Chapter 9: Risk, Uncertainty and the Market for - PowerPoint Presentation

Slide1

Chapter 9: Risk, Uncertainty and the Market for Insurance

“The policy of being too cautious is the greatest risk of all”

Attributed to Jawaharlal Nehru (former Prime Minister of India)

“Risk comes from not knowing what you're doing.”

Warren BuffettSlide2

Concepts: Risk and Uncertainty

Definitions:

Risk

exists when there is a known probability of random variance in the outcome of a given action. For example, we know that the flip of a fair coin involves an equal chance of seeing a “heads” or a “tails, but risk implies that the outcome will vary randomly from one flip to the next.

Uncertainty

is defined as imperfect information about the outcome of a given action, whether or not the outcome involves risk. Slide3

Expected Value

The

expected Value of a single outcome

equals the probability that the outcome will occur times the payoff received if it does occur. If the probability of outcome

i

equals

p

i

, and the payoff equals

M

i

, then the expected value equals

p

i

•M

i

Expected Value (EV) of a risky decision

equals the sum of the expected values of all possible outcomes of the decision. In symbols,

EV =

Σ

i

p

i

•

M

i

, where

Σ

i

equals the sum of the

i

outcomes,

p

i

equals the probability of outcome

i

, and

M

i

equals the money payoff (positive or negative) from outcome

i

.

For

a case with three possible outcomes (

i

=1 through 3), this formula would equal

p

1

M

1

+ p

2

M

2

+ p

3

M

3

,

where

p

1

+ p

2

+ p

3

=

1. Slide4

Example: The daily number

Assume you buy a

one dollar ticket with a three digit number. Each digit is chosen separately and can equal 0 through 9. The resulting number can be any number between 000 and 999, giving 1,000 equally likely outcomes.

If you win,

the payoff is $500 minus the one dollar cost of

your ticket. Otherwise you lose $1.

The

expected value of this daily number bet equals 1/1000 x (500 -1) + 999/1,000 x (-1) = - 0.5. Slide5

Example: RouletteRoulette

involves

rolling a ball around a moving wheel and betting on where that ball lands.

A

U.S. roulette wheel includes the numbers 1-36 which are half black and half red, plus 0 and 00 slots which are green.

You lose if the green slots

A simple bet could be “high/low”, or “red/black” or “even/odd”

The payoff for winning is equal to the bet, so that if someone bets $10 she ends up with $10 more. Let’s use this information to answer the following questions: Slide6

Example: Roulette1. What are the odds of winning a single roll for

betting on “red”?

2. What is the expected value of a $1 bet on “red

”?Slide7

Example: Roulette1. What are the odds of winning a single roll for

betting on “red”?

18/38 or .4737

2. What is the expected value of a $1 bet on “red

”?

18/38 *$10 + 20/38*-$10 = -$.0526, or -5 ¼ centsSlide8

Concept: A “Fair Gamble”Concept: A fair gamble is a risky decision with an expected value of zero

.

This concept relates to our analysis of

attitudes

toward risk to follow

Example: Roll a 6 sided die (dice). You win $5 if it lands on your number, and you lose $1 if it doesn’t.

EV of Rolling a die: [1/6 • $5] + [5/6 • (-$1)] = $5/6 - $5/6, or 0

. Slide9

Decision TreesA decision tree is basically a chart of the possible outcomes of one or more related risky decisions, arranged in a branching pattern.

Diagrams

of risky decisions must include the odds and the payoffs for each possible outcome. A decision tree must also identify any places where decisions are made along with various points of risk. Slide10

Decision Tree Concepts

The

square at the left of the tree in Figure 9-2 represents a decision point and the circles represent points where either probabilities or payoffs are identified

.

Payoffs are given as dollar amounts on the right, and overall the tree provides a visual representation of an expected value problem.

Slide11

Reading a decision tree

The tree may be read from left to right.

The

choice presented is whether or not to flip a coin.

If

the person does not flip, the payoff is zero with complete certainty.

If

she does flip the coin, there is a ½ chance of winning $1.10 and ½ chance of losing $1. Slide12

Solving a decision tree problem

Solving a decision tree involves starting from the right and working backwards, keeping only those choices with the highest expected values or utilities, until an answer is reached.

The expected value of

the coin flip is ½•$1.10 + ½•-$1,

or $.

05, not flipping has a payoff of 0.

Replace the risky choice with the expected value, then choose whether to flip.

This person should flip. Slide13

Example: The “Iraq Game”

The decision is whether

or not to invade Iraq in

order to find and destroy

weapons of mass

destruction (WMD).

Assume a 50 percent

chance that these

weapons exist.

To solve, start at the

right, find any expected

Values, and solve the

d

ecisions on the right.Slide14

Solving the Iraq Game

After solving the set

of decisions on the right, the

Problem reduces to this graph.

Now calculate the final expected values for invading and not invading Iraq.

Based on these numbers

what should be done? Slide15

Example: Party or Study?

Your Turn 9-3

: Assume that you have a choice of 2 study strategies in the three nights before your next

exam;

(1) study seriously for all three nights, or (2) cram the night before the test. If you study seriously, you have a 75 percent chance of an A grade and a 25 percent chance of a B. Assign your own payoff values to these alternatives. If you cram on the last night, you will have 2 extra nights to party, but there is a 50 percent chance you will get a B grade and a 50 percent chance you will freeze and get a D. An A will be out of the question.

A. Draw a decision tree for this problem.

B. Using the values you assigned for A, B, and D grades, find the expected value for each of the two choices. Add a value for the two nights or partying to the cramming choice if you wish.

C. Which choice is best for you? Discuss how different values for partying affect your choice. Slide16

The Limits of Expected Value

Expected

value ignores people’s tastes regarding risk.

A person’s wealth also affects his/her choice of whether to accept a risky decision. Slide17

Two fair gambles (9-2) EV ($1 bet) = ½ • $

1

+ ½ •($-1)

=$0

(

9-3) EV ($1 million bet) = ½ •$

1,000,000

+ ½ •($-1,000,000)

= $0

Would you accept the first gamble? The second? If you have different answers, why? Slide18

Expected UtilitySlide19

Expected UtilityThe expected utility model adds two components to the expected value model.

First

, the values of each payoff are measured in terms of utility rather than dollars.

Secondly, it measures payoffs in terms of a person’s total wealth after an outcome occurs, rather than the value of the winning or losing payoff

itself, so an initial level of wealth (Wo) is added.Slide20

Modeling risk attitudesA person’s preferences toward risk may fit into one of three categories, risk averse, risk neutral, and risk preferring

.

Each will be represented by a relatively basic utility function.Slide21

Risk Aversion

In Figure 9-6, a risk averse person prefers a fixed outcome such as

Wo

over a fair gamble even though the expected values are equal.

A risk averse utility function

for wealth is

Slide22

Ronda: A risk averse person

Rhonda

has

$1,000 of initial wealth,

and may place

a $100 bet that a fair coin flip will come up heads. The odds are ½ of winning and ½ of losing, and the payoffs are + $100 and -$100.

Her utility function is

If she wins, she ends up with $1,100, and if she loses she ends up with $900.

Putting all the pieces of the expected utility model together, the expected utility of this fair gamble would be

….Slide23

Rhonda’s Decision

Since the utility of the initial wealth is higher, she will refuse the gamble and keep the initial wealth.Slide24

Risk Preferring Decisions

For a risk preferring person

the

utility of the initial wealth is less than the

expected utility

of the

gamble

because the added utility from winning (the thrill of victory?) is greater than the reduction in utility from losing.

A risk-loving utility of wealth

is

U(W

) = (Wealth)

2

Slide25

Randy, A risk lover

Randy

has initial wealth of $1,000, and is considering a $100 fair gamble on a coin flip. His expected utility for this bet is

EU=½ ($1000 + $100)

2

+ ½ ($1000 - $100)

2

= 605,000+405,000=1,010,000

utils

For comparison, the utility of his initial wealth is (1,000)

2

, or 1,000,000

utils

, less than the utility of the gamble.

So Randy will prefer a fair gamble over not gambling. Slide26

Risk Neutrality: Indifferent to accepting a fair gamble

A risk neutral utility function is a linear function of wealth, as in U = a W, where a is any constant.

Example:

Nat Neutral is considering the same $100 bet with even odds as Randy and Rhonda did above, and Nat has the same $1,000 in initial wealth. If Nat’s utility function is U = W, his expected utility problem will be

(9-9) EU = ½ (1000 + 100) – ½ (1000 - 100) = ½ (1,100) – ½ (900) = 1,000

utils

The utility of the initial wealth is also 1,000

utils

(check),

so Nat will be indifferent between accepting or refusing the fair gamble. Slide27

Your Turn 9-4: Some gambles

Assume you have $100 in spending money for the next week. A classmate offers a straight $20 bet on a coin flip. The expected value of the flip is zero.

A. Would you take the bet? Does this mean you are risk averse, risk preferring, or risk neutral?

B. Assume that you are risk

averse and

that the same classmate offers $22 if you win the bet. You would still lose $20. Calculate the expected utility using the square root utility function. Would you take the bet now according to the expected utility model? Would you take the bet in reality?

C. Now assume that you are risk preferring, with a utility function of U = (Wealth)

2

. If the same classmate offered you $18 if you win, would you still take the bet according to the model? In reality? Slide28

Loss Aversion: Yet another utility function regarding risk

Under

Loss Aversion an

individual will be risk averse

toward

gains from an initial starting point, but will be have a risk preferring function

for losses

(

see Figure 9-8).

Therefore individuals

will lose significantly more utility from a small loss than they will gain from an equally small gain. Slide29

The Market For Insurance

For a market to exist, consumers must be willing to pay somewhat more than the cost of claims to the insurance companies.Slide30

Expected Loss to the Insurance Company

Expected loss

equals the odds of a loss times its dollar value. If a group of reckless drivers has a 1/10 chance of suffering a $2,000 fender-bender, the company’s expected loss would equal 1/10 x $2,000, or $200.

Expected loss is equal to expected value for a gamble where only losses occur. In this case the EV would equal

EV = 1/10 x -$2,000 + 9/10 x $0 = -$200

Only the sign differs between the expected loss and the expected value, but the word “loss” implies a negative value. Slide31

The individual’s maximum willingness to pay for insurance

A risk averse consumer will be willing to pay more to reduce risk than the expected cost of the risk itself. The concept of expected utility will be used to demonstrate this. Slide32

Rudy’s Insurance Demand

For Rudy Averse (Rhonda’s father), this problem would include

initial

wealth of $10,000, a 1/10 chance of suffering a $2,000 accident, and a risk

averse (square root)

utility function

. His expected utility without insurance would beSlide33

Certainty Equivalent

Concept: A Certainty Equivalent

is the amount of money with no risk which produces the same amount of utility as the expected utility of a gamble.

Since Rudy’s utility = the square root of his wealth or income, one can find the certainty equivalent by squaring his expected utility.

If Rudy has $9,789.92 with no risk of loss from an accident, he would be just as happy as he is with his current risk. Slide34

Maximum Willingness to Pay for Insurance

Assuming that Rudy is able to buy insurance that will cover 100% of his loss, the most he would be willing to pay is the difference between his initial wealth and his certainty equivalent.

Concept:

Maximum Willingness to Pay for Insurance

: A person’s maximum willingness to pay for insurance is the difference between a person’s initial wealth and her certainty equivalent.

T

he

maximum Rudy would be willing to pay for insurance is

$10,000 - $9,789.92, or $210.08.

A bill of that size would leave him indifferent between buying the insurance and facing the financial risk of an accident.Slide35

Your Turn

Rhonda

has a 1/10 chance of being in a wreck, and

her

financial loss

would

be $1,900. Assume that her initial wealth equals $10,000, and that she is also risk averse with a utility function of U

= .

A. Find Rhonda’s expected loss.

B. Find her expected utility without insurance.

C. Find her certainty equivalent.

D. Find her maximum willingness to pay

for insurance

. Slide36

Concepts and Issues in Insurance Markets

Concepts:

Community Rating

is the requirement that insurance providers charge the same rate to all clients, regardless of the client’s level of risk.

Adjusted Community Rating

limits rate differences to selected categories, and also sometimes limits the size of the rate differences that are allowed.

Adverse Selection:

Given equal insurance prices for individuals, those with low expected losses will often choose not to buy insurance coverage. Given perfect information, only the less healthy or less safe are likely to buy insuranceSlide37

An example using 2 different groups

All persons have initial wealth of $40,000 and a utility function U

=

. Members of the older group have a 1/5 (.2) chance of suffering $3,900

in

health care costs per year, while the young group on average has a 1/20 (.05) chance of suffering a $1,975

health cost.

STEPS:

Find the maximum willingness to pay and expected loss for each group

For

community rating

, take an average of the expected losses (or perhaps the maximums), and assume both groups will have to pay this average.

If insurance is optional

, compare the average cost to each group’s maximum willingness to pay. Which groups will buy and which will not? This is an example of adverse selection.

. Slide38

Mandated (required) insuranceIf both parties must buy insurance at the community (average) rate you calculated earlier, they will now have no risk but also less wealth because of the purchase.

Find the utility of their remaining wealth. Since both groups have the same initial wealth in this case, their new utility will be equal.

Compare this utility to their initial expected utility without insurance to see who is better off with the community rated insurance and who is worse off.

If you are having trouble, the answers are all in the text. Slide39

Other Risk ConceptsSlide40

Expected Net Benefits

Concept: Expected Net Benefits

= the net benefits of every possible policy outcome weighted by the odds of that outcome.

Formula:

Expected Net Benefits

=

Σ

i

p

i

•

PV

i

, where

Σ

i

equals the sum of all of the

i

possible outcomes,

p

i

equals the probability of outcome

i

, and

PV

i

equals the present value of the net benefits from outcome

i

. For a policy with three possible outcomes (

i

=1 through 3), this formula would equal

p

1

PV

1

+ p

2

PV

2

+ p

3

PV

3

.Slide41

Option Price and Option Value

Option Price

= the maximum amount a person is willing to pay for a risky policy or product before knowing the outcome that will actually occur. This equals the certainty equivalent of the risky decision.

Option Value

= the difference between the option price and the expected value of the risky policy or product. This also means the maximum a person would pay to reduce risk. Option value will be positive for a risk averse person and negative for a risk preferring person. Slide42

Your Turn 9-9:

Assume

that your campus has a crime problem, particularly at night. Your college or university is considering expanding its security force to provide regular foot patrols between 9 PM and 3 AM.

How

much

would you be willing to pay for this service?

What is the average willingness to pay for your class?

Assuming that your class’s answers are typical, use

the average willingness to pay and

the total number of

students on campus to

find the aggregate student willingness to pay for this expanded security service.

If each new security officer costs $30,000 per year, how many officers could your college afford to hire for this program? Slide43

Risk and the Discount RateSlide44

Concepts for the Capital Asset Pricing Model

Diversifiable risk:

Risk that is specific to a firm, industry, or locality which can be eliminated through a diversified portfolio of assets.

Non-diversifiable or systematic risk:

System-wide or market risk that cannot be eliminated through diversification.

Beta

(

β

):

The variance in an individual rate of return divided by the variance in the market rate of return.Slide45

Values of BetaA

beta greater than one means that the variance of the individual stock is greater than that of the market as a whole, while a stock with a beta less than one has a lower variance than the market. If an asset is negatively related to the movement of the market, the beta will be negative. A risk free asset will have a beta of zero. Slide46

The Capital Asset Pricing Formula

Capital Asset Pricing Formula:

Required

rate of return = risk free rate + [

Β

●

(market

rate-risk

free rate

)]

 Slide47

Uncertainty and Policy AnalysisSlide48

Two methods of including uncertainty in the analysis

Sensitivity analysis

involves assigning a set of different values to each uncertain variable and then calculating whether changes in these values change the policy decision

.

Quasi option value

is

the maximum a person would pay for new information that reduces uncertainty, or the minimum a person would accept to face added uncertainty. Slide49

Sensitivity Analysis Example: The Job Corps ProgramSlide50

Job Corps ExampleThe results for the 7 percent interest rate are given below. Your job is to find the net benefits for each benefit stream using the 3 percent discount rate. Under which assumptions is this program worth the money?

If the odds are ½ for each of the two benefit time periods, what is the expected value of the 40 year and 10 year benefit streams for each interest rate? Is the expected value of the benefits greater than the cost for either interest rate? Slide51

Quasi-Option Value Example

The Iraq problem displayed in Figures 9-4 and 9-5 can be used to provide an example of the benefits of waiting for new information

.

Waiting for

the

information that there were no WMDs before invading

Iraq would

have changed the problem from that presented in 9-4 to that shown in Figure 9-11.Slide52

The Effect of New Iraq Information

With the new information invading Iraq has an expected value of -100 and not invading has an expected value of

0.

The

quasi-option value

of waiting for the WMD information equals the difference between the value of the best decision with the information ($0) and the value of the best decision without the information

(-

100, as in Figure 9-5), which equals +$100.

Another

version of the Iraq game could be constructed by adding a second piece of new information, the fact that George W. Bush won the 2004 election despite invading. If that information is known, invading

again becomes

the best option. Slide53

Your Turn 9-14: Global Warming

Assume

that three

possible rates

of global temperature increase

are

1 degree per century, 3 degrees per century, and 8 degrees per century. Each has a probability of 1/3.

One choice

is to adopt a policy now that could prevent all future increases at a cost of 5 trillion dollars. The dollar benefits of this policy would equal 1/2 trillion dollars for one degree, 3 trillion dollars for 3 degrees, and 9 trillion dollars for 8 degrees.

Find

the expected net benefits of this policy.

Assume that by investing in new technology we can decrease the cost of the cleanup to 3 trillion dollars. Find the new expected value with this lower cost technology.

The quasi-option value of waiting for this new technology is the change in the net benefits produced by the policy. What is this value? Slide54

ConclusionThis chapter contains several analytical tools, most of which relate to two basic concepts, expected value and expected utility.

This chapter also analyzed issues associated with insurance markets such as the

relation

between equal prices (community rating) and adverse selection.

Our analysis of uncertainty also

teaches us that it is wise to be aware of the limitations of our knowledge about the net benefits of a policy, both now and in the future.