managerial economics - unit 9: risk analysis · managerial economics unit 9: risk analysis rudolf...

Managerial Economics

Unit 9: Risk Analysis

Rudolf Winter-Ebmer

Johannes Kepler University Linz

Winter Term 2015

Managerial Economics: Unit 9 - Risk Analysis 1 / 49

Objectives

Explain how managers should make strategic decisions when facedwith incomplete or imperfect information

◮ Study how economists make predictions about individual’s or firm’schoices under uncertainty

◮ Study the standard assumptions about attitudes towards risk


Management tools

Expected value

Decision trees

Techniques to reduce uncertainty

Expected utility


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 . . .


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: John’s monthly consumption:

◮ AC3000 if he does not get ill

◮ AC500 if he gets ill (so he cannot work)


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


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


Probability

Frequency definition of probability: An event’s 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 theevents that can occur in a given situation, the sum of the probabilitiesof the events must be equal to one.


Probability

Subjective definition of probability: The degree of a manager’sconfidence or belief that the event will occur

Probability distribution: A table that lists all possible outcomes andassigns the probability of occurrence to each outcome


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


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 “AC6”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 ofthe prizes


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


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.


Road map to decisions

Decision tree: A diagram that helps managers visualize their strategicfuture

Figure 15.1: Decision Tree, Jones Corporation


Constructing a decision tree


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


Expected value of perfect information

Expected Value of Perfect Information (EVPI)

◮ The increase in expected profit from completely accurate informationconcerning 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?


Simple decision rule

Use expected value of a project

How do people really decide?


Is the expected value a good criterion to decide between

lotteries?

Does this criterion provide reasonable predictions? Let’s 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 areequivalent. 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).


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 formanagers conditional on their attitude toward risk


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


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 derivedfrom 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 theoutcome’s occurrence

The individual will choose the lottery with the highest expected utility


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?

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 withprobability 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


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 onewith a certain outcome


Attitudes toward risk


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?

What’s the effect of a managers’ risk attitude?


Example

A risk averse person gets Y1 or Y2 with probability of 0.5

Expected Utility < Utility of expected value


Measure of Risk: Standard deviation and Coefficient of

Variation

as a measure of risk we often use the standard deviation

◮ σ = (N∑

i=1

pi [xi − E (x)]2)0.5

to consider changes in the scale of projects, use the coefficient ofvariation

◮ V = σ/E (x)

Figure 15.4: Probability Distribution of the Profit from an Investmentin a New Plant


How can we measure risk?

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


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 identifywhether the manager is a risk averter, lover, or risk neutral.


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

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

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

Indifference curves

◮ Figure 15.5: Manager’s 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.


Manager’s Indifference Curve


Definition of risk premium

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


Risk Premium


Examples of commonly used Utility functions for risk

averse individuals

U(x) = ln(x)

U(x) =√x

U(x) = xa where 0 < a < 1

U(x) = −exp(−a ∗ x) where a > 0


Measuring Risk Aversion

The most commonly used risk aversion measure was developed byPratt

◮ r(X ) = −U′′(X )

U′(X )

For risk averse individuals, U ′′(X ) < 0

◮ r(X ) will be positive for risk averse individuals


Risk Aversion

If utility is logarithmic in consumption◮ U(X ) = ln(X ) where X > 0

Pratt’s risk aversion measure is

◮ r(X ) = −U′′(X )

U′(X ) = 1X

Risk aversion decreases as wealth increases


Risk Aversion

If utility is exponential

◮ U(X ) = −e−aX = −exp(−aX ) where a is a positive constant

Pratt’s risk aversion measure is

◮ r(X ) = −U′′(X )

U′(X ) = a2e−aX

ae−aX = a

Risk aversion is constant as wealth increases


Example

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 withprobability 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

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 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


Willingness to Pay for Insurance

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

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 cont’d

No, there are no probabilities given.

Mr. Lamb believes that the probability of finding oil if he drills on thispiece of land is 1

4 , 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 cont’d

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.


managerial economics - unit 9: risk analysis · managerial economics unit 9: risk analysis rudolf...

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