Beslut under risk

Question 1

Decision under risk and uncertainty

Answer 1

Definitions
Risk - known probabilities (main focus)
Uncertainty - unknown probabilities (e.g., probability of an accident
with a new type of airplane or occurrence of natural disasters)

In real-life many decisions involve risk or uncertainty
Income (e.g., employment)
Prices (e.g., energy prices)
Quality of products (e.g., used car)

Factors affecting risk and uncertainty
State of the individual (e.g., health)
State of nature (e.g., flooding)
Markets and institutions (e.g., liability and guarantees, and
unemployment insurance)

Question 2

Risk reduction

Answer 2

People often would like to avoid risk and are willing to pay for risk reduction

Risk reduction
Insurance
Risky (or uncertain) outcomes can be sold to an insurance company
(e.g., fire insurance or unemployment insurance)
Risky (or uncertain) prices (e.g., the price of agricultural commodities) can
be insured against using financial markets (e.g., options and futures)
Behavior (e.g., wearing a helmet)
Diversification (”not all eggs in the same basket”)
Risk pooling (e.g., informal insurance schemes)

Question 3

Asymmetric information

Answer 3

Adverse selection
Moral hazard (ex-ante and ex-post)

Question 4

Decision under certainty

Answer 4

Utility maximization problem
Maximize utility subject to budget constraint
Certainty about income, prices, quality of goods and state of the world

Question 5

Decision under risk (known probabilities)

Answer 5

Utility maximization problem
Maximize expected utility subject to budget constraint
Risky in terms of the state of the world, but outcomes in a given state
is certain (common approach)

Question 6

Example lottery

Answer 6

Which lottery option would you choose?
Lottery: 50% of winning 5000 and 50% of winning 1000
For sure: 3000

You are offered a summer job selling ice cream. Which of the
following two payment schemes would you choose?
Your income depends on the sale. If there is sunshine your income will
be 5000 and if it rains it will be 1000. There is a 50% probability of
sunshine and a 50% probability of rain
For sure: 3000

Some people choose the lottery (risk lovers), and some are indifferent (risk
neutral) and some choose for the sure amount (risk averse) (note that the
the expected value of the lottery is equal to 3000)

The first-choice situation can be described as a context-free gamble.

Question 7

Definitions of the lottery

Answer 7

Probability - likelihood that a discrete event will occur
n possible events

pi is the probability that event I happens
p1 + p2 + …+ pn = 1
0 ≤pi ≤1

Probabilities are modeled as objective (sometimes modeled as
subjective, e.g., own assessment of the probability of an accident)

Outcome
xi is the outcome if event I happens (e.g., income)

Expected value-weighted average of outcomes
EV = p1 ×x1 + p2 ×x2 + …+ pn ×xn

The expected value of a lottery with two possible events
EV = p1 ×x1 + (1 −p1) ×x2

Question 8

Risk aversion and demand for insurance

Answer 8

In general, people dislike risks
If an individual is risk averse, then she prefers a for a sure amount
rather than the same amount as the expected value from a lottery

An individual prefers 3000 for sure over a lottery with 50% of winning
5000 and 50% of winning 1000

Insurance is a large industry since people are risk-averse and would
like to sell their risks to receive a sure income

Question 9

Expected Utility

Answer 9

Expected utility
EU = p1 ×u(x1) + p2 ×u(x2) + …+ pn ×u(xn )
, where u(xi ) is the utility of outcome i occurs

EU (=expected utility) is the sum of the utility of outcomes times
their probabilities

Choices affect consequences and only consequences matter for utility, for example, choices are assumed to be context independent

When the state of nature is realized, then an individual will experience
a certain level of utility depending on the outcome in that state

Question 10

Risk averse, risk neutral and risk loving

Answer 10

Question 11

Summary of risk types

Answer 11

Question 12

Utility of wealth - Risk averse and risk neutral

Answer 12

Question 13

Utility of wealth - Risk averse and risk loving

Answer 13

Question 14

Expected utility - Example risk averse

Answer 14

Adam has the following utility function u = √W (assume initial wealth=0)

Adam is offered a lottery with a 50% chance of winning 5000 and a 50%
chance of winning 1000 OR 3000 for sure. What will he choose?

For sure
(sure) = U (3000) = √3000 = 54.8

Lottery
EU = 0.5 ×√1000 + 0.5 ×√5000 = 0.5 ×31.6 + 0.5 ×70.7 = 51.2

Adam chooses the for-sure amount since
U (Sure) >EU (Lottery )(54.8 >51.2)

If we did not know Adam’s risk preferences, we could conclude that he is risk averse since he prefers 3000 for sure over a lottery where
EV = 0.5 ×1000 + 0.5 ×5000 = 3000

Question 15

Expected utility - Example risk neutral and risk loving

Answer 15

Bill is risk neutral and has the following utility function U = 0.1 ×W

For sure
U (3000) = 0.1 ×3000 = 300

Lottery
EU = 0.5 ×0.1 ×1000 + 0.5 ×0.1 ×5000 = 300

Bill is indifferent between the lottery and the for a sure amount

Anna is risk-loving and has the following utility function
U = 0.01 ×W 2

For sure
U (3000) = 0.01 ×30002 = 90000

Lottery
EU = 0.5 ×0.01 ×10002 + 0.5 ×0.01 ×50002 = 130000
Anna chooses the lottery
Decision under risk and uncertainty

Question 16

Behaviour and risk

Answer 16

Bounded rationality and heuristics
Herbert Simon argued that people make “good enough” decisions,
that is decisions that are satisfaction
To make satisfying decisions, people use heuristics (simple and efficient
rules) rather than complex optimization. Satisfying decisions
sometimes deviate from what is expected from utility maximization, and the deviation is labeled “bias”.

Inconsistent choices with expected utility
Certainty effect
Representativness
Anchoring
Gains and losses
Risk as a feeling

Question 17

Gains and losses

Answer 17

Summary of the lotteries
A=Lottery: 50% 2000 OR 50% 1000=C
B=For sure: 1500=D

People are risk averse in gains and risk loving in looses

Expected utility theory cannot explain these choices

Tversky and Kahneman developed prospect theory based on these
findings that individuals seem to make different risky decisions
depending on if it is a potential gain (risk averse) or loss (risk loving) relative to their specific situation (the reference point).

Question 18

Factors influencing the perception of risk and uncertainty

Answer 18

Risk as a feeling (Slovic, 1987)
Dread (examples)
Not Fatal - Fatal
Not Global Impact - Global Impact
Voluntary - Involuntary
Controllable - Uncontrollable
Individuals Only - Catastrophic
Easily Reduced - Not Easily Reduced

Familiarity (examples)
Observable - Not Observable
Immediate Effect - Delayed Effect
Old Risk - New Risk
Known to Science - Unknown to Science

Question 19

Risk pooling and law of large numbers

Answer 19

People make an agreement to share (gains and) losses equally to
reduce risk

Law of large numbers
As the number of members in a group increases, the predictions about
the group’s average (gains and) losses become more accurate, which is
closer to the expected value (population means (or proportion)).
Note members are assumed to be independent (if one member faces a (gain or loss) does not affect the probability of others facing a (gain or loss).

Example - Flipping a fair coin (expected value of heads is 0.5)

Question 20

Insurance

Answer 20

Insurance is a transfer of the risk of a loss in exchange for a premium

An insurer (insurance company) sells the insurance policy

An insured (policyholder) is the buyer of the insurance policy

There are many types of insurance policies
Car insurance
Life insurance
Health care insurance
Fire insurance
Maritime insurance

An insurance policy can have different coverage
Full coverage
Co-payment
Deductible (policy covers all but first X SEK of damages)

Question 21

Certainty equivalent - Example

Answer 21

Certainty equivalent: The sum of money that would make an individual indifferent between the lottery and that sum of money.

Adam has the following utility function u = √W (assume initial wealth=0)

Adam is offered a lottery with a 50% chance of winning 5000 and a 50%
chance of winning 1000 OR 3000 for sure.

Adam’s EU from the lottery is 51.2. Which for sure monetary amount does this correspond to?

U = √x = 51.2
U = √x 2 = 51.22 = 2621.4

Adam is indifferent between 2621.4 and the lottery
Risk premium: The amount a risk-averse individual is willing to pay to avoid risk (EV = 3000)

3000-2621.4=378.6

Question 22

Insurance premium - Example

Answer 22

How much is Adam maximum willing to pay for an insurance policy that compensates him fully if a bad outcome?

Full compensation means that he will either win 5000 in the lottery OR win 1000 but he also receives 4000 from the insurance policy as compensation for the bad outcome.

The certainty equivalent is 2621.4

The maximum insurance premium Adam is willing to pay for the insurance policy is:

5000-Insurance premium=2621.4

Insurance premium=2378.6 = Expected loss + Risk premium = 2000+378.6

After purchasing the insurance and after the lottery, Adam will have 5000 (or 1000+4000 if bad outcome) - 2378.6 = 2621.4 on his bank
account, which is equal to the certainty equivalent.

Question 23

Insurance premium - Example Context

Answer 23

Adam is a farmer and there is a 50% chance that harvest yields an income of 5000 and 50% chance of 1000.

The harvest situation can be described in terms of a lottery
Adam’s EU for such a lottery is 51.2
Adam is indifferent between the lottery and a sum of 2621.4

How much is Adam maximum willing to pay for an insurance that compensates him fully if a bad harvest?

Full compensation means that he will either harvest during good conditions which yields 5000 or in bad conditions which yield 1000 and 4000 from the insurance company.

The certainty equivalent is 2621.4

The maximum insurance premium Adam is willing to pay is:
5000-Insurance premium=2621.4
Insurance premium=2378.6

Question 24

Example - Smoke alarm (self-insurance)

Answer 24

Suppose your flat is worth 3000000
There is a chance (5%) that a fire will damage your flat and generate a loss of 50000 (expected loss is 0.05 ×50000 = 2500)

Your utility function can be described as: u(w ) = log (w )

How much is your maximum willingness to pay for a smoke alarm that would save your flat from any damages in case of a fire?

Without a smoke alarm
Expected value
E (W ) = (1 −p) ×w + p ×(w −l )
E (w ) = 0.95 ×(3000000) + 0.05 ×(3000000 −50000) = 2997500

Expected utility
EU = (1 −p) ×log (w ) + p ×log (w −l ) =
EU = 0.95 ×log (3000000) + 0.05 ×log (3000000 −50000) = 6.477

With a smoke alarm that costs C

Value
w −C =
3000000 −C

Utility
U = log (w −C )
U = log (3000000 −C )

How much is your maximum willingness to pay for the smoke alarm?
Utility without a smoke alarm is: EU = 6.477
Utility with the smoke alarm is: U = log (w −C )

Maximum price
log (3000000 −C ) = 6.477
3000000 −C = 106.477
3000000 −C = 2997480
C = 2520

Willingness to pay for the smoke alarm is a maximum of 2520 (and
certainty equivalent is 2997480).

If we observe an individual buying a smoke alarm only when the price is below 2500 (=risk lover), maximum of 2500 (=risk neutral), and above 2500 (=risk averse) based on their revealed preferences.

Other explanations include liquidity constraints and behavioral aspects.

Question 25

Example - Maximum willingness to pay for an insurance

Answer 25

Ann earns 10000. There is a 10% chance of a car accident, which results in a cost of 10000.

Ann has the following utility function: U = √W

How much is Ann’s maximum willingness to pay for insurance coverage?

In other words by how much can her income be reduced to be
indifferent with a sure income and facing a 10% chance of a car accident resulting in a loss of 10000.

Without insurance
EV = 0.9 ×10000 + 0.1 ×0 = 9000
EU = 0.9 ×√10000 + 0.1 ×√0 = 90

Maximum willingness to pay for an insurance (=c)
U = √(10000 −c ) = 90
10000 −c = 902
10000 −c = 8100
1900 = c
Ann’s maximum willingness to pay for insurance is 1900

Question 26

Insurance company

Answer 26

The profit function is:
Profit = Insurance Premium −Loss −Operating costs

Insurance premium = Amount paid for the policy by the insured

Loss = Claims by the insured

Operating costs = Cost to run the insurance company including
processing claims

Actuary fair premium = Expected loss

Characteristics of the good to be insured

A large homogenous group exposed to the same risk

A quantifiable loss in monetary terms (and not subject to catastrophic losses)

Risks must be independent
Systematic risk - risks are uncorrelated (e.g. cars)
Idiosyncratic risk - risks are correlated (e.g. houses along the river)

The loss must be accidental (compare moral hazard)

Risk management by the insurance company

Risk pooling and the law of large numbers (compare adverse selection)

Diversification by spreading the risks

Reinsurance by selling some of the risks to a re-insurance company, which aggregates the risk at a higher level (e.g., insure house along rivers across the world) (e.g., Lloyds and MunichRe)

Question 27

Example - Insurance company

Answer 27

An actuary fair premium is 1000 (the probability of an accident is 10% and the cost incurred is 10000).

Assume that the insurance company insured many individuals similar to Ann

Each ”Ann” is willing to pay 1900 and the expected loss is 1000.
Assume operating cost=0

Expected profit (per policy ) = Insurance Premium −Expected loss

Expected profit (per policy ) = 1900 −10000 ∗0.1 = 900

The expected profit per policy is 900 and it is due to each ”Ann” being risk averse (corresponding to ”Ann’s” risk premium).

Insurance companies rely on the law of large numbers to spread risks, that is to insure many ”Anns”.

If the insurance company only insures Ann, the financial outcome is either a profit of 1900 (if no claim) or a loss of 8100 (if a claim)

The insurance company faces operating costs so profit is:
Profit (per policy ) = Premium −Loss −Operating costs

Assume that the average operating cost per policy is 200 and shareholders would like a profit of 50.

What is the lowest insurance premium the insurance company can set?

Lowest premium
Profit (per policy ) = Insurance Premium−Expected loss −Operating costs

50 = Insurance Premium −1000 −200

Insurance Premium = 1250

How is Ann affected by this insurance premium level?
Ann’s utility with an insurance policy with a premium of 1250
U = √(10000 −1250) = 93.54

The utility with an insurance policy with an insurance premium of 1250
is 93.54.

This is a higher utility than without insurance or with insurance
with a premium of 1900 since both yield a utility of 90.

Question 28

Adverse selection

Answer 28

People are heterogeneous in terms of the risk of a loss as well as in terms of the monetary cost of a loss

If the insurer cannot identify the risk and the cost parameters at the individual level, the insurer offers the same insurance policy to everyone using the expected value.

The “very good” risks will not purchase the insurance (unless they are extremely risk-averse). Thus, if the low-risk people drop out, then the insurer has to re-calculate the insurance premium which as a result will increase.

This process will continue until only the high-risk individuals remain insured. Thus, many people will not have any insurance coverage.

Markets for lemons - Akerlof

Question 29

Moral hazard

Answer 29

Ex-ante moral hazard
An individual changes behavior when insured such that the probability of a loss is changed (e.g., the decision where to park the car)

Ex-post moral hazard
When insured, an individual requires the best compensation (e.g., more expensive health treatments than otherwise).