Question

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 ): \(42-71\).

Short answer

In summary, risk aversion and elasticity of substitution capture an individual's attitude towards facing uncertain outcomes and their readiness to reallocate wealth to cope with those uncertainties. The polar cases of R represent extreme scenarios in terms of risk aversion and the elasticity of substitution. With an increase in the price of contingent claims in "bad" times, individuals need to reallocate their budget, causing substitution and income effects. To find the value of R consistent with the given behavior and the required average return, set up the CRRA utility function and solve for R and the average return, respectively.

Step-by-step solution

a. Risk aversion and willingness to substitute wealth

Risk aversion is a measure of how much an individual prefers to avoid risk while making decisions on uncertain outcomes. In the context of wealth, risk-averse individuals prefer a less risky option that guarantees them a certain payoff over a more risky option with uncertain payoffs even if the expected value is the same. Elasticity of substitution, on the other hand, measures how easily an individual is willing to substitute wealth between different states of the world or scenarios. A higher elasticity of substitution means an individual is more willing to move wealth between states, while a lower elasticity indicates less willingness. Both risk aversion and the elasticity of substitution capture the individual's attitude towards facing uncertain outcomes and their readiness to reallocate wealth to cope with those uncertainties. When the measures are reciprocals of each other, it implies that individuals with higher risk aversion have a lower willingness to substitute wealth between states of the world, thus highlighting their preference for safer and more predictable outcomes.

b. Interpretation of polar cases R=1 and R=-∞

The polar cases for R represent extreme scenarios in terms of risk aversion and the elasticity of substitution: 1. For R = 1: Risk aversion (1-R) becomes 0, and the elasticity of substitution (1/(1-R)) becomes undefined or infinite. This represents an individual with no risk aversion, who is indifferent to all risky outcomes and not at all concerned with reallocating wealth between different scenarios. 2. For R = -∞: Risk aversion (1-R) approaches infinity, and the elasticity of substitution (1/(1-R)) becomes 0. This represents an individual with infinite risk aversion, who values safety more than any potential gain and is entirely unwilling to substitute wealth between different states of the world.

c. Effects of rising price of contingent claims in "bad" times

With an increase in the price of contingent claims in "bad" times \((p_{b})\), individuals need to reallocate their budget between \(W_{g}\) and \(W_{b^*}\), causing substitution and income effects. The income effects will cause a reduction in demand for both goods, as the individual is left with less income to purchase both. The substitution effect depends on the degree of risk aversion. If the individual has a higher risk aversion, they might increase the demand for safe or less risky goods (\(W_g\)) and reduce the demand for "bad" claims (\(W_{b^*}\)) to avoid possible negative outcomes. In contrast, if the individual exhibits lower risk aversion, they might be willing to accept more risk and allocate more wealth to \(W_{b^*}\), which might result in a decrease in \(W_g\).

d. Calculating value of R and required average return

(1) To find the value of R consistent with this behavior, we need to set up the utility function according to the given information. Let \(W_0\) be the initial wealth, and the individual gets the same utility from \(W_0\) and an even bet on \(1.055 W_0\) and \(0.955 W_0\). The CRRA utility function is given by: \(U(W) = \frac{W^{1-R}}{1-R}\) We have: \(\frac{W_0^{1-R}}{1-R} = \frac{1}{2} \left(\frac{(1.055 W_0)^{1-R}}{1-R} + \frac{(0.955 W_0)^{1-R}}{1-R}\right)\) Solve for R (approximately) to get the value consistent with the given behavior. (2) To find the required average return for a 50-50 chance of gaining or losing 10 percent, set up the utility function similarly: \(\frac{W_0^{1-R}}{1-R} = \frac{1}{2} \left(\frac{(1.1 W_0)^{1-R}}{1-R} + \frac{(0.9 W_0)^{1-R}}{1-R}\right)\) Solve for the average return, given the previously calculated value of R, to get the required average return for the 50-50 chance of gaining or losing 10 percent.

Key concepts

CRRA Utility Function

The Constant Relative Risk Aversion (CRRA) utility function is a crucial concept in understanding how individuals make decisions when faced with uncertainty. This function is represented mathematically as

\[U(W) = \frac{W^{1-R}}{1-R},\]
where \(W\) is wealth and \(R\) is the coefficient of relative risk aversion. What's essential about CRRA is that it captures how the utility or satisfaction derived from wealth changes with different levels of risk aversion.

The CRRA utility function assumes that a person's relative risk aversion is constant. This implies that as their wealth changes, they would consistently avoid or approach risk in the same relative manner. However, the absolute amount of risk they're willing to take may increase as their wealth grows.

For individuals deeply averse to risk, the function shows they would prefer a certain, but potentially lower, level of wealth over taking a gamble, even if statistically the gamble could produce a higher level of expected wealth. On the flip side, someone with a lower risk aversion would be more open to taking such risks, indicating a willingness to substitute wealth between secure and risky assets. This characteristic directly links to their well-being as they weigh the potential outcomes of uncertain investments.

Risk Aversion

Risk aversion is a fundamental trait in financial decision-making. It represents the extent to which an individual prefers certainty over uncertainty. In essence, it's about how much a person dislikes risk. Risk-averse individuals derive less utility or satisfaction from an uncertain investment compared to a certain one, even if the uncertain option has the same expected return.

In the context of the exercise, the degree of risk aversion is measured by \(1-R\), where a high value denotes higher risk aversion. A person with high risk aversion would require a higher expected return to compensate for the additional risk, whereas someone with low risk aversion might accept lower returns for risky investments.

It's intriguing to note that risk aversion does not operate in a vacuum. It impacts an individual's entire financial decision process, affecting not just their investment choices but also their reactions to price changes in the market, which in turn lead to substitution and income effects.

Substitution and Income Effects

The concepts of substitution and income effects address how individuals adjust their consumption in response to changes in prices or their income. These reactions are especially evident when studying how investors deal with the price changes of contingent claims that pay off in 'bad' times.

• Substitution Effect: This occurs when an increase in the price of one good causes consumers to purchase a substitute good, assuming their preference remains constant. For instance, if the price of 'bad' contingent claims increases, a person may shift their allocation towards 'good' claims, thus changing the distribution of their wealth between risky and secure states.
• Income Effect: Contrary to substitution, the income effect entails a change in consumption as a result of an individual's purchasing power changing, not due to their preferences changing. An increase in the price of contingent claims effectively reduces the individual’s ability to purchase other goods, possibly leading to a decrease in demand for both 'good' and 'bad' claims.

In the scope of the exercise, the combination of both effects shows how an individual's allocation between \(W_g\) and \(W_{b^*}\) will adjust based on their degree of risk aversion. High-risk aversion may mean a pivot towards more secure assets due to a pronounced income effect and a desire to avoid losses. In contrast, a person with low risk aversion may exhibit a stronger substitution effect, opting for 'bad' claims even as their prices rise, trying to maximize expected returns.