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Decisions Under Risk and Uncertainty
Chapter 11
• Very few business decisions involve certainty in which an action
invariably leads to a specific outcome.
• Most decisions involve a gamble, where the probabilities can be
known or unknown and outcomes can be known or unknown
• Risk – exits whenever all the possible outcomes and the
probabilities on these outcomes are known
» e.g., Roulette Wheel or Dice
• Uncertainty – exits whenever the possible outcomes or the
probabilities are unknown
» e.g., Drilling for oil in an unknown field
• Unknowingness – when each action has unknown
consequences so that the outcomes cannot be determined.
» e.g., the chance and impact of a worldwide nuclear war
2005 South-Western College Publishing
Slide 1
Subjective and Informal Decision Approach
• Often, probabilities and outcomes are not
precisely measurable; they are more subjective.
• Managers who must select between two
mutually exclusive projects of about the same
net returns will tend to pick the one that "feels"
less risky.
• When one investment has substantially higher
expected return and higher risk, decision-makers
must subjectively determine if the additional risk
is offset sufficiently by the higher return.
» These trade-offs can be seen in two continuous probability
distributions for risky project B, and less risky project A.
Slide 2
Continuous Probability
Distributions
• Expected valued is the mode for symmetric distributions
Project A
Project B is riskier,
but it has a higher
expected value at
rB than does Project A.
Project B
Expected Values 
-r
A
-r
B
See also Figure 2.2, page 42
Slide 3
Methods to Compare Different Risks
1.
2.
3.
4.
5.
1.
Expected Marginal Utility Approach
Prospect Theory
Decision Tree Approach
Risk Adjusted Discount Rate Approach
Simulation Approach
EXPECTED MARGINAL UTILITY APPROACH
• Dollar payoffs don’t always measure the pain or joy
of outcomes
• An example of how people seem not to use purely the
money payoffs is the St. Petersburg Paradox.
Slide 4
The St. Petersburg Paradox
• The St. Petersburg Paradox is a gamble of
tossing a fair coin, where the payoff doubles for
every consecutive heads that appears.
• The expected monetary value of this gamble is:
$2·(.5) + $4·(.25) + $8·(.125) + $16·(.0625) + ...
= 1 + 1 + 1 + ... =  or infinity.
• But no one would be willing to wager all he or
she owns to get into this bet.
• It must be that people make decisions by criteria
other than maximizing expected monetary payoff.
Slide 5
Expected Utility
• When P's are probabilities and R's are outcomes,
then the expected monetary payoff is:
E(U) = Si=1 Pi·Ri
[19.1]
• If each monetary outcome, Ri , were assigned
a utility (or happiness) value, Ui = U(Ri), then
an alternative objective is expected utility
maximization, which is E(U).
• In the St. Petersburg paradox, the utility of
money rises at a declining rate. utility
money
Slide 6
Payoff Table for Investment Problem
Figure 19. 1
• An investor can choose to
invest or not
• The state of nature may
end up being successful
or not successful.
• The expected monetary
payoff for investing is
zero
States of Nature
Success Failure
Invest 160,000 -40,000
Not
Invest
0
0
Prob. .20
of
occurrence
.80
» .2(160,000)+.8(-40,000) = 0
• The expected monetary
payoff for not investing is
also zero.
» .2(0) + .8(0) = 0
Slide 7
Utility Function with
• Very likely, the impact of
Diminishing Marginal Utility losing $40,000 and gaining
Figure 19.2
Utility
.375
Money
-40,000
+160,000
-.50
$160,000 are treated
differently by investors.
• If the utility of $160,000 is
.375, and the utility of
-40,000 is -.50, then the
expected utility of investing
is:
E(U) = .2(.375) + .8(-.50) = -.325
• Hence, the investment
should not be taken
according the expected
utility approach.
• This is a risk averse person.
Slide 8
Utility Function with
• Suppose instead that
Increasing Marginal Utility utility increased at an
Figure 19.3
Utility
.65
Money
-40,000
+160,000
-.25
increasing rate, as in
Figure 19.3.
• If the utility of $160,000
is .65, and the utility of
-$40,000 is -.25, then
expected utility of
investing is:
E(U) = .2(.65) + .8(-.25) = +.05
• The investment should be
taken according the
expected utility approach.
• This person is a risk
preferrer or a risk lover.
Slide 9
Utility Function with
• If the utility function is a
Constant Marginal Utility
straight line, then the
Figure 19.4
Utility
.50
Money
-40,000
+160,000
-.125
marginal utility of money
is constant.
• If the utility of $160,000
is .50, and the utility of
-$40,000 is -.125, the
expected utility of
investing is:
E(U) = .2(.5) + .8(-.125) = -.0
• In this case, whether to
invest or not has the same
payoff.
• When the marginal utility
of is constant, the person
is said to be risk neutral.
Slide 10
Should You Move
Your Firm to Mexico?
• If successful, the payoff is $10 million with a 68%
chance of success.
• If unsuccessful, the payoff is a loss of $8.8 million.
• The utility of success and failure, respectively, is:
+2000 and -4800.
• The expected value of the Mexican move is .68(10
million) + .32(-8.8 million) = +$3.984 million.
• But the expected utility of the move is .68(2000) +
.32(-4800) = - 176.
• According to expected marginal utility approach, the
firm should not and will not make the move because
it leads to a negative expected utility.
Slide 11
2. Prospect Theory
• Sometimes investors are both risk averse (so that
they purchase insurance) and risk preferring
(since they like to go to casinos).
• Kahneman and Tversky have suggested that
people are risk averse in positively viewed
gambles and risk preferring when the gamble is
viewed as losing.
• Prospect Theory draws an S-shaped utility
function as in Figure 19.5.
Slide 12
Prospect Theory &
Automobiles
Figure 19.5
Utility
140
100
LX
SE
17,000 22,000 27,000
-160
Base
minivan
• Suppose a buyer of a minivan
views the SE as the starting
point.
• The base minivan is viewed as
a decline in status
• An LX and a Town & Country
are improvements
Town &
Country • For $5,000 more than the SE,
the buyer gets 100 utiles
Money • For $10,000 more, she gets
only 140 more utiles
32,0000
• For $5,000 less than the SE,
the buyer loses 160 utiles
• Offering a full-line of models,
the buyer is likely to stay at
SE or move up to the LX
Slide 13
Prospect Theory & Full Line Forcing
• If a firm offers two qualities of its product (good and better), the
buyer picks between them. Let’s suppose 50:50
• If a firm offers third quality (good, better, best), more of the buyers
move up, to better and some to best.
• Also as people move through their life cycle, the tend to trade up to
the best.
» In Figure 19.5 on minivans, suppose that utility of the SE is 320.
» A move down to the base model loses 160 out 320, or half the utility for a
savings of $5,000. This doesn’t seem worth it.
» A move up to the LX gains 100 utiles beyond 320, or a 31% increase in
utility for just $5,000.
• Prospect Theory also explains why CEOs like to lump all of the
bad news in one quarterly report. Two separate notices of bad
news is worse than one announcement of both bad outcomes.
Slide 14
3. Decision Trees
• A decision tree helps to consider the various
outcomes over time.
• Each diamond is a decision node, and circles with N
are used to designate the impact of the state of nature
on that decision.
• Suppose that you must decide whether or not to
invest using the same payoffs as Figure 19.1 (also on
slide 7).
• This can be written in a decision tree.
Slide 15
Decision Trees
PROB
NCF
EMV=P•NCF
Success .20
$160,000
$32,000
Failure
-$40,000
-$32,000
.20
N
Invest
.80
.80
1
$0
Do
Not
Invest
.20
Success .20
$0
$0
.80
Failure
$0
$0
N
.80
This is a one period decision tree, and neither decision is better.
$0
Slide 16
Multi-period Decision Trees
• Over time, there are often interactions among outcomes
• Suppose that you want to start a business of selling ice
cream bars from a cart in your campus union
• Suppose that the initial cost for the cart and a license to
sell on campus is $7,000
• The first year probability is 60% that you do well and
40% that you do poorly. The second year, your
probabilities changed, based on whether your cart was
popular or not.
• In this example, you should open a mobile cart, as the
expected NPV is +$1,380.
Slide 17
Decision Trees
PROB
NCF’s
PV(10%)
$7,000 .48
-7000; 5000; 7000
$3,330
.20
$6,000 .12
-7000; 5000; 6000
$2,504
.40
$7,000 .16
-7000; 1000; 7000
-$306
$5,000
-7000; 1000; 5000
-$1959
.80
$5,000
.60
Open Cart
.40
$1,000
.60
.24
Expected NPV = .48 (3330) +.12 (2504) +.16 (-306) +.24 (-1959) = $1,380
Slide 18
4. Risk Adjusted Discount Rates
• Riskier projects should be discounted at higher discount
rates
NPV = S NCF t / ( 1 + k*) t -NINV
where k* varies with risk and NCFt are cash flows.
• The risk premium is the difference between the
risk-adjusted rate and the firm's cost of capital
(k* - k).
A
• k*A< k*B as B is riskier
B
Slide 19
Sources of Risk Adjusted Discount Rates
• Market-based rates
» The magnitude of k*depends on the project.
» Sometimes information on costs of capital in
industries similar to this project can be used.
» Look at equivalent risky projects, use that
rate
» Is it like a Bond, Stock, Venture Capital?
» For totally new product lines; however, the
selection of the appropriate risk-adjusted
discount rate is subjective. Analysts should
be aware that those who advocate the project
would want a low discount rate, whereas
those who are opposed will feel that a higher
rate is warranted.
• Capital Asset Pricing Model (CAPM)
» Project’s “beta” and the market return
Slide 20
5. Simulation Approach
• The risks in a decision come from variability of the
number of units sold, the price, and costs. With
computers, it is possible to estimate cash flows using
probability distributions and find the NPV, and to do it
over and over again.
NCFt = [q(p) – q(c+s) – D](1-t) + D [19.4]
» where q(p) is revenue (quantity times price);
» q(c+s) is cost (both for production, c, and for selling, s;
» and D and t are depreciation and taxes, respectively.
• If the NPV of each simulation is predominately in the
positive range, then the project should be undertaken.
If the NPV are mostly negative, then the project
should be avoided.
Slide 21
Decision Making Under Uncertainty
• When the probabilities are
unknown, we cannot calculate
States of Nature
expected values or standard
deviations.
S1
S2
• The solution is to use the
A1
10
7
maximin criterion.
• For each alternative action, the
manager determines the worst
A2
8
12
outcome in each state.
• The manager takes the action that
The manager should pick A2,
is the best among the set of worst
because the worst thing for
possible outcomes.
alternative A1 is 7 whereas the • To pick between alternatives A1
worst thing for alternative A2
and A2, two states of nature
is 8.
present themselves (S1 and S2).
PAYOFF MATRIX
Slide 22
Minimax Regret Criteria
REGRET TABLE
States of Nature
S1
S2
A1
0
-5
A2
-2
0
To minimize the maximum loss,
the manager would select A2,
since -2 is smaller than -5.
In this case, the maximin strategy
and the minimax regret criterion
lead to the same result, but it need
not always be so.
• The same decision can be
transformed into regret, called the
minimax regret criterion.
• Regret is measured as the
difference between the best
possible payoff and the actual
payoff.
• If S1 were the true state of the
world, the loss would be 2 if you
picked A2 and no loss if you
picked A1.
• If S2 were the true state of the
world, the loss would be 5 and no
loss if you picked A2.
Slide 23
Managing Risk and Uncertainty
• In risky situations, managers tend to ask for more
information.
• In part, this is a stall for time. But this may mean
trying to "test market" the product to learn more.
• It may mean hiring the expert opinion of others.
• Firms hire outside lawyers, accountants,
economists, and banking experts to gauge the
situation.
• Credit rating services, such as Moody's, sell
information to manage risk.
Slide 24
Diversification
• The expected return on a portfolio is the weighted
average of expected returns in the portfolio.
• Portfolio risk depends on the weights, standard
deviations of the securities in the portfolio, and on the
correlation coefficients between securities. The risk of a
two-security portfolio is:
sp = (WA2·sA2 + WB2·sB2 + 2·WA·WB·rAB·sA·sB )
• If the correlation coefficient, rAB, equals one, no risk
reduction is achieved.
• Whenever rAB < 1, then sp < wA·sA + wB·sB. Hence,
portfolio risk is less than the weighted average of the
standard deviations in the portfolio.
Slide 25
Diversification Example
• Suppose that the two assets (A & B) both have a
standard deviations of 20% and are both have expected
returns of 10%.
• Suppose also the correlation coefficient (rAB ) is +.5
• If you had 100% in asset A, the expected return is 10%
and the standard deviation would be 20%.
• If you had 50% of your assets in A and 50% in asset B,
the expected return is still 10%.
• But the portfolio risk would be:
sp = (.502·.202 +.502·.202 +2·.50·.50·.50·.20·.20) = .173
• While the return of this portfolio is still 10%, its risk
has been reduced to a standard deviation of 17.3%
Slide 26
Other Approaches for Managing
Risk & Uncertainty
• Hedging – limiting risk by making an offsetting
investment. Investing in securities that are negatively
correlated provides hedging benefits.
• Insurance – pay a premium to avoid bad outcomes
(fire, theft, accidents by chief officers, etc. )
• Gaining control over the operating environment –
invest in suppliers to assure continuance of service
• Limited use of firm-specific assets – nonredeployable assets are riskier.
• Scenario Planning – being ready for the unexpected
and determining your response to it
Slide 27