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1、Lecture Presentation Software to accompanyInvestment Analysis and Portfolio ManagementSeventh Editionby Frank K. Reilly & Keith C. BrownChapter 7Chapter 7 - An Introduction to Portfolio ManagementQuestions to be answered: What do we mean by risk aversion and what evidence indicates that investors ar
2、e 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?Chapter 7 - An Introduction to Portfolio Management How do you compute the expected rate of return for an ind
3、ividual risky asset or a portfolio of assets? How do 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 do you compute covariance?Chapter 7 - An Introduction to Portfolio Management What is the relation
4、ship 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, how and why do you diversify a portfol
5、io?Chapter 7 - An Introduction to Portfolio Management 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
6、 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 the returns for a given level of risk. Your portfolio includes all of your assets and liabilities The
7、relationship between the returns for assets in the portfolio is important. A good portfolio is not simply a collection of individually good investments.Risk AversionGiven a choice between two assets with equal rates of return, most investors will select the asset with the lower level of risk.Evidenc
8、e ThatInvestors are Risk Averse 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.Not all investors are risk averseRisk preference ma
9、y have to do with amount of money involved - risking small amounts, but insuring large lossesDefinition of Risk1. Uncertainty of future outcomesor2. Probability of an adverse outcomeMarkowitz Portfolio Theory Quantifies risk Derives the expected rate of return for a portfolio of assets and an expect
10、ed risk measure 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 portfolioAssumptions of Markowitz Portfolio Theory1. Investors consider each investment altern
11、ative as being presented by a probability distribution of expected returns over some holding period.Assumptions of Markowitz Portfolio Theory2. Investors minimize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.Assumptions of Markowitz Portfol
12、io Theory3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.Assumptions of Markowitz Portfolio Theory4. Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (o
13、r standard deviation) of returns only.Assumptions of Markowitz Portfolio Theory5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.Markowitz Portfolio TheoryUsing these five assumptions
14、, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.Alternative Measures of Risk Variance or standard deviation of expected
15、return Range of returns Returns below expectations 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 rateExpected Rates of Return For an individual asset - sum of
16、 the potential returns multiplied with the corresponding probability of the returns For a portfolio of assets - weighted average of the expected rates of return for the individual investments in the portfolio Computation of Expected Return for an Individual Risky Investment0.250.080.02000.250.100.02
17、500.250.120.03000.250.140.0350E(R) = 0.1100Expected Return(Percent)ProbabilityPossible Rate ofReturn (Percent)Exhibit 7.1Computation of the Expected Return for a Portfolio of Risky Assets0.200.100.02000.300.110.03300.300.120.03600.200.130.0260E(Rpor i) = 0.1150Expected PortfolioReturn (Wi X Ri) (Per
18、cent of Portfolio)Expected SecurityReturn (Ri)Weight (Wi) Exhibit 7.2iasset for return of rate expected the )E(Riasset in portfolio theofpercent theW :whereRW)E(Rii1ipor niiiVariance (Standard Deviation) of Returns for an Individual InvestmentStandard deviation is the square root of the varianceVari
19、ance is a measure of the variation of possible rates of return Ri, from the expected rate of return E(Ri)Variance (Standard Deviation) of Returns for an Individual Investmentni 1i2ii2P)E(R-R)( Variancewhere Pi is the probability of the possible rate of return, RiVariance (Standard Deviation) of Retu
20、rns for an Individual Investmentni 1i2iiP)E(R-R)(Standard DeviationVariance (Standard Deviation) of Returns for an Individual InvestmentPossible RateExpectedof Return (Ri)Return E(Ri)Ri - E(Ri)Ri - E(Ri)2PiRi - E(Ri)2Pi0.080.110.030.00090.250.0002250.100.110.010.00010.250.0000250.120.110.010.00010.2
21、50.0000250.140.110.030.00090.250.0002250.000500Exhibit 7.3Variance ( 2) = .0050Standard Deviation ( ) = .02236Variance (Standard Deviation) of Returns for a Portfolio Computation of Monthly Rates of ReturnExhibit 7.4ClosingClosingDatePriceDividendReturn (%)PriceDividendReturn (%)Dec.0060.93845.688 J
22、an.0158.000-4.82%48.2005.50%Feb.0153.030-8.57%42.500 -11.83%Mar.0145.1600.18-14.50%43.1000.041.51%Apr.0146.1902.28%47.1009.28%May.0147.4002.62%49.290 4.65%Jun.0145.0000.18-4.68%47.2400.04-4.08%Jul.0144.600-0.89%50.3706.63%Aug.0148.6709.13%45.9500.04-8.70%Sep.0146.8500.18-3.37%38.370-16.50%Oct.0147.8
23、802.20%38.230-0.36%Nov.0146.9600.18-1.55%46.6500.0522.16%Dec.0147.1500.40%51.0109.35%E(RCoca-Cola)= -1.81%E(Rhome Depot)=E(RExxon)= 1.47%Covariance of Returns A measure of the degree to which two variables “move together relative to their individual mean values over time Covariance of ReturnsFor two
24、 assets, i and j, the covariance of rates of return is defined as:Covij = ERi - E(Ri)Rj - E(Rj)Covariance and Correlation The correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations Covariance and CorrelationCorrelation coef
25、ficient varies from -1 to +1jtitiijR ofdeviation standard theR ofdeviation standard thereturns oft coefficienn correlatio ther:whereCovrjjiijijCorrelation Coefficient It can vary only in the range +1 to -1. A value of +1 would indicate perfect positive correlation. This means that returns for the tw
26、o assets move together in a completely linear manner. A value of 1 would indicate perfect correlation. This means that the returns for two assets have the same percentage movement, but in opposite directions Portfolio Standard Deviation Formulajiijijij2iiportn1in1iijjn1ii2i2iportrCov wherej, and i a
27、ssetsfor return of rates ebetween th covariance theCoviasset for return of rates of variancetheportfolio in the valueof proportion by the determined are weights whereportfolio, in the assets individual theof weightstheWportfolio theofdeviation standard the:whereCovwwwPortfolio Standard Deviation Cal
28、culation Any asset of a portfolio may be described by two characteristics: The expected rate of return The expected standard deviations of returns The correlation, measured by covariance, affects the portfolio standard deviation Low correlation reduces portfolio risk while not affecting the expected
29、 returnCombining Stocks with Different Returns and Risk Case Correlation Coefficient Covariance a +1.00 .0070 b +0.50 .0035 c 0.00 .0000 d -0.50 -.0035 e -1.00 -.0070 W)E(R Asset ii2ii1 .10 .50 .0049 .07 2 .20 .50 .0100 .10Combining Stocks with Different Returns and Risk Assets may differ in expecte
30、d rates of return and individual standard deviations Negative correlation reduces portfolio risk Combining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equalConstant Correlationwith Changing WeightsCaseW1W2E(Ri)f0.001.
31、000.20 g0.200.800.18 h0.400.600.16 i0.500.500.15 j0.600.400.14 k0.800.200.12 l1.000.000.10 )E(R Asset i1 .10 rij = 0.00 2 .20 Constant Correlationwith Changing WeightsCaseW1W2E(Ri)E(port)f0.001.000.20 0.1000g0.200.800.18 0.0812h0.400.600.16 0.0662i0.500.500.15 0.0610j0.600.400.14 0.0580k0.800.200.12
32、 0.0595l1.000.000.10 0.0700Portfolio Risk-Return Plots for Different Weights-0.050.100.150.200.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12Standard Deviation of ReturnE(R)Rij = +1.0012With two perfectly correlated assets, it is only possible to create a two asset portfolio with ris
33、k-return along a line between either single assetPortfolio Risk-Return Plots for Different Weights-0.050.100.150.200.00 0.01 0.02 0.03 0.04 0.050.06 0.07 0.08 0.09 0.10 0.11 0.12Standard Deviation of ReturnE(R)Rij = 0.00Rij = +1.00fghijk12With uncorrelated assets it is possible to create a two asset
34、 portfolio with lower risk than either single assetPortfolio Risk-Return Plots for Different Weights-0.050.100.150.200.00 0.01 0.02 0.03 0.04 0.050.06 0.07 0.08 0.09 0.10 0.11 0.12Standard Deviation of ReturnE(R)Rij = 0.00Rij = +1.00Rij = +0.50fghijk12With correlated assets it is possible to create
35、a two asset portfolio between the first two curvesPortfolio Risk-Return Plots for Different Weights-0.050.100.150.200.00 0.01 0.02 0.03 0.04 0.050.06 0.07 0.08 0.09 0.10 0.11 0.12Standard Deviation of ReturnE(R)Rij = 0.00Rij = +1.00Rij = -0.50Rij = +0.50fghijk12 With negatively correlated assets it
36、is possible to create a two asset portfolio with much lower risk than either single assetPortfolio Risk-Return Plots for Different Weights-0.050.100.150.200.00 0.01 0.02 0.03 0.04 0.050.06 0.07 0.08 0.09 0.10 0.11 0.12Standard Deviation of ReturnE(R)Rij = 0.00Rij = +1.00Rij = -1.00Rij = +0.50fghijk1
37、2With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no riskRij = -0.50Exhibit 7.13Estimation Issues Results of portfolio allocation depend on accurate statistical inputs Estimates of Expected returns Standard deviation Correlation coefficient Among
38、 entire set of assets With 100 assets, 4,950 correlation estimates Estimation risk refers to potential errorsEstimation 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:i
39、miiiRbaRbi = the slope coefficient that relates the returns for security i to the returns for the aggregate stock marketRm = the returns for the aggregate stock marketEstimation IssuesIf all the securities are similarly related to the market and a bi derived for each one, it can be shown that the co
40、rrelation coefficient between two securities i and j is given as:marketstock aggregatefor the returns of variancethe wherebbr2mi2mjiijjThe 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 fo
41、r every level of return Frontier will be portfolios of investments rather than individual securities Exceptions being the asset with the highest return and the asset with the lowest riskEfficient Frontier for Alternative PortfoliosEfficient FrontierABCExhibit 7.15E(R)Standard Deviation of ReturnThe
42、Efficient Frontier and Investor Utility An individual investors 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 These two interactions will determine the particular portfolio
43、 selected by an individual investorThe Efficient Frontier and Investor Utility 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 utilitySelecting an Optimal Risky Portfolio)E(port)E(RportXYU3U2U1U3U2U1Exhibit 7.16The InternetInvestments OnlineFuture topicsChapter 8 Capital Market Theory Capital Asset Pricing Model Beta Expected Return and Risk Arbitrage Pricing Theory