Investment analysis and portfolio management 8th reilly and brown chapter 07

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Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Eighth Edition by Frank K Reilly & Keith C Brown Chapter Chapter - An Introduction to Portfolio Management Questions to be answered: • What we mean by risk aversion and what evidence indicates that investors are generally risk averse? • What are the basic assumptions behind the Markowitz portfolio theory? • What is meant by risk and what are some of the alternative measures of risk used in investments? • How you compute the expected rate of return for an individual risky asset or a portfolio of assets? • How you compute the standard deviation of rates of return for an individual risky asset? • What is meant by the covariance between rates of return and how you compute covariance? Chapter - An Introduction to Portfolio Management • What is the relationship between covariance and correlation? • What is the formula for the standard deviation for a portfolio of risky assets and how does it differ from the standard deviation of an individual risky asset? • Given the formula for the standard deviation of a portfolio, why and how you diversify a portfolio? • What happens to the standard deviation of a portfolio when you change the correlation between the assets in the portfolio? • What is the risk-return efficient frontier? • Is it reasonable for alternative investors to select different portfolios from the portfolios on the efficient frontier? • What determines which portfolio on the efficient frontier is selected by an individual investor? Background Assumptions • As an investor, you want to maximize return for a given level of risk • Your portfolio includes all of your assets and liabilities, not just your traded securities • The relationship between the returns of the assets in the portfolio is important • 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: – – – – Life Automobile Health Disability Insurance is one of the few things we buy which we know has a negative NPV • The insured trades a known cost (the premium) for an unknown risk of loss • The required yield on bonds increases with risk classifications from AAA to AA to A… But Not Totally Risk Averse • Risk preferences may have to with the amount of money involved – we are willing to risk small amounts, but we insure against large losses – People buy lottery tickets (negative expected value but the potential loss is small) – But also buy insurance (negative expected value but the potential loss is large) Which Definition of Risk? • Uncertainty of future outcomes – – • Risk involves both positive & negative outcomes What we measure with standard deviation Probability of an adverse outcome – – Ignore outcomes that are better than expected Investors only really care about negative surprises They like positive surprises Rates of Return 1900-2003 Percentage Return Stock Market Index Returns Year Source: Ibbotson Associates Actual market returns exhibit significant fluctuation around the mean return Measure the size of the fluctuations with variance & standard deviation Measuring Risk Histogram of Annual Stock Market Returns # of Years Return % Each bar shows the number of years that the annual return was within that range out a total of 102 years of data Example: Solution RPortfolio = x1 R1 + x2 R2 = ( 0.40 ) ( 14% ) + ( 0.60 ) ( 8% ) = 10.0% σ Portfolio = x A2σ A2 + xB2σ B2 + x A xB ρ ABσ Aσ B = ( 0.4 ) ( 484 ) + ( 0.6 ) ( 196 ) + ( 0.4 ) ( 0.6 ) ( 0.20 ) ( 22 ) ( 14 ) = 177.6 σ Portfolio = Variance = 177.6 = 13.3% Example: Solution Stock Stock Stock Stock Many Risky Assets Portfolio • Return on the portfolio is simply a weighted average of the returns of the assets within the portfolio RPortfolio = X 1R1 + X R2 + + X N RN Xi = Proportion in asset i Ri = Return on asset i Risk: Many Risky Assets • To calculate the variance of the portfolio, use a variancecovariance matrix Asset Asset Asset Asset Asset Variance of Asset Covariance of Asset & Covariance of Asset & Covariance of Asset & Asset Covariance of Asset & Variance of Asset Covariance of Asset & Covariance of Asset & Asset Covariance of Asset & Covariance of Asset & Variance of Asset Covariance of Asset & Asset Covariance of Asset & Covariance of Asset & Covariance of Asset & Variance of Asset Variance-Covariance Matrix • The variance-covariance matrix shows that the influence of individual asset risk quickly diminishes as the size of the portfolio grows, whereas the influence of covariance grows quickly • For a portfolio of N assets, there are N variance terms and N2 – N covariance terms Contribution to Portfolio Risk σ Portfolio = 1 ( Average variance ) +  - ÷( Average Covariance ) N N  As N, the number of securities in the portfolio, increases, portfolio variance approaches the average covariance Thus the risk of a well-diversified portfolio depends on the market risk of the securities in the portfolio Market risk is measured by Beta Measuring Risk Portfolio risk falls rapidly as the number of securities in the portfolio rises A 1970 study by Fama & Lorie found that 80% of the unique risk is diversified away with stocks; 95% with 32 stocks & 99% with 128 stocks Canadian studies have found that substantially more stocks are required in Canada to achieve good diversification, due to the heavy concentration of resource stocks on the TSX Estimation Issues • Results of portfolio allocation depend on accurate statistical inputs • Estimates of – Expected returns – Standard deviation – Correlation coefficient • Among entire set of assets • With 100 assets, 4,950 correlation estimates • 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 • Single index market model: R i = a i + bi R m + ε i bi = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market Rm = the returns for the aggregate stock market έi = error term (lower case Greek letter epsilon) Estimation Issues If all the securities are similarly related to the market and a bi derived for each one, it can be shown that the correlation coefficient between two securities i and j is given as: σ m2 ρ ij = bi bj σ iσ j Where : ρ ij = the correlation between asset i and asset j σ m2 = the variance of returns for the aggregate stock market b i = the slope coefficient that relates the returns for security i to 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 • Frontier will be portfolios of investments rather than individual securities – An exception is the asset with the highest return Efficient Frontier for Alternative Portfolios E(R) Efficient Frontier A B C 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 – the more risk averse the individual, the steeper the slope of his/her utility curve • The slope of the efficient frontier decreases steadily as you move upward • These two interactions will determine the particular portfolio selected by an individual investor • The optimal portfolio has the highest utility for a given investor • It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility Selecting an Optimal Risky Portfolio E(R port ) U3’ U2’ More Risk Averse Investor Less Risk Averse Investor U1’ Y U3 U2 X U1 E(σ port ) The Internet Investments Online http://www.pionlie.com http://www.investmentnews.com http://www.ibbotson.com http://www.styleadvisor.com http://www.wagner.com http://www.effisols.com http://www.efficientfrontier.com ... covariance? Chapter - An Introduction to Portfolio Management • What is the relationship between covariance and correlation? • What is the formula for the standard deviation for a portfolio of... risky assets and how does it differ from the standard deviation of an individual risky asset? • Given the formula for the standard deviation of a portfolio, why and how you diversify a portfolio? .. .Chapter - An Introduction to Portfolio Management Questions to be answered: • What we mean by risk aversion and what evidence indicates that investors
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