Managerial Economics
Unit 9: Risk Analysis
Rudolf Winter-Ebmer
Johannes Kepler University Linz
Winter Term 2015
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Objectives
Explain how managers should make strategic decisions when faced
with incomplete or imperfect information
◮
◮
Study how economists make predictions about individual’s or firm’s
choices under uncertainty
Study the standard assumptions about attitudes towards risk
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Management tools
Expected value
Decision trees
Techniques to reduce uncertainty
Expected utility
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Uncertainty
Consumer and firms are usually uncertain about the payoffs from their
choices.
Some examples . . .
Example 1: A farmer chooses to cultivate either apples or pears
◮
When she takes the decision, she is uncertain about the profits that she
will obtain. She does not know which is the best choice.
◮
This will depend on rain conditions, plagues, world prices . . .
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Uncertainty
Example 2: playing with a fair dice
◮
We will win A
C2 if 1, 2, or 3
◮
We neither win nor lose if 4, or 5
◮
We will lose A
C6 if 6
Example 3: John’s monthly consumption:
◮
A
C3000 if he does not get ill
◮
A
C500 if he gets ill (so he cannot work)
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Lottery
Economists call a lottery a situation which involves uncertain payoffs:
◮
Cultivating apples is a lottery
◮
Cultivating pears is another lottery
◮
Playing with a fair dice is another one
◮
Monthly consumption another
Each lottery will result in a prize
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Risk and probability
Risk: Hazard or chance of loss
Probability: likelihood or chance that something will happen
The probability of a repetitive event happening is the relative
frequency with which it will occur
◮
probability of obtaining a head on the fair-flip of a coin is 0.5
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Probability
Frequency definition of probability: An event’s limit of frequency in a
large number of trials
◮
Probability of event A = P(A) = r /R
⋆
R = Large number of trials
⋆
r = Number of times event A occurs
Rules of probability
◮
Probabilities may not be less than zero nor greater than one.
◮
Given a list of mutually exclusive, collectively exhaustive list of the
events that can occur in a given situation, the sum of the probabilities
of the events must be equal to one.
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Probability
Subjective definition of probability: The degree of a manager’s
confidence or belief that the event will occur
Probability distribution: A table that lists all possible outcomes and
assigns the probability of occurrence to each outcome
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Probability
If a lottery offers n distinct prizes and the probabilities of winning the
prizes are pi (i = 1, . . . , n) then
◮
n
P
pi = p1 + p2 + . . . + pn = 1
i =1
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Expected value of a lottery
The expected value of a lottery is the average of the prizes obtained if
we play the same lottery many times
◮
If we played 600 times the lottery in Example 2
◮
We obtained a “1” 100 times, a “2” 100 times . . .
◮
We would win “A
C2” 300 times, win “A
C0” 200 times, and lose “A
C6”
100 times
◮
Average prize = (300 ∗ 2 + 200 ∗ 0 − 100 ∗ 6)/600
◮
Average prize = (1/2) ∗ 2 + (1/3) ∗ 0 − (1/6) ∗ 6 = 0
◮
Notice, we have the probabilities of the prizes multiplied by the value of
the prizes
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Expected Value. Formal definition
For a lottery (X ) with prizes x1 , x2 , . . . , xn and the probabilities of
winning p1 , p2 , . . . pn , the expected value of the lottery is
◮
◮
E (X ) = p1 x1 + p2 x2 + . . . + pn xn
n
P
pi xi
E (X ) =
i =1
The expected value is a weighted sum of the prizes
◮
the weights are the respective probabilities
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Comparisons of expected profit
Example: Jones Corporation is considering a decision involving pricing
and advertising. The expected value if they raise price is
Profit
Probability
(Probability)(Profit)
$ 800,000
0.50
$ 400,000
-600,000
0.50
-300,000
Expected Profit = $ 100,000
◮
The payoff from not increasing price is $ 200,000, so that is the
optimal strategy.
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Road map to decisions
Decision tree: A diagram that helps managers visualize their strategic
future
Figure 15.1: Decision Tree, Jones Corporation
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Constructing a decision tree
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Remarks
Decision fork: a juncture representing a choice where the decision
maker is in control of the outcome
Chance fork: a juncture where “chance” controls the outcome
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Expected value of perfect information
Expected Value of Perfect Information (EVPI)
◮
The increase in expected profit from completely accurate information
concerning future outcomes.
◮
Jones Example (Figure 15.1)
⋆
Given perfect information, the company will increase price if the
campaign will be successful and will not increase price if the campaign
will not be successful.
⋆
Expected profit = $500, 000 so
EVPI = $500, 000 − $200, 000 = $300, 000
Why is this useful?
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Simple decision rule
Use expected value of a project
How do people really decide?
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Is the expected value a good criterion to decide between
lotteries?
Does this criterion provide reasonable predictions? Let’s examine a
case . . .
◮
Lottery A: Get A
C3125 for sure (i.e. expected value = A
C3125)
◮
Lottery B: get A
C4000 with probability 0.75, and get A
C500 with
probability 0.25 (i.e. expected value also A
C3125)
Probably most people will choose Lottery A because they dislike risk
(risk averse).
However, according to the expected value criterion, both lotteries are
equivalent. The expected value does not seem a good criterion for
people that dislike risk.
If someone is indifferent between A and B it is because risk is not
important for him/her (risk neutral).
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Measuring attitudes toward risk: the utility approach
Another example
◮
◮
A small business is offered the following choice:
1
A certain profit of $2,000,000
2
A gamble with a 50-50 change of $4,100,000 profit or a $60,000 loss.
The expected value of the gamble is $2,020,000.
If the business is risk averse, it is likely to take the certain profit.
Utility function: Function used to identify the optimal strategy for
managers conditional on their attitude toward risk
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Expected Utility: The standard criterion to choose among
lotteries
Individuals do not care directly about the monetary values of the
prizes
◮
they care about the utility that the money provides
U(x) denotes the utility function for money
We will always assume that individuals prefer more money than less
money, so:
U ′ (xi ) > 0
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Expected Utility: The standard criterion to choose among
lotteries
The expected utility is computed in a similar way to the expected
value
However, one does not average prizes (money) but the utility derived
from the prizes
◮
EU =
n
P
pi U(xi ) = p1 U(x1 ) + p2 U(x2 ) + . . . + pn U(xn )
i =1
The sum of the utility of each outcome times the probability of the
outcome’s occurrence
The individual will choose the lottery with the highest expected utility
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Can we construct a utility function? Example
Utility function is not unique:
◮
you can add a constant term
◮
you can multiply by a constant factor
◮
the general shape is important
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How do you get these points?
Start with any values: e.g. U(−90) = 0, U(500) = 50
Then ask the decision maker questions about indifference cases
◮
Find value for 100
◮
Do you prefer the certainty of a $100 gain to a gamble of $500 with
probability P and $-90 with probability (1 − P)?
◮
Try several values of P until the respondent is indifferent
◮
Suppose outcome is P = 0.4
⋆
Then it follows
⋆
U(100) = 0.4U(500) + 0.6U(−90)
⋆
→ U(100) = 0.4(50) + 0.6(0) = 20
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Attitudes towards risk
Risk-averse: expected utility of lottery is lower than utility of
expected profit - the individual fears a loss more than she values a
potential gain
Risk-neutral: the person looks only at expected value (profit), but
does not care if the project is high- or low-risk.
Risk-seeking: expected utility is higher than utility of expected profit
- the individual prefers a gamble with a less certain outcome to one
with a certain outcome
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Attitudes toward risk
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Attitudes towards risk
What attitude towards risk do most people have? (maybe you want to
differentiate between long-term investment and, say, Lotto)
What attitude towards risk should a manager of a big (publicly
traded) company have?
What’s the effect of a managers’ risk attitude?
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Example
A risk averse person gets Y1 or Y2 with probability of 0.5
Expected Utility < Utility of expected value
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Measure of Risk: Standard deviation and Coefficient of
Variation
as a measure of risk we often use the standard deviation
◮
σ=(
N
P
i =1
pi [xi − E (x)]2 )0.5
to consider changes in the scale of projects, use the coefficient of
variation
◮
V = σ/E (x)
Figure 15.4: Probability Distribution of the Profit from an Investment
in a New Plant
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How can we measure risk?
Probability Distributions of the Profit from an Investment in a New Plant
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Adjusting for risk
Certainty equivalent approach:
◮
When a manager is indifferent between a certain payoff and a gamble,
the certainty equivalent (rather than the expected profit) can identify
whether the manager is a risk averter, lover, or risk neutral.
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Definition of certainty equivalent
The certainty equivalent of a lottery m, ce(m), leaves the individual
indifferent between playing the lottery m or receiving ce(m) for
certain.
◮
U(ce(m)) = E [U(m)]
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Adjusting for risk
Certainty equivalent approach
◮
If the certainty equivalent is less than the expected value, then the
decision maker is risk averse.
◮
If the certainty equivalent is equal to the expected value, then the
decision maker is risk neutral.
◮
If the certainty equivalent is greater than the expected value, then the
decision maker is risk loving.
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Adjusting for risk
The present value of future profits, which managers seek to maximize,
can be adjusted for risk by using the certainty equivalent profit in
place of the expected profit.
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Adjusting for risk
Indifference curves
◮
Figure 15.5: Manager’s Indifference Curve between Expected Profit and
Risk
◮
With expected value on the horizontal axis, the horizontal intercept of
an indifference curve is the certainty equivalent of the risky payoffs
represented by the curve.
◮
If a decision maker is risk neutral, indifference curves will be vertical.
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Manager’s Indifference Curve
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Definition of risk premium
Risk premium = E [m] − ce(m)
The risk premium is the amount of money that a risk-averse person
would sacrifice in order to eliminate the risk associated with a
particular lottery.
In finance, the risk premium is the expected rate of return above the
risk-free interest rate.
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Lottery m. Prizes m1 and m2
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Risk Premium
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Examples of commonly used Utility functions for risk
averse individuals
U(x) = ln(x)
√
U(x) = x
U(x) = x a where 0 < a < 1
U(x) = −exp(−a ∗ x) where a > 0
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Measuring Risk Aversion
The most commonly used risk aversion measure was developed by
Pratt
′′
◮
(X )
r (X ) = − UU ′ (X
)
For risk averse individuals, U ′′ (X ) < 0
◮
r (X ) will be positive for risk averse individuals
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Risk Aversion
If utility is logarithmic in consumption
◮
U(X ) = ln(X ) where X > 0
Pratt’s risk aversion measure is
′′
◮
(X )
r (X ) = − UU ′ (X
) =
1
X
Risk aversion decreases as wealth increases
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Risk Aversion
If utility is exponential
◮
U(X ) = −e −aX = −exp(−aX ) where a is a positive constant
Pratt’s risk aversion measure is
′′
◮
(X )
r (X ) = − UU ′ (X
) =
a2 e −aX
ae −aX
=a
Risk aversion is constant as wealth increases
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Example
Lotteries A and B
◮
Lottery A: Get A
C3125 for sure (i.e. expected value = A
C3125)
◮
Lottery B: get A
C4000 with probability 0.75, and get A
C500 with
probability 0.25 (i.e. expected value also A
C3125)
Suppose also that the utility function is
◮
U(X ) = sqrt(X ) where X > 0
◮
→ U(A) = 55.901699
certainty equivalent:
◮
◮
E(U(B)) = 0.75*U(4000) + 0.25*U(500) = 53.024335
→ (53.024335)2 = 2811.5801 = U(ce(B)))
risk premium: 3125 - 2811.5801 = 313.41991
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Willingness to Pay for Insurance
Consider a person with a current wealth of A
C100,000 who faces a
25% chance of losing his car worth A
C20,000
Suppose also that the utility function is
◮
U(X ) = ln(X ) where X > 0
the person’s expected utility will be
◮
E(U) = 0.75U(100,000) + 0.25U(80,000)
◮
E(U) = 0.75 ln(100,000) + 0.25 ln(80,000)
◮
E(U) = 11.45714
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Willingness to Pay for Insurance
What is the maximum insurance premium the individual is willing to
pay?
◮
E(U) = U(100,000 - y) = ln(100,000 - y) = 11.45714
◮
100,000 - y = exp(11.45714)
◮
y= 5,426
The maximum premium he is willing to pay is A
C5,426.
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Example
Roy Lamb has an option on a particular piece of land, and must
decide whether to drill on the land before the expiration of the option
or give up his rights.
If he drills, he believes that the cost will be $200,000.
If he finds oil, he expects to receive $1 million; if he does not find oil,
he expects to receive nothing.
◮
a) Can you tell wether he should drill on the basis of the available
information? Why or why not?
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Example cont’d
No, there are no probabilities given.
Mr. Lamb believes that the probability of finding oil if he drills on this
piece of land is 41 , and the probability of not finding oil if he drills here
is 34 .
◮
◮
◮
b) Can you tell wether he should drill on the basis of the available
information. Why or why not?
c) Suppose Mr. Lamb can be demonstrated to be a risk lover. Should
he drill? Why?
d) Suppose Mr. Lamb is risk neutral. Should he drill or not. Why?
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Example cont’d
b) 1/4(800) − 3/4(200) = 50 > 0, so a person who is risk neutral
would drill. However, if very risk averse, the person would not want to
drill.
c) Yes, since the project has both a positive expected value and
contains risk, Mr. Lamb will be doubly pleased.
d) Yes, Mr. Lamb cares only about expected value, which is positive
for this project.
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