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

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

Step-by-step solution

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

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)#

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.

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)#

Define the utility function

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

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#

Define the risk premium formula

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

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 \(k = 2, W = 100\): \(p = 0.5 \cdot (2^2) \cdot \frac{1}{100} = \frac{1}{100}\)

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.

Key concepts

Expected Value

The concept of expected value is a fundamental idea in probability and statistics. It provides a way to take into account all possible outcomes of a random variable and weigh them by their probabilities to find an average or expected outcome. Expected value is often used in situations involving risk and uncertainty to determine the average result of an endeavor if it were to be repeated many times.

In our exercise, we have a fair gamble, which involves a 50% chance of winning or losing $1. The expected value of the squared outcome, written as \(E(v^2)\), can be calculated using the formula:

• For winning: outcome squared is 1, probability is 0.5
• For losing: outcome squared is also 1 (since \((-1)^2 = 1\)), probability is 0.5

Thus, \(E(v^2) = (0.5 \times 1) + (0.5 \times 1) = 1\). This calculation helps us to understand the variance or dispersion of results in a gamble scenario.

Utility Function

Utility functions are a crucial concept in economics and decision theory. They represent the relationship between an individual's wealth and their level of satisfaction or happiness derived from it. Different individuals can have different utility functions, reflecting varying degrees of risk aversion.

A utility function can be used to quantify preferences, often with money, by mapping every possible wealth level to a specific level of utility. In our example, a logarithmic utility function, \(U(W) = \ln(W)\), is used. This function implies that as wealth increases, the additional satisfaction gained from each additional dollar decreases. It's a common form used to model risk-averse behavior.

To find a person's risk aversion from their utility function, economists often calculate the absolute risk aversion, denoted by \( r(W) \). This is given by:

• First derivative of utility \( U'(W) = \frac{1}{W} \)
• Second derivative of utility \( U''(W) = -\frac{1}{W^2} \)
• Absolute risk aversion \( r(W) = -\frac{U''(W)}{U'(W)} = \frac{1}{W} \)

Thus, the degree of risk aversion declines as wealth increases.

Fair Gamble

A fair gamble is a scenario where the expected payoff is neutral. In other words, the probability-weighted gains are equal to the probability-weighted losses. Thus, on average, you neither gain nor lose money from participating in a fair gamble. This concept is essential in risk management and decision-making.

In mathematical terms, a fair gamble with outcomes \( v = 1\) (gain) and \( v = -1\) (loss) each with a probability of 0.5, yields an expected payoff of 0. This is because the positive and negative outcomes cancel each other out. For example, \(E(v) = 0.5 \times 1 + 0.5 \times (-1) = 0\).

The idea of a fair gamble helps to analyze individuals' choices when they are risk-averse. By comparing the expected payoff of zero with the risk involved, we can understand what amount of money people might be willing to pay to avoid such uncertainty. This amount is commonly known as the risk premium.

Logarithmic Utility

Logarithmic utility functions are particularly useful for modeling risk-averse behavior, where individuals prefer certain outcomes over uncertain ones with the same expected value.

The logarithmic utility function \( U(W) = \ln(W) \) indicates that as wealth \( W \) increases, utility increases at a decreasing rate. This reflects the principle of diminishing marginal utility: each additional unit of wealth provides less additional satisfaction than the previous one. This is a common feature in real-world scenarios, as people generally feel less excitement about gaining a million dollars when they already have ten million, compared to when they have only one million.

Using this function, we can calculate a person's risk premium they would pay to avoid uncertainty, based on their wealth \( W \) and the size of the gamble \( k \). Risk premium reflects the cost of smoothing out the gamble's risk according to individuals’ wealth level and helps to contrast different people's attitudes toward risk. As the exercise demonstrates, wealthier individuals with logarithmic utility functions are generally less risk-averse.