Principles of Finance Lecture 5: Utility Theory

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Questions and Answers

What is the end-of-period wealth for investment 1 if the investor gains $10?

The end-of-period wealth for investment 1 would be $2010.

What is the expected utility for investment 2 given its probabilities and outcomes?

The expected utility for investment 2 is approximately 7.601.

What is the Pratt-Arrow risk premium for investment 1?

The Pratt-Arrow risk premium for investment 1 is $0.

Calculate the expected wealth for investment 2 considering the probabilities and outcomes.

The expected wealth for investment 2 is $1750. Signup and view all the answers

What key difference exists between the Pratt-Arrow risk premium and Markowitz's risk premium?

The Pratt-Arrow risk premium focuses on utility while the Markowitz risk premium considers mean and variance. Signup and view all the answers

What is the variance of end-of-period wealth used for in investor choice theory?

The variance of end-of-period wealth is used to measure the risk associated with investment outcomes. Signup and view all the answers

What does the term 'stochastic dominance' refer to in investor choice theory?

Stochastic dominance refers to a situation where one investment is preferred over another under uncertainty. Signup and view all the answers

How does the normality assumption simplify decision-making for investors?

The normality assumption simplifies decision-making by allowing the use of mean and variance for investment analysis. Signup and view all the answers

What is the formula for the Pratt-Arrow measure of a local risk premium?

The formula is $π = \frac{σ_{Z}^2}{2}U''(W) - U(W)$. Signup and view all the answers

Define Absolute Risk Aversion (ARA).

ARA is defined as $ARA = -\frac{U''(W)}{U'(W)}$. Signup and view all the answers

What does Relative Risk Aversion (RRA) measure?

RRA measures the level of risk aversion relative to wealth, defined as $RRA = -\frac{U''(W)}{W U'(W)}$. Signup and view all the answers

For a power utility function $U(W) = -W^{-α}$, what can be inferred about ARA and RRA?

Both ARA and RRA are constant in $W$ for power utility functions. Signup and view all the answers

Given the logarithmic utility function $U(W) = \ln(W)$, how do ARA and RRA behave?

ARA decreases and RRA is constant as $W$ increases. Signup and view all the answers

What does the expected utility formula represent in terms of wealth?

It represents the expected wealth at the end of a period, considering the risk profile of the investor. Signup and view all the answers

How do utility functions differ among individuals?

Utility functions are specific to individuals but changes in marginal utility can be compared across them. Signup and view all the answers

Explain how uncertainty impacts rational decision-making in investment.

Uncertainty requires investors to assess risk, influencing their choice based on expected utility. Signup and view all the answers

What is the relationship between the local risk premium and investor choice?

The local risk premium affects how investors value risky assets and their willingness to take on risk. Signup and view all the answers

Define a risk-averse investor.

A risk-averse investor dislikes risk and seeks compensation for taking it. Signup and view all the answers

What role does the uncertainty of returns play in the investor's judgment?

Uncertainty of returns compels investors to evaluate potential risks versus returns in their decisions. Signup and view all the answers

What is characteristic of a risk-neutral investor's decision-making process?

A risk-neutral investor makes decisions solely based on expected end-of-period wealth, ignoring risk. Signup and view all the answers

What motivates a risk lover in investing?

A risk lover enjoys taking risks and seeks compensation for avoiding risk. Signup and view all the answers

How can we express the expected utility of end-of-period wealth mathematically?

The expected utility can be expressed as $E[U(W)] = \sum_{i} p_i U(W_i)$, where $p_i$ is the probability of wealth $W_i$. Signup and view all the answers

In the context of a fair gamble, what factors influence an investor's decision to invest or not?

An investor's initial wealth and their risk profile significantly influence the decision to invest. Signup and view all the answers

What is the role of risk attitude in rational decision making under uncertainty?

Risk attitude determines how an individual evaluates potential decisions amidst uncertainty, influencing their choices. Signup and view all the answers

How does a risk-averse investor typically respond to uncertain outcomes?

A risk-averse investor prefers to avoid risk and will often choose a guaranteed outcome over uncertain potential gains. Signup and view all the answers

What is meant by 'certainty equivalent wealth' in investment choices?

Certainty equivalent wealth refers to the guaranteed amount of wealth an investor would accept instead of taking a risk with uncertain outcomes. Signup and view all the answers

What characterizes a risk-loving investor's utility function?

A risk-loving investor has a convex utility function, indicating they derive higher utility from risky choices compared to certain outcomes. Signup and view all the answers

How does the expected utility differ for a risk-neutral investor?

For a risk-neutral investor, the expected utility equals the utility of the expected wealth, as their utility function is linear. Signup and view all the answers

What does the risk premium represent in investment decisions?

The risk premium represents the difference between the expected wealth and the certainty equivalent wealth. Signup and view all the answers

In the provided investment scenario, what does it mean when expected end-of-period wealth is equal for both alternatives?

It means that, on average, both investment options provide the same amount of wealth at the end of the period, but not necessarily the same level of utility. Signup and view all the answers

What is the relationship between expected wealth and utility for risk-averse investors?

Risk-averse investors typically find that the expected utility of a certain outcome is greater than the utility of expected wealth, indicating they value certainty more. Signup and view all the answers

Describe how attitudes towards risk can influence investment choices.

Attitudes towards risk can significantly affect investment choices, with risk-averse investors favoring safer options, while risk-tolerant investors may seek higher returns through riskier investments. Signup and view all the answers

In the context of normally distributed random variables, what two parameters fully describe a normal distribution?

The mean and variance. Signup and view all the answers

How is the expected return $R̄$ defined in terms of asset returns?

It is defined as $E(R) = rac{E(W_f) - W_0}{W_0}$. Signup and view all the answers

What is the relationship between end-of-period wealth and returns under the normality assumption?

Returns, $R$, can be expressed as $R = rac{W_f - W_0}{W_0}$, where $W_f$ is the end-of-period wealth. Signup and view all the answers

What does the variance of returns, $ heta_R^2$, represent?

It represents the variability or risk of asset returns. Signup and view all the answers

What does it mean when we say returns are normally distributed, denoted as $R ∼ N(R̄, heta_R^2)$?

It means that returns follow a normal distribution characterized by mean $R̄$ and variance $ heta_R^2$. Signup and view all the answers

Explain the implication of rational decision-making based solely on mean and variance in investment contexts.

It implies that investors make decisions based on expected return and risk, ignoring other factors. Signup and view all the answers

What formula represents the calculation for the variance of returns using end-of-period wealth?

The variance of returns is calculated as $ heta_R^2 = E[(R - E(R))^2]$. Signup and view all the answers

In the equation for expected return, what does $W_0$ represent?

It represents the initial wealth at the beginning of the investment period. Signup and view all the answers

What significance does the mean, $ar{W}$, hold in investment decision-making?

The mean represents the average expected wealth at the end of the investment horizon. Signup and view all the answers

How do you express the variance of wealth, $ heta_W^2$, in terms of expected wealth?

It can be expressed as $ heta_W^2 = E[W_f^2] - (E[W_f])^2$. Signup and view all the answers

What are the two key definitions of risk aversion introduced by Pratt and Arrow?

Absolute risk aversion and relative risk aversion. Signup and view all the answers

If an investor's utility function is defined as U(W) = W^2, how would they generally respond to a risky investment?

They would likely avoid the investment due to risk aversion, preferring safer options. Signup and view all the answers

What does the expected utility theory suggest about the relationship between risk and utility?

It suggests that individuals choose investments to maximize their expected utility, often leading to risk aversion. Signup and view all the answers

How is the certainty equivalent (CE) calculated in the context of risky investments?

CE is calculated as CE = E(W + Z) - π(W, Z), where π is the risk premium. Signup and view all the answers

Explain the significance of the risk premium in relation to end-of-period wealth.

The risk premium represents the amount above the expected value of a gamble that an investor requires to accept the risk. Signup and view all the answers

What are the implications of an actuarially neutral gamble on the expected wealth of an investor?

An actuarially neutral gamble has an expected value of zero, meaning it neither adds nor subtracts from expected wealth. Signup and view all the answers

In the context of risk aversion, what does a utility function of U(W) = ln(W) indicate?

It indicates decreasing marginal utility, implying that as wealth increases, the additional satisfaction from each extra dollar decreases. Signup and view all the answers

Describe how Taylor series expansion is used in the context of investment utility functions.

Taylor series expansion approximates the utility of wealth by taking into account the variance and higher moments of the utility function. Signup and view all the answers

What is the expected value of a gamble with outcomes z1 and z2 with probabilities p1 = p2 = 0.5 when E(z1 + z2) = 0?

The expected value is zero, as the outcomes balance each other out. Signup and view all the answers

How does the level of risk aversion affect investment decisions?

Higher levels of risk aversion lead to more conservative investment choices, favoring lower-risk options. Signup and view all the answers

Flashcards

Risk Premium

The amount an individual will pay to avoid risk. It measures how much an investor values certainty over uncertainty.

Logarithmic Utility Function

A utility function where the increase in utility decreases as wealth increases. This means that the individual is risk-averse.

Pratt-Arrow Risk Premium

A type of risk premium that is based on the expected utility of a certain outcome and the expected utility of the risky outcome. It measures the difference in utility between the two.

Markowitz Risk Premium

The risk premium based on the classic mean-variance framework. It measures the difference between the expected return of the risky investment and the return of a risk-free investment.