“Differential Information and Performance
Measurement Using a Security Market Line”
by Philip H. Dybvig and Stephen A. Ross
Presented by Jane Zhao
Motivation
 The 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.
Main Results
I. 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.
Background
Sharpe ratio and Jensen's alpha
Sharpe 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 it
Jensen’s alpha
• a positive value for Jensen's alpha means a fund manager has "beat the
market" with his or her stock picking skills
Background
Efficient Frontier With Risky Assets Only
• Mean-variance rule: Investment A dominates Investment B if it has a higher
expected return and a lower variance
• Those portfolios that have the greatest expected return for each level of risk
(SD) make up the efficient frontier in the mean-variance framework
Background
Capital 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 framework
• Different risky portfolios form different CALs
Background
Efficient Frontier with Riskless and Risky Assets
• Assume 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).
Background
Security Market Line (SML)
• Capital market theory assumes diversification is free (perfect market); thus
only systematic/market risk will be compensated
• The 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:
Background
Sharpe Ratio & Jensen’s Alpha: Practice VS Theory
• Practice: 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 return
• Theory: 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
Approach
Model Set Up
• Two 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
• Unconditional abnormal return
or
Approach
Case of market timing
• Abnormal return viewed by the observer
• Informed manager appears inferior based on SML analysis
Approach
Case of security-specific information
Approach
Case of security-specific information
Approach
What 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 s
• When 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
Main Results
Differential information disrupts the validity of SML
analysis
 When 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
Appendix
The assumptions of mean-variance theory
For the simple decision problem, the assumptions are: Single-period
model
Preferences depend only on the mean and variance of payoffs
At a given mean, lower variance is preferred
At a given variance, a higher mean is preferred
Price-taking with no taxes or transaction costs
For the equilibrium model (CAPM): we have the above assumptions
and no information asymmetry
competitive equilibrium
The assumptions of no taxes, transaction costs, or information
asymmetry are sometimes known collectively as the assumption of
“perfect capital markets”.