Demand Estimation and Forecasting Finance 30210: Managerial Economics.
Finance 30210: Managerial Economics Risk, Uncertainty, and Information.
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Transcript of Finance 30210: Managerial Economics Risk, Uncertainty, and Information.
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Finance 30210: Managerial Economics
Risk, Uncertainty, and Information
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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
Question #2: You are offered the following choice
A. A lottery ticket with an 11% chance of winning $1M
B. A lottery ticket with a 10% chance of winning $5M
Question #3: You are offered the following choice
A. $1M in Cash
B. A lottery ticket with a 10/11 chance of winning $5M
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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)
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Probability of Event i
Payout of Event i
Note: if all the probabilities are equal, then the expected value is the average.
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1
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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 $1M and a 1% chance of winning nothing
MMAE 1$)1)($1()(
MMMBE 39.1$)0)($01(.)1)($89(.)5)($10(.)(
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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.
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iii xEVpxSD
1
2)()(
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
0)1$1)($1()( 2 MMASD
207.1)39.10)(01(.)39.11)(89(.)39.15)(10(.)( 222 BSD
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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.
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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
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Util
ity
Income
Suppose that this individual were to choose between:
$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 )(
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Preferences towards risk
Suppose that you have a utility function defined as follows:
2)( IIU (Convex in Income)
U
I
Suppose that this individual were to choose between:
$100 with certainty
A 50% chance of earning $200
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U
I
Suppose that this individual were to choose between:
$100 with certainty
A 50% chance of earning $200
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
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Preferences towards risk
Suppose that you have a utility function defined as follows:
IIU )( (Concave in Income)
U
I
Suppose that this individual were to choose between:
$100 with certainty
A 50% chance of earning $200
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U
I
Suppose that this individual were to choose between:
$100 with certainty
A 50% chance of earning $200
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 )(
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Preferences towards risk
Type Condition Preference
Risk Loving Choice B
Risk Neutral Indifferent
Risk Averse Choice A
0)('' IU0)('' IU
0)('' IU
Suppose that this individual were to choose between:
$100 with certainty (Choice A)
A 50% chance of earning $200 (Choice B)
)()( BEAE )()( BSDASD
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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?
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Question #2: You are offered the following choice
A. A lottery ticket with an 11% chance of winning $1M
B. A lottery ticket with a 10% chance of winning $5M
Question #3: You are offered the following choice
A. $1M in Cash
B. A lottery ticket with a 10/11 chance of winning $5M
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
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Question #3: You are offered the following choice
A. $1M in Cash
B. A lottery ticket with a 10/11 chance of winning $5M
Lets write these a bit differently
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
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:
A: 100% Chance of a win ($1M)
B: 10/11 Chance of a win ($5M)
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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:
A: 100% Chance of a win ($1M)
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
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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
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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”
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Who Plays the Lottery?
* Clotfelter, Charles, et al , “Report to the National Gambling Impact Study Commission”
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Who Plays the Lottery?
* Clotfelter, Charles, et al , “Report to the National Gambling Impact Study Commission”
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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!!
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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)
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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
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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)
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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!
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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)
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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
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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
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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
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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?
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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!!
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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
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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
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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