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Chapter 7: Problem 8

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?

Short Answer

Expert verified
Answer: The general expression for absolute risk aversion with a logarithmic utility function is \(r(W) = \frac{1}{W}\).

Step by step solution

01

Define the fair gamble

The fair gamble \(v\) is defined as winning or losing \(\$1\) with equal probabilities. Mathematically, we can represent it with a probability distribution such that: P(v = 1) = 0.5 P(v = -1) = 0.5
02

Compute the expected value

Now we will compute the expected value of the square of \(v\). The formula for expected value is: \(E(v^2) = \sum{P(v^i) \cdot (v^i)^2}\) Plugging in the values: \(E(v^2) = (0.5 \cdot (1)^2) + (0.5 \cdot (-1)^2) = 0.5 + 0.5 = 1\) So the expected value of \(E(v^2)\) is 1. #b. Expected value of the square of the gamble multiplied by a constant (k)#
03

Define the gamble multiplied by a constant

The scaled gamble is defined as \(h = kv\), where \(k\) is a positive constant. We need to find the expected value of \(h^2\) in this case.
04

Compute the expected value

The expected value formula is the same, but for \(h^2\): \(E(h^2) = \sum{P(h^i) \cdot (h^i)^2}\) Since \(h = kv\), \(E((kv)^2) = \sum{P(kv^i) \cdot (kv^i)^2}\) Plugging in the values and factoring out the constant \(k^2\): \(E((kv)^2) = k^2 \cdot [(0.5 \cdot (1)^2) + (0.5 \cdot (-1)^2)] = k^2 \cdot (0.5 + 0.5) = k^2\) So the expected value of \(E(h^2)\) is \(k^2\). #c. General expression for absolute risk aversion, r(W)#
05

Define the utility function

Given the logarithmic utility function \(U(W) = \ln(W)\).
06

Derive the expression for absolute risk aversion

The measure of absolute risk aversion is given by the negative of the quotient of the second and the first derivative of the utility function: \(r(W) = -\frac{U''(W)}{U'(W)}\) First, we find the derivatives of the utility function: \(U'(W) = \frac{1}{W}\) \(U''(W) = -\frac{1}{W^2}\) Plugging the derivatives into our equation: \(r(W) = -\frac{-\frac{1}{W^2}}{\frac{1}{W}} = \frac{1}{W}\) So the general expression for absolute risk aversion is \(r(W) = \frac{1}{W}\). #d. Compute the risk premium for different values of k and W#
07

Define the risk premium formula

The risk premium \(p\) is given by the formula: \(p = 0.5 \cdot E(h^2) \cdot r(W)\)
08

Compute the risk premium for different values of k and W

We will compute the risk premium for \(k = 0.5, 1, 2\) and \(W = 10, 100\) using the formula and the expressions for \(E(h^2)\) and \(r(W)\) we derived in previous parts. For \(k = 0.5, W = 10\): \(p = 0.5 \cdot (0.5^2) \cdot \frac{1}{10} = \frac{1}{80}\) For \(k = 0.5, W = 100\): \(p = 0.5 \cdot (0.5^2) \cdot \frac{1}{100} = \frac{1}{800}\) For \(k = 1, W = 10\): \(p = 0.5 \cdot (1^2) \cdot \frac{1}{10} = \frac{1}{20}\) For \(k = 1, W = 100\): \(p = 0.5 \cdot (1^2) \cdot \frac{1}{100} = \frac{1}{200}\) For \(k = 2, W = 10\): \(p = 0.5 \cdot (2^2) \cdot \frac{1}{10} = \frac{1}{10}\) For \(k = 2, W = 100\): \(p = 0.5 \cdot (2^2) \cdot \frac{1}{100} = \frac{1}{100}\)
09

Conclude

Comparing the six values, the risk premium decreases as wealth increases, reflecting diminishing absolute risk aversion. Additionally, the risk premium increases as the gamble size (k) increases, reflecting the individual's higher valuation for avoiding more significant risks.

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Most popular questions from this chapter

Show that if an individual's utility-of-wealth function is convex then he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence?

The CARA and CRRA utility functions are both members of a more general class of utility functions called harmonic absolute risk aversion (HARA) functions. The general form for this function is \(U(W)=\theta(\mu+W / \gamma)^{1-\gamma}\), where the various parameters obey the following restrictions: \(\bullet$$\gamma \leq 1\) \(\bullet$$\mu+W / \gamma > 0\) \(\bullet$$\theta[(1-\gamma) / \gamma] > 0\) The reasons for the first two restrictions are obvious; the third is required so that \(U^{\prime} > 0\) a. Calculate \(r(W)\) for this function. Show that the reciprocal of this expression is linear in \(W\). This is the origin of the term harmonic in the function's name. b. Show that when \(\mu=0\) and \(\theta=[(1-\gamma) / \gamma]^{\gamma-1},\) this function reduces to the CRRA function given in Chapter 7 (see footnote 17 ). c. Use your result from part (a) to show that if \(\gamma \rightarrow \infty\), then \(r(W)\) is a constant for this function. d. Let the constant found in part (c) be represented by \(A\). Show that the implied form for the utility function in this case is the CARA function given in Equation 7.35 e. Finally, show that a quadratic utility function can be generated from the HARA function simply by setting \(\gamma=-1\) f. Despite the seeming generality of the HARA function, it still exhibits several limitations for the study of behavior in uncertain situations. Describe some of these shortcomings.

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. \(^{25}\)

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 \(U(W-f)=U(W)-f U^{\prime}(W)+\left(f^{2} / 2\right) U^{\prime \prime}(W)\)

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 ): \(42-71\).
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