1. 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

2. 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

3. 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

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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

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

18. 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

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

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

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

22. 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

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

24. 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?

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

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

27. 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

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

29. 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

30. 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

31. 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:

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

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