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Managerial economics strategy by m perloff and brander chapter 14 decision making uncertainty

Chapter 14
Managerial
Decision Making
Under
Uncertainty


Table of Contents


14.1 Assessing Risk



14.2 Attitudes Toward Risk



14.3 Reducing Risk




14.4 Investing Under Uncertainty



14.5 Behavioral Economics and Uncertainty

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Introduction
• Managerial Problem



The 2010 BP oil spill may cost the firm above $40 billion. However, there is a 1990
law that capped cleanup costs up to $75 million for a rig spill.
How does a cap on liability affect a firm’s willingness to make a risky investment or
to invest less than the optimal amount in safety? How does a cap affect the amount
of risk that the firm and others in society bear? How does a cap affect the amount of
insurance against the costs of an oil spill that a firm buys?

• Solution Approach


We need to focus on how uncertainty affects consumption decisions made by
individuals and business decisions made by firms.

• Empirical Methods





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Probability, expected value and variance are tools for assessing risk.
Depending on their attitudes about risk, people and firms try to reduce risk.

Investing under uncertainty requires assessing and reducing risk.
How psychological factors influence risk assessment is evaluated by behavioral
economics.

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14.1 Assessing Risk
• A Risk Problem
– Gregg, a promoter, is considering whether to schedule an outdoor concert
on July 4th. Booking the concert is a gamble: He stands to make a tidy
profit if the weather is good, but he’ll lose a substantial amount if it rains.

• Quantifying Risk
– This particular event has two possible outcomes: either it rains or it does
not rain.
– To schedule the concert, first, Gregg quantifies how risky each outcome is
using a probability. Second, he uses these probabilities to determine
expected earnings.

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14.1 Assessing Risk
• Probability
– Probability: number between 0 and 1 that indicates the likelihood that a
particular outcome will occur.
– If an outcome cannot occur, probability = 0. If the outcome is sure to
happen, probability = 1. If it rains one time in four on July 4th, probability =
¼ or 25%.
– Weather outcomes are mutually exclusive (either it rains or it doesn’t) and
exhaustive (no other outcome is possible). So probabilities must add up to
100%.

• Calculating Probability using Frequency
– Gregg has data about raining on July 4th : number of years that it rained (n)
and the total number of years (N).
– Frequency: Ө = n/N

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14.1 Assessing Risk
• Using Subjective Probability
– If Gregg wants to refine the observed frequency, he can use a weather
forecaster’s experience to form a subjective probability: best estimate of
the likelihood that the outcome will occur—that is, our best, informed
guess.
– Subjective probability may also be used in the absence of frequency data.

• Probability Distributions
– A probability distribution relates the probability of occurrence to each
possible outcome.
– Panel a of Figure 14.1 shows a probability distribution over five possible
outcomes: zero to four days of rain per month in a relatively dry city.
– These weather outcomes are mutually exclusive and exhaustive, so
exactly one of these outcomes will occur, and the probabilities must add
up to 100%.

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14.1 Assessing Risk
Figure 14.1 Probability Distributions

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14.1 Assessing Risk
• Expected Value, EV = Pr1V1 + Pr2V2 + … + PrnVn
– The expected value, EV, is the weighted average of the values of the
outcomes, sum of the product of the probability and the value of each
outcome.
– Concert and rain example: EV = [0.5 *15]+[0.5*(-5)] = 5
– Gregg will earn an average of 5 per concert over a long period of time.

• Variance, σ2= Pr1(V1 – EV)2 + Pr2(V2 – EV)2 + … + Prn(Vn – EV)2
– Variance, σ2: measures the spread of the probability distribution;
probability-weighted average of the squares of the differences between
the observed outcome and the expected value.
– Concert and rain example: Variance = [0.5 (15-5)2] + [0.5 (-5-5)2] = 100
– Another measure of risk is the standard deviation, σ: square root of the
variance.
– For the outdoor concert, the values are σ2 = 100 and σ = 10.

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14.2 Attitudes Towards Risk
• Expected Utility: EU = Pr1 x U (V1) + Pr2 x U (V2) + … + Prn x U
(Vn)
– Expected utility: probability-weighted average of the utility from each
possible outcome.
– For example, Gregg’s EU = [0.5 U(15)] + [0.5 U(-5)]
– EU and EV have similar math form. The key difference is that the EU
captures the tradeoff between risk and value, whereas the EV considers
only value.



Fair Bet and Risk Attitudes
– We can classify people based on their willingness to make a fair bet: a bet
with an expected value of zero.
– Someone who is unwilling to make a fair bet is risk averse.
– A person who is indifferent about making a fair bet is risk neutral.
– A person who will make a fair bet is risk preferring.

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14.2 Attitudes Towards Risk
• Risk Aversion




A concave utility function shows diminishing marginal utility of wealth: The extra
pleasure from each extra dollar of wealth is smaller than the extra pleasure from the
previous dollar.
A concave utility function shows risk aversion: unwillingness to take a fair bet.

• Unwillingness to Take a Fair Bet





Irma’s utility function is concave, her subjective probability is 50% (Figure 14.2).
Irma can keep $40 (secure action) or buy a stock (riskier action). Buying the stock is
a fair bet because its EV = (0.5 x $70 + 0.5 x $10) = $40 (same as keeping $40).
However, Irma does not accept the fair bet (risk averse). The secure utility at point b
is higher than EU of buying the stock at point d (120 > 105).
A risk averse person picks the less risky choice if both choices have the same EV.

• Using Calculus: Diminishing Marginal Utility of Wealth



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Irma’s utility from Wealth is U(W); Marginal Utility = dU(W)/dW > 0
Diminishing marginal utility cocondition (concavity): d 2U(W)/dW2 < 0

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14.2 Attitudes Towards Risk
Figure 14.2 Risk Aversion

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14.2 Attitudes Towards Risk
• Risk Premium
– The risk premium is the maximum amount that a decision-maker would
pay to avoid taking a risk.
– Equivalently, the risk premium is the minimum extra compensation
(premium) that a decision-maker would require to willingly incur a risk.

• Risk Premium Calculation
– Risk premium: the difference between the expected value of the uncertain
prospect and the certainty equivalent
– Certainty Equivalent: if held with certainty, would yield the same utility as
the uncertain prospect (point e in Figure 14.2).
– The risk premium in Figure 14.2 is 14 (40 – 26).
– Irma would accept to buy the stock (fair bet) if offered $14 to do it.

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14.2 Attitudes Towards Risk
• Risk Neutrality
– A risk neutral person is indifferent about taking a fair bet.
– Such a person has a constant marginal utility of wealth: Each extra dollar
of wealth raises utility by the same amount as the previous dollar.
– The utility function is a straight line in a graph of utility against wealth.
– As a consequence, a risk-neutral person’s utility depends only on wealth
and not on risk.

• Risk Neutrality Calculation
– In panel a of Figure 14.3, Irma’s utility is a straight line, constant marginal
utility of wealth (35/30).
– Her expected utility of a fair bet (between points a and c) exactly equals
her utility with certain wealth of 40 (at point b).
– A risk-neutral person chooses the option with the highest expected value,
because maximizing expected value maximizes utility. The risk premium is
zero.

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14.2 Attitudes Towards Risk
Figure 14.3 Risk Neutrality and Risk
Preference

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14.2 Attitudes Towards Risk
• Risk Preference
– An individual with an increasing marginal utility of wealth is risk preferring:
willing to take a fair bet.
– Increasing marginal utility of wealth: the extra pleasure from each extra
dollar of wealth is bigger than the extra pleasure from the previous dollar.
– The utility function is convex.

• Risk Preference Calculation and Risk Premium
– If Irma has the utility function in panel b of Figure 14.3, she is risk
preferring. Her expected utility from buying the stock, 105 at b, is higher
than her certain utility if she does not buy the stock, 82 at d. Therefore,
she buys the stock.
– A risk-preferring person is willing to pay for the right to make a fair bet (a
negative risk premium). Irma would like to pay no more than $18 (40 –
58).

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14.2 Attitudes Towards Risk
• Risk Attitudes of Managers
– Some shareholders want managers to be risk neutral and to maximize
expected profit. However, managers may make risk-averse or riskpreferring decisions.

• Managers Act in Their Own Interests (Principal-Agent Problem)
– If a manager is worried about being fired if the firm has large losses, the
manager may act to avoid the possibility of such losses. The firm is made
to pay a risk premium (reduced expected profit) to avoid an extreme risk
of losing money.
– If the manager’s compensation is based on the firm’s short-run profit and
there is no penalty in the event of a very bad outcome, the manager may
prefer risk.

• Shareholders May Want Managers to be Risk Averse
– If bankruptcy would impose large costs on shareholders, they might prefer
that managers try to avoid bankruptcy even at the expense of reduced
expected profit.

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14.3 Reducing Risk
• All Individuals and Firms Want to Reduce Risk
– Risk-averse, risk-neutral, and risk-preferring people all reduce risk of
unfair bets.
– Individuals can avoid optional risky activities, but often they can’t escape
risk altogether.

• Abstaining as an Easy Way of Reducing Risk
– The simplest way to avoid risk is to abstain from optional risky activities
(lottery, stock market, high-risk jobs).
– Even when you can’t avoid risk altogether, you can take precautions to
reduce the probability of bad events or the magnitude of any loss that
might occur (closing doors at home, installing fire alarms, servicing a car
regularly).

• Strategies to Reduce Risk
– Three most common ways: obtaining information, diversification, and
insurance

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14.3 Reducing Risk
• Obtaining Information
– Collecting accurate information before acting is one of the most important
ways in which people can reduce risk and increase expected value and
expected utility.

• An Application to Bond Ratings
– Investors look at Moody’s and Standard & Poor’s ratings. Their letter-grade
ratings reflect whether a bond’s issuer has made timely payments in the
past, whether the issuer is in danger of becoming bankrupt, and other
problems.
– Investment grade bonds (Moody Baa through Aaa or Standard and Poor
BBB- through AAA) are said to be suitable for purchase by an institutional
money manager (legally obligated to be a prudent investor).
– Lower-ranked bonds (Moody C through Ba and Standard and Poor D
through BB+) are riskier and must offer higher rates of return to attract
buyers.
– The lowest ranked, junk bonds are generally issued by new firms.

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14.3 Reducing Risk
• “Don’t Put All Your Eggs in One Basket”
– This common advice is called technically risk pooling or diversification

• Correlation and Diversification
– How much diversification reduces risk depends on the degree to which the
payoffs of various investments are correlated (move in the same direction).
– If the performances of two investments move independently—do not move
together in a predictable way—their payoffs are uncorrelated.

• Perfect Negative Correlation to Eliminate Risk
– Diversification can eliminate risk if the returns to two investments are perfectly
negatively correlated.
– Case: Two firms compete for a government contract with equal chance of
winning. You can buy 2 shares of stock in either firm for $20. The stock of the firm
that wins the contract will be worth $40, whereas the stock of the loser will be
worth $10 (perfect negative correlation). Why diversify?
– Buying two shares of the same stock: EV = 0.5 x 80 + 0.5 x 20 = 50 and σ2 = 900
– Buying one share of each firm: EV = 0.5 x 50 + 0.5 x 50 = 50 but σ2 = 0. Zero
risk.

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14.3 Reducing Risk
• No Correlation and Risk Reduction
– Diversification reduces risk even if the two investments are uncorrelated
or imperfectly positively correlated, but cannot be eliminated.
– Case: Same case as before but whether one firm wins (W) a contract does
not affect whether the other firm loses (L). There are 4 scenarios (WW, WL,
LW, LL), each with probability ¼ or 25%.
– Buying one share of each firm: EV = 0.25 x 80 + 0.5 x 50 + 0.25 x 20 =
50, σ2 = 450. Same EV but the risk has been reduced by half.

• Positive Correlation and Risk Reduction
– Diversification can reduce risk even if the investments are positively
correlated provided that the correlation is not perfect.
– Diversification does not reduce risk if two investments have a perfect
positive correlation.
– Case: Same case as before but both firms win or lose together (perfectly
positively correlated). The EV and the σ2 are the same whether you buy
two shares of one firm or one share of each firm.

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14.3 Reducing Risk
• Diversification Through Mutual Funds
– One way to effectively own shares in a number of companies at once and
diversify is by buying shares in a mutual fund. For instance the S&P 500,
which is a value-weighted average of 500 large firms’ stocks.
– Mutual funds allow investors to reduce the risk associated with
uncorrelated price movements across stocks. Random, firm-specific risks
are reduced.

• Diversification Cannot Eliminate Systemic Market Risk
– A stock mutual fund has a market-wide risk, a risk that is common to the
overall market. Prices of almost all stocks tend to rise when the economy
is expanding and to fall when the economy is contracting.
– No investor can avoid the systematic risks associated with shifts in the
economy that have a similar effect on most stocks even if you buy a
diversified stock mutual fund. Even the global economy has a global
market-wide risk.

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14.3 Reducing Risk
• Insurance
– Many individuals and firms buy insurance to shift some or all of the risk they
face to an insurance company. Global insurance revenues exceeded $4.59
trillion in 2011, approximately 6% of world GDP.
– A risk-averse person or firm pays an insurance premium to the insurance
company in exchange for an amount of money if a bad outcome occurs.

• Determining the Amount of Insurance to Buy
– Case: Scott is risk averse, wants to insure his store that is worth 500. Probability
of fire is 20%. If a fire occurs, the store will be worth nothing.
– With no insurance, the EV = 0.8 x 500 + 0.2 x 0 = 400, and σ2 = 10,000
– Fair Insurance: a contract between an insurer and a policyholder in which the
expected value of the contract to the policyholder is zero (fair bet).
– With fair insurance, for every 1 dollar of insurance premium paid, the company
will pay Scott 5 dollars to cover the damage if the fire occurs, so that he has 1
dollar less if the fire does not occur, but 4 dollars more if it does occur.
– Scott’s premium x to make his wealth 400 in either case: 500 – x = 4x, x = 100.
– With insurance, the EV = 400 and σ2 = 0. Zero risk.

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14.3 Reducing Risk
• Fairness and Insurance
– Because insurance firms have operating expenses, we expect that realworld insurance companies offer unfair insurance: the expected payout to
policyholders is less than the premiums paid by policyholders.
– A risk-averse person fully insures if offered a fair insurance. If not, buy
less.
– A monopoly insurance company could charge an amount up to the risk
premium a person is willing to pay to avoid risk. If there were more firms,
it would be lower but still enough to cover operating expenses.

• Insurance and Diversifiable Risks
– By pooling the risks of many people, the insurance company can lower its
risk much below that of any individual.
– Insurance companies generally try to protect themselves from insolvency
(going bankrupt) by selling policies only for risks that they can adequately
diversify.
– Because wars are nondiversifiable risks, insurance companies normally do
not offer policies insuring against wars.

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14.4 Investing Under Uncertainty
• Risk-Neutral Investing
– Chris is a risk-neutral monopoly owner. Because she is risk neutral, she
invests if the EV of the firm rises due to the investment. Any action that
increases her EV must also increase her EU because she is indifferent to
risk.
– In panel a of Figure 14.4, if Chris does not open the new store, she makes
$0. If she does open the new store, she expects to make $200 (thousand)
with 80% probability and to lose $100 (thousand) with 20% probability.

• Solution
– Chris decides whether to invest or not at the decision node (rectangle).
The circle, a chance node, denotes that a random process determines the
outcome.
– Chris decides to invest because EV = 140 > 0. She prefers an EV of 140 to
a certain one of 0.

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14.4 Investing Under Uncertainty
Figure 14.4
Investment
Decision Trees
with Uncertainty

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