Utility Functions and Expected Utility Theorem

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Questions and Answers

What is the form of the power utility function?

$U(w) = w^{rac{1}{eta}}$, $w > 0$

$U(w) = rac{w^{-eta}}{eta}$, $w > 0$ (correct)

$U(w) = w^{-eta}$, $w > 0$

$U(w) = rac{w^{rac{1}{eta}} - 1}{eta}$, $w > 0$

What condition must the parameter value $eta$ satisfy for risk-aversion to hold in the power utility function?

$eta = 1$

$eta > 1$

$eta < 0$

$eta < 1$ (correct)

Which type of risk aversion does the power utility function exhibit?

Decreasing absolute risk aversion (correct)

Variable absolute risk aversion

Increasing absolute risk aversion

Constant absolute risk aversion

What is the relative risk aversion of the power utility function?

Constant relative risk aversion Signup and view all the answers

Which of the following defines an iso-elastic utility function?

A function that has a defining property of $R'(w) = 0$ Signup and view all the answers

Why are iso-elastic utility functions considered useful?

They simplify optimal strategy determinations for multi-period investments Signup and view all the answers

Which of the following is an example of an iso-elastic utility function?

Logarithmic utility function Signup and view all the answers

What does decreasing absolute risk aversion imply about an individual's utility?

Utility increases with wealth but at a decreasing rate Signup and view all the answers

What are the two main categories of tests of the semi-strong form of the EMH?

Tests of informational efficiency and tests of excessive volatility Signup and view all the answers

Which of the following is an example of over-reaction to events that contradict informational efficiency?

Market over-reacting to poor past performance Signup and view all the answers

What characterizes an excessively volatile market?

Market value changes cannot be explained by new information Signup and view all the answers

How did Shiller test for excessive volatility in equity markets?

By assessing the justification of market value changes against new information Signup and view all the answers

Which of the following represents an under-reaction to events that contradict informational efficiency?

Stock prices continuing to respond to earnings announcements over time Signup and view all the answers

Which accounting ratio might be used incorrectly to predict stock performance, suggesting over-reaction?

Price-to-earnings ratio Signup and view all the answers

What is a common result following a de-merger that indicates an under-reaction?

Abnormal excess returns for the resulting companies Signup and view all the answers

Which of the following situations suggests that the market may be exhibiting excessive volatility?

Dramatic price swings without corresponding news Signup and view all the answers

What is systematic risk related to?

The market as a whole Signup and view all the answers

What is the equation for the covariance of returns on securities i and j under the single-index model?

Cij = βiβjVM Signup and view all the answers

How does the single-index model improve data efficiency compared to mean-variance portfolio theory?

It reduces data items for covariances significantly. Signup and view all the answers

What does Var[Sn] represent in the context of investments?

The variance of the accumulated amount at time n Signup and view all the answers

Which of the following best describes specific risk?

It depends on factors peculiar to the individual security. Signup and view all the answers

Which equation correctly expresses the expected value E[Sn] based on the provided content?

E[Sn] = (1 + j) + s Signup and view all the answers

Which of the following is NOT required in the single-index model?

N(N – 1)/2 covariances Signup and view all the answers

What does the variable 'j' signify in the given equations?

The interest rate on the investment Signup and view all the answers

Which of the following is a key component in calculating the variance Var[Sn]?

The independence of investment returns Signup and view all the answers

In the formula for Var[Sn], what does the term (1 + j) + s signify?

The accumulation of interest and additional factors Signup and view all the answers

What is the significance of the product notation ⨕ in the expression for E[Sn]?

It represents the cumulative effect of varying interest rates Signup and view all the answers

How is the variance Var[Sn] calculated using the expected value?

Var[Sn] = E[Sn] - (E[Sn])^2 Signup and view all the answers

What is the role of the variable 's' in the variance formula Var[Sn]?

It represents the standard deviation of returns Signup and view all the answers

What is assumed about the interest rates 'i_t' in the cumulative product for E[Sn]?

They are statistically independent Signup and view all the answers

Which statement describes the correct relationship between mean and variance for the investment?

Both mean and variance increase together Signup and view all the answers

What does the expected utility formula E[U] = ∑ pi U(wi) represent in terms of investment?

The anticipated return from investing in a risky asset Signup and view all the answers

Which axiom states that an investor can express a preference for all available certain outcomes?

Comparability Signup and view all the answers

If an investor prefers outcome A over B and outcome B over C, what is the implication according to the transitivity axiom?

A is preferred over C Signup and view all the answers

What concept describes the indifference of an investor between two outcomes and their associated gambles?

Independence Signup and view all the answers

What does certainty equivalence establish in the context of risk and preference?

A unique probability linking certain outcomes with mixed outcomes Signup and view all the answers

Which axiom implies that if A is preferred to B, the investor remains indifferent when introducing a third option C?

Independence Signup and view all the answers

In the context of an investor's utility function U(w), what does non-satiation refer to?

The idea that more wealth always leads to higher utility Signup and view all the answers

What is the outcome of applying the independence axiom in investment decisions?

An increase in preference stability across different scenarios Signup and view all the answers

Study Notes

Power Utility Function

The power utility function is defined as: U(w) = (w^(γ-1))/γ where w > 0.
The risk-aversion condition for this function requires γ < 1.
This function exhibits decreasing absolute risk aversion (ARA) as A'(w) < 0 and constant relative risk aversion (RRA) as R'(w) = 0.

Iso-Elastic Utility Function

Iso-elastic utility functions are characterized by constant relative risk aversion (RRA) where R'(w) = 0.
These functions simplify multi-period investment decisions by allowing for "myopic" decisions, focusing only on the current period's outcomes.
Log and power utility functions are examples of iso-elastic functions.

Expected Utility Theorem

The expected utility theorem is formally derived from four axioms: comparability, transitivity, independence, and certainty equivalence.

Comparability

Investors can rank and compare all available certain outcomes.

Transitivity

If outcome A is preferred to B and B is preferred to C, then A is preferred to C.

Independence

An investor indifferent between two outcomes, A and B, will remain indifferent when facing gambles where A and B are paired with a third outcome C, with the same probabilities.

Certainty Equivalence

If A is preferred to B and B is preferred to C, there's a unique probability, p, that makes the investor indifferent between B and a gamble offering A with probability p and C with probability (1-p).

Non-Satiation

Investors' utility functions, U(w), are monotonically increasing, meaning they prefer more wealth to less.

Systematic and Specific Risk

Systematic risk is the market-wide risk component affecting all securities. It cannot be diversified.
Specific risk is the risk unique to an individual security. It can be diversified away.

Single-Index Model: Covariance of Returns

The single-index model's covariance formula is: Cij = βiβjVM
This equation shows that correlation between two securities is driven solely by their common response to market movements (represented by VM).

Single-Index Model: Reduction in Data Requirements

Mean-variance portfolio theory requires N means, N variances, and N(N-1)/2 covariances.
The single-index model reduces this to N αi's, N βi's, N Vεi's, along with the market's mean and variance, representing a total of 3N+2 data items.

Single-Index Model: Variance of Portfolio Returns

The single-index model simplifies the portfolio return variance calculation by taking into account both systematic (market) and specific (non-market) components.

Weak Form Efficient Market Hypothesis (EMH)

Recent econometric research challenges the weak form EMH by suggesting short-term momentum (trending in the same direction) and medium-term mean-reversion (trending in the opposite direction).

Tests of the Semi-Strong Form EMH

Informational efficiency focuses on how quickly and accurately market prices incorporate new information.
Excessive volatility examines whether market price fluctuations exceed the justified level based on new information.

Over-Reactions Violating Informational Efficiency

Markets may overreact to past performance, leading to mispricing.
Certain accounting ratios may incorrectly appear to hold predictive power.
New share issues may underperform in the long-term despite initial overvaluation.

Under-Reactions Violating Informational Efficiency

Stock prices might continue to respond to earnings announcements even long after the initial release.
Abnormal excess returns might follow de-mergers.
Abnormal negative returns might follow mergers.

Excessively Volatile Market and Shiller's Test.

An excessively volatile market exhibits price swings unjustified by the arrival of new information.
Shiller's test for excessive volatility examines price fluctuations compared to the changes warranted by fundamentals.

Expected Utility in a Risky Asset

The expected utility yielded by investment in a risky asset is calculated as: E[U] = Σ(piU(wi)), where pi is the probability of each outcome wi.

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Description

This quiz covers essential concepts related to power utility functions, iso-elastic utility functions, and the expected utility theorem. Learn about risk aversion and how these functions aid in investment decisions. Test your understanding of these foundational theories in economics.