FRM Notes 2008 Unit IA What is Risk

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Version: January 3, 2008 FIN 7400: Financial Risk Managemen

Unit IA Notes Page 1 of 22 D. M. Chance, LSU

I. CONCEPTS OF RISK MANAGEMENT

This is a course on risk management, primarily financial risk management. It is

designed for MBAs’, MS students, and students of accounting, math, statistics,engineering, and economics who want to know more about measuring andmanaging risk. The course is not designed to make you a risk manager. To dothat, you have to study risk at a fairly advanced level and learn a great deal aboutfinance and probability. Banks and major investment firms, which offer risk management products and services, have these types of people on their staffs.Corporations, pension funds, and government agencies typically cannot afford thistype of expertise. But increasingly we find that these organizations are being hurt

by not managing risk and realizing that they have to start doing it.

We know risk management is important and has reached the mainstream when itstarts being mentioned in sit-coms. In a Seinfeld episode called “The Fatigues,”which first aired on October 31, 1996, George Costanza’s boss, New York Yankees Owner George Steinbrenner, asks George C. to give a lecture to theYankees’ staff on risk management. George C. is no expert but gets a book on risk management and reads a little. He records himself saying some things about risk management into a tape recorder and plays it back:

“In order to understand risk, we must first define risk.”

So, taking the advice of George Constanza, let’s begin.

A. What is Risk?

As a general and very simplistic definition:

Risk is the potential that an event will have an outcome different from the

outcome that is expected to occur.

For example, we might expect that

• the stock market will go up 10% in a year

• we will make a B in this course





Each of these outcomes is the result of an event. The actual outcome may differ from what we expect. Risk deals with unexpected outcomes, defined obviously asthose that occur that were not expected. Ironically, if we know there is risk, we

know that an unexpected outcome is the most likely result. That is, we expect anunexpected outcome. We simply do not know what specific outcome to expect.

Some events have outcomes that are both desirable and undesirable. Each of theevents above has unexpected outcomes that are both desirable and undesirable.Sometimes this phenomenon is called good risk and bad risk . But some eventshave only bad outcomes and a few have only good outcomes. For example,suppose your supervisor has told you that due to financial difficulties in thecompany, your job will either be terminated or you will have a salary reduction,which will be determined at a later meeting. There are no truly good outcomeshere, but good is relative. One outcome is better than the other.

Some events have highly skewed outcomes. For example, you go on a car trip.Consider the event to be whether you return safely. Returning safely is clearly thegood outcome. Returning unsafely encompasses a whole host of outcomes, someof which are only moderately bad and some of which are terrible. Credit risk issomewhat like this. A creditor either pays you back the amount promised or defaults and pays back a lesser amount.

It is technically possible to identify events in which there are no outcomes that are particularly undesirable. For example, you enter a photograph in a contest. Theworst outcome is that you lose, but there is not much pain in losing. But suchevents are not really of much interest to us in this course.

The one defining factor in all risky events that would interest us is that there is at

least one undesirable outcome, with undesirability defined on a relative basis.That means there is at least one other outcome that is desirable or less undesirable.So let us redefine risk with a definition that best suits our purposes:

Risk is the exposure to an event in which the outcomes are uncertain and at

least one of the outcomes is undesirable relative to at least one other

outcome.

Now let us examine the elements of this definition of risk.





Elements of a Definition of Risk

Exposure to an event

Investing in the market, taking a drive in an automobile, going out on a date withsomeone you do not know very well. These are all somewhat risky events. Life ingeneral is a sequence of risky events.

But some exposures can be avoided, so the event is not a source of risk. The risk of dying in a plane crash is zero if the person chooses not to fly (we’redisregarding the infinitesimal probability of being killed on the ground by acrashing plane). The risk of losing money at the race track is zero if we never beton the horses. The risk of being paralyzed in a diving accident is zero if we never get in the pool.

Of course, there can be opportunity losses from not taking risk. We might never lose money in the stock market if we keep our money in bank CDs, but we willlose the opportunity to make money from the market (and we run the risk of losing

purchasing power due to inflation). We would never enjoy the benefits of certainactivities if we do not take some risk and engage in them.

Exposure exists in degrees. Individuals in large cities who commute are at far

greater risk of being in an automobile accident than those who take the train towork and drive their cars only occasionally. But everyone who rides in a car is atsome risk of being in an automobile accident.

Uncertain outcomes

There is no risk if the outcome is known. If a suicidal person fires a loaded gun athis head, the event of whether he will be hurt is not a risky event (I am assumingthe gun cannot misfire.) The nature of the damage is a risky event, but damage of some form is certain.

In finance, the return on a zero coupon government bond is a riskless event, at leastin terms of nominal returns (i.e., not considering inflation). There is no defaultrisk. Changing interest rates affect the opportunity cost. But the actual return iscertain.





At least one of the outcomes must be undesirable relative to at least one other

outcome. We covered this above.

Probability as a Useful but (Sometimes) Misleading Measure of Risk

Risk is oftentimes associated with probability. While mathematicians have muchmore formal definitions, probability is casually defined as a measure of the

likelihood of occurrence of the outcome of an event . This definition does not helpus much, because then we have to backtrack and define likelihood . Then let usredefine probability as the relative frequency with which the outcomes of an event

occur . (This definition defines likelihood as relative frequency, a notion that weassume people understand as being the number of times an outcome occurs divided

by the total number of possible outcomes.) A 5% probability of an event occurringon a daily basis means that it would be expected to occur every 20 days. Looking

back over a large number of days, we would count the number of days it occurredrelative to the total number of days. If this ratio is not pretty close to 5%, then 5%was probably not the correct probability. (Statisticians would tell us that there arerigorous procedures that one would have to follow to definitively reject 5% as the

probability.)

Probability is just a means of quantifying risk . It is not the risk itself . One doesnot have to know the probability for the risk to exist. For example, who knows the

probability that your car will break down on the way home?

But probability can be a misleading way to quantify risk, and people are not verygood at processing probability information. Professor Gerd Gergenzee of Germany covers this idea very well in his book Calculated Risks. I repeat aslightly modified example from his book.

One out of 100 forty-year old women has breast cancer. Mammography isused to detect the presence of suspicious tissue. A positive mammogramusually leads to a biopsy. But mammography is not perfect. It will detect90% of breast cancers but it will indicate the presence of cancer 9% of thetime in women who do not have breast cancer.

A 40-year old woman gets a positive mammogram. What is the probabilityshe has breast cancer? Gergenzee gave 48 physicians this scenario and





asked them this question. The median estimate was 70%. A third estimatedmore than 90%. What is the correct answer?

Consider 10,000 forty-year old women, of whom 100 (1%) have breastcancer. Of these 100 women, 90 (90%) will have a positive mammogram.Of the 9,900 who do not have breast cancer, 891 (9%) will also have a

positive mammogram. Therefore, 90 + 891 = 981 women will have a positive mammogram, but only 90 have breast cancer. Therefore, the probability that the woman has breast cancer is only 90/891 = 0.092.

Of course, without having the mammogram, she might assume that she has a1% chance of having cancer. With the mammogram, her odds are aboutnine times higher. But they are nowhere near the typical estimates of thetrue probability made by physicians and others. (Obviously each case isdifferent and other factors such as family history can affect these odds.)

Statisticians will recognize this point as an application of the notion of conditional probability and Bayes’s Rule.

Gergenzee gives another interesting example.

A cholesterol-lowering drug is advertised as giving a 22% reduction in heart

attacks for people with high cholesterol. What does this mean? Most peoplethink that of 1,000 people with high cholesterol, 220 will not die if they takethe drug, making the drug sound relatively effective. What it really means isquite different.

One thousand people with high cholesterol were given the drug and another 1,000 people with high cholesterol were given a placebo. Of the grouptaking the drug, 32 died from a heart attack. Of the group given the placebo,41 died from a heart attack. The 22% figure comes from the ratio 9/41 =0.22.

What is the correct interpretation? Assuming the drug is effective, nine people lived who would have died had they not taken the drug. Thus, only0.9% (9 out of 1,000) benefited from the drug. It would require 111 people(1/(.009)) taking the drug for one life to be saved. Now the drug does notsound so effective.





Probability, even correctly stated, can be misleading. Gergenzee argues for the useof relative frequencies. When doctors were given the breast cancer problem using

relative frequencies, their estimates of the probability were much more accurate.

Conditional and Unconditional Notions of Risk

The breast cancer example above illustrates the notion of conditional risk compared to unconditional risk. Even though the probability of having cancer if she has a positive mammography is much smaller than most people expect, shenonetheless has a much greater probability of breast cancer than a woman withouta positive mammography. Here’s another interesting one from Gergenzee.

The celebrated O. J. Simpson trial brought out the best and worst in the legal profession. Little known is the fact that it brought out the worst in the use of risk information, which probably contributed to the acquittal. The

prosecution argued that Simpson’s history of spousal abuse constituted a pattern and a motive, increasing the likelihood that he was the killer.Famous criminal defense attorney and Harvard law professor AlanDershowitz argued against this claim by noting several facts. As many asfour million women are battered annually by husbands and boyfriends in theU. S. In 1992 the FBI reported that 913 women were killed by husbands and

519 by boyfriends. So, Dershowitz claims that of four million incidents,only 1,432 ended in homicides. Therefore, only about 1 in 2,500 incidentsof abuse ends in murder.

It was compelling evidence for that jury, and the prosecution was unable toshow the fallacy of those numbers, true as they were. Nicole Simpson had

been battered and murdered. That requires a different analysis. If 1 in 2,500abused women is murdered, we would expect about 40 murders in every100,000 battered women. We need to know the numbered of batteredwomen killed each year by someone other than a husband or boyfriend.Assuming this is the same as for the general population, statistics show thatthis is about five out of 100,000.

So consider 100,000 battered women. We would expect 40 (1 in 2,500) to be murdered by their batterers and five to be murdered by someone else.





Thus, if a woman is murdered, the probability she was murdered by her batterer is 40/45 = 0.89, about 89%.

In this case, the notion is conditional probability. Given that she was battered andmurdered, what is the probability that she was murdered by her batterer?

Later in this course, we will see how the concept of Value-at-Risk can be a sourceof similar confusion with respect to probability.

Probability can be misleading, but if used properly, it can be very helpful.

Risk and Uncertainty

The famous economist Frank Knight was well known for making a distinction between risk and uncertainty. Risk , he claimed, exists when the probabilities areknown. Uncertainty is when the probabilities are not known. In reality, nearly allsituations would fall under his classification of uncertainty. We rarely know the

probabilities. Even when loaded with statistical data (such as the probability of breast cancer in a 40-year old woman), we do not know how special circumstancesaffect that probability. (She might have a family history of breast cancer, but sheexercises regularly, is not overweight, and has other low-risk characteristics). In afew cases, we might know the true probabilities (drawing an ace out of full deck),

but most of these are not particularly important. So in reality, we typically do notknow the probabilities. We just estimate them and use them, sometimes withoutmuch regard for the notion of Knightian uncertainty.

A related notion is the fact that probabilities can change randomly. We mightknow the probabilities today but they might change tomorrow. In a casino, thatcould be something like the first hand of a blackjack game where the odds arecertain. Once some cards are played, the odds change. The volatility of the stock market can be greater today than it was yesterday, which changes the probabilities.

This point just goes to show that another risk is the risk of risk . In finance, this isoftentimes called the volatility of volatility.

How People Deal with Risk





Risk is only part of the picture. We must deal with how we feel about risk andhow we react to it. One person might stress over a financial loss of $100 to the

point of requiring anti-depressive medication, while another might regard it as a

risk worth taking that just did not happen to work out this time. Economists studyhow we view risk using a concept called utility.

Utility

Utility is a measure of satisfaction. Satisfaction comes in a variety of forms suchas consumption of goods and services, love and companionship of family andfriends, art, literature, and music, or pride from success. Our concern here is utilityarising from consumption. But ultimately this utility derives from financial wealth.

In the eighteenth century, Swiss mathematician Nicholas Bernoulli asked hiscousin, the mathematician Daniel Bernoulli, a puzzling question. Suppose Peter tosses a coin and continues tossing until it comes up heads. He offers to give Paulone ducat if it lands on heads the first time, two if it lands on heads the secondtime, four if it lands on heads the third time, doubling the payoff each time it failsto land on heads. What is Paul’s expected payoff? How much would Paul bewilling to pay Peter to play the game? This problem is called the Petersburg

paradox because it had been introduced at a meeting of mathematicians inPetersburg, Russia.

A simple calculation reveals that the expected payoff is infinite:

20(1/2) + 21(1/2)2 + 22(1/2)3 + …→ ∞

Would Paul pay an infinite amount to play? Or more appropriately, would you payan infinite amount to play? Most people would not, even though on average the

payoff is infinite.

Daniel Bernoulli proposed a solution to this problem by applying a theory of risk based on how people feel about risk, that is, their expected utility. He proposedevaluating utility in terms of the logarithm of utility, where the risk associated withtrying to achieve greater wealth exacts a penalty that shows up in the subjectiveevaluation of the worth of money: the utility. Bernoulli’s views have stayed withus for years. Bernoulli’s solution showed that a person should pay two units of thecurrency. Thus, if the first payoff is $1, a person should pay $2 to play.





The modern view is that financial wealth cannot be achieved by single-mindedly

pursuing financial opportunities that promise favorable outcomes. Competitionforces markets to reward higher returns with higher risks required to earn thosereturns. Were it any other way, the most attractive financial opportunities would

be bid up in price until they were less attractive.

Utility reflects return and risk. Two investments of equal expected return butunequal risk will not be viewed equally. The one with the lower risk will be takenand the other will be ignored, after which its price will fall and make it morecompetitively attractive. All assets must offer the same expected return per unit of risk.

From the works of Daniel Bernoulli, economists have come to model utility in theform of a utility function. Let us look at the log utility function. If the investor can

be characterized by log utility, he would make decisions based on maximizing theexpected utility. A log utility function is of the form U(W) = ln(W). That is, for wealth of 100, utility is ln(100) = 4.61, sometimes casually referred to in terms of 4.61 utils of satisfaction. Theoretically, one can calculate the expected utility froma risky choice. Consider the following two possibilities for a coin toss:

Outcome Payoff Heads $10Tails $5

The expected outcome is 0.5($10) + 0.5($5) = $7.50. What would you pay to playthis game? If you say $7.5, you are risk neutral. That means that if you paid $7.50

played the game repeatedly, you would earn nothing. You would be as satisfied asif you had not played. Risk averse investors would not be satisfied. If they pay$7.50 and win $2.50 and then play again and lose $2.50, they would not be ashappy as if they had not played.

What is a reasonable price to pay? The utilities of the two outcomes are

U(10) = ln(10) = 2.30U(5) = ln(5) = 1.61





The expected utility is 0.5(2.30) + 0.5(1.61) = 1.955. A risk averse person wouldsee this game as generating expected utility of 1.955. How much money wouldthis person take for certain and be as satisfied as playing the game? We can invert

the utility function as follows:

U(W) = ln(W) = 1.955e1.955 = $7.06.

This person would be satisfied to receive $7.06 instead of playing the game.Hence, this person would be willing to pay $7.06 to play. You can think of this

person as requiring a risk premium of $7.50 - $7.06 = $0.44 to play the game.

The value $7.06 is called the certainty equivalent . It is the amount a person wouldtake for certain to avoid the risk. It is the value that the person would say the riskyopportunity is worth.

Economists use many different utility functions, most of which are morecomplicated than the log function. And economists have devised many modelsthat capture the prices of assets by incorporating utility assumptions indirectly, butwhich ultimately end up not requiring the utility function. The Capital Asset

Pricing Model is one such example.

One other noted criterion for decision making under risk is stochastic dominance,whereby the cumulative probabilities of outcomes lead to a decision rule.Stochastic dominance exists in degrees. Consider investment M and investment N.If the probability of earning a return of x% or less is always greater for M than for

N for all x, then N is preferred to M. This is called first-order stochasticdominance. Not all comparisons are, however, this easy. Second-order stochasticdominance considers the difference in the cumulative probability distributions of the two choices. Second-order stochastic dominance turns out to be consistentwith maximizing expected utility if the decision maker is risk averse. Stochasticdominance is quite flexible. Sometimes third- and higher-orders are required tomake decisions.

Another model of risky decision making is state preference theory, in which riskyopportunities are viewed as the equivalent of combinations of certain fundamentalsecurities, called pure securities, that pay off $1 in various states and nothingotherwise. For example, consider an investment that pays $10 if one outcome





occurs (called it state 1) and $5 if the other (state 2) occurs. We can think of thisinvestment as consisting of two pure securities. One pure security pays $1 if state1 occurs and nothing if state 2 occurs. The other pure security pays $1 if state 2

occurs and nothing if state 1 occurs. The original security can be viewed as acombination of 10 units of pure security 1 and five units of pure security 2. The

prices of these pure securities reflect the fundamental risk. State pricing theory can be shown to be closely related to arbitrage theory. While it is useful as atheoretical device, its practical implications are quite limited.

In the world of risk management, we shall benefit from the ability to price risk management instruments, such as financial derivatives, without resorting todetermining the risk premiums. If we hold a stock and assume the risk of thestock, a derivative contract that insures against loss on the stock will cost anamount that is a function of the risk of the stock, not the risk of the derivative. Thederivative is merely a tool that alters the stockholder’s risk but is not a source of risk itself. The derivative will allow transferring the risk but does not create anyrisk that did not already exist as a result of the stock. Derivatives serve as a formof insurance, transferring risks from those who do not want it to those willing to

bear it. The price of this insurance is a function of the amount of risk beingtransferred, which is a function of the amount of underlying risk and the amount of this risk that the risk take is willing to bear. As a result, we will be able to treat

people as though they are risk neutral, when in fact, they are risk averse. Their risk

aversion manifests in the prices and volatilities of the underlying sources of risk,not in the derivative contracts used to transfer that risk. By treating people asthough they are risk neutral, we simplify the pricing. But remember, we are notassuming people are risk neutral. We are just ignoring their risk aversion, becauseit is captured in the price of the underlying source of risk, like a stock or bond priceor interest rate.

Some Observations about How People Deal with Risk

As noted, people deal with risk in different ways. They process information aboutrisk in a variety of ways, many of which are wrong, but some of which are right or at least not terribly wrong. People behave in inconsistent ways with respect to risk.It is worthwhile to look at some of the ways people behave with respect to risk.

People confuse ex post with ex ante. After the fact (ex post) there is no risk.The event has already happened. The risk exists before the fact (ex ante).





Ex post analysis of the outcomes of events can be useful, however, inevaluating the risk going forward, but only if the risk remains constant. Weshall say a good bit more about this later in this unit.

People frequently cannot differentiate good risk management from luck .Good risk management is often thought of as skill. You buy a stock, watchit go up, and then proclaim how skilled you are. Sell a stock, watch it godown, and think you have cleverly staved off an undesirable event.Consider this hypothetical Internet experiment. One million people go to aweb site, pay $1, and get to predict the outcome of a coin toss. The coin istossed (perhaps with illustrative graphics). Those who call the toss correctlycontinue playing. The losers have to quit. After 10 coin tosses, those whohave called all 10 tosses correctly split the $1 million. How many peoplewould you expect to call all 10 tosses correctly? The probability of one

person calling 10 straight correctly is 0.510

= 0.00097656. Assuming theycall independently, we would expect 1,000,000(0.00097656) = 976.56 to getall 10 tosses correct. Let’s just round that to 1,000. So each would get$1,000. Would it be skill? Of course not. Would football teams line up tohire these people, enlisting their aid in helping to call the coin tosses? Of course not.

True skill is a difficult thing to measure and can require an unbelievable

amount of experience to measure. Some experts have calculated thatdemonstrating true investment skill would require more years of trading thanin a normal human life span.

A corollary to the above rule is that people view good outcomes as the result

of skill and bad outcomes as the result of bad luck. Investment managerswho make money think they’re good and when they lose money, they think it’s not their fault.

People often perceive the risk to be greater when they have less information

about the probabilities. As noted earlier, knowledge of true probabilities israre but oftentimes we have more information than others. Initial publicofferings are fraught with uncertainty about the probability distribution of astock. Casino gambling, however, is relatively well-defined risk. Perhapsthis partially explains why casino gambling is so popular.





People fear risk a lot less if the benefit justifies the risk . This is similar tothe financial notion of a risk-return tradeoff. The expected gain justifies therisk. An individual that lives along an earthquake fault-line in California but

enjoys moderate temperature, sunny skies, and low but sufficient rainfall believes the gain justifies the risk. This notion is really just the risk-returntradeoff. If the return (or reward) justifies the risk, people will accept therisk.

People fear risk less if information about the risk is more trustworthy.When we have good information about the risk, we seem more willing totake it. For some reason, people tend to trust government more than privateindustry in providing information about risk, though there is no reason to

believe that government is better able to accurately measure risk.

People fear risk less the further back in history the adverse outcomes occur .Fear of terrorist attacks, plane crashes, and large drops in the stock marketfade from people’s minds the further back they go.

People are more afraid of risks that affect them more directly than those that

affect them less directly and affect others more directly. For example, whitecollar workers have less fear of the risk of a weakening economy than bluecollar workers because it affects them less directly.

People are often irrational about risks, sometimes weighing the risk of less

probable events with severe consequences more than more probable events

with less severe consequences. As a result, sometimes people take strange

precautions against highly unlikely events.

Which is riskier for children? A house with a handgun or a house with aswimming pool? A child in a home with a swimming pool is 100 timesmore likely to drown than to be killed by the gun.

Many people worry more about being killed in a plane crash than beingkilled in an automobile accident. (We’re not understating the potentialseverity of a car accident but simply saying that the expected loss from beingin a car accident is far less than the expected loss from being in a planecrash.) Another example: following September 11, many people refused tofly and many corporations made their people drive rather than fly, at least





for a period of time. Gergenzee calculated that far more people died as aresult of automobile accidents than would be expected under normalcircumstances. And certainly it was far more than died over that time due to

terrorist events, plane crashes, etc.

Another example: Bicycles, beds, lawnmowers and wheelchairs are four household items that are sometimes associated with injury. Place them inorder from highest risk to lowest risk of causing injury (data compiled by U.S. Census Bureau). I will tell you the answer in class.

The media plays a major role in conveying risk information, a role that is

oftentimes unhelpful, misleading, and incorrect . Media stories of increasedrisk of some malady from taking a taking a certain pharmaceutical designedto treat another malady are often misleading. Suppose 1,000 women try anew drug for hormone replacement therapy and 1,000 other women usenothing. Of the women who take the existing drug, five get ovarian cancer.Of the women who take the new drug, 10 get ovarian cancer. Otherwise, thenew drug is almost completely effective in treating the problem for which itwas designed. The media says that this new drug designed to treat post-menopausal symptoms doubles the risk of ovarian cancer. Is this helpfulinformation? Perhaps. But the details are never reported. Five women outof 1,000 may have gotten ovarian cancer from this drug. Would you take a

drug designed to address a problem that has near 100 % effectiveness butone in 200 will get ovarian cancer. Some, perhaps many, would be willingto bear that risk. But the negative publicity may well kill the drug.

ABC journalist John Stossel likes to tell this story. Suppose the mediareports that a new form of energy had been developed. Large companieswould be given licenses to pump this energy into our homes. It would burncleanly and would be relatively inexpensive. But it would be expected tokill 200 Americans a year. Would this energy likely win praise from themedia and government approval? Probably not. But then, it won’t need to.We already use it. It’s called natural gas.

If automobiles were introduced today would the media (and many people)see the risk as worth the reward? Over 35,000 people a year are killed inautomobile crashes. It’s hard to imagine this scenario, because automobilesare such an ingrained part of our lives. But they might not be approved.





People feel differently about risk as a function of their age, responsibility,

experiences, wealth, and gender. Young people are notoriously less averse

to risk, perhaps due to less responsibility and less experience with adverseoutcomes. Greater wealth is often thought to lead to less risk aversion, butsometimes wealthier people are more risk averse. That’s partially how theyachieved their wealth, by avoiding bad financial events. Gender, albeit asensitive subject, is nonetheless a determinant of how we react to risk.Women are more risk averse than men. Men engage in much riskier

behavior than women. Even in investment decisions young men will tend totake far more risks than young women.

Finally, let us note that taking risks in light of expected returns is not uniquelyhuman. Place a piece of cheese in front of cat. A mouse with a full stomach maywell decide the cheese isn’t worth the risk. But a mouse with an empty stomachmight take the risk and go for the cheese.

Risk and Arbitrage

One of the most important concepts in studying risk management is arbitrage.Arbitrage is defined as a condition in which two identical assets or combinations of

assets sell at different prices. Arbitrage leads to a strategy (sometimes called

arbitrage itself) in which the cheaper assets are purchased and the more expensiveassets are sold. If the assets are identical, their risks offset, but the difference intheir prices is captured as a risk-free gain. Indeed arbitrage is thought of as anopportunity to earn a return without bearing any risk or committing any funds.

Consider the following simple example. Suppose you have an opportunity to playa game in which you earn $100 if a coin toss comes up heads and $0 if it comes uptails. If it costs nothing to play this game, you would clearly play. Even thoughhalf the time, you would walk away with no gain or loss, the other half of the time

you would walk away with a gain. Since there is no cost, there is no possibility of a loss and some possibility of a gain. Such an example, while theoretically possible, is not realistic. Surely no one would offer to play with you withoutcharging a price. Nonetheless, this point will play a role later in defining the limitsof the prices of options.





Let us now look at a more realistic possibility, such as a stock selling for twodifferent prices. Later in the course, we shall see that one combination of assetsand derivatives on the asset could sell for a different price than another

combination of assets and derivatives that produces the same results. If this pricediscrepancy exists, investors will buy the cheaper combination and sell the moreexpensive one, thereby eliminating the risk and earning the difference in prices.The combined effects of a large number of investors engaging in this type of trading will force the price of the expensive combination down and the price of thecheap combination up until the prices are equal and no further gains are made bytrading. There can be only one price. This is sometimes called the law of one

price.

Arbitrage manifests in slightly different ways. Suppose combination A offers areturn of $100 if a coin toss comes up heads and $0 if a coin toss comes up tails.Combination B offers a return of $50 if the same coin toss comes up heads and $0if it comes up tails. You have to pay money to play the game. If the two gamesare offered at the same price, the first is underpriced or the second is overpriced,relative to the other. Let us say it costs $35 to play either game. What would youdo? You would pay $35 to play the first game and offer the second game for $35.You would receive $35 and pay $35, netting nothing. But if heads comes up, youreceive $100 and pay $50 for a net gain. If tails comes up, you net nothing. Youhave everything to gain and nothing to lose by playing. Everyone would do this

and the price of the first game would have to rise and/or the second fall.

When situations can be analyzed using arbitrage as a basis, there is no need toworry about a person’s risk aversion. Every person, whether risk averse or not,would take advantage of arbitrage opportunities. It turns out that managing risk can be nearly always analyzed this way. Hence, treating people as though they arerisk averse but willing to engage in arbitrage opportunities is perfectly acceptableand does not mean they are not risk averse.

Risk and Financial Market Efficiency

We oftentimes talk about financial markets being efficient. Sometimes we saymarkets are a random walk, meaning that prices are unpredictable. Efficientmarkets are markets in which participants expect to earn returns that arecommensurate with the risks taken. Risk-free investments should earn the risk-freerate. Risky investments expect to earn a risk premium, as described above. In an





inefficient market, an investor earns more return than expected. Arbitrage would be a case in which a risk-free investment earns more than the risk-free rate. Sincefunds can theoretically be borrowed at the risk-free rate (or perhaps a little higher),

the return requires no commitment of funds. Arbitrage is the easiest violation toobserve in the market, because we do not have to know what the risk premiumwas. That does not mean it occurs frequently. In fact, arbitrage opportunities arethe least frequent violations of the rule of market efficiency. For riskyinvestments, we have to know what risk premium was expected to know if thereturn is truly in excess of what would be normal for the level of risk.

The notion of earning an abnormal return (“beating the market”) is a long-runconcept. Just earning an occasional abnormal return is not sufficient proof thatmarkets are inefficient. Short-run abnormal returns are common. In an efficientmarket, these returns are spread among investors. No one can consistently capturethem over the long run in an efficient market.

Are markets truly efficient? We not know. But we should operate with the belief that they are highly competitive and probably close to efficient. The good side of this is that this means prices of assets and derivatives should be fare. But we mustnot take this idea so seriously that we assume markets are always efficient. If everyone did this, they would stop processing information rapidly and marketswould become inefficient.

Thus, markets can be efficient only if people are somewhat skeptical about whether they are efficient.

Risk and the Law

Risk comes into play in the legal profession often, if not almost always. Wealready saw the example in the O. J. Simpson case. Plea bargains in criminal casesand out-of-court settlements in civil cases are excellent examples of certaintyequivalents. Taking the certain outcome and avoiding the risk associated with atrial is common. In fact, experts tell us that in civil litigation an out-of-courtsettlement is the preferred outcome.

Economists who study the criminal mind also theorize that criminal behavior ismerely a form of risky decision making. Choosing to commit a crime, in the faceof the risk that injury or loss will occur during the commission of the crime or





societal punishment will occur later, is just a decision that the benefit justifies therisk. That may be oversimplifying but criminals are mostly interested in self-

preservation (though some may have a death wish) and in many cases probably

weigh the risks before committing the crime.

The law is also used to regulate risk-taking behavior. Society often prohibitsactivities that will endanger people. It is certainly common and widely acceptedthat society can enact laws to prohibit risk-taking behavior that endangers peopleother than or in addition to the risk-taker, such as speed laws. But we alsoroutinely enact laws primarily designed to protect people from their own risky

behavior, such as seat-belt and motorcycle helmet laws.

Risk, Government, and Politics

Without trying to take a political position, let us think about President Bush’sdecision to invade Iraq in terms of the risk and rewards. As leader of this country,President Bush bore responsibility for the safety and security of the U. S. andarguably many other countries of the world, particularly Israel. The informationavailable was that Iraq had used weapons of mass destruction (WMD) against theKurds following the Gulf War of 1991 (a fact documented by the U.N.).Intelligence provided by the CIA, Great Britain, Russia, and several middle easternnations stated that Iraq currently had WMD. U. N. resolutions requesting

permission to give inspectors free rein in Iraq and to allow private interviews withIraqi scientists were denied by Iraq.

Consider the following decision matrix.

OutcomeDecision

Iraq has WMDs Iraq has no WMDs

Attack Likely capture and stopWMDs

Severe political fallout

Do not attack Iraq may use WMDs Nothing

(Obviously this is somewhat oversimplified. There are other consequences and benefits of attacking or not attacking. We’re focusing only on the WMD issue.)

From a risk-reward perspective, how should this decision be analyzed? Consider that, given the information that Iraq likely has WMDs, the ex ante probability is





highly tilted toward there being WMD. The consequences of not attacking are thatIraq uses these weapons. The gain if there are no WMDs is nothing. Theconsequences of attacking are that Iraq has no WMDs and Bush suffers severe

political fallout, possibly leading to loss of election. The gain if there are WMDsis stopping them. When the consequences are weighed against the gain, thedecision was made to attack. We are not saying this was right or wrong, but thiswas the decision, which was made conditional on the available information, theresponsibility assumed, and the risk of an adverse outcome.

Of course, we all know what happened. But far too much has been analyzed aboutthis decision ex post . Media and individual criticism that this decision was wrongare misguided and confuse ex post with ex ante. A decision was correct ex post if that same decision would be made in the same ex ante circumstances with the sameinformation. It is difficult to imagine being faced with that decision again and

being able to forget this ex post result, but all risky decisions should be analyzedunder their own unique circumstances, making proper use but not abuse of historical information. And ex post decision making (in sports frequently called

Monday morning quarterbacking) is of little use, except in how it can be used toimprove the quality of information available for future decision making. Thereinlies the only real legitimate concern: that substantial amounts of corroborativeinformation were inaccurate. Of course, decisions are rarely made in the presenceof perfect information. If so, there is no risk and the decision is an easy one. A

computer could be programmed to make that decision.

Ex post criticism of decision makers also ignores the element of responsibility.When citizens were asked (before the invasion) “Should the U. S. invade?”, themost common answers were “yes” and “no.” The only correct answer was “I don’tknow.” If a citizen believes we should not go to war, but Iraq has and uses WMD,no one will hold that citizen responsible. Likewise, a citizen who believes weshould go to war is not held responsible when we do and no WMD are found. Thedecision maker bears the biggest risk, because that person has the responsibility.

Moreover, only the decision maker has access to the full set of information,however imperfect or misleading it might be.

In short, decision makers are those who possess the information, are charged withmaking the decision, and bear the primary risks of the outcomes of the decision.





(For those who might view this material as politically biased, let me add that asimilar analysis of President Clinton’s decision to bomb Kosovo could have beendone and might well have led to the same decision.)

A great business example is Coca-Cola’s introduction of “new Coke” in 1984.Coca-Cola took away traditional Coke and replaced it with a sweeter Coke thattasted more like Pepsi and have proven more popular with Coke drinkers in blindtaste tests. The data said that Coke needed to make this change or continue to loseground to Pepsi. But when the change was made, Coke drinkers were angry anddemanded their product back. Coke then re-introduced the original Coke (CokeClassic) though some drinkers are not sure it was precisely the same as the originalCoke. Coke was heavily criticized for this mistake, but based on the data it hadwhen it made the decision, this change was appropriate.

The intelligence business, sports, business decision making, and life are just asequence of risky decisions. We must not confuse the past with the future. AsYogi Berra allegedly said “Predicting is very hard, especially when it involves thefuture.” We might add, “Predicting is very easy when it involves the past.” Ingovernment or business, we select leaders to make decisions under risk. We mustkeep in mind what is reasonable to expect of decision makers.

Observing Risk Taking Behavior

As noted above, lawsuits and criminal prosecutions are an excellent example of how risk is analyzed and certainty equivalents are determined. Another excellentexample is game shows. As silly as some of them may seem, game shows offer anopportunity to observe people, typically of average wealth, with opportunities totake risks and make more money. In some cases, these individuals have theopportunity to take a certain amount and go no further. The next time you watch agame show, think about it in terms of the analysis of risk and reward.

What Managing Risk Means for Businesses and Non-profits

The focus in this course is how businesses and non-profits manage risk. Allorganizations face risk; indeed risk taking is what life in general is all about.Without risk taking, no one could expect much success. Managing risk is the

process of defining the risk an organization wants, measuring the risk it has, andensuring that the former equals the latter. Managing risk sometimes means





adjusting the risk upward. Sometimes it means adjusting the risk downward, a process usually called hedging. The focus of this course, and indeed most of therisk management process, is about hedging.

Stating this more formally, managing risk is about taking actions that alter thelikelihood of undesirable outcomes. Here are the three general ways in which risk is managed.

• Diversifying a portfolio eliminates the risk that undesirable events associatedwith one component of the portfolio will have a significant effect on thewealth of the holder of the portfolio.

• Purchasing a contract that passes on the risk to another party. Thisarrangement is equivalent to insurance, but in our world of financial risk,this will usually take the form of options.

• Giving up the opportunity to benefit from favorable outcomes in return for protection against unfavorable outcomes. This form of risk managementcharacterizes forward contracts, futures contracts, and swaps.

We will not spend much time on diversification, because it is well-covered ininvestments courses, and also because it is a very simple and inexpensive way to

manage risk. Our focus is on the use of forwards, futures, swaps, and options. But before we can begin examining these instruments, we have to study more aboutwhat risk means to a business or nonprofit organization.

Some References

These notes were drawn from personal observations and a number of references.Some suggested readings if you are interested in learning more.

Bernstein, Peter L. Against the Gods: The Remarkable Story of Risk . New York:Wiley (1996).

Dembo, Ron S. and Andrew Freeman. Seeing Tomorrow: Rewriting the Rules of

Risk . New York: Wiley (1998).





Gigenrenzer, Gerd. Calculated Risks: How to Know When Numbers Deceive You. New York: Simon and Schuster (2002).

Peterson, Ivars. The Jungles of Randomness. New York: Wiley (1998).

Ropeik, David and George Gray. Risk: A Practical Guide for Deciding What’s

Really Safe and What’s Really Dangerous in the World Around You. Boston:Houghton Mifflin (2002).

Ross, John F. The Polar Bear Strategy: Reflections on Risk in Modern Life.Reading, Massachusetts: Perseus Books (1999).

Taleb, Nassim Nicholas. Fooled by Randomness: The Hidden Role of Chance in

the Markets and in Life. New York: Texere (2001).

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FRM Notes 2008 Unit IA What is Risk

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