1. Optimal Risky Portfolio, CAPM, and APT
Diversification
Portfolio of Two Risky Assets
Asset Allocation with Risky and Risk-free Assets
Markowitz Portfolio Selection Model
CAPM
APT (arbitrage pricing theory)

2. Diversification Effect

3. Systematic risk v. Nonsystematic Risk
Systematic risk, (nondiversifiable risk or market risk), is the risk that remains after extensive diversifications
Nonsystematic risk (diversifiable risk, unique risk, firm-specific risk) – risks can be eliminated through diversifications

4. Two-Security Portfolio: Return
rp = W1r1 + W2r2
W1 = Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2

5. Two-Security Portfolio: Risk
p2 = w1212 + w2222 + 2W1W2 Cov(r1r2)
12 = Variance of Security 1
22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2

6. Covariance
Cov(r1r2) = 1,212
1,2 = Correlation coefficient of returns
1 = Standard deviation of
returns for Security 1
2 = Standard deviation of
returns for Security 2

7. Correlation Coefficients: Possible Values
Range of values for 1,2
> r > -1.0
If r= 1.0, the securities would be perfectly positively correlated
If r= - 1.0, the securities would be perfectly negatively correlated

8. Expected Return and Portfolio Weights

9. Expected Return and Standard Deviation
Look at ρ=-1, 0 or 1.
Minimum Variance Portfolio

10. The Effect of Correlation
The relationship depends on correlation coefficient.
-1.0 <  < +1.0
The smaller the correlation, the greater the risk reduction potential.
If r = +1.0, no risk reduction is possible.

11. Capital Asset Line
A graph showing all feasible risk-return combinations of a risky and risk-free asset.
See page 206 for possible CAL
Optimal CAL – what is the objective function in the optimization?

12. Optimal CAL and the Optimal Risky Portfolio
Equation 7.13, page 207

13. Example
A pension fund manager is considering 3 mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as following:
Exp(ret) Std Dev
Stock fund % %
bond Fund 12% %
The correlation between the fund returns is 0.10
Answer Problem 4 through 6, page 223.
Also see Example 7.2 (optimal risky portfolio) on page 207

14. Determination of the Optimal Overall Portfolio

15. Markowitz Portfolio Selection
Generalize the portfolio construction problem to the case of many risky securities and a risk-free asset
Steps
Get minimum variance frontier
Efficient frontier – the part above global MVP
An optimal allocation between risky and risk-free asset

16. Minimum-Variance Frontier

17. Capital Allocation Lines and Efficient Frontier

18. Capital Asset Pricing Model (CAPM)
Harry Markowitz laid down the foundation of modern portfolio theory in The CAPM was developed by William Sharpe, John Lintner, Jan Mossin in mid 1960s.
It is the equilibrium model that underlies all modern financial theory.
Derived using principles of diversification with simplified assumptions.

19. Assumptions Individual investors are price takers.
Single-period investment horizon.
Investments are limited to traded financial assets.
No taxes and transaction costs.
Information is costless and available to all investors.
Investors are rational mean-variance optimizers.
There are homogeneous expectations.

20. Resulting Equilibrium Conditions
All investors will hold the same portfolio for risky assets – market portfolio
Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value
Risk premium on the market depends on the average risk aversion of all market participants
Risk premium on an individual security is a function of its covariance with the market

21. Figure 9.1 The Efficient Frontier and the Capital Market Line

22. CAPM
E(R)=Rf+β*(Rm-Rf)

23. Security Market line and a Positive-Alpha Stock

24. CAPM Applications: Index Model
To move from expected to realized returns—use the index model in excess return form:
Ri=αi+βiRM+ei
The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship
What would be the testable implication?
See page 293

25. Estimates of Individual Mutual Fund Alphas

26. CAPM Applications: Market Model
Ri-rf=αi+βi(RM-rf)+ei
Test implication: αi=0

27. Is CAMP Testable? Is the CAPM testable
Proxies must be used for the market portfolio
CAPM is still considered the best available description of security pricing and is widely accepted

28. Other CAPM Models: Multiperiod Model
Page 303
Considering CAPM in the multi-period setting
Other than comovement with the market portfolio, uncertainty in investment opportunity and changes in prices of consumption goods may affect stock returns
Equation (9.14)

29. Other CAPM Models: Consumption Based Model
No longer consider the comovements in returns of individual securities with returns of market portfolios
Key intuition: investors balance between today’s consumption and the saving and investments that will support future consumption
Page 304; Equation (9.15)

30. Liquidity and CAPM
Liquidity – the ease and speed with which an asset can be sold at fair market value.
Illiquidity Premium
The discount in security price that results from illiquidity is large
Compensation for liquidity risk – inanticipated change in liquidity
Research supports a premium for illiquidity. Amihud and Mendelson and Acharya and Pedersen

31. Illiquidity and Average Returns

32. APT Arbitrage Pricing Theory
This is a multi-factor approach in pricing stock returns. See chapter 10

33. Fama-French Three-Factor Model
The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums (page 336)
rit=αi+βiMRMt+βiSMBSMBt+βiHMLHMLt+eit
Where:
SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks
HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio