Portfolio Theory 0 What and how do you select any investment opportunity? Investors are concerned with Returns Risk How to choose the optimal portfolio? 1 Mean-Variance Portfolio Theory •We consider a risk-based approach and select an investment portfolio by minimizing risk for a given level of expected return by maximizing expected return for a given level of risk. •We estimate different measure of risk. Variance is one of them and enjoys desirable mathematical properties. •The problem here is how can we select an investment opportunity, which minimizes risk measured or described by variance? 2 Mean-Variance Portfolio Theory Harry Markowitz, a Nobel Laureate in Economics, develops the celebrated mean-variance portfolio selection model, which is also known as the modern portfolio theory in some finance textbooks and literature. 3 Mean-Variance Portfolio Theory The key idea of the Markowitz mean-variance portfolio selection model is Diversification, which is just the same as an old saying “Don't put all of your eggs in a basket”. In the context of finance, diversification means that an investor can reduce the risk from his/her portfolio by holding a portfolio of assets, which are not perfectly correlated. Diversification also allows for the same expected return from the portfolio with reduced risk. 4 What are investment returns? Investment returns measure the financial results of an investment. Returns may be historical or prospective (anticipated). Returns can be expressed in: Dollar terms. Percentage terms. 5 What is the return on an investment that costs $1,000 and is sold after 1 year for $1,100? Dollar return: $ Received - $ Invested $1,100 $1,000 = $100. Percentage return: $ Return/$ Invested $100/$1,000 = 0.10 = 10%. 6 What is investment risk? Typically, investment returns are not known with certainty. Investment risk pertains to the probability of earning a return less than that expected return. The greater the chance of a return far below the expected return, the greater the risk. 7 Probability distribution Stock X Stock Y -20 0 15 50 Rate of return (%) Which stock is riskier? Why? 8 Individual Securities The characteristics of individual securities that are of interest are the: Expected Return Variance and Standard Deviation Covariance and Correlation 9 Calculate the expected rate of return on each alternative. r = expected rate of return. n r= ri Pi . i =1 There are n situations. ri means the returns when the situation is at i and its probability is P . 10 What is the standard deviation of returns for each alternative? Standard deviation Variance 2 n 2P . r r i i i 1 11 Standard deviation measures the stand-alone risk of an investment. The larger the standard deviation, the higher the probability that returns will be far below the expected return. Coefficient of variation is an alternative measure of stand-alone risk. 12 Expected Return, Variance, and Covariance Rate of Return Scenario Probability Stock fund Bond fund Recession 33.3% -7% 17% Normal 33.3% 12% 7% Boom 33.3% 28% -3% Consider the following two risky asset world. There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund. 13 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% E ( rS ) 1 (7%) 1 (12%) 1 ( 28%) 3 3 3 E ( rS ) 11% 14 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% E ( rB ) 1 (17%) 1 (7%) 1 ( 3%) 3 3 3 E ( rB ) 7% 15 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (7% 11%) 3.24% 2 16 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (12% 11%) .01% 2 17 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (28% 11%) 2.89% 2 18 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% 1 2.05% (3.24% 0.01% 2.89%) 3 19 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% 14.3% 0.0205 20 The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% Note that stocks have a higher expected return than bonds and higher risk. 21 The Return and Risk for Portfolios Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% Scenario Recession Normal Boom Expected return Variance Standard Deviation 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks. The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: rP w B rB w S rS 5% 50% ( 7%) 50 % (17%) 22 The Return and Risk for Portfolios Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% Scenario Recession Normal Boom Expected return Variance Standard Deviation 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: rP w B rB w S rS 9 .5% 50% (12%) 50% (7 %) 23 The Return and Risk for Portfolios Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% Scenario Recession Normal Boom Expected return Variance Standard Deviation 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: rP w B rB w S rS 12.5% 50% ( 28 %) 50 % ( 3 %) 24 The Return and Risk for Portfolios Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% Scenario Recession Normal Boom Expected return Variance Standard Deviation 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio. E ( rP ) w B E ( rB ) w S E ( rS ) 9% 50% (11%) 50 % (7 %) 25 The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% The variance of the rate of return on the two risky assets portfolio is σ P2 (w B σ B ) 2 (w S σ S ) 2 2(w B σ B )(w S σ S )ρ BS where BS is the correlation coefficient between the returns on the stock and bond funds. 26 The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation. 27 % in stocks Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50.00% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.08% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.00% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolio Return The Efficient Set for Two Assets Portfolo Risk and Return Combinations 12.0% 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 0.0% 100% stocks 100% bonds 5.0% 10.0% 15.0% 20.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds … 28 % in stocks Risk Return 0% 0% 5% 5% 10% 10% 15% 15% 20% 20% 25% 25% 30% 30% 35% 35% 40% 40% 45% 45% 50% 50% 55% 55% 60% 60% 65% 65% 70% 70% 75% 75% 80% 80% 85% 85% 90% 90% 95% 95% 100% 100% 8.2% 8.2% 7.0% 7.0% 5.9% 5.9% 4.8% 4.8% 3.7% 3.7% 2.6% 2.6% 1.4% 1.4% 0.4% 0.4% 0.9% 0.9% 2.0% 2.0% 3.1% 3.1% 4.2% 4.2% 5.3% 5.3% 6.4% 6.4% 7.6% 7.6% 8.7% 8.7% 9.8% 9.8% 10.9% 10.9% 12.1% 12.1% 13.2% 13.2% 14.3% 14.3% 7.0% 7.0% 7.2% 7.2% 7.4% 7.4% 7.6% 7.6% 7.8% 7.8% 8.0% 8.0% 8.2% 8.2% 8.4% 8.4% 8.6% 8.6% 8.8% 8.8% 9.0% 9.0% 9.2% 9.2% 9.4% 9.4% 9.6% 9.6% 9.8% 9.8% 10.0% 10.0% 10.2% 10.2% 10.4% 10.4% 10.6% 10.6% 10.8% 10.8% 11.0% 11.0% Portfolio Return The Efficient Set for Two Assets Portfolo Risk and Return Combinations 12.0% 11.0% 100% stocks 10.0% 9.0% 8.0% 7.0% 6.0% 100% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds … 29 % in stocks Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.1% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolio Return The Efficient Set for Two Assets Portfolo Risk and Return Combinations 12.0% 11.0% 10.0% 100% stocks 9.0% 8.0% 7.0% 6.0% 100% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These compromise the efficient frontier. 30 Two-Security Portfolios with Various return Correlations 100% stocks = -1.0 100% bonds = 1.0 = 0.2 31 Portfolio Risk/Return Two Securities: Correlation Effects 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 32 Mean-Variance Example With Excel A case of Three Stock 33 What would happen to the risk of portfolio as more randomly selected stocks were added? p would decrease because the added stocks would not be perfectly correlated, but rp would remain relatively constant. 34 Prob. Large 2 1 0 15 Return 1 35% ; 2 20%. 35 Portfolio Risk as a Function of the Number of Stocks in the Portfolio In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n Thus diversification can eliminate some, but not all of the risk of individual securities. 36 Total Risk Total risk = systematic risk + unsystematic risk The standard deviation of returns is a measure of total risk. For well-diversified portfolios, unsystematic risk is very small. Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk. Stand-alone risk = Market risk + Diversifiable risk Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification. Firm-specific, or diversifiable, risk is that part of a security’s stand-alone risk that can be eliminated by diversification. 38 Can an investor holding one stock earn a return commensurate with its risk? No. Rational investors will minimize risk by holding portfolios. They bear only market risk, so prices and returns reflect this lower risk. The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk. 39 Conclusions As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio. In general p falls very slowly after about 40 stocks are included. By forming well-diversified portfolios, investors can eliminate part of the riskiness of owning a single stock. 40

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