio if interest rates rise. VanHusen states:
a. “Selling a bond futures contract will generate positive cash flow in a rising interest rate envi-

ronment prior to the maturity of the futures contract.”
b. “The cost of carry causes bond futures contracts to trade for a higher price than the spot price

of the underlying bond prior to the maturity of the futures contract.”
Comment on the accuracy of each of VanHusen’s two statements.

E$INVESTMENTS EXERCISES
Go to the Chicago Mercantile Exchange site at www.cme.com. From the Trading tab, select the
link to Equity Index, and then link to the NASDAQ-%&& E-mini contract. Now find the tab for Con-
tract Specifications.

%. What is the contract size for the futures contract?

!. What is the settlement method for the futures contract?

#. For what months are the futures contracts available?

‘. Click the link to view Price Limits and then U.S. Equity Price Limits. What is the current value
of the “% down limit for the S&P (&& contract?

(. Click on!Calendar. What is the settlement date of the shortest-maturity outstanding contract?
The longest-maturity contract?

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!”# P A R T V I Options, Futures, and Other Derivatives

bod77178_ch22_747-774.indd 748 04/08/17 06:01 PM

The miller who must purchase wheat for processing faces a risk management problem
that is the mirror image of the farmer’s. He is subject to profit uncertainty because of the
unpredictable cost of the wheat.

Both parties can hedge their risk by entering into a forward contract calling for the
farmer to deliver the wheat when harvested at a price agreed upon now, regardless of the
market price at harvest time. No money need change hands at this time. A forward contract
is simply a deferred-delivery sale of some asset with the sales price agreed on now. All
that is required is that each party must be willing to lock in the ultimate delivery price. The
contract protects each party from future price fluctuations.

Futures markets formalize and standardize forward contracting. Buyers and sellers trade
in a centralized futures exchange. The exchange standardizes the types of contracts that may
be traded: It establishes contract size, the acceptable grade of commodity, contract delivery
dates, and so forth. Although standardization eliminates much of the flexibility available
in forward contracting, it offers the offsetting advantage of liquidity because many traders
will concentrate on the same small set of contracts. Futures contracts also differ from for-
ward contracts in that they call for a daily settling up of any gains or losses on the contract.
By contrast, no money changes hands in forward contracts until the delivery date.

The centralized market, standardization of contracts, and depth of trading in each con-
tract allows futures positions to be liquidated easily rather than renegotiated with the other
party to the contract. Because the exchange guarantees the performance of each party,
costly credit checks on other traders are not necessary. Instead, each trader simply posts a
good-faith deposit, called the margin, to guarantee contract performance.

The Basics of Futures Contracts
The futures contract calls for delivery of a commodity at a specified delivery or maturity
date, for an agreed-upon price, called the futures price, to be paid at contract maturity.
The contract specifies precise requirements for the commodity. For agricultural commodi-
ties, the exchange sets allowable grades (e.g., No. 2 hard winter wheat or No. 1 soft red
wheat). The place and means of delivery of the commodity are specified as well. Delivery
of agricultural commodities is made by transfer of warehouse receipts issued by approved
warehouses. For financial futures, delivery may be made by wire transfer; for index futures,
delivery may be accomplished by a cash settlement procedure such as those for index
options. Although the futures contract technically calls for delivery of an asset, delivery
rarely occurs. Instead, parties to the contract much more commonly close out their posi-
tions before contract maturity, taking gains or losses in cash.

Because the futures exchange specifies all the terms of the contract, the traders need bar-
gain only over the futures price. The trader taking the long position commits to purchasing
the commodity on the delivery date. The trader who takes the short position commits to
delivering the commodity at contract maturity. The trader in the long position is said to
“buy” a contract; the short-side trader “sells” a contract. The words buy and sell are figura-
tive only, because a contract is not really bought or sold like a stock or bond; it is entered
into by mutual agreement. At the time the contract is entered into, no money changes hands.

Figure 22.1 shows prices for several futures contracts as they appear in The Wall Street
Journal. The boldface heading lists in each case the commodity, the exchange where the
futures contract is traded, the contract size, and the pricing unit. The first agricultural contract
listed is for corn, traded on the Chicago Board of Trade (CBT). (The CBT merged with the
Chicago Mercantile Exchange in 2007 but still maintains a separate identity.) Each contract
calls for delivery of 5,000 bushels, and prices in the entry are quoted in cents per bushel.

The next two! rows detail price data for contracts expiring on various dates. The July
2016 maturity corn contract, for example, opened during the day at a futures price of

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!”#

bod77178_ch22_747-774.indd 752 04/08/17 06:01 PM

necessarily incomplete. The nearby box discusses some comparatively fanciful futures
markets, sometimes called prediction markets, in which payoffs may be tied to the winner
of presidential elections, the box office receipts of a particular movie, or anything else in
which participants are willing to take positions.

Prediction Markets
W

O
R

D
S

F
R

O
M

T
H

E
S

T
R

E
E

T

If you find S&P “$$ or T-bond contracts a bit dry, perhaps you’d
be interested in futures contracts with payoffs that depend on
the winner of the next presidential election, or the severity of
the next influenza season, or the host city of the #$#% Olym-
pics. You can now find “futures markets” in these events and
many others.

For example, both Iowa Electronic Markets (www.biz.uiowa
.edu/iem) and the Politics page of BetFair (www.betfair.com)
maintain presidential futures markets. In September #$&’,
you could have purchased a contract that would pay off $& in
November if Hillary Clinton won the presidential race but noth-
ing if she lost. The contract price (expressed as a percentage
of face value) therefore may be viewed as the probability of a
Clinton victory, at least according to the consensus view of mar-
ket participants at the time. If you believed in September that
the probability of a Clinton victory was “”%, you would have
been prepared to pay up to $.”” for the contract. Alternatively,
if you had wished to bet against Clinton, you could have(sold
the contract. Similarly, you could have bet on (or against) a
Donald Trump victory using his contract. (When there are only

two relevant parties, betting on one is equivalent to betting
against the other, but in other elections, such as primaries
where there are several viable candidates, selling one candi-
date’s contract is not the same as buying another’s.)

The accompanying figure shows the price of Democratic and
Republican contracts from November #$&) through Election Day.
The price clearly tracks each party’s perceived prospects. You
can see Clinton’s price rise to above $.*$ in the week just before
the election as the polls increasingly suggested she would win.
Her price then declined substantially when the FBI announced
it was reopening its investigation into her e-mail server. By the
day before the election, with the investigation again apparently
closed, her price had rebounded to $.%$: Her victory seemed
nearly inevitable, at least until the votes were counted.

Interpreting prediction market prices as probabilities actu-
ally requires a caveat. Because the contract payoff is risky, the
price of the contract may reflect a risk premium. Therefore, to be
precise, these probabilities are actually risk-neutral probabilities
(see Chapter #&). In practice, however, it seems unlikely that the
risk premium associated with these contracts is substantial.

Prediction markets for the !”#$ presidential election. Contract on each party pays $& if the party wins the
election. Price is in cents.

!

Date

“!

#!

$!

%!

C
o

nt
ra

ct
P

ric
e

(c
e

nt
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(!

)!

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Democrat

Republican

Source: Iowa Electronic Markets, downloaded November &’, #$&’.

Final PDF to printer

!”# P A R T I V Fixed-Income Securities

bod77178_ch15_467-494.indd 482 04/18/17 05:31 PM

Given that an upward-sloping yield curve implies a forward rate higher than the spot,
or current, yield to maturity, we ask next what can account for that higher forward rate.
The challenge is that there always are two possible answers to this question. Recall that the
forward rate can be related to the expected future short rate according to:

fn!=!E(rn )!+!Liquidity premium (15.8)

where the liquidity premium might be necessary to induce investors to hold bonds of
maturities that do not correspond to their preferred investment horizons.

By the way, the liquidity premium need not be positive, although that is the position
generally taken by advocates of the liquidity premium hypothesis. We showed previously
that if most investors have long-term horizons, the liquidity premium in principle could
be negative.

In any case, Equation 15.8 shows that there are two reasons that the forward rate could
be high. Either investors expect rising interest rates, meaning that E(rn) is high, or they
require a large premium for holding longer-term bonds. Although it is tempting to infer
from a rising yield curve that investors believe that interest rates will eventually increase,
this does not necessarily follow. Indeed, Panel A in Figure 15.4 provides a simple
counterexample. There, the short rate is expected to stay at 5% forever. Yet there is a
constant 1% liquidity premium so that all forward rates are 6%. The result is that the
yield curve continually rises, starting at a level of 5% for 1-year bonds, but eventually
approaching 6% for long-term bonds as more and more forward rates at 6% are averaged
into the yields to maturity.

Therefore, while expectations of increases in future interest rates can result in a ris-
ing yield curve, the converse is not true: A rising yield curve does not in and of itself
imply expectations of higher future interest rates. Potential liquidity premiums confound
any simple attempt to extract expectations from the term structure. But estimating the
market’s expectations is crucial because only by comparing your own expectations to those
reflected in market prices can you determine whether you are relatively bullish or bearish
on interest rates.

One very rough approach to deriving expected future spot rates is to assume that
liquidity premiums are constant. An estimate of that premium can be subtracted from the
forward rate to obtain the market’s expected interest rate. For example, again making use
of the example plotted in Panel A of Figure 15.4, the researcher would estimate from
historical data that a typical liquidity premium in this economy is 1%. After calculating the
forward rate from the yield curve to be 6%, the expectation of the future spot rate would be
determined to be 5%.

This approach has little to recommend it for two reasons. First, it is next to impos-
sible to obtain precise estimates of a liquidity premium. The general approach to doing
so would be to compare forward rates and eventually realized future short rates and to
calculate the average difference between the two. However, the deviations between the
two values can be quite large and unpredictable because of unanticipated economic events
that affect the realized short rate. The data are too noisy to calculate a reliable estimate of
the expected premium. Second, there is no reason to believe that the liquidity premium
should be constant. Figure 15.5 shows the rate of return variability of prices of long-term
Treasury bonds since 1971. Interest rate risk fluctuated dramatically during the period.
So we should expect risk premiums on various maturity bonds to fluctuate, and empirical
evidence suggests that liquidity premiums do in fact fluctuate over time.

Still, very steep yield curves are interpreted by many market professionals as warn-
ing signs of impending rate increases. In fact, the yield curve is a good predictor of the

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C H A P T E R !” The Term Structure of Interest Rates #$%

bod77178_ch15_467-494.indd 487 04/18/17 05:31 PM

KEY TERMSterm structure of interest rates
yield curve
bond stripping
bond reconstitution

pure yield curve
on-the-run yield curve
spot rate
short rate

forward interest rate
liquidity premium
expectations hypothesis
liquidity preference theory

Forward rate of interest: 1!+!fn!=!
(1!+!yn)n ____________

(1!+!yn ” 1)n ” 1

Yield to maturity given sequence of forward rates: 1 + yn = [ (1 + r1) (1 + f2) (1 + f3)!#!#!#!(1 + fn) ] 1/n

Liquidity premium = Forward rate ” Expected short rate

KEY EQUATIONS

liquidity premium remained reasonably stable over time. However, both empirical and theoretical
considerations cast doubt on the constancy of that premium.

6. Forward rates are market interest rates in the important sense that commitments to forward
(i.e., deferred) borrowing or lending arrangements can be made at these rates.

1. What is the relationship between forward rates and the market’s expectation of future short rates?
Explain in the context of both the expectations hypothesis and the liquidity preference theory of
the term structure of interest rates.

2. Under the expectations hypothesis, if the yield curve is upward-sloping, the market must expect
an increase in short-term interest rates. True/false/uncertain? Why?

3. Under the liquidity preference theory, if inflation is expected to be falling over the next few
years, long-term interest rates will be higher than short-term rates. True/false/uncertain?
Why?

4. If the liquidity preference hypothesis is true, what shape should the term structure curve have in a
period where interest rates are expected to be constant?
a. Upward sloping.
b. Downward sloping.
c. Flat.

5. Which of the following is true according to the pure expectations theory? Forward rates:
a. Exclusively represent expected future short rates.
b. Are biased estimates of market expectations.
c. Always overestimate future short rates.

6. Assuming the pure expectations theory is correct, an upward-sloping yield curve implies:
a. Interest rates are expected to increase in the future.
b. Longer-term bonds are riskier than short-term bonds.
c. Interest rates are expected to decline in the future.

7. The following is a list of prices for zero-coupon bonds of various maturities.!
a. Calculate the yield to maturity for a bond with a maturity of (i) one year; (ii) two years;

(iii) three years; (iv) four years.
b. Calculate the forward rate for (i) the second year; (ii) the third year; (iii) the fourth year.

Maturity (years) Price of Bond

! $”#$.#%
& ‘”‘.#(
$ ‘#(.)&
# (“&.!)

PROBLEM SETS

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bod77178_ch15_467-494.indd 488 04/18/17 05:31 PM

!”” P A R T I V Fixed-Income Securities

8. a. Assuming that the expectations hypothesis is valid, compute the expected price of the 4-year
bond in Problem 7 at the end of (i) the first year; (ii) the second year; (iii) the third year;
(iv) the fourth year.

b. What is the rate of return of the bond in years 1, 2, 3, and 4? Conclude that the expected
return equals the forward rate for each year.

9. Consider the following $1,000 par value zero-coupon bonds:
Bond Years to Maturity YTM(%)

A ! “%
B # $
C % $.”
D & ‘

According to the expectations hypothesis, what is the market’s expectation of the yield curve
one year from now? Specifically, what are the expected values of next year’s yields on bonds
with maturities of (a) one year? (b) two years? (c) three years?

10. The term structure for zero-coupon bonds is currently:
Maturity (years) YTM (%)

! &%
# “(
% $(

Next year at this time, you expect it to be:
Maturity (years) YTM (%)

! “%
# $
% ‘

a. What do you expect the rate of return to be over the coming year on a 3-year zero-coupon
bond?

b. Under the expectations theory, what yields to maturity does the market expect to observe on
1- and 2-year zeros at the end of the year?

c. Is the market’s expectation of the return on the 3-year bond greater or less than yours?
11. The yield to maturity on 1-year zero-coupon bonds is currently 7%; the YTM on 2-year zeros is

8%. The Treasury plans to issue a 2-year maturity coupon bond, paying coupons once per year
with a coupon rate of 9%. The face value of the bond is $100.
a. At what price will the bond sell?
b. What will the yield to maturity on the bond be?
c. If the expectations theory of the yield curve is correct, what is the market expectation of the

price for which the bond will sell next year?
d. Recalculate your answer to part (c) if you believe in the liquidity preference theory and you

believe that the liquidity premium is 1%.
12. Below is a list of prices for zero-coupon bonds of various maturities.

Maturity (years)
Price of $#,$$$ Par Bond

(zero-coupon)

! $)&%.&*
# +’%.”#
% +!$.%’

a. An 8.5% coupon $1,000 par bond pays an annual coupon and will mature in three years.
What should the yield to maturity on the bond be?

b. If at the end of the first year the yield curve flattens out at 8%, what will be the 1-year
holding-period return on the coupon bond?

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C H A P T E R !” The Term Structure of Interest Rates #$%

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13. Prices of zero-coupon bonds reveal the following pattern of forward rates:
Year Forward Rate

! “%
# $
% &

In addition to the zero-coupon bond, investors also may purchase a 3-year bond making annual
payments of $60 with par value $1,000.
a. What is the price of the coupon bond?
b. What is the yield to maturity of the coupon bond?
c. Under the expectations hypothesis, what is the expected realized compound yield of the

coupon bond?
d. If you forecast that the yield curve in one year will be flat at 7%, what is your forecast for the

expected rate of return on the coupon bond for the 1-year holding period?
14. You observe the following term structure:

& Effective Annual YTM

!-year zero-coupon bond ‘.!%
#-year zero-coupon bond ‘.#
%-year zero-coupon bond ‘.%
(-year zero-coupon bond ‘.(

a. If you believe that the term structure next year will be the same as today’s, calculate the
return on (i) the 1-year zero and (ii) the 4-year zero.

b. Which bond provides a greater expected 1-year return?
c. Redo your answers to parts (a) and (b) if you believe in the expectations hypothesis.

15. The yield to maturity (YTM) on 1-year zero-coupon bonds is 5%, and the YTM on 2-year
zeros is 6%. The YTM on 2-year-maturity coupon bonds with coupon rates of 12% (paid
annually) is 5.8%.
a. What arbitrage opportunity is available for an investment banking firm?
b. What is the profit on the activity?

16. Suppose that a 1-year zero-coupon bond with face value $100 currently sells at $94.34, while a
2-year zero sells at $84.99. You are considering the purchase of a 2-year-maturity bond making
annual coupon payments. The face value of the bond is $100, and the coupon rate is 12% per year.
a. What is the yield to maturity of the 2-year zero?
b. What is the yield to maturity of the 2-year coupon bond?
c. What is the forward rate for the second year?
d. According to the expectations hypothesis, what are (i) the expected price of the coupon bond

at the end of the first year and (ii) the expected holding-period return on the coupon bond
over the first year?

e. Will the expected rate of return be higher or lower if you accept the liquidity preference
hypothesis?

17. The current yield curve for default-free zero-coupon bonds is as follows:
Maturity (years) YTM (%)

! !)%
# !!
% !#

a. What are the implied 1-year forward rates?
b. Assume that the pure expectations hypothesis of the term structure is correct. If market

expectations are accurate, what will be the yield to maturity on 1-year zero-coupon bonds
next year?

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!”# P A R T I V Fixed-Income Securities

c. What about the yield on 2-year zeros?
d. If you purchase a 2-year zero-coupon bond now, what is the expected total rate of return over

the next year?!(Hint: Compute the current and expected future prices.) Ignore taxes.
e. What is the expected total rate of return over the next year on a 3-year zero-coupon bond?
f. What should be the current price of a 3-year maturity bond with a 12% coupon rate paid

annually?
g. If you purchased the coupon bond at the price you computed in part (f ), what would your

total expected rate of return be over the next year (coupon plus price change)? Ignore
taxes.

18. Suppose that the prices of zero-coupon bonds with various maturities are given in the following
table. The face value of each bond is $1,000.

Maturity (years) Price

! $”#$.”%
# &$%.%”
% ‘&#.”#
( ‘!$.))
$ *$).))

a. Calculate the forward rate of interest for each year.
b. How could you construct a 1-year forward loan beginning in year 3? Confirm that the rate on

that loan equals the forward rate.
c. Repeat part (b) for a 1-year forward loan beginning in year 4.

19. Use the data from Problem 18. Suppose that you want to construct a 2-year maturity forward
loan commencing in 3 years.
a. Suppose that you buy today one 3-year maturity zero-coupon bond. How many 5-year matu-

rity zeros would you have to sell to make your initial cash flow equal to zero?
b. What are the cash flows on this strategy in each year?
c. What is the effective 2-year interest rate on the effective 3-year-ahead forward loan?
d. Confirm that the effective 2-year forward interest rate equals (1 + f4) ” (1 + f5) # 1. You

therefore can interpret the 2-year loan rate as a 2-year forward rate for the last two years.
Alternatively, show that the effective 2-year forward rate equals

(1!+!y5)
5
________

(1!+!y3)3
!#!1

1. Briefly explain why bonds of different maturities have different yields in terms of the expecta-
tions and liquidity preference hypotheses. Briefly describe the implications of each hypothesis
when the yield curve is (1) upward-sloping and (2) downward-sloping.

2. Which one of the following statements about the term structure of interest rates is true?
a. The expectations hypothesis indicates a flat yield curve if anticipated future short-term rates

exceed current short-term rates.
b. The expectations hypothesis contends that the long-term rate is equal to the anticipated short-

term rate.
c. The liquidity premium theory indicates that, all else being equal, longer maturities will have

lower yields.
d. The liquidity preference theory contends that lenders prefer to buy securities at the short end

of the yield curve.

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490 PART IV Fixed-Income Securities

c. What about the yield on 2-year zeros?

d. If you purchase a 2-year zero-coupon bond now, what is the expected total rate of return over

the next year? (Hint: Compute the current and expected future prices.) Ignore taxes.

e. What is the expected total rate of return over the next year on a 3-year zero-coupon bond?

f. What should be the current price of a 3-year maturity bond with a 12% coupon rate paid

annually?

g. If you purchased the coupon bond at the price you computed in part (f ), what would your

total expected rate of return be over the next year (coupon plus price change)? Ignore

taxes.

18. Suppose that the prices of zero-coupon bonds with various maturities are given in the following

table. The face value of each bond is $1,000.

Maturity (years)Price

1$925.93

2 853.39

3 782.92

4 715.00

5 650.00

a. Calculate the forward rate of interest for each year.

b. How could you construct a 1-year forward loan beginning in year 3? Confirm that the rate on

that loan equals the forward rate.

c. Repeat part (b) for a 1-year forward loan beginning in y ear 4.

19. Use the data from Problem 18. Suppose that you want to construct a 2-year maturity forward

loan commencing in 3 y ears.

a. Suppose that you buy today one 3-year maturity zero-coupon bond. How many 5-year matu-

rity zeros would you have to sell to make your initial cash flow equal to zero?

b. What are the cash flows on this strategy in each year?

c. What is the effective 2-year interest rate on the effective 3-year-ahead forward loan?

d. Confirm that the effective 2-year forward interest rate equals (1 + f

4

) × (1 + f

5

) − 1. You

therefore can interpret the 2-year loan rate as a 2-year forward rate for the last two years.

Alternatively, show that the effective 2-year forward rate equals

(1 + y

5

)

5

________

(1 + y

3

)

3

− 1

1. Briefly explain why bonds of different maturities have different yields in terms of the expecta-

tions and liquidity preference hypotheses. Briefly describe the implications of each hypothesis

when the yield curve is (1) upward-sloping and (2) downward-sloping.

2. Which one of the following statements about the term structure of interest rates is true?

a. The expectations hypothesis indicates a flat yield curve if anticipated future short-term rates

exceed current short-term rates.

b. The expectations hypothesis contends that the long-term rate is equal to the anticipated short-

term rate.

c. The liquidity premium theory indicates that, all else being equal, longer maturities will have

lower yields.

d. The liquidity preference theory contends that lenders prefer to buy securities at the short end

of the yield curve.

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3. The following table shows yields to maturity of zero-coupon Treasury securities.
Term to Maturity (years) Yield to Maturity (%)

! “.#$%
% &.#$
” #.$$
& #.#$
# ‘.$$

!$ ‘.’$

a. Calculate the forward 1-year rate of interest for year 3.
b. Describe the conditions under which the calculated forward rate would be an unbiased

estimate of the 1-year spot rate of interest for that year.
c. Assume that a few months earlier, the forward 1-year rate of interest for that year had been sig-

nificantly higher than it is now. What factors could account for the decline in the forward rate?
4. The 6-month Treasury bill spot rate is 4%, and the 1-year Treasury bill spot rate is 5%. What is the

implied 6-month forward rate for six months from now?
5. The tables below show, respectively, the characteristics of two annual-pay bonds from the same

issuer with the same priority in the event of default, and spot interest rates. Neither bond’s price
is consistent with the spot rates. Using the information in these tables, recommend either bond A
or bond B for purchase.

Bond Characteristics

Bond A Bond B

Coupons Annual Annual
Maturity ” years ” years
Coupon rate !$% ‘%
Yield to maturity !$.’#% !$.(#%
Price )*.&$ **.”&

Spot Interest Rates

Term (years) Spot Rates (zero-coupon)

! #%
% *
” !!

6. Sandra Kapple is a fixed-income portfolio manager who works with large institutional clients.
Kapple is meeting with Maria VanHusen, consultant to the Star Hospital Pension Plan, to discuss
management of the fund’s approximately $100 million Treasury bond portfolio. The current U.S.
Treasury yield curve is given in the following table. VanHusen states, “Given the large differen-
tial between 2- and 10-year yields, the portfolio would be expected to experience a higher return
over a 10-year horizon by buying 10-year Treasuries, rather than buying 2-year Treasuries and
reinvesting the proceeds into 2-year T-bonds at each maturity date.”

Maturity Yield Maturity Yield

! year %.$$% ‘ years &.!#%
% %.)$ ( &.”$
” “.#$ * &.&#
& “.*$ ) &.’$
# &.$$ !$ &.($

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!”# P A R T I V Fixed-Income Securities

a. Indicate whether VanHusen’s conclusion is correct, based on the pure expectations hypothesis.
b. VanHusen discusses with Kapple alternative theories of the term structure of interest rates and

gives her the following information about the U.S. Treasury market:
Maturity (years) ! ” # $ % & ‘ ( )*
Liquidity premium (%) *.$$ *.$$ *.%$ *.&$ *.(* ).)* ).!* ).$* ).%*

Use this additional information and the liquidity preference theory to determine what the slope of
the yield curve implies about the direction of future expected short-term interest rates.

7. A portfolio manager at Superior Trust Company is structuring a fixed-income portfolio to meet the
objectives of a client. The portfolio manager compares coupon U.S. Treasuries with zero-coupon
stripped U.S. Treasuries and observes a significant yield advantage for the stripped bonds:

Term
Coupon

U.S. Treasuries
Zero-Coupon Stripped

U.S. Treasuries

” years $.$*% $.’*%
& %.&$+ &.!$+

)* &.!$+ &.%*+
“* &.&$+ ‘.!*+

Briefly discuss why zero-coupon stripped U.S. Treasuries could have higher yields to maturity
than coupon U.S. Treasuries with the same final maturity.

8. The shape of the U.S. Treasury yield curve appears to reflect two expected Federal Reserve
reductions in the federal funds rate. The current short-term interest rate is 5%. The first reduction
of approximately 50 basis points (bp) is expected six months from now and the second reduction
of approximately 50 bp is expected one year from now. The current U.S. Treasury term premiums
are 10 bp per year for each of the next three years (out through the 3-year benchmark).

However, the market also believes that the Federal Reserve reductions will be reversed in a
single 100-bp increase in the federal funds rate 2 1 ⁄ 2 years from now. You expect liquidity premi-
ums to remain 10 bp per year for each of the next three years (out through the 3-year benchmark).

Describe or draw the shape of the Treasury yield curve out through the 3-year benchmark.
Which term structure theory supports the shape of the U.S. Treasury yield curve you’ve described?

9. U.S. Treasuries represent a significant holding in many pension portfolios. You decide to analyze
the yield curve for U.S. Treasury notes.
a. Using the data in the table below, calculate the 5-year spot and forward rates assuming annual

compounding. Show your calculations.
U.S. Treasury Note Yield Curve Data

Years to Maturity
Par Coupon

Yield to Maturity
Calculated
Spot Rates

Calculated
Forward Rates

) $.** $.** $.**
! $.!* $.!) $.#!
” %.** %.*$ &.&$
# &.** &.)% )*.$%
$ &.** ? ?

b. Define and describe each of the following three concepts:
i. Short rate
ii. Spot rate
iii. Forward rate

Explain how these concepts are related.
c. You are considering the purchase of a zero-coupon U.S. Treasury note with four years to

maturity. On the basis of the above yield-curve analysis, calculate both the expected yield
to maturity and the price for the security. Show your calculations.

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10. The spot rates of interest for five U.S. Treasury securities are shown in the following exhibit.
Assume all securities pay interest annually.

Spot Rates of Interest

Term to Maturity Spot Rate of Interest

! year !”.##%
$ !$.##
” !!.##
% !#.##
& ‘.##

a. Compute the 2-year implied forward rate for a deferred loan beginning in three years.
b. Compute the price of a 5-year annual-pay Treasury security with a coupon rate of 9%

by using the information in the exhibit.

E&INVESTMENTS EXERCISES
Go to’stockcharts.com/freecharts/yieldcurve.php where you will find a dynamic or “living” yield
curve, a moving picture of the yield curve over time. Hit the Animate button to start the demon-
stration. Is the yield curve usually upward- or downward-sloping? What about today’s yield curve?
How much does the slope of the curve vary? Which varies more: short-term or long-term rates?
Can you explain why this might be the case?

SOLUTIONS TO CONCEPT CHECKS
1. The price of the 3-year bond paying a $40 coupon is

40 ____
1.05

!+! 40 _____
1.062

!+! 1040 _____
1.073

!=!38.095!+!35.600!+!848.950!=!$922.65

At this price, the yield to maturity is 6.945% [n = 3; PV = (“)922.65; FV = 1,000; PMT = 40].
This bond’s yield to maturity is closer to that of the 3-year zero-coupon bond than is the yield to
maturity of the 10% coupon bond in Example 15.1. This makes sense: This bond’s coupon rate is
lower than that of the bond in Example 15.1. A greater fraction of its value is tied up in the final
payment in the third year, and so it is not surprising that its yield is closer to that of a pure 3-year
zero-coupon security.

2. We compare two investment strategies in a manner similar to Example 15.2:

Buy!and!hold!4-year!zero!=!Buy!3-year!zero;!roll!proceeds!into!1-year!bond
(1!+!y4)4!=!(1!+!y3)3!#!(1!+!r4)

1.084!=!1.073!#!(1!+!r4)

which implies that r4 = 1.084/1.073 ” 1 = .11056 = 11.056%. Now we confirm that the yield on
the 4-year zero reflects the geometric average of the discount factors for the next 3 years:

1!+!y4!=![(1!+!r1)!#!(1!+!r2)!#!(1!+!r3)!#!(1!+!r4)]1/4

1.08!=![1.05!#!1.0701!#!1.09025!#!1.11056]1/4

3. The 3-year bond can be bought today for $1,000/1.073 = $816.30. Next year, it will have a
remaining maturity of two years. The short rate in year 2 will be 7.01% and the short rate in year 3

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!”! P A R T I V Fixed-Income Securities

will be 9.025%. Therefore, the bond’s yield to maturity next year will be related to these short rates
according to

(1!+!y2)2!=!1.0701!”!1.09025!=!1.1667

and its price next year will be $1,000/(1 + y2)2 = $1,000/1.1667 = $857.12. The 1-year holding-
period rate of return is therefore ($857.12 # $816.30)/$816.30 = .05, or 5%.

4. The n-period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n periods.
The short rate for period n is the one-period interest rate that will prevail in period n. Finally, the
forward rate for period n is the short rate that would satisfy a “break-even condition” equating
the total returns on two n-period investment strategies. The first strategy is an investment in an
n-period zero-coupon bond; the second is an investment in an n # 1 period zero-coupon bond
“rolled over” into an investment in a one-period zero. Spot rates and forward rates are observable
today, but because interest rates evolve with uncertainty, future short rates are not. In the special
case in which there is no uncertainty in future interest rates, the forward rate calculated from the
yield curve would equal the short rate that will prevail in that period.

5. 7% # 1% = 6%.
6. The risk premium will be zero.
7. If issuers prefer to issue long-term bonds, they will be willing to accept higher expected interest

costs on long bonds over short bonds. This willingness combines with investors’ demands for
higher rates on long-term bonds to reinforce the tendency toward a positive liquidity premium.

8. In general, from Equation 15.5, (1 + yn)n = (1 + yn#1)n#1 ” (1 + fn). In this case, (1 + y4)4 =
(1.07)3 ” (1 + f4). If f4 = .07, then (1 + y4)4 = (1.07)4 and y4 = .07. If f4 is greater than .07, then
y4 also will be greater, and conversely if f4 is less than .07, then y4 will be as well.

9. The 3-year yield to maturity is ( 1,000 ______ 816.30 )
1/3

!#!1!=!.07!=!7.0%

The forward rate for the third year is therefore

f3!=!
(1!+!y3)3 ________
(1!+!y2)2

!#!1!=! 1.07
3
_____

1.062
!#!1!=!.0903!=!9.03%

(Alternatively, note that the ratio of the price of the 2-year zero to the price of the 3-year zero
is 1 + f3 = 1.0903.) To construct the synthetic loan, buy one 2-year maturity zero, and sell 1.0903
3-year maturity zeros. Your initial cash flow is zero, your cash flow at time 2 is +$1,000, and
your cash flow at time 3 is #$1,090.30, which corresponds to the cash flows on a 1-year forward
loan commencing at time 2 with an interest rate of 9.03%.

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Calculate the yield to maturity of the coupon bond in Example 15.1, and you may be sur-
prised. Its yield to maturity is 6.88%; so while its maturity matches that of the 3-year zero
in Table 15.1, its yield is a bit lower.1 This reflects the fact that the 3-year coupon bond may
usefully be thought of as a portfolio of three implicit zero-coupon bonds, one corresponding
to each cash flow. The yield on the coupon bond is then an amalgam of the yields on each
of the three components of the “portfolio.” Think about what this means: If their coupon
rates differ, bonds of the same maturity generally will not have the same yield to maturity.

What then do we mean by “the” yield curve? In fact, in practice, traders refer to sev-
eral yield curves. The pure yield curve refers to the curve for stripped, or zero-coupon,

payment from a whole Treasury bond as a separate cash flow. For example, a 1-year
maturity T-bond paying semiannual coupons can be split into a 6-month maturity zero
(by selling the first coupon payment as a stand-alone security) and a 12-month zero (cor-
responding to payment of final coupon and principal). Treasury stripping suggests exactly
how to value a coupon bond. If each cash flow can be (and in practice often is) sold off
as a separate security, then the value of the whole bond should equal the total value of its
cash flows bought piece by piece in the STRIPS market.

What if it weren’t? Then there would be easy profits to be made. For example, if invest-
ment bankers ever noticed a bond selling for less than the amount at which the sum of its
parts could be sold, they would buy the bond, strip it into stand-alone zero-coupon securi-
ties, sell off the stripped cash flows, and profit by the price difference. If the bond were
selling for more than the sum of the values of its individual cash flows, they would run
the process in reverse: buy the individual zero-coupon securities in the STRIPS market,
reconstitute (i.e., reassemble) the cash flows into a coupon bond, and sell the whole bond
for more than the cost of the pieces. Both bond stripping and bond reconstitution offer
opportunities for arbitrage—the exploitation of mispricing among two or more securities
to clear a riskless economic profit. Any violation of the Law of One Price, that identical
cash flow bundles must sell for identical prices, gives rise to arbitrage opportunities.

To value each stripped cash flow, we simply look up its appropriate discount rate in
The Wall Street Journal. Because each coupon payment matures at a different time, we
discount by using the yield appropriate to its particular maturity—this is the yield on a
Treasury strip maturing at the time of that cash flow. We can illustrate with an example.

Suppose the yields on stripped Treasuries are as given in Table !”.!, and we wish to value
a !#% coupon bond with a maturity of three years. For simplicity, assume the bond makes
its payments annually. Then the first cash flow, the $!## coupon paid at the end of the first
year, is discounted at “%; the second cash flow, the $!## coupon at the end of the second
year, is discounted for two years at $%; and the final cash flow consisting of the final coupon
plus par value, or $!,!##, is discounted for three years at %%. The value of the coupon bond
is therefore

!##

____
!.#”

!+!
!##

____
!.#$&

!+!
!,!##

____
!.#%’

!=!(“.&’)!+!)(.###!+!)(%.(&)!=!$!,#)&.!%

Example 15.1 Valuing Coupon Bonds

1Remember that the yield to maturity of a coupon bond is the single interest rate at which the present value
of cash flows equals market price. To calculate the coupon bond’s yield to maturity on your calculator or
spreadsheet, set n = 3;!price = “1,082.17; future value = 1,000;!payment = 100. Then compute the interest rate.

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