9.1 Options and Futures

Overview

This is the last module of class and will close with derivatives. Derivatives are assets that get their value from other assets. So, for example, with stock options, the option’s value is dependent, among other things, on the trading price of the stock on which the option was written (by written, we mean sold).

Options are complicated because the underlying stock price can fluctuate up and down, thus changing the price of the option in real-time. Futures are less complicated because they settle at maturity and the settlement price is the Spot Price (that is, the market price for that asset on the date of maturity) minus the Futures Price (the price that was agreed to when the futures contract was signed.)

Tech companies often use options to pay employees. The concept is that employees will feel like their pay will rise if the company performs well – in other words, the incentives of the employee and the company are perfectly aligned. Options are also used by speculators betting that stock prices will rise or fall. It’s cheaper to bet on these fluctuations by using options than by actually buying the stock itself.

Futures are used mostly for hedging. So, for example, a farmer who is expecting to harvest 100,000 bushels of wheat can sell the harvest in advance and lock the price using a futures contract. The settlement is done financially and not physically (that is, the farmer does not deliver the wheat to the counter-party of the futures contract; instead, they settle via an electronic fund transfer.) The farmer sells the wheat to the processor directly at the spot price come harvest time.

So, after all is said and done, the farmer collects 1) the spot price for the wheat and 2) the futures price minus the spot price. If the futures price is higher than the spot price, the farmer has a gain. If, however, the futures price is below the spot price, the farmer takes the loss. The farmer’s bottom line, then, is = spot price received at the silo + gain/(loss) of the futures contract.

Final Exam

The final exam covers the following material from the course:

• Discounted cash flows
• Free Cash Flow (FCF) valuation of a firm
• Bond prices and yields
• Options pricing using the Black-Scholes Model, Futures profits/losses, and a fixed/floating interest rate swap

The questions and parameters are all contained in the Excel spreadsheet below.

the 1st file is a case study to read through and the 2nd file is the excel template to answer.

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Corporate Finance (Symbiosis International University)
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Introduction to Derivative Instruments
A derivative is a financial instrument, or contract, between two parties that derives its value
from some other underlying asset or underlying reference price, interest rate, or index. Common
derivatives include options, forward contracts, futures contracts, and swaps. Common underlying
assets include interest rates, exchange rates, commodities, stocks, stock indices, bonds, and bond
indices. Derivatives are created and traded in two interlinked markets—organized exchanges at the
national and regional level, and an international network of dealers and end-users in which
transactions are executed privately, that is, over the counter (OTC).
Over recent decades, financial markets have been marked by increased volatility. As foreign
exchange rates, interest rates, and commodity prices continue to experience sharp and unexpected
movements, it has become increasingly important that corporations exposed to these risks be
equipped to manage them effectively. Risk management, the managerial process that is used to
control such price volatility, has consequently risen to the top of financial agendas. And in the hot
spot are these so-called derivatives. Furthermore, as these instruments have become more readily
available, their application has extended beyond traditional risk management to the more
opportunistic realm of speculation. In both applications, derivatives represent powerful tools by
which institutions and individuals alike can significantly affect their financial security and viability.
Derivatives are used by a variety of entities such as corporations, commercial banks, and
individual and institutional investors to reduce or “lay off” various risks including the
aforementioned interest rate risk, foreign currency risk, commodity price risk, and investment risk.
Exhibit 1 provides results of a survey on the uses of derivatives by chief financial officers. For
example, a chief financial officer of a company heavily exposed to foreign exchange fluctuations often
exploits the foreign exchange forward market to shield the company’s balance sheet from currency
depreciation. Similarly, a grain producer might use a forward contract to hedge against price
depreciation in, say, wheat or soybeans. Through the use of a put option, an investor can establish a
limit on the potential loss on an investment. On the other end of the application spectrum, an entity
can trade derivatives for purely speculative purposes. Broadly, holders of derivatives securities, as
well as their counterparties, can achieve goals ranging from risk management to speculation. The
derivatives themselves help allocate economic risks efficiently by transferring risks between parties
such that each holds the risk it is better able or more willing to bear.
This note provides a conceptual basis for understanding the fundamental properties and
applications of common derivative products that give rise to their use in financial management. Each
of three major families of derivative instruments—options, forwards and futures, and swaps—is
discussed in the separate sections that follow.
Research Associate Kendall Backstrand wrote this note under the supervision of Professor W. Carl Kester as the basis for
class discussion rather than to illustrate either effective or ineffective handling of an administrative situation.
Copyright © 1995 by the President and Fellows of Harvard College. To order copies or request permission to
reproduce materials, call 1-800-545-7685, write Harvard Business School Publishing, Boston, MA 02163, or go to
http://www.hbsp.harvard.edu. No part of this publication may be reproduced, stored in a retrieval system,
used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying,
recording, or otherwise—without the permission of Harvard Business School.
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Introduction to Derivative Instruments
Options
Common Terminology
Options are derivative instruments that can be used as a means of speculation or investment
as well as hedging or risk management. Options written on both financial and physical assets have
been traded for many years in dealer-markets. However, it was not until 1973, when the Chicago
Board of Trade formed the Chicago Board Options Exchange (CBOE), that organized public markets
for options began to appear. Exchanges were then established to trade options written on assets such
as individual stocks, stock indices, commodities, foreign currencies, and Treasury bonds.
An option is a contract between the buyer (or holder) of the option and the seller (or writer)
of the option. This contract gives the buyer of the option the right to buy (or sell) an asset from (to)
the seller of the option. The seller, on the other hand, is obligated under the terms of the option
contract to perform. Plainly stated, an option contract defines the rights of the buyer and the
obligations of the seller. The option to buy an asset is known as a call option, and the option to sell an
asset is known as a put option. An example of a call and put option written on a particular company’s
common stock, that of Microsoft Corporation, is provided in Table A below.
Table A
Options traded on Microsoft’s stock, November 30, 1994 (dollars per share)
Stock (asset) price
Exercise price
Maturity date
Call option price (premium)
Put option price (premium)
$64.125
$60
April 15, 1995
$7.50
$2.125
The specified asset involved in the option contract is referred to as the underlying asset on
which the option is written. The specified price at which the asset may be bought or sold in the future
is known as the exercise or strike price. Purchasing or selling the asset in the future through the option
contract is referred to as exercising the option, and the specified date on or before which the option
may be exercised is called the expiration date or maturity date. So-called American-style options are
contracts that may be exercised at any time prior to maturity, whereas European-style options are
contracts that may be exercised only at maturity.
The options on Microsoft’s stock shown in Table A were American options. A holder of the
call option could have purchased Microsoft’s stock at $60 per share by exercising the call option on or
before April 15, 1995. Likewise, a holder of the put option could have sold Microsoft’s stock at $60
per share by exercising the put option on or before April 15, 1995.
Option contracts have a market or premium value, and an intrinsic value. The market value of
the option contract is simply the price at which a buyer and seller are willing to enter into an option
contract. More specifically, it is the up-front cash premiums that the buyer must pay the seller in
order to claim the rights of that particular option contract. As shown in Table A, the market value of
the call option on Microsoft’s stock was $7.50 per option as of the end of trading on November 30,
1994. Likewise, the market value of the put option on Microsoft’s stock was $2.125. Because standard
option contracts are contracts to buy or sell 100 shares at a time, an investor would actually have had
to pay $750.00 to buy a standard call option contract on Microsoft’s stock and $212.50 to buy a
standard put contract.
The intrinsic value of an option can be thought of as the price a rational investor would pay
for an option if it were about to mature instantly. Because an option contract gives the holder the
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right to exercise but not the obligation, the intrinsic value of an option can never be less than zero.
This is true because if the option is never exercised by the holder, it simply expires worthless.
If, for instance, the price of Microsoft’s stock had fallen to $55 per share, the owner of the call
option described in Table A would not have elected to exercise the option to buy at $60 per share. An
investor wishing to own Microsoft’s stock, in this case, would have been better off buying it directly
on the stock exchange at $55 per share. Thus, at a stock price of $55 per share, the intrinsic value of a
call option with an exercise price of $60 would have been zero, representing a worthless position for
the holder of the call.
In general, the intrinsic value of a call option is always the greater of zero and the difference
between the current market price of the underlying asset and the option’s exercise price. In the case
of a call option, this intrinsic value will be positive when the market price of the asset exceeds the
exercise price of the option, and zero otherwise. At $64.125 per share, the call option on Microsoft’s
stock had a positive intrinsic value of $64.125 less $60, or $4.125. The call option holder could have
bought Microsoft’s stock for less than its actual market value. The opposite is true in the case of a put
option: sensible investors would not sell a put option’s underlying asset at the put’s exercise price
unless that exercise price were above the asset’s market value. Thus, the intrinsic value of a put
option is always the greater of zero and the difference between the put’s exercise price and the
current market price of the underlying asset.
An option is said to be in-the-money when its intrinsic value is positive and out-of-the-money
when it is zero. That is, a call option is in-the-money when its underlying asset’s market price is
above the exercise price and out-of-the-money when the opposite occurs. The converse is true for a
put option: when the exercise price is above (below) the underlying asset’s market price at maturity
the put is considered in-the-money (out-of-the-money). As the term suggests, an at-the-money option
describes an option when its exercise price exactly equals the underlining asset’s market price. Again
using the Microsoft example, the terms described in Table A constitute an in-the-money call option
and an out-of-the-money put option. If the exercise price were $64.125, or the stock price were $60,
both options would be at-the-money. If the market price of an underlying asset is far above (below)
the exercise price of a call (put) option, then the option is said to be deep-in-the-money. If the opposite
is true, it is said to be deep-out-of-the-money. A deep-in-the-money position at maturity is the most
desirable outcome for either a call or put option owner.
Graphical representation of an option’s intrinsic value is useful to illustrate its total payoff.
Payoff diagrams for both put and call options written on the same underlying asset with the same
exercise price are provided in Figure 1 below.
Figure 1
Payoff Diagrams
A. Total Payoff on a Call Option
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Introduction to Derivative Instruments
B. Total Payoff on a Put Option
where
K = exercise price
P = premium
Determinants of Option Value
Notice in Table A that each option’s market value is greater than its intrinsic value. This will
always be true for options that have some time remaining before maturity. A graph of a call option’s
market, or premium, value relative to intrinsic value is shown in Figure 2 below.
Figure 2
Call Option Premium in Relation to Intrinsic Value
where
K = exercise price
How much greater the premium value is over intrinsic value depends on several factors. In
general, for generic American-style call and put options, the premium value depends upon the
following six determinants: underlying asset price, exercise price, the risk-free rate of return,
volatility of the asset price, time to expiration, and expected cash distributions, if any. Their
respective effects on option value are briefly described below.
Asset price For an American or European call option, the higher the price of the underlying asset,
the greater the option’s intrinsic value and the more likely it will remain above the option’s exercise
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price at expiration. Hence, the higher the asset price, the greater will be the call option’s premium,
other things held constant. The opposite is true for American and European put options: the higher
the value of the underlying asset, the lower will be a put option’s intrinsic value and premium, other
things held constant.
Exercise price An increase in a call option’s exercise price decreases both the intrinsic value of the
option and the likelihood that the option will be worth anything at maturity. Consequently, the
higher an American or European call option’s exercise price, the lower will be its premium value,
other things held constant. Again, the opposite is true of a put option: a higher exercise price
increases the put option’s intrinsic value, other things held constant, and would be reflected in a
higher premium.
Interest rates Because buyers of options do not pay or receive the option’s exercise price until later,
if ever, interest rates play a role in the determination of option premium value. Specifically, an
increase in the interest rate lowers the present value of the cash exercise price expected to be paid or
received in the future. For a call option, a rise in interest rates means the future cash payment of its
exercise price is worth less in present value terms, implying greater option value for the holder.
Hence, the value of an American or European call option increases as interest rates rise, holding other
factors constant. In contrast, a rise in interest rates lowers the present value of the cash that a put
option holder might receive in the future upon exercising the put. Consequently, American and
European put option premiums decline as interest rates rise, other things being equal.
Volatility of the asset price Other factors held constant, the more volatile the underlying asset
price, the more valuable the option. Again, this is true because of the asymmetrical construct between
an option’s potential upside gains and downside losses (see Figure 1). The holder of a call option
experiences unlimited potential gains as the price of the asset increases. At the same time, however,
the call option holder effectively limits loss by simply not exercising the option if the asset’s price falls
below the option’s exercise price. The holder of a put, although only experiencing limited potential
gains (the maximum gain being obtained when the asset price is zero upon maturity, implying an
intrinsic value exactly equal to the option’s exercise price), can also limit loss by simply not exercising
the put if the asset’s price rises above the put’s exercise price. In short, the more volatile the asset
price, the greater the chance the holder of either a put or call option has of realizing a large gain
without equally increasing the chance of incurring a large loss. Thus, higher expected volatility in the
underlying asset’s price enhances both American and European option values, other things being
equal.
Time to expiration American and European call options increase in value when the time remaining
to expiration is further away. This positive influence derives from two sources. First, in connection
with the interest rate effect, the longer the time before expiration when the exercise payment will be
made, the lower the discounted present value of that cash payment. Second, in connection with the
volatility effect, the more time there is before expiration, the more likely it is that a large price change
will occur and dramatically increase the value of the option. Consequently, so long as there is time
remaining before expiration, an option’s premium will exceed its intrinsic value. Provided there are
no cash distributions to owners of a call option’s underlying asset (see below), it follows that a call
option should not be exercised before maturity since doing so would sacrifice the value attributable to
time.
American put option value is also positively affected by time to expiration. Because of the
asymmetry between potential gains and losses from holding a put option, more time before
expiration increases the chance that the put will mature in the money. Although the proceeds to be
received from the future exercise of the put will have a lower present value as time to expiration
increases, other things constant, this negative influence will not generally outweigh the positive
influence associated with price volatility unless interest rates are high. When this is so, American put
option holders might find it in their best interests to exercise their puts prematurely and reinvest the
cash proceeds.
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For European put options, the time to expiration can have either a positive or negative
influence on prices depending on which of two effects dominate. When a European put is in the
money, a longer time to expiration will tend to have a negative influence on premium value because
the expected receipt of cash proceeds from exercising the put is further in the future. However, if the
European put is deep out of the money, a long time to expiration will tend to enhance option value.
This is because more time provides a greater opportunity for the stock price to drop far enough to
make the put valuable at expiration. Of course, the stock price could rise as well, but, as in the case of
call options, losses on the downside can be limited by simply not exercising the put.
Cash distributions Some assets, notably many common stocks, have cash distributions
associated with them. A cash dividend paid on an underlying stock decreases the value of a call
option, other things held constant. The reason is that cash dividends reduce the market price of the
stock on the day the stock goes ex dividend (i.e., begins to trade without rights to any cash dividends
previously declared on the stock; shareholders of record just prior to the ex dividend date are entitled
to the cash dividends, but holders of call options on that stock are not). As the price of a stock
declines when it goes ex dividend, so too will the value of a call option on the stock, other things
remaining constant. The opposite is true for a put option: the holder of the put option, as well as the
owner of the stock, benefit from cash dividends in that the stock owner receives a cash payout and the
put holder obtains increased option value when the stock’s price declines upon going ex dividend.
A summary of the effect each of the preceding factors has on American option value is
illustrated in Table B below.
Table B
Summary of Factors Determining American Option Valuea
Asset price
Exercise price
Interest rates
Volatility of the asset price
Time to maturity
Cash distributions
Call option
Put option
+
+
+
+
–
+
+
+b
+
aThe + and – signs indicate the nature of the effect each factor has on the value of the option.
bAs discussed above, time to maturity could have either a positive or negative influence on European put option value.
Put-Call Parity
Consider again the Microsoft put and call options described in Table A. Notice that, in
addition to being written on the same stock, these options had identical exercise prices and maturity
dates. Given their similar characteristics, it seems logical that the market values of the call and put
would have been related to one another in a predictable way. That is, as the price of Microsoft’s stock
changed, the prices of the options should also have changed, but in such a way that an astute investor
could not have bought one and sold another so as to lock in a virtually riskless profit. Should such an
arbitrage opportunity develop, the very act of exploiting it ought to set buy and sell transactions in
motion that will ultimately ensure a kind of parity between put and call prices.
This is, in fact, the case. A condition known as put-call parity describes the relationship that a
put and call option written on the same stock with the same exercise price and maturity date must
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sustain if there are to be no riskless arbitrage opportunities.1 Specifically, put-call parity states that
the difference in price between a call option and a put option with the same terms should equal the
price of the underlying asset less the present discounted value of the exercise price. This relationship
can be expressed as follows:
Vc-Vp = Pa- X
where: Vc
Vp
Pa
X
=
=
=
=
the price of a call option
the price of a put option
the price of the underlying asset
present discounted value of the underlying asset’s exercise price
Another way to interpret this relationship is to say that someone owning a call option while
having simultaneously written (sold) a comparable put option on the same asset should, at all times,
be in a position equivalent to someone who purchased the underlying asset with a pure-discount (i.e.,
zero coupon) loan having a face value equal to the option’s exercise price and maturing at the
option’s expiration date. The value of these two positions must be equal because each investor would
realize identical payoffs at the time of maturity. You can demonstrate this to yourself by constructing
payoff diagrams such as those shown in Figure 1 for each of these two positions. As you will observe,
the payoff in both cases is equivalent to owning stock purchased on “margin” (that is, purchased
partly with borrowed proceeds).
Consider what could be done if this relationship were not true. For illustrative purposes,
assume that the options on Microsoft’s stock shown in Table A were European options. Suppose,
further, that the call option on Microsoft’s stock shown in Table A actually sold for $8.50 instead of
$7.50. At the time, short-term interest rates were about 6% annually (equivalent to a compound daily
rate of 1.6 basis points, or 0.016%). Under these conditions, strict put-call parity would not have held:
($8.50 – $ 2.125) > ($64.125 – $58.709), or
$6.375 > $5.414
where
$8.50 =
$2.125 =
$64.125 =
$58.709 =
assumed market value of the call option
market value of the put option
market value of Microsoft’s stock
current value of a pure-discount loan maturing on April 15 at a value of 60
1Strictly speaking, put-call parity as described above applies only to European options because, unlike American
options, they cannot be exercised prior to the expiration date.
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Upon observing such a discrepancy, an astute trader would have executed the following transactions:
November 30, 1994
1. Write (sell) a call option on Microsoft’s stock
2. Buy a put option on Microsoft’s stock
3. Borrow $58.709 at a daily compound rate of interest of 0.016%
4. Purchase Microsoft’s stock at $64.125
Net Proceeds
April 15, 1995
a. If Microsoft’s stock was worth more than $60 per share, then:
1. Deliver the stock to the call option owner
2. Receive $60 from the call option owner
3. Use the proceeds from the exercise of the call option to repay the loan
Net proceeds
b.
If Microsoft’s stock was worth less than $60 per share, then:
1. Exercise the put by delivering the stock to the put writer
2. Receive $60 from the put writer
3. Use the proceeds from exercising the put to repay the loan
Net proceeds
Per share
cash proceeds
$8.50
($2.125)
$58.709
($64.125)
$0.959
C
$60.00
($60.00)
$0.00
C
$60.00
($60.00)
$0.00
Notice that regardless of what happened to the price of Microsoft’s stock, the trader would
have received $60 on April 15, 1995, which is exactly sufficient to repay the loan with interest. Thus,
the residual net proceeds of $0.959 per share from the November 30, 1994, transactions represent an
immediate, riskless profit involving no commitment of the trader’s own capital. Notice too that such
an arbitrage profit would have been virtually immaterial at the call option’s actual price of $7.50. If
call or put option prices deviated substantially from levels dictated by the put-call parity relationship,
transactions similar to those described above would drive prices up or down until the arbitrage
opportunity was eliminated.
Applications
Options can be used to insure against various risks as well as to bet on various market
movements. Risk management, or insurance, is often achieved through, for example, the purchase of
put options. Assume a company expects to receive some foreign currency and is concerned that the
currency will depreciate against its home currency. To limit its losses, the company might elect to
purchase an at-the-money put option written on the exposed currency. Buying such a put option
would, in effect, limit the company’s loss associated with currency depreciation to the amount of the
put premium. In effect, by buying a put option, the company buys insurance against currency
depreciation. The cost of this insurance is the put premium. By insuring against loss in this way,
however, the company also gives up some of the potential gains it might realize from currency
appreciation in that it must pay a cash premium to buy the put.
Speculative positions can also be achieved by using options. A directional position is taken
when a company or individual uses options to bet on a belief that the underlying asset price will
move in one particular direction. If an entity believes that the British pound will appreciate, for
example, then it could buy a call option written on the pound (i.e., go “long” British pounds).
Because the currency could easily move in the “wrong” direction (i.e., contrary to one’s prior beliefs),
buying currency call options does not secure a profit, nor does this transaction cover an already
exposed position. But still, because of the inherent asymmetry of potential upside gains and
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downside losses, the holder stands to gain quite a bit while potentially losing only the amount of the
premium paid for the call option. This would be a more powerful way to speculate on the pound’s
movement than simply buying the currency in the spot market because, for a given amount of dollars,
considerably more currency can be controlled through the purchase of relatively inexpensive option
contracts than can be done by buying pounds outright on the spot foreign exchange market (a
standard option contract on British pounds would provide an investor with a call option on ,62,500
for a price in the vicinity of $1500; the same amount of currency might cost $95,000 to $100,000 on the
spot market).
Forwards and Futures
Forwards and futures, like options, are derivative securities that can be used as a means of
hedging or risk management, as well as to speculate. Predating any other derivative instrument, the
privately traded forward contract serves as the foundation for its more standardized exchange-traded
variant, the futures contract. While these two contracts are viewed and traded quite differently, they
both operate under the same essential framework. Specifically, both the forward and the futures
contract are defined by an obligation of the buyer and the seller both to perform under the specified
terms of the contract. In this respect, forward and futures contracts differ fundamentally from option
contracts. Because options give the owner the right but not the obligation to exercise the option,
option contracts provide owners with asymmetric payoff patterns that are well suited to insuring
against loss under certain circumstances. Because forwards and futures provide an obligation to
transact at a prespecified future price, they are better suited for true “hedging” activities in which
transacting parties wish to lock in future prices without risk. Figure 3 provides a payoff diagram of a
generic forward contract to illustrate and distinguish these particular forms of derivative securities
from options.
Figure 3
Total Payoff on a Forward Contract
where: F = forward price
Forward Contracts
In contrast to exchange-traded derivatives, forward contracts are not standardized products.
Instead, forward contracts are OTC derivatives that can be tailored to meet specific user needs. The
underlying assets of these contracts include traditional agricultural or physical commodities,
currencies (referred to as foreign exchange forwards), and interest rates (referred to as forward rate
agreements or FRAs). A forward transaction typically involves a contract, most often with a bank,
under which both the buyer (or holder) of the contract and the seller (or writer) of the contract are
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obligated to execute a transaction at a prespecified price on a prespecified date. That is, the seller is
obligated to deliver a specified asset to the buyer on a specified date in the future. Likewise, the buyer
is obligated to pay the seller a specified price (the forward price) upon delivery. If, at maturity, the
actual spot market price is higher than the forward contract’s exercise price, the contract holder
makes a profit and the seller suffers a loss; if the spot price is lower, the contract seller makes a profit
and the buyer suffers a loss. In any event, one party’s gain is the other party’s loss.
Normally, a forward contract’s exercise price is fixed at inception at a level that makes the
contract’s value zero in the eyes of both the buyer and the seller. That is, ignoring risk aversion, both
sides of the transaction would be roughly indifferent between entering into the contract at the
specified exercise price or remaining unhedged. However, as the value of the underlying asset
changes throughout the life of the contract, the value of the forward contract as seen by the buyer and
the seller also changes. Specifically, the value changes for the benefit of one party and at the expense
of the other. This property of the forward contract makes it a “zero-sum game” for the buyer and the
seller.
To illustrate this zero-sum characteristic, consider a forward contract written on some
specified asset with a forward exercise price for the asset of $50. Now imagine how a sudden
upswing in the asset’s price to $55 will affect both parties’ views of the value of the contract. The
party on the sell side of the forward contract views the contract to have lost value because the price at
which he or she is obligated to sell the asset ($50) is now below that which could be received in the
spot market ($55). In contrast, the party on the buy side of the contract sees this change as positive
because, as the spot price of the asset increases, there is a better chance that the forward exercise price
will be below the prevailing spot market price in the future when the forward contract matures and
the asset is to be delivered. If this market condition persists until the specified delivery date, the
seller’s loss of $5 ($55 – $50) equals the buyer’s gain.
To summarize, both the buyer and the seller of a forward contract view their positions as
having zero initial value. The agreed-upon forward price for the underlying asset is the contract price
that fulfills this initial condition; that is, the forward price is determined so as to eliminate any initial
value for either party. Subsequent changes in the spot market price of the underlying asset will lead
to equal but opposite gains on the part of the buyer and seller.
Cost of carry, arbitrage, and forward prices To understand how the correct forward price is
determined, one must first appreciate the concepts of cost of carry and, again, arbitrage. Simply stated,
the cost of carry is the opportunity cost that would be borne by an investor if the asset underlying a
forward contract were actually bought and held rather than the forward contract itself. In the
simplest possible case, this would essentially be the cost of money; that is, the opportunity cost of
tying up one’s money in the asset in question, thereby foregoing its use in other investments. For
some underlying assets, however, ownership requires storage and the incurrence of storage costs (e.g.,
rental of space in a grain silo, rental of vault space, insurance costs). Storage costs, if any, add to the
cost of carry.
Offsetting some of the cost of carry are cash payouts on the underlying asset (e.g., cash
interest payments on debt securities or cash dividend payments on shares of stock) and so-called
convenience yields. A convenience yield is the value that might be associated with actually owning,
and therefore being able to use, the asset in question rather than simply having a future claim on that
asset. A manufacturer that uses a lot of copper, for example, might wish to own a fairly sizable
inventory of copper to assure that shortages are not experienced as demand for output fluctuates.
Likewise, heavy users of fuel oil will often prefer to own oil itself rather than oil futures to safeguard
against unanticipated interruptions in supply.
Consider now an asset such as gold, provides no cash payouts, and capital market conditions
in which the 1-year yield on Treasury bills is 10%. For simplicity, assume further that under current
market conditions, the convenience yield on gold equals storage costs. Under these simplified
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conditions, the cost of carry on gold is simply the cost of money. If someone were to purchase gold
with cash in the spot market for $375 per ounce and hold it for a year, money would be tied up for a
year, thereby imposing an opportunity cost on the investors of 10%, or $37.50—resulting in a total
cost of $412.50 per ounce of gold by the time it is used or sold 1 year later.
This opportunity cost could be avoided if the investor elected instead to enter into a forward
contract that would oblige him or her to pay cash for gold a year later, but not before. What would be
a fair price to agree to pay 1 year later? In principle, the investor should be happy to pay any price
less than or equal to $412.50, for at such prices, the investor should be no worse off, and possibly
better off, than buying gold and holding it for a year. Similarly, the party writing the forward
contract should be happy to sell the contract at any price equal to or greater than $412.50, for such
prices would permit the writer to buy and hold gold for a year, thus eliminating the risk of future
price changes in the spot market, while also at least covering his or her cost of carry. The interests of
both the buyer and the seller can be met at their mutual breakeven price of $412.50 = $375 x (1 + .10).
This pricing equilibrium implies the following simple formula for determining the forward
price of an asset:
Fn = S (1 + c)
n
where
Fn =
S =
c =
n =
the forward price of an asset n years into the future
the current spot price for the asset
the annual cost of carry, expressed as a fraction of the asset’s spot price (e.g., .01, .05, etc.)
years to maturity
Because c is composed of several different costs and yields, the forward price can also be expressed
more fully as:
n
F = S (1 + rf + s – i – v) ,
where
rf
s
i
v
=
=
=
=
the riskless rate of return
storage costs
cash yield
convenience yield
all expressed as annual costs or yields as a fraction of the spot price.
Forward contracts in which the forward price is established at inception, according to the
above formula, will have an initial value of zero. Notice that any other forward price would lead to a
potential arbitrage opportunity. Suppose, for example, that a forward contract on gold such as that
described above was struck at a below-market forward price of $400 per ounce. This being the case,
and assuming ample supplies of gold in storage, arbitrageurs could lock in a riskless profit by
simultaneously buying that which is relatively “cheap” (gold in the forward market) and selling that
which is relatively “expensive” (gold in the spot market).
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Specifically, an arbitrageur would:
Per ounce
cash proceeds
1.
2.
3.
Borrow some gold and sell it, (i.e., “short” gold)
Invest the proceeds of the sale for one year at 10%
Enter into a 1-year forward contract to purchase gold at $400.00/oz.
Net proceeds
$375.00
($375.00)
—
$0.00
One year later, the same arbitrageur would:
1.
2.
3.
Collect the proceeds from the one-year investment
Use the proceeds to execute the forward agreement to buy gold at $400/oz.
Deliver the gold to the party from whom it was originally borrowed
Net proceeds
$412.50
($400.00)
—
$12.50
In effect, market arbitrageurs would make a riskless profit of $12.50 per ounce of gold on zero
net investment. This arbitrage opportunity arises because the forward price is too low given the
current spot price and the cost of carry. To eliminate this arbitrage opportunity, forward and/or spot
prices for gold must adjust until the forward price formula shown above is satisfied.
Notice that if a forward contract’s underlying asset does not have a significant cash payout
relative to the cost of money, and/or if storage costs significantly exceed convenience yields, the cost
of carry will be positive and the current forward price will be greater than the spot price. This
premium of the forward price over the spot price is known as contango. Typical examples of assets
with low or no cash payouts are stock indices and foreign exchange.2 The opposite will be true if
there are large cash payouts or when the convenience yield is especially high (a common occurrence
for many commodities when supply conditions in the spot market become quite tight). Under these
conditions, the forward price will be below the spot price, a condition known as backwardation. Notice
too that, regardless of how high or low the forward price is relative to the spot price at the time the
forward contract is established, the forward price eventually converges with the spot price as the time
to delivery shortens to zero. This is because the cost of carry in an asset necessarily becomes less as
the time to delivery approaches.
2In the particular case of foreign exchange, the forward price must take account of two interest rates because
two currencies are involved. “Shorting” one currency implies borrowing it at prevailing interest rates in that
currency, while investment in the other currency will take place at that other currency’s prevailing interest rates.
The formula for determining the forward exchange rate between a domestic currency (D) and a foreign currency
(F) is as follows:
D
F
F = S x (1 + R )/(1 + R )
where
F = forward rate of exchange, expressed as units of domestic currency per unit of foreign.
S = spot market rate of exchange, expressed as units of domestic currency per units of foreign.
D
R = domestic interest rate.
F
R = foreign interest rate.
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Futures Contracts
Futures contracts, unlike forwards, trade on organized exchanges. They are traded in three
primary areas: agricultural commodities, metals and petroleum, and financial assets. While
commodity futures have been traded since the 1860s, financial futures were first traded in 1972 with
the advent of the foreign currency future. Since then, financial futures have been established for
various debt instruments, stock market indices, and foreign currencies.
The basic form of the futures contract mirrors that of the forward contract: both parties are
obligated under the terms of the contract either to deliver a specified asset or pay the specified price
of the asset on the contract maturity date. In addition, the futures contract entails the following two
obligations, both of which help to minimize the default (or credit) risk inherent in forward contracts.
1) The value of the futures contract is “settled” (i.e., paid or received) at the end of
each trading day. In the language of the futures markets, the futures contract is
cash settled, or marked-to-market, daily.
The marked-to-market provision
effectively reduces the performance period of the contract to a day, thereby
minimizing the risk of default.
2) Both buyers and sellers are required to post a performance bond called margin.
At the end of each trading day, gains and losses are added to and taken away
from the margin account, respectively. The margin account must remain above
an agreed-upon minimum or the account will be closed. The margin provision
prevents the depletion of accounts, which, in turn, largely eliminates the risk of
default.
With these additional features in mind, a futures contract can be thought of as a connected
series of 1-day forward contracts in which the forwards are settled and restruck daily until the
specified maturity date. By definition, a futures contract is an agreement between the seller of the
contract and the buyer of the contract in which the seller is obligated to deliver a specified asset to the
buyer on a specified date in the future and the buyer is obligated to pay the seller the then prevailing
futures price upon delivery. The nature of marked-to-market defines the “then prevailing futures
price” simply as the then prevailing spot price. Therefore, upon final settlement of a futures contract
that has reached maturity, the only profit and loss incurred is that associated with the last day’s
market movement.
Applications
The two generic uses for forwards and futures are speculation and hedging. As an example
of forward market speculation assume an investor expects the dollar price of the Japanese yen to fall
dramatically over the next 90 days. Foreign currency markets allow such an investor to bet on his or
her expectations. First, the investor sells yen forward at the prevailing forward spot rate. After 90
days, assuming the yen depreciated as expected, the investor then purchases yen in the spot market
for delivery on the forward contract. If all goes well, the forward price at which the investor sells yen
will exceed the future spot price at which he or she buys, and a profit will result from the difference.
Of course, if the opposite is true and the yen strengthens against the dollar, the investor will lose the
difference between the future spot rate and the forward price.
Hedging, unlike speculation, is a tactic used to avoid or limit risk. Forward and futures
contracts are commonly used for this purpose. For example, assume an investor will hold some
specified asset for one year and is fearful of price depreciation over the holding period. To hedge
against price depreciation by locking in a known value today, the investor could sell a forward
contract written on the asset; that is, sell the asset forward, just as the investor in the previous
speculation example sold the yen forward. In doing so, the investor covers his or her “long” position
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in the asset with a “short” position (the forward sale). Losses that might occur on the long position
will be offset by gains on the short position, and vice versa. In this way, uncertainty about the future
market value of the asset in question can be eliminated.
Swaps
A swap is any agreement to a future exchange of one asset for another, one liability for
another, or more specifically, one stream of cash flows for another. The most common swaps include
currency swaps, in which one currency is exchanged for another at prespecified terms on one or more
prespecified future dates, and interest rate swaps, in which one type of interest payment (e.g., interest
payments that float with LIBOR3) is exchanged for another (e.g., fixed interest payments) at one or
more prespecified future dates. Like other derivative securities, these swaps (as well as more
sophisticated swaps not addressed in this note) are used by various entities such as corporations,
banks, and investors to hedge risk or to speculate in the expectation of making a profit. As a tool of
risk management, swaps offer considerable flexibility and cost savings to their users. The boom in
swaps transactions since the early 1980s is testament to the growing demand for flexible and
standardized risk management products.
Although its origins can be traced back to the 1970s, the swap market did not publicly exist
until 1981 when currency swaps were first introduced. U.S. interest rate swaps followed in 1982 as
rising interest-rate volatility necessitated a flexible means by which companies with floating interestrate exposures could hedge such risk. As swap markets grew, swaps became common adjuncts to
financings, particularly cross-border financings, as a way to help companies lower their funding costs.
They did so by enabling companies to source capital in whatever market or currency it was found to
be cheapest (e.g., floating-rate Swiss francs), and then to convert the resulting liability into whatever
form made most sense (e.g., fixed-rate dollars). Today it is a common practice of major borrowers to
analyze funding opportunities in light of relative pricings for new debt issues and swaps across
global markets.
Like a forward or futures contract, a swap is a private agreement between two parties in
which both parties are obligated to exchange some specified cash flows at periodic intervals for a fixed
period of time. In contrast to a forward or futures contract, a swap agreement generally involves
multiple future points of exchange. The cash flows of a swap may be fixed in advance, or adjusted
for each settlement date by reference to some specified interest rate, such as LIBOR, or other market
yield. Although it is convenient to describe swaps as involving an outright exchange of cash flows at
the so-called settlement dates, in practice, it is generally the case that a difference check is simply paid by
whichever party in the swap is obligated to pay more cash than is to be received at that settlement
date. For example, consider a fixed-for-floating interest rate swap agreement that requires one party
to pay a fixed rate of interest of 9% a year on $100 million of principal in exchange for receiving from
a counterparty interest equal to LIBOR plus % on $100 million. If, at the first settlement date,
LIBOR is equal to 7.5%, the party paying a fixed rate would owe the floating-rate counterparty a net
payment of $1 million: (.09 – (.075 + .005)) x $100 million. If, at the next settlement date, LIBOR had
risen to 9%, the fixed-rate party would receive a net cash payment of $.5 million from the floating rate
counterparty: (.09 – (.09 + .005)) x $100 million. All of these settlements would be carried out by a
financial intermediary such as an investment or commercial bank.
2
Also, like forward or futures contracts, swaps are priced so as to have zero value at inception.
As interest rates or exchange rates change, the swap agreement then takes on positive value for
whichever party becomes a net recipient of cash, and negative value for the counterparty that is the
3LIBOR stands for the London Interbank Offered Rate. It is the interest rate offered by banks for dollar deposits
in the London market. It is frequently used as a base interest rate for dollar loans.
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net payer of cash. In a sense, a swap agreement can be thought of as a prepackaged bundle of
forward contracts, and its cash flows can be decomposed into the equivalent cash flows of these
individual forward contracts.
Currency Swaps
In its simplest form, a currency swap is an agreement between two parties to exchange a
given amount of one currency for another and to repay these currencies with interest in the future.
As an example, consider one party, Global Enterprises, Inc. (Global) that has borrowed 200 million
Swiss francs (SF) at 6% and wishes to transform this liability into dollars. At the same time, the
World Financial Institution (WFI), which actively manages the currency mix of its debt portfolio in
light of changing economic conditions, wishes to convert a $100 million obligation bearing 8% interest
into a Swiss franc liability. Both companies’ obligations have a 4-year maturity and are rated AAA.
The prevailing spot exchange rate between the Swiss franc and the U.S. dollar is SF 2.00/$1.
Given these “matching,” or opposite, hedging needs, a mutually satisfactory swap could be
arranged in which Global agrees to pay 8% dollar interest to WFI for 4 years plus $100 million at
maturity, and WFI agrees to pay Global 6% Swiss franc interest for 4 years plus SF 200 million at
maturity. In this way, each borrower would have its debt service to its respective lender exactly
covered, and would be left with a payment stream in the currency of its choice. Figure 4 below
illustrates this arrangement and the cash flows entailed.
Figure 4
FX Swap Illustration
A. Swap Diagram
B. Swap Cash Flow Diagram (millions)
In practice, one party in a swap agreement seldom makes payments directly to the
counterparty. When parties to a swap are matched directly, a financial institution usually
intermediates the arrangement, guaranteeing each party that payments in the needed currency will
continue uninterrupted even if the counterparty defaults. The intermediary is paid a fee for acting as
guarantor.
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The most common swap arrangement is one in which the intermediary itself acts as the swap
counterparty to its corporate clients. Major international banks make a market in currency swaps by
quoting bid and offer rates for payments in various currencies for various maturities. The bid rate
and the offer rate are the fixed rates of return in a specified currency that a bank is willing to pay a
corporate client in exchange for receiving six-month dollar LIBOR, or to receive from a corporate
client in exchange for paying six-month dollar LIBOR. For example, on December 16, 1985, foreign
currency swap rates being quoted by Morgan Guaranty, Ltd. in London were as follows:
Table C
Selected Swap Ratesa (December 16, 1985)
5 Years
3 Years
U.S. Dollar
British sterling
Japanese yen
Swiss francs
Deutsche marks
Pay
Receive
Pay
Receive
8.79%
11.49
7.12
5.10
5.80
8.97%
11.70
7.28
5.35
6.10
9.42%
11.45
7.02
5.35
6.45
9.58%
11.66
7.17
5.60
6.75
aAll quotes are fixed annual rates against six-month dollar LIBOR, and quoted from the swap
dealer’s perspective. That is, the bank is willing to pay British sterling at a fixed annual rate of
11.49% in exchange for receiving six-month dollar LIBOR, and to receive British sterling at a fixed
annual rate of 11.70% in exchange for paying six-month dollar LIBOR.
The bank earns a profit on swap transactions by realizing the spread between its bid and offer rates
on six-month dollar LIBOR. Notice that by relating any two quotes to dollar LIBOR, fixed swap rates
can be quoted between any two currencies. For instance, using the quotes in Table C above, the bank
would be willing to pay yen for 3 years at a fixed annual rate of 7.12% in exchange for receiving
deutsche marks for 3 years at a fixed annual rate of 6.10%, and to receive yen at a fixed annual rate of
7.28% in exchange for paying deutsche marks at a fixed annual rate of 5.80%.
Applications
Currency swaps, like other derivative instruments, are often used by corporations, banks, and
government entities to hedge foreign exchange risk on both assets and liabilities. In this capacity, a
currency swap functions much like a series of long-dated forward foreign exchange contracts.
One of the most common applications of currency swaps is their use in conjunction with debt
issues. Sometimes companies find that they can source capital especially cheaply by selling debt
denominated in a foreign currency. At the same time, however, they may wish to avoid the exchange
rate risk associated with such foreign currency debt. A currency swap allows such companies to
capture the low-cost capital while avoiding exchange rate risk. In effect, currency swaps allow
corporate financial officers to uncouple the market in which financial execution takes place from the
currency of the liability that they ultimately incur. In addition to transforming new debt, swaps are
also flexible tools for companies to transform the currency denomination of existing debt. To cite a
well known example of such an application, the World Bank pursues a swap program to fine-tune its
liability structure by actively swapping into and out of different currencies to achieve the lowest
possible debt costs.
Interest Rate Swaps
An interest rate swap is a derivative transaction in which an asset or liability with a floating
rate of interest can be converted into a fixed rate, or vice versa. Like a currency swap, an interest rate
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swap is a counterparty transaction in which the respective positions of two counterparties with equal
but opposite needs are exchanged.
Principal payments are not exchanged in interest rate swaps. This is because the dollar value
of the principal remains the same throughout the contract for both the fixed-rate asset or liability and
the floating-rate asset or liability. The agreed “notional” principal is only used as a basis for
calculating the fixed- and floating-rate payment streams. These payments are made, or more
commonly netted by the use of a difference check, on specified periodic settlement dates. While the
fixed rate of interest is set for the life of the contract, the floating interest rate is set at the beginning of
each interval and typically based on three- or six-month LIBOR.
An example of a typical U.S. dollar-denominated interest rate swap might involve a company
that wants to convert a portion of its fixed-rate debt to floating rate, perhaps because it has acquired
some assets generating cash flows that will vary directly with short-term interest rates. To achieve
this conversion, the company’s treasurer could call a swap dealer at a major bank to obtain quotes on
interest rate swaps. As with currency swaps, dealers in interest rate swaps typically make a market in
six-month LIBOR. That is, swap dealers quote a bid rate, which is the fixed rate of interest the bank
will pay in exchange for receiving six-month LIBOR (i.e., the “price” at which the bank stands ready
to “buy” six-month LIBOR), and an offer rate, which is the fixed rate of interest the bank is willing to
accept as payment in exchange for paying six-month LIBOR (i.e., the “price” at which the bank stands
ready to “sell” six-month LIBOR). Swap rate quotes made in London by Morgan Guaranty, Ltd. on
December 16, 1985, were as follows:
Table D
Interest Rate Swap Quotesa
Years
Bid
Offer
3
5
7
8.79%
9.21
9.48
8.97%
9.36
9.63
aRates are quoted from the bank’s perspective.
Thus, the bank is willing to pay a fixed-rate of interest
of 8.79% in exchange for receiving six-month LIBOR
for 3 years, and to receive 8.97% in exchange for
paying six-month LIBOR for 3 years.
Given these quotes, a company wishing to get out of fixed-rate debt into floating-rate debt for, say, 5
years could do so by agreeing to pay the bank six-month LIBOR in exchange for receiving fixed rate
payments of 9.21%, which could then be used to cover a portion of the interest on its outstanding
fixed-rate debt obligations.4
Applications
Interest rate risk is the leading reason why corporations use swaps. They are typically used
to insure against loss in value of existing corporate liabilities and assets due to unexpected changes in
interest rates. For example, a corporation that has recently taken on a substantial amount of debt
might want to adjust the duration of its debt to match better the duration of its expected cash inflows,
thereby reducing the exposure of the corporation’s market value to interest rate risk.
4In practice, the bid rate by the bank may not cover precisely the fixed rate of interest that the company must
pay to its debt holders. When this occurs, an adjustment is made by adding or subtracting an appropriate
number of basis points to the fixed rate paid and six-month LIBOR received.
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In addition to hedging, corporations often use interest rate swaps to reduce debt costs. There
are three principal ways by which these swaps might provide cost savings: speculating on market
movements, exploiting arbitrage opportunities, and reducing transactions costs. A corporation can
speculate on the direction of interest rates by swapping in and out of fixed- and floating-rate
agreements in hopes of achieving lower borrowing rates. Of course, this sort of speculation can result
in higher borrowing costs if interest rates were to move in an adverse direction.
A corporation might also reduce borrowing costs by exploiting arbitrage opportunities
arising from an ability to source either fixed- or floating-rate debt at particularly attractive rates in
one market compared to another. A company wishing to issue fixed-rate debt might, for example,
discover that it can command unusually low rates in the Eurodollar floating-rate note market. The
company can exploit this opportunity by issuing the floating-rate notes, thus securing the low-cost
funds, and then entering into an interest rate swap that would convert the floating-rate debt to fixed
rate. In this respect, like currency swaps, interest rate swaps enable corporate treasurers to uncouple
the market in which they source funds from the desired interest rate structure of their debt
obligations. In the early days of the swap market, funding could be obtained at savings of as much as
50 basis points given the significant arbitrage opportunities that were then available. Today, due to
more integrated capital markets, arbitrage savings are rarer and more commonly below 20 basis
points.
Finally, transaction costs of an interest rate swap are relatively lower than those of its
predecessor, the interest rate forward contract (forward rate agreements), due to the standardized
nature of the swap market. Thus, interest rate swaps represent an attractive risk management and
cost savings tool for an increasingly wide range of market participants.
Basis Rate Swaps
A basis rate swap is essentially an interest rate swap in which both interest rates are floating.
In effect, a basis rate swap allows a borrower or investor to exchange cash flows determined by one
floating interest rate for cash flows determined by another floating interest rate. For example, a
corporation could transform a loan based on six-month LIBOR to the same loan based on one-month
commercial paper rates.
A basis rate swap can be thought of as two interest rate swaps paired together. One of the
pair would be a floating-for-fixed swap, and the other would be an exchange of the fixed rate with
another floating rate. For example, a company could swap a six-month LIBOR obligation for a fixed
rate, and then swap the fixed rate with another counterparty for another floating-rate obligation
based upon commercial paper rates. The basis rate swap conveniently rolls into one transaction what
would otherwise be two using conventional fixed-for-floating interest rate swaps.
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Exhibit 1
A.
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Survey of the Use of Derivatives by CFOs
Percent of affirmative answers to the question: What kind of derivatives, if any, does your company use?
Derivative Instrument
B.
Foreign exchange forwards
64.2%
Interest rate swaps
78.9
Foreign exchange options
Oil and energy-linked swaps
Other commodity-linked swaps
40.4
11.9
14.7
Exchange-traded interest rate futures and options
Exchange-traded foreign exchange futures and options
Exchange-traded equity futures and options
OTC interest rate futures and options
Equity-linked swaps
Equity swaps
29.4
11.0
10.1
13.8
4.6
2.8
Percent of affirmative answers to question: For what purpose does your company use derivatives?
Purpose
To hedge floating rate debt
To hedge commercial paper issuance
To create synthetic floating-rate debt at a lower cost
To create synthetic fixed-rate debt at a lower cost
To access capital markets globally
To hedge investments overseas
To achieve strategic liability management
52.7%
23.2
35.7
43.8
15.2
36.6
40.2
Source: Institutional Investor, CFO forum, February 1993
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Glossary
American option
see Option.
Arbitrage
profiting from price differences on the same security, currency, or commodity traded in two or
more markets.
At-the-money
term used to describe an option contract whose exercise price is equal to the current market price
of the underlying asset.
Backwardation
pricing situation in which forward and futures prices are higher for those contracts expiring in
the near future than those expiring farther out.
Bid/ask spread
difference between the bid price (the highest price a prospective buyer is prepared to pay for a
particular security) and the ask price (the lowest price a prospective seller is willing to accept for
the same security).
Call option
see Option.
Contango
pricing situation in which forward and futures prices get progressively higher as maturities get
progressively longer.
Cost of carry
out-of-pocket costs incurred while an investor has an investment position.
Deep-in/out-of-the-money
call option whose exercise price is well below the current market price of the underlying asset
(deep-in-the-money) or well above the current market price of the underlying asset (deep-out-ofthe-money). The situation would be exactly opposite for a put option.
Default (credit) risk
financial risk that a debtor will fail to make timely payments of interest and principal as they
come due, or to meet some other provision of a financial agreement.
Derivative instrument
financial instrument whose value is based on that of another underlying security.
Difference check
form of direct, one-way payment upon settlement of a financial contract.
European option
see Option.
Ex-dividend
the absence of the right to receive a cash dividend payment already declared on a stock.
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Exercise price
price at which some security underlying a derivative instrument can be purchased or sold on or
before the contract’s maturity date.
Expiration date
see Maturity date.
Forward contract
privately traded contract to buy or sell a specific amount of some underlying asset at a specified
price on a specified future date.
Futures contract
standardized exchange-traded contract to buy or sell a specific amount of some underlying asset
at a specified price on a specified future date.
Guarantor
entity that takes on a contingent liability by assuming the responsibility for payment of a debt or
performance of some obligation if the party primarily liable fails to perform.
Hedging
the reduction of risk by eliminating the possibility of future gains or losses (e.g., by buying or
selling forward and futures contracts).
Insurance
the reduction of risk by the purchase of contingent claims (e.g., put options, call options,
guarantees, insurance policies) that offset future losses by paying off under those circumstances
in which losses are expected to be incurred.
In-the-money
term used to describe an option contract whose exercise price is below the current market price of
an underlying asset in the case of a call option, and above the current market price of the
underlying asset in the case of a put option.
Intrinsic value
for call options, the greater of zero and the difference between the market value of the call’s
underlying asset and its Exercise price. For put options, the greater of zero and the difference
between the put’s Exercise price and the market value of its underlying asset.
London Interbank Offered Rate (LIBOR)
rate that the most creditworthy international banks dealing in Eurodollars charge each other for
large loans.
Margin
amount of cash an investor deposits with a broker when borrowing from the broker to buy
securities. If the price of the security purchased “on margin” falls, the broker will require the
investor to put up more “margin” by making additional cash deposits.
Mark-to-market
adjust the recorded value of a security or portfolio to reflect actual current market values.
Market value (or price)
the price at which willing buyers and sellers trade similar items in a free and open market.
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Maturity date
date on which payment on some financial contract becomes due and payable. In the case of
options, the maturity date is the final date on which the option owner can buy or sell the
underlying asset.
Option
the right, but not the obligation, to buy or sell some specified underlying asset for a specified
price on (or before) a specified date.
Call option
gives its buyer the right to buy some underlying asset at a fixed price on or before a specified date
in the future.
Put option
gives its buyer the right to sell some underlying asset at a fixed price on or before a specified date
in the future.
American option
option that can be exercised on or before the expiration date.
European option
option that can be exercised only on the expiration date.
Option premium
price an option buyer must pay an option seller for an option contract.
Out-of-the-money
term used to describe an option contract whose exercise price is above the current market price of
the underlying asset in the case of a call option, and below the current price of the underlying
asset in the case of a put option.
Over-the-counter (OTC)
market in which securities transactions are conducted through a telephone and computer
network connecting dealers in stocks and bonds, rather than on the floor of an organized
exchange.
Put-call parity
relationship between put and call option prices that, if held in parity, prevents arbitrage
opportunities.
Put option
see Option.
Settlement date
date by which an executed order must be settled, either by a buyer paying for the securities with
cash or by a seller delivering the securities and receiving the proceeds of the sale for them.
Speculation
assumption of risk in anticipation of gain, but often implying a higher than average possibility of
loss.
Spot price
current delivery price of some physical commodity or financial asset traded in the spot market.
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Strike price
see Exercise price.
Swap
exchange of one asset or liability with particular terms and conditions for another asset or liability
with different terms and conditions for a specified period of time.
Transaction costs
cost of buying or selling a security, which consists mainly of the brokerage commission, the dealer
markdown or markup, or fee (as would be charged by a bank).
Zero-coupon security
security that makes no periodic interest payments but instead is sold at a deep discount from its
face value.
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JTM is considering purchasing a new 3-D printer costing $682,000.
The manufacturer is offering a payment plan in which JTM pays
33% down and finances the rest over 12 months at $40,500/month.
What is the implicit financing rate? If JTM’s WACC is 7.5%, should
it accept the financing offer?
Price
Implicit cost of credit
Cash price
Down payment
Monthly payment
Cash Flows
Cash price minus down payment
Mo. 1
Mo. 2
Mo. 3
Mo. 4
Mo. 5
Mo. 6
Mo. 7
Mo. 8
Mo. 9
Mo. 10
Mo. 11
Mo. 12
ABC Inc. is looking to buy another firm in its industry. It is being offered an established business whose own
gave you the numbers and asked you to calculate its price. The CFO asks that you use the data below. What s
Valuation
Revenue growth
Costs (% of revenue):
Wages and benefits
Aircraft and fuel costs
General and administrative
Rates:
Tax
Discount
Results
PV of NCF (incl. TV)
+ Cash
– Debt
Value of equity
Yrs. 1-5
2.7%
45%
39%
10%
21.0%
7.0%
12.0
18.0
TV
2.6%
Sales
Expenses:
Wages and benefits
Aircraft and fuel costs
General and administrative
Operating Income
Taxes
Net Income after Tax
Cash flow adjustments:
Working Capital
Capital Expenditures
Net Cash Flows
1
482.1
(4.0)
(9.0)
Cash Flows
2
(4.1)
(9.1)
d business whose owner wants to sell. ABC’s CFO
he data below. What should ABC consider paying?
Cash Flows (in ‘000 $)
3
4
(4.2)
(9.2)
(4.3)
(9.3)
5
(4.4)
(9.4)
TV
Given the coupons, par values, market rates and market prices below, please calculate the prices for bonds A
Coupon
Par value
Market rate
Cash flows:
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
Price
A
B
C
D
4.6%
5.0%
6.0%
4.7%
5.2%
5.9%
0.0%
1,000
5.5%
Coupon
Par Value
Cash flows:
Market price
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
Yield
ate the prices for bonds A-D and the yields for bonds D-G.
D
E
F
G
6.00%
6.70%
5.40%
0.00%
1,000
(757)
(813)
(799)
(505)
JTM pays its C-suite officers with stock
options. Treasury asked you to price them.
JTM’s stock trades at $17.50/share; U.S.
Treasurys, aka the risk-free rate, yield 3.40%
and stock’s volatility is 20%. The details of the
stock option offers are below. What are the
prices of the options?
a. Option Pricing
CEO
CFO
Exercise price
31.00
38.00
Maturity
11.0
8.0
Stock price
Risk free rate
Volatility
BS calculations:
d1
#NUM! #NUM!
N(d1)
#NUM! #NUM!
d2
#NUM! #NUM!
N(d2)
#NUM! #NUM!
Price of call
#NUM! #NUM!
JTM’s treasury unit bought jet fuel futures to
hedge its expenses. It uses 61M gallons/yr.
Each contract runs 42,000 gallons. The
contract price locked JTM at $4.7912/gal. At
maturity, JTM found that the spot price was
$4.7699/gal. In effect, had they not taken the
futures, they’d have paid less. What was the
profit/(loss) on the contract?
b. Futures Prices
CIO
37.00 Gallons
8.0 Gallons/contract
# of contracts
Contract price
Spot price
Profit/(Loss)
#NUM!
#NUM!
#NUM!
#NUM!
#NUM!
C. JTM’s can swap its bonds as per the data below. You are asked to show
the cash flows for the fixed and floating scenarios and the net difference
each year plus the net overall difference undiscounted and discounted
using a 7% discount rate. Show all fixed and floating payments as negative
cash flows. Net benefits use the formula: floating payments minus fixed
payments. State whether the swap is better or worse in the space provided.
c. Interest Rate Swap
Bond outstanding
Maturity (yrs.)
Fixed rate
Spread over BSTBY
BSTBY:
Years 1-2
Years 3-4
Years 5-7
900
30 Year 1
4.25% Year 2
1.25% Year 3
Year 4
3.00% Year 5
3.25% Year 6
3.55% Year 7
Cash Flows
Fixed
Floating
Undiscounted Net
Discounted Net
Net
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