# 3.9: Risk Attitudes - Expected Utility Theory and Demand for Hedging(Exercises)

- Page ID
- 33033

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- What is risk? How is it philosophically different from uncertainty?
- What is asymmetric information? Explain how it leads to market failures in an otherwise perfectly competitive market.
- Explain the difference between moral hazard and adverse selection. Can one exist without the other?
- What externalities are caused in the insurance market by moral hazard and adverse selection? How are they overcome in practice?
- Do risk-averse individuals outnumber risk-seeking ones? Give an intuitive explanation.
- Provide examples that appear to violate expected utility theory and risk aversion.
- Give two examples that tell how the framing of alternatives affects peoples’ choices under uncertainty.
- Suppose you are a personal financial planner managing the portfolio of your mother. In a recession like the one in 2008, there are enormous losses and very few gains to the assets in the portfolio you suggested to your mother. Given the material covered in this chapter, suggest a few marketing strategies to minimize the pain of bad news to your mother.
- Distinguish, through examples, between sunk cost, availability bias, and anchoring effect as reasons for departure from the expected utility paradigm.
- Suppose Yuan Yuan wants to purchase a house for investment purposes. She will rent it out after buying it. She has two choices. Either buy it in an average location where the lifetime rent from the property will be $700,000 with certainty or buy it in an upscale location. However, in the upscale neighborhood there is a 60 percent chance that the lifetime income will equal $1 million and 40 percent chance it will equal only $250,000. If she has a utility function that equals \(U(W)= W\), Where would she prefer to buy the house?
- What is the expected value when a six-sided fair die is tossed?
- Suppose Yijia’s utility function is given by \(LN(W)\) and her initial wealth is $500,000. If there is a 0.01 percent chance that a liability lawsuit will reduce her wealth to $50,000, how much premium will she be willing to pay to get rid of the risk?
- Your professor of economics tells you, “The additional benefit that a person derives from a given increase of his stock of a thing decreases with every increase in the stock he already has.” What type of risk attitude does such a person have?
- Ms. Frangipani prefers Pepsi to Coke on a rainy day; Coke to Pepsi on a sunny one. On one sunny day at the CNN center in Atlanta, when faced with a choice between Pepsi, Coke, and Lipton iced tea, she decides to have a Pepsi. Should the presence of iced teas in the basket of choices affect her decision? Does she violate principles of utility maximization? If yes, which assumptions does she violate? If not, then argue how her choices are consistent with the utility theory.
- Explain why a risk-averse person will purchase insurance for the following scenario: Lose $20,000 with 5 percent chance or lose $0 with 95 percent probability. The premium for the policy is $1,000.
- Imagine that you face the following pair of concurrent decisions. First examine both decisions, then indicate the options you prefer:

- Decision (i) Choose between
- a sure gain of $240,
- 25 percent chance to gain $1,000, and 75 percent chance to gain nothing.

- Decision (ii) Choose between:
- a sure loss of $750,
- 75 percent chance to lose $1,000 and 25 percent chance to lose nothing.

- Indicate which option you would choose in each of the decisions
and why.This problem has been adopted from D. Kahneman and D.
Lovallo, “Timid Choices and Bold Forecasts: A Cognitive Perspective
on Risk Taking,”
*Management Science*39, no. 1 (1993): 17–31.

- Consider the following two lotteries:
Which of these lotteries will you prefer to play?

Now, assume somebody promises you sure sums of money so as to induce you to not play the lotteries. What is the sure sum of money you will be willing to accept in case of each lottery: a or b? Is your decision “rational”?

- Gain of $100 with probability 0.75; no gain ($0 gain) with probability 0.25
- Gain of $1,000 with probability 0.05; no gain ($0 gain) with probability 0.95

- Partial insurance:Challenging problem. This problem is designed
to illustrate why partial insurance (i.e., a policy that includes
deductibles and coinsurance) may be optimal for a risk-averse
individual.
Suppose Marco has an initial wealth of $1,000 and a utility function given by \(U(W)= W\). He faces the following loss distribution:

Prob Loss 0.9 0 0.1 500 - If the price per unit of insurance is $0.10 per dollar of loss, show that Marco will purchase full insurance (i.e., quantity for which insurance is purchased = $500).
- If the price per unit of insurance is $0.11 per dollar of loss, show that Marco will purchase less than full insurance (i.e., quantity for which insurance is purchased is less than $500). Hint: Compute \(E(U)\) for full $500 loss and also for an amount less than $500. See that when he insures strictly less than $500, the EU is higher.

- Otgo has a current wealth of $500 and a lottery ticket that pays $50 with probability 0.25; otherwise, it pays nothing. If her utility function is given by \(U(W)= W^2\), what is the minimum amount she is willing to sell the ticket for?
- Suppose a coin is tossed twice in a row. The payoffs associated
with the outcomes are
Outcome Win (+) or loss (−) H-H +15 H-T +9 T-H −6 T-T −12 If the coin is unbiased, what is the fair value of the gamble?

- If you apply the principle of framing to put a favorable spin to
events in your life, how would you value the following gains or
losses?
- A win of $100 followed by a loss of $20
- A win of $20 followed by a loss of $100
- A win of $50 followed by a win of $60
- A loss of $50 followed by a win of $60

- Explain in detail what happens to an insurer that charges the same premium to teenage drivers as it does to the rest of its customers.
- Corporations are risk neutral, yet they hedge. Why?