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Chapter 7: Problem 13

Investment in risky assets can be examined in the state-preference framework by assuming that \(W^{*}\) dollars invested in an asset with a certain return \(r\) will yield \(W^{*}(1+r)\) in both states of the world, whereas investment in a risky asset will yield \(W^{+}\left(1+r_{g}\right)\) in good times and \(W^{*}\left(1+r_{b}\right)\) in bad times (where \(r_{g}>r>r_{b}\) ). a. Graph the outcomes from the two investments. b. Show how a "mixed portfolio" containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals' attitudes toward risk will determine the mix of risk- free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual's utility takes the constant relative risk aversion form (Equation 7.42 ), explain why this person will not change the fraction of risky assets held as his or her wealth increases. \(^{25}\)

Short Answer

Expert verified
Answer: An individual's attitude towards risk influences the mix of risk-free and risky assets they hold in their investment portfolio. The more risk-tolerant they are, the larger the fraction of wealth they allocate towards risky assets. Conversely, the less risk-tolerant they are, the larger the fraction of wealth they allocate towards risk-free assets. Extremely risk-averse individuals may prefer to invest their entire wealth in risk-free assets.

Step by step solution

01

Graph the outcomes of the two investments

To graph the outcomes of the two investments, plot the investment in risk-free assets on the x-axis and the investment in risky assets on the y-axis. The risk-free asset has a return of \(r\), which yields \(W^{*}(1+r)\) in both states. Therefore, its payoff is a straight line with a slope of \((1+r)\). On the other hand, the risky asset has differing returns (\(r_g\) in good times and \(r_b\) in bad times), yielding payoffs of \(W^{+}\left(1+r_{g}\right)\) and \(W^{*}\left(1+r_{b}\right)\). Consequently, the payoff will be two separate points on the graph corresponding to the good and bad states.
02

Illustrate a mixed portfolio containing both risk-free and risky assets

The mixed portfolio can be represented as a line segment connecting the risk-free asset and the risky asset on the graph. Each point on the line segment represents a different mix of the risk-free and risky assets. The fraction of wealth invested in the risky asset can be shown by the distance of a specific point on the line segment from the risk-free asset point, divided by the total length of the line segment.
03

Determine the mix of risk-free and risky assets based on individuals' attitudes towards risk

An individual's risk tolerance will influence how much they invest in risk-free and risky assets. The more risk-tolerant they are, the larger the fraction of wealth they will allocate towards risky assets. The less risk-tolerant they are, the larger the fraction of wealth they will allocate towards risk-free assets. In the case where a person holds no risky assets, it means they are extremely risk-averse and prefer to invest their entire wealth in risk-free assets.
04

Explain the constant relative risk aversion form of an individual's utility

According to the constant relative risk aversion (CRRA) form of utility - Equation 7.42, an individual's utility function takes the following form: \(u(W) = \frac{W^{1-\rho}}{1-\rho}\), where \(\rho\) is the CRRA coefficient (a measure of risk aversion). In this case, the elasticity of the utility function remains constant as wealth increases, meaning the proportion of wealth allocated to risky assets does not change. This is because the individual is assumed to maintain the same level of risk aversion, regardless of changes in wealth.

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Most popular questions from this chapter

For the CRRA utility function (Equation 7.42), we showed that the degree of risk aversion is measured by 1 \(-R\). In Chapter 3 we showed that the elasticity of substitution for the same function is given by \(1 /(1-R)\). Hence the measures are reciprocals of each other. Using this result, discuss the following questions. a. Why is risk aversion related to an individual's willingness to substitute wealth between states of the world? What phenomenon is being captured by both concepts? b. How would you interpret the polar cases \(R=1\) and \(R=-\infty\) in both the risk-aversion and substitution frameworks? c. A rise in the price of contingent claims in "bad" times \(\left(p_{b}\right)\) will induce substitution and income effects into the demands for \(W_{g}\) and \(W_{b^{*}}\) If the individual has a fixed budget to devote to these two goods, how will choices among them be affected? Why might \(W_{g}\) rise or fall depending on the degree of risk aversion exhibited by the individual? d. Suppose that empirical data suggest an individual requires an average return of 0.5 percent before being tempted to invest in an investment that has a \(50-50\) chance of gaining or losing 5 percent. That is, this person gets the same utility from \(W_{0}\) as from an even bet on \(1.055 \mathrm{W}_{0}\) and \(0.955 \mathrm{W}_{0}\) (1) What value of \(R\) is consistent with this behavior? (2) How much average return would this person require to accept a \(50-50\) chance of gaining or losing 10 percent? Note: This part requires solving nonlinear equations, so approximate solutions will suffice. The comparison of the riskreward trade-off illustrates what is called the equity premium puzzle in that risky investments seem actually to earn much more than is consistent with the degree of risk aversion suggested by other data. See N. R. Kocherlakota, "The Equity Premium: It's Still a Puzzle," Journal of Economic Literature (March 1996 ): \(42-71\).

Two pioneers of the field of behavioral economics, Daniel Kahneman and Amos Tversky (winners of the Nobel Prize in economics in 2002 , conducted an experiment in which they presented different groups of subjects with one of the following two scenarios: Scenario 1: In addition to \(\$ 1,000\) up front, the subject must choose between two gambles. Gamble \(A\) offers an even chance of winning \(\$ 1,000\) or nothing. Gamble \(B\) provides \(\$ 500\) with certainty. Scenario 2: In addition to \(\$ 2,000\) given up front, the subject must choose between two gambles. Gamble \(C\) offers an even chance of losing \(\$ 1,000\) or nothing. Gamble \(D\) results in the loss of \(\$ 500\) with certainty. a. Suppose Standard Stan makes choices under uncertainty according to expected utility theory. If Stan is risk neutral, what choice would he make in each scenario? b. What choice would Stan make if he is risk averse? c. Kahneman and Tversky found 16 percent of subjects chose \(A\) in the first scenario and 68 percent chose \(C\) in the second scenario. Based on your preceding answers, explain why these findings are hard to reconcile with expected utility theory. d. Kahneman and Tversky proposed an alternative to expected utility theory, called prospect theory, to explain the experimental results. The theory is that people's current income level functions as an "anchor point" for them. They are risk averse over gains beyond this point but sensitive to small losses below this point. This sensitivity to small losses is the opposite of risk aversion: \(A\) risk-averse person suffers disproportionately more from a large than a small loss. (1) Prospect Pete makes choices under uncertainty according to prospect theory. What choices would he make in Kahneman and Tversky's experiment? Explain. (2) Draw a schematic diagram of a utility curve over money for Prospect Pete in the first scenario. Draw a utility curve for him in the second scenario. Can the same curve suffice for both scenarios, or must it shift? How do Pete's utility curves differ from the ones we are used to drawing for people like Standard \(\operatorname{Stan} ?\)

In Example 7.3 we showed that a person with a CARA utility function who faces a Normally distributed risk will have expected utility of the form \(E[U(W)]=\mu_{W}-(A / 2) \sigma_{W}^{2},\) where \(\mu_{W}\) is the expected value of wealth and \(\sigma_{W}^{2}\) is its variance. Use this fact to solve for the optimal portfolio allocation for a person with a CARA utility function who must invest \(k\) of his or her wealth in a Normally distributed risky asset whose expected return is \(\mu_{r}\) and variance in return is \(\sigma_{r}^{2}\) (your answer should depend on \(A\) ). Explain your results intuitively.

In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is \(p\) and that the fine for receiving the ticket is \(f\). Suppose that all individuals are risk averse (i.e., \(U^{\prime \prime}(W)<0\), where \(W\) is the individual's wealth). Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking? Hint: Use the Taylor series approximation \(U(W-f)=U(W)-f U^{\prime}(W)+\left(f^{2} / 2\right) U^{\prime \prime}(W)\)

In Equation 7.30 we showed that the amount an individual is willing to pay to avoid a fair gamble \((h)\) is given by \(p=0.5 E\left(h^{2}\right) r(W),\) where \(r(W)\) is the measure of absolute risk aversion at this person's initial level of wealth. In this problem we look at the size of this payment as a function of the size of the risk faced and this person's level of wealth. a. Consider a fair gamble ( \(v\) ) of winning or losing \(\$ 1 .\) For this gamble, what is \(E\left(v^{2}\right) ?\) b. Now consider varying the gamble in part (a) by multiplying each prize by a positive constant \(k\). Let \(h=k v\). What is the value of \(E\left(h^{2}\right) ?\) c. Suppose this person has a logarithmic utility function \(U(W)=\ln W\). What is a general expression for \(r(W) ?\) d. Compute the risk premium ( \(p\) ) for \(k=0.5,1\), and 2 and for \(W=10\) and \(100 .\) What do you conclude by comparing the six values?
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