Financial Economics Bocconi Lecture5

7/30/2019 Financial Economics Bocconi Lecture5

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INVESTMENTS| BODIE, KANE, MARCUSCopyri ght 2011 by The McGraw-H il l Companies, Inc. All ri ghts reserved.McGraw-Hill/Irwin

Lecture 5: Optimal Risky

PortfoliosChapter 7, BKM

Part 1


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The Investment Decision

1.Capital allocation between the risky portfolio

and risk-free asset

a. Determines the investors exposure to risk.

b. The optimal capital allocation is determined by

risk aversion & expectations for the riskreturn

trade-off of the optimal risky portfolio.

2.Asset allocation across broad asset classes

3.Security selection of individual assets within

each asset class


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Lecture Plan

1. Illustrate the potential gains from simple

diversification into many assets

2.Efficient diversification

Two risky assets

Two risky assets and a risk-free

The entire universe of available risky securities.


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Diversification and Portfolio Risk

Market risk

Systematic or nondiversifiable

E.g. conditions in the general economy, such

as the business cycle, inflation, interest rates,and exchange rates

Firm-specific risk

Diversifiable or nonsystematic

E.g. firms success in research and

development, personnel changes


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Figure 7.1 Portfolio Risk as a Function of theNumber of Stocks in the Portfolio


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Figure 7.2 Portfolio Diversification


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Efficient Diversification:The Two-Assets Case

Now consider efficient diversification,

whereby we construct risky portfolios

to provide the lowest possible risk forany given level of expected return.

Suppose for now that we have only

two assets in which to invest: a bondmutual fund (denoted by D) and a

stock mutual fund (denoted by E).


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Two-Security Portfolio: Return

Portfolio Return

Bond Weight

Bond Return

Equity Weight

Equity Return

p D ED EP

D

D

E

E

r

r

w

r

w

r

w wr r

( ) ( ) ( )p D D E E

E r w E r w E r


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= Variance of Security D= Variance of Security E

= Covariance of returns for

Security D and Security E

Two-Security Portfolio: Risk

2 2 2 2 2p

2 2 2 2 2

p

2 ,

2 ,

D D E E D E D E

D D E E D E D E D E

w w w w Cov r r

w w w w Corr r r

2

E

2

D

( , )D E

Cov r r

= Correlation of returns for

Security D and Security E

( , )D E

Corr r r


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D,E = Correlation coefficient ofreturns

Cov(rD,rE) = DEDE

D = Standard deviation ofreturns for Security D

E = Standard deviation ofreturns for Security E

Covariance


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1. The formula for the portfolio variance

reveals that variance is reduced if the

covariance term is negative.

2. Even if the covariance term is positive, theportfolio standard deviation is less than the

weighted average of the individual security

standard deviations, unless the two

securities are perfectly positively correlated.

Portfolio Variance


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Correlation Coefficients

When DE = 1, there is no diversification:

the standard deviation of the portfolio with perfect positive

correlation is just the weighted average of the component

standard deviations.

In all other cases, the correlation coefficient is less than 1,making the portfolio standard deviation less than the

weighted average of the component standard deviations.

DDEEPww


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Correlation Coefficients

When DE = -1, a perfect hedge is possible:

The portfolio standard deviation is zero.

D

ED

D

Eww

1


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Portfolios offer better risk-returntradeoffs

Because the portfolios expected return is the weighted

average of its component expected returns, whereas its

standard deviation is less than the weighted average of

the component standard deviations, portfolios of less than

perfectly correlated assets always offer better riskreturnopportunities than the individual component securities on

their own.

The lower the correlation between the assets, the greater

the gain in efficiency.


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Bond-stock portfolio: Data

Expected return:

Stock 13%

Bond 8%

Standard deviation:

Stock 20%

Bond 12%

Correlation coefficient: 0.3

What are the expected return and variance of the

portfolio as functions of the weight on stocks?


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Portfolio Expected Return as a Function ofInvestment Proportions


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Portfolio Standard Deviation as a Functionof Investment Proportions


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The Minimum Variance Portfolio

The minimum varianceportfolio is the portfoliocomposed of the riskyassets that has thesmallest standarddeviation, the portfoliowith least risk.

When correlation isless than +1, theportfolio standarddeviation may besmaller than that of

either of the individualcomponent assets.

When correlation isequal to -1, the

standard deviation ofthe minimum varianceportfolio is zero.

2

2 2

( , )( )

2 ( , )( ) 1 ( )

D D E

MinVar

D E D E

MinVar MinVar

Cov r r w E

Cov r r

w D w E

7 19


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Figure 7.5 Portfolio Expected Return as aFunction of Standard Deviation

7 20


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The amount of possible risk reductionthrough diversification depends on the

correlation.

The risk reduction potential increases as

the correlation approaches -1.

If = +1.0, no risk reduction is possible.

If = 0, P

may be less than the standard

deviation of either component asset.

If = -1.0, a riskless hedge is possible.

Correlation Effects

Financial Economics Bocconi Lecture5

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