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?

    <p>Constant relative risk aversion</p> Signup and view all the answers

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

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

    Why are iso-elastic utility functions considered useful?

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

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

    <p>Logarithmic utility function</p> Signup and view all the answers

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

    <p>Utility increases with wealth but at a decreasing rate</p> Signup and view all the answers

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

    <p>Tests of informational efficiency and tests of excessive volatility</p> Signup and view all the answers

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

    <p>Market over-reacting to poor past performance</p> Signup and view all the answers

    What characterizes an excessively volatile market?

    <p>Market value changes cannot be explained by new information</p> Signup and view all the answers

    How did Shiller test for excessive volatility in equity markets?

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

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

    <p>Stock prices continuing to respond to earnings announcements over time</p> Signup and view all the answers

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

    <p>Price-to-earnings ratio</p> Signup and view all the answers

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

    <p>Abnormal excess returns for the resulting companies</p> Signup and view all the answers

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

    <p>Dramatic price swings without corresponding news</p> Signup and view all the answers

    What is systematic risk related to?

    <p>The market as a whole</p> 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?

    <p>Cij = βiβjVM</p> Signup and view all the answers

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

    <p>It reduces data items for covariances significantly.</p> Signup and view all the answers

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

    <p>The variance of the accumulated amount at time n</p> Signup and view all the answers

    Which of the following best describes specific risk?

    <p>It depends on factors peculiar to the individual security.</p> Signup and view all the answers

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

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

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

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

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

    <p>The interest rate on the investment</p> Signup and view all the answers

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

    <p>The independence of investment returns</p> Signup and view all the answers

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

    <p>The accumulation of interest and additional factors</p> Signup and view all the answers

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

    <p>It represents the cumulative effect of varying interest rates</p> Signup and view all the answers

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

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

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

    <p>It represents the standard deviation of returns</p> Signup and view all the answers

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

    <p>They are statistically independent</p> Signup and view all the answers

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

    <p>Both mean and variance increase together</p> Signup and view all the answers

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

    <p>The anticipated return from investing in a risky asset</p> Signup and view all the answers

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

    <p>Comparability</p> 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?

    <p>A is preferred over C</p> Signup and view all the answers

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

    <p>Independence</p> Signup and view all the answers

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

    <p>A unique probability linking certain outcomes with mixed outcomes</p> 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?

    <p>Independence</p> Signup and view all the answers

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

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

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

    <p>An increase in preference stability across different scenarios</p> 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.

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