chapter 6

Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition

Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-1Slide 6-1

Chapter 6

Portfolio Portfolio SelectionSelection



Chapter Summary

Objective:To present the basics of modern portfolio selection process

Capital allocation decision Two-security portfolios and extensions The Markowitz portfolio selection model



Possible to split investment funds between safe and risky assets

Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio)

Allocating Capital Between Risky & Risk Free Assets



Examine risk/return tradeoff Demonstrate how different degrees of

risk aversion will affect allocations between risky and risk free assets

Allocating Capital Between Risky & Risk Free Assets



The Risk-Free Asset

Perfectly price-indexed bond – the only risk free asset in real terms;

T-bills are commonly viewed as “the” risk-free asset;

Money market funds - the most accessible risk-free asset for most investors.



Portfolios of One Risky Asset

and One Risk-Free Asset

Assume a risky portfolio P defined by :

E(rp) = 15% and p = 22%

The available risk-free asset has:

rf = 7% and rf = 0%

And the proportions invested:

y% in P and (1-y)% in rf



E(rc) = yE(rp) + (1 - y)rf

rc = complete or combined portfolio

If, for example, y = .75

E(rc) = .75(.15) + .25(.07)

= .13 or 13%

Expected Returns for Combinations



rf

pc=

Since

y

= 0, then

* Rule 4 in Chapter 5

*

Variance on the Possible Combined Portfolios



Possible Combinations

E(r)

E(rp) = 15%

rf = 7%

22%0

P

F

c

E(rc) = 13%C



c= .75(.22) = .165 or 16.5%

If y = .75, then

c= 1(.22) = .22 or 22%

If y = 1

c=0(.22) = .00 or 0%

If y = 0

Combinations Without Leverage



CAL (Capital Allocation Line)

E(r)

E(rp) = 15%

rf = 7%

p = 22%0

P

F

) S = 8/22E(rp) - rf = 8%



Borrow at the Risk-Free Rate and invest in stock

Using 50% Leverage

rc = (-.5) (.07) + (1.5) (.15) = .19

c = (1.5) (.22) = .33

Using Leverage with Capital Allocation Line



Indifference Curves and Risk Aversion

Certainty equivalent of portfolio P’s expected return for two different investors

P

E(r)

rf=7%

A = 4

A = 2

p = 22%

E(rp)=15%



Greater levels of risk aversion lead to larger proportions of the risk free rate

Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets

Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

Risk Aversion and Allocation



CAL with Risk Preferences

P

E(r)

7%Lender

Borrower

p = 22%



CAL with Higher Borrowing Rate

E(r)

9%

7%) S = .36

) S = .27P

p = 22%



Risk Reduction with Diversification

Number of Securities

St. Deviation

Market Risk

Unique Risk



Summary Reminder





w1 = proportion of funds in Security 1w2 = proportion of funds in Security 2r1 = expected return on Security 1r2 = expected return on Security 2

1wn

1ii

Two-Security Portfolio: Return

2211P rwrwr



12 = variance of Security 1

22 = variance of Security 2

Cov(r1,r2) = covariance of returns for Security 1 and Security 2

Two-Security Portfolio: Risk

)r,r(Covww2ww 21212

22

22

12

12

p



1,2 = Correlation coefficient of returns

1 = Standard deviation of returns for Security 12 = Standard deviation of returns for

Security 2

Covariance

212,121 )r,r(Cov



Range of values for 1,2

+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly positively correlated

If = - 1.0, the securities would be perfectly negatively correlated

Correlation Coefficients: Possible Values



Three-Security Portfolio

332211p rwrwrwr

)r,r(Covww2

)r,r(Covww2

)r,r(Covww2

www

3232

3131

2121

23

23

22

22

21

21

2p



Generally, for an n-Security Portfolio:

n

1i

iip rwr

n

kj1k,j

kjkj

n

1i

2i

2i

2p )r,r(Covww2w



Returning to the Two-Security Portfolio

2211p rwrw)r(E

)r,r(Covww2ww 21212

22

22

12

12

p

and

, or

)r,r(Covww2ww 21212

22

22

12

1p

Question: What happens if we use various securities’ combinations, i.e. if we vary ?



Two-Security Portfolios with Different Correlations

= 1

13%

%8E(r)

St. Dev12% 20%

= .3

= -1

= -1



Relationship depends on correlation coefficient

-1.0 < < +1.0 The smaller the correlation, the greater

the risk reduction potential If= +1.0, no risk reduction is possible

Portfolio of Two Securities: Correlation Effects



Minimum-Variance Combination

Suppose our investment universe comprises the following two securities:

A B A,B

E(r) 10% 14%0.2

15% 20%

What are the weights of each security in the minimum-variance portfolio?



Minimum-Variance Combination: = .2

)r,r(Cov2

)r,r(Covw

BA2

B2

A

BA2

BA

Solving the minimization problem we get:

Numerically:

6733.0)2.0)(20)(15(2)15()20(

)2.0)(20)(15()20(w 22

2

A

3267.0w1w AB



Minimum -Variance: Return and Risk with = .2

Using the weights wA and wB we determine minimum-variance portfolio’s characteristics:

%31.11%)14)(3267.0(%)10)(6733.0(rP

09.171)2.0)(15)(20)(3267.0)(6733.0(2

)20()3267.0()15()6733.0( 22222P

%08.1309.171P



Minimum -Variance Combination: = -.3

Using the same mathematics we obtain:wA = 0.6087

wB = 0.3913 While the corresponding minimum-

variance portfolio’s characteristics are:rP = 11.57% and

sP = 10.09%



Summary Reminder





The optimal combinations result in lowest level of risk for a given return

The optimal trade-off is described as the efficient frontier

These portfolios are dominant

Extending Concepts to All Securities



The Minimum-Variance Frontier of Risky Assets

E(r)

Efficientfrontier

Globalminimum

varianceportfolio Minimum

variancefrontier

Individualassets



The set of opportunities again described by the CAL

The choice of the optimal portfolio depends on the client’s risk aversion

A single combination of risky and riskless assets will dominate

Extending to Include A Riskless Asset



Alternative CALs

M

E(r)

CAL (Globalminimum variance)

CAL (A)CAL (P)

P

A

F

P P&F M

A

G

P

M



Portfolio Selection & Risk Aversion

E(r)

Efficientfrontier ofrisky assets

Morerisk-averseinvestor

U’’’ U’’ U’

Q

PS

Lessrisk-averseinvestor



Efficient Frontier with Lending & Borrowing

F

P

E(r)

rf

A

Q

BCAL

s

chapter 6

Documents