Utility Functions and Expected Utility Theorem

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.