chapter 6 an introduction to portfolio management

Chapter 6An Introduction to

Portfolio Management

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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

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The relationship between the returns for assets in the portfolio is important.A good portfolio is not simply a collection of individually good investments.

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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.

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Evidence ThatEvidence ThatInvestors are Risk AverseInvestors 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….

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Definition of RiskDefinition of Risk

1. Uncertainty of future outcomesor2. Probability of an adverse

outcome

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Markowitz Portfolio TheoryMarkowitz Portfolio Theory

Quantifies riskDerives the expected rate of return and expected risk for a portfolio of assets.

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Shows 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

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Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory

1. Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.

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Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory

2. Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.

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Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory

3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.

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Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory

4. 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.

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Assumptions of Assumptions of Markowitz Portfolio TheoryMarkowitz Portfolio Theory

5. 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.

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Markowitz Portfolio TheoryMarkowitz 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.

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Alternative Measures of RiskAlternative Measures of RiskVariance or standard deviation of expected returnRange of returnsReturns 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 rate

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Expected ReturnExpected Return for an for an IndividualIndividual Risky InvestmentRisky Investment

0.25 0.08 0.02000.25 0.10 0.02500.25 0.12 0.03000.25 0.14 0.0350

E(R) = 0.1100

Expected Return(Percent)Probability

Possible Rate ofReturn (Percent)

Exhibit 6.1

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Expected Return for a Expected Return for a PortfolioPortfolio of Risky Assets of Risky Assets

0.20 0.10 0.02000.30 0.11 0.03300.30 0.12 0.03600.20 0.13 0.0260

E(Rpor i) = 0.1150

Expected Portfolio

Return (Wi X Ri) (Percent of Portfolio)

Expected Security

Return (Ri)

Weight (Wi)

Exhibit 6.2

iasset for return of rate expected the )E(Riasset in portfolio theofpercent theW

:where

RW)E(R

i

i

1ipor

n

iii

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Variance (Standard Deviation) of Variance (Standard Deviation) of Returns for an Returns for an IndividualIndividual Investment Investment

n

i 1i

2ii

2 P)]E(R-R[)( Variance

where Pi is the probability of the possible rate of return, Ri

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Variance (Standard Deviation) of Variance (Standard Deviation) of Returns for an Returns for an IndividualIndividual Investment Investment

Possible Rate Expected

of Return (Ri) Return E(Ri) Ri - E(Ri) [Ri - E(Ri)]2 Pi [Ri - E(Ri)]

2Pi

0.08 0.11 0.03 0.0009 0.25 0.0002250.10 0.11 0.01 0.0001 0.25 0.0000250.12 0.11 0.01 0.0001 0.25 0.0000250.14 0.11 0.03 0.0009 0.25 0.000225

0.000500

Exhibit 6.3

Variance ( 2) = .0050

Standard Deviation ( ) = .02236

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Variance (Standard Deviation) of Returns Variance (Standard Deviation) of Returns for a for a PortfolioPortfolio

Computation of Monthly Rates of ReturnExhibit 6.4

Closing ClosingDate Price Dividend Return (%) Price Dividend Return (%)

Dec.00 60.938 45.688 Jan.01 58.000 -4.82% 48.200 5.50%Feb.01 53.030 -8.57% 42.500 -11.83%Mar.01 45.160 0.18 -14.50% 43.100 0.04 1.51%Apr.01 46.190 2.28% 47.100 9.28%May.01 47.400 2.62% 49.290 4.65%Jun.01 45.000 0.18 -4.68% 47.240 0.04 -4.08%Jul.01 44.600 -0.89% 50.370 6.63%

Aug.01 48.670 9.13% 45.950 0.04 -8.70%Sep.01 46.850 0.18 -3.37% 38.370 -16.50%Oct.01 47.880 2.20% 38.230 -0.36%Nov.01 46.960 0.18 -1.55% 46.650 0.05 22.16%Dec.01 47.150 0.40% 51.010 9.35%

E(RCoca-Cola)= -1.81% E(Rhome Depot)=E(RExxon)= 1.47%

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Covariance of ReturnsCovariance of Returns

A measure of the degree to which two variables “move together” relative to their individual mean values over time

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Covariance of ReturnsCovariance of Returns

For two assets, i and j, the covariance of rates of return is defined as:

Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}

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Computation of Covariance of Returns for Coca Computation of Covariance of Returns for Coca cola and Home Depot: 2001cola and Home Depot: 2001

Exhibit 6.7

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Covariance and Covariance and CorrelationCorrelation

Correlation coefficient varies from -1 to +1

jt

iti

ij

R ofdeviation standard the

R ofdeviation standard the

returns oft coefficienn correlatio ther

:where

Covr

j

ji

ijij

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Portfolio Standard Deviation Portfolio Standard Deviation FormulaFormula

ji

ijij

ij

2i

i

port

n

1i

n

1iijj

n

1ii

2i

2iport

rCov where

j, and i assetsfor return of rates ebetween th covariance theCov

iasset for return of rates of variancethe

portfolio in the valueof proportion by the determined are weights

whereportfolio, in the assets individual theof weightstheW

portfolio theofdeviation standard the

:where

Covwww

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Portfolio Standard Deviation Portfolio Standard Deviation CalculationCalculation

Any asset of a portfolio may be described by two characteristics:

The expected rate of returnThe expected standard deviations of returns

The correlation, measured by covariance, affects the portfolio standard deviationLow correlation reduces portfolio risk while not affecting the expected return

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Time Patterns of Returns for Two Assets with Time Patterns of Returns for Two Assets with Perfect NegativePerfect Negative Correlation Correlation

Exhibit 6.10

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Risk-Return Plot for Portfolios with Risk-Return Plot for Portfolios with Equal Returns and Standard Equal Returns and Standard DeviationsDeviations but Different Correlations but Different Correlations (page 182-183)(page 182-183)

Exhibit 6.11Correlation affects portfolio risk

A: correlation=1B: correlation=0.5C: correlation=0D: correlation=-0.5E: correlation=-1

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Combining Stocks with Combining Stocks with Different Different Returns and RiskReturns 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 ii2

ii 1 .10 .50 .0049 .07

2 .20 .50 .0100 .10

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Combining Stocks with Different Combining Stocks with Different Returns and RiskReturns and Risk

Negative correlation reduces portfolio riskCombining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equal.

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Risk-Return Plot for Portfolios with Different Risk-Return Plot for Portfolios with Different Returns, Standard Deviations, and CorrelationsReturns, Standard Deviations, and Correlations

Exhibit 6.12

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ConstantConstant Correlation Correlationwith Changing Weightswith Changing Weights

Case W1 W2 E(Ri)

f 0.00 1.00 0.20 g 0.20 0.80 0.18 h 0.40 0.60 0.16 i 0.50 0.50 0.15 j 0.60 0.40 0.14 k 0.80 0.20 0.12 l 1.00 0.00 0.10

)E(R Asset i

1 .10 rij = 0.00

2 .20

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Constant CorrelationConstant Correlationwith Changing Weightswith Changing Weights

Case W1 W2 E(Ri) E( port)

f 0.00 1.00 0.20 0.1000g 0.20 0.80 0.18 0.0812h 0.40 0.60 0.16 0.0662i 0.50 0.50 0.15 0.0610j 0.60 0.40 0.14 0.0580k 0.80 0.20 0.12 0.0595l 1.00 0.00 0.10 0.0700

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Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = +1.00

1

2With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk-return along a line between either single asset

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Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

f

gh

ij

k1

2With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset

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Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

Rij = +0.50

f

gh

ij

k1

2With correlated assets it is possible to create a two asset portfolio between the first two curves

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Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

Rij = -0.50

Rij = +0.50

f

gh

ij

k1

2

With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset

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Portfolio Risk-Return Plots for Different Weights

-

0.05

0.10

0.15

0.20

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Standard Deviation of Return

E(R)

Rij = 0.00

Rij = +1.00

Rij = -1.00

Rij = +0.50

f

gh

ij

k1

2

With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk

Rij = -0.50

Exhibit 6.13

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Estimation IssuesEstimation Issues

Results of portfolio allocation depend on accurate statistical inputsEstimates of

Expected returns Standard deviationCorrelation coefficient

n(n-1)/2 correlation estimates: with 100 assets, 4,950 correlation estimates

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Estimation IssuesEstimation Issues

Single index market model:

imiii RbaR

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

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Estimation IssuesEstimation Issues

The correlation coefficient between two securities i and j is given as:

marketstock aggregate

for the returns of variancethe where

bbr

2m

i

2m

jiij

j

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The Efficient FrontierThe 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 returnFrontier will be portfolios of investments rather than individual securities

Exceptions being the asset with the highest return and the asset with the lowest risk

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Efficient Frontier Efficient Frontier for Alternative Portfoliosfor Alternative Portfolios

Efficient Frontier

A

B

C

Exhibit 6.15

E(R)

Standard Deviation of Return

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The Efficient Frontier The Efficient Frontier and Investor Utilityand Investor Utility

An individual investor’s 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

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The Efficient Frontier The Efficient Frontier and Investor Utilityand Investor Utility

The 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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Selecting an Optimal Risky PortfolioSelecting an Optimal Risky Portfolio

)E( port

)E(R port

X

Y

U3

U2

U1

U3’

U2’ U1’

Exhibit 6.16