1® 1999 South-Western College Publishing

Chapter 7Portfolio Mean And Variance

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Portfolioof

Assets

Weight Asset 1

Weight Asset 2

Weight Asset 3

Weights Sum to 100%

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Measuring Portfolio Risk

• VarianceVariance

• Standard DeviationStandard Deviation

• Degree of DependencyDegree of Dependency

– PositivePositive

– NegativeNegative

• Lower riskLower risk

• Lower risk premiumLower risk premium

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What Is The Risk Premium?

• Required to Compensate Investors for RiskRequired to Compensate Investors for Risk

• Higher the Variance or Standard Deviation, the Higher the Variance or Standard Deviation, the Higher the Required Risk PremiumHigher the Required Risk Premium

• Degree of Dependency Affects the Risk PremiumDegree of Dependency Affects the Risk Premium

– The more negative the degree of dependencyThe more negative the degree of dependency

– The lower the risk of the portfolioThe lower the risk of the portfolio

– The lower the required risk premium The lower the required risk premium

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What Does A Risk-Averse Investor Require?

• A Risk PremiumA Risk Premium

– Requires a risk premium that decreases as Requires a risk premium that decreases as the degree of dependency decreasesthe degree of dependency decreases

– The required risk premium is a function The required risk premium is a function of the asset’s variance and its dependency of the asset’s variance and its dependency with other assetswith other assets

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Summary• One AssetOne Asset

– Variance is measurer of riskVariance is measurer of risk– Higher the variance, the higher the required risk Higher the variance, the higher the required risk

premiumpremium• More Than One AssetMore Than One Asset

– Risk is a function of bothRisk is a function of both• The asset’s varianceThe asset’s variance• The degree of dependencyThe degree of dependency

– Portfolio’s variance Portfolio’s variance (Key factor)(Key factor)• Larger the varianceLarger the variance• Higher the risk premiumHigher the risk premium• Larger the risk premium on each assetLarger the risk premium on each asset

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Expected Rate Of Return On The Portfolio E(R)

• Calculate All Possible Return on the Calculate All Possible Return on the Portfolio, and Then Calculate its Portfolio, and Then Calculate its E(R)E(R)

• Use Equation 7.2Use Equation 7.2

WW1 1 • E(R• E(R11) + W) + W22 • • E(RE(R22) = E(R) = E(Rpp))

• Both Methods Provide the Same ResultsBoth Methods Provide the Same Results

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Covariance

Expected value of the Product of Expected value of the Product of Deviations From the MeanDeviations From the Mean

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Covariance

• Measures the Degree of Dependency of 2 AssetsMeasures the Degree of Dependency of 2 Assets

– PositivePositive

• Rates of return moves togetherRates of return moves together

– ZeroZero

• Rates of return have independent movementsRates of return have independent movements

– NegativeNegative

• Rates of return move in opposite directionsRates of return move in opposite directions

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Correlation Coefficient• Correlation Coefficient of 2 StocksCorrelation Coefficient of 2 Stocks

Cov(RCov(RAA, R, RBB))A,BA,B = =

AA BB

• Strength of DependencyStrength of Dependency

– +1 perfectly positive+1 perfectly positive– 0 no correlation0 no correlation– - 1 perfectly negative- 1 perfectly negative

• Correlations are Directly ComparableCorrelations are Directly Comparable no units or $no units or $

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Portfolio Variance

• Direct Method (easy to calculate)Direct Method (easy to calculate)

– Calculate rate of returnCalculate rate of return

– Calculate varianceCalculate variance

• Indirect Method (demonstrates relationship)Indirect Method (demonstrates relationship)

– Equation based on variance & covarianceEquation based on variance & covariance

– Sheds light on factors affecting risk Sheds light on factors affecting risk reductionreduction

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Low Correlation

• Reduce Portfolio FluctuationReduce Portfolio Fluctuation

• Achieved by DiversificationAchieved by Diversification

• Attractive to Risk-Averse InvestorsAttractive to Risk-Averse Investors

13® 1999 South-Western College Publishing

Set 1 Bonus Questions for Ch. 1