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Managerial Economics
Unit 9: Risk Analysis
Rudolf Winter-Ebmer
Johannes Kepler University Linz
Winter Term 2012
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Objectives
Explain how managers should make strategic decisions when facedwith incomplete or imperfect information
Study how economists make predictions about individuals or firmschoices 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 theirchoices.
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 shewill 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 AC2 if 1, 2, or 3
We neither win nor lose if 4, or 5
We will lose AC6 if 6
Example 3: Johns monthly consumption:
AC3000 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 relativefrequency 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 events limit of frequency in alarge 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 probabilitiesof the events must be equal to one.
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Probability
Subjective definition of probability: The degree of a managersconfidence or belief that the event will occur
Probability distribution: A table that lists all possible outcomes andassigns the probability of occurrence to each outcome
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Probability
If a lottery offers n distinct prizes and the probabilities of winning theprizes are pi(i= 1, . . . ,n) then
n
i=1
pi=p1+p2+ . . . +pn = 1
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Expected value of a lottery
The expected value of a lottery is the average of the prizes obtained ifwe 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 AC2 300 times, win AC0 200 times, and lose AC6100 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 ofwinning p1, p2, . . . pn, the expected value of the lottery is
E(X) =p1x1+p2x2+ . . . +pnxn
E(X) =n
i=1
pixi
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 pricingand 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 theoptimal strategy.
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Road map to decisions
Decision tree: A diagram that helps managers visualize their strategicfuture
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 decisionmaker 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 thecampaign will be successful and will not increase price if the campaignwill not be successful.
Expected profit = $500, 000 soEVPI= $500, 000 $200, 000 = $300, 000
Why is this useful?
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S l d l
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Simple decision rule
Use expected value of a project
How do people really decide?
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I h d l d i i d id b
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Is the expected value a good criterion to decide between
lotteries?
Does this criterion provide reasonable predictions? Lets examine acase . . .
Lottery A: Get AC3125 for sure (i.e. expected value = AC3125)
Lottery B: get AC4000 with probability 0.75, and get AC500 withprobability 0.25 (i.e. expected value also AC3125)
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 forpeople that dislike risk.
If someone is indifferent between A and B it is because risk is notimportant for him/her (risk neutral).
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M i i d d i k h ili h
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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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E t d Utilit Th t d d it i t h
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Expected Utility: The standard criterion to choose among
lotteries
Individuals do not care directly about the monetary values of theprizes
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 lessmoney, so:
U(xi)> 0
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E t d Utilit Th t d d it i t h
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Expected Utility: The standard criterion to choose among
lotteries
The expected utility is computed in a similar way to the expectedvalue
However, one does not average prizes (money) but the utility derived
from the prizes EU=
n
i=1
piU(xi) =p1U(x1) +p2U(x2) + . . .+pnU(xn)
The sum of the utility of each outcome times the probability of the
outcomes occurrenceThe individual will choose the lottery with the highest expected utility
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Can e constr ct a tilit f nction? E ample
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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?
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How do you get these points?
Start with any values: e.g. U(90) = 0, U(500) = 50Then 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 Pand $-90 with probability (1 P)? Try several values ofPuntil 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
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Attitudes towards risk
Risk-averse:expected utility of lottery is lower than utility ofexpected profit - the individual fears a loss more than she values apotential gain
Risk-neutral: the person looks only at expected value (profit), butdoes 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 toward risk
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Attitudes towards risk
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Attitudes towards risk
What attitude towards risk do most people have? (maybe you want todifferentiate between long-term investment and, say, Lotto)
What attitude towards risk should a manager of a big (publiclytraded) company have?
Whats the effect of a managers risk attitude?
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Example
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Example
A risk averse person gets Y1 orY2 with probability of 0.5
Expected Utility< Utility of expected value
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Measure of Risk: Standard deviation and Coefficient of
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Measure of Risk: Standard deviation and Coefficient of
Variation
as a measure of risk we often use the standard deviation
= (N
i=1
Pi[i E()]2)0.5
to consider changes in the scale of projects, use the coefficient ofvariation
V =/E()
Figure 15.4: Probability Distribution of the Profit from an Investmentin a New Plant
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How can we measure risk?
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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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Definition of certainty equivalent
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Definition of certainty equivalent
The certainty equivalent of a lottery m, ce(m), leaves the individualindifferent between playing the lottery m or receiving ce(m) forcertain.
U(ce(m)) =E[U(m)]
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Adjusting for risk
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j g
Certainty equivalent approach
If the certainty equivalent is less than the expected value, then thedecision maker is risk averse.
If the certainty equivalent is equal to the expected value, then thedecision maker is risk neutral.
If the certainty equivalent is greater than the expected value, then thedecision maker is risk loving.
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Adjusting for risk
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j g
The present value of future profits, which managers seek to maximize,can be adjusted for risk by using the certainty equivalent profit inplace of the expected profit.
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Adjusting for risk
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j g
Indifference curves
Figure 15.5: Managers Indifference Curve between Expected Profit andRisk
With expected value on the horizontal axis, the horizontal intercept ofan indifference curve is the certainty equivalent of the risky payoffsrepresented by the curve.
If a decision maker is risk neutral, indifference curves will be vertical.
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Managers Indifference Curve
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g
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Definition of risk premium
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Risk premium=E[m] ce(m)
The risk premium is the amount of money that a risk-averse personwould sacrifice in order to eliminate the risk associated with aparticular lottery.
In finance, the risk premium is the expected rate of return above therisk-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
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averse individuals
U(x) =ln(x)
U(x) = xU(x) =xa where 0
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The most commonly used risk aversion measure was developed byPratt
r(X) =
U(X)
U(X)
For risk averse individuals, U(X)
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If utility is logarithmic in consumption U(X) =ln(X) where X>0
Pratts risk aversion measure is
r(X) = U(X)
U(X) = 1X
Risk aversion decreases as wealth increases
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Risk Aversion
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If utility is exponential
U(X) = eaX = exp(aX) where a is a positive constant
Pratts risk aversion measure is r(X) = U
(X)U(X) =
a2eaX
aeaX =a
Risk aversion is constant as wealth increases
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Example
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Lotteries A and B
Lottery A: Get AC3125 for sure (i.e. expected value = AC3125) Lottery B: get AC4000 with probability 0.75, and get AC500 with
probability 0.25 (i.e. expected value also AC3125)
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
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Consider a person with a current wealth of AC100,000 who faces a25% chance of losing his car worth AC20,000
Suppose also that the utility function is
U(X) =ln(X) where X>0
the persons 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
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What is the maximum insurance premium the individual is willing topay?
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 AC5,426.
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Example
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Roy Lamb has an option on a particular piece of land, and mustdecide whether to drill on the land before the expiration of the optionor 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 availableinformation? Why or why not?
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Example contd
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No, there are no probabilities given.
Mr. Lamb believes that the probability of finding oil if he drills on thispiece of land is 14 , 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 availableinformation. Why or why not?
c) Suppose Mr. Lamb can be demonstrated to be a risk lover. Shouldhe drill? Why?
d) Suppose Mr. Lamb is risk neutral. Should he drill or not. Why?
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Example contd
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b) 1/4(800) 3/4(200) = 50>0, so a person who is risk neutralwould drill. However, if very risk averse, the person would not want todrill.
c) Yes, since the project has both a positive expected value andcontains risk, Mr. Lamb will be doubly pleased.
d) Yes, Mr. Lamb cares only about expected value, which is positivefor this project.
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