by Philip H Dybvig and Stephen A Ross Presented by Jane Zhao Motivation The Sharpe ratio and Jensens alpha are two measures that are commonly used in practice to evaluate portfolio managers ID: 1029937
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1. “Differential Information and Performance Measurement Using a Security Market Line” by Philip H. Dybvig and Stephen A. RossPresented by Jane Zhao
2. MotivationThe Sharpe ratio and Jensen's alpha are two measures that are commonly used in practice to evaluate portfolio managers. These measures are motivated intuitively by the CAPM.But if the CAPM is true there is no superior performance and statistical measurement error is the only source of variation in the measures. This paper introduces superior performance into the CAPM and considers to what extent the measures accurately capture performance.
3. Main ResultsI. If we extend the CAPM to allow superior performance, the Sharpe measure and the Jensen measure may identify superior performance (market timer) as inferior.II. However, if the information is about individual assets and not the market as a whole, these measures get it right.
4. BackgroundSharpe ratio and Jensen's alphaSharpe ratio is often used to compare the change in a portfolio's overall risk-return characteristics when a new asset or asset class is added to itJensen’s alphaa positive value for Jensen's alpha means a fund manager has "beat the market" with his or her stock picking skills
5. BackgroundEfficient Frontier With Risky Assets OnlyMean-variance rule: Investment A dominates Investment B if it has a higher expected return and a lower varianceThose portfolios that have the greatest expected return for each level of risk (SD) make up the efficient frontier in the mean-variance framework
6. BackgroundCapital Allocation Line (CAL)The line of possible portfolio risk and return combinations given the risk-free rate and the risk and return of a portfolio of risky assets in the mean-variance frameworkDifferent risky portfolios form different CALs
7. BackgroundEfficient Frontier with Riskless and Risky AssetsAssume investors have homogeneous expectations (thus the same efficient frontier)All investors who hold any risky assets will use the same risky portfolio (market portfolio) and have the same optimal capital allocation line that is tangent to efficient frontier. It is the capital market line (CML).
8. BackgroundSecurity Market Line (SML)Capital market theory assumes diversification is free (perfect market); thus only systematic/market risk will be compensatedThe sensitivity of an asset’s return to the return on the market index in the market model is referred to as its beta (Covariance of asset’s return with the market return divided by variance of the market return)Formally, the SML is stated as:
9. BackgroundSharpe Ratio & Jensen’s Alpha: Practice VS TheoryPractice: use Jensen’s alpha to measure a single asset’s contribution to the whole portfolio; use Sharpe Ratio to evaluate a portfolio’s risk-adjusted returnTheory: In a CAPM world, all properly priced securities and portfolios will plot on the SML, so all portfolio should have the same Sharpe Ratio, and the Jensen’s Alpha should always be zero
10. ApproachModel Set UpTwo assets: one risk-free with return r and one risky with return Manager-observed signal term and unobserved noise term are independently normally distributed; Manager’s information is useful ;Manager’s information is not complete ;Utility function Goal: choose portfolio that maximize or Unconditional abnormal return
11. ApproachCase of market timingAbnormal return viewed by the observerInformed manager appears inferior based on SML analysis
12. ApproachCase of security-specific information
13. ApproachCase of security-specific information
14. ApproachWhat makes the difference between the two cases?In , the term is generally nonlinear and not the average of the conditional values. So generally is not a simple average of across information states sWhen the information is on the whole market, the manager’s portfolio return has a chi-squared distribution term, which is skewed to the right and bounded below, and is no longer normally distributed However, if s does not have any information about the market (the case of security-specific superior information), then conditional beta is the same as unconditional beta, the unconditional value of the term is the average of the conditional values, which makes
15. Main ResultsDifferential information disrupts the validity of SML analysisWhen the superior information is “security specific”, SML analysis for the evaluation of performance is valid;When the superior information is about the market as a whole (market timer), SML performance measurement can fail
16. AppendixThe assumptions of mean-variance theoryFor the simple decision problem, the assumptions are: Single-period modelPreferences depend only on the mean and variance of payoffsAt a given mean, lower variance is preferredAt a given variance, a higher mean is preferredPrice-taking with no taxes or transaction costsFor the equilibrium model (CAPM): we have the above assumptions and no information asymmetrycompetitive equilibriumThe assumptions of no taxes, transaction costs, or information asymmetry are sometimes known collectively as the assumption of “perfect capital markets”.