Chapter 8Chapter 8

Portfolio SelectionPortfolio Selection

State three steps involved in building a State three steps involved in building a portfolio.portfolio.

Apply the Markowitz efficient portfolio Apply the Markowitz efficient portfolio selection model.selection model.

Describe the effect of risk-free borrowing Describe the effect of risk-free borrowing and lending on the efficient frontier.and lending on the efficient frontier.

Discuss the separation theorem and its Discuss the separation theorem and its importance to modern investment theory.importance to modern investment theory.

Separate total risk into systematic and Separate total risk into systematic and non-systematic risk.non-systematic risk.

Learning ObjectivesLearning Objectives

Diversification is key to optimal risk Diversification is key to optimal risk managementmanagement

Analysis required because of the Analysis required because of the infinite number of portfolios of risky infinite number of portfolios of risky assetsassets

How should investors select the How should investors select the best risky portfolio?best risky portfolio?

How could riskless assets be used?How could riskless assets be used?

Portfolio SelectionPortfolio Selection

Step 1Step 1: Use the Markowitz : Use the Markowitz portfolio selection model to identify portfolio selection model to identify optimal combinationsoptimal combinations

Step 2Step 2: Consider borrowing and : Consider borrowing and lending possibilitieslending possibilities

Step 3Step 3: Choose the final portfolio : Choose the final portfolio based on your preferences for based on your preferences for return relative to riskreturn relative to risk

Building a PortfolioBuilding a Portfolio

Optimal diversification takes into Optimal diversification takes into account all available informationaccount all available information

Assumptions in portfolio theoryAssumptions in portfolio theory A single investment period (one year)A single investment period (one year) Liquid position (no transaction costs)Liquid position (no transaction costs) Preferences based only on a Preferences based only on a

portfolio’s expected return and riskportfolio’s expected return and risk

Portfolio TheoryPortfolio Theory

Smallest portfolio risk for a given Smallest portfolio risk for a given level of expected returnlevel of expected return

Largest expected return for a given Largest expected return for a given level of portfolio risklevel of portfolio risk

From the set of all possible From the set of all possible portfoliosportfolios Only locate and analyze the subset Only locate and analyze the subset

known as the known as the efficient setefficient setLowest risk for given level of returnLowest risk for given level of return

An Efficient PortfolioAn Efficient Portfolio

All other portfolios in attainable set All other portfolios in attainable set are dominated byare dominated by efficient setefficient set

Global minimum variance portfolioGlobal minimum variance portfolio Smallest risk of the efficient set of Smallest risk of the efficient set of

portfoliosportfolios

Efficient setEfficient set Segment of the minimum variance Segment of the minimum variance

frontier above the global minimum frontier above the global minimum variance portfoliovariance portfolio

An Efficient PortfolioAn Efficient Portfolio

Bx

A

Cy

Risk =

E(R)

• Efficient frontier or Efficient set (curved line from A to B)

• Global minimum variance portfolio (represented by point A)

Efficient PortfoliosEfficient Portfolios

Assume investors are risk averseAssume investors are risk averse

Indifference curves help select from Indifference curves help select from efficient setefficient set Description of preferences for risk and Description of preferences for risk and

returnreturn Portfolio combinations which are Portfolio combinations which are

equally desirableequally desirable Greater slope implies greater risk Greater slope implies greater risk

aversionaversion

Selecting an Optimal Selecting an Optimal Portfolio of Risky AssetsPortfolio of Risky Assets

Markowitz portfolio selection modelMarkowitz portfolio selection model Generates a frontier of efficient Generates a frontier of efficient

portfolios which are equally goodportfolios which are equally good Does not address the issue of riskless Does not address the issue of riskless

borrowing or lendingborrowing or lending Different investors will estimate the Different investors will estimate the

efficient frontier differentlyefficient frontier differentlyElement of uncertainty in applicationElement of uncertainty in application

Selecting an Optimal Selecting an Optimal Portfolio of Risky AssetsPortfolio of Risky Assets

Another way to use the Markowitz Another way to use the Markowitz model is with asset classesmodel is with asset classes Allocation of portfolio assets to broad Allocation of portfolio assets to broad

asset categoriesasset categoriesAsset class rather than individual security Asset class rather than individual security decisions most important for investorsdecisions most important for investors

Different asset classes offers various Different asset classes offers various returns and levels of riskreturns and levels of risk

Correlation coefficients may be quite lowCorrelation coefficients may be quite low

Selecting Optimal Asset Selecting Optimal Asset ClassesClasses

Optimal Risky PortfoliosOptimal Risky Portfolios

Investor Utility Function

Efficient Frontier

E (R)

*

Risk-free assets Risk-free assets Certain-to-be-earned expected return, Certain-to-be-earned expected return,

zero variancezero variance No correlation with risky assetsNo correlation with risky assets Usually proxied by a Treasury BillUsually proxied by a Treasury Bill

Amount to be received at maturity is free of Amount to be received at maturity is free of default risk, known with certaintydefault risk, known with certainty

Adding a risk-free asset extends and Adding a risk-free asset extends and changes the efficient frontierchanges the efficient frontier

Borrowing and Lending Borrowing and Lending PossibilitiesPossibilities

Riskless assets can Riskless assets can be combined with be combined with any portfolio in the any portfolio in the efficient set ABefficient set AB Z implies lendingZ implies lending

Set of portfolios on Set of portfolios on line RF to T line RF to T dominates all dominates all portfolios below itportfolios below it

B

A

T

Risk

E(R)

RF

L

Z X

Risk-Free LendingRisk-Free Lending

If wIf wRFRF placed in a risk-free asset: placed in a risk-free asset: Expected portfolio returnExpected portfolio return

))E(Rw-(1 RF w )E(R XRFRFp ))E(Rw-(1 RF w )E(R XRFRFp

▪ Risk of the portfolio

XRFp )w-(1 XRFp )w-(1

• Expected return and risk of the portfolio with lending is a weighted average

Impact of Risk-Free LendingImpact of Risk-Free Lending

Investor no longer restricted to own Investor no longer restricted to own wealthwealth

Interest paid on borrowed moneyInterest paid on borrowed money Higher returns sought to cover expenseHigher returns sought to cover expense Assume borrowing at RFAssume borrowing at RF

Risk will increase as the amount of Risk will increase as the amount of borrowing increasesborrowing increases Financial leverageFinancial leverage

Borrowing PossibilitiesBorrowing Possibilities

Risk-free investing and borrowing Risk-free investing and borrowing creates a new set of expected creates a new set of expected return-risk possibilitiesreturn-risk possibilities

Addition of risk-free asset results inAddition of risk-free asset results in A change in the efficient set from an A change in the efficient set from an

arc to a straight line tangent to the arc to a straight line tangent to the feasible set without the riskless assetfeasible set without the riskless asset

Chosen portfolio depends on Chosen portfolio depends on investor’s risk-return preferencesinvestor’s risk-return preferences

The New Efficient SetThe New Efficient Set

The more conservative the investor, The more conservative the investor, the more that is placed in risk-free the more that is placed in risk-free lending and the less in borrowinglending and the less in borrowing

The more aggressive the investor, The more aggressive the investor, the less that is placed in risk-free the less that is placed in risk-free lending and the more in borrowinglending and the more in borrowing Most aggressive investors would use Most aggressive investors would use

leverage to invest more in portfolio Tleverage to invest more in portfolio T

Portfolio ChoicePortfolio Choice

Investors use their preferences Investors use their preferences (reflected in an indifference curve) to (reflected in an indifference curve) to determine their optimal portfoliodetermine their optimal portfolio

Separation TheoremSeparation Theorem The investment decision regarding which The investment decision regarding which

risky portfolio to hold is separate from the risky portfolio to hold is separate from the financing decisionfinancing decision

Allocation between risk-free asset and Allocation between risk-free asset and risky portfolio separate from choice of risky portfolio separate from choice of risky portfolio, Trisky portfolio, T

The Separation TheoremThe Separation Theorem

All investorsAll investors Invest in the same portfolioInvest in the same portfolio Attain any point on the straight line RF-T-Attain any point on the straight line RF-T-

L by either borrowing or lending at the L by either borrowing or lending at the rate RF, depending on their preferencesrate RF, depending on their preferences

Risky portfolios are not tailored to Risky portfolios are not tailored to each individual’s tasteeach individual’s taste

Separation TheoremSeparation Theorem

Investors should focus on risk that Investors should focus on risk that cannot be managed by diversificationcannot be managed by diversification

Total risk =Total risk = Systematic (non-diversifiable) riskSystematic (non-diversifiable) risk

+ + Non-systematic (diversifiable) riskNon-systematic (diversifiable) risk

Implications of Portfolio Implications of Portfolio SelectionSelection

Systematic riskSystematic risk Variability in a security’s total returns Variability in a security’s total returns

directly associated with economy-wide directly associated with economy-wide eventsevents

Common to virtually all securitiesCommon to virtually all securities

Systematic riskSystematic risk

Non-Systematic RiskNon-Systematic Risk Variability of a security’s total return not Variability of a security’s total return not

related to general market variabilityrelated to general market variability Diversification decreases this riskDiversification decreases this risk

The relevant risk of an individual stock The relevant risk of an individual stock is its contribution to the riskiness of a is its contribution to the riskiness of a well-diversified portfoliowell-diversified portfolio Portfolios rather than individual assets Portfolios rather than individual assets

most importantmost important

Non-Systematic RiskNon-Systematic Risk

p %

35

20

0

Number of securities in portfolio

10 20 30 40 ...... 100+

Total risk

Systematic Risk

Diversifiable Risk

Portfolio Risk and Portfolio Risk and DiversificationDiversification