Amazon Managerial Decision Making Research and Analysis 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 fortheir objective function. We argued that the firm’s objective will be to maximize profits, and that managers must make decisions underconditions 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 thefuture, the appropriate decision criterion is the expected net present value (ENPV). We also argued in Chapter 1 that decisions will usually haveboth 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) ofnonmonetary costs is greater than the psychic gain (utility) of the nonmonetary benefits, we say there is net disutility associated with thedecision and the decision maker will require additional profit to compensate for the net disutility associated with the decision. Conversely, if thenonmonetary benefits exceed the nonmonetary costs, there is net utility associated with the decision, and the decision maker will be willing togive 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 compensated for bearing risk. In this chapter we integraterisk analysis into the decision-making process and consider several decision criteria that adjust the monetary outcomes of a decision for the riskthat is associated with those returns. In Chapter 1, we noted that the ENPV criterion is only really appropriate if the manager continually makesthe same type of decision in the same environment, such that the manager could reasonably expect that over many trials the aggregateoutcome would be approximately equal to the sum of the individual ENPVs. Approximately half the time the actual outcome will be higher thanthe ENPV and the other half of the time the actual outcome will be below the ENPV. Although facing risk in each specific decision, therepetitiveness of the decision allows the chances of below-average outcomes to be offset by above-average outcomes, and over many similardecisions 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 repeatedoften enough to make the ENPV criterion an appropriate decision criterion since it does not adjust for differing degrees of risk associated withindividual decision problems. In this chapter, we recognize th.

1. Amazon Managerial Decision Making Research and Analysis
Introduction
In Chapter 1, we saw that managers wanting to make decisions t
hat best serve their objective functions will need to first define t
he metric fortheir objective function. We argued that the firm’s
objective will be to maximize profits, and that managers must m
ake decisions underconditions 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 t
he norm, and where expenses and revenues are incurred and rec
eived both in the present period and into thefuture, the appropri
ate decision criterion is the expected net present value (ENPV).
We also argued in Chapter 1 that decisions will usually havebot
h monetary and nonmonetary outcomes that are of interest, or co
ncern, to the decision maker. Accordingly, decision makers will
make trade-
offs against profit to compensate for the nonmonetary costs or b
enefits that are associated with the decision. If the psychic pain
(disutility) ofnonmonetary costs is greater than the psychic gain
(utility) of the nonmonetary benefits, we say there is net disutil
ity associated with thedecision and the decision maker will requ
ire additional profit to compensate for the net disutility associat
ed with the decision. Conversely, if thenonmonetary benefits ex
ceed the nonmonetary costs, there is net utility associated with t
he decision, and the decision maker will be willing togive up so
me profit to compensate for the nonmonetary aspects of the deci
sion.
Risk causes disutility for most business decision makers and so
they will want to be compensated for bearing risk. In this chapte
r we integraterisk analysis into the decision-
making process and consider several decision criteria that adjust

2. the monetary outcomes of a decision for the riskthat is associat
ed with those returns. In Chapter 1, we noted that the ENPV crit
erion is only really appropriate if the manager continually make
sthe same type of decision in the same environment, such that th
e manager could reasonably expect that over many trials the agg
regateoutcome would be approximately equal to the sum of the i
ndividual ENPVs. Approximately half the time the actual outco
me will be higher thanthe ENPV and the other half of the time t
he actual outcome will be below the ENPV. Although facing ris
k in each specific decision, therepetitiveness of the decision all
ows the chances of below-
average outcomes to be offset by above-
average outcomes, and over many similardecisions the total prof
it outcome would approximate the sum of the ENPVs of all the
decisions.
But in many business situations the manager faces a variety of d
ifferent decisions from day to day and most types of decision ar
e not repeatedoften enough to make the ENPV criterion an appr
opriate decision criterion since it does not adjust for differing d
egrees of risk associated withindividual decision problems. In th
is chapter, we recognize that the decision maker will want to inc
orporate risk analysis into the decision-
making process for those decisions that are not repeated frequen
tly and will want to adjust each decision to take account of the
degree of riskinvolved in each particular decision.
How Is Risk Measured?
Standard Deviation
Risk can be expressed as a measure of the chance that the value
of the ENPV of adecision will not be the actual outcome. To cal
culate a measure of variability of thepotential outcomes relative
to the ENPV we need to understand the statistical conceptknow
n as the standard deviation, which is a measure of the deviations
of the possibleoutcomes from the central tendency (or mean) of
those possible outcomes. First notethat the ENPV is a measure
of central tendency of the potential outcomes—
indeedthe ENPV is the weighted mean (where the weights are th

3. e probabilities of occurring)of the possible outcomes. The mean
of any series of numbers has an associatedstandard deviation, w
hich indicates the extent to which the mean value isrepresentati
ve of all the data points that enter the calculation of that mean.
Thestandard deviation is higher if the possible outcomes are mo
re widely dispersedaround the mean value, or is lower if the act
ual outcomes lie relatively close to themean value. To calculate
the standard deviation, the deviations of each data pointfrom the
mean are squared and then summed to find the variance of the d
istribution,and the standard deviation is then simply the square r
oot of the variance. In effectthe standard deviation indicates the
average absolute deviation of the outcomes fromthe mean outco
me. Thus, the standard deviation provides a suitable measure of
therisk 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 s
preadsheet totype (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 thecomputer can calculat
e the standard deviation. Note that it is more complex to calcula
te the standard deviation of a probability distribution. Weneed t
o (1) find the ENPV of that distribution; (2) subtract each outco
me from the ENPV to find the deviations from the mean; (3) wei
ght eachdeviation by its probability of occurring; (4) sum these
weighted deviations to find the variance; and (5) take the square
root of the variance tofind the standard deviation. This is demo
nstrated for a simple case in Table 2.1, in which we suppose tha
t three outcomes are possible (column1) with probabilities as sh
own (in column 5), which you can verify gives an expected valu
e of 10 as shown (in column 2). For simplicity here, weassume t
he cash flows all take place in the present period such that ENP
V = EV.
Table 2.1: Calculation of the standard deviation of a probability
distribution
Possibleoutcome($)
Expectedvalue ($)

4. Deviation of possible outcomefrom the mean (EV) outcome
Squared deviation fromthe mean outcome
Probability of eachpossible outcome
Weighteddeviation fromEV
−10
10
30
10
10
10
−20
0
20
400
0
400
0.25
0.5
0.25
100
0
100
Variance =
200
Std. Deviation =
14.14
© Photodisc/Thinkstock

5. To find whether the wins offset the losses,variance or standard
deviation can beused to measure the risk associated withuncerta
in outcomes.
Half of the variance around the ENPV is quite desirable, and of
course I am referring to the outcomesthat are greater than the E
NPV. This part of the risk is known as the upside risk and repre
sents betteroutcomes than can generally be expected in multiple
trials of this decision. Some risk analysts havesuggested that no
one is worried about these positive deviations from the mean be
cause we would"laugh all the way to the bank" if one of these w
ere to occur. On the other hand, the downside riskrepresents the
outcomes that are worse than the ENPV, and we definitely worr
y about these.Accordingly, some analysts have suggested that w
e calculate only the semi-
variance of the outcomes byincluding only those outcomes with
negative deviation from the ENPV to measure only the downsid
erisk. Although intuitively appealing, the semi-
variance approach is not commonly used because it ignoresthe u
pside risk; after all, if "sometimes you win and sometimes you l
ose," you need to know the extentof the wins to see whether the
y would offset the losses. Thus, we tend to use the standard devi
ation asour measure of the risk associated with uncertain outco
mes. So, now we have a measure of risk, butbefore we adjust for
risk, we need to consider the decision maker’s attitude toward r
isk.
Attitudes Toward Risk
People have different attitudes toward risk. Some people seem t
o enjoy doing risky things, while othersare extremely unhappy t
o 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 riskneutrality. Ri
sk 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 equallypr
ofitable investments. This is only rational behavior if the indivi
dual’s objective function is to maximizerisk rather than to maxi

6. mize profit, or if the person is so rich that he or she places very
little value on the money he or she might lose whileplacing mor
e value on the thrill he or she will get by taking the risk; perhap
s high-
roller gamblers might fit this profile. Risk preferers might getlu
cky and have a series of wins (despite the odds) but sooner or la
ter will be bankrupted if they continue to make large ENPV deci
sions thisway.
Risk aversion means that the individual prefers less risk to more
risk, other things being equal, and will therefore choose the les
s risky of twoequally profitable investments. Risk-
averse individuals may choose the more risky alternative if they
expect to be adequately compensated forthe additional risk und
ertaken. They weigh up the monetary trade-
off of the extra income against the psychic dissatisfaction of ad
ditional riskbearing 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 canoccur due to a genuin
e lack of concern for risk and subsequent losses (which makes it
almost as dangerous to your wealth as risk preference),or due t
o the repetition of the same or similar decision many times, so t
hat on any one occasion the decision makers can act as if they a
re riskneutral. Thus, as we have noted, the ENPV measure of pr
ofit is appropriate if the same (or sufficiently similar) decision i
s to be repeated manytimes. In this situation the decision maker
s can act as if they are risk neutral for any one of those decision
s.
© iStockphoto/Thinkstock
Entrepreneurs, skydivers, and motorcyclists all voluntarily take
riskswith the expectation that the utility of the reward will outw
eighthe disutility of the risk.
We need to clarify a few more terms that are commonly used in
discussionsof risk. People often talk about entrepreneurs, for ex

7. ample, being risk seekers.Risk seekers seek to do risky things (l
ike entrepreneurship, skydiving, andmotor racing), because they
expect that risk and return are positivelycorrelated: The higher
the risk the higher the return. Whether the return issimply mone
tary, or is both monetary and nonmonetary (i.e., includes psychi
csatisfaction), most risk seekers only take the risk if they expect
the payoff tobe greater to compensate them for risk bearing. Ri
sk seekers are therefore notrisk preferers but are actually risk-
averse.1 Another term that probably needsclarifying is risk take
r. We are all risk takers, like it or not. Every day we aresubject
to the risks of global warming, asteroids, tsunamis, earthquakes,
globalfinancial crises, traffic accidents, and physical violence,
to name just a fewsources of the risks we continually take. What
is important is not that we takerisks but what our attitude is to
ward taking risks. As you now know, ourattitude either will be r
isk preference, risk aversion, or risk neutrality,depending on our
prior knowledge of the situation and our cognitiveprocessing of
the psychic costs and benefits associated with taking specificris
ks. We should also distinguish between voluntary risk taking an
dinvoluntary risk taking. The everyday risks listed earlier in thi
s paragraph areimposed on us by nature or by our fellow man an
d are borne involuntarily. Risk seekers, however, take risk volu
ntarily in the expectation thatthe utility of the reward will outw
eigh the disutility of the risk. Thus entrepreneurs, skydivers, rac
ing drivers, and business decision makersvoluntarily undertake r
isky projects and make risky decisions even though they are ave
rse to risk.
Degrees of Risk Aversion
Risk aversion can range from almost zero degrees of risk aversi
on (i.e., being almost risk neutral) to being extremely risk-
averse. For someonewho is slightly risk-
averse, bearing risk causes relatively little psychic dissatisfactio
n. We say they are highly tolerant of risk. People like this willre
quire only a relatively small amount of monetary compensation
for bearing additional risk. For others, bearing risk causes much
more psychicdissatisfaction. We say they are highly intolerant

8. of risk—
they will try to avoid voluntary risk taking as much as possible.
For these people, itwill require much greater monetary compens
ation to induce them to accept additional risk. Since different de
cision makers exhibit differentdegrees of risk aversion (or conv
ersely, risk tolerance), the extent to which they will want to adj
ust their decision for risk will differ. Accordingly,we must take
into account the decision maker’s degree of risk aversion as wel
l as the extent of risk involved in any particular decision.
Risk Perception
Similarly, each person might perceive risk differently. Individua
ls perceive the risk in a decision situation more or less accuratel
y depending ontheir prior knowledge and their cognitive biases.
Greater prior knowledge of the situation, or greater information
search activity,2 may providethe decision maker with useful inf
ormation that others do not have, such that she might (correctly)
say the situation is not very risky whileothers might say it is hi
ghly risky because of their ignorance of the situation. The old s
aying 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 knowledg
e.Next, a cognitive bias such as overconfidence may cause one p
erson to overlook risks that a less confident person might percei
ve because thelatter looks more carefully into the situation or sp
ends more time and money on information search activity to rev
eal the hidden dangers.Another cognitive bias is the tendency of
decision makers to use heuristics, or simplistic decision rules.
While economizing on time and searchcosts, heuristics could act
ually increase the decision maker’s exposure to risk, since they
consider only some of the information that ispotentially availabl
e. For example, entrepreneurs have been shown to be more over
confident and to use heuristics more than employedmanagers of
firms (Busenitz & Barney, 1997).3 When others see entrepreneu
rs taking extraordinary risks they often presume that theseentrep
reneurs must be highly tolerant of risks, when in fact many entr
epreneurs are highly risk-

9. averse; they do indeed take greater risks, butthis may be becaus
e 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, a
vailable with certainty, that a person would consider equivalent
to theexpected value of a risky decision. In this section, we will
introduce risk–return trade-
off curves and show how these differ according to thedecision m
aker’s degree of risk aversion. This will allow us to demonstrate
that different individuals typically have different certainty equi
valents.
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 differe
nt degrees of riskaversion will require different amounts of com
pensation to induce them to bear an additional quantum of risk.
Using simple graphical analysiswe 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 decisionmaker, Mr. X.
Suppose that Mr. X must decide where (at which location) he wi
ll build a new restaurant. The points A, B, and C shown in Figur
e 2.1 relate tothree different risk–
return combinations that represent different restaurant locations.
We depict these three decision alternatives with riskmeasured b
y standard deviation (SD) and return measured by ENPV. Their
risk and return outcomes differ because of differences in popula
tiondensity, passing traffic, proximity to public transport, and s
o on. As you can see, decision A has ENPV = 100 and SD = 50;

10. decision B has ENPV =100 and SD = 30; and decision C has EN
PV = 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 eq
ually profitable but has more risk (higher SD) than B. Similarly,
all risk averters willprefer B to C, because these two options ha
ve the same amount of risk but B is more profitable (higher ENP
V) 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 haveprejudged the answer by drawing th
e risk–
return (RR) curves such that A and C lie on the same RR curve (
shown as RR2) so the answer is thatthey are both equal in the ca
se depicted (i.e., reflecting Mr. X’s feelings about risk and retur
n). Each RR curve depicts those combinations of riskand return
that give the same level of utility. These curves are more comm
only known as indifference curves, which are lines drawn to pas
sthrough combinations of variables among which the decision m
aker is indifferent, that is, receives the same amount of utility.4
Thus, Mr. X willbe indifferent between A and C, or indeed any
other combination of risk and return that lies on RR2. Now, sinc
e point B is preferred to bothpoint A and C, it follows that ever
y combination of risk and return on RR3 is preferred to any com
bination on RR2. Similarly, any risk–
returncombination on RR1 is considered inferior to any combina
tion on any higher indifference curve. Thus, we can say that any
point on a higherindifference curve will be preferred to any poi
nt on a lower indifference curve and that the direction of prefere
nce is shown by the arrow; morereturn is preferred when risk is
the same, or conversely, less risk is preferred when return is the
same, and the decision maker preferscombinations that have bot
h more return and less risk. Note that we do not need to know th
e actual value to the utility represented by theRR1, RR2, and R
R3 curves, we just need to know the order of preference—
thus indifference curve analysis is concerned with ordinal (i.e.,

11. simplyin order) preferences rather than cardinal (i.e., measurabl
e) preference differences.
The RR indifference curves demonstrate the decision maker’s tr
ade-off between risk and return. This trade-
off is also known as the marginalrate of substitution (MRS) bet
ween risk and return, which is equal to the amount of risk the de
cision maker will accept for an additionalmeasure of return. Thi
s trade-
off is indicated by the slope of the RR curve, which is equal to t
he "rise over the run." In Figure 2.1, we saw thatthe decision ma
ker considers points A and C to be equivalent. Now if Mr. X wa
s asked to change from C to A, we can see that he wants 40more
units of return (the rise from 60 to 100) to compensate for the 2
0 extra units of risk (the run of 30 to 50). Thus, the slope of the
RR2indifference 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 combinationsin the vicinity of decisions A, B, and C.5
Figure 2.2: Differing degrees of risk aversion for twodifferent d
ecision 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 avers
ion, and thusquite different marginal rates of substitution. These
people are considering restaurant locations A and C, because lo
cation B has already beentaken by Mr. X. Note that Mr. Y prefe
rs decision A because, for him, it lies on a higher RR indifferen
ce curve. Conversely, Ms. Z prefers decisionC 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 r
equire only about $5 more (from $60 to about $65) to compensa
te for the 20additional units of risk. Thus, his MRS for return an
d risk is 5/20 = 0.25. Because decision A offers $40 more return
for those 20 extra units ofrisk, it is utility maximizing for him t

12. o take decision A rather than decision C. Conversely, the MRS f
or Ms. Z, moving along RRZ2, is about 4 (i.e.,a $40 change in r
eturn from $60 to $100 is necessary to compensate for a 10 chan
ge in risk from 30 to about 40, along RRZ2), so the additionalri
sk associated with decision A (20 units) is not compensated for
by the additional 40 units of return offered by A, and thus Ms. Z
prefersdecision C.
What we have demonstrated is that risk-
averse decision makers will make different decisions according
to their degree of risk aversion. Notethat both Mr. Y and Ms. Z
would have preferred restaurant location B if it were still availa
ble, since its risk–
return outcomes would fall on ahigher indifference curve for bot
h of them. Once that decision option was gone, they had differe
nt preferences for the remaining two options.We saw that Mr. X
moved first and chose his preferred alternative, which was loca
tion B. Subsequently, Mr. Y and Ms. Z chose differently,becaus
e their risk–return trade-
offs were different. Mr. Y, being less risk-
averse, chose location A, while Ms. Z, being more risk-
averse, choselocation C.
The Certainty Equivalent as a Decision Criterion
The analysis above allows us to consider the certainty equivalen
t (CE) of a risky decision. The CE of a risky decision (or gambl
e) is the amount ofreturn, available with certainty (i.e., zero risk
), that the decision maker will consider is equivalent to the risk
–
return combination of the riskydecision. Looking at Figures 2.1
and 2.2, you will see that the vertical axis in each figure represe
nts a series of levels of return (ENPV) that havezero risk attach
ed to them. Now notice that each of the indifference curves term
inates at the vertical axis, at a point of zero risk, thus revealingt
he CEs for all of the risk–
return combinations on each indifference curve.
© Stockbyte/Thinkstock

13. Put in simple terms, the certaintyequivalent factor, which is the
ratio of theperceived value of the risk-
free alternativeto the risky alternative, expresses howmany cent
s in the dollar a decision makerwould consider to be equivalent
to therisky decision.
In Figure 2.1 for example, for Mr. X the CE of decision B seems
to be about 80, while the CEs of bothdecision A and C seem to
be about 40 (i.e., where the indifference curves hit the vertical a
xis). Thus,for Mr. X the CE of decision B is much greater than t
he CE for either A or C, so he prefers option Bover the other tw
o options. In Figure 2.2 we see that the CE for Mr. Y seems to b
e about 85 fordecision A and 55 for decision C. Finally, for Ms.
Z, the CE is about 22 for decision C and much lowerfor decisio
n A. In each case the individual prefers the decision alternative
with the highest certaintyequivalent.
The Certainty Equivalent Factor
The Certainty Equivalent Factor (CEF) is the ratio of the percei
ved monetary value of the risk-
freealternative (i.e., the CE) to the risky alternative (i.e., the E
NPV). In the case of Mr. X, the CEF fordecision B is 80/100 = 0
.8. The CEF effectively tells us what proportion of the risky EN
PV would beconsidered equivalent to the risky ENPV, if it were
risk-
free. Put another way, the CEF tells us howmany cents in the do
llar, available with certainty, the decision maker will consider t
o be equivalent tothe risky decision. Thus, Mr. X values decisio
n B at 80% of the dollar value of the ENPV. So, the CEcriterion
will tell us not only which is the preferred alternative but will a
lso tell us how many cents inthe dollar would be just sufficient t
o trade for the risky decision, which tells us just how risk-
averse thedecision maker is. In this case Mr. X is willing to take
a 30% reduction in monetary value tocompensate for the risk in
volved in decision B.6
Notice that the CE values are different for each individual—
we cannot compare the psychic value of either risk or return acr
oss people. That iswhy the RR curves are labeled differently for

14. the three people depicted: Each person makes his or her own, p
ersonal, internal psychic evaluationof the disutility of risk and t
he 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 d
ecision they willchoose. The graphical model of the decision-
making process that we have utilized here is primarily intended
to facilitate your learning aboutrisk–return trade-
offs in decision making. But note that the model has brought us
to the point of a rather simple decision rule for decisionmaking
under risk and uncertainty; namely, risk-
averse managers should choose the decision alternative that has
the highest certaintyequivalent. A little introspection on the part
of decision makers will lead them to an intuitive preference for
one decision alternative over theothers, 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, th
ey can make their business decisions this way, but ifthey are em
ployed managers of firms that are owned by other shareholders t
hey will have to be more accountable to those shareholders fort
he decisions they make, and, accordingly, will have to adopt a
more transparent decision rule than, "I made that decision becau
se it made mefeel better." Thus, we need to consider some decis
ion rules that can be argued somewhat more objectively by man
agers 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 thattake the risk associated with differe
nt decisions into account.7 One such decision rule is maximin—
that is, choose the alternative that has thehighest (maximal) wor

15. st (minimum) outcome. In the examples discussed earlier, we w
ere concerned with the standard deviation of the potentialoutco
mes associated with restaurant locations A, B, and C. The maxi
min rule is concerned with only one of those potential outcomes
for each ofA, 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
arisky decision? Shareholders will not want the manager to mak
e decisions that could possibly bankrupt the firm (and cause sha
reholders to losetheir investment), so they may put pressure on
managers to make relatively conservative decisions.
© Neil Leslie/Getty Images
Simple and transparent, the maximincriterion is appropriate for
risk-
aversepeople making risky decisions and is aneffective way to a
void incurring losses.
As an example of the maximin decision rule, consider the choic
e between an investment Project A andan investment Project B.
For Project A the initial investment is $1 million in year 1 with
possible finaloutcomes in year 2 ranging from a loss of $200,00
0 to a profit of $5 million. Project B has an initialinvestment co
st of $2 million with possible outcomes in year 2 ranging from a
loss of $500,000 to aprofit of $10 million. For the maximin dec
ision rule, we simply compare the two minimum outcomes andth
erefore choose Project A because its worst outcome is a loss of
$200,000 compared to Project B’sworst outcome, which is a los
s of $500,000. If the worst outcome were to occur, the firm wou
ld bebetter 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 thatare not repeated enoug
h times to allow the law of averages to work out in the firm’s fa
vor. Thisdecision-
making rule is designed to be a simple and very transparent way
to avoid incurring losses thatcannot be tolerated by the firm an
d its shareholders. But, by refusing to consider the other potenti

16. aloutcomes it may be a very poor decision criterion. What if the
probability of Project B’s worst outcomeoccurring was only 10
% and the probability of Project A’s worst outcome was 40%? I
n that case, bytaking decision A the managers have chosen to ris
k a worst outcome with four times the chance ofoccurring than t
he worst outcome of Option B. Or, what if the other possible ou
tcomes for Project Awere positive but relatively small while the
other outcomes for Project B were positive and relativelylarge?
The maximin criterion does not consider these other outcomes a
t all.8 So let’s look at somedecision rules that do.
Coefficient of Variation Decision Rule
The coefficient of variation (CV) is a statistic of a probability d
istribution and is calculated as the ratio of the standard deviatio
n 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 =0.5; f
or 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 thedecision 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 butinferior to location B, thereby agreeing wit
h his certainty-equivalent-
based decision. For Mr. Y and Ms. Z, the CV rule would say the
tworemaining options are equal, but we saw that Mr. Y preferre
d option A (higher CE for him) while Ms. Z preferred option C (
higher CE for her).Thus, the CV criterion does not take into acc
ount the differing degrees of risk aversion that individuals may
have and is, therefore, an inferiordecision rule for individuals m
aking decisions when taking into account only their own risk–
return preferences. But for managers makingdecisions on behalf
of shareholders, the CV criterion may be more suitable because
some shareholders (like Mr. Y) will be less risk-
averse whileothers (like Ms. Z) will be more risk-
averse. On average, shareholders might be happy enough with th
e CV decision rule, and they can always selltheir share in this fi
rm and buy shares in a more (or less) conservative alternative b

17. usiness if they want to.
In Figure 2.3 we show the CV decision rule as it applies to the r
estaurant location decision problem. The CVs associated with d
ecision A and Care equal to 0.5 in both cases, and the CV associ
ated with decision B is 0.3. Notice that the slope of the CV line
s emanating from the origin(these lines are known as rays) are e
ach equal to the reciprocal of the CV value, since the slope is eq
ual to the rise (ENPV) over the run (SD),while CV is equal to th
e run (SD) divided by the rise (ENPV). Also note that in effect t
he CV rays are like indifference curves since everycombination
on a particular CV ray is equally preferred. These CV rays have
constant MRS between risk and return, but as we have seen,indi
viduals do not. Their risk–
return indifference curves are concave from above, exhibiting in
creasing MRS as more and more risk is taken on.As we saw in t
he case of our three restaurateurs, individual preferences might
agree with the CV criteria (as Mr. X did) or not. While both Mr.
Yand 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 fo
r individual decision making.
Figure 2.3: The coefficient of variation decision criterion
As a more complex application of the CV criterion, let us now r
econsider the investment Project A and Project B decision intro
duced above. InTables 2.2a and 2.2b we show the probability di
stribution of outcomes associated with these projects and calcul
ate the ENPV, SD, and CV foreach project (behind the scenes I
have used an Excel spreadsheet to calculate these numbers). Yo
u will see that I have assumed a discount rateof 10% and that th
e initial cost is paid at the end of year 1 while the possible outc
omes (cash inflows and outflows) are realized at the end ofyear
2. Whereas earlier we selected Project A using the maximin deci
sion criterion, by applying the CV criterion we find that Project
B ispreferred. Although it is riskier (SD = 1.4374 compared to 1
.0414), its ENPV is much higher ($3.5124 million compared to

18. $0.7603 million) suchthat the CV ratio is only 0.4092 for Projec
t B compared with 1.3697 for Project A.
Table 2.2a: Calculating the coefficient of variation for Project
A
(1) Initial Costyear 1(millions)
(2) Presentvalue of initialcost
(3) Year 2outcomes(millions)
(4) Present valueof year 2 outcomes
(5) Net presentvalue (millions)
(6) Probability ofyear 2 outcomes
(7) ENPV ofoutcomes(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

19. −2.2314
0.1
−0.2231
ENPV =
3.5124
SD =
1.4374
CV =
0.4092
Table 2.2b: Calculating the coefficient of variation for Project
B
(1) Initial costyear 1(millions)
(2) Presentvalue of initialcost

20. (3) Year 2outcomes(millions)
(4) Present valueof year 2 outcomes
(5) Net presentvalue (millions)
(6) Probability ofyear 2 outcomes
(7) ENPV ofoutcomes(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

21. ENPV =
0.7603
SD =
1.0414
CV =
1.3697
Thus, the CV decision criterion is an extension of the ENPV pro
fit-
maximizing rule and is appropriate when (1) outcomes are uncer
tain; (2) cashflows occur beyond the current time period; (3) si
milar decisions are not made repeatedly; and (4) managers are ri
sk-
averse (hopefully reflectingtheir shareholder’s preferences). Not
e that in this example the CV criterion agrees with the ENPV cri
terion but disagrees with the maximincriterion, which neglected
most of the information available and made the decision based s
imply on the best of the worst outcomes. The CVcriterion is thu

22. s a more sophisticated decision criterion that, while not generall
y suitable for individual decision making, does have value forde
cisions made by managers of firms where the decision made mu
st be justified to risk-
averse shareholders on some objective basis. Moreover,it can be
argued that in the context of managerial decision making withi
n the firm, the linear indifference rays implied by the CV criteri
on mightbe a sufficient approximation of shareholders’ preferen
ces in aggregate since these shareholders are free to build a port
folio of shares (indifferent companies) that best serves their ove
rall risk–
return preferences. The CV decision rule is transparent and easil
y communicable toshareholders. 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 sharesin another firm that better fits their risk preferenc
es. And, in any case, they can buy shares in a variety of firms s
uch that the overall riskexposure of their investment portfolio b
etter suits their risk and return preferences.
The ENPV Criteria Using Risk Premiums
Another commonly used method to adjust uncertain cash flows f
or risk is to adjust the discount factor to reflect the degree of ris
k. We do thisby adding a risk premium to the discount factor su
ch that projects with higher risk are discounted at higher opport
unity discount rates. A riskpremium is an additional amount that
is proportionate to the additional risk perceived. Recall that we
defined the opportunity discount factoras the rate of interest th
at could be earned on an alternative opportunity of equal risk.
When the decision alternatives are clearly not equallyrisky it fol
lows that their opportunity discount rate should not be the same
for each alternative.
© iStockphoto/Thinkstock
Government bonds are regarded as the ultimate risk-
free securitybecause repayment is absolutely certain.
At this point it is appropriate to examine the two main compone

23. nt parts ofthe opportunity discount rate (ODR). The first main p
art is the risk-
free rate ofreturn that one could earn on a loan that was absolut
ely certain to be repaid,for example, the purchase of governmen
t bonds. Although the governmentmay change from time to time
, the newly elected politicians would respectthe previous govern
ment’s obligation to repay lenders who had boughtgovernment b
onds, so government bonds are regarded as the ultimate risk-
free security.9 The risk-
free rate is made up of two subparts, the real rate ofinterest and
the premium for expected inflation. The real rate of interest isth
e rate that would cause the supply and demand for loanable fund
s to beequal in a market for funds without risk and without infla
tion. But whenlenders expect inflation to occur, they expect the
purchasing power of thefunds returned (after the loan is settled)
to be lower than the amount loaned.For example, if prices are e
xpected to rise by 5% over a year, the goods andservices that co
uld be purchased with $100 at the start of the year willprobably
cost about $105 at the end of the year. Thus, the inflation premi
umcharged needs to be about 5% to compensate the lender for th
e loss of purchasing power due to inflation, in this case where t
he expected rateof 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 tha
t the borrowermight default on the loan and not pay the money b
ack. To estimate the appropriate risk premium, the lender must
ask, "What is the probabilitythat the borrower will not repay the
loan?" To answer this question, the lender (like the insurance m
anager in Chapter 1) must consider whatproportion of people, in
roughly the same risky situations, have previously defaulted on
their loans. Suppose the answer is 20%. That meansthat 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.Be
cause the lender cannot tell in advance which one in every five
borrowers will be unable to repay the loan, the lender must set a
riskpremium on all loans that is high enough to allow the funds

24. received back (from borrowers who do in fact repay their loans
) to compensate thelender for the funds lost due to borrowers w
ho cannot repay the loan. In this case, since only four of the fiv
e are expected to repay, all will becharged a 25% risk premium t
o ensure that the four borrowers repaying the loan allow the len
der to recoup 4 x 25% = 100% of the loanadvanced to the borro
wer who ultimately defaults. The formula for the risk premium i
s thus the ratio of the probability of default (PD) to itscomplem
ent, the probability of repayment (PR). That is, PD/PR. In Table
2.3, we show the risk premiums for a range of default probabili
ties, andyou can see that the risk premium increases exponential
ly as the probability of default increases.
Table 2.3: Default risk and calculation of the applicable risk pre
mium
Probability of Default PD
Probability of Repayment PR
Risk Premium PD/PR
5%
95%
5.26%
10%
90%
11.11%
20%
80%
25.00%

25. 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 b
y finding the risk-per-dollar-of-
return and we selected Project B despite it being more risky (hig
her 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. Supposethat 10% was indeed the correct ODR for Project
A, being the real rate of interest (say 2%) plus an expected infla
tion (say 3%), plus a riskpremium of 5%. Also suppose that for
Project B the appropriate risk premium is about 15%, causing th
e ODR to be 20%. In Table 2.4 werecalculate the ENPV for Proj
ect B.
Table 2.4: Recalculating the ENPV for Project B, with ODR = 2
0%
(1) Initial CostYear 1 (millions)

26. (2) Presentvalue of initialcost
(3) Year 2outcomes(millions)
(4) Present value ofYear 2 outcomes
(5) Net presentvalue (millions)
(6) Probabilityof outcomes
(7) ENPV ofoutcomes(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

27. ENPV =
2.8125
SD =
1.2078
CV =
0.4295
Now compare the ENPVs of the two Projects A and B. Of cours
e the ENPV is still $760,300 for Project A, but it is now about $
2.8 million forProject B (down from $3.5 million) due to being
more heavily discounted. So, Project B is still the preferred alte
rnative using the risk-
adjustedENPV decision criteria. Also note that while the CV for
Project B has increased due to the higher ODR, the CV criterio
n also still favors Project Bover Project A.
A Simplification for More Complex Situations
In practice, we are typically confronted by more complex situati

28. ons than the above examples. Fortunately, we can simplify these
examples byassuming only three outcomes (high, medium, and
low) in each year and by assigning what seem to be reasonable "
guesstimates" of thedifferent monetary outcomes and of the pro
babilities of these different outcomes occurring. If these estimat
es are inaccurate, scrutiny by otherswho have different informat
ion will lead us to revise them to more accurately reflect the co
nsensus of opinion about what the values shouldmore likely be.
If we suppose that the possible outcomes are symmetric around t
he medium outcome each year, and also suppose the probability
distributionsare symmetric around the medium outcome, then a
useful simplification becomes possible. To find the ENPV of the
decision alternative weneed only add the medium NCF outcome
s in each year and subtract the initial cost outlay. This is becaus
e the high outcomes would be exactlyoffset by the low outcomes
when they are symmetric around the medium outcome. This sim
plification is hardly necessary for the relativelysimple two-
and three-
year time horizons that we have considered, but think about a de
cision with a five-year horizon—
with a NCF streamstretching over five years with high, medium,
and low outcomes in each year. This would involve 35 = 243 te
rminal branches on the decisiontree! Although one could build a
very large spreadsheet or write a computer program to do all th
e hard work, it is generally not necessary to doso. This simplific
ation will give a sufficiently robust indication of the ENPV as l
ong as there is not substantial asymmetry of the high, medium,a
nd low outcomes. For the most part, asymmetry one way (e.g., t
oward the high outcome) in one year will be offset by asymmetr
y the otherway (toward the low outcome) in another year. Next,
the future outcomes and their probabilities are estimates anyway
, and these estimates areincreasingly like guesswork in the "out
years" (i.e., beyond the present period), so it is false accuracy t
o place too much credence on the precisevalue we find for the E
NPV.
Thus, it is generally a sufficient approximation to consider only

29. the medium NCF outcome for each of the out years when the ti
me horizon isthree to five years or longer. For decision alternati
ves that have longer time horizons the "sum of the medium outc
omes" approach is likely tobe 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 mediumoutcome as being r
epresentative of the middle part of the probability distribution,
and similarly view the high outcome as being representativeof t
he upside-
risk side of the probability distribution and the low outcome as
being representative of the downside-
risk side of the probabilitydistribution. Thus, the point estimate
s of high, medium, and low should be viewed as representative
of the three regions of the probabilitydistribution.
The Normal Distribution
We can illustrate these scenarios in the context of the special ca
se of the normal distribution.11 The first property of a normal d
istribution isthat it is symmetric around its mean value. It is oft
en called a bell curve because it looks somewhat like an old-
fashioned bell, as displayed inFigure 2.4. As you would guess, t
he mean value of the normal distribution is also the median (or t
he 50th percentile) value. That is, 50% of theobservations lie ab
ove, and 50% lie below, the mean value. The standard deviation
of a normal distribution 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 distributi
on is thatthe bell curve is just tall enough to cause 99.7% of all
outcomes to lie within plus or minus three SDs from the mean.

30. Moreover, the shape ofthe bell curve is such that 95% of all out
comes will lie within plus or minus two SDs from the mean, and
68% will lie within plus or minus oneSD from the mean.
Now notice that the mean value (i.e., the ENPV) of the probabil
ity distribution effectively represents the middle part of the dist
ribution, or 68%of all outcomes. It is most likely (with 68% pro
bability) that the actual outcome will fall within the range of pl
us or minus one SD from themean outcome. With repeated trials
of the same decision we would expect the actual outcome to so
metimes be more than the mean, andsometimes less, such that th
e 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 extrem
e right-
hand side of the probabilitydistribution. Instead it is representat
ive of the range of outcomes that are more than one SD above th
e mean. In Figure 2.4, we have depictedthe 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 SDa
bove the mean. Similarly, the worst-
case scenario is not the worst possible outcome at the extreme l
eft of the probability distribution, butrepresents all the outcome
s that have values more than one SD below the mean outcome, s
o we position it at approximately the point thatbisects the area u
nder the curve to the left of the outcome that is one SD below th
e mean outcome.
Figure 2.4: The properties of a normal distribution
Since the most-
likely scenario (MLS) represents 68% of the outcomes (when th
e outcomes are normally distributed) the best-
case scenario (BCS)must represent half of the remainder (i.e., 1
6%) and the worst-
case scenario (WCS) must represent the other half of the remain
der (16%). Beaware that these specific probabilities for the high
, medium, and low outcomes may or may not be appropriate for

31. particular business decisionproblems. If you have information t
hat indicates that the outcomes seem likely to be approximately
normally distributed around the mean, thenthese probabilities w
ill be appropriate. On the other hand, you might have informatio
n that indicates that the probability distribution isdefinitely not
normally distributed and so you should use the probabilities that
seem to be more appropriate.
Skewness of the Probability Distribution
© Biwa Studio/Getty Images
Skewness refers to the degree ofasymmetry of the probability di
stribution.A distribution that is perfectly symmetricis said to be
nonskewed.
Skewness refers to the degree of asymmetry of the probability d
istribution. A distribution that isperfectly symmetric is said to b
e nonskewed. But if the bell shape is distorted with more outco
mes lyingto the left of the mean outcome, with a longer tail stre
tching out to the right-
hand side, the distributionis said to be positively skewed. In this
case the median outcome (the 50th percentile outcome) will liet
o the left of (below) the mean outcome. The modal outcome, wh
ich is the single outcome with thehighest probability of occurrin
g, will lie to the left of both the mean and the median outcome,
as shownin Figure 2.5a. A negatively skewed distribution will h
ave the bulge on the right-
hand side of thedistribution with a long tail to the left. The med
ian outcome will lie to the right of the mean outcome,and the m
odal outcome will lie to the right of both the mean and the medi
an outcomes, as shown inFigure 2.5b.
Figure 2.5a: Positively skewed probability distribution
Figure 2.5b: Negatively skewed probability distribution
The implication of positive skewness for managerial decision m
aking is that while the majority of the possible outcomes will be
below the mean,there will be a significant number of upside-

32. risk outcomes that lie more than three standard deviations above
the mean (ENPV). Repeated trialsof such decisions would not g
enerate an average outcome equal to the mean outcome, but wou
ld 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 bedifferent to the normal distribution, of co
urse. Something like 10%, 60%, and 30% might be more approp
riate for the worst-case, most-likely-case, and the best-
case scenarios, respectively.
Conversely, when the distribution is negatively skewed, the maj
ority of the possible outcomes will be above the mean, but there
will besignificant number of downside-
risk outcomes that lie more than three standard deviations below
the mean. Repeated trials of decisions withnegatively skewed d
istributions would tend to result in an average outcome that is a
bove the weighted mean (ENPV) outcome. The probabilitiesasso
ciated with the worst-case, most-likely scenario, and best-
case scenario would be different to the normal distribution, of c
ourse. Somethinglike 30%, 60%, and 10% might be more approp
riate 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 di
stribution, specifically its height. A distribution that is taller tha
n a normaldistribution is said to be leptokurtic and would have
more than 68% of the outcomes falling within one SD each side
of the mean. Conversely, adistribution that is platykurtic (like a
plate, i.e., flatter) would have less than 68% of the outcomes wi
thin one SD each side of the mean.Kurtosis of probability distri
butions is demonstrated in Figure 2.6.
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 outcomeslying within one SD from the mean will be substanti

33. ally above 68%, perhaps 80% in the example shown. This means
that the probabilities of thebest-case and worst-
case scenarios are relatively small, about 10% each in the situati
on depicted. Conversely, for the platykurtic distribution, thepro
portion 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 inFigure 2.6.
What is the point of all this for the managerial decision maker?
First, the decision maker needs to consider whether the probabil
ity distributionof outcomes is likely to be approximately normal
or not. Remember that many decision situations are likely to de
liver approximately normaldistributions of outcomes since the i
ndependent actions of many people (e.g., buyers) typically resul
t in a normal distribution. Second, if thedecision maker has reas
on to believe that the distribution will be skewed to one side or
the other, or taller or flatter than a normal distributiondue to fac
tors that he or she suspects are characteristic of the population o
r sample, the probabilities need to be adjusted in the direction t
hatreflects the decision maker’s best estimate of the actual shap
e of the probability distribution. In the absence of any informati
on to indicate thatthe normal distribution is not appropriate, it i
s usually a good first approximation to assume normality of the
distribution. It is useful toremember that unless you have data o
n the distribution of prior outcomes of similar decisions, assigni
ng probabilities to possible outcomes isan art, not a science—
the decision maker needs to think about the situation and go wit
h his or her best guesses. As a decision maker, it isuseful to che
ck your own best guesses against the opinions of others who hav
e knowledge of the decision scenario. Their scrutiny may reveal
information that was not known to you and allow a more accurat
e probability distribution to underlie your decision. And finally,
if this type ofdecision scenario is repeated again and again, dat
a will build up and the probability distribution can be corrected

34. subsequently.