Frm Risk Management Derivatives

Chapter 1: Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Section 1.1. The basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Section 1.2. Models and derivatives markets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12Section 1.3. Using derivatives the right way. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Section 1.4. Nineteen steps to using derivatives the right way. . . . . . . . . . . . . . . . . . . .19Literature Note. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Figure 1.1. Payoff of derivative which pays the 10m times the excess of the square of the

decimal interest rate over 0.01.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24Figure 1.2. Hedging with forward contract.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

Panel A. Income to unhedged exporter.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25Panel B. Forward contract payoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25Panel C. Hedged firm income. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25

Figure 1.3. Payoff of share and call option strategies. . . . . . . . . . . . . . . . . . . . . . . . . .26Figure 1.3.A. Payoff of buying one share of Amazon.com at $75. . . . . . . . . . .26Figure 1.3.B. Payoff of buying a call option on one share of Amazon.com with

exercise price of $75 for a premium of $10.. . . . . . . . . . . . . . . . . . . . . .26

Chapter 1: Introduction

August 5, 1999

Ren M. Stulz 1997, 1999

Chapter 2, page 1

Throughout history, the weather has determined the fate of nations, businesses, and

individuals. Nations have gone to war to take over lands with a better climate. Individuals have

starved because their crops were made worthless by poor weather. Businesses faltered because the

goods they produced were not in demand as a result of unexpected weather developments. Avoiding

losses due to inclement weather was the dream of poets and the stuff of science fiction novels - until

it became the work of financial engineers, the individuals who devise new financial instruments and

strategies to enable firms and individuals to better pursue their financial goals. Over the last few years,

financial products that can be used by individuals and firms to protect themselves against the financial

consequences of inclement weather have been developed and marketed. While there will always be

sunny and rainy days, businesses and individuals can now protect themselves against the financial

consequences of unexpectedly bad weather through the use of financial instruments. The introduction

of financial instruments that help firms and individuals to deal with weather risks is just one example

of the incredible growth in the availability of financial instruments for managing risks. Never in the

course of history have firms and individuals been able to mitigate the financial impact of risks as

effectively through the use of financial instruments as they can now.

There used to be stocks and bonds and not much else. Knowing about stocks and bonds was

enough to master the intricacies of financial markets and to choose how to invest ones wealth. A

manager had to know about the stock market and the bond market to address the problems of his

firm. Over the last thirty years, the financial instruments available to managers have become too

numerous to count. Not only can managers now protect their firms against the financial consequences

of bad weather, there is hardly a risk that they cannot protect their firm against if they are willing to

1 The American Heritage Dictionary of the English Language, Third Edition (1982),Houghton Mifflin Company, Boston.

Chapter 2, page 2

pay the appropriate price or a gamble that they cannot take through financial instruments. Knowing

stocks and bonds is therefore not as useful as it used to be. Attempting to know all existing financial

instruments is no longer feasible. Rather than knowing something about a large number of financial

instruments, it has become critical for managers to have tools that enable them to evaluate which

financial instruments - existing or to be invented - best suit their objectives. As a result of this

evolution, managers and investors are becoming financial engineers.

Beyond stocks and bonds, there is now a vast universe of financial instruments called

derivatives. In chemistry, a derivative is a compound derived from another and containing essential

elements of the parent substance.1 Derivatives in finance work on the same principle as in chemistry.

They are financial instruments whose payoffs are derived from something else, often but not

necessarily another financial instrument. It used to be easier to define the world of derivatives. Firms

would finance themselves by issuing debt and equity. Derivatives would then be financial instruments

whose payoffs would be derived from debt and equity. For instance, a call option on a firms stock

gives its owner the right to buy the stock at a given price, the exercise price. The call option payoff

is therefore derived from the firms stock. Unfortunately, defining the world of derivatives is no

longer as simple. Non-financial firms now sell derivatives to finance their activities. There are also

derivatives whose value is not derived from the value of financial instruments directly. Consider a

financial instrument of the type discussed in chapter 18 that promises its holder a payment equal to

ten million dollars times the excess of the square of the decimal interest rate over 0.01 in 90 days

where the interest rate is the London Interbank Offer Rate (LIBOR) on that day as reported by the

Chapter 2, page 3

British Bankers Association. Figure 1.1. plots the payoff of that financial instrument. If the interest

rate is 10%, this instrument therefore pays nothing, but if the interest rate is 20%, this instrument pays

$300,000 (i.e., 0.20 squared minus 0.01 times ten million). Such an instrument does not have a value

that depends directly on a primitive asset such as a stock or a bond.

Given the expansion of the derivatives markets, it is hard to come up with a concise definition

of derivatives that is more precise than the one given in the previous paragraph. A derivative is a

financial instrument with contractually specified payoffs whose value is uncertain when the contract

is initiated and which depend explicitly on verifiable prices and/or quantities. A stock option is a

derivative because the payoff is explicitly specified as the right to receive the stock in exchange of the

exercise price. With this definition of a derivative, the explicit dependence of payoffs on prices or

quantities is key. It distinguishes derivatives from common stock. The payoffs of a common stock are

the dividend payments. Dividends depend on all sorts of things, but this dependence is not explicit.

There is no formula for a common stock that specifies the size of the dividend at one point in time.

The formula cannot depend on subjective quantities or forecasts of prices: The payoff of a derivative

has to be such that it can be determined mechanically by anybody who has a copy of the contract.

Hence, a third party has to be able to verify that the prices and/or quantities used to compute the

payoffs are correct. For a financial instrument to be a derivative, its payoffs have to be determined

in such a way that all parties to the contract could agree to have them defined by a mathematical

equation that could be enforced in the courts because its arguments are observable and verifiable.

With a stock option, the payoff for the option holder is receiving the stock in exchange of paying the

exercise price. The financial instrument that pays $300,000 if the interest rate is 20% is a derivative

by this definition since the contractually specified payoff in ninety days is given by a formula that

Chapter 2, page 4

depends on a rate that is observable. A judge could determine the value of the contracts payoff by

determining the appropriate interest rate and computing the payoff according to the formula. A share

of IBM is not a derivative and neither is a plain vanilla bond issued by IBM. With this definition, the

variables that determine the payoffs of a derivative can be anything that the contracting parties find

useful including stock prices, bond prices, interest rates, number of houses destroyed by hurricanes,

gold prices, egg prices, exchange rates, and the number of individual bankruptcies within a calendar

year in the U.S.

Our definition of a derivative can be used for a weather derivative. Consider a financial

contract that specifies that the purchaser will receive a payment of one hundred million dollars if the

temperature at La Guardia at noon exceeds 80 degrees Fahrenheit one hundred days or more during

a calendar year. If one thinks of derivatives only as financial instruments whose value is derived from

financial assets, such a contract would not be called a derivative. Such a contract is a derivative with

our definition because it specifies a payoff, one hundred million dollars, that is a function of an

observable variable, the number of days the temperature exceeds 80 degrees at La Guardia at noon

during a calendar year. It is easy to see how such a weather derivative would be used by a firm for

risk management. Consider a firm whose business falls dramatically at high temperatures. Such a firm

could hedge itself against weather that is too hot by purchasing such a derivative. Perhaps the

counterparty to the firm would be an ice cream manufacturer whose business suffers when

temperatures are low. But it might also be a speculator who wants to make a bet on the weather.

At this point, there are too many derivatives for them to be counted and it is beyond

anybodys stamina to know them all. In the good old days of derivatives - late 1970s and early 1980s

- a responsible corporate manager involved in financial matters could reasonably have detailed

Chapter 2, page 5

knowledge of all economically relevant derivatives. This is no longer possible. At this point, the key

to success is being able to figure out which derivatives are appropriate and how to use them given

ones objectives rather than knowing a few of the many derivatives available. Recent history shows

that this is not a trivial task. Many firms and individuals have faced serious problems using derivatives

because they were not well equipped to evaluate their risks and uses. Managers that engaged in

poorly thought out derivatives transactions have lost their jobs, but not engaging in derivatives

transactions is not an acceptable solution. While derivatives used to be the province of finance

specialists, they are now intrinsic to the success of many businesses and businesses that do not use

them could generally increase their shareholders wealth by using them. A finance executive who

refuses to use derivatives because of these difficulties is like a surgeon who does not use a new

lifesaving instrument because some other surgeon made a mistake using it.

The remainder of this chapter is organized as follows. We discuss next some basic ideas

concerning derivatives and risk management. After explaining the role of models in the analysis of

derivatives and risk management, we discuss the steps one has to take to use derivatives correctly.

We then turn to an overview of the book.

Section 1.1. The basics.

Forward contracts and options are often called plain vanilla derivatives because they are

the simplest derivatives. A forward contract is a contract where no money changes hands when the

contract is entered into but the buyer promises to purchase an asset or a commodity at a future date,

the maturity date, at a price fixed at origination of the contract, the forward price, and where the

seller promises to deliver this asset or commodity at maturity in exchange of the agreed upon price.

Chapter 2, page 6

An option gives its holder a right to buy an asset or a commodity if it is a call option or to sell an

asset or a commodity if it is a put option at a price agreed upon when the option contract is written,

the exercise price. Predating stocks and bonds, forward contracts and options have been around a

long time. We will talk about them throughout the book. Lets look at them briefly to get a sense of

how derivatives work and of how powerful they can be.

Consider an exporter who sells in Europe. She will receive one million euros in ninety days.

The dollar value of the payment in ninety days will be one million times the dollar price of the euro.

As the euro becomes more valuable, the exporter receives more dollars. The exporter is long in the

euro, meaning that she benefits from an increase in the price of the euro. Whenever cash flow or

wealth depend on a variable - price or quantity - that can change unexpectedly for reasons not under

our control, we call such a variable a risk factor. Here, the dollar price of a euro is a risk factor. In

risk management, it is always critical to know what the risk factors are and how their changes affect

us. The sensitivity of cash flow or wealth to a risk factor is called the exposure to that risk factor.

The change in cash flow resulting from a change in a risk factor is equal to the exposure times the

change in the risk factor. Here, the risk factor is the dollar price of the euro and the cash flow impact

of a change in the dollar price of the euro is one million times the change in the price of the euro. The

exposure to the euro is therefore one million euros. We will see that measuring exposure is often

difficult. Here, however, it is not.

In ninety days, the exporter will want to convert the euros into dollars to pay her suppliers.

Lets assume that the suppliers are due to receive $950,000. As long as the price of the euro in ninety

days is at least 95 cents, everything will be fine. The exporter will get at least $950,000 and therefore

can pay the suppliers. However, if the price of the euro is 90 cents, the exporter receives $900,000

Chapter 2, page 7

dollars and is short $50,000 to pay her suppliers. In the absence of capital, she will not be able to pay

the suppliers and will have to default, perhaps ending the business altogether. A forward contract

offers a solution for the exporter that eliminates the risk of default. By entering a forward contract

with a maturity of 90 days for one million euros, the exporter promises to deliver one million euros

to the counterparty who will in exchange pay the forward price per euro times one million. For

instance, if the forward price per euro is 99 cents, the exporter will receive $990,000 in ninety days

irrespective of the price of the euro in ninety days. With the forward contract, the exporter makes

sure that she will be able to pay her suppliers.

Panel A of figure 1.2. shows the payoff of the position of the exporter in the cash market, i.e.,

the dollars the exporter gets for the euros it sells on the cash market if she decides to use the cash

market to get dollars in 90 days. Panel B of figure 1.2. shows the payoff of a short position in the

forward contract. A short position in a forward contract benefits from a fall in the price of the

underlying. The underlying in a forward contract is the commodity or asset one exchanges at

maturity of the forward contract. In our example, the underlying is the euro. The payoff of a short

position is the receipt of the forward price per unit times the number of units sold minus the value of

the price of the underlying at maturity times the number of units delivered. By selling euros through

a forward contract, our exporter makes a bigger profit from the forward contract if the dollar price

of the euro falls more. If the euro is at $1.1 at maturity, our exporter agreed to deliver euros worth

$1.1 per unit at the price of $0.99, so that she loses $0.11 per unit or $110,000 on the contract. In

contrast, if the euro is at $0.90 at maturity, the exporter gets $0.99 per unit for something worth $0.9

per unit, thereby gaining $90,000 on the forward contract. With the forward contract, the long -, i.e.,

the individual who benefits from an increase in the price of the underlying - receives the underlying

Chapter 2, page 8

at maturity and pays the forward price. His profit is therefore the cash value of the underlying at

maturity minus the forward price times the size of the forward position. The third panel of Figure

1.1., panel C, shows how the payoff from the forward contract added to the payoff of the cash

position of the exporter creates a risk-free position. A financial hedge is a financial position that

decreases the risk resulting from the exposure to a risk factor. Here, the cash position is one million

euros, the hedge is the forward contract. In this example, the hedge is perfect - it eliminates all the

risk so that the hedged position, defined as the cash position plus the hedge, has no exposure to the

risk factor.

Our example has three important lessons. First, through a financial transaction, our exporter

can eliminate all her risk without spending any cash to do so. This makes forward contracts

spectacularly useful. Unfortunately, life is more complicated. Finding the best hedge is often difficult

and often the best hedge is not a perfect hedge.

Second, to eliminate the risk of the hedged position, one has to be willing to make losses on

derivatives positions. Our exporter takes a forward position such that her hedged cash flow has no

uncertainty - it is fixed. When the euro turns out to be worth more than the forward price, the

forward contract makes a loss. This is the case when the price of the euro is $1.1. This loss exactly

offsets the gain made on the cash market. It therefore makes no sense whatsoever to consider

separately the gains and losses of derivatives positions from the rest of the firm when firms use

derivatives to hedge. What matters are the total gain and loss of the firm.

Third, when the exporter enters the forward contract, she agrees to sell euros at the forward

price. The counterparty in the forward contract therefore agrees to buy euros at the forward price.

No money changes hands except for the agreed upon exchange of euros for dollars. Since the

Chapter 2, page 9

counterparty gets the mirror image of what the exporter gets, if the forward contract has value at

inception for the exporter in that she could sell the forward contract and make money, it has to have

negative value for the counterparty. In this case, the counterparty would be better off not to enter the

contract. Consequently, for the forward contract to exist, it has to be that it has no value when

entered into. The forward price must therefore be the price that insures that the forward contract has

no value at inception.

Like the forward contract, many derivatives have no value at inception. As a result, a firm can

often enter a derivatives contract without leaving any traces in its accounting statements because no

cash is used and nothing of value is acquired. To deal with derivatives, a firm has to supplement its

conventional accounting practices with an accounting system that takes into account the risks of

derivatives contracts. We will see how this can be done.

Lets now consider options. The best-known options are on common stock. Consider a call

option on Amazon.com. Suppose the current stock price is $75 and the price of a call option with

exercise price of $75 is $10. Such an option gives the right to its holder to buy a fixed number of

shares of Amazon.com stock at $75. Options differ as to when the right can be exercised. With

European options, the right can only be exercised at maturity. In contrast, American options are

options that can be exercised at maturity and before. Consider now an investor who believes that

Amazon.com is undervalued by the market. This individual could buy Amazon.com shares and could

even borrow to buy such shares. However, this individual might be concerned that there is some

chance he will turn out to be wrong and that something bad might happen to Amazon.com. In this

case, our investor would lose part of his capital. If the investor wants to limit how much of his capital

he can lose, he can buy a call option on Amazon.com stock instead of buying Amazon.com shares.

Chapter 2, page 10

In this case, the biggest loss our investor would make would be to lose the premium he paid to

acquire the call option. Figure 1.3. compares the payoff for our investor of holding Amazon.com

shares and of buying a call option instead at maturity of the option assuming it is a European call

option. If the share price falls to $20, the investor who bought the shares loses $55 per share bought,

but the investor who bought the call option loses only the premium paid for the call of $10 per share

since he is smart enough not to exercise a call option that requires him to pay $75 for shares worth

$20. If the share price increases to $105, the investor who bought the shares gains $30 per share, but

the investor who bought options gains only $20 since he gains $30 per share when he exercises the

call option but had paid a premium of $10 per share. Our investor could have used a different

strategy. He could have bought Amazon.com shares and protected himself against losses through the

purchase of put options. A put option on a stock gives the right to sell shares at a fixed price. Again,

a put option can be a European or an American option.

With our example of a call option on Amazon.com, the investor has to have cash of $75 per

share bought. He might borrow some of that cash, but then his ability to invest depends on his credit.

To buy a call, the investor has to have cash of $10. Irrespective of which strategy the investor uses,

he gets one dollar for each dollar that Amazon.com increases above $75. If the share price falls below

$75, the option holder loses all of the premium paid but the investor in shares loses less as long as the

stock price does not fall by $10 or more. Consider an investor who is limited in his ability to raise

cash and can only invest $10. This investor can get the same gain per dollar increase in the stock price

as an investor who buys a share if he buys the call. If this investor uses the $10 to buy a fraction of

a share, he gets only $0.13 per dollar increase in the share price. To get a gain of one dollar from a

one dollar increase in the share price, our investor with $10 would have to borrow $65 to buy one

Chapter 2, page 11

share. In other words, he would have to borrow $6.5 for each dollar of capital. Option strategies

therefore enable the investor to lever up his resources without borrowing explicitly. The same is true

for many derivatives strategies. This implicit leverage can make the payoff of derivatives strategies

extremely volatile. The option strategy here is more complicated than a strategy of borrowing $65

to buy one share. This is because the downside risk is different between the borrowing strategy and

the option strategy. If the stock price falls to $20, the loss from the call strategy is $10 but the loss

from the borrowing strategy is $55. The option payoff is nonlinear. The gain for a one dollar increase

in the share price from $75 is not equal to minus the loss for a one dollar decrease in the share price

from $75. This nonlinearity is typical of derivatives. It complicates the analysis of the pricing of these

financial instruments as well as of their risk.

Call and put options give their holder a right. Anybody who has the right but not the

obligation to do something will choose to exercise the right to make himself better off. Consequently,

a call option is never exercised if the stock price is below the exercise price and a put option is never

exercised if the stock price is above the exercise price. Whoever sells an option at initiation of the

contract is called the option writer. The call option writer promises to deliver shares for the exercise

price and the put option writer promises to receive shares in exchange of the exercise price. When

an option is exercised, the option writer must always deliver something that is worthwhile. For the

option writer to be willing to deliver something worthwhile upon exercise, she must receive cash

when she agrees to enter the option contract. The problem is then to figure out how much the option

writer should receive to enter the contract.

Chapter 2, page 12

Section 1.2. Models and derivatives markets.

To figure out the price of an option, one has to have a model. To figure out whether it is

worthwhile to buy an option to hedge a risk, one has to be able to evaluate whether the economic

benefits from hedging the risk outweigh the cost from purchasing the option. This requires a model

that allows us to quantify the benefits of hedging. Models therefore play a crucial role in derivatives

and risk management. Models are simplified representations of reality that attempt to capture what

is essential. One way to think of models is that they are machines that allow us to see the forest rather

than only trees. No model is ever completely right because every model always abstracts from some

aspects of the real world. Since there is no way for anybody to take into account all the details of the

real world, models are always required to guide our thinking. It is easy to make two mistakes with

models. The first mistake is to think that a model is unrealistic because it misses some aspect of the

real world. Models do so by necessity. The key issue is not whether models miss things, but rather

whether they take enough things into account that they are useful. The second mistake is to believe

that if we have a model, we know the truth. This is never so. With a good model, one knows more

than with a bad model. Good models are therefore essential. Things can still go wrong with good

models because no model is perfect.

Stock options were traded in the last century and much of this century without a satisfactory

model that allowed investors and traders to figure out a price for these options. Markets do not have

to have a model to price something. To obtain a price, an equilibrium for a product where demand

equals supply is all that is required. Operating without a model is like flying a plane without

instruments. The plane can fly, but one may not get where one wants to go. With options, without

a model, one cannot quantify anything. One can neither evaluate a market price nor quantify the risk

Chapter 2, page 13

of a position. Lack of a model to price options was therefore a tremendous impediment to the growth

of the option market. The lack of a model was not the result of a lack of trying. Even Nobel

prizewinners had tried their hand at the problem. People had come up with models, but they were just

not very useful because to use them, one had to figure out things that were not observable. This lasted

until the early 1970s. At that time, two financial economists in Boston developed a formula that

revolutionized the field of options and changed markets for derivatives forever. One, Fischer Black,

was a consultant. The other one, Myron Scholes, was an assistant professor at MIT who had just

earned a Ph.D. in finance from the University of Chicago. These men realized that there was a trading

strategy that would yield the same payoff as an option but did not use options. By investing in stocks

and bonds, one could obtain the same outcome as if one had invested in options. With this insight and

the help of a third academic, Robert Merton, they derived a formula that was instantly famous -

except with the editors of academic journals who, amazingly, did not feel initially that it was

sufficiently useful to be publishable. This formula is now called the Black-Scholes formula for the

pricing of options. With this formula, one could compute option prices using only observable

quantities. This formula made it possible to assess the risk of options as well as the value of portfolios

of options.

There are few achievements in social sciences that rival the Black-Scholes formula. This

formula is tremendously elegant and represents a mathematical tour-de-force. At the same time, and

more importantly, it is so useful that it has spawned a huge industry. Shortly after the option pricing

formula was discovered, the Chicago Board of Trade started an options exchange. Business on this

exchange grew quickly because of the option pricing formula. Traders on the exchange would have

calculators with the formula programmed in them to conduct business. When Fischer Black or Myron

Chapter 2, page 14

Scholes would show up at the exchange, they would receive standing ovations because everybody

knew that without the Black-Scholes formula, business would not be what it was.

The world is risky. As a result, there are many opportunities for trades to take place where

one party shifts risks to another party through derivatives. These trades must be mutually beneficial

or otherwise they would not take place. The purchaser of a call option wants to benefit from stock

price increases but avoid losses. He therefore pays the option writer to provide a hedge against

potential losses. The option writer does so for appropriate compensation. Through derivatives,

individuals and firms can trade risks and benefit from these trades. Early in the 1970s, this trading of

risks took place through stock options and forward transactions. However, this changed quickly. It

was discovered that the Black-Scholes formula was useful not only to price stock options, but to

price any kind of financial contract that promises a payoff that depends on a price or a quantity.

Having mastered the Black-Scholes formula, one could price options on anything and everything.

This meant that one could invent new instruments and find their value. One could price exotic

derivatives that had little resemblance to traditional options. Exotic derivatives are all the derivatives

that are not plain vanilla derivatives or cannot be created as a portfolio of plain vanilla derivatives.

The intellectual achievements involved in the pricing of derivatives made possible a huge industry.

Thirty years ago, the derivatives industry had no economic importance. We could produce countless

statistics on its current importance. Measuring the size of the derivatives industry is a difficult

undertaking. However, the best indicator of the growth and economic relevance of this industry is

that observers debate whether the derivatives markets are bigger and more influential than the

markets for stocks and bonds and often conclude that they are.

Because of the discovery of the Black-Scholes formula, we are now in a situation where any

Chapter 2, page 15

type of financial payoff can be obtained at a price. If a corporation would be better off receiving a

large payment in the unlikely event that Citibank, Chase, and Morgan default in the same month, it

can go to an investment bank and arrange to enter the appropriate derivatives contract. If another

corporation wants to receive a payment which is a function of the square of the yen/dollar exchange

rate if the volatility of the S&P500 exceeds 35% during a month, it can do so. There are no limits to

the type of financial contracts that can be written. However, anybody remembers what happened in

their youth when suddenly their parents were not watching over their shoulders. Without limits, one

can do good things and one can do bad things. One can create worthwhile businesses and one can

destroy worthwhile businesses. It is therefore of crucial importance to know how to use derivatives

the right way.

Section 1.3. Using derivatives the right way.

A corporate finance specialist will see new opportunities to take positions in derivatives all

the time. He will easily think of himself as a master of the universe, knowing which instruments are

too cheap, which are too expensive. As it is easy and cheap to take positions in derivatives, this

specialist can make dramatic changes in the firms positions in instants. With no models to measure

risks, he can quickly take positions that can destroy the firm if things go wrong. Not surprisingly,

therefore, some firms have made large losses on derivatives and some firms have even disappeared

because of derivatives positions that developed large losses.

The first thing to remember, therefore, is that there are few masters of the universe. For every

corporate finance specialist who thinks that a currency is overvalued, there is another one who thinks

with the same amount of conviction that currency is undervalued. It may well be that a corporate

Chapter 2, page 16

finance specialist is unusually good at forecasting exchange rates, but typically, that will not be the

case. To beat the market, one has to be better than the investors who have the best information - one

has to be as good as George Soros at the top of his game. This immediately disqualifies most of us.

If mutual fund managers whose job it is to beat the market do not do so on average, why could a

corporate finance specialist or an individual investor think that they have a good enough chance of

doing so that this should direct their choice of derivatives positions? Sometimes, we know something

that has value and should trade on it. More often, though, we do not.

A firm or an individual that take no risks hold T-bills and earn the T-bill rate. Rockfeller had

it right when he said that one cannot get rich by saving. If one is to become rich, one has to take risks

to exploit valuable opportunities. Valuable opportunities are those where we have a comparative

advantage in that they are not as valuable to others. Unfortunately, the ability to bear risk for

individuals or firms is limited by lack of capital. An individual who has lost all his wealth cannot go

back to the roulette table. A firm that is almost bankrupt cannot generally take advantage of the same

opportunities as a firm that is healthy. This forces individuals and firms to avoid risks that are not

profitable so that they can take on more risks that are advantageous. Without derivatives, this is often

impossible. Derivatives enable individuals and firms to shed risks and take on risks cheaply.

To shed risks that are not profitable and take on the ones that are profitable, it is crucial to

understand the risks one is exposed to and to evaluate their costs and benefits. Risks cannot be

analyzed without statistics. One has to be able to quantify risks so that one can understand their costs

and so that one can figure out whether transactions decrease or increase risk and by how much. When

one deals with risks, it is easy to let ones biases take charge of ones decisions. Individuals are just

not very good at thinking about risks without quantitative tools. They will overstate the importance

Chapter 2, page 17

of some risks and understate the importance of others. For instance, individuals put too much weight

on recent past experience. If a stock has done well in the recent past, they will think that it will do

unusually well in the future so that it has little risk. Yet, a quick look at the data will show them that

this is not so. They will also be reluctant to realize losses even though quantitative analysis will show

that it would be in their best interest to do so. Psychologists have found many tendencies that people

have in dealing with risk that lead to behavior that cannot be justified on quantitative grounds. These

tendencies have even led to a new branch of finance called behavioral finance. This branch of finance

attempts to identify how the biases of individuals influence their portfolio decisions and asset returns.

To figure out which risks to bear and which risks to shed, one therefore must have models

that allow us to figure out the economic value of taking risks and shedding risks. Hence, to use

derivatives in the right way, one has to be able to make simple statements like the following: If I keep

my exposure to weather risk, the value of my firm is X; if I shed my exposure to weather risk, the

value of my firm after purchasing the appropriate financial instruments is Y; if Y is greater than X,

I shed the weather risk. For individuals, it has to be that the measure of their welfare they focus on

is affected by a risk and they can establish whether shedding the risk makes them better off than

bearing it. To figure out the economic value of taking risks and shedding risks, one has to be able to

quantify risks. This requires statistics. One has to be able to trace out the impact of risks on firm value

or individual welfare. This requires economic analysis. Finally, one must know how a derivative

position will affect the risks the firm is exposed to. This requires understanding the derivatives and

their pricing. A derivative could eliminate all of a risk, but it may be priced so that one is worse off

without the risk than with. A derivatives salesperson could argue that a derivative is the right one to

eliminate a risk we are concerned about, but a more thorough analysis might reveal that the derivative

Chapter 2, page 18

actually increases our exposure to other risks so that we would be worse off purchasing it.

To use derivatives the right way, one has to define an objective function. For a firm, the

objective function is generally to maximize shareholder wealth. For an investor, there will be some

measure of welfare that she focuses on. Objective functions are of little use unless we can measure

the impact of choices on our objectives. We therefore have to be able to quantify how various risks

affect our objective function. Doing so, we will find some risks that make us worse off and, possibly,

others that make us better off. Having figured out which risks are costly, we need to investigate

whether there are derivatives strategies that can be used to improve our situation. This requires us

to be able to figure out the impact of these strategies on our objective function. The world is not

static. Our exposures to risks change all the time. Consequently, derivatives positions that were

appropriate yesterday may not be today. This means that we have to be able to monitor these

positions and monitor our risk exposures to be able to make changes when it is appropriate to do so.

This means that we must have systems in place that make it possible to monitor our risk exposures.

Using derivatives the right way means that we look ahead and figure out which risks we

should bear and how. Once we have decided which risks we should bear, nature has to run its course.

In our example of the exporter to Europe, after she entered the forward contract, the euro ended up

either above or below the forward price of $0.99. If it ended up above, the exporter actually lost

money on the contract. The temptation would be to say that she made a poor use of derivatives since

she lost money on a derivative. This is simply not the way to think about derivatives use. When the

decision was made to use the derivative, the exporter figured out that she was better off hedging the

currency risk. She had no information that allowed her to figure out that the price of the euro was

going to appreciate and hence could not act on such information. At that time, it was as likely that

Chapter 2, page 19

the exchange rate would fall and that she would have gained from her forward position. If a derivative

is bought to insure against losses, it is reasonable to think that about half the time, the losses one

insures against will not take place and the derivative will therefore not produce a gain to offset losses.

The outcome of a derivatives transactions does not tell us whether we were right or wrong in entering

the transaction any more than whether our house burns down or not tells us whether we were right

or wrong to buy fire insurance. Houses almost never burn down, so that we almost always make a

loss on fire insurance. We buy the insurance because we know ex ante that we are better off shedding

the financial risk of having to replace the house.

Section 1.4. Nineteen steps to using derivatives the right way.

This book has nineteen chapters. Each chapter will help you to understand better how to use

derivatives the right way. Because of the biases in decision making in the absence of quantitative

evaluations, risk has to be evaluated using statistical tools that are not subject to the hidden biases

of the human mind. We therefore have to understand how to measure risk. In chapters 2 through 4,

we investigate how to measure risk and how risk affects firm value and the welfare of individuals. A

crucial issue in risk measurement is that lower tail risks - risks that things can go wrong in a serious

way - affect firm value and individual welfare in ways that are quite different from other risks. Small

cash flow fluctuations around their mean generally have little impact on firm value. Extreme outcomes

can mean default and bankruptcy. It is therefore essential to have quantitative measures of these lower

tail risks. We therefore introduce such measures in chapter 4. These measures enable us to assess the

impact of derivatives strategies on risk and firm value. As we consider different types of derivatives

throughout the book, we will have to make sure that we are able to use our risk measures to evaluate

Chapter 2, page 20

these derivatives.

After chapter 4, we will have the main framework of risk management and derivatives use in

place. We will know how to quantify risks, how to evaluate their costs and benefits, and how to make

decisions when risk matters. This framework is then used throughout the rest of the book to guide

us in figuring out how to use derivatives to manage risk. As we progress, however, we learn about

derivatives uses but also learn more about how to quantify risk. We start by considering the uses and

pricing of plain vanilla derivatives. In chapter 5, we therefore discuss the pricing of forward contracts

and of futures contracts. Futures contracts are similar to forward contracts but are traded on

organized exchanges. Chapters 6 through 9 discuss extensively how to use forward and futures

contracts to manage risk. We show how to set up hedges with forwards and futures. Chapter 8

addresses many of the issues that arise in estimating foreign exchange rate exposures and hedging

them. Chapter 9 focuses on interest rate risks.

After having seen how to use forwards and futures, we turn our attention on options. Chapter

10 shows why options play an essential role in risk management. We analyze the pricing of options

in chapters 11 and 12. Chapter 12 is completely devoted to the Black-Scholes formula. Unfortunately,

options complicate risk measurement. The world is full of options, so that one cannot pretend that

they do not exist to avoid the risk measurement problem. Chapter 13 therefore extends our risk

measurement apparatus to handle options and more complex derivatives. Chapter 14 covers fixed

income options. After chapter 14, we will have studied plain vanilla derivatives extensively and will

know how to use them in risk management. We then move beyond plain vanilla derivatives. In chapter

15, we address the tradeoffs that arise when using derivatives that are not plain vanilla derivatives.

We then turn to swaps in chapter 16. Swaps are exchanges of cash flows: one party pays well-defined

Chapter 2, page 21

cash flows to the other party in exchange for receiving well-defined cash flows from that party. The

simplest swap is one where one party promises to pay cash flows corresponding to the interest

payments of fixed rate debt on a given amount to a party that promises to pay cash flows

corresponding to the payments of floating rate debt on the same amount. We will see that there are

lots of different types of swaps. In chapter 17, we discuss the pricing and uses of exotic options.

Credit risks are important by themselves because they are a critical source of risk for firms. At the

same time, one of the most recent growth areas in derivative markets involves credit derivatives,

namely derivatives that can be used to lay off credit risks. In chapter 18, we analyze credit risks and

show how they can be eliminated through the uses of credit derivatives.

After developing the Black-Scholes formula, eventually Fischer Black, Robert Merton, and

Myron Scholes all played major roles in the business world. Fischer Black became a partner at

Goldman Sachs, dying prematurely in 1996. Robert Merton and Myron Scholes became partners in

a high-flying hedge fund company named Long-term Capital Management that made extensive use

of derivatives. In 1997, Robert Merton and Myron Scholes received the Nobel Memorial Prize in

Economics for their contribution to the pricing of options. In their addresses accepting the prize, both

scholars focused on how derivatives can enable individuals and firms to manage risks. In September

1998, the Federal Reserve Bank of New York arranged for a group of major banks to lend billions

of dollars to that hedge fund. The Fed intervened because the fund had lost more than four billion

dollars of its capital, so it now had less than half a billion dollars to support a balance sheet of more

than $100 billion. Additional losses would have forced the long-term capital fund to unwind positions

in a hurry, leading regulators to worry that this would endanger the financial system of the Western

world. The press was full of articles about how the best and the brightest had failed. This led to much

Chapter 2, page 22

chest-beating about our ability to manage risk. Some thought that if Nobel prizewinners could not

get it right, there was little hope for the rest of us. James Galbraith in an article in the Texas Observer

even went so far as to characterize their legacy as follows: They will be remembered for having tried

and destroyed, completely, utterly and beyond any redemption, their own theories.

In the last chapter, we will consider the lessons of this book in the light of the LTCM

experience. Not surprisingly, we will discover that the experience of LTCM does not change the main

lessons of this book and that, despite the statements of James Galbraith, the legacy of Merton and

Scholes will not be the dramatic losses of LTCM but their contribution to our understanding of

derivatives. This book shows that risks can be successfully managed with derivatives. For those who

hear about physicists and mathematicians flocking to Wall Street to make fortunes in derivatives, you

will be surprised to discover that this book is not about rocket science. If there was a lesson from

LTCM, it is that derivatives are too important to be left to rocket scientists. What makes financial

markets different from the experiments that physicists focus on in their labs, it is that financial history

does not repeat itself. Markets are new every day. They surprise us all the time. There is no

mathematical formula that guarantees success every time.

After studying this book, all you will know is how, through careful use of derivatives, you can

increase shareholder wealth and improve the welfare of individuals. Derivatives are like finely tuned

racing cars. One would not think of letting an untutored driver join the Indianapolis 500 at the wheel

of a race car. However, if the untutored driver joins the race at the wheel of a Ford Escort, he has

no chance of ever winning. The same is true with derivatives. Untutored users can crash and burn.

Nonusers are unlikely to win the race.

Chapter 2, page 23

Literature Note

Bernstein (1992) provides a historical account of the interaction between research and practice in the

history of derivatives markets. The spectacular growth in financial innovation is discussed in Miller

(1992). Finnerty (1992) provides a list of new financial instruments developed since the discovery of

the Black-Scholes formula. Black () provides an account of the discovery of the Black-Scholes

formula. Allen and Gale (), Merton (1992), and Ross (1989) provide an analysis of the determinants

of financial innovation and Merton (1992).

Chapter 2, page 24

5 10 15 20 25 30

0.2

0.4

0.6

0.8

Figure 1.1. Payoff of derivative which pays the 10m times the excess of the square of thedecimal interest rate over 0.01.

Million dollars

Interest rate in percent

Chapter 2, page 25

Exchange rate

Income to firm

999,000

Unhedged income

0.990.90

900,000

Exchange rate

Income to firm

Forwardgain

Forward loss

0.99

0.9

99,000

Exchange rate

Income to firm

Forwardgain

Forward loss999,000

Unhedged income

Hedged income

0.99

Figure 1.2. Hedging with forward contract. The firms income is in dollars and the exchange rateis the dollar price of one euro.

Panel A. Income to unhedged exporter. The exporter receives euro 1m in 90 days, so that thedollar income is the dollar price of the euro times 1m if the exporter does not hedge.

Panel B. Forward contract payoff. Theforward price for the euro is $0.99. If the spotexchange rate is $0.9, the gain from theforward contract is the gain from selling euro1m at $0.99 rather than $0.9.

Panel C. Hedged firm income. The firm sellsits euro income forward at a price of $0.99 pereuro. It therefore gets a dollar income of$990,000 for sure, which is equal to theunhedged firm income plus the forwardcontract payoff.

Chapter 2, page 26

Amazon.com price

105

Payoff

20 75

-55

+30

Amazon.com price

105

Payoff

75

0

20

-10

20

Figure 1.3. Payoff of share and call option strategies.

Figure 1.3.A. Payoff of buying one share of Amazon.com at $75.

Figure 1.3.B. Payoff of buying a call option on one share of Amazon.com with exercise priceof $75 for a premium of $10.

Chapter 2, page 27

Chapter 2: Investors, Derivatives, and Risk ManagementChapter objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Section 2.1. Evaluating the risk and the return of individual securities and portfolios.. . .3

Section 2.1.1. The risk of investing in IBM.. . . . . . . . . . . . . . . . . . . . . . . . . . . .4Section 2.1.2. Evaluating the expected return and the risk of a portfolio.. . . . .11

Section 2.2. The benefits from diversification and their implications for expected returns.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

Section 2.2.1. The risk-free asset and the capital asset pricing model. . . . . . . . .21Section 2.2.2. The risk premium for a security.. . . . . . . . . . . . . . . . . . . . . . . . .24

Section 2.3. Diversification and risk management.. . . . . . . . . . . . . . . . . . . . . . . . . . . .33Section 2.3.1. Risk management and shareholder wealth. . . . . . . . . . . . . . . . . .36Section 2.3.2. Risk management and shareholder clienteles. . . . . . . . . . . . . . . .41Section 2.3.3. The risk management irrelevance proposition. . . . . . . . . . . . . . .46

1) Diversifiable risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .462) Systematic risk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463) Risks valued by investors differently than predicted by the CAPM. . .46Hedging irrelevance proposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

Section 2.4. Risk management by investors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47Section 2.5. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50Literature Note. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52Key concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54Review questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55Questions and exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57Figure 2.1. Cumulative probability function for IBM and for a stock with same return and

twice the volatility.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60Figure 2.2. Normal density function for IBM assuming an expected return of 13% and a

volatility of 30% and of a stock with the same expected return but twice thevolatility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61

Figure 2.3. Efficient frontier without a riskless asset. . . . . . . . . . . . . . . . . . . . . . . . . . .62Figure 2.4. The benefits from diversification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63Figure 2.5. Efficient frontier without a riskless asset.. . . . . . . . . . . . . . . . . . . . . . . . . .64Figure 2.6. Efficient frontier with a risk-free asset.. . . . . . . . . . . . . . . . . . . . . . . . . . . .65Figure 2.7. The CAPM.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66Box: T-bills.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67Box: The CAPM in practice.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68SUMMARY OUTPUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

Chapter 2: Investors, Derivatives, and Risk Management

December 1, 1999

Ren M. Stulz 1997, 1999

1Chapter objectives

1. Review expected return and volatility for a security and a portfolio.

2. Use the normal distribution to make statements about the distribution of returns of a portfolio.

3. Evaluate the risk of a security in a portfolio.

4. Show how the capital asset pricing model is used to obtain the expected return of a security.

4. Demonstrate how hedging affects firm value in perfect financial markets.

5. Show how investors evaluate risk management policies of firms in perfect financial markets.

6. Show how investors can use risk management and derivatives in perfect financial markets to make

themselves better off.

2During the 1980s and part of the 1990s, two gold mining companies differed dramatically in

their risk management policies. One company, Homestake, had a policy of not managing its gold price

risk at all. Another company, American Barrick, had a policy of eliminating most of its gold price risk

using derivatives. In this chapter we investigate whether investors prefer one policy to the other and

why. More broadly, we consider how investors evaluate the risk management policies of the firms

in which they invest. In particular, we answer the following question: When does an investor want

a firm in which she holds shares to spend money to reduce the volatility of its stock price?

To find out how investors evaluate firm risk management policies, we have to study how

investors decide to invest their money and how they evaluate the riskiness of their investments. We

therefore examine first the problem of an investor that can invest in two stocks, one of them IBM and

the other a fictional one that we call XYZ. This examination allows us to review concepts of risk and

return and to see how one can use a probability distribution to estimate the risk of losing specific

amounts of capital invested in risky securities. Throughout this book, it will be of crucial importance

for us to be able to answer questions such as: How likely is it that the value of a portfolio of securities

will fall by more than 10% over the next year? In this chapter, we show how to answer this question

when the portfolio does not include derivatives.

Investors have two powerful risk management tools at their disposal that enable them to

invest their wealth with a level of risk that is optimal for them. The first tool is asset allocation. An

investors asset allocation specifies how her wealth is allocated across types of securities or asset

classes. For instance, for an investor who invests in equities and the risk-free asset, the asset

allocation decision involves choosing the fraction of his wealth invested in equities. By investing less

in equities and more in the risk-free asset, the investor reduces the risk of her invested wealth. The

3second tool is diversification. Once the investor has decided how much to invest in an asset class,

she has to decide which securities to hold and in which proportion. A portfolios diversification is the

extent to which the funds invested are distributed across securities to reduce the dependence of the

portfolios return on the return of individual securities. If a portfolio has only one security, it is not

diversified and the investor always loses if that security performs poorly. A diversified portfolio can

have a positive return even though some of its securities make losses because the gains from the other

securities can offset these losses. Diversification therefore reduces the risk of funds invested in an

asset class. We will show that these risk management tools imply that investors do not need the help

of individual firms to achieve their optimal risk-return takeoff. Because of the availability of these risk

management tools, investors only benefit from a firms risk management policy if that policy

increases the present value of the cash flows the firm is expected to generate. In the next chapter, we

demonstrate when and how risk management by firms can make investors better off.

After having seen the conditions that must be met for a firms risk management policy to

increase firm value, we turn to the question of when investors have to use derivatives as additional

risk management tools. We will see that derivatives enable investors to purchase insurance, to hedge,

and to take advantage of their views more efficiently.

Section 2.1. Evaluating the risk and the return of individual securities and portfolios.

Consider the situation of an investor with wealth of $100,000 that she wants to invest for one

year. Her broker recommends two common stocks, IBM and XYZ. The investor knows about IBM,

but has never heard of XYZ. She therefore decides that first she wants to understand what her wealth

will amount to after holding IBM shares for one year. Her wealth at the end of the year will be her

4initial wealth times one plus the rate of return of the stock over that period. The rate of return of the

stock is given by the price appreciation plus the dividend payments divided by the stock price at the

beginning of the year. So, if the stock price is $100 at the beginning of the year, the dividend is $5,

and the stock price appreciates by $20 during the year, the rate of return is (20 + 5)/100 or 25%.

Throughout the analysis in this chapter, we assume that the frictions which affect financial

markets are unimportant. More specifically, we assume that there are no taxes, no transaction costs,

no costs to writing and enforcing contracts, no restrictions on investments in securities, no differences

in information across investors, and investors take prices as given because they are too small to affect

prices. Financial economists call markets that satisfy the assumptions we just listed perfect financial

markets. The assumption of the absence of frictions stretches belief, but it allows us to zero in on

first-order effects and to make the point that in the presence of perfect financial markets risk

management cannot increase the value of a firm. In chapter 3, we relax the assumption of perfect

financial markets and show how departures from this assumption make it possible for risk

management to create value for firms. For instance, taxes make it advantageous for firms and

individuals to have more income when their tax rate is low and less when their tax rate is high. Risk

management with derivatives can help firms and individuals achieve this objective.

Section 2.1.1. The risk of investing in IBM.

Since stock returns are uncertain, the investor has to figure out which outcomes are likely and

which are not. To do this, she has to be able to measure the likelihood of possible returns. The

statistical tool used to measure the likelihood of various returns for a stock is called the stocks

return probability distribution. A probability distribution provides a quantitative measure of the

5likelihood of the possible outcomes or realizations for a random variable. Consider an urn full of balls

with numbers on them. There are multiple balls with the same number on them. We can think of a

random variable as the outcome of drawing a ball from the urn. The urn has lots of different balls, so

that we do not know which number will come up. The probability distribution specifies how likely

it is that we will draw a ball with a given number by assigning a probability for each number that can

take values between zero and one. If many balls have the same number, it is more likely that we will

draw a ball with that number so that the probability of drawing this number is higher than the

probability of drawing a number which is on fewer balls. Since a ball has to be drawn, the sum of the

probabilities for the various balls or distinct outcomes has to sum to one. If we could draw from the

urn a large number of times putting the balls back in the urn after having drawn them, the average

number drawn would be the expected value. More precisely, the expected value is a probability

weighted average of the possible distinct outcomes of the random variable. For returns, the

expected value of the return is the return that the investor expects to receive. For instance, if a stock

can have only one of two returns, 10% with probability 0.4 and 15% with probability 0.6, its expected

return is 0.4*10% + 0.6*15% or 13%.

The expected value of the return of IBM, in short IBMs expected return, gives us the

average return our investor would earn if next year was repeated over and over, each time yielding

a different return drawn from the distribution of the return of IBM. Everything else equal, the investor

is better off the greater the expected return of IBM. We will see later in this chapter that a reasonable

estimate of the expected return of IBM is about 13% per year. However, over the next year, the

return on IBM could be very different from 13% because the return is random. For instance, we will

find out that using a probability distribution for the return of IBM allows us to say that there is a 5%

6chance of a return greater than 50% over a year for IBM. The most common probability distribution

used for stock returns is the normal distribution. There is substantial empirical evidence that this

distribution provides a good approximation of the true, unknown, distribution of stock returns.

Though we use the normal distribution in this chapter, it will be important later on for us to explore

how good this approximation is and whether the limitations of this approximation matter for risk

management.

The investor will also want to know something about the risk of the stock. The variance

of a random variable is a quantitative measure of how the numbers drawn from the urn are spread

around their expected value and hence provides a measure of risk. More precisely, it is a probability

weighted average of the square of the differences between the distinct outcomes of a random variable

and its expected value. Using our example of a return of 10% with probability 0.6 and a return of

15% with probability 0.4, the decimal variance of the return is 0.4*(0.10 - 0.13)2 + 0.6*(0.15 - 0.13)2

or 0.0006. For returns, the variance is in units of the square of return differences from their expected

value. The square root of the variance is expressed in the same units as the returns. As a result, the

square root of the return variance is in the same units as the returns. The square root of the variance

is called the standard deviation. In finance, the standard deviation of returns is generally called the

volatility of returns. For our example, the square root of 0.0006 is 0.0245. Since the volatility is in

the same units as the returns, we can use a volatility in percent or 2.45%. As returns are spread

farther from the expected return, volatility increases. For instance, if instead of having returns of 10%

and 15% in our example, we have returns of 2.5% and 20%, the expected return is unaffected but the

volatility becomes 8.57% instead of 2.45%.

If IBMs return volatility is low, the absolute value of the difference between IBMs return

7and its expected value is likely to be small so that a return substantially larger or smaller than the

expected return would be surprising. In contrast, if IBMs return volatility is high, a large positive or

negative return would not be as surprising. As volatility increases, therefore, the investor becomes

less likely to get a return close to the expected return. In particular, she becomes more likely to have

low wealth at the end of the year, which she would view adversely, or really high wealth, which she

would like. Investors are generally risk-averse, meaning that the adverse effect of an increase in

volatility is more important for them than the positive effect, so that on net they are worse off when

volatility increases for a given expected return.

The cumulative distribution function of a random variable x specifies, for any number X,

the probability that the realization of the random variable will be no greater than X. We denote the

probability that the random variable x has a realization no greater than X as prob(x # X). For our urn

example, the cumulative distribution function specifies the probability that we will draw a ball with

a number no greater than X. If the urn has balls with numbers from one to ten, the probability

distribution function could specify that the probability of drawing a ball with a number no greater than

6 is 0.4. When a random variable is normally distributed, its cumulative distribution function depends

only on its expected value and on its volatility. A reasonable estimate of the volatility of the IBM

stock return is 30%. With an expected return of 13% and a volatility of 30%, we can draw the

cumulative distribution function for the return of IBM. The cumulative distribution function for IBM

is plotted in Figure 2.1. It is plotted with returns on the horizontal axis and probabilities on the

vertical axis. For a given return, the function specifies the probability that the return of IBM will not

exceed that return. To use the cumulative distribution function, we choose a value on the horizontal

axis, say 0%. The corresponding value on the vertical axis tells us the probability that IBM will earn

8less than 0%. This probability is 0.32. In other words, there is a 32% chance that over one year, IBM

will have a negative return. Such probability numbers are easy to compute for the normal distribution

using the NORMDIST function of Excel. Suppose we want to know how likely it is that IBM will

earn less than 10% over one year. To get the probability that the return will be less than 10%, we

choose x = 0.10. The mean is 0.13 and the standard deviation is 0.30. We finally write TRUE in the

last line to obtain the cumulative distribution function. The result is 0.46. This number means that

there is a 46% chance that the return of IBM will be less than 10% over a year.

Our investor is likely to be worried about making losses. Using the normal distribution, we

can tell her the probability of losing more than some specific amount. If our investor would like to

know how likely it is that she will have less than $100,000 at the end of the year if she invests in IBM,

we can compute the probability of a loss using the NORMDIST function by noticing that a loss means

a return of less than 0%. We therefore use x = 0 in our above example instead of x = 0.1. We find that

there is a 33% chance that the investor will lose money. This probability depends on the expected

return. As the expected return of IBM increases, the probability of making a loss falls.

Another concern our investor might have is how likely it is that her wealth will be low enough

that she will not be able to pay for living expenses. For instance, the investor might decide that she

needs to have $50,000 to live on at the end of the year. She understands that by putting all her wealth

in a stock, she takes the risk that she will have less than that amount at the end of the year and will

be bankrupt. However, she wants that risk to be less than 5%. Using the NORMDIST function, the

probability of a 50% loss for IBM is 0.018. Our investor can therefore invest in IBM given her

objective of making sure that there is a 95% chance that she will have $50,000 at the end

of the year.

9 The probability density function of a random variable tells us the change in prob(x # X)

as X goes to its next possible value. If the random variable takes discrete values, the probability

density function tells us the probability of x taking the next higher value. In our example of the urn,

the probability density function tells us the probability of drawing a given number from the urn. We

used the example where the urn has balls with numbers from one through ten and the probability of

drawing a ball with a number no greater than six is 0.4. Suppose that the probability of drawing a ball

with a number no greater than seven is 0.6. In this case, 0.6 - 0.4 is the probability density function

evaluated at seven and it tells us that the probability of drawing the number seven is 0.2. If the

random variable is continuous, the next higher value than X is infinitesimally close to X.

Consequently, the probability density function tells us the increase in prob(x # X) as X increases by

an infinitesimal amount, say ,. This corresponds to the probability of x being between X and X + ,.

If we wanted to obtain the probability of x being in an interval between X and X + 2,, we would add

the probability of x being in an interval between X and X + , and then the probability of x being in

an interval between X + , to X + 2,. More generally, we can also obtain the probability that x will

be in an interval from X to X by adding upthe probability density function from X to X, so that

the probability that x will take a value in an interval corresponds to the area under the probability

density function from X to X.

In the case of IBM, we see that the cumulative distribution function first increases slowly, then

more sharply, and finally again slowly. This explains that the probability density function of IBM

shown in Figure 2.2. first has a value close to zero, increases to reach a peak, and then falls again.

This bell-shaped probability density function is characteristic of the normal distribution. Note that this

bell-shaped function is symmetric around the expected value of the distribution. This means that the

10

cumulative distribution function increases to the same extent when evaluated at two returns that have

the same distance from the mean on the horizontal axis. For comparison, the figure shows the

distribution of the return of a security that has twice the volatility of IBM but the same expected

return. The distribution of the more volatile security has more weight in the tails and less around the

mean than IBM, implying that outcomes substantially away from the mean are more likely. The

distribution of the more volatile security shows a limitation of the normal distribution: It does not

exclude returns worse than -100%. In general, this is not a serious problem, but we will discuss this

problem in more detail in chapter 7.

When interpreting probabilities such as the 0.18 probability of losing 50% of an investment

in IBM, it is common to state that if our investor invests in IBM for 100 years, she can expect to lose

more than 50% of her beginning of year investment slightly less than two years out of 100. Such a

statement requires that returns of IBM are independent across years. Two random variables a and

b are independent if knowing the realization of random variable a tells us nothing about the realization

of random variable b. The returns to IBM in years i and j are independent if knowing the return of

IBM in year i tells us nothing about the return of IBM in year j. Another way to put this is that,

irrespective of what IBM earned in the previous year (e.g., +100% or - 50%), our best estimate of

the mean return for IBM is 13%.

There are two good reasons for why it makes sense to consider stock returns to be

independent across years. First, this seems to be generally the case statistically. Second, if this was

not roughly the case, there would be money on the table for investors. To see this, suppose that if

IBM earns 100% in one year it is likely to have a negative return the following year. Investors who

know that would sell IBM since they would not want to hold a stock whose value is expected to fall.

11

By their actions, investors would bring pressure on IBMs share price. Eventually, the stock price will

be low enough that it is not expected to fall and that investing in IBM is a reasonable investment. The

lesson from this is that whenever security prices do not incorporate past information about the history

of the stock price, investors take actions that make the security price incorporate that information.

The result is that markets are generally weak-form efficient. The market for a security is weak-form

efficient if all past information about the past history of that security is incorporated in its price. With

a weak-form efficient market, technical analysis which attempts to forecast returns based on

information about past returns is useless. In general, public information gets incorporated in security

prices quickly. A market where public information is immediately incorporated in prices is called a

semi-strong form efficient market. In such a market, no money can be made by trading on

information published in the Wall Street Journal because that information is already incorporated

in security prices. A strong form efficient market is one where all economically relevant information

is incorporated in prices, public or not. In the following, we will call a market to be efficient when

it is semi-strong form efficient, so that all public information is incorporated in prices.

Section 2.1.2. Evaluating the expected return and the risk of a portfolio.

To be thorough, the investor wants to consider XYZ. She first wants to know if she would

be better off investing $100,000 in XYZ rather than in IBM. She finds out that the expected return

of XYZ is 26% and the standard deviation of the return is 60%. In other words, XYZ has twice the

expected return and twice the standard deviation of IBM. This means that, using volatility as a

summary risk measure, XYZ is riskier than IBM. Figure 2.2. shows the probability density function

of a return distribution that has twice the volatility of IBM. Since XYZ has twice the expected return,

12

its probability density function would be that distribution moved to the right so that its mean is 26%.

It turns out that the probability that the price of XYZ will fall by 50% is 10.2%. Consequently, our

investor cannot invest all her wealth in XYZ because the probability of losing $50,000 would exceed

5%.

We now consider the volatility and expected return of a portfolio that includes both IBM and

XYZ shares. At the end of the year, the investors portfolio will be $100,000 times one plus the

return of her portfolio. The return of a portfolio is the sum of the return on each security in the

portfolio times the fraction of the portfolio invested in the security. The fraction of the portfolio

invested in a security is generally called the portfolio share of that security. Using wi for the portfolio

share of the i-th security in a portfolio with N securities and Ri for the return on the i-th security, we

have the following formula for the portfolio return:

(2.1.)w R Portfolio Returni ii 1

N

==

Suppose the investor puts $75,000 in IBM and $25,000 in XYZ. The portfolio share of IBM is

$75,000/$100,000 or 0.75. If, for illustration, the return of IBM is 20% and the return on XYZ is -

10%, applying formula (2.1.) gives us a portfolio return in decimal form of:

0.75*0.20 + 0.25*(-0.10) = 0.125

In this case, the wealth of the investor at the end of the year is 100,000(1+0.125) or $125,000.

At the start of the year, the investor has to make a decision based on what she expects the

13

distribution of returns to be. She therefore wants to compute the expected return of the portfolio and

the return volatility of the portfolio. Denote by E(x) the expected return of random variable x, for any

x. To compute the portfolios expected return, it is useful to use two properties of expectations. First,

the expected value of a random variable multiplied by a constant is the constant times the expected

value of the random variable. Suppose a can take value a1 with probability p and a2 with probability

1-p. If k is a constant, we have that E(k*a) = pka1 + (1-p)k a2 = k[pa1 + (1-p)a2] = kE(a).

Consequently, the expected value of the return of a security times its portfolio share, E(wiRi) is equal

to wiE(Ri). Second, the expected value of a sum of random variables is the sum of the expected values

of the random variables. Consider the case of random variables which have only two outcomes, so

that a1 and b1 have respectively outcomes a and b with probability p and a2 and b2 with probability

(1-p). With this notation, we have E(a + b) = p(a1 + b1) + (1-p)(a2 + b2) = pa1 + (1-p)a2+ pb1 +(1-

p)b2 = E(a) + E(b). This second property implies that if the portfolio has only securities 1 and 2, so

that we want to compute E(w1R1 + w2R2), this is equal to E(w1R1) + E(w2R2), which is equal to

w1E(R1) + w2E(R2) because of the first property. With these properties of expectations, the expected

return of a portfolio is therefore the portfolio share weighted average of the securities in the portfolio:

(2.2.)w E(R ) Portfolio Expected Returni ii 1

N

= =

Applying this formula to our problem, we find that the expected return of the investors portfolio in

decimal form is:

0.75*0.13 + 0.25*0.26 = 0.1625

Our investor therefore expects her wealth to be 100,000*(1 + 0.1625) or $116,250 at the end of the

14

year.

Our investor naturally wants to be able to compare the risk of her portfolio to the risk of

investing all her wealth in IBM or XYZ. To do that, she has to compute the volatility of the portfolio

return. The volatility is the square root of the variance. The variance of a portfolios return is the

expected value of the square of the difference between the portfolios return and its expected return,

E[Rp - E(Rp)]2. To get the volatility of the portfolio return, it is best to first compute the variance and

then take the square root of the variance. To compute the variance of the portfolio return, we first

need to review two properties of the variance. Denote by Var(x) the variance of random variable x,

for any x. The first property is that the variance of a constant times random variable a is the constant

squared times the variance of a. For instance, the variance of 10 times a is 100 times the variance of

a. This follows from the definition of the variance of a as E[a - E(a)]2. If we compute the variance

of ka, we have E[ka - E(ka)]2. Since k is not a random variable, we can remove it from the

expectation to get the variance of ka as k2E[a - E(a)]2. This implies that Var(wiRi) = wi2Var(Ri).

To obtain the variance of a + b, we have to compute E[a+b - E(a + b)]2. Remember that the square

of a sum of two terms is the sum of each term squared plus two times the cross-product of the two

numbers (the square of 5 + 4 is 52 + 42 + 2*5*4, or 81). Consequently:

Var(a + b) = E[a + b - E(a + b)]2

= E[a - E(a) + b - E(b)]2

= E[a - E(a)]2 + E[b - E(b)]2 + 2E[a - E(a)][b - E(b)]

= Var(a) + Var(b) + 2Cov(a,b)

15

The bold term is the covariance between a and b, denoted by Cov(a,b). The covariance is a measure

of how a and b move together. It can take negative as well as positive values. Its value increases as

a and b are more likely to exceed their expected values simultaneously. If the covariance is zero, the

fact that a exceeds its expected value provides no information about whether b exceeds its expected

value also. The covariance is closely related to the correlation coefficient. The correlation coefficient

takes values between minus one and plus one. If a and b have a correlation coefficient of one, they

move in lockstep in the same direction. If the correlation coefficient is -1, they move in lockstep in

the opposite direction. Finally, if the correlation coefficient is zero, a and b are independent. Denote

by Vol(x) the volatility of x, for any x, and by Corr(x,y) the correlation between x and y, for any x

and y. If one knows the correlation coefficient, one can obtain the covariance by using the following

formula:

Cov(a,b) = Corr(a,b)*Vol(a)*Vol(b)

The variance of a and b increases with the covariance of a and b since an increase in the covariance

makes it less likely that an unexpected low value of a is offset by an unexpected high value of b. It

therefore follows that the Var(a + b) increases with the correlation between a and b. In the special

case where a and b have the same volatility, a + b has no risk if the correlation coefficient between

a and b is minus one because a high value of one of the variables is always exactly offset by a low

value of the other, insuring that the sum of the realizations of the random variables is always equal

to the sum of their expected values. To see this, suppose that both random variables have a volatility

of 0.2 and a correlation coefficient of minus one. Applying our formula for the covariance, we have

16

w Var(R) w w Cov(R,R ) Variance of Portfolio Returni2

i i j i jj i

N

i 1

NN

+ ===

i 1

that the covariance between a and b is equal to -1*0.2*0.2, which is -0.04. The variance of each

random variable is the square of 0.2, or 0.04. Applying our formula, we have that Var(a + b) is equal

to 0.04 + 0.04 - 2*0.04 = 0. Note that if a and b are the same random variables, they have a

correlation coefficient of plus one, so that Cov(a,b) is Cov(a,a) = 1*Vol(a)Vol(a), which is Var(a).

Hence, the covariance of a random variable with itself is its variance.

Consider the variance of the return of a portfolio with securities 1 and 2, Var(w1R1 + w2R2).

Using the formula for the variance of a sum, we have that Var(w1R1 + w2R2) is equal to Var(w1R1)

+ Var(w2R2) +2Cov(w1R1,w2R2). Using the result that the variance of ka is k2Var(a), we have that

w12Var(R1) + w2

2Var(R2) +2w1w2Cov(R1,R2). More generally, therefore, the formula for the variance

of the return of a portfolio is:

(2.3.)

Applying the formula to our portfolio of IBM and XYZ, we need to know the covariance between

the return of the two securities. Lets assume that the correlation coefficient between the two

securities is 0.5. In this case, the covariance is 0.5*0.30*0.60 or 0.09. This gives us the following

variance:

0.752*0.32 + 0.252*0.62 +2*0.25*0.75*0.5*0.3*0.6 = 0.11

The volatility of the portfolio is the square root of 0.11, which is 0.33. Our investor therefore

discovers that by investing less in IBM and investing some of her wealth in a stock that has twice the

volatility of IBM, she can increase her expected return from 13% to 16.25%, but in doing so she

17

increases the volatility of her portfolio from 30% to 32.70%. We cannot determine a priori which

of the three possible investments (investing in IBM, XYZ, or the portfolio) the investor prefers. This

is because the portfolio has a