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Chapter 7: Problem 9

Return to Example \(7.5,\) in which we computed the value of the real option provided by a flexible-fuel car. Continue to assume that the payoff from a fossil-fuel-burning car is \(A_{1}(x)=1-x\). Now assume that the payoff from the biofuel car is higher, \(A_{2}(x)=2 x\). As before, \(x\) is a random variable uniformly distributed between 0 and 1 , capturing the relative availability of biofuels versus fossil fuels on the market over the future lifespan of the car. a. Assume the buyer is risk neutral with von Neumann-Morgenstern utility function \(U(x)=x\). Compute the option value of a flexible-fuel car that allows the buyer to reproduce the payoff from either single-fuel car. b. Repeat the option value calculation for a risk-averse buyer with utility function \(U(x)=\sqrt{x}\) c. Compare your answers with Example \(7.5 .\) Discuss how the increase in the value of the biofuel car affects the option value provided by the flexible- fuel car.

Short Answer

Expert verified
Question: Briefly discuss the effects of the increased value of the biofuel car on the option value of the flexible-fuel car for both risk-neutral and risk-averse buyers. Answer: The increased value of the biofuel car results in higher option values for both risk-neutral and risk-averse buyers. This is because the flexible-fuel car allows buyers to reproduce the payoffs of both single-fuel cars, making it more valuable as the value of the biofuel car increases.

Step by step solution

01

Calculate the expected payoffs for the two individual fuel types

For the fossil fuel car, the payoff is A1(x)=1-x. For the biofuel car, the payoff is A2(x)=2x. Since x is uniformly distributed between 0 and 1, we can calculate the expected payoff for each type of car by integrating each function over the range [0,1] and dividing by the interval length. Expected Payoff Fossil = \(E[A1] = \int_0^1{(1-x)dx} = \left[x-\frac{x^2}{2}\right]_0^1 = 1-\frac{1}{2} = \frac{1}{2}\) Expected Payoff Biofuel = \(E[A2] = \int_₀^1(2x)dx = \left[x^2\right]_0^1 = 1^2 - 0^2 = 1\)
02

Calculate the option value of the flexible-fuel car for a risk-neutral buyer

Since the buyer is risk-neutral, their utility function is U(x)=x. So, we can calculate the option value of the flexible-fuel car as the sum of the expected payoffs of both the fossil fuel car and the biofuel car. Option value (Risk-neutral) = Expected Payoff Fossil + Expected Payoff Biofuel = \(\frac{1}{2} + 1 = \frac{3}{2}\) #Calculating the option values for risk-averse buyer#
03

Calculate the expected utility for the individual fuel types

For the risk-averse buyer, their utility function is U(x)=sqrt(x). We will calculate the expected utility for each type of car by integrating the utility function of each payoff over the range [0,1] and dividing by the interval length. Expected Utility Fossil = \(E[U(A1)] = \int_0^1\sqrt{1-x}dx = \left[-\frac{2}{3}(1-x)^{\frac{3}{2}}\right]_0^1 = \frac{2}{3}\) Expected Utility Biofuel = \(E[U(A2)] = \int_0^1\sqrt{2x}dx = \frac{4}{3}\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_0^1 = \frac{4}{3}\)
04

Calculate the option value of the flexible-fuel car for a risk-averse buyer

Since the buyer is risk-averse, we want to calculate the option value as the sum of the expected utilities for both the fossil fuel car and the biofuel car. Option value (Risk-averse) = Expected Utility Fossil + Expected Utility Biofuel = \(\frac{2}{3} + \frac{4}{3} = 2\) #Comparing with Example 7.5#
05

Discussion of the results

In Example 7.5, the option values for both the risk-neutral buyer and the risk-averse buyer were 1. Here, the option value for the risk-neutral buyer is \(\frac{3}{2}\), which is higher than in the previous example. The option value for the risk-averse buyer is 2, also higher than before. This increase in option values can be explained by the higher payoff of the biofuel car, A2(x) = 2x. Since the biofuel car is more valuable in this example, the flexible-fuel car that allows the buyer to reproduce either single-fuel car's payoffs also becomes more valuable. This demonstrates that the increased value of the biofuel car boosts the option value provided by the flexible-fuel car for both types of buyers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Risk Neutrality
Risk neutrality is an important concept in decision-making and finance. When a person is risk-neutral, it means they evaluate choices purely based on expected outcomes, not caring about any potential variability or uncertainty in those outcomes. In the context of utility, a risk-neutral individual's utility function is linear, such as \( U(x) = x \).
For example, if a risk-neutral buyer is choosing between a flexible-fuel car and other options, they would calculate the option value based on expected payoffs. In the problem we discussed, the risk-neutral buyer looks at the expected payoffs from both fossil fuel and biofuel options:
  • Fossil Fuel Car: \( E[A_1] = \frac{1}{2} \)
  • Biofuel Car: \( E[A_2] = 1 \)
The option value of the flexible-fuel car becomes the sum of these expected payoffs, which amounts to \( \frac{3}{2} \). This indicates that making decisions under risk-neutral conditions focuses solely on maximizing expected returns, ignoring any ambiguity or risk involved.
Risk Aversion
Risk aversion describes a preference for certainty over uncertainty. A risk-averse individual prefers a guaranteed outcome over a gamble that has the same expected payoff. In terms of utility, a risk-averse person's utility function is concave, like \( U(x) = \sqrt{x} \), meaning it increases at a decreasing rate.
In the example, the risk-averse buyer would evaluate the flexible-fuel car's option value based on expected utilities:
  • Fossil Fuel Car Expected Utility: \( E[U(A_1)] = \frac{2}{3} \)
  • Biofuel Car Expected Utility: \( E[U(A_2)] = \frac{4}{3} \)
The sum of these expected utilities gives the flexible-fuel car an option value of \( 2 \). Here, risk aversion emphasizes the importance of how fluctuations in payoffs are perceived and valued, influencing the perceived attractiveness of uncertain but potentially high-reward investments.
von Neumann-Morgenstern Utility
The von Neumann-Morgenstern utility theorem provides a foundation for understanding decisions under risk. It describes how individuals can order preferences using a utility function, which then reflects the expected utility of outcomes. This utility aids in decision-making, especially when outcomes are uncertain.
In our scenario, we applied von Neumann-Morgenstern utility functions to different buyer profiles:
  • Risk-neutral buyer: \( U(x) = x \)
  • Risk-averse buyer: \( U(x) = \sqrt{x} \)
By employing these utility functions, we can compute the expected utility arising from uncertain scenarios, guiding decisions between competing options like fossil and biofuel cars. This concept illustrates how individual preferences for risk impact their utility calculations and, consequently, their choices.
Expected Payoff
The expected payoff is a key concept in evaluating investments and making decisions under uncertainty. It refers to the average outcome expected when a decision is repeated multiple times, accounting for all possible scenarios weighted by their probabilities.
In our problem, we calculated expected payoffs for two types of cars:
  • Fossil Fuel Car: The payoff is simply integrated over the probability distribution to get \( \frac{1}{2} \).
  • Biofuel Car: Similarly integrated to yield an expected payoff of \( 1 \).
These calculations provide a baseline for assessing the value of the flexible-fuel car. By evaluating the likely outcomes under different scenarios, expected payoffs offer a quantitative method to compare and decide upon investment alternatives with inherent uncertainties.

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Most popular questions from this chapter

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. \(^{25}\)

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 \(\operatorname{Stan} ?\)

An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take two trips with 6 eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly?

A farmer believes there is a \(50-50\) chance that the next growing season will be abnormally rainy. His expected utility function has the form \\[ \text { expected utility }=\frac{1}{2} \ln Y_{N R}+\frac{1}{2} \ln Y_{R} \\] where \(Y_{N R}\) and \(Y_{R}\) represent the farmer's income in the states of "normal rain" and "rainy," respectively. a. Suppose the farmer must choose between two crops that promise the following income prospects: $$\begin{array}{lcc} \text { Crop } & Y_{N R} & Y_{R} \\ \hline \text { Wheat } & \$ 28,000 & \$ 10,000 \\ \text { Corn } & \$ 19,000 & \$ 15,000 \end{array}$$ Which of the crops will he plant? b. Suppose the farmer can plant half his field with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance-which is available to farmers who grow only wheat and which costs \(\$ 4,000\) and pays off \(\$ 8,000\) in the event of a rainy growing season-cause this farmer to change what he plants?

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.
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