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
• 1M in Cash
• 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 lottery ticket with an 11 chance of winning
  1M
• A lottery ticket with a 10 chance of winning 5M

• Question 3 You are offered the following
  choice
• 1M in Cash
• 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)
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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• Question 1 You are offered the following
  choice
• 1M in Cash
• A lottery ticket with a 10 chance of winning
  5M, an 89 chance of winning 1M and a 1 chance
  of winning nothing

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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.
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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• Question 1 You are offered the following
  choice
• 1M in Cash
• A lottery ticket with a 10 chance of winning
  5M, an 89 chance of winning 1M and a 1 chance
  of winning nothing

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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
(Linear in Income)
Utility

• Suppose that this individual were to choose
  between
• 100 with certainty
• A 50 chance of earning 200

Income
9

• Suppose that this individual were to choose
  between
• 100 with certainty
• A 50 chance of earning 200

Utility
Choice A gives this individual 200 units of
utility with certainty
400
200
Choice B gives this individual a 50 chance at
400 units of utility E(Utility) 200
0
Income
0
100
200
We would describe this individual as Risk
Neutral
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Preferences towards risk
Suppose that you have a utility function defined
as follows
(Convex in Income)
U

• Suppose that this individual were to choose
  between
• 100 with certainty
• A 50 chance of earning 200

I
11

• Suppose that this individual were to choose
  between
• 100 with certainty
• A 50 chance of earning 200

U
Choice A gives this individual 10,000 units of
utility with certainty
40,000
20,000
Choice B gives this individual a 50 chance at
40,000 units of utility E(Utility) 20,000
10,000
0
I
0
100
200
We would describe this individual as Risk Loving
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Preferences towards risk
Suppose that you have a utility function defined
as follows
(Concave in Income)
U

• Suppose that this individual were to choose
  between
• 100 with certainty
• A 50 chance of earning 200

I
13

• Suppose that this individual were to choose
  between
• 100 with certainty
• A 50 chance of earning 200

U
Choice A gives this individual 10 units of
utility with certainty
14
10
7
Choice B gives this individual a 50 chance at 14
units of utility E(Utility) 7
0
I
0
100
200
We would describe this individual as Risk
Adverse
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Preferences towards risk

• Suppose that this individual were to choose
  between
• 100 with certainty (Choice A)
• A 50 chance of earning 200 (Choice B)

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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 lottery ticket with an 11 chance of winning
  1M
• A lottery ticket with a 10 chance of winning 5M

• Question 3 You are offered the following
  choice
• 1M in Cash
• 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 1 You are offered the following
  choice
• 1M in Cash
• A lottery ticket with a 10 chance of winning
  5M, an 89 chance of winning 1M and a 1 chance
  of winning nothing

• Question 3 You are offered the following
  choice
• 1M in Cash
• A lottery ticket with a 10/11 chance of winning
  5M

Lets write these a bit differently
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 cant 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?
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
In other words, those who play the lottery are
precisely those who shouldnt!!
Income
Risk Averse
Risk Loving
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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
632

• You are choosing between
• Income Premium (with certainty)
• A 1 chance of earning 0, and a 99 chance of
  earning 400K

0
I
0
400K
(Income Premium)
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Risk Aversion and the Value of insurance
Expected Utility without insurance
What income level generates 625 units of
happiness?
U
632
625
You would pay 9,000 for this policy
0
I
0
391K
400K
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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
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
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)
Unsafe (Cost 8,000)
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?
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U
632
625
The safe driver would pay 9,000 for this policy
0
I
0
391K
400K
U
632
619
The unsafe driver would pay 16,000 for this
policy
0
I
0
384K
400K
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If the insurance agent can tell them apart, the
each is charged a price according to their risk
Safe
Unsafe
Value 9,000
Value 16,000
Cost 4,000
Cost 8,000
What If the insurance agent cant tell them apart?
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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 cant tell good drivers from
bad drivers)
Suppose that the agent knows there are an equal
number of good drivers and bad drivers
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With a 6,000 premium to both groups, the safe
driver is penalized while the unsafe driver
benefits
Safe
Unsafe
Value 9,000
Value 16,000
Cost 4,000
Cost 8,000
What would happen if the unsafe driver had a 4
chance of getting in a wreck?
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As before, we can calculate the expected cost to
the insurance company of the unsafe driver
If, again, the agent knows there are an equal
number of good drivers and bad drivers
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 lack
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.
However, what if the safe driver chooses to
become an unsafe driver (after all, hes insured!)
Safe (Cost 4,000)
Moral Hazard refers to situations where, after a
deal is made, one party lack information about
the behavior of the other party
Unsafe (Cost 8,000)
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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