Risk & Uncertainty in Decision Making. Topics to be discussed Describing & Differentiating Risk from Uncertainty Role of Risk in Crime Deterrence Preferences.

Risk & Uncertainty in Risk & Uncertainty in Decision MakingDecision Making

Topics to be discussed

• Describing & Differentiating Risk from Uncertainty

• Role of Risk in Crime Deterrence

• Preferences Toward Risk

• Mechanisms for Reducing Risk

• Demand for Risky Assets (optional)

Describing & Interpreting Risk

• To measure risk one must know:

1) All of the possible outcomes.

2) The likelihood that each outcome will occur (its probability).

• In case of uncertainty, either (1) or even both (1) and (2) are unknown.

• Objective Interpretation– Based on the observed frequency of past events

• Subjective Interpretation– Based on perception or experience with or without an

observed frequency– Different information or different abilities to process the

same information can influence the subjective probability

Describing Risk: Expected Value

• The weighted average of the payoffs or values resulting from all possible outcomes.

– The probabilities of each outcome are used as weights

– Expected value measures the central tendency; the payoff or value expected on average

– E(X) = p1x1 + p2x2 + …..+ pnxn

• While the expected values are the same, the variability is not.

• Greater variability from expected values signals greater risk.

• Deviations or differences between expected payoff and actual payoff are important

– The standard deviation measures the square root of the average of the squares of the deviations of the payoffs associated with each outcome from their expected value.

Standard deviation measuring risk

Why SD important?

f(X), g(Y) = prob. densities

X, Y

E(X)=E(Y), but sd(X) >sd(Y)

Which distribution would you prefer, if you are risk-averse?

Example of a risky situation

• Job 1 is a job in which the income ranges from $1000 to $2000 in increments of $100 that are all equally likely.

• Job 2 is a job in which the income ranges from $1300 to $1700 in increments of $100 that, also, are all equally likely.

– Job 1: greater spread & standard deviation

– Job 2: peaked distribution - extreme payoffs are less likely

• Decision Making

– A risk avoider would choose Job 2: same expected income as Job 1, but with less risk.

Role of Risk in Crime Deterrence• Suppose, Extent of crime=r• Gain per unit of crime=g• Probability of detection & conviction =P(r)• Cost to guilty if detected & convicted=C• Hence, optimizing exercise of the criminal is to

– Maximize E(net gain) = g.r – P(r).C w.r.t. r– FOC=> g – P’(r).C = 0 => g = P’(r).C => r = r (g, C, P)

(+) (-) (-)

– SOC=> -P’’(r)<0 => P’’(r)>0 => you need some curb to keep crime rate r at finite level (check C(r) with C’’(r)>0 and/or g(r) with g’’(r)<0 can serve similar purpose)

– Observations: (i) A rise in P and drop in C or a drop in P with rise in C can serve the same purpose; (ii) With very high C, E(net gain) < 0 even with low P(r) close to zero; (iii) Societal cost to make P(r) = 1 or even close to 1 is very high; (iv) If P(r) = 0 then E(net gain) >0 for all values of r & r tends to be infinite.

– Why then capital punishment sometimes doesn’t work as deterrent to even heinous crimes? (look into text of example 6.2 for answer)

Role of Risk in Crime Deterrence: Figure

r

g.r, P.C

g.r

P(r).C with P’(r)>0 & P’’(r)>0

r*=equilibrium level of crime

Preferences Toward Risk

• Assumptions for choosing among risky alternatives– Consumption of a single commodity– The consumer knows all probabilities– Payoffs measured in terms of utility– Utility function given

• Defining risk averseness – i.e., preferences toward risk– A person who prefers a certain given income to a

risky income with the same expected value.– A person is considered risk averse if he has a

diminishing marginal utility of income– The use of insurance demonstrates risk aversive

behavior– If an individual gets more utility from his present

lower-paid job than a higher-paid risky job, he is said to be risk averse

Income ($1,000)

Utility The consumer is riskaverse because she

would prefer a certainincome of $20,000 to a

gamble with a 0.5 probabilityof $10,000 and a 0.5

probability of $30,000.

E

10

10 15 20

1314

16

18

0 16 30

AB

C

D

Risk Averse Preferences Toward Risk

Risk averseness meansgain valued less thanloss at the margin

Income ($1,000)10 20

Utility

0 30

6A

E

C

12

18

The consumer is riskneutral as he is indifferent

between certain eventsand uncertain events

with the same expected income.

Risk Neutral Preferences Toward Risk

A person is said to be risk neutral if he shows no preference between a certain income, and an uncertain one with the same expected value. Risk neutrality means

gain valued same asloss at the margin

Income ($1,000)

Utility

0

3

10 20 30

A

E

C8

18The consumer is riskloving because she

would prefer the gamble to a certain income.

Risk Loving Preferences

A person is said to be risk loving if he shows a preference toward an uncertain income over a certain income with the same expected value. Examples: gambling, criminal activity

Risk loving behavior meansgain valued more thanloss at the margin

Read example 6.3 carefully to understand why the US is becoming a nation of gamblers & poor nations too are prone to thesame virus!

Notion of Risk Premium

• The risk premium is the amount of money that a risk-averse person would pay to avoid taking a risk.

• R = U[E(Y)] – E[U(Y)] in utility terms

• It can also be measured along the horizontal axis in terms of money

• The greater the variability – i.e., the spread of values of the random variable around mean, the greater the risk premium.

• The greater the concavity of the utility curve, the greater the risk premium – it means a much higher weight is assigned to a decline in income as compared to an equal amount of rise in income

Income ($1,000)

Utility

0 10 16

Here , the risk premiumis $4,000 because a

certain income of $16,000gives the person the same

expected utility as the uncertain income that

has an expected value of $20,000.

10

18

30 40

20

14

A

CE

G

20

F

Risk Premium in money terms

Estimating Risk Premium

Risk premium in utility terms

Meaning of High Degree of Risk Aversion

Standard Deviation of Income

ExpectedIncome

Here increase in Standard deviation requires a large increase in income to maintainsatisfaction.

U1

U2

U3

Meaning of Low Degree of Risk Aversion

Standard Deviation of Income

ExpectedIncome

Slightly Risk Averse =>A large increase in standarddeviation requires only a small increase in incometo maintain satisfaction.

U1

U2

U3

Indifference curves of risk-neutral & risk-loving

people• Risk-neutrality means horizontal

indifference curve showing zero premium the consumer is willing to pay to buy hedge against risk (i.e., he doesn’t mind bearing more risk)

• Risk-lover willing to pay (i.e., make sacrifice in terms expected income) to enjoy the thrill of greater risk-bearing – thus making indifference curves usual downward sloping

Utility function of a person who buys insurance and gambles

too!Total utility

Income

A

Near point of inflection, A (MUat its minimum), a rise averter will spend small amount to buy insurance against small chance of large loss, while as a risk seekerspend a small amount (on lottery)to seize small chance of a largegain.

TU curve

Concave to horizontal axis

Convex to horizontal axis

Risk Reduction Strategies

• Three ways consumers attempt to reduce risk are:

1) Diversification

2) Insurance

3) Obtaining more information

• Firms can reduce risk by diversifying among a variety of activities that are not closely related.

Reducing Risk thro’ Diversification

• Diversification– Reducing risk by allocating resources to a

variety of activities whose outcomes are not closely related

• Example: – Suppose a firm has a choice of selling air

conditioners, heaters, or both– The probability of it being hot or cold is 0.5– How does a firm decide what to sell?

Income from Sales of Appliances

Hot Weather

Cold Weather

Air conditioner sales

$30,000 $12,000

Heater sales

12,000 30,000

Diversification – Example

• If the firm sells only heaters or air conditioners their income will be either $12,000 or $30,000

• Their expected income would be:– 1/2($12,000) + 1/2($30,000) = $21,000

• If the firm divides their time evenly between appliances, their air conditioning and heating sales would be half their original values

• If it were hot, their expected income would be $15,000 from air conditioners and $6,000 from heaters, or $21,000

• If it were cold, their expected income would be $6,000 from air conditioners and $15,000 from heaters, or $21,000

• With diversification, expected income is $21,000 with no risk• Better off diversifying to minimize risk• Firms can reduce risk by diversifying among a variety of

activities that are not closely related

Reducing Risk – The Stock Market

• If invest all money in one stock, then take on a lot of risk– If that stock loses value, you lose all

your investment value

• Can spread risk out by investing in many different stocks or investments– Ex: Mutual funds

Choosing an Investment Portfolio

Rate of return(%)

Risk

Highest possible indifference curve

E= less return, less risk investment

F=high return, high risk investment

C=optimum portfolio choice

Risk-return frontier

Reducing Risk – Insurance

• Risk averse are willing to pay to avoid risk• If the cost of insurance equals the expected loss, risk averse

people will buy enough insurance to recover fully from a potential financial loss

• For the risk averse consumer, guarantee of same income regardless of outcome has higher utility than facing the probability of risk

• Expected utility with insurance is higher than without

• Purchases of insurance transfers wealth and increases expected utility

– Actual Fairness: When the insurance premium = expected payout

Value of Title InsuranceWhen Buying a House : An

Example

• In the absence of title insurance, a risk averse buyer would pay much less for the house

• By reducing risk, title insurance thus increases the value of the house by an amount far greater than the premium.

The Decision to Insure – An example

How does insurance work? The Law of Large

Numbers• Insurance companies know that although single

events are random and largely unpredictable, the average outcome of many similar events can be predicted

• When insurance companies sell many policies, they face relatively little risk

• Insurance companies can be sure total premiums paid will equal total money paid out

• Companies set the premiums so money received will be enough to pay expected losses

Why no insurance against certain events?• Some events with very little probability

of occurrence such as floods and earthquakes are no longer insured privately– Cannot calculate true expected values and

expected losses– Governments have had to create insurance

for these types of events• Ex: National Flood Insurance Program

The Value of Information

• Risk often exists because we don’t know all the information surrounding a decision

• Because of this, information is valuable and people are willing to pay for it

• The value of complete information– The difference between the expected value

of a choice with complete information and the expected value when information is incomplete

The Value of Information – Example

• Per capita milk consumption has fallen over the years• The milk producers engaged in market research to develop

new sales strategies to encourage the consumption of milk • Milk advertising increases sales most in the spring• Allocating advertising based on this information in New York

increased profits by 9% or $14 million• The cost of the information was relatively low, while the

value was substantial (increased profits)• Findings

– Milk demand is seasonal with the greatest demand in the spring

– Price elasticity of demand is negative and small– Income elasticity is positive and large

• Assets: Something that provides a flow of money or services to its owner. However, the flow of money or services can be explicit (dividends) or implicit (capital gain).

• Capital Gain: An increase in the value of an asset, while a decrease is a capital loss.

• Risky Asset: Provides an uncertain flow of money or services to its owner. Examples: apartment rent, capital gains, corporate bonds, stock prices

• Risk-free Asset: Provides a flow of money or services that is known with certainty. Examples: short-term government bonds, short-term certificates of deposit

• Return on an Asset: The total monetary flow of an asset as a fraction of its price.

• Real Return of an Asset: The simple (or nominal) return less the rate of inflation.

• Asset Return: Return = Money flow / bond price

Demand for Risky Assets

Trade-Off Between Risk and Return

• Higher returns are associated with greater risk.• The risk-averse investor must balance risk relative to

return • An investor is choosing between T-Bills and stocks:

– T-bills (risk-free) versus Stocks (risky)– Rf = the return on risk free T-bills

• Expected return equals actual return when there is no risk

– Rm = the expected return on stocks– rm = the actual returns on stock

• The risky asset will have a higher expected return than the risk free asset (Rm > Rf).

• Otherwise, risk-averse investors would buy only T-bills.

The Investment Portfolio

• How to allocate savings:

b = fraction of savings in stock market

1 - b = fraction in T-bills

• Expected Return:

Rp: weighted average of the expected return on the two assets

Rp = bRm + (1-b)Rf

If Rm = 12%, Rf = 4%, and b = ½, then

Rp = 1/2(.12) + 1/2(.04) = 8%

The Investment Portfolio

• Risk (standard deviation) of the portfolio is the fraction of the portfolio invested in the risky asset times the standard deviation of that asset: σp = bσm

• Rp = bRm + (1-b)Rf => Rp = Rf + b(Rm-Rf)

=> Rp = Rf + (σp/σm)(Rm-Rf), which is a budget line describing the trade-off between risk (σp) and expected return (Rp).

• Observations:

– This is a straight line with Rp, Rf and σm as constants;

– Its slope= (Rm-Rf)/σm;

– Expected return, RP, increases as σp - i.e., risk increases.

Choosing betweenRisk and Return

0 p Return, of

Deviation Standard

ExpectedReturn,Rp

U2 is the optimal choice of those

obtainable, since it gives the highest

return for agiven risk and istangent to the

budget line.

Rf

Budget Line

m

Rm

R*

U2U1

U3

Rf

Budget line

The Choices ofTwo Different Investors

0

ExpectedReturn,Rp

p Return, of

Deviation Standard

Given the same budgetline, investor A chooses

low return-low risk, while investor B

chooses high return-high risk.

UA

RA

A

UB

RB

m

Rm

B

M

Can equilibrium occur at a point to the right of point M? Under what circumstances?

How will this figurechange for a risk-lover? Indifference curves will have the usual shape for two goods, rather than one good and one bad.

Hedging against Foreign Exchange Risk

• Read the example in pp.181-82 for details about– Forward contract &– Futures contract (standardized

forward contract for predetermined quantities & selected calendar dates)

– What happens if US$ & Yen continues to depreciate vis-à-vis Indian rupee?

Investing in the Stock Market: Example

• Observations– Percent of American families who had directly or

indirectly invested in the stock market• 1989 = 32%• 1995 = 41%

– Share of wealth in the stock market• 1989 = 26%• 1995 = 40%

– Participation in the stock market by age• Less than 35

– 1989 = 23%– 1995 = 29%

• More than 35– Small increase

Behavioral Economics• Sometimes individuals’ behavior

contradicts basic assumptions of consumer choice– More information about human behavior

might lead to better understanding– This is the objective of behavioral

economics• Behavioral economics is a developing field to help

explain and elaborate on situations not well explained by the basic consumer model

• Improving understanding of consumer choice by incorporating more realistic and detailed assumptions regarding human behavior

Behavioral Economics• There are a number of examples of

consumer choice contradictions– You take at trip and stop at a

restaurant that you will most likely never stop at again. You still think it fair to leave a 15% tip rewarding the good service.

– You choose to buy a lottery ticket even though the expected value is less than the price of the ticket