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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)

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Diversification Effect

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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

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rp = W1r1 + W2r2

W1 = Proportion of funds in Security 1W2 = Proportion of funds in Security 2r1 = Expected return on Security 1r2 = Expected return on Security 2

1

n

1iiw

Two-Security Portfolio: Return

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p2 = w1

212 + w2

222 + 2W1W2 Cov(r1r2)

12 = Variance of Security 1

22 = Variance of Security 2

Cov(r1r2) = Covariance of returns for Security 1 and Security 2

Two-Security Portfolio: Risk

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1,2 = Correlation coefficient of returns

Cov(r1r2) = 1,212

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2

Covariance

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Range of values for 1,2

+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly positively correlated

If = - 1.0, the securities would be perfectly negatively correlated

Correlation Coefficients: Possible Values

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Expected Return and Portfolio Weights

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Expected Return and Standard Deviation

Look at ρ=-1, 0 or 1.

Minimum Variance Portfolio

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The relationship depends on correlation coefficient.

-1.0 < < +1.0

The smaller the correlation, the greater the risk reduction potential.

If = +1.0, no risk reduction is possible.

The Effect of Correlation

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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?

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Optimal CAL and the Optimal Risky Portfolio

Equation 7.13, page 207

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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 DevStock 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

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Determination of the Optimal Overall Portfolio

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Markowitz Portfolio Selection

Generalize the portfolio construction problem to the case of many risky securities and a risk-free asset

StepsGet minimum variance frontier

Efficient frontier – the part above global MVP

An optimal allocation between risky and risk-free asset

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Minimum-Variance Frontier

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Capital Allocation Lines and Efficient Frontier

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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.

Capital Asset Pricing Model (CAPM)

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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.

Assumptions

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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

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Figure 9.1 The Efficient Frontier and the Capital Market Line

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CAPM

2

),(

M

Mii

rrCov

E(R)=Rf+β*(Rm-Rf)

)(RER

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Security Market line and a Positive-Alpha Stock

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CAPM Applications: Index ModelTo 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

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Estimates of Individual Mutual Fund Alphas

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CAPM Applications: Market Model

Market model

Ri-rf=αi+βi(RM-rf)+ei

Test implication: αi=0

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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

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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)

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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)

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Liquidity and CAPM

Liquidity – the ease and speed with which an asset can be sold at fair market value.

Illiquidity PremiumThe 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

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Illiquidity and Average Returns

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APT

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

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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