Finance 30210: Managerial Economics Risk, Uncertainty, and Information

Finance 30210: Managerial Economics

Risk, Uncertainty, and Information

Rationality Quiz: Which would you prefer

Question #1: You are offered the following choice

A. $1M in Cash

B. A lottery ticket with a 10% chance of winning $5M, an 89% chance of winning $1M and a 1% chance of winning nothing


A. A lottery ticket with an 11% chance of winning $1M

B. A lottery ticket with a 10% chance of winning $5M


A. $1M in Cash

B. A lottery ticket with a 10/11 chance of winning $5M

When dealing with, uncertain events, we need a way to characterize the level of “risk” that you face.

Expected Value refers to the “most likely” outcome (i.e. the average)

N

iiiVpxE

1

)(

Probability of Event i

Payout of Event i

Note: if all the probabilities are equal, then the expected value is the average.

N

iiiVpxE

1

)(


A. $1M in Cash


MMAE 1$)1)($1()(

MMMBE 39.1$)0)($01(.)1)($89(.)5)($10(.)(

When dealing with, uncertain events, we need a way to characterize the level of “risk” that you face.

Standard Deviation measures the “spread” around the mean – this is what we mean by risk.

N

iii xEVpxSD

1

2)()(

Probability of Event i

Squared difference between each event and the expected value

Note: Standard Deviation is the (square root of) the expected value of squared differences from the mean.

N

iii xEVpxSD

1

2)()(


A. $1M in Cash


0)1$1)($1()( 2 MMASD

207.1)39.10)(01(.)39.11)(89(.)39.15)(10(.)( 222 BSD

Choice A Choice B

Question Expected Value

Standard Deviation

Expected Value

Standard Deviation

#1 $1M 0 $1.39M 1.207

#2 $110K 1.317 $500K 1.744

#3 $1M 0 $4.5M 3.45

Risk versus Return

We can calculate the expected payout and the standard deviation for each choice.

Preferences towards risk

Suppose that you have a utility function defined as follows: IIU 2)( (Linear in Income)

Util

ity

Income

Suppose that this individual were to choose between:

$100 with certainty

A 50% chance of earning $200

Util

ity

Income


$100 with certainty

A 50% chance of earning $200IIU 2)(

Choice A gives this individual 200 units of utility with certainty

$0

200

0$100 $200

400

Choice B gives this individual a 50% chance at 400 units of utility

E(Utility) = 200

We would describe this individual as “Risk Neutral” IEUIUE )(


Suppose that you have a utility function defined as follows:

2)( IIU (Convex in Income)

U

I


$100 with certainty


U

I


$100 with certainty


Choice A gives this individual 10,000 units of utility with certainty

$0

10,000

0$100 $200

40,000

Choice B gives this individual a 50% chance at 40,000 units of utility

E(Utility) = 20,000

We would describe this individual as “Risk Loving” IEUIUE )(

20,000

2)( IIU


Suppose that you have a utility function defined as follows:

IIU )( (Concave in Income)

U

I


$100 with certainty


U

I


$100 with certainty


Choice A gives this individual 10 units of utility with certainty

$0

10

0$100 $200

14

Choice B gives this individual a 50% chance at 14 units of utility

E(Utility) = 7

We would describe this individual as “Risk Adverse” IEUIUE )(

7

IIU )(


Type Condition Preference

Risk Loving Choice B

Risk Neutral Indifferent

Risk Averse Choice A

0)('' IU0)('' IU

0)('' IU


$100 with certainty (Choice A)

A 50% chance of earning $200 (Choice B)

)()( BEAE )()( BSDASD

Choice A Choice B

Question Expected Value

Standard Deviation

Expected Value

Standard Deviation

#1 $1M 0 $1.39M 1.207

#2 $110K 1.317 $500K 1.744

#3 $1M 0 $4.5M 3.45

Back to our Quiz…

How would a rational, risk neutral individual answer this quiz?


A. A lottery ticket with an 11% chance of winning $1M

B. A lottery ticket with a 10% chance of winning $5M


A. $1M in Cash


If you look closely, you will see that both of the choices in Question #2 are 11% of the values in Question #3 (Question #2 gives you an 11% chance of obtaining the choices in question #3)

Your answer to Question #2 = Your answer to Question #3


A. $1M in Cash


Lets write these a bit differently


A. $1M in Cash


Question #1:

A: 11% Chance of a win ($1M)

(Consolation Prize of $1M)

B: 11% Chance of a 10/11 Chance of a win ($5M) (Consolation prize of $1M)

Question #3:


B: 10/11 Chance of a win ($5M)

Question #1:


(Consolation Prize of $1M)

B: 11% Chance of a 10/11 chance of a win ($5M)

(Consolation prize of $1M)

Question #3:


B: 10/11 Chance of a win ($5M)

If you look closely, you will see that both of the choices in Question #1 are 11% of the values in Question #3 – with the addition of a $1M consolation prize in the event of a loss

Your answer to Question #1 = Your answer to Question #3

Did you pass (Are you rational)?

Possibility #1: You are Risk Loving

Risk lovers prefer situations with more risk. Therefore, a risk loving person would always choose B

Possibility #2: You are Risk Neutral

Risk neutral people ignore risk and only look at expected payouts. Therefore, a risk neutral person would always choose B

Possibility #3: You are Risk Averse

Risk averse people try to avoid risk. Note that in each case, choice B offers a higher expected payout, but higher risk. Therefore, we can’t say which choice a risk averse person would make – we can only say that they will either always choose A or always choose B

What are the expected returns from the Lottery?

Lottery Dates Highest Jackpot

Expected Payout (per $1)

Powerball 4/92-1/03 $315M $.73

The “Big Game” 9/96-5/99 $190M $.78

California Lotto 10/86-1/02 $141M $.71

Florida Lotto 5/88-7/01 $106M $.95

Texas Lotto 11/92-1/03 $85M $.97

Ohio Super Lotto 1/91-7/01 $54M $1.00

NY Lotto 4/99-8/01 $45M $.69

Mass. “Millions” 11/97-8/01 $30M $1.15

* Grote, Kent and Victor Matheson, ‘In Search of a Fair Bet in the Lottery”

Who Plays the Lottery?

* Clotfelter, Charles, et al , “Report to the National Gambling Impact Study Commission”

Who Plays the Lottery?

* Clotfelter, Charles, et al , “Report to the National Gambling Impact Study Commission”

Lottery Data and Risk Aversion

The data on Lottery Participation Suggests that at low levels of income, utility is convex (low income individuals are risk loving), but becomes concave at higher levels of income.

Utility

Income

)(IU

Risk AverseRisk Loving

In other words, those who play the lottery are precisely those who shouldn’t!!

Risk Aversion and the Value of insurance

Suppose that the probability of being involved in a traffic accident is 1%. Further, the average damage from an accident is $400,000. How much would you be willing to pay for insurance? (For Simplicity, assume that you earn $400,000 per year

U

I$00

$400K

632 You are choosing between:

Income – Premium (with certainty)

A 1% chance of earning $0, and a 99% chance of earning $400K

IIU )(

(Income – Premium)

Risk Aversion and the Value of insurance

U

I$00

$391K $400K

632 IIU )(625

625001.63299.))(( IUEExpected Utility without insurance

What income level generates 625 units of happiness?

KI

I

391$625

6252

You would pay $9,000 for this policy

Can we make a deal?

We have already determined that you would pay up to $9,000 for this policy

Is it worthwhile for the insurance company to offer you this policy?

Expected Payout

000,4$000,400$01.0$99.)( PayoutE

The insurance company should be willing to sell this policy for any price above $4,000 (ignoring other costs)

Insurance Markets Rely on Risk Aversion to make mutually beneficial agreements

000,4$000,400$01.0$99.)( PayoutE

Risk Loving Risk Neutral Risk Averse

Consumer would pay price <$4,000

Consumer would pay price =$4,000

Consumer would pay price >$4,000

This is why there are mandatory insurance laws!

Suppose that there are two types of drivers (safe and unsafe). Safe drivers have a 1% chance of an accident ($400,000) cost while unsafe drivers have a 2% chance ($400,000) cost.

Safe (Cost = $4,000)

If the insurance agent can tell them apart, he charges each an amount (at least) equal to their expected cost. Would both policies be sold?

Unsafe (Cost = $8,000)

U

I$00

$391K $400K

632 IIU )(625

KI

I

391$625

6252

The safe driver would pay $9,000 for this policy

U

I$00

$384K $400K

632

619

IIU )(

KI

I

384$619

6192

The unsafe driver would pay $16,000 for this policy

Cost = $4,000 Cost = $8,000

Value = $9,000 Value = $16,000

If the insurance agent can tell them apart, the each is charged a price according to their risk

What If the insurance agent can’t tell them apart?

Safe Unsafe

000,4$P 000,8$P

Problems With Asymmetric Information

Adverse selection refers to situations where, prior to a deal being made, one party lacks information about the other that would be useful (The insurance agent can’t tell good drivers from bad drivers)

Suppose that the agent knows there are an equal number of good drivers and bad drivers

000,6$000,8$5.000,4$5.)( CostEP

Cost = $4,000 Cost = $8,000

Value = $9,000 Value = $16,000

With a $6,000 premium to both groups, the safe driver is penalized while the unsafe driver benefits

Safe Unsafe

000,6$P

What would happen if the unsafe driver had a 4% chance of getting in a wreck?

If, again, the agent knows there are an equal number of good drivers and bad drivers

000,10$000,16$5.000,4$5.)( CostEP

As before, we can calculate the expected cost to the insurance company of the unsafe driver

000,16$000,400$04.0$96.)( PayoutE

Value = $9,000

With a $10,000 premium, the safe drivers get priced out of the market!!

Adverse selection refers to situations where, prior to a deal being made, one party lacks information about the other that would be useful

How can we deal with adverse selection?

Signaling involves using visible data to classify individuals

Sports cars cost more to insure than sedans of equal value

Smokers pay more for life/health insurance

Banks use credit scoring to assess credit risk

Regulation attempts to eliminate the risk that adverse selection creates

“Lemon Laws” protect used car buyers

FDIC protects bank depositors

Mandatory car insurance keeps insurance prices from exploding

If the safe and unsafe drivers can be identified by the insurance company, they will be charged a rate according to their risk.

Safe (Cost = $4,000)

Unsafe (Cost = $8,000)

However, what if the safe driver chooses to become an unsafe driver (after all, he’s insured!)

Moral Hazard refers to situations where, after a deal is made, one party lack information about the behavior of the other party

How can we deal with moral hazard?

Optimal Contracting involves the structuring of deals to align individual incentives

Car insurance policies have a deductible

Banks add restrictive covenants to bank loans

Collaterals

Monitoring attempts to directly observe the other party

Regulatory agencies monitor banks

Some employers use timecards

Moral Hazard refers to situations where, after a deal is made, one party lack information about the behavior of the other party

Documents

Finance 30210: Managerial Economics Risk, Uncertainty, and Information