BUS 640 Week 1 Responses

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Decision Making Under Risk and Uncertainty

Learning Objectives

A�er reading this chapter, you should be able to:

Explain that risk is measured by the varia�on of poten�al returns around the mean or expected value of the poten�al returns. Describe what is meant by the expressions risk taking, risk seeking, risk aversion, risk preference, and risk neutrality. Discuss why the degree of risk aversion (or conversely risk tolerance) can lead different people to different decisions. Explain how risk-return "indifference curves" can be used to demonstrate why different people make different choices in the same decision scenarios. Explain several decision criteria that allow individuals to adjust for risk in their decision making. Describe how the shape of the distribu�on of possible outcomes will change the probabili�es associated with the most-likely scenario and the worst-case scenario.

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Standard Devia�on

Introduction

In Chapter 1, we saw that managers wan�ng to make decisions that best serve their objec�ve func�ons will need to first define the metric for their objec�ve func�on. We argued that the firm’s objec�ve will be to maximize profits, and that managers must make decisions under condi�ons of either certainty or uncertainty, and might foresee returns that accrue in the current period or in future periods. In the real world, where risk and uncertainty is the norm, and where expenses and revenues are incurred and received both in the present period and into the future, the appropriate decision criterion is the expected net present value (ENPV). We also argued in Chapter 1 that decisions will usually have both monetary and nonmonetary outcomes that are of interest, or concern, to the decision maker. Accordingly, decision makers will make trade-offs against profit to compensate for the nonmonetary costs or benefits that are associated with the decision. If the psychic pain (disu�lity) of nonmonetary costs is greater than the psychic gain (u�lity) of the nonmonetary benefits, we say there is net disu�lity associated with the decision and the decision maker will require addi�onal profit to compensate for the net disu�lity associated with the decision. Conversely, if the nonmonetary benefits exceed the nonmonetary costs, there is net u�lity associated with the decision, and the decision maker will be willing to give up some profit to compensate for the nonmonetary aspects of the decision.

Risk causes disu�lity for most business decision makers and so they will want to be compensated for bearing risk. In this chapter we integrate risk analysis into the decision-making process and consider several decision criteria that adjust the monetary outcomes of a decision for the risk that is associated with those returns. In Chapter 1, we noted that the ENPV criterion is only really appropriate if the manager con�nually makes the same type of decision in the same environment, such that the manager could reasonably expect that over many trials the aggregate outcome would be approximately equal to the sum of the individual ENPVs. Approximately half the �me the actual outcome will be higher than the ENPV and the other half of the �me the actual outcome will be below the ENPV. Although facing risk in each specific decision, the repe��veness of the decision allows the chances of below-average outcomes to be offset by above-average outcomes, and over many similar decisions the total profit outcome would approximate the sum of the ENPVs of all the decisions.

But in many business situa�ons the manager faces a variety of different decisions from day to day and most types of decision are not repeated o�en enough to make the ENPV criterion an appropriate decision criterion since it does not adjust for differing degrees of risk associated with individual decision problems. In this chapter, we recognize that the decision maker will want to incorporate risk analysis into the decision-making process for those decisions that are not repeated frequently and will want to adjust each decision to take account of the degree of risk involved in each par�cular decision.

How Is Risk Measured?

Risk can be expressed as a measure of the chance that the value of the ENPV of a decision will not be the actual outcome. To calculate a measure of variability of the poten�al outcomes rela�ve to the ENPV we need to understand the sta�s�cal concept known as the standard devia�on, which is a measure of the devia�ons of the possible outcomes from the central tendency (or mean) of those possible outcomes. First note that the ENPV is a measure of central tendency of the poten�al outcomes—indeed the ENPV is the weighted mean (where the weights are the probabili�es of occurring) of the possible outcomes. The mean of any series of numbers has an associated standard devia�on, which indicates the extent to which the mean value is representa�ve of all the data points that enter the calcula�on of that mean. The standard devia�on is higher if the possible outcomes are more widely dispersed around the mean value, or is lower if the actual outcomes lie rela�vely close to the mean value. To calculate the standard devia�on, the devia�ons of each data point from the mean are squared and then summed to find the variance of the distribu�on, and the standard devia�on is then simply the square root of the variance. In effect the standard devia�on indicates the average absolute devia�on of the outcomes from the mean outcome. Thus, the standard devia�on provides a suitable measure of the risk that the ENPV will not be a�ained.

Any good calculator can instantly deliver the standard devia�on of a series of simple numbers. Similarly, it is easy in an Excel spreadsheet to type (for example) = stdev(c2-c24) into a vacant cell to indicate the range of data points (cells c2 to c24 in this example) over which the computer can calculate the standard devia�on. Note that it is more complex to calculate the standard devia�on of a probability distribu�on. We need to (1) find the ENPV of that distribu�on; (2) subtract each outcome from the ENPV to find the devia�ons from the mean; (3) weight each devia�on by its probability of occurring; (4) sum these weighted devia�ons to find the variance; and (5) take the square root of the variance to find the standard devia�on. This is demonstrated for a simple case in Table 2.1, in which we suppose that three outcomes are possible (column 1) with probabili�es as shown (in column 5), which you can verify gives an expected value of 10 as shown (in column 2). For simplicity here, we assume the cash flows all take place in the present period such that ENPV = EV.

Table 2.1: Calcula�on of the standard devia�on of a probability distribu�on

Possible outcome ($)

Expected value ($)

Devia�on of possible outcome from the mean (EV) outcome

Squared devia�on from the mean outcome

Probability of each possible outcome

Weighted devia�on from EV

−10  10  30

10 10 10

−20  0 20

400   0 400

0.25 0.5  0.25

100   0 100

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To find whether the wins offset the losses, variance or standard devia�on can be used to measure the risk associated with uncertain outcomes.

© Photodisc/Thinkstock

Entrepreneurs, skydivers, and motorcyclists all voluntarily take risks with the expecta�on that the u�lity of the reward will outweigh the disu�lity of the risk.

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Variance = 200

Std. Devia�on = 14.14

Half of the variance around the ENPV is quite desirable, and of course I am referring to the outcomes that are greater than the ENPV. This part of the risk is known as the upside risk and represents be�er outcomes than can generally be expected in mul�ple trials of this decision. Some risk analysts have suggested that no one is worried about these posi�ve devia�ons from the mean because we would "laugh all the way to the bank" if one of these were to occur. On the other hand, the downside risk represents the outcomes that are worse than the ENPV, and we definitely worry about these. Accordingly, some analysts have suggested that we calculate only the semi-variance of the outcomes by including only those outcomes with nega�ve devia�on from the ENPV to measure only the downside risk. Although intui�vely appealing, the semi-variance approach is not commonly used because it ignores the upside risk; a�er all, if "some�mes you win and some�mes you lose," you need to know the extent of the wins to see whether they would offset the losses. Thus, we tend to use the standard devia�on as our measure of the risk associated with uncertain outcomes. So, now we have a measure of risk, but before we adjust for risk, we need to consider the decision maker’s a�tude toward risk.

Attitudes Toward Risk

People have different a�tudes toward risk. Some people seem to enjoy doing risky things, while others are extremely unhappy to be exposed to risk, and, of course, there are those who do not seem to care. Indeed, individuals will have one of three a�tudes toward risk: risk preference, risk aversion, or risk neutrality. Risk preference means that the individual prefers more risk to less risk, with other things (such as reward or profit) being equal. A risk preferer would therefore choose the riskier of two equally profitable investments. This is only ra�onal behavior if the individual’s objec�ve func�on is to maximize risk rather than to maximize profit, or if the person is so rich that he or she places very li�le value on the money he or she might lose while placing more value on the thrill he or she will get by taking the risk; perhaps high-roller gamblers might fit this profile. Risk preferers might get lucky and have a series of wins (despite the odds) but sooner or later will be bankrupted if they con�nue to make large ENPV decisions this way.

Risk aversion means that the individual prefers less risk to more risk, other things being equal, and will therefore choose the less risky of two equally profitable investments. Risk-averse individuals may choose the more risky alterna�ve if they expect to be adequately compensated for the addi�onal risk undertaken. They weigh up the monetary trade-off of the extra income against the psychic dissa�sfac�on of addi�onal risk bearing and make their decision between less-risky-but-less-rewarding investments and more-risky-but-more-rewarding investments. Finally, some individuals are risk neutral, not caring about risk, in which case they do not need to adjust their decisions for risk. Risk neutrality can occur due to a genuine lack of concern for risk and subsequent losses (which makes it almost as dangerous to your wealth as risk preference), or due to the repe��on of the same or similar decision many �mes, so that on any one occasion the decision makers can act as if they are risk neutral. Thus, as we have noted, the ENPV measure of profit is appropriate if the same (or sufficiently similar) decision is to be repeated many �mes. In this situa�on the decision makers can act as if they are risk neutral for any one of those decisions.

We need to clarify a few more terms that are commonly used in discussions of risk. People o�en talk about entrepreneurs, for example, being risk seekers. Risk seekers seek to do risky things (like entrepreneurship, skydiving, and motor racing), because they expect that risk and return are posi�vely correlated: The higher the risk the higher the return. Whether the return is simply monetary, or is both monetary and nonmonetary (i.e., includes psychic sa�sfac�on), most risk seekers only take the risk if they expect the payoff to be greater to compensate them for risk bearing. Risk seekers are therefore not

risk preferers but are actually risk-averse.1

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#footernote1) Another term that probably needs clarifying is risk taker. We are all risk takers, like it or not. Every day we are subject to the risks of global warming, asteroids, tsunamis, earthquakes, global financial crises, traffic accidents, and physical violence, to name just a few sources of the risks we con�nually take. What is important is not that we take risks but what our a�tude is toward taking risks. As you now know, our a�tude either will be risk preference, risk aversion, or risk neutrality, depending on our prior knowledge of the situa�on and our cogni�ve processing of the psychic costs and benefits associated with taking specific risks. We should also dis�nguish between voluntary risk taking and involuntary risk taking. The everyday risks listed earlier in this paragraph are imposed on

us by nature or by our fellow man and are borne involuntarily. Risk seekers, however, take risk voluntarily in the expecta�on that the u�lity of the reward will outweigh the disu�lity of the risk. Thus entrepreneurs, skydivers, racing drivers, and business decision makers voluntarily undertake risky projects and make risky decisions even though they are averse to risk.

Degrees of Risk Aversion

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Risk aversion can range from almost zero degrees of risk aversion (i.e., being almost risk neutral) to being extremely risk-averse. For someone who is slightly risk-averse, bearing risk causes rela�vely li�le psychic dissa�sfac�on. We say they are highly tolerant of risk. People like this will require only a rela�vely small amount of monetary compensa�on for bearing addi�onal risk. For others, bearing risk causes much more psychic dissa�sfac�on. We say they are highly intolerant of risk—they will try to avoid voluntary risk taking as much as possible. For these people, it will require much greater monetary compensa�on to induce them to accept addi�onal risk. Since different decision makers exhibit different degrees of risk aversion (or conversely, risk tolerance), the extent to which they will want to adjust their decision for risk will differ. Accordingly, we must take into account the decision maker’s degree of risk aversion as well as the extent of risk involved in any par�cular decision.

Risk Percep�on

Similarly, each person might perceive risk differently. Individuals perceive the risk in a decision situa�on more or less accurately depending on their prior

knowledge and their cogni�ve biases. Greater prior knowledge of the situa�on, or greater informa�on search ac�vity,2

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#footernote2) may provide the decision maker with useful informa�on that others do not have, such that she might (correctly) say the situa�on is not very risky while others might say it is highly risky because of their ignorance of the situa�on. The old saying that "fools rush in where angels fear to tread" reflects the percep�on of li�le or no risk by those who have less knowledge about the situa�on compared to those who have more knowledge. Next, a cogni�ve bias such as overconfidence may cause one person to overlook risks that a less confident person might perceive because the la�er looks more carefully into the situa�on or spends more �me and money on informa�on search ac�vity to reveal the hidden dangers. Another cogni�ve bias is the tendency of decision makers to use heuris�cs, or simplis�c decision rules. While economizing on �me and search costs, heuris�cs could actually increase the decision maker’s exposure to risk, since they consider only some of the informa�on that is poten�ally available. For example, entrepreneurs have been shown to be more overconfident and to use heuris�cs more than employed managers of firms (Busenitz &

Barney, 1997).3 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#footernote3) When others see entrepreneurs taking extraordinary risks they o�en presume that these entrepreneurs must be highly tolerant of risks, when in fact many entrepreneurs are highly risk-averse; they do indeed take greater risks, but this may be because they have be�er informa�on, have stronger desire for income, or they did not perceive some of the risks in the first place.

1. Probably most gamblers are risk-averse because they know that the odds are stacked in favor of the house or person offering the gamble. If they con�nue to take the same gamble, they know that the ENPV of repeated gambles (e.g., rolling the dice) is ul�mately nega�ve, but they hope to get lucky and experience one of the upside-risk outcomes. When be�ng on horses or sports teams, gamblers may believe they possess be�er informa�on than the person accep�ng the bet. Decision making is akin to gambling, of course, since the decision maker must choose one decision out of two or more possible decisions and then wait for the roll of the dice to see which of the possible outcomes actually happens. Thus, we o�en use the word "gamble" (as a noun) to refer to the decision-making problem facing a business decision maker. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#return1) ]

2. As we saw in Chapter 1, informa�on search ac�vity is the purposeful search for informa�on to inform the decision making process. It includes gathering data and analyzing that data to reduce the decision maker’s ignorance of the factors and issues that might otherwise cause the outcome of the decision to vary. In Chapters 4 and 6 we look into informa�on search ac�vity rela�ve to consumer demand and to cost levels, respec�vely. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#return2) ]

3. Entrepreneurs tend to anchor their es�mates on past outcomes and to not revise their es�mates on the basis of new informa�on. Second, they tend to base their decision making on the most recently acquired or most easily recalled informa�on, this being known as the availability heuris�c, but of course such data may not be representa�ve of the range of outcomes that should be expected. Third, the representa�ve heuris�c is where people base decisions on a rela�vely small number of observa�ons (rather than a representa�ve sample), which introduces risk because the limited sample might not be representa�ve of the range of probable outcomes. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/ch02introduc�on#return2) ]

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2.1 Adjusting for Risk Using the Certainty Equivalent

The certainty equivalent of a decision is the amount of money, available with certainty, that a person would consider equivalent to the expected value of a risky decision. In this sec�on, we will introduce risk–return trade-off curves and show how these differ according to the decision maker’s degree of risk aversion. This will allow us to demonstrate that different individuals typically have different certainty equivalents.

Risk–Return Trade-off Curves

As noted, risk causes disu�lity to be incurred by the risk-averse decision maker. We have argued that people with different degrees of risk aversion will require different amounts of compensa�on to induce them to bear an addi�onal quantum of risk. Using simple graphical analysis we can depict the risk– return trade-off curves of a par�cular risk-averse individual (whom we shall call Mr. X) shown in Figure 2.1.

Figure 2.1: Risk–return trade-off curve for risk-averse decision maker, Mr. X.

Suppose that Mr. X must decide where (at which loca�on) he will build a new restaurant. The points A, B, and C shown in Figure 2.1 relate to three different risk–return combina�ons that represent different restaurant loca�ons. We depict these three decision alterna�ves with risk measured by standard devia�on (SD) and return measured by ENPV. Their risk and return outcomes differ because of differences in popula�on density, passing traffic, proximity to public transport, and so on. As you can see, decision A has ENPV = 100 and SD = 50; decision B has ENPV = 100 and SD = 30; and decision C has ENPV = 60 and SD = 30. It should be immediately clear that Mr. X, and indeed any risk-averse profit-maximizing decision maker, will prefer B to A, because A is equally profitable but has more risk (higher SD) than B. Similarly, all risk averters will prefer B to C, because these two op�ons have the same amount of risk but B is more profitable (higher ENPV) than C.

We now know that decision B is the best choice for Mr. X, but which would he consider to be the second-best loca�on? In fact, I have prejudged the answer by drawing the risk–return (RR) curves such that A and C lie on the same RR curve (shown as RR2) so the answer is that they are both equal in the case

depicted (i.e., reflec�ng Mr. X’s feelings about risk and return). Each RR curve depicts those combina�ons of risk and return that give the same level of u�lity. These curves are more commonly known as indifference curves, which are lines drawn to pass through combina�ons of variables among which the

decision maker is indifferent, that is, receives the same amount of u�lity.4 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#footernote4) Thus, Mr. X will be indifferent between A and C, or indeed any other combina�on of risk and return that lies on RR2. Now, since point B is preferred to both point A and C, it

follows that every combina�on of risk and return on RR3 is preferred to any combina�on on RR2. Similarly, any risk–return combina�on on RR1 is considered

inferior to any combina�on on any higher indifference curve. Thus, we can say that any point on a higher indifference curve will be preferred to any point on a lower indifference curve and that the direc�on of preference is shown by the arrow; more return is preferred when risk is the same, or conversely, less risk is preferred when return is the same, and the decision maker prefers combina�ons that have both more return and less risk. Note that we do not need to know the actual value to the u�lity represented by the RR1, RR2, and RR3 curves, we just need to know the order of preference—thus indifference curve

analysis is concerned with ordinal (i.e., simply in order) preferences rather than cardinal (i.e., measurable) preference differences.

The RR indifference curves demonstrate the decision maker’s trade-off between risk and return. This trade-off is also known as the marginal rate of subs�tu�on (MRS) between risk and return, which is equal to the amount of risk the decision maker will accept for an addi�onal measure of return. This trade-off is indicated by the slope of the RR curve, which is equal to the "rise over the run." In Figure 2.1, we saw that the decision maker considers points A and C to be equivalent. Now if Mr. X was asked to change from C to A, we can see that he wants 40 more units of return (the rise from 60 to 100) to compensate for the 20 extra units of risk (the run of 30 to 50). Thus, the slope of the RR2 indifference curve between points C and A is 40/20 = 2 and this

value is rather typical of this individual’s MRS at other risk–return combina�ons in the vicinity of decisions A, B, and C.5

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#footernote5)

Figure 2.2: Differing degrees of risk aversion for two different decision makers

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In Figure 2.2, we show the RR curves of two other individuals (Mr. Y and Ms. Z) who have quite different degrees of risk aversion, and thus quite different marginal rates of subs�tu�on. These people are considering restaurant loca�ons A and C, because loca�on B has already been taken by Mr. X. Note that Mr. Y prefers decision A because, for him, it lies on a higher RR indifference curve. Conversely, Ms. Z prefers decision C because, for her, it is on a higher indifference curve.

Looking carefully at Mr. Y’s indifference curves we no�ce that his risk–return trade-off (i.e., his MRS) is rela�vely low in the vicinity of point C; to move from

30 units to 50 units of risk (along RRY1) he would require only about $5 more (from $60 to about $65) to compensate for the 20 addi�onal units of risk.

Thus, his MRS for return and risk is 5/20 = 0.25. Because decision A offers $40 more return for those 20 extra units of risk, it is u�lity maximizing for him to

take decision A rather than decision C. Conversely, the MRS for Ms. Z, moving along RRZ2, is about 4 (i.e., a $40 change in return from $60 to $100 is

necessary to compensate for a 10 change in risk from 30 to about 40, along RRZ2), so the addi�onal risk associated with decision A (20 units) is not

compensated for by the addi�onal 40 units of return offered by A, and thus Ms. Z prefers decision C.

What we have demonstrated is that risk-averse decision makers will make different decisions according to their degree of risk aversion. Note that both Mr. Y and Ms. Z would have preferred restaurant loca�on B if it were s�ll available, since its risk–return outcomes would fall on a higher indifference curve for both of them. Once that decision op�on was gone, they had different preferences for the remaining two op�ons. We saw that Mr. X moved first and chose his preferred alterna�ve, which was loca�on B. Subsequently, Mr. Y and Ms. Z chose differently, because their risk–return trade-offs were different. Mr. Y, being less risk-averse, chose loca�on A, while Ms. Z, being more risk-averse, chose loca�on C.

The Certainty Equivalent as a Decision Criterion

The analysis above allows us to consider the certainty equivalent (CE) of a risky decision. The CE of a risky decision (or gamble) is the amount of return, available with certainty (i.e., zero risk), that the decision maker will consider is equivalent to the risk–return combina�on of the risky decision. Looking at Figures 2.1 and 2.2, you will see that the ver�cal axis in each figure represents a series of levels of return (ENPV) that have zero risk a�ached to them. Now no�ce that each of the indifference curves terminates at the ver�cal axis, at a point of zero risk, thus revealing the CEs for all of the risk–return combina�ons on each indifference curve.

In Figure 2.1 for example, for Mr. X the CE of decision B seems to be about 80, while the CEs of both decision A and C seem to be about 40 (i.e., where the indifference curves hit the ver�cal axis). Thus, for Mr. X the CE of decision B is much greater than the CE for either A or C, so he prefers op�on B over the other two op�ons. In Figure 2.2 we see that the CE for Mr. Y seems to be about 85 for decision A and 55 for decision C. Finally, for Ms. Z, the CE is about 22 for decision C and much lower for decision A. In each case the individual prefers the decision alterna�ve with the highest certainty equivalent.

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Put in simple terms, the certainty equivalent factor, which is the ra�o of the perceived value of the risk-free alterna�ve to the risky alterna�ve, expresses how many cents in the dollar a decision maker would consider to be equivalent to the risky decision.

© Stockbyte/Thinkstock

The Certainty Equivalent Factor

The Certainty Equivalent Factor (CEF) is the ra�o of the perceived monetary value of the risk-free alterna�ve (i.e., the CE) to the risky alterna�ve (i.e., the ENPV). In the case of Mr. X, the CEF for decision B is 80/100 = 0.8. The CEF effec�vely tells us what propor�on of the risky ENPV would be considered equivalent to the risky ENPV, if it were risk-free. Put another way, the CEF tells us how many cents in the dollar, available with certainty, the decision maker will consider to be equivalent to the risky decision. Thus, Mr. X values decision B at 80% of the dollar value of the ENPV. So, the CE criterion will tell us not only which is the preferred alterna�ve but will also tell us how many cents in the dollar would be just sufficient to trade for the risky decision, which tells us just how risk-averse the decision maker is. In this case Mr. X is willing to take a 30% reduc�on in monetary value to

compensate for the risk involved in decision B.6 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#footernote6)

No�ce that the CE values are different for each individual—we cannot compare the psychic value of either risk or return across people. That is why the RR curves are labeled differently for the three people depicted: Each person makes his or her own, personal, internal psychic evalua�on of the disu�lity of risk and the u�lity of income and makes his own decision accordingly.

Of course, it is unrealis�c to think we would plot out risk–return indifference curves for all decision makers to see which decision they will choose. The graphical model of the decision-making process that we have u�lized here is primarily intended to facilitate your learning about risk–return trade-offs in decision making. But note that the

model has brought us to the point of a rather simple decision rule for decision making under risk and uncertainty; namely, risk-averse managers should choose the decision alterna�ve that has the highest certainty equivalent. A li�le introspec�on on the part of decision makers will lead them to an intui�ve preference for one decision alterna�ve over the others, which will reflect their personal risk–return trade-off.

4. Indifference curve analysis is used widely in economic analysis. We will encounter them again in Chapter 3 when we examine the decision-making process of consumers as they allocate their limited incomes among compe�ng goods and services. Indifference in an economics context means not preferring any one combina�on over the others, in contrast to vernacular usage where it might mean you do not care about any of the combina�ons. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#return4) ]

5. Although the slopes of Mr. X’s indifference curves are generally about 2, you can see that they are convex from below (or equivalently, concave from above) and thus the MRS value ranges from below 2 (where the curves are slightly fla�er) to above 2 (where they are slightly steeper). The convexity of these risk–return indifference curves reflects the individual’s increasing MRS for risk and return as more risk is undertaken. That is, the individual will require an increasing quantum of return to compensate for constant increments of risk, as the total amount of risk borne is increased. Increasing MRS of risk and return reflects the individual’s diminishing marginal u�lity of income and increasing marginal disu�lity of risk, which seems characteris�c of most human beings. We shall revisit these concepts in Chapter 3, in the context of consumer behavior. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#return5) ]

6. No�ce that for Mr. X, the CEF for loca�on A is 40/100 = 0.4; while for loca�on C the CEF = 40/60 = 0.67, and thus appears to rank loca�on C ahead of loca�on A, which is not what the indifference curves and the CE criterion previously told us (and we assume that they are the true reflec�on of Mr. X’s preferences). Instead the CEF provides an insight into the rela�ve degree of risk aversion exhibited by the individual for the op�on selected. For Mr. Y, the CEF of loca�on A is about 85/100 = 0.85 and for Ms. Z the CEF of loca�on C is about 22/60 = 0.37. Thus, as we noted earlier, Mr. Y is less risk-averse, followed by Mr. X, and then followed by Ms. Z, who is the least risk-tolerant of the three. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.1#return6) ]

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Simple and transparent, the maximin criterion is appropriate for risk-averse people making risky decisions and is an effec�ve way to avoid incurring losses.

© Neil Leslie/Ge�y Images

2.2 More Transparent Decision Rules for Managers

If decision makers are self-employed and are the sole owner of their own business firms, they can make their business decisions this way, but if they are employed managers of firms that are owned by other shareholders they will have to be more accountable to those shareholders for the decisions they make, and, accordingly, will have to adopt a more transparent decision rule than, "I made that decision because it made me feel be�er." Thus, we need to consider some decision rules that can be argued somewhat more objec�vely by managers to shareholders.

The Maximin Decision Rule

When the shareholders of a firm are risk-averse, as we expect they are, they will want managers to adopt decision-making rules or policies that take the risk

associated with different decisions into account.7 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote7) One such decision rule is maximin—that is, choose the alterna�ve that has the highest (maximal) worst (minimum) outcome. In the examples discussed earlier, we were concerned with the standard devia�on of the poten�al outcomes associated with restaurant loca�ons A, B, and C. The maximin rule is concerned with only one of those poten�al outcomes for each of A, B, and C—the worst one. It is based on the principle of affordable loss—can the firm afford to suffer the worst outcome associated with a risky decision? Shareholders will not want the manager to make decisions that could possibly bankrupt the firm (and cause shareholders to lose their investment), so they may put pressure on managers to make rela�vely conserva�ve decisions.

As an example of the maximin decision rule, consider the choice between an investment Project A and an investment Project B. For Project A the ini�al investment is $1 million in year 1 with possible final outcomes in year 2 ranging from a loss of $200,000 to a profit of $5 million. Project B has an ini�al investment cost of $2 million with possible outcomes in year 2 ranging from a loss of $500,000 to a profit of $10 million. For the maximin decision rule, we simply compare the two minimum outcomes and therefore choose Project A because its worst outcome is a loss of $200,000 compared to Project B’s worst outcome, which is a loss of $500,000. If the worst outcome were to occur, the firm would be be�er off taking a hit of $200,000 rather than $500,000.

As you can see, the maximin criterion is appropriate for risk-averse people making risky decisions that are not repeated enough �mes to allow the law of averages to work out in the firm’s favor. This decision-making rule is designed to be a simple and very transparent way to avoid incurring losses that cannot be tolerated by the firm and its shareholders. But, by refusing to consider the other poten�al outcomes it may be a very poor decision criterion. What if the probability of Project B’s worst outcome occurring was only 10% and the probability of Project A’s worst outcome was 40%? In that case, by taking decision A the managers have chosen to risk a worst outcome with four �mes the chance of occurring than the worst outcome of Op�on B. Or, what if the other possible outcomes for Project A were posi�ve but rela�vely small while the other outcomes for Project B were posi�ve and rela�vely large?

The maximin criterion does not consider these other outcomes at all.8

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote8) So let’s look at some decision rules that do.

Coefficient of Variation Decision Rule

The coefficient of varia�on (CV) is a sta�s�c of a probability distribu�on and is calculated as the ra�o of the standard devia�on to the mean. Going back to the example of restaurant loca�on A, B, and C in the earlier decision-making problem, we can calculate the CV of A as 50/100 = 0.5; for B it is 30/100 = 0.3; and for C it is 30/60 = 0.5. In effect the CV criterion provides a measure of the risk per dollar of return, and the decision rule is to choose the op�on that has the smallest CV. So according to this rule, Mr. X would consider loca�ons C and A as equal but inferior to loca�on B, thereby agreeing with his certainty- equivalent-based decision. For Mr. Y and Ms. Z, the CV rule would say the two remaining op�ons are equal, but we saw that Mr. Y preferred op�on A (higher CE for him) while Ms. Z preferred op�on C (higher CE for her). Thus, the CV criterion does not take into account the differing degrees of risk aversion that individuals may have and is, therefore, an inferior decision rule for individuals making decisions when taking into account only their own risk– return preferences. But for managers making decisions on behalf of shareholders, the CV criterion may be more suitable because some shareholders (like Mr. Y) will be less risk-averse while others (like Ms. Z) will be more risk-averse. On average, shareholders might be happy enough with the CV decision rule, and they can always sell their share in this firm and buy shares in a more (or less) conserva�ve alterna�ve business if they want to.

In Figure 2.3 we show the CV decision rule as it applies to the restaurant loca�on decision problem. The CVs associated with decision A and C are equal to 0.5 in both cases, and the CV associated with decision B is 0.3. No�ce that the slope of the CV lines emana�ng from the origin (these lines are known as rays) are each equal to the reciprocal of the CV value, since the slope is equal to the rise (ENPV) over the run (SD), while CV is equal to the run (SD) divided by the rise (ENPV). Also note that in effect the CV rays are like indifference curves since every combina�on on a par�cular CV ray is equally preferred. These CV rays have constant MRS between risk and return, but as we have seen, individuals do not. Their risk–return indifference curves are concave from above, exhibi�ng increasing MRS as more and more risk is taken on. As we saw in the case of our three restaurateurs, individual preferences might agree with the CV criteria (as Mr. X did) or not. While both Mr. Y and Ms. Z agreed that loca�on B was the best loca�on, Mr. Y ranked loca�on A superior to C while Ms. Z ranked loca�on C superior to A. Thus, the CV criterion is not generally suitable for individual decision making.

Figure 2.3: The coefficient of varia�on decision criterion

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As a more complex applica�on of the CV criterion, let us now reconsider the investment Project A and Project B decision introduced above. In Tables 2.2a and 2.2b we show the probability distribu�on of outcomes associated with these projects and calculate the ENPV, SD, and CV for each project (behind the scenes I have used an Excel spreadsheet to calculate these numbers). You will see that I have assumed a discount rate of 10% and that the ini�al cost is paid at the end of year 1 while the possible outcomes (cash inflows and ou�lows) are realized at the end of year 2. Whereas earlier we selected Project A using the maximin decision criterion, by applying the CV criterion we find that Project B is preferred. Although it is riskier (SD = 1.4374 compared to 1.0414), its ENPV is much higher ($3.5124 million compared to $0.7603 million) such that the CV ra�o is only 0.4092 for Project B compared with 1.3697 for Project A.

Table 2.2a: Calcula�ng the coefficient of varia�on for Project A

(1) Ini�al Cost year 1 (millions)

(2) Present value of ini�al cost

(3) Year 2 outcomes (millions)

(4) Present value of year 2 outcomes

(5) Net present value (millions)

(6) Probability of year 2 outcomes

(7) ENPV of outcomes (millions)

DF = 0.9091 DF = 0.8264

 10 8.2644 6.4463 0.4 2.5785

−2 −1.8182   5 4.1322 2.3140 0.5 1.1570

−0.5 −0.4132 −2.2314 0.1 −0.2231

ENPV = 3.5124

SD = 1.4374

CV = 0.4092

Table 2.2b: Calcula�ng the coefficient of varia�on for Project B

(1) Ini�al cost year 1 (millions)

(2) Present value of ini�al cost

(3) Year 2 outcomes (millions)

(4) Present value of year 2 outcomes

(5) Net present value (millions)

(6) Probability of year 2 outcomes

(7) ENPV of outcomes (millions)

DF = 0.9091 DF = 0.8264

5 4.1322 3.2231 0.3 0.9669

−1 −0.9091 2 1.6529 0.7438 0.3 0.2231

−0.2 −0.1653 −1.0744 0.4 −0.4298

ENPV = 0.7603

SD = 1.0414

CV = 1.3697

Thus, the CV decision criterion is an extension of the ENPV profit-maximizing rule and is appropriate when (1) outcomes are uncertain; (2) cash flows occur beyond the current �me period; (3) similar decisions are not made repeatedly; and (4) managers are risk-averse (hopefully reflec�ng their shareholder’s preferences). Note that in this example the CV criterion agrees with the ENPV criterion but disagrees with the maximin criterion, which neglected most of the informa�on available and made the decision based simply on the best of the worst outcomes. The CV criterion is thus a more sophis�cated decision criterion that, while not generally suitable for individual decision making, does have value for decisions made by managers of firms where the decision made

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Government bonds are regarded as the ul�mate risk-free security because repayment is absolutely certain.

© iStockphoto/Thinkstock

must be jus�fied to risk-averse shareholders on some objec�ve basis. Moreover, it can be argued that in the context of managerial decision making within the firm, the linear indifference rays implied by the CV criterion might be a sufficient approxima�on of shareholders’ preferences in aggregate since these shareholders are free to build a por�olio of shares (in different companies) that best serves their overall risk–return preferences. The CV decision rule is transparent and easily communicable to shareholders. If they want to hold shares in a more- or less-risk-taking firm they are usually free to sell their shares in this firm and buy shares in another firm that be�er fits their risk preferences. And, in any case, they can buy shares in a variety of firms such that the overall risk exposure of their investment por�olio be�er suits their risk and return preferences.

The ENPV Criteria Using Risk Premiums

Another commonly used method to adjust uncertain cash flows for risk is to adjust the discount factor to reflect the degree of risk. We do this by adding a risk premium to the discount factor such that projects with higher risk are discounted at higher opportunity discount rates. A risk premium is an addi�onal amount that is propor�onate to the addi�onal risk perceived. Recall that we defined the opportunity discount factor as the rate of interest that could be earned on an alterna�ve opportunity of equal risk. When the decision alterna�ves are clearly not equally risky it follows that their opportunity discount rate should not be the same for each alterna�ve.

At this point it is appropriate to examine the two main component parts of the opportunity discount rate (ODR). The first main part is the risk-free rate of return that one could earn on a loan that was absolutely certain to be repaid, for example, the purchase of government bonds. Although the government may change from �me to �me, the newly elected poli�cians would respect the previous government’s obliga�on to repay lenders who had bought government bonds, so government bonds are regarded as the

ul�mate risk-free security.9

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote9) The risk-free rate is made up of two subparts, the real rate of interest and the premium for expected infla�on. The real rate of interest is the rate that would cause the supply and demand for loanable funds to be equal in a market for funds without risk and without infla�on. But when lenders expect infla�on to occur, they expect the purchasing power of the funds returned (a�er the loan is se�led) to be lower than the amount loaned. For example, if prices are expected to rise by 5% over a year, the goods and services that could be purchased with $100 at the start of the year will probably cost about $105 at the end of the year. Thus,

the infla�on premium charged needs to be about 5% to compensate the lender for the loss of purchasing power due to infla�on, in this case where the

expected rate of infla�on is 5% per annum.10 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#footernote10)

The second main part of the ODR is the risk premium, which is the addi�onal return the lender will require to cover the risk that the borrower might default on the loan and not pay the money back. To es�mate the appropriate risk premium, the lender must ask, "What is the probability that the borrower will not repay the loan?" To answer this ques�on, the lender (like the insurance manager in Chapter 1) must consider what propor�on of people, in roughly the same risky situa�ons, have previously defaulted on their loans. Suppose the answer is 20%. That means that one out of five borrowers did not pay the lender back the loaned funds and the interest that should have been earned on those funds. Because the lender cannot tell in advance which one in every five borrowers will be unable to repay the loan, the lender must set a risk premium on all loans that is high enough to allow the funds received back (from borrowers who do in fact repay their loans) to compensate the lender for the funds lost due to borrowers who cannot repay the loan. In this case, since only four of the five are expected to repay, all will be charged a 25% risk premium to ensure that the four borrowers repaying the loan allow the lender to recoup 4 x 25% = 100% of the loan advanced to the borrower who ul�mately defaults. The formula for the risk premium is thus the ra�o of the probability of default (PD) to its complement, the probability of repayment (PR). That is, PD/PR. In Table 2.3, we show the risk premiums for a range of default

probabili�es, and you can see that the risk premium increases exponen�ally as the probability of default increases.

Table 2.3: Default risk and calcula�on of the applicable risk premium Probability of Default PD Probability of Repayment PR Risk Premium PD/PR

 5% 95%  5.26%

10% 90% 11.11%

20% 80% 25.00%

30% 70% 42.85%

40% 60% 66.67%

50% 50% 100.00%

Now let’s revisit the Project A versus Project B decision that we considered above. Using the CV criterion we adjusted for risk by finding the risk-per-dollar- of-return and we selected Project B despite it being more risky (higher SD). But note that the expected cash flows of both projects were discounted by the same 10% discount factor. Now that we know Project B is more risky, we should discount it by a higher rate. Suppose that 10% was indeed the correct ODR

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for Project A, being the real rate of interest (say 2%) plus an expected infla�on (say 3%), plus a risk premium of 5%. Also suppose that for Project B the appropriate risk premium is about 15%, causing the ODR to be 20%. In Table 2.4 we recalculate the ENPV for Project B.

Table 2.4: Recalcula�ng the ENPV for Project B, with ODR = 20%

(1) Ini�al Cost Year 1 (millions)

(2) Present value of ini�al cost

(3) Year 2 outcomes (millions)

(4) Present value of Year 2 outcomes

(5) Net present value (millions)

(6) Probability of outcomes

(7) ENPV of outcomes (millions)

DF = 0.8333 DF = 0.6944

 10 0.6944 5.2778 0.4 2.1111

−2.000 −1.6667   5 3.4722 1.8056 0.5 0.9028

−0.5 −0.3472 −2.0139 0.1 −0.2014

ENPV = 2.8125

SD = 1.2078

CV = 0.4295

Now compare the ENPVs of the two Projects A and B. Of course the ENPV is s�ll $760,300 for Project A, but it is now about $2.8 million for Project B (down from $3.5 million) due to being more heavily discounted. So, Project B is s�ll the preferred alterna�ve using the risk-adjusted ENPV decision criteria. Also note that while the CV for Project B has increased due to the higher ODR, the CV criterion also s�ll favors Project B over Project A.

A Simplification for More Complex Situations

In prac�ce, we are typically confronted by more complex situa�ons than the above examples. Fortunately, we can simplify these examples by assuming only three outcomes (high, medium, and low) in each year and by assigning what seem to be reasonable "guess�mates" of the different monetary outcomes and of the probabili�es of these different outcomes occurring. If these es�mates are inaccurate, scru�ny by others who have different informa�on will lead us to revise them to more accurately reflect the consensus of opinion about what the values should more likely be.

If we suppose that the possible outcomes are symmetric around the medium outcome each year, and also suppose the probability distribu�ons are symmetric around the medium outcome, then a useful simplifica�on becomes possible. To find the ENPV of the decision alterna�ve we need only add the medium NCF outcomes in each year and subtract the ini�al cost outlay. This is because the high outcomes would be exactly offset by the low outcomes when they are symmetric around the medium outcome. This simplifica�on is hardly necessary for the rela�vely simple two- and three-year �me horizons that we have considered, but think about a decision with a five-year horizon—with a NCF stream stretching over five years with high, medium, and low

outcomes in each year. This would involve 35 = 243 terminal branches on the decision tree! Although one could build a very large spreadsheet or write a computer program to do all the hard work, it is generally not necessary to do so. This simplifica�on will give a sufficiently robust indica�on of the ENPV as long as there is not substan�al asymmetry of the high, medium, and low outcomes. For the most part, asymmetry one way (e.g., toward the high outcome) in one year will be offset by asymmetry the other way (toward the low outcome) in another year. Next, the future outcomes and their probabili�es are es�mates anyway, and these es�mates are increasingly like guesswork in the "out years" (i.e., beyond the present period), so it is false accuracy to place too much credence on the precise value we find for the ENPV.

Thus, it is generally a sufficient approxima�on to consider only the medium NCF outcome for each of the out years when the �me horizon is three to five years or longer. For decision alterna�ves that have longer �me horizons the "sum of the medium outcomes" approach is likely to be a sufficient approxima�on for the ENPV of the decision alterna�ve.

7. Shareholders may buy stock in a variety of firms and thus build up a por�olio of risky stocks in which the downward varia�ons in one stock’s price are likely to be offset by the upward varia�ons in another stock’s price. Notwithstanding this, risk-averse individuals will want each firm in which they hold stock to make decisions that adjust for risk. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return7) ]

8. Other decision rules of the maximin type are maximax, which says you should select the alterna�ve that has the highest of the maximal outcomes, and minimax, which says you should choose the alterna�ve with the smallest of the largest outcomes. These rules similarly focus on only the largest, or the smallest, outcomes and do not consider other possible outcomes or their probabili�es. These rules seem even less applicable in real-world business situa�ons although maximax may be useful for risk-preferring decision makers and minimax might be useful for people trying to reduce their taxable income in the current year, for example. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return8) ]

9. This may not apply to governments in all countries, of course. If a revolu�on or military takeover were to overthrow a government, the incoming government may decide not to honor the borrowings of the prior government. Also, in countries with excessive government debt (such as in the European debt crisis that erupted in 2011) those governments at risk of defaul�ng on their debts need to add a risk premium to induce investors to be willing to take on any further debt issues. But generally, for poli�cally stable na�ons without excessive government debt, it is safe to regard government bonds as a risk-free investment. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return9) ]

10. Strictly, the infla�on premium is calculated as k/(1–k), where k is the expected rate of infla�on. Thus, for example, if expected infla�on is 5%, the infla�on premium is 0.05/0.95 = 5.26%. For expected infla�on of 10%, the infla�on premium is 0.1/0.9 = 11.11%. It can be seen that as the rate of expected infla�on increases, the infla�on premium increases faster. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.2#return9) ]

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2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios

What we have been calling the medium outcome is alterna�vely called the most-likely scenario. You will note that it had the highest probability in each year, so it is indeed more likely to occur than the high (best-case) or the low (worst-case) scenario. Now we can view the medium outcome as being representa�ve of the middle part of the probability distribu�on, and similarly view the high outcome as being representa�ve of the upside-risk side of the probability distribu�on and the low outcome as being representa�ve of the downside-risk side of the probability distribu�on. Thus, the point es�mates of high, medium, and low should be viewed as representa�ve of the three regions of the probability distribu�on.

The Normal Distribution

We can illustrate these scenarios in the context of the special case of the normal distribu�on.11

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.3#footernote11) The first property of a normal distribu�on is that it is symmetric around its mean value. It is o�en called a bell curve because it looks somewhat like an old-fashioned bell, as displayed in Figure 2.4. As you would guess, the mean value of the normal distribu�on is also the median (or the 50th percen�le) value. That is, 50% of the observa�ons lie above, and 50% lie below, the mean value. The standard devia�on of a normal distribu�on is such that almost all (about 99.7%) of the outcomes lie within plus or minus three standard devia�ons from the mean. Thus, the second property of a normal distribu�on is that the bell curve is just tall enough to cause 99.7% of all outcomes to lie within plus or minus three SDs from the mean. Moreover, the shape of the bell curve is such that 95% of all outcomes will lie within plus or minus two SDs from the mean, and 68% will lie within plus or minus one SD from the mean.

Now no�ce that the mean value (i.e., the ENPV) of the probability distribu�on effec�vely represents the middle part of the distribu�on, or 68% of all outcomes. It is most likely (with 68% probability) that the actual outcome will fall within the range of plus or minus one SD from the mean outcome. With repeated trials of the same decision we would expect the actual outcome to some�mes be more than the mean, and some�mes less, such that the average outcome over many trials would be the ENPV.

Note that what we call the best-case scenario is not the absolute best outcome out at the extreme right-hand side of the probability distribu�on. Instead it is representa�ve of the range of outcomes that are more than one SD above the mean. In Figure 2.4, we have depicted the best-case scenario as the outcome that roughly bisects the area under the curve to the right of the outcome that is more than one SD above the mean. Similarly, the worst-case scenario is not the worst possible outcome at the extreme le� of the probability distribu�on, but represents all the outcomes that have values more than one SD below the mean outcome, so we posi�on it at approximately the point that bisects the area under the curve to the le� of the outcome that is one SD below the mean outcome.

Figure 2.4: The proper�es of a normal distribu�on

Since the most-likely scenario (MLS) represents 68% of the outcomes (when the outcomes are normally distributed) the best-case scenario (BCS) must represent half of the remainder (i.e., 16%) and the worst-case scenario (WCS) must represent the other half of the remainder (16%). Be aware that these specific probabili�es for the high, medium, and low outcomes may or may not be appropriate for par�cular business decision problems. If you have informa�on that indicates that the outcomes seem likely to be approximately normally distributed around the mean, then these probabili�es will be appropriate. On the other hand, you might have informa�on that indicates that the probability distribu�on is definitely not normally distributed and so you should use the probabili�es that seem to be more appropriate.

Skewness of the Probability Distribution

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Skewness refers to the degree of asymmetry of the probability distribu�on. A distribu�on that is perfectly symmetric is said to be nonskewed.

© Biwa Studio/Ge�y Images

Skewness refers to the degree of asymmetry of the probability distribu�on. A distribu�on that is perfectly symmetric is said to be nonskewed. But if the bell shape is distorted with more outcomes lying to the le� of the mean outcome, with a longer tail stretching out to the right-hand side, the distribu�on is said to be posi�vely skewed. In this case the median outcome (the 50th percen�le outcome) will lie to the le� of (below) the mean outcome. The modal outcome, which is the single outcome with the highest probability of occurring, will lie to the le� of both the mean and the median outcome, as shown in Figure 2.5a. A nega�vely skewed distribu�on will have the bulge on the right-hand side of the distribu�on with a long tail to the le�. The median outcome will lie to the right of the mean outcome, and the modal outcome will lie to the right of both the mean and the median outcomes, as shown in Figure 2.5b.

Figure 2.5a: Posi�vely skewed probability distribu�on

Figure 2.5b: Nega�vely skewed probability distribu�on

The implica�on of posi�ve skewness for managerial decision making is that while the majority of the possible outcomes will be below the mean, there will be a significant number of upside-risk outcomes that lie more than three standard devia�ons above the mean (ENPV). Repeated trials of such decisions would not generate an average outcome equal to the mean outcome, but would tend to average the median outcome (i.e., below the weighted mean outcome). The probabili�es associated with the worst-case, most-likely scenario, and best-case scenario would be different to the normal distribu�on, of course. Something like 10%, 60%, and 30% might be more appropriate for the worst-case, most-likely-case, and the best-case scenarios, respec�vely.

Conversely, when the distribu�on is nega�vely skewed, the majority of the possible outcomes will be above the mean, but there will be significant number of downside-risk outcomes that lie more than three standard devia�ons below the mean. Repeated trials of decisions with nega�vely skewed distribu�ons would tend to result in an average outcome that is above the weighted mean (ENPV) outcome. The probabili�es associated with the worst-case, most-likely scenario, and best-case scenario would be different to the normal distribu�on, of course. Something like 30%, 60%, and 10% might be more appropriate for the worst-case, most-likely-case, and the best-case scenarios, respec�vely.

Kurtosis of the Probability Distribution

Kurtosis refers to another aspect of the shape of a probability distribu�on, specifically its height. A distribu�on that is taller than a normal distribu�on is said to be leptokur�c and would have more than 68% of the outcomes falling within one SD each side of the mean. Conversely, a distribu�on that is platykur�c (like a plate, i.e., fla�er) would have less than 68% of the outcomes within one SD each side of the mean. Kurtosis of probability distribu�ons is demonstrated in Figure 2.6.

Figure 2.6: Kurtosis of probability distribu�ons

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You can see from the shapes of the two probability distribu�ons in Figure 2.6 that for a leptokur�c distribu�on, the propor�on of outcomes lying within one SD from the mean will be substan�ally above 68%, perhaps 80% in the example shown. This means that the probabili�es of the best-case and worst-case scenarios are rela�vely small, about 10% each in the situa�on depicted. Conversely, for the platykur�c distribu�on, the propor�on of the outcomes lying within one SD of the mean would be substan�ally below 68%, perhaps only 40–50% in the example shown. Thus, the probabili�es of the best- and worst- case scenarios might be rela�vely large, about 30% each in the platykur�c distribu�on shown in Figure 2.6.

What is the point of all this for the managerial decision maker? First, the decision maker needs to consider whether the probability distribu�on of outcomes is likely to be approximately normal or not. Remember that many decision situa�ons are likely to deliver approximately normal distribu�ons of outcomes since the independent ac�ons of many people (e.g., buyers) typically result in a normal distribu�on. Second, if the decision maker has reason to believe that the distribu�on will be skewed to one side or the other, or taller or fla�er than a normal distribu�on due to factors that he or she suspects are characteris�c of the popula�on or sample, the probabili�es need to be adjusted in the direc�on that reflects the decision maker’s best es�mate of the actual shape of the probability distribu�on. In the absence of any informa�on to indicate that the normal distribu�on is not appropriate, it is usually a good first approxima�on to assume normality of the distribu�on. It is useful to remember that unless you have data on the distribu�on of prior outcomes of similar decisions, assigning probabili�es to possible outcomes is an art, not a science—the decision maker needs to think about the situa�on and go with his or her best guesses. As a decision maker, it is useful to check your own best guesses against the opinions of others who have knowledge of the decision scenario. Their scru�ny may reveal informa�on that was not known to you and allow a more accurate probability distribu�on to underlie your decision. And finally, if this type of decision scenario is repeated again and again, data will build up and the probability distribu�on can be corrected subsequently.

11. It is called the normal distribu�on because this par�cular distribu�on of outcomes occurs in thousands of situa�ons, across many different fields of the natural and the social sciences. It does not imply that distribu�ons that are not normally distributed around their mean are abnormal in any pejora�ve sense. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec2.3#return11) ]

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When data is not available on the normal distribu�on of prior outcomes of similar decisions, the decision maker must rely on best guesses.

© Paul Taylor/Ge�y Images

Summary

In this chapter, we introduced adjustment for risk into our decision-making toolkit. Since business managers typically must make decisions under condi�ons of risk and uncertainty, we needed to incorporate a measure of the dispersion of the possible outcomes as well as the expected net present value (ENPV) of the probability distribu�on into the decision rule. We noted that risk is best measured by the standard devia�on of the probability distribu�on of the possible outcomes that might follow a decision. But we also needed to consider how decision makers feel about risk, since they might be risk-averse, risk preferring, or unconcerned by risk (risk neutral). We concluded that decision makers need to be risk-takers but are most likely risk-averse, preferring less risk when all other things (such as ENPV) are equal. For repe��ve decisions of the same type and in the same context, decision makers can act as if they are risk neutral. Similarly, if shareholders of firms hold stock in a large number of different firms, they would have a por�olio of stocks that would not be suscep�ble to wide swings in total value (except in a global financial crisis) and could act as if they were risk neutral with respect to any par�cular firm’s decisions. Notwithstanding that, these shareholders will want the managers of each of the firms (in which they hold stock) to adjust their decisions for risk.

Because risk causes the risk-averse owners of businesses to experience disu�lity, a trade- off is made against the monetary value of the decision (i.e., the ENPV) to compensate for bearing risk. For individuals making decisions on their own behalf (including sole owners of businesses) the certainty equivalent (CE) criterion is applicable. We introduced risk– return indifference curves to demonstrate that the individual will take addi�onal risk for addi�onal return, or conversely would give up part of the poten�al return in order to avoid risk. The certainty equivalent (CE) is the amount of money available with certainty that the decision maker will feel is equivalent to the weighted average of the risky gamble (i.e., the ENPV). Without plo�ng out all risk–return indifference curves, the individual decision maker simply needs a moment of introspec�on to conclude which one of the possible alterna�ve decisions makes him or her feel most comfortable, taking into account both the expected u�lity from money and the expected disu�lity from risk.

This process of introspec�on and personal choice is not usually sufficient for managers of other peoples’ money. So managers of firms in which there are other shareholders must u�lize decision rules that can be jus�fied on some objec�ve basis. The coefficient of varia�on (CV) decision criterion is the ra�o of the standard devia�on to the weighted

mean (i.e., ENPV) of the probability distribu�on and the CV ra�o effec�vely measures the risk-per-dollar-of-return. This is a transparent decision rule that shareholders can easily understand, such that they can invest in a different firm if the firm in which they have previously invested is taking (in their view) too much risk-per-dollar of poten�al return. Another transparent adjustment to the ENPV criterion to explicitly take risk into account is to use an appropriate risk premium as part of the opportunity discount rate (ODR). Choosing a risk premium that is based on the expected prior probability of failure of the investment or other decision causes the ENPV to be smaller, the higher the risk premium is. Riskier gambles are discounted more heavily, reducing their ENPV to lower levels. So, when comparing decision alterna�ves with different risk profiles, the decision maker should choose the one with the highest ENPV remaining a�er the net cash flow streams have been discounted by an ODR containing the appropriate risk premium.

Many decision problems can be reduced to rela�vely simple terms, such as three poten�al outcomes (high, medium, and low) in each of one or two years. Decision tree analysis can be u�lized for these rela�vely simple problems, where the number of terminal branches is equal to the number of outcomes (in each year) to the power of the number of years. But decisions that have outcomes con�nuing into the third and subsequent years have too many terminal branches to be realis�cally solved using a decision tree. Instead, we can simplify these more complex decisions in one way or another and s�ll arrive at a risk-adjusted-ENPV valua�on of the decision alterna�ve that should be sufficiently reliable (given that we are guess�ma�ng about future outcomes in any case). One way is to treat the decision tree as being approximately symmetrical, with the high outcomes being balanced out by the low outcomes. If this is a sufficiently realis�c assump�on, we can simply add up the net present value (NPV) of the medium case outcomes in each year to approximate the ENPV of the en�re probability distribu�on. We then argued that the medium outcome is actually representa�ve of the most-likely scenario—that is, it is the midpoint of a rela�vely narrow range of outcomes (plus or minus one SD from the mean) and that the actual outcome will fall within that range about two thirds (68%) of the �me; falling above that range (best-case scenario) about one sixth of the �me; and falling below that range about one sixth of the �me. Thus, we can safely make our decisions based on the ENPV as being the best predictor of the actual outcome, as long as the probability distribu�on is approximately a normal distribu�on.

If the distribu�on is skewed to one side or the other, the ENPV is no longer the most probable (or modal) outcome. If the distribu�on is posi�vely skewed (with a long tail to the right) the most likely scenario will have probability less than two thirds and the best-case scenario with have probability more than one sixth. Conversely, if the skew is nega�ve with a long tail to the le�, the most likely scenario will have probability less than two thirds and the worst-case scenario with have probability more than one sixth. Further, if the probability distribu�on exhibits leptokurtosis, being taller than a normal distribu�on, the most-likely scenario will have probability significantly higher than two thirds and oppositely, if it exhibits platykurtosis, the most-likely scenario will have probability significantly lower than two thirds. It is up to the decision maker to ensure that the use of the normal distribu�on is appropriate before proceeding on that simple but rela�vely reliable assump�on.

So, in the first two chapters of this book we have paved the way for decision making in the real world where decisions are made in the context of risk and uncertainty and where there are cost and revenue implica�ons that stretch beyond the present period. We have established that we need to use both expected value analysis to account for risk and uncertainty, and net present value analysis to account for the future value of revenues and costs. Combining these we have expected net present value (ENPV) as our measure of the firm’s objec�ve func�on. Then we recognized that people tend to be risk-averse

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and incur disu�lity from risk and uncertainty and thus, will wish to make risk-adjusted decisions that reflect the monetary trade-off they are prepared to make to avoid risk, or, conversely, the monetary gains they expect to receive if they have to bear risk. Having spent this �me learning the ground rules, we are now ready to proceed to learn more about the costs and revenue sides of the decisions that managers must make. In the following chapters, we will focus on the demand side of markets to gain a strong understanding of the revenue implica�ons of decisions.

Ques�ons for Review and Discussion

Click on each ques�on to reveal the answer.

1. Explain why the standard devia�on of a probability distribu�on is an appropriate measure of the risk involved in a real-world decision problem. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

A measure of risk needs to measure the extent by which the actual outcome is likely to diverge from the expected value (EV) of a probability distribu�on. The standard devia�on (SD) is a measure of the average absolute devia�on (i.e., both above and below the EV of a probability distribu�on). The SD measures how far to one side or the other of the EV the actual outcome would fall, on average over many trials, and is thus a suitable measure of the risk because the SD is higher when the risk of an outcome not being the EV is higher. Conversely, the SD is lower when the risk of the actual outcome not being the EV is lower.

2. Define risk preference, risk aversion, risk seeking, and risk taking. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Risk preference means that other things being equal (such as rewards) the person will prefer the op�on with the highest risk. Oppositely, risk aversion means that other things being equal, the person will prefer the op�on with the lowest risk. Risk seeking means the person is mo�vated to seek out risky situa�ons either because the person is risk preferring, or if risk averse, because the person knows that higher-risk situa�ons generally offer higher rewards. Risk taking means that the person takes risks, either voluntarily or involuntarily, as we all do.

3. Explain why a risk averter facing two choices might prefer to take the higher risk alterna�ve. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

When facing two choices, one with higher risk than the other, the risk averter will choose the higher-risk alterna�ve if the u�lity from the reward associated with the higher-risk alterna�ve is greater than the disu�lity associated with the risk of the higher-risk alterna�ve.

4. How might a risk-averse decision maker adjust the expected net present value of a decision to take into account its risk? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

Several methods were suggested. (i) The person might introspec�vely consider his or her risk-return trade-offs in risk-reward space and choose the alterna�ve that lies on the highest indifference curve. (ii) The certainty equivalent (CE) is the sum of money available with certainty that the person will consider equal to the EV of the risky decision—the person would choose the alterna�ve with the highest CE. (iii) The CE factor (CEF) is the ra�o of the CE to the ENPV and effec�vely measures how many cents in the dollar, available with certainty, that the person would accept instead of taking the gamble. (iv) The maximin rule avoids the worst outcome by choosing the alterna�ve with the largest of the smallest outcomes. (v) The coefficient of varia�on (CV) rule is to choose the alterna�ve with the highest ra�o of SD/ENPV and effec�vely measures risk per dollar of return. (vi) Finally, if the ENPV is calculated using opportunity discount factors reflec�ve of the appropriate risk premium, the risk-averse decision maker will choose the alterna�ve with the highest ENPV.

5. Some risk-adjus�ng decision rules work well for the individual but are not sufficient for decisions made on behalf of others, such as shareholders of the firm. Please explain. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The subjec�ve criteria that require the person to reflect on his or her personal risk aversion and profit preferences (such as risk-reward curves and certainty equivalents) cannot easily be explained or jus�fied to other shareholders or stakeholders of the firm. Criteria that are more objec�ve, such as the CV rule and the ENPV using risk premiums, are more readily explained and defended to shareholders and stakeholders.

6. What is the certainty equivalent of a decision alterna�ve? Use this rule to explain your choice between a 70% chance of winning $10 and a 50% chance of winning $20. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The certainty equivalent (CE) of a decision alterna�ve is the sum of money available with certainty that the person will consider to be equal to the EV of the risky decision. The EV of $10 with probability 70% is $7.00 and similarly the 50% chance of $20 is $10. The la�er gamble has higher EV, but is more risky (i.e., 50% chance of not happening vs. 30%). If you plot these in risk-return space you would see that your indifference curves through these points would need to be very flat for the $20 alterna�ve to be preferred (i.e., to lie on a higher indifference curve than the $10 gamble). If so, this would mean you would have a very low marginal rate of subs�tu�on (MRS)—i.e., the amount of risk you would accept for an addi�onal unit of return is very low, meaning you are highly risk averse. Conversely, if your MRS is higher, the indifference curves will be steeper and the CE of the $20 gamble will be lower than the CE of the $10 gamble.

7. What are the main two components of the opportunity discount rate, and how do you find the value of these components? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The opportunity discount rate (ODR) is composed of the risk-free rate and the risk premium. The risk-free rate is the sum of the "real" rate of interest and the expected rate of infla�on. The real rate is the "price" of money (expressed as a percentage rate of interest) determined in the market for loanable funds by the intersec�on of the demand for loanable funds (i.e., loans demanded by individuals and organiza�ons) and the supply of loanable funds (i.e., deposits put in banks by individuals and organiza�ons). The expected rate of infla�on is the market consensus regarding the expected rate of decline (expressed as a percentage) of the purchasing power of the dollar over a specified period. The risk-free rate is found by checking the rate of return on government bonds, since these are generally considered to be risk free. The risk premium is calculated as the ra�o of the probability of default over the probability of repayment, and this must be es�mated based on the experience of similar loans or investments in the recent past.

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8. Why would a symmetric decision tree analysis of a complex decision problem give the same result as a simple summa�on of the medium net cash flows associated with that decision problem? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

A symmetric decision tree means that the upside variance is the same in absolute terms as the downside variance, such that posi�ve devia�ons from the expected value are exactly offset by nega�ve devia�ons from the expected value. If the decision tree is symmetric, the mean, median and model values will coincide, and thus the expected value will be the median outcome.

9. Define the most-likely, best-case, and worst-case scenarios in terms of a probability distribu�on of the outcomes of a decision alterna�ve. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

The most-likely outcome is the modal outcome, the one with the highest probability of occurring. In decision analysis, we use the expected value (equal to the modal outcome if the probability distribu�on is symmetric) to be representa�ve of the range of outcomes that extends from one standard devia�on (SD) either side of the expected value (EV) outcome. The best-case scenario is a number chosen to represent the outcomes that are more than one SD higher than the EV. The worst-case scenario is a number chosen to represent the outcomes that are more than one SD below the EV.

10. Explain how the probabili�es associated with a probability distribu�on would need to be adjusted if the probability distribu�on was (i) platykur�c, and (ii) nega�vely skewed. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/boo

If the distribu�on is significantly platykur�c (i.e., fla�er than a normal distribu�on), the percentage of observa�ons that lie within plus or minus one SD from the EV will exceed 68% and the probability of the best and worst case scenarios will be less than 16% each. Conversely, if the distribu�on is significantly leptokur�c (i.e., taller than a normal distribu�on), the percentage of observa�ons that lie within plus or minus one SD from the EV will be less than 68% and the probability of the best and worst case scenarios will exceed 16% each.

Decision Problems

1. While comple�ng your studies you want to try out as an entrepreneur and are considering two different entrepreneurial opportuni�es that you have iden�fied. On the one hand, you could invest all your money in a project to print T-shirts and sell them to visitors to the county fair in your home town. You expect that no one else would be doing that, yet you are fairly sure there will be a market for these T-shirts for people a�ending the fair and wan�ng a commemora�ve keepsake or a gi� for others. Your es�mate of ENPV for this project is $3,000 with standard devia�on of $1,000. On the other hand, you could invest all your money in a project to sell umbrellas outside the football stadium at your old high school for the annual East versus West game, which would be in high demand if it is raining, but would sell very poorly if it is not raining. You es�mate that the ENPV of this is alterna�ve is $4,000 with standard devia�on of $2,000.

a. Apply the coefficient-of-varia�on decision criterion to these alterna�ves to find which is preferred using that method of risk adjustment. b. Apply the maximin criterion, supposing that the worst outcome for the T-shirt alterna�ve is that you would make only $500 profit while for the

umbrella op�on the worst outcome (no rain) would cause you to lose $1,000. c. Indulge in some introspec�on and es�mate your personal certainty equivalent for these two projects.

2. The Sounds True Music company is considering the introduc�on of a new memory storage device to compete in the market for mini devices that allow recorded music to be replayed via Bluetooth through smart TVs and stereo systems. Sounds True Music has been taking losses due to heavy compe��on from rivals and is concerned about making the best choice among two alterna�ves being considered. Op�on A is to make a minor faceli� to an exis�ng product, while op�on B is to introduce a totally new product. The managers have determined that the net cash flow outcomes for the current year will depend on the state of the economy, as shown in the following table. They es�mate that the probabili�es of these macroeconomic outcomes are 30% for a downturn, 50% for constant, and 20% for an upturn.

State of the economy Op�on A – minor faceli� (NCF) Op�on B – new product (NCF)

Downturn $10 $−20

Constant $30 $20

Upturn $80 $150

a. Calculate the expected value, standard devia�on, and coefficient of varia�on for each decision alterna�ve (use a spreadsheet and the embedded formula for standard devia�on).

b. Apply the expected value, coefficient of varia�on, and maximin decision criteria to this decision problem. c. Which alterna�ve is likely to have the greater certainty equivalent, and why?

3. The Express Delivery Company operates a courier and parcel delivery service between the major ci�es in southern California. Business has been booming and it needs to add another very large truck to the fleet. Management is considering whether to lease or to buy the addi�onal truck. Careful analysis of costs and the poten�al demand situa�on has led to the following es�mates of net cash flow for each alterna�ve. The applicable opportunity discount rate is 15%.

NCF Year 1 (millions) Probability Year 1 NCF Year 2 (millions) Probability Year 2

LEASE OPTION

−5 0.25  5 0.30

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 5 0.40 10 0.50

15 0.35 15 0.20

BUY OPTION

−10 0.20 10 0.30

 0 0.50 15 0.50

10 0.30 20 0.20

a. Using decision tree analysis, find the expected net present value (ENPV) of each alterna�ve. b. Calculate a measure of risk for each alterna�ve. (See Table 2.1 for the basic method, which you will need to modify to accommodate the two-year

scenario here.) c. Apply several decision criteria and make your recommenda�on to management. d. State any qualifica�ons or reserva�ons you would want to add to your recommenda�on.

4. Your firm is considering the introduc�on of a new product, and you are required to set the price. You are considering three price strategies: high ($6), medium ($4), and low ($2.50). Your market research team has provided the following es�mates of sales at each price level over the next two years. The ini�al investment will be $22,000 and your costs per unit of output will be $1, regardless of volume. Your finance manager says that you could otherwise invest these funds at comparable risk in a forthcoming bond issue at 12.5% per annum.

HIGH PRICE MEDIUM PRICE LOW PRICE

Sales volume Probability Sales volume Probability Sales volume Probability

First year First year First year

3,500 0.1 5,000 0.2 10,000 0.4

2,500 0.3 4,000 0.5  7,500 0.3

1,500 0.6 3,000 0.3  5,000 0.3

Second year Second year Second year

5,000 0.2 8,000 0.3 12,000 0.3

4,000 0.3 6,500 0.4  9,000 0.5

3,000 0.5 5,000 0.3  5,000 0.2

a. Using decision tree analysis, find which alterna�ve promises the highest ENPV over the two-year period. b. Should the investment funds be used to buy the bonds instead? Why? c. Rank the alterna�ves in order of their risk, and explain the basis for your ranking. d. Rank the alterna�ves in order of their risk-adjusted expected present value.

5. The manager of the Highfields Real Estate Development Company is faced by a complex decision problem that has outcomes stretching over five years with mul�ple outcomes possible each year. There are two alterna�ve plans under considera�on for a large parcel of land that the company owns. Plan A is to seek city approval to subdivide the land into small housing lots, install all the necessary u�li�es and other infrastructure, and offer the lots for sales to individual homeowners wan�ng to build their own homes. Plan B is to set up the required infrastructure and then sell the land to a major home-building company that would develop the land into a new execu�ve suburb with higher quality homes of similar appearance on larger lots. In both cases the revenues would come to Highfields only as the lots are sold (plan A) or as completed houses are sold by the major home-building company (plan B). Highfields would need to borrow at a risk premium that depends on which project is undertaken. Suppose the current risk-free five-year bond rate is 3%, and that plan A carries a probability of a low outcome causing bankruptcy of 25%, whereas plan B has a probability of a low outcome (and bankruptcy) of only 20%. Highfields considers the probability distribu�ons to be symmetric in both cases and has es�mated the most-likely outcomes for each plan as shown in the following table.

Plan A – net cash flows in the most-likely scenario ($m) Plan B – net cash flows in the most-likely scenario ($m)

Year 1 −20 −10

Year 2 10 5

Year 3 40 30

Year 4 60 80

Year 5 30 100

a. Calculate the ENPV of each plan to find the preferred alterna�ve by that criterion.

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b. What addi�onal informa�on would be required to allow applica�on of the maximin criterion? c. What addi�onal informa�on would be required to allow applica�on of the coefficient of varia�on criterion? d. Given that there is no more informa�on available, make your recommenda�on and support it with reasoning, and list any other informa�on that you

suggest Highfields should seek before taking ac�on.

Key Terms

Click on each key term to see the defini�on.

best-case scenario (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A scenario that is representa�ve of the range of outcomes that are more than one standard devia�on above the mean of all the possible scenarios.

cardinal (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A system of measurement that allows the differences between things, such as the height of two fences, to be compared using a linear scales (such as inches), versus ordinal measures that simply rate one situa�on as being greater (higher) or less (lower) than another without specifying how much higher or lower one fence is.

certainty equivalent (CE) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The amount of money available with certainty (that is, without risk) that the decision maker feels is equivalent to the risky gamble (the ENPV with an associated standard devia�on).

Certainty Equivalent Factor (CEF) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The ra�o of the perceived monetary value of the risk-free alterna�ve (the CE) to the risky alterna�ve (the ENPV).

coefficient of varia�on (CV) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of the risk-per-dollar-of-return, derived by dividing the standard devia�on by the expected net present value.

direc�on of preference (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The direc�on that a person would want to go to increase total u�lity or to achieve higher return and lower risk.

downside risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The risk that the outcome that occurs will be less than the expected net present value.

highly intolerant of risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Those who are in this category try to avoid voluntary risk taking as much as possible. Risk-intolerant people need much greater monetary compensa�on to induce them to accept addi�onal risk.

highly tolerant of risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Are those who are slightly risk-averse. Bearing risk causes them rela�vely li�le psychic dissa�sfac�on, so they will take on addi�onal risk for rela�vely small increases in expected value.

indifference curves (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Lines that represent combina�ons of goods and services that give a consumer an iden�cal quan�ty of u�lity or represent a set of choices among which the buyer is indifferent.

infla�on premium (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

That part of the opportunity discount rate that is intended to compensate the lender for the expected fall in the purchasing power of the monetary unit (the dollar) during the period of the loan. It is calculated by the equa�on: k/(1 − k) where k is the expected rate of infla�on.

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involuntary risk taking (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Everyday risks that are imposed on us by nature, or other external circumstances, and which cannot easily be avoided.

kurtosis (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The height of a probability distribu�on; the degree to which it is peaked or flat.

leptokur�c (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A feature of a probability or frequency distribu�on that is evident as a rela�vely tall and narrow peak surrounding the mean because the data values have rela�vely li�le variance and are mostly similar to the average value.

marginal rate of subs�tu�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Reflected by the slope of the indifference curve and represents the rate of subs�tu�on between two compe�ng items that leaves a person at the same level of u�lity.

maximin (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A decision rule, where a consumer chooses the alterna�ve that has the largest (maximal), of the smallest (minimum) outcomes.

median outcome (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The 50th percen�le outcome, calculated by ranking the outcomes in order of magnitude and selec�ng the outcome that falls at 50% of the total number of outcomes.

modal outcome (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The single outcome with the highest probability of occurring, or the outcome that repeats the most amount of �mes in a given set of data.

most-likely scenario (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A scenario representa�ve of the middle range of the probability distribu�on, usually defined as the range of outcomes that are plus or minus one standard devia�on from the mean outcome.

nega�vely skewed (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A distribu�on pa�ern that has a bulge on the right-hand side of the distribu�on with a long tail stretching to the le�.

normal distribu�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A symmetrical distribu�on with a peak in the center of the range of possible outcomes, and with shape such that 68% of the observa�ons lie within plus or minus (+/–) distribu�ons that are not one standard devia�on from the mean; 95% lie within +/– two standard devia�ons; and 99.7% lie within +/– three standard devia�ons.

opportunity discount factor (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The rate of interest that could be earned on an alterna�ve opportunity of equal risk.

ordinal (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An evalua�on system that relies on rela�ve loca�on (such as higher or lower, larger or smaller), as opposed to cardinal measures that would place a value on the differences (such as how much higher or how much larger).

platykur�c (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A distribu�on that has rela�vely wide dispersion around the mean value and thus has a wider and fla�er distribu�on surrounding the mean.

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posi�vely skewed (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

When results are posi�vely skewed the mean is greater than the median, which is greater than the mode, and therefore the right tail of the distribu�on is stretched. In this type of distribu�on, the expected outcome is likely to be below the mean, but there is a higher chance of a few extremely posi�ve results.

real rate of interest (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of how much it really costs to borrow money when the infla�on rate is subtracted from the actual nominal interest rate of a given loan.

risk aversion (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A condi�on in which an investor or other decision maker exhibits a preference for less risk, other things (such as expected returns) being equal.

risk neutral (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A state where investors are unconcerned by the amount of risk associated with a given investment and don’t allow risk to enter their investment decisions.

risk preference (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A state where the individual prefers more risk to less risk, with other things (such as reward or profit) being equal. A risk preferer would therefore choose the riskier of two equally profitable investments.

risk premium (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

This is the amount by which the opportunity discount rate (ODR) is increased to reflect the higher risk of a risky decision, compared to one that is devoid of risk.

risk seeker (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An investor that deliberately seeks out investments that have higher risk, generally because they also have higher expected returns.

risk taker (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

While risk takers do not go out of their way to deliberately find risky ventures like risk seekers do, they are ready to accept extra risk if the an�cipated returns are sufficiently higher than for a less risky investment.

risk-free rate (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Government bonds issued by a federal Treasury, according to standard economic theory, are examples of risk-free investments because they offer a very small rate of return in exchange of a lack of default risk.

semi-variance (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A calcula�on of the variance of a distribu�on including only those data points that fall below the mean.

skewness (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

Indicated by asymmetry in the distribu�on curve around the mean and can be either nega�ve (skewed to the le�) or posi�ve (skewed to the right).

slope (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of the incline or decline of a surface, or a line such as the risk–return (RR) curve, which is equal to the "rise over the run," or the length versus height of the line over a specific distance.

standard devia�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The extent of dispersion of the observa�ons in a data set from the data’s mean value. Standard devia�on increases when the data observa�ons are spread more widely around the mean value.

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upside risk (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

The measure of the probability that value of the actual outcome will be higher than the expected present value of the prior distribu�on of possible outcomes.

variance (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure that evaluates the spread of data observa�ons in comparison to the loca�on of the mean for the given data set.

voluntary risk taking (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

An ac�vity engaged in by investors and individuals with the expecta�on that the u�lity of the reward will outweigh the disu�lity of the risk.

weighted mean (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A measure of central tendency that weights each observa�on by its probability of occurring, such as the ENPV of a probability distribu�on of poten�al outcomes.

worst-case scenario (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AU

A situa�on that represents a range of outcomes that lie more than one standard devia�on below the mean or ENPV outcome.