Bài giảng Investment - chapter 7: Optimal Risky Portfolios

CHAPTER 7Optimal Risky PortfoliosDiversification and Portfolio RiskMarket riskSystematic or nondiversifiableFirm-specific riskDiversifiable or nonsystematicFigure 7.1 Portfolio Risk as a Function of the Number of Stocks in the PortfolioFigure 7.2 Portfolio DiversificationCovariance and CorrelationPortfolio risk depends on the correlation between the returns of the assets in the portfolioCovariance and the correlation coefficient provide a measure of the way returns two assets varyTwo-Security Portfolio: Return = Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security ETwo-Security Portfolio: RiskTwo-Security Portfolio: Risk ContinuedAnother way to express variance of the portfolio:D,E = Correlation coefficient of returns Cov(rD,rE) = DEDED = Standard deviation of returns for Security DE = Standard deviation of returns for Security ECovarianceRange of values for 1,2+ 1.0 > r > -1.0If r = 1.0, the securities would be perfectly positively correlatedIf r = - 1.0, the securities would be perfectly negatively correlatedCorrelation Coefficients: Possible ValuesTable 7.1 Descriptive Statistics for Two Mutual Funds 2p = w1212+ w2212+ 2w1w2 Cov(r1,r2)+ w3232 Cov(r1,r3)+ 2w1w3 Cov(r2,r3)+ 2w2w3Three-Security PortfolioTable 7.2 Computation of Portfolio Variance From the Covariance MatrixTable 7.3 Expected Return and Standard Deviation with Various Correlation CoefficientsFigure 7.3 Portfolio Expected Return as a Function of Investment ProportionsFigure 7.4 Portfolio Standard Deviation as a Function of Investment ProportionsMinimum Variance Portfolio as Depicted in Figure 7.4Standard deviation is smaller than that of either of the individual component assetsFigure 7.3 and 7.4 combined demonstrate the relationship between portfolio riskFigure 7.5 Portfolio Expected Return as a Function of Standard Deviation The relationship depends on the correlation coefficient-1.0 <  < +1.0The smaller the correlation, the greater the risk reduction potentialIf r = +1.0, no risk reduction is possibleCorrelation EffectsFigure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALsThe Sharpe RatioMaximize the slope of the CAL for any possible portfolio, pThe objective function is the slope:Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky PortfolioFigure 7.8 Determination of the Optimal Overall PortfolioFigure 7.9 The Proportions of the Optimal Overall PortfolioMarkowitz Portfolio Selection ModelSecurity SelectionFirst step is to determine the risk-return opportunities availableAll portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinationsFigure 7.10 The Minimum-Variance Frontier of Risky AssetsMarkowitz Portfolio Selection Model ContinuedWe now search for the CAL with the highest reward-to-variability ratioFigure 7.11 The Efficient Frontier of Risky Assets with the Optimal CALMarkowitz Portfolio Selection Model ContinuedNow the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8Figure 7.12 The Efficient Portfolio SetCapital Allocation and the Separation PropertyThe separation property tells us that the portfolio choice problem may be separated into two independent tasksDetermination of the optimal risky portfolio is purely technicalAllocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preferenceFigure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient SetThe Power of DiversificationRemember:If we define the average variance and average covariance of the securities as: We can then express portfolio variance as:Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated UniversesRisk Pooling, Risk Sharing and Risk in the Long RunConsider the following:1 − p = .999p = .001Loss: payout = $100,000No Loss: payout = 0Risk Pooling and the Insurance PrincipleConsider the variance of the portfolio:It seems that selling more policies causes risk to fallFlaw is similar to the idea that long-term stock investment is less riskyRisk Pooling and the Insurance Principle ContinuedWhen we combine n uncorrelated insurance policies each with an expected profit of $ , both expected total profit and SD grow in direct proportion to n:Risk SharingWhat does explain the insurance business?Risk sharing or the distribution of a fixed amount of risk among many investors