Question

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

Question: Based on the given step-by-step solution, explain how the risk premium changes as the gamble amount and person's initial level of wealth change. Answer: As the gamble amount (k) increases, the risk premium increases, demonstrating that a higher gamble amount leads to a higher risk premium. On the other hand, as the person's initial level of wealth (W) increases, the risk premium decreases, implying that an individual with a higher level of wealth is more risk-tolerant than someone with a lower wealth level.

Step-by-step solution

Part a: Find the expected value of \(v^2\) for winning or losing \(1

The gamble has two outcomes: winning \)1 or losing \(1 with equal probabilities. Therefore, \)E\left(v^2\right) = \frac{1}{2}(1^2) + \frac{1}{2}(-1)^2 = \frac{1}{2} + \frac{1}{2} = 1.$

Part b: Find the expected value of \(h^2\) for \(h=kv

Substitute \)h = kv\( into the expression for \)E\left(h^2\right)\(. Since winning or losing probabilities don't change, we get \)E\left(h^2\right) = \frac{1}{2}(k^2v^2) + \frac{1}{2}(k^2v^2) = k^2E\left(v^2\right)\(. We've found that \)E\left(v^2\right) =1\( in part a. So, \)E\left(h^2\right) = k^2.$

Part c: Find the general expression for \(r(W)\) for a logarithmic utility function

We're given that the utility function is \(U(W) = \ln W\). To find the measure of absolute risk aversion \(r(W)\), we need to compute the second derivative of \(U(W)\) with respect to W and divide it by the first derivative of \(U(W)\) with respect to W. \(U'(W) = \frac{1}{W}, U''(W) = -\frac{1}{W^2}.\) Now, we can calculate \(r(W) = -\frac{U''(W)}{U'(W)} = -\frac{-\frac{1}{W^2}}{\frac{1}{W}} = \frac{1}{W}\).

Part d: Compute the risk premium for different values of \(k\) and \(W\) and compare the results

We will calculate risk premium \((p)\) using the given formula \(p=0.5E\left(h^2\right)r(W)\) and the derived expressions \(E\left(h^2\right) = k^2\) and \(r(W) = \frac{1}{W}\). We need to calculate risks for \(k=0.5,1,2\) and for \(W=10\) and \(100.\) Case 1: \(k=0.5, W=10\) \(p = 0.5 \cdot (0.5^2) \cdot \frac{1}{10} = \frac{1}{80}\) Case 2: \(k=0.5, W=100\) \(p = 0.5 \cdot (0.5^2) \cdot \frac{1}{100} = \frac{1}{800}\) Case 3: \(k=1, W=10\) \(p = 0.5 \cdot (1^2) \cdot \frac{1}{10} = \frac{1}{20}\) Case 4: \(k=1, W=100\) \(p = 0.5 \cdot (1^2) \cdot \frac{1}{100} = \frac{1}{200}\) Case 5: \(k=2, W=10\) \(p = 0.5 \cdot (2^2) \cdot \frac{1}{10} = \frac{1}{5}\) Case 6: \(k=2, W=100\) \(p = 0.5 \cdot (2^2) \cdot \frac{1}{100} = \frac{1}{50}\) By comparing these six values, we can conclude that as the person's initial level of wealth increases, the risk premium decreases, and as the gamble amount (k) increases, the risk premium increases. It shows that decreasing the person's wealth makes them more risk-averse, while increasing the gamble amount makes the risk premium higher.

Key concepts

Expected Value

The expected value of a random variable offers a measure of the central tendency or the average of a probability distribution. It accounts for the various possible outcomes and the likelihood of each. When considering a gamble or uncertain situation like the one presented in the exercise, expected value helps in understanding the long-term expectation from repetitive participation in that gamble.

For instance, in the exercise with the fair gamble of winning or losing \(1, we calculated the expected value of the square of the gamble, represented as \( E(v^2) \). The value is derived by squaring the outcomes, winning \)1 (which becomes 1) and losing $1 (which becomes 1 as well because \( (-1)^2 = 1 \)), and then taking the average, since each outcome has a 50% probability. The result is an expected value of 1. When the gamble is scaled by a positive constant \( k \), this measure scales by \( k^2 \), reflecting how the gamble's volatility impacts the potential outcomes.

Utility Function

Utility functions in microeconomics represent the satisfaction or preferences of consumers for goods, services, or wealth. They are a cornerstone concept for understanding decision-making under uncertainty. In the exercise, we're particularly interested in a logarithmic utility function, \( U(W) = \ln(W) \), where \( W \) is the wealth variable.

This type of utility function implies that a person's satisfaction increases with wealth, but at a decreasing rate. It models a common real-world observation: as people become wealthier, an extra dollar brings progressively less additional satisfaction. Such models are crucial for exploring how individuals might behave when faced with risky financial decisions, because depending on their wealth, the same gamble can represent very different levels of risk to their well-being.

Absolute Risk Aversion

The concept of absolute risk aversion measures an individual's reluctance to accept risk. It quantifies this aversion by evaluating the curvature of the utility function. With a steeper curvature, the individual is seen to have higher risk aversion. To calculate absolute risk aversion, economists use the second derivative of the utility function divided by the first derivative, which in finance is called the Arrow-Pratt measure of absolute risk aversion.

In the exercise, we derived the absolute risk aversion of \( r(W) = \frac{1}{W} \) using the logarithmic utility function. The result shows that as wealth \( W \) increases, absolute risk aversion decreases; wealthy individuals are less sensitive to the same level of risk compared to less wealthy individuals. This finding is fundamental in assessing how much an individual is willing to pay to avoid a gamble, known as the risk premium, reinforcing the idea that wealth impacts attitudes towards risk.