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hapter 6

Efficient

Diversificatio

n

Copyright 2010 by The McGraw-Hill Companies, Inc. ll rights reser!e".McGraw-Hill#Irwin

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Outline

Diversification and Portfolio Risk

Asset Allocation With Two Risky

Assets The Optimal Risky Portfolio With A

Risk-Free Asset

Sinle !nde" #odel

$-2

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$%& Diversification and Portfolio Risk

'Review% (overed in F#)

$-%

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6.1 Diversification (Review)

When we put stocks in a

portfolio* p+ ,ecause

the firm-specific risk or uniue

risk is diversified away%

The risk that remains is called

systematic risk or market risk%

'Wii)

n = # securities in the portfolio

$-&

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$%. Asset Allocation With Two Risky

Assets

$-'

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E(rp) = W1r1 +W2r2

W1=

W2=

=

=

Two!ecurit" ortfolio Return

E(rp) = $.6(%.2&') + $.(11.%') = 1$.*6'

Wi = ' of total one"

investe, in securit" i

$.6

$.

%.2&'

11.%'

r1

r2

$-$

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ortfolio -ariance an, !tan,ar,

Deviation

-ariance of a Two !toc ortfolio/

$-(

= =

=/

&!

/

&0

0!0!.

p )1r*(ov'rW2W3

portfoliotheinstocksofnum,ertotalThe/

lyrespective0and!stockininvestedportfoliototaltheofPercentaeW*W 0!

=

=

0Stockand!Stockofreturnstheof(ovariance)r*(ov'r 0! =

)r*r'(ov)r*(ov'r43)r*'r(ovthen0!!f !00!.!0! ===

22

222121

21

21

22 ++= W)r*r'(ovWWWp

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0easurin the correlation coefficient

(ovariance does not tell us the intensityof thecomovement of the stock returns* only the

direction%

We can standardi5e the covariance and

calculate the correlation coefficientwhich tellsus the direction and provides a scaleto estimate

the deree to which the stocks move toether%

For Stock & and Stock .

$-)

.&

.&'&*.)

33

)r*(ov'r6

=

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an, ,iversification in a 2stoc portfolio

is always in the rane 7777777777 inclusive% What ,oes (12) = +1.$ ipl"8

The two areperfectly positivelycorrelated%

!f '&*.) 9 :&* then '&*.) 9 W&&: W..

What ,oes (12) = 1.$ ipl"3

The two are perfectly neativelycorrelated%

!f '&*.) 9 -&* then '&*.) 9 ;'W&&< W..)

!t is possi,le to choose W&and W.such that

'&*.) 9 =%

1.$ to +1.$

$-*

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an, ,iversification in a 2stoc portfolio

!f -& + '&*.) + & then

There are some diversification ,enefits from com,inin

stocks & and . into a portfolio%

p2

= W121

2+ W222

2+ 2W1W2 4ov(r1r2)

And since (ov'r&r

.) 9

1,2

&

.

p2

= W121

2+ W222

2+ 2W1W21212

'&*.) 9 '.*&)

'&*&) 9 :&%= ,y definition The covariance ,etween any stock such as Stock & and

itself is the variance of Stock &

$-10

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p2 =W1

21

2+ 2W1W2 4ov(r1r2) + W222

2

5et W1= 6$' an, W2= $' (!toc 1 = 748 !toc 2 = 9:;)

p2

= $.*6($.1

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The effects of correlation 4

covariance on diversification

sset an, 7 > ortfolio 7

sset 4 an, D > ortfolio 4D

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= +1

= .*

E(r)

1*'

&'

12' 2$'!t. Dev

TW?!E4@RAT: ?RTB?5A?! WATC

DABBERET 4?RRE5TA?!

!toc !toc 7

W= $'

W7= 1$$'

W= 1$$'

W7= $'

= $

= 1

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Summary> Portfolio Risk?Return

Two Security Portfolio

Amount of risk reduction depends criticallyon 7777777777777777777777777%

Addin securities with correlations 77777will result in risk reduction%

!f risk is reduced ,y more than e"pectedreturn* what happens to the return per unit

of risk 'the Sharpe ratio)8

correlations covariances

1

$-1&

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6. Efficient Diversification With

0an" Ris" ssets

$-1'

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@"tendin (oncepts to All Securities

4onsi,er all possiFle coFinations of securitieswith all possiFle ,ifferent weihtins an, eep trac

of coFinations that provi,e ore return for less

ris or the least ris for a iven level of return an,

raph the result.

The set of portfolios that provi,e the optial tra,e

offs are ,escriFe, as the efficient frontier.

The efficient frontier portfolios are ,oinantor the

Fest ,iversifie, possiFle coFinations.

ll investors shoul, want a portfolio on the efficient

frontierG until we a,, the risless asset

$-1$

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E(r)

The iniuvariance frontier of

ris" assets Efficient Brontier is the Fest ,iversifie, set ofinvestents with the hihest returns

$-1(

HloFaliniu

variance

portfolio

Efficientfrontier

An,ivi,ual

assets

0iniu

variancefrontier

!t. Dev.

Boun, F" forinportfolios of securities

with the lowest

covariances at a iven

E(r) level.

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E(r)The EB an, asset allocation

Efficient

frontier

!t. Dev.

2$' !tocs

&$' 7on,s

1$$' !tocs

EB inclu,in

international Ialternative

investents&$' !tocs

2$' 7on,s6$' !tocs

$' 7on,s

$' !tocs

6$' 7on,s

$-1)

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Recap with a -i,eo

u55 word> @fficient Frontier http>??www%,loom,er%com?video?,o,-s-daily-,u55word-

efficient-frontier-BC@EwGRCuAympdfhnHw%html

http://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.htmlhttp://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.htmlhttp://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.htmlhttp://www.bloomberg.com/video/bob-s-daily-buzzword-efficient-frontier-YXELG3w5RXuAympdfqhnUw.html

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$% The Optimal Risky Portfolio With

A Risk-Free Asset

$-20

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E(r)

otential 4apital llocation 5ines

E(r)

45 ()

B

Ris Bree

EfficientBrontier

H

E(r)

45 ()

IB

E(rIB)

$-22

45 (H)

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The 405 I !harpe Ratio

I The 4apital 0aret 5ine (405)is the dominant (A,ecause it has the larest slope%

I !t provides the ,est rewar,tovolatilit"trade off%

I The reward-to-volatility trade off is called the !harpe

0easure!harpe Ratio>

!harpe Ratio = (E(rp) J rf) p

I Reardless of risk preferences* some com,inations of P 4 Fhas an eual or hiher Sharpe ratio than P alone%

$-2&

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ractical Aplications for sset

llocation

A financial planner identifies the optial ris portfolioon the efficient frontier 'call it P)%

I The optial ris portfolio () is the tanency portfolio on

the efficient frontier that connects the risk-free rate%

I may include funds* stocks* ,onds* international and otheralternative investments%

This tanenc" portfolio will serve as the startin point

for alltheir clients%

I The planner varies the asset allocation ,etween the risky

portfolio and risk-free investment accordin to risk tolerance

of client to create a complete portfolio%

$-2'

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$%G A Sinle !nde" Asset #arket

Prelude to (hapter J% (AP#%

$-2$

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An,ivi,ual !ecurities

!t is ,eneficial to hold individual securities in aportfolio to diversify away idiosyncratic risk%

A well,iversifie, portfolio has no idiosyncratic

risk% The remainin risk is called s"steatic

risaret ris%

!"steatic ris arises from events that effect

the entire economy% These include chanes in

interest rates?EDP or financial crises% Kow do we measure a stockLs systematic risk8

$-2(

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!ndividual securities

Kow do we measure a stockLs systematicrisk8

Returns

!toc

Returns

well

,iversifie,

portfolio

K interest rates

K HD

K consuer spen,in

etc.

!"steatic Bactors

$-2)

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An,eL 0o,els/ ?verview

An,eL o,els are statistical models desined toestimate the two coponents of risk for a

particular security or portfolio%

Separates the actual return of a security into macro

's"steatic) and micro 'fir-specific) components%

,vantaes/ Practicality> Reduces the num,er of inputs needed to

account for diversification ,enefits%

(onvenient> @asy reference point for understandin stock

risk usin the measure M%

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!inle Bactor An,eL 0o,el

$-%0

Ri9 MiRm: Ni : iRi9Actuale"cessreturn9 ri< rf

Rm 9 rm< rf 'e"cess return)% Systematic factor or pro"y

in this case # is unanticipated movement in a well-diversified,road market inde" like the S4PG==

Mi9 sensitivity of a securityLs particular return to the

systematic factor

Ni 9 unanticipated firm specific events

i9 @"pected return ,eyond any return induced ,y

movements in the market inde"% Typically 9=%

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An,eL 0o,el/ !catter lot

ELcess Returns (i)

!ecurit"

4haracteristic

5ine.. ..

..

.. ..

..

.. ..

.. ....

.. .. .... ..

.. ....

....

..

.. ..

.. ....

......

.. ..

....

..

.. .. .... ..

..

.. ...... .... .... ..

ELcess returns

on aret in,eL

-ariation in Ri eLplaine, F" the line isthe stocMs NNNNNNNNNNNNN

-ariation in Riunrelate, to the aret

(the line) is NNNNNNNNNNNNNNNNuns"steatic ris

Ri= i+ OiR+ ei

s"steatic ris

$-%1

!lope of !45 = Feta"intercept = alpha

..

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@n,erstan,in 7eta (P)

7etais the sensitivity of a securityLs returns to the

systematic or market factor%

I P > 1 indicates a stock with reater sensitivity to the

economy that the averae stock in the market%I P 1indicates a stock with ,elow-averae sensitivityto

the economy%

I P = 1. y definition* the market portfolio has a ,eta of &%

Qideo> http>??www%youtu,e%com?watch8v9&($A@u,Ew

$-%2

http://www.youtube.com/watch?v=14C6AEubGNwhttp://www.youtube.com/watch?v=14C6AEubGNwhttp://www.youtube.com/watch?v=14C6AEubGNw

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4oponents of RisDecompose stock variance into two components%

Qariance 'Ri) 9 Qariance 'MiRm: i: i)

9 Qariance 'Mi Rm) : Qariance'i)

9 M.i3.

m : 3.'

i)

9 Systematic risk : Firm-specific risk

!"steatic ris of each stock depends on ,oth Q2and the

sensitivity Pi.

Birspecific ris is measured ,y

i. !t has an e"pected valueof 5ero as the impact of unanticipated events must averae

out to 5ero%

-ariance(i) is 5ero%

$-%%

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( i S it

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(omparin Security

(haracteristic ines

Descri,e

e

for each%