me unit9 risk

7/25/2019 ME Unit9 Risk

1/49

Managerial Economics

Unit 9: Risk Analysis

Rudolf Winter-Ebmer

Johannes Kepler University Linz

Winter Term 2012

Managerial Economics: Unit 9 - Risk Analysis 1 / 1


2/49

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



3/49

Management tools

Expected value

Decision trees

Techniques to reduce uncertainty

Expected utility



4/49

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 . . .



5/49

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)



6/49

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



7/49

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



8/49

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.



9/49

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



10/49

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



11/49

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



12/49

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



13/49

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.



14/49

Road map to decisions

Decision tree: A diagram that helps managers visualize their strategicfuture

Figure 15.1: Decision Tree, Jones Corporation



15/49

Constructing a decision tree



16/49

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



17/49

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?


S l d l


18/49

Simple decision rule

Use expected value of a project

How do people really decide?


I h d l d i i d id b


19/49

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).


M i i d d i k h ili h


20/49

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


E t d Utilit Th t d d it i t h


21/49

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


E t d Utilit Th t d d it i t h


22/49

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


Can e constr ct a tilit f nction? E ample


23/49

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


How do you get these points?


24/49

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


Attitudes towards risk


25/49


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


Attitudes toward risk


26/49

Attitudes toward risk




27/49


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?


Example


28/49

Example

A risk averse person gets Y1 orY2 with probability of 0.5

Expected Utility< Utility of expected value


Measure of Risk: Standard deviation and Coefficient of


29/49

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


How can we measure risk?


30/49

How can we measure risk?

Probability Distributions of the Profit from an Investment in a New Plant



31/49

Definition of certainty equivalent


32/49

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)]


Adjusting for risk


33/49

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.


Adjusting for risk


34/49

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.


Adjusting for risk


35/49

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.


Managers Indifference Curve


36/49

g


Definition of risk premium


37/49

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.


Lottery m. Prizes m1 and m2


38/49


Risk Premium


39/49


Examples of commonly used Utility functions for risk


40/49

averse individuals

U(x) =ln(x)

U(x) = xU(x) =xa where 0


41/49

The most commonly used risk aversion measure was developed byPratt

r(X) =

U(X)

U(X)

For risk averse individuals, U(X)


42/49

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


Risk Aversion


43/49

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


Example


44/49

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


Willingness to Pay for Insurance


45/49

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


Willingness to Pay for Insurance


46/49

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.


Example


47/49

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?


Example contd


48/49

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?


Example contd


49/49

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.


me unit9 risk

Documents