Capital Market Line Line from RF to L is capital market line (CML) x = risk premium = E(RM) - RF y = risk = M Slope = x/y = [E(RM) - RF]/M y-intercept = RF L M E(RM) x RF y M Risk 6

Capital Market Line Slope of the CML is the market price of risk for efficient portfolios, or the equilibrium price of risk in the market Relationship between risk and expected return for portfolio P (Equation for CML): 7

Security Market Line CML Equation only applies to markets in equilibrium and efficient portfolios The Security Market Line depicts the tradeoff between risk and expected return for individual securities Under CAPM, all investors hold the market portfolio How does an individual security contribute to the risk of the market portfolio? 8

Security Market Line Equation for expected return for an individual stock similar to CML Equation 9

Security Market Line Beta = 1.0 implies as risky as market Securities A and B are more risky than the market Beta > 1.0 Security C is less risky than the market Beta < 1.0 SML E(R) A E(RM) B C RF 0.5 1.0 1.5 2.0 BetaM 10

Security Market Line Beta measures systematic risk Measures relative risk compared to the market portfolio of all stocks Volatility different than market All securities should lie on the SML The expected return on the security should be only that return needed to compensate for systematic risk 11

SML and Asset Values Er rf β Underpriced SML: Er = rf +  (Erm – rf) Overpriced rf β Underpriced  expected return > required return according to CAPM  lie “above” SML Overpriced  expected return < required return according to CAPM  lie “below” SML Correctly priced  expected return = required return according to CAPM  lie along SML

CAPM’s Expected Return-Beta Relationship Required rate of return on an asset (ki) is composed of risk-free rate (RF) risk premium (i [ E(RM) - RF ]) Market risk premium adjusted for specific security ki = RF +i [ E(RM) - RF ] The greater the systematic risk, the greater the required return 12

Estimating the SML Treasury Bill rate used to estimate RF Expected market return unobservable Estimated using past market returns and taking an expected value Estimating individual security betas difficult Only company-specific factor in CAPM Requires asset-specific forecast 13

Estimating Beta Market model Relates the return on each stock to the return on the market, assuming a linear relationship Rit =i +i RMt + eit 14

Test of CAPM Empirical SML is “flatter” than predicted SML Fama and French (1992) Market Size Book-to-market ratio Roll’s Critique True market portfolio is unobservable Tests of CAPM are merely tests of the mean-variance efficiency of the chosen market proxy

Arbitrage Pricing Theory Based on the Law of One Price Two otherwise identical assets cannot sell at different prices Equilibrium prices adjust to eliminate all arbitrage opportunities Unlike CAPM, APT does not assume single-period investment horizon, absence of personal taxes, riskless borrowing or lending, mean-variance decisions 17

Factors APT assumes returns generated by a factor model Factor Characteristics Each risk must have a pervasive influence on stock returns Risk factors must influence expected return and have nonzero prices Risk factors must be unpredictable to the market 18

APT Model Most important are the deviations of the factors from their expected values The expected return-risk relationship for the APT can be described as: E(Rit) =a0+bi1 (risk premium for factor 1) +bi2 (risk premium for factor 2) +… +bin (risk premium for factor n) 19

APT Model Reduces to CAPM if there is only one factor and that factor is market risk Roll and Ross (1980) Factors: Changes in expected inflation Unanticipated changes in inflation Unanticipated changes in industrial production Unanticipated changes in the default risk premium Unanticipated changes in the term structure of interest rates 19

Problems with APT Factors are not well specified ex ante To implement the APT model, the factors that account for the differences among security returns are required CAPM identifies market portfolio as single factor Neither CAPM or APT has been proven superior Both rely on unobservable expectations 20

Two-Security Case For a two-security portfolio containing Stock A and Stock B, the variance is:

Two Security Case (cont’d) Example Assume the following statistics for Stock A and Stock B: Stock A Stock B Expected return .015 .020 Variance .050 .060 Standard deviation .224 .245 Weight 40% 60% Correlation coefficient .50

Two Security Case (cont’d) Example (cont’d) What is the expected return and variance of this two-security portfolio?

Two Security Case (cont’d) Example (cont’d) Solution: The expected return of this two-security portfolio is:

Two Security Case (cont’d) Example (cont’d) Solution (cont’d): The variance of this two-security portfolio is:

Minimum Variance Portfolio The minimum variance portfolio is the particular combination of securities that will result in the least possible variance Solving for the minimum variance portfolio requires basic calculus

Minimum Variance Portfolio (cont’d) For a two-security minimum variance portfolio, the proportions invested in stocks A and B are:

Minimum Variance Portfolio (cont’d) Example (cont’d) Assume the same statistics for Stocks A and B as in the previous example. What are the weights of the minimum variance portfolio in this case?

Minimum Variance Portfolio (cont’d) Example (cont’d) Solution: The weights of the minimum variance portfolios in this case are:

Minimum Variance Portfolio (cont’d) Example (cont’d) Weight A Portfolio Variance

Correlation and Risk Reduction Portfolio risk decreases as the correlation coefficient in the returns of two securities decreases Risk reduction is greatest when the securities are perfectly negatively correlated If the securities are perfectly positively correlated, there is no risk reduction

The n-Security Case For an n-security portfolio, the variance is:

The n-Security Case (cont’d) A covariance matrix is a tabular presentation of the pairwise combinations of all portfolio components The required number of covariances to compute a portfolio variance is (n2 – n)/2 Any portfolio construction technique using the full covariance matrix is called a Markowitz model

Computational Advantages The single-index model compares all securities to a single benchmark An alternative to comparing a security to each of the others By observing how two independent securities behave relative to a third value, we learn something about how the securities are likely to behave relative to each other

Computational Advantages (cont’d) A single index drastically reduces the number of computations needed to determine portfolio variance A security’s beta is an example:

Portfolio Statistics With the Single-Index Model Beta of a portfolio: Variance of a portfolio:

Portfolio Statistics With the Single-Index Model (cont’d) Variance of a portfolio component: Covariance of two portfolio components: