dis 2 man econ
2
Decision Making Under Risk and Uncertainty
Learning Objectives
After reading this chapter, you should be able to:
• Explain that risk is measured by the variation of potential returns around the mean or expected value of the potential 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 distribution of possible outcomes will change the prob- abilities associated with the most-likely scenario and the worst-case scenario.
©Philip and Karen Smith/Getty Images
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CHAPTER 2Introduction
Introduction
In Chapter 1, we saw that managers wanting to make decisions that best serve their objective functions will need to first define the metric for their objective function. We argued that the firm’s objective will be to maximize profits, and that managers must make decisions under conditions 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 (disutility) of nonmonetary costs is greater than the psychic gain (utility) of the nonmonetary benefits, we say there is net disutility associated with the decision and the decision maker will require additional profit to compensate for the net disutility associated with the decision. Conversely, if the nonmonetary benefits exceed the nonmon- etary costs, there is net utility 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 disutility for most business decision makers and so they will want to be com- pensated for bearing risk. In this chapter we integrate risk analysis into the decision-mak- ing 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 continually 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 time the actual outcome will be higher than the ENPV and the other half of the time the actual outcome will be below the ENPV. Although facing risk in each specific decision, the repetitiveness 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 situations the manager faces a variety of different decisions from day to day and most types of decision are not repeated often 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 particular decision.
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CHAPTER 2Introduction
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 potential out- comes relative to the ENPV we need to understand the statistical concept known as the standard deviation, which is a measure of the deviations 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 potential outcomes—indeed the ENPV is the weighted mean (where the weights are the probabilities of occurring) of the possible outcomes. The mean of any series of numbers has an associated standard deviation, which indicates the extent to which the mean value is representative of all the data points that enter the cal- culation of that mean. The standard deviation is higher if the possible outcomes are more widely dispersed around the mean value, or is lower if the actual outcomes lie relatively close to the mean value. To calculate the standard deviation, the deviations of each data point from the mean are squared and then summed to find the variance of the distribu- tion, and the standard deviation is then simply the square root of the variance. In effect the standard deviation indicates the average absolute deviation of the outcomes from the mean outcome. Thus, the standard deviation provides a suitable measure of the risk that the ENPV will not be attained.
Any good calculator can instantly deliver the standard deviation of a series of simple numbers. Similarly, it is easy in an Excel spreadsheet to type (for example) 5stdev(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 deviation. Note that it is more complex to calculate the standard deviation of a probability distribution. We need to (1) find the ENPV of that distribution; (2) subtract each outcome from the ENPV to find the deviations from the mean; (3) weight each deviation by its probability of occurring; (4) sum these weighted deviations to find the variance; and (5) take the square root of the variance to find the standard deviation. This is demonstrated for a simple case in Table 2.1, in which we suppose that three outcomes are possible (column 1) with probabilities 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 5 EV.
Table 2.1: Calculation of the standard deviation of a probability distribution
Possible outcome ($)
Expected value ($)
Deviation of possible outcome from the mean (EV) outcome
Squared deviation from the mean outcome
Probability of each possible outcome
Weighted deviation from EV
–10 10 30
10 10 10
-20 0 20
400 0 400
0.25 0.5 0.25
100 0 100
Variance 5 200
Std. Deviation 5 14.14
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CHAPTER 2Introduction
Half of the variance around the ENPV is quite desirable, and of course I am referring to the out- comes that are greater than the ENPV. This part of the risk is known as the upside risk and rep- resents better outcomes than can generally be expected in multiple trials of this decision. Some risk analysts have suggested that no one is wor- ried about these positive deviations 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 includ- ing only those outcomes with negative deviation from the ENPV to measure only the downside risk. Although intuitively appealing, the semi- variance approach is not commonly used because it ignores the upside risk; after all, if “sometimes you win and sometimes 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 deviation 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 attitude toward risk.
Attitudes Toward Risk People have different attitudes 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 attitudes toward risk: risk preference, risk aversion, or risk neutrality. Risk preference means that the individ- ual 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 rational behavior if the individual’s objective function is to maximize risk rather than to maximize profit, or if the person is so rich that he or she places very little 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 prefer- ers might get lucky and have a series of wins (despite the odds) but sooner or later will be bankrupted if they continue 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 alternative if they expect to be ade- quately compensated for the additional risk undertaken. They weigh up the monetary trade-off of the extra income against the psychic dissatisfaction of additional risk bear- ing and make their decision between less-risky-but-less-rewarding investments and
©Photodisc/Thinkstock
To find whether the wins offset the losses, variance or standard deviation can be used to measure the risk associated with uncertain outcomes.
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CHAPTER 2Introduction
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 repetition of the same or similar decision many times, 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 times. In this situation 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 often 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 positively correlated: The higher the risk the higher the return. Whether the return is simply monetary, or is both monetary and nonmonetary (i.e., includes psy- chic satisfaction), 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 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 continually take. What is important is not that we take risks but what our attitude is toward taking risks. As you now know, our attitude either will be risk preference, risk aversion, or risk neutrality, depending on our prior knowledge of the situation and our cognitive pro- cessing of the psychic costs and benefits associated with taking specific risks. We should also distinguish 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
©iStockphoto/Thinkstock
Entrepreneurs, skydivers, and motorcyclists all voluntarily take risks with the expectation that the utility of the reward will outweigh the disutility of the risk.
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 continue to take the same gamble, they know that the ENPV of repeated gambles (e.g., rolling the dice) is ultimately negative, but they hope to get lucky and experience one of the upside-risk outcomes. When betting on horses or sports teams, gamblers may believe they possess better information than the person accepting the bet. Decision making is akin to gambling, of course, since the decision maker must choose one deci- sion 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 often use the word “gamble” (as a noun) to refer to the decision-making problem facing a business decision maker.
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CHAPTER 2Introduction
involuntarily. Risk seekers, however, take risk voluntarily in the expectation that the util- ity of the reward will outweigh the disutility 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
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 relatively little psychic dissatisfaction. We say they are highly tolerant of risk. People like this will require only a relatively small amount of monetary compensation for bearing additional risk. For others, bearing risk causes much more psychic dissatisfaction. 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 compensation to induce them to accept additional 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 particular decision.
Risk Perception
Similarly, each person might perceive risk differently. Individuals perceive the risk in a decision situation more or less accurately depending on their prior knowledge and their cognitive biases. Greater prior knowledge of the situation, or greater information search activity,2 may provide the decision maker with useful information that others do not have, such that she might (correctly) say the situation is not very risky while others might say it is highly risky because of their ignorance of the situation. The old saying that “fools rush in where angels fear to tread” reflects the perception of little or no risk by those who have less knowledge about the situation compared to those who have more knowledge. Next, a cognitive bias such as overconfidence may cause one person to overlook risks that a less confident person might perceive because the latter looks more carefully into the situa- tion or spends more time and money on information search activity to reveal the hidden dangers. Another cognitive bias is the tendency of decision makers to use heuristics, or simplistic decision rules. While economizing on time and search costs, heuristics could actually increase the decision maker’s exposure to risk, since they consider only some of the information that is potentially available. For example, entrepreneurs have been shown to be more overconfident and to use heuristics more than employed managers of firms (Busenitz & Barney, 1997).3 When others see entrepreneurs taking extraordinary
2. As we saw in Chapter 1, information search activity is the purposeful search for information 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 information search activity relative to consumer demand and to cost levels, respectively.
3. Entrepreneurs tend to anchor their estimates on past outcomes and to not revise their estimates on the basis of new information. Second, they tend to base their decision making on the most recently acquired or most easily recalled information, this being known as the availability heu- ristic, but of course such data may not be representative of the range of outcomes that should be expected. Third, the representative heuristic is where people base decisions on a relatively small number of observations (rather than a representative sample), which introduces risk because the limited sample might not be representative of the range of probable outcomes.
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CHAPTER 2Section 2.1 Adjusting for Risk Using the Certainty Equivalent
risks they often 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 better information, have stronger desire for income, or they did not perceive some of the risks in the first place.
2.1 Adjusting for Risk Using the Certainty Equivalent
The certainty equivalent of a decision is the amount of money, available with cer-tainty, that a person would consider equivalent to the expected value of a risky decision. In this section, 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 disutility 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 compensation to induce them to bear an additional quantum of risk. Using simple graphical analysis we can depict the risk–return trade-off curves of a particular 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.
100
80
60
40
20
0 30 50
Direction of preference
Risk (SD)
C
A
RR2
RR3
RR1
Return (ENPV)
B
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CHAPTER 2Section 2.1 Adjusting for Risk Using the Certainty Equivalent
Suppose that Mr. X must decide where (at which location) he will build a new restaurant. The points A, B, and C shown in Figure 2.1 relate to three different risk–return combina- tions that represent different restaurant locations. We depict these three decision alterna- tives with risk measured by standard deviation (SD) and return measured by ENPV. Their risk and return outcomes differ because of differences in population density, passing traf- fic, proximity to public transport, and so on. As you can see, decision A has ENPV 5 100 and SD 5 50; decision B has ENPV 5 100 and SD 5 30; and decision C has ENPV 5 60 and SD 5 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 options 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 location? 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., reflecting Mr. X’s feelings about risk and return). Each RR curve depicts those combinations of risk and return that give the same level of utility. These curves are more commonly known as indifference curves, which are lines drawn to pass through combinations of variables among which the deci- sion maker is indifferent, that is, receives the same amount of utility.4 Thus, Mr. X will be indifferent between A and C, or indeed any other combination of risk and return that lies on RR2. Now, since point B is preferred to both point A and C, it follows that every com- bination of risk and return on RR3 is preferred to any combination on RR2. Similarly, any risk–return combination on RR1 is considered inferior to any combination 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 direction 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 combinations that have both more return and less risk. Note that we do not need to know the actual value to the utility 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 substitution (MRS) between risk and return, which is equal to the amount of risk the decision maker will accept for an additional 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
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 competing goods and services. Indifference in an economics context means not preferring any one combination over the others, in contrast to vernacular usage where it might mean you do not care about any of the combinations.
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CHAPTER 2Section 2.1 Adjusting for Risk Using the Certainty Equivalent
between points C and A is 40/20 5 2 and this value is rather typical of this individual’s MRS at other risk–return combinations in the vicinity of decisions A, B, and C.5
Figure 2.2: Differing degrees of risk aversion for two different decision makers
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 substitu- tion. These people are considering restaurant locations A and C, because location 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 notice that his risk–return trade-off (i.e., his MRS) is relatively 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 com- pensate for the 20 additional units of risk. Thus, his MRS for return and risk is 5/20 5 0.25. Because decision A offers $40 more return for those 20 extra units of risk, it is utility maxi- mizing 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 additional risk associated with decision A (20 units) is not compensated for by the additional 40 units of return offered by A, and thus Ms. Z prefers decision C.
100
60
0
30 50 Risk
C
A
RR Y
1
RR Y
2
Return
100
60
0
30 50 Risk
C
A
RR Z
2 RR Z
1
Ms. Z
Return
Mr. Y
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 flatter) 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 quan- tum 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 util- ity of income and increasing marginal disutility of risk, which seems characteristic of most human beings. We shall revisit these concepts in Chapter 3, in the context of consumer behavior.
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CHAPTER 2Section 2.1 Adjusting for Risk Using the Certainty Equivalent
What we have demonstrated is that risk-averse decision makers will make different deci- sions according to their degree of risk aversion. Note that both Mr. Y and Ms. Z would have preferred restaurant location B if it were still available, since its risk–return outcomes would fall on a higher indifference curve for both of them. Once that decision option was gone, they had different preferences for the remaining two options. We saw that Mr. X moved first and chose his preferred alternative, which was location B. Subsequently, Mr. Y and Ms. Z chose differently, because their risk–return trade-offs were different. Mr. Y, being less risk- averse, chose location A, while Ms. Z, being more risk-averse, chose location 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- tion of the risky decision. Looking at Figures 2.1 and 2.2, you will see that the vertical axis in each figure represents a series of levels of return (ENPV) that have zero risk attached to them. Now notice that each of the indifference curves terminates at the vertical axis, at a point of zero risk, thus revealing the CEs for all of the risk–return combinations 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 vertical axis). Thus, for Mr. X the CE of decision B is much greater than the CE for either A or C, so he prefers option B over the other two options. 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 indi- vidual prefers the decision alternative with the highest certainty equivalent.
The Certainty Equivalent Factor
The Certainty Equivalent Factor (CEF) is the ratio of the perceived monetary value of the risk-free alternative (i.e., the CE) to the risky alternative (i.e., the ENPV). In the case of Mr. X, the CEF for decision B is 80/100 5 0.8. The CEF effectively tells us what proportion 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 alternative 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% reduction in monetary value to compensate for the risk involved in decision B.6
6. Notice that for Mr. X, the CEF for location A is 40/100 5 0.4; while for location C the CEF 5 40/60 5 0.67, and thus appears to rank location C ahead of location A, which is not what the indifference curves and the CE criterion previously told us (and we assume that they are the true reflection of Mr. X’s preferences). Instead the CEF provides an insight into the relative degree of risk aversion exhibited by the individual for the option selected. For Mr. Y, the CEF of location A is about 85/100 5 0.85 and for Ms. Z the CEF of location C is about 22/60 5 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.
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
Notice 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 dif- ferently for the three people depicted: Each person makes his or her own, personal, inter- nal psychic evaluation of the disutility of risk and the utility of income and makes his own decision accordingly.
Of course, it is unrealistic 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 uti- 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 alternative that has the highest certainty equivalent. A little introspection on the part of decision makers will lead them to an intuitive preference for one decision alternative over the others, which will reflect their personal risk–return trade-off.
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 bet- ter.” Thus, we need to consider some decision rules that can be argued somewhat more objectively 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
©Stockbyte/Thinkstock
Put in simple terms, the certainty equivalent factor, which is the ratio of the perceived value of the risk-free alternative to the risky alternative, expresses how many cents in the dollar a decision maker would consider to be equivalent to the risky decision.
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
different decisions into account.7 One such decision rule is maximin—that is, choose the alternative that has the highest (maximal) worst (minimum) outcome. In the examples discussed earlier, we were concerned with the standard deviation of the potential out- comes associated with restaurant locations A, B, and C. The maximin rule is concerned with only one of those potential 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 deci- sions that could possibly bankrupt the firm (and cause shareholders to lose their invest- ment), so they may put pressure on managers to make relatively conservative 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 initial 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 initial 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 better off taking a hit of $200,000 rather than $500,000.
As you can see, the maximin criterion is appropri- ate for risk-averse people making risky decisions that are not repeated enough times 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 incur- ring losses that cannot be tolerated by the firm and its shareholders. But, by refusing to consider the other potential 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 out- come was 40%? In that case, by taking decision A the managers have chosen to risk a worst out- come with four times the chance of occurring than the worst outcome of Option B. Or, what if the other possible outcomes for Project A were posi- tive but relatively small while the other outcomes
©Neil Leslie/Getty Images
Simple and transparent, the maximin criterion is appropriate for risk-averse people making risky decisions and is an effective way to avoid incurring losses.
7. Shareholders may buy stock in a variety of firms and thus build up a portfolio of risky stocks in which the downward variations in one stock’s price are likely to be offset by the upward varia- tions 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.
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
for Project B were positive and relatively large? The maximin criterion does not consider these other outcomes at all. 8 So let’s look at some decision rules that do.
Coefficient of Variation Decision Rule The coefficient of variation (CV) is a statistic of a probability distribution and is calcu- lated as the ratio of the standard deviation to the mean. Going back to the example of restaurant location A, B, and C in the earlier decision-making problem, we can calculate the CV of A as 50/100 5 0.5; for B it is 30/100 5 0.3; and for C it is 30/60 5 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 option that has the smallest CV. So according to this rule, Mr. X would consider locations C and A as equal but inferior to location B, thereby agreeing with his certainty-equivalent-based decision. For Mr. Y and Ms. Z, the CV rule would say the two remaining options are equal, but we saw that Mr. Y preferred option A (higher CE for him) while Ms. Z preferred option 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, there- fore, 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) conservative alternative business if they want to.
In Figure 2.3 we show the CV decision rule as it applies to the restaurant location 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. Notice that the slope of the CV lines emanating 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 combination on a particular 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, exhibiting increasing MRS as more and more risk is taken on. As we saw in the case of our three restaurateurs, indi- vidual preferences might agree with the CV criteria (as Mr. X did) or not. While both Mr. Y and Ms. Z agreed that location B was the best location, Mr. Y ranked location A superior to C while Ms. Z ranked location C superior to A. Thus, the CV criterion is not generally suitable for individual decision making.
8. Other decision rules of the maximin type are maximax, which says you should select the alterna- tive that has the highest of the maximal outcomes, and minimax, which says you should choose the alternative 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- ties. These rules seem even less applicable in real-world business situations 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.
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
Figure 2.3: The coefficient of variation decision criterion
As a more complex application 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 distribution 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 initial cost is paid at the end of year 1 while the possible outcomes (cash inflows and outflows) 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 5 1.4374 compared to 1.0414), its ENPV is much higher ($3.5124 million compared to $0.7603 million) such that the CV ratio is only 0.4092 for Project B compared with 1.3697 for Project A.
100
60
0 30 50
CV = 0.3
CV = 0.5
Risk (SD)
Return (ENPV)
C
AB
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
Table 2.2a: Calculating the coefficient of variation for Project A
(1) Initial Cost year 1 (millions)
(2) Present value of initial 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 5 0.9091 DF 5 0.8264
10 8.2644 6.4463 0.4 2.5785
22 21.8182 5 4.1322 2.3140 0.5 1.1570
20.5 20.4132 22.2314 0.1 –0.2231
ENPV 5 3.5124
SD 5 1.4374
CV 5 0.4092
Table 2.2b: Calculating the coefficient of variation for Project B
(1) Initial cost year 1 (millions)
(2) Present value of initial 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 5 0.9091 DF 5 0.8264
5 4.1322 3.2231 0.3 0.9669
21 20.9091 2 1.6529 0.7438 0.3 0.2231
20.2 20.1653 21.0744 0.4 –0.4298
ENPV 5 0.7603
SD 5 1.0414
CV 5 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 time period; (3) similar decisions are not made repeatedly; and (4) managers are risk- averse (hopefully reflecting 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 information available and made the decision based simply on the best of the worst outcomes. The CV criterion is thus a more sophisticated deci- sion criterion that, while not generally suitable for individual decision making, does have
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
value for decisions made by managers of firms where the decision made must be justi- fied to risk-averse shareholders on some objective 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 approximation of shareholders’ prefer- ences in aggregate since these shareholders are free to build a portfolio of shares (in differ- ent 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 better 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 portfolio better 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 opportu- nity discount rates. A risk premium is an additional amount that is proportionate to the additional risk perceived. Recall that we defined the opportunity discount factor as the rate of interest that could be earned on an alternative opportunity of equal risk. When the decision alternatives are clearly not equally risky it follows that their opportunity dis- count rate should not be the same for each alternative.
At this point it is appropriate to examine the two main com- ponent 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 cer- tain to be repaid, for example, the purchase of government bonds. Although the govern- ment may change from time to time, the newly elected politi- cians would respect the previ- ous government’s obligation to repay lenders who had bought government bonds, so govern- ment bonds are regarded as the ultimate risk-free security.9 The
©iStockphoto/Thinkstock
Government bonds are regarded as the ultimate risk-free security because repayment is absolutely certain.
9. This may not apply to governments in all countries, of course. If a revolution or military takeover were to overthrow a government, the incoming government may decide not to honor the bor- rowings 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 defaulting 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 politically stable nations without excessive government debt, it is safe to regard government bonds as a risk-free investment.
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
risk-free rate is made up of two subparts, the real rate of interest and the premium for expected inflation. 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 inflation. But when lenders expect inflation to occur, they expect the purchasing power of the funds returned (after the loan is settled) 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 inflation premium charged needs to be about 5% to compensate the lender for the loss of purchasing power due to inflation, in this case where the expected rate of inflation is 5% per annum.10
The second main part of the ODR is the risk premium, which is the additional return the lender will require to cover the risk that the borrower might default on the loan and not pay the money back. To estimate the appropriate risk premium, the lender must ask, “What is the probability that the borrower will not repay the loan?” To answer this ques- tion, the lender (like the insurance manager in Chapter 1) must consider what propor- tion of people, in roughly the same risky situations, 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% 5 100% of the loan advanced to the borrower who ultimately defaults. The formula for the risk premium is thus the ratio 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 probabilities, and you can see that the risk premium increases exponentially as the probability of default increases.
10. Strictly, the inflation premium is calculated as k/(1–k), where k is the expected rate of inflation. Thus, for example, if expected inflation is 5%, the inflation premium is 0.05/0.95 5 5.26%. For expected inflation of 10%, the inflation premium is 0.1/0.9 5 11.11%. It can be seen that as the rate of expected inflation increases, the inflation premium increases faster.
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CHAPTER 2Section 2.2 More Transparent Decision Rules for Managers
Table 2.3: Default risk and calculation 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 for Project A, being the real rate of interest (say 2%) plus an expected inflation (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: Recalculating the ENPV for Project B, with ODR = 20%
(1) Initial Cost Year 1 (millions)
(2) Present value of initial 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 5 0.8333 DF 5 0.6944
10 0.6944 5.2778 0.4 2.1111
22.000 21.6667 5 3.4722 1.8056 0.5 0.9028
20.5 20.3472 22.0139 0.1 20.2014
ENPV 5 2.8125
SD 5 1.2078
CV 5 0.4295
Now compare the ENPVs of the two Projects A and B. Of course the ENPV is still $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 still the preferred alternative using
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CHAPTER 2Section 2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios
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 still favors Project B over Project A.
A Simplification for More Complex Situations In practice, we are typically confronted by more complex situations than the above exam- ples. 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 “guessti- mates” of the different monetary outcomes and of the probabilities of these different out- comes occurring. If these estimates are inaccurate, scrutiny by others who have different information 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 distributions are symmetric around the medium outcome, then a useful simplification becomes possible. To find the ENPV of the deci- sion alternative we need only add the medium NCF outcomes in each year and subtract the initial 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 simplification is hardly necessary for the relatively simple two- and three-year time 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 5 243 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 simplification will give a sufficiently robust indication of the ENPV as long as there is not substantial asymmetry of the high, medium, and low out- comes. 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 probabilities are estimates anyway, and these estimates 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 approximation to consider only the medium NCF outcome for each of the out years when the time horizon is three to five years or longer. For decision alternatives that have longer time horizons the “sum of the medium outcomes” approach is likely to be a sufficient approximation for the ENPV of the decision alternative.
2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios
What we have been calling the medium outcome is alternatively 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 representative of the middle part of the probability distribution, and similarly view the high outcome as being representative of the upside-risk side of the probability distribution and the low outcome as being representative of the downside-risk side of the probability distribution. Thus, the
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CHAPTER 2Section 2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios
point estimates of high, medium, and low should be viewed as representative of the three regions of the probability distribution.
The Normal Distribution We can illustrate these scenarios in the context of the special case of the normal distribution.11 The first property of a normal distribution is that it is symmetric around its mean value. It is often 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 distri- bution is also the median (or the 50th percentile) value. That is, 50% of the observations lie above, and 50% lie below, the mean value. The standard deviation of a normal distribu- tion is such that almost all (about 99.7%) of the outcomes lie within plus or minus three standard deviations from the mean. Thus, the second property of a normal distribution 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 notice that the mean value (i.e., the ENPV) of the probability distribution effectively represents the middle part of the distribution, 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 sometimes be more than the mean, and sometimes 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 distribution. Instead it is representative 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 left of the probability distribution, but represents all the outcomes that have values more than one SD below the mean outcome, so we position it at approximately the point that bisects the area under the curve to the left of the outcome that is one SD below the mean outcome.
11. It is called the normal distribution because this particular distribution of outcomes occurs in thousands of situations, across many different fields of the natural and the social sciences. It does not imply that distributions that are not normally distributed around their mean are abnormal in any pejorative sense.
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CHAPTER 2Section 2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios
Figure 2.4: The properties of a normal distribution
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 remain- der (i.e., 16%) and the worst-case scenario (WCS) must represent the other half of the remainder (16%). Be aware that these specific probabilities for the high, medium, and low outcomes may or may not be appropriate for particular business decision problems. If you have information that indicates that the outcomes seem likely to be approximately nor- mally distributed around the mean, then these probabilities will be appropriate. On the other hand, you might have information that indicates that the probability distribution is definitely not normally distributed and so you should use the probabilities that seem to be more appropriate.
Skewness of the Probability Distribution Skewness refers to the degree of asymmetry of the probability distribution. A distribu- tion that is perfectly symmetric is said to be nonskewed. But if the bell shape is distorted
99.7% of outcomes
Probability
95% of outcomes
68% of outcomes
Worst-case scenario
Most-likely scenario
Best-case scenario
-3 SD -2 SD -1 SD +1 SD +2 SD +3 SD
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CHAPTER 2Section 2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios
with more outcomes lying to the left of the mean outcome, with a longer tail stretching out to the right-hand side, the distri- bution is said to be positively skewed. In this case the median outcome (the 50th percentile outcome) will lie to the left of (below) the mean outcome. The modal outcome, which is the single outcome with the highest probability of occurring, will lie to the left of both the mean and the median outcome, as shown in Figure 2.5a. A negatively skewed distribution will have the bulge on the right-hand side of the distribution with a long tail to the left. The median out- come 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: Positively skewed probability distribution
©Biwa Studio/Getty Images
Skewness refers to the degree of asymmetry of the probability distribution. A distribution that is perfectly symmetric is said to be nonskewed.
Mode Median Mean
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CHAPTER 2Section 2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios
Figure 2.5b: Negatively skewed probability distribution
The implication of positive skewness for managerial decision making is that while the majority of the possible outcomes will be below the mean, there will be a significant num- ber of upside-risk outcomes that lie more than three standard deviations 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 probabilities associated with the worst-case, most-likely scenario, and best-case scenario would be different to the normal distribution, of course. Something like 10%, 60%, and 30% might be more appropriate for the worst-case, most- likely-case, and the best-case scenarios, respectively.
Conversely, when the distribution is negatively skewed, the majority of the possible out- comes will be above the mean, but there will be significant number of downside-risk outcomes that lie more than three standard deviations below the mean. Repeated trials of decisions with negatively skewed distributions would tend to result in an average out- come that is above the weighted mean (ENPV) outcome. The probabilities associated with the worst-case, most-likely scenario, and best-case scenario would be different to the nor- mal distribution, of course. Something like 30%, 60%, and 10% might be more appropriate for the worst-case, most-likely-case, and the best-case scenarios, respectively.
Kurtosis of the Probability Distribution Kurtosis refers to another aspect of the shape of a probability distribution, specifically its height. A distribution that is taller than a normal distribution is said to be leptokurtic and would have more than 68% of the outcomes falling within one SD each side of the mean. Conversely, a distribution that is platykurtic (like a plate, i.e., flatter) would have less than 68% of the outcomes within one SD each side of the mean. Kurtosis of probability distribu- tions is demonstrated in Figure 2.6.
Mean Median Mode
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CHAPTER 2Section 2.3 Most-Likely Scenario and the Best- and Worst-Case Scenarios
Figure 2.6: Kurtosis of probability distributions
You can see from the shapes of the two probability distributions in Figure 2.6 that for a leptokurtic distribution, the proportion of outcomes lying within one SD from the mean will be substantially above 68%, perhaps 80% in the example shown. This means that the probabilities of the best-case and worst-case scenarios are relatively small, about 10% each in the situation depicted. Conversely, for the platykurtic distribution, the proportion of the outcomes lying within one SD of the mean would be substantially below 68%, perhaps only 40–50% in the example shown. Thus, the probabilities of the best- and worst-case scenarios might be relatively large, about 30% each in the platykurtic distribution 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 distribution of outcomes is likely to be approxi- mately normal or not. Remember that many decision situations are likely to deliver approx- imately normal distributions of outcomes since the independent actions of many people (e.g., buyers) typically result in a normal distribution. Second, if the decision maker has reason to believe that the distribution will be skewed to one side or the other, or taller or flatter than a normal distribution due to factors that he or she suspects are characteristic of the population or sample, the probabilities need to be adjusted in the direction that reflects the decision maker’s best estimate of the actual shape of the probability distribu- tion. In the absence of any information to indicate that the normal distribution is not appro- priate, it is usually a good first approximation to assume normality of the distribution. It is useful to remember that unless you have data on the distribution of prior outcomes of similar decisions, assigning probabilities to possible outcomes is an art, not a science—the decision maker needs to think about the situation 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 scrutiny may reveal information that was not known to you and allow a more accurate probability distribution to underlie your decision. And finally, if this type of decision scenario is repeated again and again, data will build up and the probability distribution can be corrected subsequently.
Platykurtic distribution
Leptokurtic distribution
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CHAPTER 2Summary
Summary
In this chapter, we introduced adjustment for risk into our decision-making toolkit. Since business managers typically must make decisions under conditions of risk and uncer- tainty, 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 distribution into the deci- sion rule. We noted that risk is best measured by the standard deviation of the probability distribution 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 prefer- ring, 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 repetitive decisions of the same type and in the same con- text, 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 portfolio of stocks that would not be susceptible 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 particular 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 disutility, a trade-off is made against the monetary value of the decision (i.e., the ENPV) to compensate for bear- ing risk. For individuals making decisions on their own behalf (including sole owners of busi- nesses) the certainty equivalent (CE) criterion is applicable. We introduced risk–return indiffer- ence curves to demonstrate that the individual will take addi- tional risk for additional return, or conversely would give up part of the potential 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 plotting out all risk–return indifference curves, the individual decision maker simply needs a moment of introspection to conclude which one of the possible alternative decisions makes him or her feel most comfortable, taking into account both the expected utility from money and the expected disutility from risk.
©Paul Taylor/Getty Images
When data is not available on the normal distribution of prior outcomes of similar decisions, the decision maker must rely on best guesses.
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CHAPTER 2Summary
This process of introspection and personal choice is not usually sufficient for manag- ers of other peoples’ money. So managers of firms in which there are other shareholders must utilize decision rules that can be justified on some objective basis. The coefficient of variation (CV) decision criterion is the ratio of the standard deviation to the weighted mean (i.e., ENPV) of the probability distribution and the CV ratio effectively 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 potential 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 alternatives with different risk profiles, the decision maker should choose the one with the highest ENPV remaining after the net cash flow streams have been discounted by an ODR containing the appropriate risk premium.
Many decision problems can be reduced to relatively simple terms, such as three potential outcomes (high, medium, and low) in each of one or two years. Decision tree analysis can be utilized for these relatively 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 continuing into the third and subsequent years have too many terminal branches to be realistically solved using a decision tree. Instead, we can simplify these more complex decisions in one way or another and still arrive at a risk- adjusted-ENPV valuation of the decision alternative that should be sufficiently reliable (given that we are guesstimating about future outcomes in any case). One way is to treat the decision tree as being approximately symmetrical, with the high outcomes being bal- anced out by the low outcomes. If this is a sufficiently realistic assumption, we can simply add up the net present value (NPV) of the medium case outcomes in each year to approxi- mate the ENPV of the entire probability distribution. We then argued that the medium outcome is actually representative of the most-likely scenario—that is, it is the midpoint of a relatively 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 time; falling above that range (best-case scenario) about one sixth of the time; and falling below that range about one sixth of the time. 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 distribution is approximately a normal distribution.
If the distribution is skewed to one side or the other, the ENPV is no longer the most prob- able (or modal) outcome. If the distribution is positively 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 negative with a long tail to the left, 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 distribution exhibits leptokurtosis, being taller than a normal distribu- tion, 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
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CHAPTER 2Questions for Review and Discussion
of the normal distribution is appropriate before proceeding on that simple but relatively reliable assumption.
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 implications 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 objective function. Then we recognized that people tend to be risk-averse and incur disutility 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 time 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 implications of decisions.
Questions for Review and Discussion
1. Explain why the standard deviation of a probability distribution is an appropriate measure of the risk involved in a real-world decision problem.
2. Define risk preference, risk aversion, risk seeking, and risk taking. 3. Explain why a risk averter facing two choices might prefer to take the higher risk
alternative. 4. How might a risk-averse decision maker adjust the expected net present value of a
decision to take into account its risk? 5. Some risk-adjusting decision rules work well for the individual but are not suffi-
cient for decisions made on behalf of others, such as shareholders of the firm. Please explain.
6. What is the certainty equivalent of a decision alternative? Use this rule to explain your choice between a 70% chance of winning $10 and a 50% chance of winning $20.
7. What are the main two components of the opportunity discount rate, and how do you find the value of these components?
8. Why would a symmetric decision tree analysis of a complex decision problem give the same result as a simple summation of the medium net cash flows associated with that decision problem?
9. Define the most-likely, best-case, and worst-case scenarios in terms of a probability distribution of the outcomes of a decision alternative.
10. Explain how the probabilities associated with a probability distribution would need to be adjusted if the probability distribution was (i) platykurtic, and (ii) negatively skewed.
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CHAPTER 2Decision Problems
Decision Problems
1. While completing your studies you want to try out as an entrepreneur and are con- sidering two different entrepreneurial opportunities that you have identified. 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 peo- ple attending the fair and wanting a commemorative keepsake or a gift for others. Your estimate of ENPV for this project is $3,000 with standard deviation 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 estimate that the ENPV of this is alternative is $4,000 with standard deviation of $2,000.
a. Apply the coefficient-of-variation decision criterion to these alternatives 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 alternative is that you would make only $500 profit while for the umbrella option the worst outcome (no rain) would cause you to lose $1,000.
c. Indulge in some introspection and estimate your personal certainty equivalent for these two projects.
2. The Sounds True Music company is considering the introduction 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 competition from rivals and is concerned about making the best choice among two alternatives being considered. Option A is to make a minor facelift to an existing product, while option 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 follow- ing table. They estimate that the probabilities of these macroeconomic outcomes are 30% for a downturn, 50% for constant, and 20% for an upturn.
State of the economy
Option A – minor facelift (NCF)
Option B – new product (NCF)
Downturn $10 $220
Constant $30 $20
Upturn $80 $150
a. Calculate the expected value, standard deviation, and coefficient of variation for each decision alternative (use a spreadsheet and the embedded formula for stan- dard deviation).
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CHAPTER 2Decision Problems
b. Apply the expected value, coefficient of variation, and maximin decision criteria to this decision problem.
c. Which alternative 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 cities 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 additional truck. Careful analysis of costs and the potential demand situation has led to the following estimates of net cash flow for each alternative. The applicable opportunity discount rate is 15%.
NCF Year 1 (millions)
Probability Year 1
NCF Year 2 (millions)
Probability Year 2
LEASE OPTION
25 0.25 5 0.30
5 0.40 10 0.50
15 0.35 15 0.20
BUY OPTION
210 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 alternative.
b. Calculate a measure of risk for each alternative. (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 recommendation to management. d. State any qualifications or reservations you would want to add to your
recommendation.
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CHAPTER 2Decision Problems
4. Your firm is considering the introduction 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 estimates of sales at each price level over the next two years. The initial 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 alternative 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 alternatives in order of their risk, and explain the basis for your
ranking. d. Rank the alternatives 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 multiple outcomes possible each year. There are two alternative plans under consideration 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 utilities and other infrastructure, and offer the lots for sales to individual homeowners wanting 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- tive 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 under- taken. 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
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CHAPTER 2Key Terms
probability distributions to be symmetric in both cases and has estimated 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 1 220 210
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 alternative by that criterion.
b. What additional information would be required to allow application of the maxi- min criterion?
c. What additional information would be required to allow application of the coef- ficient of variation criterion?
d Given that there is no more information available, make your recommendation and support it with reasoning, and list any other information that you suggest Highfields should seek before taking action.
Key Terms
best-case scenario A scenario that is representative of the range of outcomes that are more than one standard devia- tion above the mean of all the possible scenarios.
cardinal 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 situation as being greater (higher) or less (lower) than another with- out specifying how much higher or lower one fence is.
certainty equivalent (CE) 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 deviation).
Certainty Equivalent Factor (CEF) The ratio of the perceived monetary value of the risk-free alternative (the CE) to the risky alternative (the ENPV).
coefficient of variation (CV) A measure of the risk-per-dollar-of-return, derived by dividing the standard deviation by the expected net present value.
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CHAPTER 2Key Terms
direction of preference The direction that a person would want to go to increase total utility or to achieve higher return and lower risk.
downside risk The risk that the outcome that occurs will be less than the expected net present value.
highly intolerant of risk Those who are in this category try to avoid voluntary risk taking as much as possible. Risk-intolerant people need much greater monetary compensation to induce them to accept additional risk.
highly tolerant of risk Are those who are slightly risk-averse. Bearing risk causes them relatively little psychic dissatisfac- tion, so they will take on additional risk for relatively small increases in expected value.
indifference curves Lines that represent combinations of goods and services that give a consumer an identical quantity of utility or represent a set of choices among which the buyer is indifferent.
inflation premium That part of the opportunity discount rate that is intended to compensate the lender for the expected fall in the purchasing power of the mon- etary unit (the dollar) during the period of the loan. It is calculated by the equation: k/(1 2 k) where k is the expected rate of inflation.
involuntary risk taking Everyday risks that are imposed on us by nature, or other external circumstances, and which cannot easily be avoided.
kurtosis The height of a probability dis- tribution; the degree to which it is peaked or flat.
leptokurtic A feature of a probability or frequency distribution that is evident as a relatively tall and narrow peak surround- ing the mean because the data values have relatively little variance and are mostly similar to the average value.
marginal rate of substitution Reflected by the slope of the indifference curve and represents the rate of substitution between two competing items that leaves a person at the same level of utility.
maximin A decision rule, where a con- sumer chooses the alternative that has the largest (maximal), of the smallest (mini- mum) outcomes.
median outcome The 50th percentile out- come, calculated by ranking the outcomes in order of magnitude and selecting the outcome that falls at 50% of the total num- ber of outcomes.
modal outcome The single outcome with the highest probability of occurring, or the outcome that repeats the most amount of times in a given set of data.
most-likely scenario A scenario represen- tative of the middle range of the prob- ability distribution, usually defined as the range of outcomes that are plus or minus one standard deviation from the mean outcome.
negatively skewed A distribution pattern that has a bulge on the right-hand side of the distribution with a long tail stretching to the left.
normal distribution A symmetrical distribution with a peak in the center of the range of possible outcomes, and with shape such that 68% of the observations lie within plus or minus (1/–) distribu- tions that are not one standard deviation from the mean; 95% lie within 1/– two standard deviations; and 99.7% lie within 1/– three standard deviations.
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CHAPTER 2Key Terms
opportunity discount factor The rate of interest that could be earned on an alter- native opportunity of equal risk.
ordinal An evaluation system that relies on relative location (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.
platykurtic A distribution that has rela- tively wide dispersion around the mean value and thus has a wider and flatter distribution surrounding the mean.
positively skewed When results are posi- tively skewed the mean is greater than the median, which is greater than the mode, and therefore the right tail of the distribu- tion is stretched. In this type of distribu- tion, the expected outcome is likely to be below the mean, but there is a higher chance of a few extremely positive results.
real rate of interest A measure of how much it really costs to borrow money when the inflation rate is subtracted from the actual nominal interest rate of a given loan.
risk aversion A condition in which an investor or other decision maker exhibits a preference for less risk, other things (such as expected returns) being equal.
risk neutral 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 A state where the indi- vidual prefers more risk to less risk, with other things (such as reward or profit) being equal. A risk preferer would there- fore choose the riskier of two equally profitable investments.
risk premium 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 An investor that deliberately seeks out investments that have higher risk, generally because they also have higher expected returns.
risk taker While risk takers do not go out of their way to deliberately find risky ven- tures like risk seekers do, they are ready to accept extra risk if the anticipated returns are sufficiently higher than for a less risky investment.
risk-free rate Government bonds issued by a federal Treasury, according to stan- dard 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 A calculation of the vari- ance of a distribution including only those data points that fall below the mean.
skewness Indicated by asymmetry in the distribution curve around the mean and can be either negative (skewed to the left) or positive (skewed to the right).
slope 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 deviation The extent of disper- sion of the observations in a data set from the data’s mean value. Standard devia- tion increases when the data observations are spread more widely around the mean value.
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CHAPTER 2Key Terms
upside risk The measure of the prob- ability that value of the actual outcome will be higher than the expected present value of the prior distribution of possible outcomes.
variance A measure that evaluates the spread of data observations in comparison to the location of the mean for the given data set.
voluntary risk taking An activity engaged in by investors and individuals with the expectation that the utility of the reward will outweigh the disutility of the risk.
weighted mean A measure of central tendency that weights each observation by its probability of occurring, such as the ENPV of a probability distribution of potential outcomes.
worst-case scenario A situation that rep- resents a range of outcomes that lie more than one standard deviation below the mean or ENPV outcome.
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