5. Uncertainty and Consumer Behavior


• Describing Risk

• Preferences Toward Risk

• Reducing Risk

• The Demand for Risky Assets

Chapter Outline


Introduction

To examine ways that people can compare and choose among

risky alternatives, we take the following steps:

1. In order to compare the riskiness of alternative choices, we need

to quantify risk.

2. We will examine people’s preferences towards risk.

3. We will see how people can sometimes reduce or eliminate risk.

4. In some situations, people can choose the amount of risk they

wish to bear.

5. Sometimes demand for a good is driven partly or entirely by

speculation—people buy the good because they think its price will

increase.


Describing Risk

• Probability

– Likelihood that a given outcome will occur.

• Objective Interpretation

– Is based on the observed frequency of the

occurrence of past events.


Describing Risk

• Interpretation of probability

– Subjective probability

is the perception that an outcome will occur.

Different information or different methods of processing

the same information can influence the subjective

probability.


Describing Risk

• Expected Value

– Probability-weighted average of the payoffs associated with all possible outcomes.

The probability of the respective payoffs are used as weighted averages.

The expected value measures the central tendency, i.e., the payoff or value that we would expect on average.


Describing Risk

Example:

– Investment in an off-shore oil drilling project:

– Two results are possible:

Success – the stock price increases from €30 to €40/

share.

Failure – the stock price decreases from €30 to €20/

share.

– Objective probability

Out of 100 drillings, 25 are successful and 75 fail.

Probability of success (Pr) = 1/4 and probability of

failure = ¾.


Describing Risk

• An example:

EV Pr(success)(€40/share) Pr(failure)(€20/share)

EV 1 4(€40/share) 3 4(€20/share)

EV €25/share

Expected Value (EV)


Describing Risk

• Generally, the expected value is written as follows:

nn2211 XPr...XPrXPr E(X)


Describing Risk

•The Variability

Extent to which possible outcomes of an

uncertain event differ.

• Consider the following, henceforth called

Scenario A: Let's say we choose between two part-time

sales jobs with the same expected income (€

1,500).

The first job is entirely based on commission

payments.

The second job is remunerated with a salary.

Variability


Describing Risk

Scenario A

For the first job, there are two equally likely

(commission based) results for one’s efforts:

€2,000 and €1,000.

In the second job, the remuneration is usually

€1,510 (0.99 probability), but you will only earn

€510 if the company goes out of business (0.01

probability).

Variability


Describing Risk

Scenario A

• Income from sales jobs

Outcome 1 Outcome 2

Probability Income

(€)

Probability Income

(€)

Expected

Income (€)

Job 1:

Commission

0.5 2,000 0.5 1,000 1,500

Job 2:

Fixed Salary

0.99 1,510 0.01 510 1,500


Describing Risk

• Job 1 – expected income

€15000.1(€510))0.99(€1510 )E(X2

• Job 2 – expected income

Scenario A

Income from sales jobs:

1500€ 5(€1000).05(€2000).0)E(X1


Describing Risk

• While the expected values are the same, this does not apply to the variability.

• Higher variability in the expected value indicates higher risk.

• Deviation

– Extent to which possible outcomes of an uncertain event differ.

Outcome 1 Deviation Outcome 2 Deviation

Job 1 2,000 500 1,000 -500

Job 2 1,510 10 510 -990

Scenario A


Describing Risk

Standard deviation: Square root of the weighted

average of the squared deviations of payoffs

associated with each outcome.

Variability

2

22

2

11 ))((Pr))((Pr XEXXEX


Describing Risk

• Computing the variance (€)

Out-

come 1

Deviation

Squared

Out-

come 2

Deviation

Squared

Weighted

Average

Deviation

Squared

Standard

Deviation

Job 1 2,000 250,000 1,000 250,000 250,000 500

Job 2 1,510 100 510 980,100 9,900 99.50

Scenario A


Describing Risk

• The standard deviations of job 1 and job 2 are not

the same:

50.99

900,9€

0)01(€980,10.00.99(€100)

500

000,250€

)5(€250,000.000)0.5(€250,0

2

2

2

1

1

1

higher risk

Scenario A


Describing Risk

• Job 1 is a workplace where income is between

€1,000 and €2,000, with increments of €100 each

having equal probability of being achieved.

• Job 2 is a workplace where income is between

€1,300 and €1,700, with increments of €100 each

having equal probability of being achieved.

Example

Scenario B


Outcome Probabilities for Two Jobs

Income

0.1

€1000 €1500 €2000

0.2

Job 1

Job 2

The distribution of payoffs associated with Job 1

have a greater variance, a greater standard

deviation, and a greater overall risk than the

distribution of payoffs associated with Job 2.

Probability

Scenario B


Describing Risk

• Decision making

– A person that does not like to take risks would choose Job

2, which has the same expected income as Job 1 but with

lower risk.

Scenario B


Income from Sales Job – modified (€)

• Suppose we add €100 to each of the payoffs from job 1, so

that the expected payoff increases from €1500 to €1600.

Outcome 1 Deviation

Squared

Outcome 2 Deviation

Squared

Expected

Income

Standard

Deviation

Job 1 2,100 250,000 1,100 250,000 1,600 500

Job 2 1,510 100 510 980.100 1,500 99.50

Scenario A: revisited


Decision Making

• Job 1: Expected income of €1,600 and a standard deviation of €500.

• Job 2: Expected income of €1,500 standard deviation of €99.50

• Which job to choose?

– Higher income or lower risk?

Decision Making


Describing Risk

• A state wants to prevent double parking.

• What are possible solutions?

• Assumptions:

1. The individual saves €5 (in terms of value of time), if he double parks instead taking the time to park properly.

2. The driver is risk neutral.

3. The costs of catching a double-parker are zero.

Example


Describing Risk

• Paying a fine of €5.01 would prevent the driver from

parking in second the row .

– The benefit from parking in the second row is (€5),

which is lower than the cost of (€5.01); this

corresponds to a net benefit (of -0.01) that is less

than 0.

Example


Describing Risk

• A policy that combines a high fine with a low

probability of apprehension is likely to reduce

enforcement costs.

• A fine of €50 with a probability of being caught of 0.1

will result in an expected penalty of €5.

• A fine of €500 with a probability of being caught of

0.01 will result in an expected penalty of €5.

• To have an effective fine, the more risk-loving the

population of drivers are, the higher the fine must

be.

Example


Preferences Toward Risk

• Choosing between risky alternatives

– Assumptions

Consumption of a single product.

The consumer knows all outcomes and their

associated probabilities.

Payoffs are measured in terms of utility.

Utility function is given.



• A woman has an income of €15,000 and a utility of

13 units.

• She is considering a new but risky sales job.

• Her income will either double to €30,000 or will fall

to €10,000. Each possibility has a probability of

50%.

• To evaluate the new job, she can calculate the

expected value of the resulting income. Because

we are measuring value in terms of her utility, we

must calculate the expected utility, E(u), that she

can obtain.

Example



Income (€ thousands)

Utility

E

10

10 15 20

13

14

16

18

0 16 30

A

B

C

D



• Expected utility: Sum of the utilities associated

with all possible outcomes, weighted by the

probability of each outcome occurring.

Example



• The expected utility can be written as follows:

– E(u) = (1/2)u(€10,000) + (1/2)u(€30,000)

= 0.5(10) + 0.5(18)

= 14

• The risky new job is thus preferred to the original job,

because the expected utility of 14 is greater than the

original utility of 13.

Example



• Individuals differ in their preferences toward risk.

• People can be risk averse, risk loving, or risk neutral.

• risk averse: The condition of preferring a certain income to an income subject to risk, with both having the same expected value.

– A consumer is risk averse when, his marginal utility diminishes as income increases.

The use of insurance illustrates the risk averse behavior of individuals.



• Consider the following scenario:

– An individual can have a job earning €20,000 with 100% probability and an expected utility of 16.

– An Individual can have a job with 0.5 probability of earning €30,000 and 0.5 probability of earning €10,000.

– Expected income = (0.5)(€30,000) + (0.5)(€10,000) = €20,000

Risk averse



• The expected income from both jobs is the same.

Thus, the risk averse person will choose the

(certain) income.

• The expected utility from the uncertain income:

– E(u) = (1/2)u(€10,000) + (1/2)u(€30,000)

– E(u) = (0.5)(10) + (0.5)(18) = 14

E(u) from job 1 is 16, which is greater than

E(u) from job 2 which is 14.

Risk averse




Utility

The consumer is risk averse,

because she would prefer an

income of €20,000 with

certainty instead of gambling

with a 0.5 probability of

earning €10,000 and a 0.5

probability of earning €30,000.

E

10

10 15 20

13

14

16

18

0 16 30

A

B

C

D

Risk averse



Risk neutral: is a condition of being indifferent

between a risky income and a certain income with the

same expected value.

Risk neutral



Income (€ thousands) 10 20

Utility

0 30

6

A

E

C

12

18 The consumer is risk

neutral and indifferent

between certain and

uncertain events with the

same expected income.

Risk neutral



Risk loving: Condition of preferring a risky income to

a certain income with the same expected value.

– E.g., Gambling, some criminal activities.

Risk loving




Utility

0

3

10 20 30

A

E

C 8

18 The consumer is risk loving;

she would prefer the

uncertain income instead of

the certain income when both

have the same expected

value.

Risk loving



Risk premium: Maximum amount of money that a

risk-averse person will pay to avoid taking a risk.

• Consider the following scenario:

– With a probability of 0.5, the consumer could earn

an income of €30,000, and with a 0.5 probability,

she could earn an income of €10,000 (expected

income = €20,000).

– The expected utility is then

E(u) = 0.5(18) + 0.5(10) = 14

RISK PREMIUM


How much would the consumer pay

to avoid risk?


Utility

0 10 16

Here the risk premium is

€4,000, because a certain

income of €16,000 gives

her the same expected

utility (14) as the

uncertain income that

has an expected value of

€20,000.

10

18

30 40

20

14

A

C E

G

20

F

RISK PREMIUM

RISK PREMIUM



• Combinations of the expected value of income and

its standard deviation with which the same expected

utility can be achieved.

Indifference Curves


Risk Aversion and Indifference Curves

Standard deviation of income

Expected

Income

Highly risk averse: an

increase in the standard

deviation of this individual’s

income requires a large

increase in expected income

if he or she is to remain

equally well off.

U1

U2

U3


Risik Aversion and Indifference Curves

Standard deviation of income

Expected

Income Slightly risk averse:

an increase in the standard

deviation of income

requires only a small

increase in expected

income if he or she is to

remain equally well off.

U1

U2

U3


Business Executives and Risk Preferences

• In one study, 464 executives were asked questions

describing their risk preferences. The results were

as follows:

– 20% were risk neutral

– 40% were risk loving.

– 20% were risk averse.

– 20% did not answer the questionnaire.

Example


Reducing Risk

• There are three methods which consumers use to

try to avoid risk:

1) Diversification

2) Insurance

3) The value of complete information


Reducing Risk

• Diversification

– Assume that a company can decide between

selling air conditioners and heaters or both.

– The probability of hot or cold weather is 0.5.

– If the company diversifies, it is probably better off.


Income from sales of appliances

• Income from sales of appliances in €s:

Hot Weather Cold Weather

Air Conditioner sales 30,000 12,000

Heater sales 12,000 30,000


Reducing Risk

• If the company sells only air conditioners or only heaters, the actual income will be either €12,000 or €30,000.

• Expected income will be

– 1/2(€12,000) + 1/2(€30,000) = €21,000

• Diversification between the products will yield half of what would have been earned from only selling heaters plus half that which would have been earned from only selling air conditioners.

Diversification


Reducing Risk

• If the weather is hot, the company will earn €15,000 from air conditioner sales and €6,000 from heater sales; thus earning a total of €21,000 in sales.

• If the weather is cold, the company will earn €6,000 from air conditioners and €15,000 from heaters earning a total income of €21,000.

Diversification


Reducing Risk

• In this instance, diversification eliminates all risk

because the certain income from diversification is

€21,000.

• Companies can reduce risk by dividing their business

into a series of activities whose outcomes are not

closely related.

Diversification


Reducing Risk

• Questions to be discussed

– How can diversification reduce the risk of

investment in the stock market?

– Can diversification eliminate the risk of investment

in the stock market?

The Stock Market


Reducing Risk

• For a risk-averse individual, losses count more (in terms of changes in utility) than gains.

• A risk-averse homeowner, therefore, will enjoy higher utility by purchasing building insurance.

Insurance


The Decision to Buy Insurance

• The decision to buy insurance (€) with insurance costs of

1,000 €.

Insurance Burglary

(Probability

0.1)

No Burglary

(Probability

0.9)

Expected

Wealth

Standard

Deviation

No 40,000 50,000 49,000 3,000

Yes 49,000 49,000 49,000 0


Reducing Risk

• While the expected payout is the same, the

expected utility is greater when having insurance.

Since, the marginal utility in the event of a loss is

greater than when no loss occurs.

• When buying insurance, we can expect to have an

insurance premium which is equal to the total

expected payout (to all insurance customers). Thus,

this increases our utility.

Insurance


Reducing Risk

• Although single events may be random and largely

unpredictable, the average outcome of many similar

events can be predicted.

• Example

– A single toss of a coin and a large number of such

coin tosses.

– The question of which driver suffers a total loss and

the amount of total damage caused by a large group

of drivers.

The Law of Large Numbers


Reducing Risk

• Assumption:

– The probability of a loss of €10,000 from home burglary is 10%.

– Expected loss = 0.10 x €10,000 = €1,000 at high risk (the probability of a loss of €10,000 amounts to 10%)

– 100 individuals will be confronted with the same risk.

Actuarial Fairness


Reducing Risk

• Hence:

– With a premium of €1,000, a fund of €100,000 will

be created from which the losses can be covered.

– Actuarial justice

Applies when: Insurance premium =

expected payout.

Actuarial Fairness


Reducing Risk

• Value of complete information

Difference between the expected value of a

choice when there is complete information

and the expected value when information is

incomplete.

• Suppose that the manager of a clothing business

must decide how many suits he has to order for

autumn:

– 100 suits cost €180/suit.

– 50 suits cost €200/suit.

– The price of the suits is €300.

The Value of Information


Reducing Risk

• Suppose that the manager of a clothing business

must decide how many suits he has to order for

autumn:

– Unsold suits can be returned at half of the original

cost.

– Without further information, the probability of

selling 100 suits is equal to 50% and the

probability of selling 50 suits is equal to 50%.



The Decision of sales of suits

• Profits from Sales of Suits (€)

• With incomplete information:

– Risk neutral: Buy 100 suits

– Risk averse: Buy 50 suits

Sales of 50

units

Sales of 100

units

Expected

Profits

Buy 50 units 5,000 5,000 5,000

Buy 100 units 1,500 12,000 6,750


Reducing Risk

• The expected profit with complete information is

€8,500.

8,500 = 0.5(5,000) + 0.5(12,000)

• The expected profit with uncertainty (buy 100 suits)

is €6,750.

• The value of complete information is equal to €

1,750 (that is, the expected value with complete

information minus the expected value with

uncertainty (buy 100 suits)).



The Demand for Risky Assets

• Assets

– Something that provides a flow of money or

service to its owner.

The flow of money or money itself can be

explicit (dividends) or implicit (capital gain).

• Capital gain

– An increase in the value of an asset is a

capital gain; a decrease is a capital loss.



• Risky Assets: – Asset that provides an uncertain flow of money or

services to its owner.

– Examples Housing, capital gains,

industrial bonds, investment prices

• Riskless (or risk-free) Assets – Assets that provide a flow of money or services that is

known with certainty.

– Examples Short term government bonds, short term money

market papers.

Risky & Riskless Assets



• Asset returns

– Return

Total monetary flow of an asset as a fraction of its

price.

– Real return

Simple (or nominal) return on an asset, less the rate of

inflation.



• Returns on Assets

𝐴𝑠𝑠𝑒𝑡 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 =𝑚𝑜𝑛𝑒𝑦 𝑓𝑙𝑜𝑤

𝑝𝑟𝑖𝑐𝑒

€100/YAsset returns 10%

selling price €1,000

payment



• Expected return:

– Return that an asset should earn on average.

• Actual return:

– Return that an asset earns.

Expected versus actual returns


Investments – Risk and Return (1926 – 1999)

• Investments – Risk and Return (1926 – 1999)

Average Real Rate

of Return (%)

Risk (Standard

Deviation, %)

Common stocks

(S&P 500)

9.2 20.1

Long-term corporate

bonds

3.1 8.5

U.S. Treasury bills 0.7 3.1

Source: Stocks, Bands, Bills, and Inflation: 2007 Year book, Morningstar, Inc.



• Higher returns are associated with higher risks.

• The risk-averse investor must adjust his risk in

terms of his earnings.

Expected and Actual Return



• The investor chooses between investing in treasury bills

and stocks.

• Treasury bills (risk-free) and stocks (risky)

– Rf = risk-free return on the Treasury bill

Because the return is risk free, the expected

and actual returns are the same

Rm = expected return from investing in the stock

market

rm = the actual return

The Trade-off between Risk and Return



• At the time of making the decision on whether or

not to invest, we know the set of possible results

and the probability of each result occurring;

however, we do not know what specific result we

will end up with.

• The risky investment has a higher expected return

than the risk-free investment (Rm > Rf).

• If this was not the case, risk averse investors would

only invest in treasury bills.

The Trade-off Between Risk and Return



Distribution of savings

• b = the fraction of savings placed in the stock market

• 1 - b = fraction used to purchase Treasury bills

The Investment Portfolio

Expected return on a portfolio of assets (total assets):

Rp: is a weighted average of the expected return from the

two assets.

Rp = bRm + (1-b)Rf



• Expected Return:

When Rm = 12%, Rf = 4%, and b = 1/2,

Rp = 1/2(0.12) + 1/2(0.04) = 8%

• Question

– How should the investor should choose the fraction

b?




• The standard deviation of the risky stock market

investment is equal to the proportion of the portfolio

invested in risky investments times the standard

deviation of this portfolio:

mp b


needs to be chosen b



• choosing b:

fmp RbbRR )1(

)( fmfp RRbRR

The decision making problem of the investor


mp b


• Choosing :

p

m

fm

fp

RRRR

)(

mpb /

The decision making problem of the investor

We have , then: b



• Note that,

this last equation

is a budget line describing the trade-off between risk

and expected return .

The Risk and the Budget line

p( )

)p

(R

p

m

fmfp σ

σ

)R(RRR



• Note that

a) the expected return on the portfolio, Rp,

increases as the standard deviation of that

return, σp, increases.

b) the slope is equal to the price of the risk or the

tradeoff between risk and return.

The Risk and the Budget line


Choosing between Risk and Return

0 p return of

deviation Standard

Expected

Return Rp

The utility-maximizing investment

portfolio is at the point where the

indifference curve U2 is tangent to

the budget line, because this point

yields the highest expected feasible

return for a given level of risk.

Rf

Budget line

m

Rm

R*

U2 U1

U3


The Choices of two different Investors

Rf

Budget line

0

Expected return Rp

p return of

deviation Standard

Using the same budget

line, investor A chooses

the combination with

lower return and lower

risk, while investor B

chooses the combination

with higher return and

higher risk.

UA

RA

A

UB

RB

m

Rm

B


Investing in the Stock Market

• Notice

– The percentage of American families that invested directly

or indirectly in the stock market:

1989 = 32%

1995 = 41%

– Their share of total assets invested in the stock market:

1989 = 26%

1995 = 40%


Investing in the Stock Market

• Note that:

– Participation in the stock market according to age

Younger than 35 years

– 1989 = 23%

– 1995 = 29%

Older than 35 year

– Slight increase


Dividend Yield and P/E Ratio for S&P 500


Concluding Remarks

• Consumers and managers often make decisions

when there is uncertainty about the future.

• Consumers and investors are concerned about the

expected value and the variability of uncertain

results.

• In a case of uncertainty, consumers maximize their

expected utility by using the respective weighted

probabilities of the different outcomes.

• A person may be risk averse, risk neutral, or risk

loving.


Concluding Remarks

• The maximum sum of money that a risk-averse

person would pay to avoid a risk is called the risk

premium.

• Risk can be reduced by diversification, the purchase

of insurance, or the procurement of additional

information.


Concluding Remarks

• The law of large numbers allows insurance companies to offer actuarially fair insurance in which the premium paid corresponds to the expected value of the insured loss.

• Consumption theory can be applied to decisions regarding investments in risky assets.

• Individual behavior is not always predictable.

5. Uncertainty and Consumer Behavior

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