Chapter 7Chapter 7

An Introduction to Portfolio An Introduction to Portfolio ManagementManagement

Why Should Capital MarketsWhy Should Capital MarketsBe Efficient?Be Efficient?

The premises of an efficient marketThe premises of an efficient market A large number of competing profit-maximizing A large number of competing profit-maximizing

participants analyze and value securities, each participants analyze and value securities, each independently of the othersindependently of the others

New information regarding securities comes to New information regarding securities comes to the market in a random fashionthe market in a random fashion

Profit-maximizing investors adjust security prices Profit-maximizing investors adjust security prices rapidly to reflect the effect of new informationrapidly to reflect the effect of new information

Conclusion: the expected returns implicit in Conclusion: the expected returns implicit in the current price of a security should reflect the current price of a security should reflect its riskits risk

AlternativeAlternativeEfficient Market HypothesesEfficient Market Hypotheses

Weak-form efficient market Weak-form efficient market hypothesishypothesis

Semistrong-form EMHSemistrong-form EMH Strong-form EMHStrong-form EMH

Efficient Capital MarketsEfficient Capital Markets Joint hypothesis problemJoint hypothesis problem

market efficiency must be tested at same market efficiency must be tested at same time as test’ of asset pricing model being time as test’ of asset pricing model being used to generate expected returnsused to generate expected returns

no one APM shown to be “the true model” no one APM shown to be “the true model” that represents how returns are generatedthat represents how returns are generated

for this reason, some believe market for this reason, some believe market efficiency is not truly able to be testedefficiency is not truly able to be tested

Fama explains that we may never be able to Fama explains that we may never be able to say for sure whether markets are efficient or say for sure whether markets are efficient or not, results of tests are still worthwhilenot, results of tests are still worthwhile

Implications of Implications of Efficient Capital MarketsEfficient Capital Markets

Overall results indicate the capital Overall results indicate the capital markets are efficient as related to markets are efficient as related to numerous sets of informationnumerous sets of information

There are substantial instances There are substantial instances where the market fails to rapidly where the market fails to rapidly adjust to public informationadjust to public information So, what techniques will or won’t work?So, what techniques will or won’t work? What do you do if you can’t beat the What do you do if you can’t beat the

market?market?

Efficient Markets Efficient Markets and Portfolio Managementand Portfolio Management

Management depends on analystsManagement depends on analysts With superior analysts, follow them and look With superior analysts, follow them and look

for opportunities in neglected stockfor opportunities in neglected stock Without superior analysts, passive Without superior analysts, passive

management may outperform active management may outperform active management by management by Build a globally diversified portfolio with a risk Build a globally diversified portfolio with a risk

level matching client preferenceslevel matching client preferences Minimize transaction costs (taxes, trading Minimize transaction costs (taxes, trading

turnover, liquidity costs)turnover, liquidity costs)

The Rationale and The Rationale and Use of Index FundsUse of Index Funds

Efficient capital markets and a lack of Efficient capital markets and a lack of superior analysts imply that many superior analysts imply that many portfolios should be managed passively portfolios should be managed passively (so their performance matches the (so their performance matches the aggregate market, minimizes the costs of aggregate market, minimizes the costs of research and trading)research and trading)

Institutions created market (index) funds Institutions created market (index) funds which duplicate the composition and which duplicate the composition and performance of a selected index seriesperformance of a selected index series

Background Background AssumptionsAssumptions

As an investor you want to maximize the As an investor you want to maximize the returns for a given level of risk.returns for a given level of risk.

Your portfolio includes all of your assets Your portfolio includes all of your assets and liabilitiesand liabilities

The relationship between the returns for The relationship between the returns for assets in the portfolio is important.assets in the portfolio is important.

A good portfolio is not simply a collection A good portfolio is not simply a collection of individually good investments.of individually good investments.

Risk AversionRisk Aversion

Given a choice between two Given a choice between two assets with equal rates of return, assets with equal rates of return, most investors will select the most investors will select the asset with the lower level of risk.asset with the lower level of risk.

Evidence ThatEvidence ThatInvestors are Risk Investors are Risk

AverseAverse Many investors purchase insurance Many investors purchase insurance

for: Life, Automobile, Health, and for: Life, Automobile, Health, and Disability Income. The purchaser Disability Income. The purchaser trades known costs for unknown risk trades known costs for unknown risk of lossof loss

Yield on bonds increases with risk Yield on bonds increases with risk classifications from AAA to AA to A….classifications from AAA to AA to A….

Not all Investors are Risk Not all Investors are Risk AverseAverse

Risk preference may have to do with Risk preference may have to do with amount of money involved - risking amount of money involved - risking small amounts, but insuring large small amounts, but insuring large losseslosses

Markowitz Portfolio TheoryMarkowitz Portfolio Theory

Quantifies riskQuantifies risk Derives the expected rate of return for a Derives the expected rate of return for a

portfolio of assets and an expected risk portfolio of assets and an expected risk measuremeasure

Shows that the variance of the rate of return Shows that the variance of the rate of return is a meaningful measure of portfolio riskis a meaningful measure of portfolio risk

Derives the formula for computing the Derives the formula for computing the variance of a portfolio, showing how to variance of a portfolio, showing how to effectively diversify a portfolioeffectively diversify a portfolio

Alternative Measures of Alternative Measures of RiskRisk

Variance or standard deviation of Variance or standard deviation of expected returnexpected return

Range of returnsRange of returns Returns below expectationsReturns below expectations

Semivariance – a measure that only Semivariance – a measure that only considers deviations below the meanconsiders deviations below the mean

These measures of risk implicitly assume These measures of risk implicitly assume that investors want to minimize the that investors want to minimize the damage from returns less than some target damage from returns less than some target raterate

Expected Rates of Expected Rates of ReturnReturn

For an individual asset - sum of the For an individual asset - sum of the potential returns multiplied with the potential returns multiplied with the corresponding probability of the corresponding probability of the returnsreturns

For a portfolio of investments - For a portfolio of investments - weighted average of the expected weighted average of the expected rates of return for the individual rates of return for the individual investments in the portfolioinvestments in the portfolio

Expected Return for an Expected Return for an Individual Risky InvestmentIndividual Risky Investment

0.35 0.08 0.02800.30 0.10 0.03000.20 0.12 0.02400.15 0.14 0.0210

E(R) = 0.1030

Expected Return(Percent)Probability

Possible Rate ofReturn (Percent)

Exhibit 7.1

Expected Return for a Expected Return for a Portfolio of Risky AssetsPortfolio 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(Rport) 0.1150

(Percent of Portfolio)

Expected Security

Return (Ri)

Weight (Wi) Expected Portfolio

Return (Wi X Ri)

Exhibit 7.2

iasset for return of rate expected the )E(R

iasset in portfolio theofpercent theW

:where

RW)E(R

i

i

1

port

n

iii

Variance of Returns for an Variance of Returns for an Individual InvestmentIndividual Investment

Variance is a measure of the variation Variance is a measure of the variation of possible rates of return Rof possible rates of return Rii, from the , from the expected rate of return [E(Rexpected rate of return [E(Rii)])]

Standard deviation is the square root Standard deviation is the square root of the varianceof the variance

Variance of Returns for an Variance of Returns for an Individual InvestmentIndividual Investment

n

i 1i

2ii

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

where Pwhere Pii is the probability of the is the probability of the possible rate of return, Rpossible rate of return, R ii

Standard Deviation of Standard Deviation of Returns for an Individual Returns for an Individual

InvestmentInvestment

n

i 1i

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

Standard DeviationStandard Deviation

Standard Deviation of Standard Deviation of Returns for an Individual Returns for an Individual

InvestmentInvestmentPossible Rate Expected

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

2Pi

0.08 0.103 -0.023 0.0005 0.35 0.0001850.10 0.103 -0.003 0.0000 0.30 0.0000030.12 0.103 0.017 0.0003 0.20 0.0000580.14 0.103 0.037 0.0014 0.15 0.000205

0.000451

Exhibit 7.3

Variance ( 2) = .000451

Standard Deviation ( ) = .021237

Covariance of ReturnsCovariance of Returns

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

Covariance of ReturnsCovariance of Returns

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

CovCovijij = sum{[R = sum{[Rii - E(R - E(Rii)] [R)] [Rjj - E(R - E(Rjj)]p)]pii}} Correlation coefficient varies from -1 to Correlation coefficient varies from -1 to

+1+1

ji

ijij

Covr

Portfolio Standard Portfolio Standard Deviation FormulaDeviation Formula

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

Combining Stocks with Combining Stocks with Different Returns and RiskDifferent Returns and Risk

Case Correlation Coefficient CovarianceCase Correlation Coefficient Covariance

a +1.00 .0070a +1.00 .0070

b +0.50 .0035b +0.50 .0035

c 0.00 .0000c 0.00 .0000

d -0.50 -.0035d -0.50 -.0035

e -1.00 -.0070e -1.00 -.0070

W)E(R Asset ii2

ii 1 .10 .50 .0049 .07

2 .20 .50 .0100 .10

Constant CorrelationConstant Correlationwith Changing Weightswith Changing Weights

Case W1 W2E(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 r ij = 0.00

2 .20

Portfolio Risk-Return Portfolio Risk-Return Plots for Different Plots for Different

WeightsWeights

-

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

Portfolio Risk-Return Portfolio Risk-Return Plots for Different Plots for Different

WeightsWeights

-

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

Portfolio Risk-Return Portfolio Risk-Return Plots for Different Plots for Different

WeightsWeights

-

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

Portfolio Risk-Return Portfolio Risk-Return Plots for Different Plots for Different

WeightsWeights

-

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

Portfolio Risk-Return Portfolio Risk-Return Plots for Different Plots for Different

WeightsWeights

-

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 7.13

Estimation IssuesEstimation Issues

Results of portfolio allocation depend on Results of portfolio allocation depend on accurate statistical inputsaccurate statistical inputs

Estimates ofEstimates of Expected returns Expected returns Standard deviationStandard deviation Correlation coefficient Correlation coefficient

Among entire set of assetsAmong entire set of assets With 100 assets, 4,950 correlation estimatesWith 100 assets, 4,950 correlation estimates

Estimation risk refers to potential errorsEstimation risk refers to potential errors

Estimation IssuesEstimation Issues

With assumption that stock returns With assumption that stock returns can be described by a single market can be described by a single market model, the number of correlations model, the number of correlations required reduces to the number of required reduces to the number of assetsassets

Single index market model: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

The Efficient FrontierThe Efficient Frontier The efficient frontier represents that The efficient frontier represents that

set of portfolios with the maximum set of portfolios with the maximum rate of return for every given level of rate of return for every given level of risk, or the minimum risk for every risk, or the minimum risk for every level of returnlevel of return

Frontier will be portfolios of Frontier will be portfolios of investments rather than individual investments rather than individual securitiessecurities Exceptions being the asset with the Exceptions being the asset with the

highest return and the asset with the highest return and the asset with the lowest risklowest risk

Efficient Frontier Efficient Frontier for Alternative Portfoliosfor Alternative Portfolios

Efficient Frontier

A

B

C

Figure 8.9

E(R)

Standard Deviation of Return

The Efficient Frontier The Efficient Frontier and Investor Utilityand Investor Utility

An individual investor’s utility curve An individual investor’s utility curve specifies the trade-offs he is willing to specifies the trade-offs he is willing to make between expected return and riskmake between expected return and risk

The slope of the efficient frontier curve The slope of the efficient frontier curve decreases steadily as you move upwarddecreases steadily as you move upward

These two interactions will determine These two interactions will determine the particular portfolio selected by an the particular portfolio selected by an individual investorindividual investor

Selecting an Optimal Risky Selecting an Optimal Risky PortfolioPortfolio

)E( port

)E(R port

X

Y

U3

U2

U1

U3’

U2’ U1’

Figure 8.10