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What is the form of the power utility function?
What is the form of the power utility function?
What condition must the parameter value $eta$ satisfy for risk-aversion to hold in the power utility function?
What condition must the parameter value $eta$ satisfy for risk-aversion to hold in the power utility function?
Which type of risk aversion does the power utility function exhibit?
Which type of risk aversion does the power utility function exhibit?
What is the relative risk aversion of the power utility function?
What is the relative risk aversion of the power utility function?
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Which of the following defines an iso-elastic utility function?
Which of the following defines an iso-elastic utility function?
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Why are iso-elastic utility functions considered useful?
Why are iso-elastic utility functions considered useful?
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Which of the following is an example of an iso-elastic utility function?
Which of the following is an example of an iso-elastic utility function?
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What does decreasing absolute risk aversion imply about an individual's utility?
What does decreasing absolute risk aversion imply about an individual's utility?
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What are the two main categories of tests of the semi-strong form of the EMH?
What are the two main categories of tests of the semi-strong form of the EMH?
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Which of the following is an example of over-reaction to events that contradict informational efficiency?
Which of the following is an example of over-reaction to events that contradict informational efficiency?
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What characterizes an excessively volatile market?
What characterizes an excessively volatile market?
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How did Shiller test for excessive volatility in equity markets?
How did Shiller test for excessive volatility in equity markets?
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Which of the following represents an under-reaction to events that contradict informational efficiency?
Which of the following represents an under-reaction to events that contradict informational efficiency?
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Which accounting ratio might be used incorrectly to predict stock performance, suggesting over-reaction?
Which accounting ratio might be used incorrectly to predict stock performance, suggesting over-reaction?
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What is a common result following a de-merger that indicates an under-reaction?
What is a common result following a de-merger that indicates an under-reaction?
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Which of the following situations suggests that the market may be exhibiting excessive volatility?
Which of the following situations suggests that the market may be exhibiting excessive volatility?
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What is systematic risk related to?
What is systematic risk related to?
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What is the equation for the covariance of returns on securities i and j under the single-index model?
What is the equation for the covariance of returns on securities i and j under the single-index model?
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How does the single-index model improve data efficiency compared to mean-variance portfolio theory?
How does the single-index model improve data efficiency compared to mean-variance portfolio theory?
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What does Var[Sn] represent in the context of investments?
What does Var[Sn] represent in the context of investments?
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Which of the following best describes specific risk?
Which of the following best describes specific risk?
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Which equation correctly expresses the expected value E[Sn] based on the provided content?
Which equation correctly expresses the expected value E[Sn] based on the provided content?
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Which of the following is NOT required in the single-index model?
Which of the following is NOT required in the single-index model?
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What does the variable 'j' signify in the given equations?
What does the variable 'j' signify in the given equations?
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Which of the following is a key component in calculating the variance Var[Sn]?
Which of the following is a key component in calculating the variance Var[Sn]?
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In the formula for Var[Sn], what does the term (1 + j) + s signify?
In the formula for Var[Sn], what does the term (1 + j) + s signify?
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What is the significance of the product notation ⨕ in the expression for E[Sn]?
What is the significance of the product notation ⨕ in the expression for E[Sn]?
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How is the variance Var[Sn] calculated using the expected value?
How is the variance Var[Sn] calculated using the expected value?
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What is the role of the variable 's' in the variance formula Var[Sn]?
What is the role of the variable 's' in the variance formula Var[Sn]?
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What is assumed about the interest rates 'i_t' in the cumulative product for E[Sn]?
What is assumed about the interest rates 'i_t' in the cumulative product for E[Sn]?
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Which statement describes the correct relationship between mean and variance for the investment?
Which statement describes the correct relationship between mean and variance for the investment?
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What does the expected utility formula E[U] = ∑ pi U(wi) represent in terms of investment?
What does the expected utility formula E[U] = ∑ pi U(wi) represent in terms of investment?
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Which axiom states that an investor can express a preference for all available certain outcomes?
Which axiom states that an investor can express a preference for all available certain outcomes?
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If an investor prefers outcome A over B and outcome B over C, what is the implication according to the transitivity axiom?
If an investor prefers outcome A over B and outcome B over C, what is the implication according to the transitivity axiom?
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What concept describes the indifference of an investor between two outcomes and their associated gambles?
What concept describes the indifference of an investor between two outcomes and their associated gambles?
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What does certainty equivalence establish in the context of risk and preference?
What does certainty equivalence establish in the context of risk and preference?
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Which axiom implies that if A is preferred to B, the investor remains indifferent when introducing a third option C?
Which axiom implies that if A is preferred to B, the investor remains indifferent when introducing a third option C?
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In the context of an investor's utility function U(w), what does non-satiation refer to?
In the context of an investor's utility function U(w), what does non-satiation refer to?
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What is the outcome of applying the independence axiom in investment decisions?
What is the outcome of applying the independence axiom in investment decisions?
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Study Notes
Power Utility Function
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The power utility function is defined as: U(w) = (w^(γ-1))/γ where w > 0.
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The risk-aversion condition for this function requires γ < 1.
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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
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Iso-elastic utility functions are characterized by constant relative risk aversion (RRA) where R'(w) = 0.
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These functions simplify multi-period investment decisions by allowing for "myopic" decisions, focusing only on the current period's outcomes.
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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
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Systematic risk is the market-wide risk component affecting all securities. It cannot be diversified.
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Specific risk is the risk unique to an individual security. It can be diversified away.
Single-Index Model: Covariance of Returns
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The single-index model's covariance formula is: Cij = βiβjVM
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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
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Mean-variance portfolio theory requires N means, N variances, and N(N-1)/2 covariances.
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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
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Informational efficiency focuses on how quickly and accurately market prices incorporate new information.
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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.
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An excessively volatile market exhibits price swings unjustified by the arrival of new information.
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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.