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)

Diversification Effect

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

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

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

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

Correlation Coefficients: Possible Values Range of values for 1,2 + 1.0 > 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

Expected Return and Portfolio Weights

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

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.

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?

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

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 20% 30% bond Fund 12% 15% 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

Determination of the Optimal Overall Portfolio

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

Minimum-Variance Frontier

Capital Allocation Lines and Efficient Frontier

Capital Asset Pricing Model (CAPM) Harry Markowitz laid down the foundation of modern portfolio theory in 1952. 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.

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.

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

Figure 9.1 The Efficient Frontier and the Capital Market Line

CAPM E(R)=Rf+β*(Rm-Rf)

Security Market line and a Positive-Alpha Stock

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

Estimates of Individual Mutual Fund Alphas

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

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

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)

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)

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

Illiquidity and Average Returns

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

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