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Chapter 14Risk Analysis

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 he takes the decision, he is uncertain about the profits that he will obtain. He does not know which is the best choiceThis will depend on rain conditions, plagues, world prices…

Uncertainty

Example 2: playing with a fair dieWe will win €2 if 1, 2, or 3, We neither win nor lose if 4, or 5We will lose €6 if 6

Example 3: John’s monthly consumption:€3000 if he does not get ill€500 if he gets ill (so he cannot work)

Our objectives in this part

Study how economists make predictions about individual’s or firm’s choices under uncertaintyStudy the standard assumptions about attitudes towards risk

Definition

Economists call a lottery a situation which involves uncertain payoffs:

Cultivating apples is a lotteryCultivating pears is another lotteryPlaying with a fair die is another oneMonthly consumption

Each lottery will result in a prize

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Probability

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

If a lottery offers n distinct prizes and the probabilities of winning the prizes are pi(i=1,…,n) then

1 21

... 1n

i ni

p p p p=

= + + + =∑

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An important concept: Expected Value

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 2We obtained a “1” 100 times, a “2” 100 times…We would win “€2” 300 times, win “€0” 200 times, and lose “€6” 100 timesAverage prize=(300*2+200*0-100*6)/600Average prize=(1/2)*2+(1/3)*0-(1/6)*6=0Notice, 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 valueof the lottery is

1( )

n

i ii

E X p x=

=∑

1 1 2 2( ) ... n nE X p x p x p x= + + +

The expected value is a weighted sum of the prizes the weights the respective probabilities

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Constructing a decision tree

Decision fork: a juncture representing a choice where the decision maker is in control of the outcomeChance fork: a juncture where “chance”(some call it “nature”) controls the outcome

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Constructing a decision tree

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A small game….

How much do you pay for an envelope?

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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 €3125 for sure (i.e. expected value= €3125)Lottery B: get €4000 with probability 0.75,

and get €500 with probability 0.25(i.e. expected value also €3125)

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 riskIf someone is indifferent between A and B it is because risk is not important for him (risk neutral)

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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 moneyWe will always assume that individuals prefer more money than less money, so:

'( ) 0iU x >

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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 valueHowever, one does not average prizes (money) but the utility derived from the prizes

1 1 2 21

( ) ( ) ( ) ... ( )n

i i n ni

EU pU x pU x p U x p U x=

= = + + +∑The individual will choose the lottery with the highest expected utility

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Can we construct a utility function:Example: Tomco Oil Corporation

•Utility function is not unique: •you can add a constant term•You can multipy by a constant factor

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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 aboutindifference cases

Find value for 100Do 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 indifferentSuppose outcome is P=0.4

Then it followsU(100) = 0.4U(500) + 0.6U(-90)

==> U(100) = 0.4 (50) + 0.6(0) = 20

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Expected utilityfor Tomco Oil

EU = .6U($-90) + .15U(100) + .15U(300) + .10 U(500)

=.6 (0) + .15(20) + .15(40) + .10(50) = 14

≠ E(Profits) = .6(-90) + .15 (100) + .15(300) + .10(500) = 56

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Probability Distribution

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Attitudes toward risk

Utility

Profit

Risk Seeking Risk

Neutral

Risk Averse

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Attitudes towards riskRisk-averse: expected utility 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: …

What attitude towards risk do most people have? (maybe you wanna 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 the managers’ risk attitude?

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Example: A risk averse person gets Y1 or Y2 with probability of ½Expected Utility < Utility of expected value

Utility

Profit0

Risk Averse

•

•

Y1 Y2E(Y1, Y2) =

= ½ Y1 + ½ Y2

U(Y1)

U(Y2)

Utility of exp. Profit

Expected Utility = ½ U(Y1) + ½ U(Y2)

•

•

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Examples of commonly used Utility functions for risk averse individuals

( ) ln( )

( )( ) 0 1( ) exp( * ) 0

a

U x x

U x xU x x where aU x a x where a

=

=

= < <= − − >

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Measuring Risk Aversion

The most commonly used risk aversion measure was developed by Pratt

"( )( ) '( )

U Xr XU X

= −

For risk averse individuals, U” (X) < 0r (X) will be positive for risk averse individuals

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Risk Aversion

If utility is logarithmic in consumptionU(X) = ln (X )

where X> 0Pratt’s risk aversion measure is

"( ) 1( ) '( )

U Xr XU X X

= − =

Risk aversion decreases as wealth increases

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Risk Aversion

If utility is exponentialU(X) = -e-aX = -exp (-aX)

where a is a positive constantPratt’s risk aversion measure is

2"( )( ) '( )

aX

aX

U X a er X aU X ae

−

−= − = =

Risk aversion is constant as wealth increases

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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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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.

•29 Money

U(.)

Lottery m. Prizes m1 and m2

m1 m2

U(m1)

U(m2)

E[m]

E[U(m)]

ce(m)

Risk premium

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Willingness to Pay for Insurance

Consider a person with a current wealth of €100,000 who faces a 25% chance of losing his car worth €20,000

Suppose also that the utility function is

U(X) = ln (x)

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Willingness to Pay for Insurance

The person’s expected utility will beE(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 max. insurance premium the individual is willing to pay?

E(U) = U(100,000 - y) = ln(100,000 - y) = 11.45714

100,000 - y = e11.45714

y= 5,426

The maximum premium he is willing to pay is €5,426

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Manager‘s Indifference Curve

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Risk Premium

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Problem 6The CEO of a publishing company says she is indifferent between thecertainty of receiving $7,500 and a gamble where there is 0.5 chanceof receiving $5,000 and a 0.5 chance of reveiving $10,000. Also, shesays she is indifferent between the certainty of reveiving $10,000 and a gamble where there is a 0.5 chance of receiving $7,500 and a 0.5 chance of receiving $12,500.

a) Draw (on a piece of graph paper) four points on the utility functionof this publishing executive.

b) Does she seem to be a risk averter, a risk lover, or risk neutral?

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Problem 10Roy Lamb has an option on a particular piece of land, and must decide whether to drill on the land before the expirationof the option or give up his rights. If he drills, he believes thatthe cost will be $200,000. If he finds oil, he expects to receive$1 million; if he does not find oil, he expects to receivenothing.

a) Can you tell wether he should drill on the basis of theavailable information? Why or why not?

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Solution Problem 10

a) No, there are no probabilities given.

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Problem 10 – Part 2Mr. Lamb believes that the probability of finding oil if he drills on this piece of land is ¼, and the probability of not finding oil if he drills here is ¾.

b) Can you tell wether he should drill on the basis of theavailable 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. Shoul he drill or not. Why?

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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.

Solution Problem 10 – Part 2