Lecture Presentation Software to accompanyInvestment Analysis and Portfolio ManagementSeventh Editionby Frank K. Reilly & Keith C. BrownChapter 7

Background AssumptionsAs an investor you want to maximize the returns for a given level of risk.Your portfolio includes all of your assets and liabilitiesThe 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.

Evidence ThatInvestors are Risk AverseMany investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of lossYield on bonds increases with risk classifications from AAA to AA to A.

Not all investors are risk averseRisk preference may have to do with amount of money involved - risking small amounts, but insuring large losses

Definition of Risk1. Uncertainty of future outcomesor2. Probability of an adverse outcome

Markowitz Portfolio TheoryQuantifies riskDerives the expected rate of return for a portfolio of assets and an expected risk measureShows that the variance of the rate of return is a meaningful measure of portfolio riskDerives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio

Assumptions of Markowitz Portfolio Theory1. Investors consider each investment alternative 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 Portfolio 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 (or 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, 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 RiskVariance or standard deviation of expected returnRange of returnsReturns below expectationsSemivariance a measure that only considers deviations below the meanThese measures of risk implicitly assume that investors want to minimize the damage from returns less than some target rate

Expected Rates of ReturnFor an individual asset - sum of the potential returns multiplied with the corresponding probability of the returnsFor 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 InvestmentExhibit 7.1

Table 6.1

Table 6.1Computation of Expected Return for an Individual Risky Asset

Possible Rate ofExpected Return

ProbabilityReturn (Percent)(Percent)

0.250.080.0200

0.250.100.0250

0.250.120.0300

0.250.140.0350

E(R) =0.1100

Computation of the Expected Return for a Portfolio of Risky AssetsExhibit 7.2

Table 6.2

Table 6.2Computation of the Expected Return for a Portfolio of Risky Assets

Expected SecurityExpected Portfolio

(Percent of Portfolio)

0.200.100.0200

0.300.110.0330

0.300.120.0360

0.200.130.0260

0.1150

Variance (Standard Deviation) of Returns for an Individual InvestmentStandard deviation is the square root of the varianceVariance 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 Investmentwhere Pi is the probability of the possible rate of return, Ri

Variance (Standard Deviation) of Returns for an Individual InvestmentStandard Deviation

Variance (Standard Deviation) of Returns for an Individual InvestmentExhibit 7.3Variance ( 2) = .0050Standard Deviation ( ) = .02236

Table 6.3

Table 6.3Computation of the Variance for an Individual of Risky Asset

Possible RateExpected

0.080.110.030.00090.250.000225

0.100.110.010.00010.250.000025

0.120.110.010.00010.250.000025

0.140.110.030.00090.250.000225

0.000500

Variance (Standard Deviation) of Returns for a Portfolio Computation of Monthly Rates of ReturnExhibit 7.4

Table 8.4

Table 6.4Computation of Monthly Rates of Return

Coca - ColaHome Depot

ClosingClosing

DatePriceDividendReturn (%)PriceDividendReturn (%)

Dec.0060.93845.688

Jan.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%49.2904.65%

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.0147.8802.20%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 ReturnsA measure of the degree to which two variables move together relative to their individual mean values over time

Covariance of ReturnsFor two assets, i and j, the covariance of rates of return is defined as:Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}

Covariance and CorrelationThe correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations

Covariance and CorrelationCorrelation coefficient varies from -1 to +1

Correlation CoefficientIt can vary only in the range +1 to -1. A value of +1 would indicate perfect positive correlation. This means that returns for the two 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 Formula

Portfolio Standard Deviation CalculationAny asset of a portfolio may be described by two characteristics:The expected rate of returnThe expected standard deviations of returnsThe correlation, measured by covariance, affects the portfolio standard deviationLow correlation reduces portfolio risk while not affecting the expected return

The Efficient FrontierThe 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 returnFrontier will be portfolios of investments rather than individual securitiesExceptions being the asset with the highest return and the asset with the lowest risk

Efficient Frontier for Alternative PortfoliosEfficient FrontierABCExhibit 7.15E(R)Standard Deviation of Return

The Efficient Frontier and Investor UtilityAn individual investors utility curve specifies the trade-offs he is willing to make between expected return and riskThe slope of the efficient frontier curve decreases steadily as you move upwardThese two interactions will determine the particular portfolio selected by an individual investor

The Efficient Frontier and Investor UtilityThe optimal portfolio has the highest utility for a given investorIt lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

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