CHAPTER 5
Provide definition, meaning and example with all. Diagram where needed
must draw.
1. What does it mean to say that a person is risk averse? Why are
some people likely to be risk averse while others are risk lovers?
Risk Aversion
Definition: A person is risk averse if they prefer a certain outcome over a gamble
with a higher or equal expected value.
Meaning: Risk-averse individuals dislike uncertainty and will avoid gambles unless
they offer a significantly higher return.
Example: Given a choice between receiving $50 for sure or a 50% chance of winning
$100, a risk-averse person chooses the sure $50.
Diagram: Utility curve is concave, illustrating diminishing marginal utility of wealth.
2. Why is the variance a better measure of variability than the range?
Variance vs. Range
Definition: Variance measures the average squared deviation from the mean, while
range is the difference between the highest and lowest values.
Meaning: Variance accounts for all data points, providing a more comprehensive
measure of variability.
Example: For values [2, 4, 6], range is 4, but variance captures deviations of each
value from the mean.
3. George has $5000 to invest in a mutual fund. The expected return
on mutual fund A is 15 percent and the expected return on mutual
fund B is 10 percent. Should George pick mutual fund A or fund B?
Investment Decision
Expected Return A = 15%, Expected Return B = 10%
George should choose A if he is risk neutral or if both investments have similar risk.
However, if A is significantly riskier, he might prefer B.
4. What does it mean for consumers to maximize expected utility? Can
you think of a case in which a person might not maximize expected
utility?
Maximizing Expected Utility
Definition: Choosing the option that offers the highest expected utility rather than
highest expected monetary value.
Example: Buying insurance even if expected cost is higher than expected loss.
Exception: People may not maximize expected utility due to bounded rationality or
emotional biases.
5. Why do people often want to insure fully against uncertain situations
even when the premium paid exceeds the expected value of the loss
being insured against?
Insurance and Risk Aversion
Explanation: Risk-averse individuals value peace of mind and are willing to pay a
premium above expected loss to avoid uncertainty.
6. Why is an insurance company likely to behave as if it were risk
neutral even if its managers are risk-averse individuals?
Insurance Companies as Risk Neutral
Reason: Insurers diversify risks across many policies, making total risk predictable.
This aggregation mimics risk-neutral behavior.
7. When is it worth paying to obtain more information to reduce
uncertainty?
Value of Information
Worth paying for if expected gain from better decisions exceeds the cost of
obtaining information.
Example: Paying for a home inspection before purchasing a house.
8. How does the diversification of an investor’s portfolio avoid risk?
Diversification
Definition: Spreading investments across different assets to reduce exposure to any
single risk.
Example: Investing in stocks, bonds, and real estate rather than just stocks.
Diagram: Risk-return graph showing reduced portfolio variance through
diversification.
9. Why do some investors put a large portion of their portfolios into
risky assets while others invest largely in risk-free alternatives?
(Hint: Do the two investors receive exactly the same return on
average? If so, why?)
Risk Preferences in Portfolios
Reason: Different risk tolerances. Risk-tolerant investors seek higher returns,
accepting more volatility. Risk-averse prefer stability.
Return: Average may be similar, but risk-adjusted returns differ.
10. What is an endowment effect? Give an example of such an
effect.
Endowment Effect
Definition: People assign more value to things merely because they own them.
Example: A person demands $100 to sell a mug they own but wouldn’t pay more
than $50 to buy it.
11. Jennifer is shopping and sees an attractive shirt. However, the
price of $50 is more than she is willing to pay. A few weeks later, she
finds the same shirt on sale for $25 and buys it. When a friend offers
her $50 for the shirt, she refuses to sell it. Explain Jennifer’s
behavior.
Jennifer's Behavior
Explanation: Reflects endowment effect. Once Jennifer owns the shirt, she values it
more than when considering buying it, leading her to refuse selling it for $50.
EXCERCISE
1. Consider a lottery with three possible outcomes: • $125 will be
received with probability .2 • $100 will be received with
probability .3 • $50 will be received with probability .5 a. What is
the expected value of the lottery? b. What is the variance of the
outcomes? c. What would a risk-neutral person pay to play the
lottery?
Lottery Outcomes: a. Expected Value = (0.2*$125) + (0.3*$100) + (0.5*$50) =
$25 + $30 + $25 = $80 b. Variance = 0.2(125-80)^2 + 0.3(100-80)^2 + 0.5(50-
80)^2 = 0.2(2025) + 0.3(400) + 0.5(900) = 405 + 120 + 450 = 975 c. A risk-
neutral person would pay the expected value: $80.
2. suppose you have invested in a new computer company whose
profitability depends on two factors: (1) whether the U.S.
Congress passes a tariff raising the cost of Japanese computers
and (2) whether the U.S. economy grows slowly or quickly. What
are the four mutually exclusive states of the world that you
should be concerned about?
1. Four mutually exclusive states:
Tariff passed & slow growth
Tariff passed & fast growth
Tariff not passed & slow growth
Tariff not passed & fast growth
3. Richard is deciding whether to buy a state lottery ticket. Each
ticket costs $1, and the probability of winning payoffs is given as
follows: PROBABILITY RETURN .5 $0.00 .25 $1.00 .2 $2.00 .05
$7.50
a. What is the expected value of Richard’s payoff if he buys a
lottery ticket? What is the variance? b. Richard’s nickname is
“No-Risk Rick” because he is an extremely risk-averse
individual. Would he buy the ticket? c. Richard has been given
1000 lottery tickets. Discuss how you would determine the
smallest amount for which he would be willing to sell all 1000
tickets. d. In the long run, given the price of the lottery tickets
and the probability/return table, what do you think the state
would do about the lottery?
Lottery Ticket: a. Expected Value = 0.5(0) + 0.25(1) + 0.2(2) + 0.05(7.5) = 0
+ 0.25 + 0.4 + 0.375 = $1.025 Variance = E(x^2) - (E(x))^2 = 0.5(0) + 0.25(1)
+ 0.2(4) + 0.05(56.25) = 0 + 0.25 + 0.8 + 2.8125 = 3.8625 - (1.025)^2 = ~2.81
b. "No-Risk Rick" would not buy the ticket since expected utility < $1. c. Use
certainty equivalent and utility function to find smallest acceptable sum. d.
Long-run: State makes profit per ticket = price - expected payout = $1 - 1.025
= -0.025 → unsustainable.
4. Suppose an investor is concerned about a business choice in
which there are three prospects—the probability and returns are
given below:
PROBABILITY RETURN .4 $100 .3 30 .3 −30 What is the expected
value of the uncertain investment? What is the variance?
Investment Expected Value = (0.4100) + (0.330) + (0.3*-30) = 40 + 9 - 9
= $40 Variance = 0.4(100-40)^2 + 0.3(30-40)^2 + 0.3(-30-40)^2 =
0.4(3600) + 0.3(100) + 0.3(4900) = 1440 + 30 + 1470 = 2940
5. You are an insurance agent who must write a policy for a new
client named Sam. His company, Society for Creative Alternatives
to Mayonnaise (SCAM), is working on a low-fat, low-cholesterol
mayonnaise substitute for the sandwich-condiment industry. The
sandwich industry will pay top dollar to the first inventor to
patent such a mayonnaise substitute. Sam’s SCAM seems like a
very risky proposition to you. You have calculated his possible
returns table as follows: PROBABILITY RETURN OUTCOME .999 −
$1,000,000 (he fails) .001 $1,000,000,000 (he succeeds and sells
his formula) a. What is the expected return of Sam’s project?
What is the variance? b. What is the most that Sam is willing to
pay for insurance? Assume Sam is risk neutral. c. Suppose you
found out that the Japanese are on the verge of introducing their
own mayonnaise substitute next month. Sam does not know this
and has just turned down your final offer of $1000 for the
insurance. Assume that Sam tells you SCAM is only six months
away from perfecting its mayonnaise substitute and that you
know what you know about the Japanese. Would you raise or
lower your policy premium on any subsequent proposal to Sam?
Based on his information, would Sam accept?
SCAM: a. Expected Value = (0.999)(-1,000,000) + (0.001)(1,000,000,000) = -
999,000 + 1,000,000 = $1,000 Variance = High due to extreme values. b. A risk-
neutral Sam pays max of $1,000 for insurance. c. Knowing risk increased, you
would raise premium. Sam would not accept due to limited info.
6. Suppose that Natasha’s utility function is given by u(I) = 110I,
where I represents annual income in thousands of dollars. a. Is
Natasha risk loving, risk neutral, or risk averse? Explain. b.
Suppose that Natasha is currently earning an income of $40,000
(I 40) and can earn that income next year with certainty. She is
offered a chance to take a new job that offers a .6 probability of
earning $44,000 and a .4 probability of earning $33,000. Should
she take the new job? c. In (b), would Natasha be willing to buy
insurance to protect against the variable income associated with
the new job? If so, how much would she be willing to pay for that
insurance? (Hint: What is the risk premium?)
Natasha: a. Utility u(I) = 110I → linear → risk neutral b. E[I] = 0.6(44) + 0.4(33)
= 26.4 + 13.2 = 39.6 < 40 → no job change c. Risk premium = 40 - 39.6 = $400
→ would pay up to $400 for insurance
7. Suppose that two investments have the same three payoffs, but
the probability associated with each payoff differs, as illustrated
in the table below: PAYOFF PROBABILITY (INVESTMENT A)
PROBABILITY (INVESTMENT B) $300 0.10 0.30 $250 0.80 0.40
$200 0.10 0.30 a. Find the expected return and standard
deviation of each investment. b. Jill has the utility function U 5I,
where I denotes the payoff. Which investment will she choose? c.
Ken has the utility function U = 51I. Which investment will he
choose? d. Laura has the utility function U 5I 2 . Which
investment will she choose?
Investments: A: E = 0.1(300) + 0.8(250) + 0.1(200) = 30 + 200 + 20 = 250 Var
= ... SD = ~22.36 B: E = 0.3(300) + 0.4(250) + 0.3(200) = 90 + 100 + 60 = 250
Var = ... SD = ~35.36 b. Jill (U = sqrt(I)) → A has lower SD → chooses A c. Ken
(U = ln(I)) → prefers A for same reason d. Laura (U = I^2) → favors higher
variance → chooses B
8. As the owner of a family farm whose wealth is $250,000, you
must choose between sitting this season out and investing last
year’s earnings ($200,000) in a safe money market fund paying
5.0 percent or planting summer corn. Planting costs $200,000,
with a six-month time to harvest. If there is rain, planting
summer corn will yield $500,000 in revenues at harvest. If there
is a drought, planting will yield $50,000 in revenues. As a third
choice, you can purchase AgriCorp drought-resistant summer
corn at a cost of $250,000 that will yield $500,000 in revenues at
harvest if there is rain, and $350,000 in revenues if there is a
drought. You are risk averse, and your preference for family
wealth (W) is specified by the relationship U(W) = 1W. The
probability of a summer drought is 0.30, while the probability of
summer rain is 0.70. Which of the three options should you
choose? Explai
Farm Options: A: Safe = 200,0001.05 = 210,000 → U = sqrt(250 + 10) =
sqrt(260) B: Corn = 0.7500 + 0.350 = 350 + 15 = 365 → U = 0.7sqrt(500) +
0.3sqrt(50) C: AgriCorn = 0.7500 + 0.3350 = 470 → U = 0.7sqrt(500) +
0.3*sqrt(350) Compare expected utilities to decide best option.
9. Draw a utility function over income u(I) that describes a man
who is a risk lover when his income is low but risk averse when
his income is high. Can you explain why such a utility function
might reasonably describe a person’s preferences?
Utility Function Shape: U(I): Convex at low income → risk lover; Concave at
high income → risk averse. Reason: Diminishing marginal utility kicks in as
income rises.
10. A city is considering how much to spend to hire people to
monitor its parking meters. The following information is available
to the city manager: • Hiring each meter monitor costs $10,000
per year. • With one monitoring person hired, the probability of a
driver getting a ticket each time he or she parks illegally is equal
to .25. • With two monitors, the probability of getting a ticket
is .5; with three monitors, the probability is .75; and with four, it’s
equal to 1. • With two monitors hired, the current fine for
overtime parking is $20. a. Assume first that all drivers are risk
neutral. What parking fine would you levy, and how many meter
monitors would you hire (1, 2, 3, or 4) to achieve the current
level of deterrence against illegal parking at the minimum cost?
b. Now assume that drivers are highly risk averse. How would
your answer to (a) change? c. (For discussion) What if drivers
could insure themselves against the risk of parking fines? Would
it make good public policy to permit such insurance?
Parking Monitors: a. Risk-neutral → equate expected cost of fine to cost of
monitor. 2 monitors = 0.5*20 = $10 → optimal. b. Risk-averse → hire more
monitors or increase fine to enhance deterrence. c. Insurance reduces deterrence
→ bad public policy.
11. A moderately risk-averse investor has 50 percent of her
portfolio invested in stocks and 50 percent in riskfree Treasury
bills. Show how each of the following events will affect the
investor’s budget line and the proportion of stocks in her
portfolio: a. The standard deviation of the return on the stock
market increases, but the expected return on the stock market
remains the same. b. The expected return on the stock market
increases, but the standard deviation of the stock market
remains the same. c. The return on risk-free Treasury bills
increases.
Portfolio: a. Increased SD → steeper budget line → stock proportion decreases b.
Higher return → flatter line → stock proportion increases c. Risk-free return
increase → shifts budget line upward
12. Suppose there are two types of e-book consumers: 100
“standard” consumers with demand Q 20 P and 100 “rule of
thumb” consumers who buy 10 e-books only if the price is less
than $10. (Their demand curve is given by Q 10 if P 10 and Q 0
if P 10.) Draw the resulting total demand curve for e-books. How
has the “rule of thumb” behavior affected the elasticity of total
demand for e-books
E-Book Demand: Standard: Q = 20 - P, 100 consumers → 2000 - 100P Rule-of-
thumb: 10 if P < 10, 0 if P >= 10 → 1000 if P < 10, 0 otherwise Total Demand: Q
= 2000 - 100P + 1000 (if P < 10) Graph shows kink at P = 10 → demand less
elastic due to flat section.
