Chapter 7 An Introduction to Portfolio Management

Why Should Capital Markets Be Efficient? The premises of an efficient market A large number of competing profit-maximizing participants analyze and value securities, each independently of the others A large number of competing profit-maximizing participants analyze and value securities, each independently of the others New information regarding securities comes to the market in a random fashion New information regarding securities comes to the market in a random fashion Profit-maximizing investors adjust security prices rapidly to reflect the effect of new information Profit-maximizing investors adjust security prices rapidly to reflect the effect of new information Conclusion: the expected returns implicit in the current price of a security should reflect its risk

Alternative Efficient Market Hypotheses Weak-form efficient market hypothesis Weak-form efficient market hypothesis Semistrong-form EMH Semistrong-form EMH Strong-form EMH Strong-form EMH

Efficient Capital Markets Joint hypothesis problem Joint hypothesis problem market efficiency must be tested at same time as test’ of asset pricing model being used to generate expected returns market efficiency must be tested at same time as test’ of asset pricing model being used to generate expected returns no one APM shown to be “the true model” that represents how returns are generated no one APM shown to be “the true model” that represents how returns are generated for this reason, some believe market efficiency is not truly able to be tested for this reason, some believe market efficiency is not truly able to be tested Fama explains that we may never be able to say for sure whether markets are efficient or not, results of tests are still worthwhile Fama explains that we may never be able to say for sure whether markets are efficient or not, results of tests are still worthwhile

Implications of Efficient Capital Markets Overall results indicate the capital markets are efficient as related to numerous sets of information Overall results indicate the capital markets are efficient as related to numerous sets of information There are substantial instances where the market fails to rapidly adjust to public information There are substantial instances where the market fails to rapidly adjust to public information So, what techniques will or won’t work? So, what techniques will or won’t work? What do you do if you can’t beat the market? What do you do if you can’t beat the market?

Efficient Markets and Portfolio Management Management depends on analysts Management depends on analysts With superior analysts, follow them and look for opportunities in neglected stock With superior analysts, follow them and look for opportunities in neglected stock Without superior analysts, passive management may outperform active management by Without superior analysts, passive management may outperform active management by Build a globally diversified portfolio with a risk level matching client preferences Build a globally diversified portfolio with a risk level matching client preferences Minimize transaction costs (taxes, trading turnover, liquidity costs) Minimize transaction costs (taxes, trading turnover, liquidity costs)

The Rationale and Use of Index Funds Efficient capital markets and a lack of superior analysts imply that many portfolios should be managed passively (so their performance matches the aggregate market, minimizes the costs of research and trading) Efficient capital markets and a lack of superior analysts imply that many portfolios should be managed passively (so their performance matches the aggregate market, minimizes the costs of research and trading) Institutions created market (index) funds which duplicate the composition and performance of a selected index series Institutions created market (index) funds which duplicate the composition and performance of a selected index series

Background Assumptions As an investor you want to maximize the returns for a given level of risk. As an investor you want to maximize the returns for a given level of risk. Your portfolio includes all of your assets and liabilities Your portfolio includes all of your assets and liabilities The relationship between the returns for assets in the portfolio is important. The relationship between the returns for assets in the portfolio is important. A good portfolio is not simply a collection of individually good investments. A good portfolio is not simply a collection of individually good investments.

Risk Aversion Given a choice between two assets with equal rates of return, most investors will select the asset with the lower level of risk.

Evidence That Investors are Risk Averse Many investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of loss Many investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of loss Yield on bonds increases with risk classifications from AAA to AA to A…. Yield on bonds increases with risk classifications from AAA to AA to A….

Not all Investors are Risk Averse Risk preference may have to do with amount of money involved - risking small amounts, but insuring large losses

Markowitz Portfolio Theory Quantifies risk Quantifies risk Derives the expected rate of return for a portfolio of assets and an expected risk measure Derives the expected rate of return for a portfolio of assets and an expected risk measure Shows that the variance of the rate of return is a meaningful measure of portfolio risk Shows that the variance of the rate of return is a meaningful measure of portfolio risk Derives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio Derives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio

Alternative Measures of Risk Variance or standard deviation of expected return Variance or standard deviation of expected return Range of returns Range of returns Returns below expectations Returns below expectations Semivariance – a measure that only considers deviations below the mean Semivariance – a measure that only considers deviations below the mean These measures of risk implicitly assume that investors want to minimize the damage from returns less than some target rate These measures of risk implicitly assume that investors want to minimize the damage from returns less than some target rate

Expected Rates of Return For an individual asset - sum of the potential returns multiplied with the corresponding probability of the returns For an individual asset - sum of the potential returns multiplied with the corresponding probability of the returns For a portfolio of investments - weighted average of the expected rates of return for the individual investments in the portfolio For a portfolio of investments - weighted average of the expected rates of return for the individual investments in the portfolio

Expected Return for an Individual Risky Investment Exhibit 7.1

Expected Return for a Portfolio of Risky Assets Exhibit 7.2

Variance of Returns for an Individual Investment Variance is a measure of the variation of possible rates of return R i, from the expected rate of return [E(R i )] Standard deviation is the square root of the variance

Variance of Returns for an Individual Investment where P i is the probability of the possible rate of return, R i

Standard Deviation of Returns for an Individual Investment Standard Deviation

Standard Deviation of Returns for an Individual Investment Exhibit 7.3 Variance ( 2 ) = Standard Deviation ( ) =

Covariance of Returns A measure of the degree to which two variables “move together” relative to their individual mean values over time A measure of the degree to which two variables “move together” relative to their individual mean values over time

Covariance of Returns  For two assets, i and j, the covariance of rates of return is defined as: Cov ij = sum{[R i - E(R i )] [R j - E(R j )]p i } Correlation coefficient varies from -1 to +1 Correlation coefficient varies from -1 to +1

Portfolio Standard Deviation Formula

Combining Stocks with Different Returns and Risk Case Correlation Coefficient Covariance Case Correlation Coefficient Covariance a a b b c c d d e e

Constant Correlation with Changing Weights 1.10 r ij =

Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk- return along a line between either single asset

Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = f g h i j k 1 2 With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset

Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = R ij = f g h i j k 1 2 With correlated assets it is possible to create a two asset portfolio between the first two curves

Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = R ij = R ij = f g h i j k 1 2 With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset

Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = R ij = R ij = f g h i j k 1 2 With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk R ij = Exhibit 7.13

Estimation Issues Results of portfolio allocation depend on accurate statistical inputs Results of portfolio allocation depend on accurate statistical inputs Estimates of Estimates of Expected returns Expected returns Standard deviation Standard deviation Correlation coefficient Correlation coefficient Among entire set of assets Among entire set of assets With 100 assets, 4,950 correlation estimates With 100 assets, 4,950 correlation estimates Estimation risk refers to potential errors Estimation risk refers to potential errors

Estimation Issues With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets Single index market model: Single index market model: b i = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market R m = the returns for the aggregate stock market

The Efficient Frontier The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return Frontier will be portfolios of investments rather than individual securities Frontier will be portfolios of investments rather than individual securities Exceptions being the asset with the highest return and the asset with the lowest risk Exceptions being the asset with the highest return and the asset with the lowest risk

Efficient Frontier for Alternative Portfolios Efficient Frontier A B C Figure 8.9 E(R) Standard Deviation of Return

The Efficient Frontier and Investor Utility An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk The slope of the efficient frontier curve decreases steadily as you move upward The slope of the efficient frontier curve decreases steadily as you move upward These two interactions will determine the particular portfolio selected by an individual investor These two interactions will determine the particular portfolio selected by an individual investor

Selecting an Optimal Risky Portfolio X Y U3U3 U2U2 U1U1 U 3’ U 2’ U 1’ Figure 8.10