cf chap2 - risk and return (2012)

8/12/2019 CF Chap2 - Risk and Return (2012)

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Topic 2: Risk and Return: Mean-

Variance Analysis & the Capital Asset

Pricing Model

Portfolio Theory, Mean Variance

Analysis and the CAPM

!" #$ternal %ntake 2'(2 ) 2'(*+

*'.2 Corporate inance


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T!PC Co/erage

20(: Portfolio Analysis

202: Mean-Variance Analysis

20*: The CAPM


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2.1 Portfolio Theory

Topic 2 Part (


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20(0( ntroduction: Risk and Return

Risk and Return

Risk A/erse Assu1p: #$posure to risk only acceptale

y in/estors if they are offered a higher e$pected rate of

return 3#%R+4

Risk refers to uncertainty

Measured in ter1s of the dispersion of possile outco1es

Measured y standard de/iation of the distriution of

possile rets around the #%R+

!C5: inancial n/est1ent


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The Value of an Investment of $1 in 1900

$1

$10

$100

$1,000

$10,000

$100,000

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

Start of Year

Dolla

rs

Common Stock

US Got !on"s

#!%lls

21,536

176

66

2007


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The Value of an Investment of $1 in 1900

$1

$10

$100

$1,000

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

Start of Year

Dollars

&'(%t%es

!on"s

!%lls

914

7)48

2)82

2007

Real Returns


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Measuring Risk

1 1

4

1012

20

17

24

13

32

0

4

8

12

16

20

24

$50

to$

40

$40

to$

30

$30

to$

20

$20t

o$

10

$10

to

0

0

to

10

10

to

20

20

to3

0

30

to

40

40t

o

50

50

to6

0

Return %

# of Years

Histogram of Annual Stock Market Returns

(1900-2006)


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20(02 Measuring Risk & Return: 5ingle

5ecurity &*+ecte" ,et(rn - The average of a probability ist! of possible returns

"in perspetive stimation of the value of an I &inluing the ' in prie (payments or ivs) alulate from a prob! ist! urve of all possible rates of ret!

"ormula for *pete Return

&R+) , 1R+1 . /R+/ . . nR+n

,

here i , probability of state i ourring

R+i , return e*pete from the I hen the eonomy is in state i &R+) , e*pete return on investment +

n , no! of possible states

i , one possible state of the n feasible outomes

+ , the partiular I being onsiere

ji

n

i

iRP=1


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20(02 Measuring Risk& Return: 5ingle

5ecurity -ar%ance . 2 the e*pe value of the s3uare of the iffs btn the possible

values of a ranom variable ( its e*pete value &i!e! average value of

s3uare eviations from mean) 4 measure of volatility

Stan"ar" De%at%on. 2 53uare root of the variane measure of volatility

measure of the e*tent to hih no!6s are sprea aroun their e*pete value!

"ormula for variane of returns

Var&R+) , +/ , &R+ 2 &R+))/ or

Var&R+) , +/ , 1&R+1 2 &R+))/ . /&R+/ 2 &R+))/ . . n&R+n 2 &R+))/

an

5tanar 7eviation , 57&R+) , +

[ ]( )/

1

=

=n

i

jjii RERP

( )jRVar=


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5ecurityNumerical Example

Assu1ing the rets on an depends si1ply on ho6 6ell anecono1y perfor1s:

#%R+ can e calculated as the 6eighted a/erage of possile

outco1es:

#%R+ 7 '0*%('+ 8 '09%2'+ 8 '0*%*'+ 7 * 8 8 .

7 2'

Anticipated rets on I depending on the state of theEcon:

State of Econ. Prob. Of State % Ret on I

Recession 0.3 10Slow Growth 0.4 20

Boom 0.3 30


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5ecurityNumerical Example (cont)

Var%R+ and 5;%R+ can e calculated as follo6s: Var%R+ 7 +/ 7 '0*%(' - 2'+2 8 '09%2' - 2'+2 8 '0*%*' - 2'+2

7 '0*%- ('+2 8 '09%'+2 8 '0*%('+2

7 *' 8 ' 8 *'

7 0>9

Anticipated rets on I depending on the state of theEcon:

State of Econ. Prob. Of State % Ret on I

Recession 0.3 10

Slow Growth 0.4 20

Boom 0.3 30


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5ecurity Proaility dist0 of possile rates of ret0 in the

pre/ious e$a1ple 6ould e a discrete dist0 as itis ased on a li1ited no0 of possile outco1es

and their associated proailities

5pecifying a large no0 of possile states & theirassoc proailities along 6ith the ret in each

state 6ould allo6 the generation of a proaility

dist appro$0 y a continuous cur/e


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5ecurityExample: Discrete Dist.

Standard Deviation VS. Expected Return

Investment A

0

2

4

6

8

10

12

14

16

18

20

50 0 50

%probability

% return


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5ecurityExample: Discrete Dist.

Standard Deviation VS. Expected ReturnInvestment B

0

2

4

6

8

10

12

14

16

18

20

50 0 50

%probability

% return


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5ecurity Continuous dists often pro/ide a etter appro$ to

the relationships under consideration

Possile to assu1e returns are normally dist

Completely described in terms of E(R ! SD(R

"#ere is$i. a %&' prob. t#at rets ill be ) one SD of t#e expec ret

ii. A *+' prob. t#at rets ill be ) to SDs of t#e expec ret

iii. a **' prob. t#at rets ill be ) t#ree SDs of t#e expec ret

,ote$ as max possible loss on any investment is -' ! potential

/ains are unlimited t#e dist can be expec to s0e to t#e ri/#t

1 lo/ normal


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20(0* Risk & Ret: Portfolios & a?/e

;i/ersification

n/estors often hold a portfolio of assets @pro/ides considerale scope for risk 1g1t

a?/e ;i/ersification: Reducing risk of portfolios

y rando1ly picking securities

#0g0: agner and "au %(.>(+ took 2'' securities traded on the B5# & constructed

portfolios consisting of t6n ( & 2' rando1ly chosen

securities

Risk @ 5; of past returns

indings: Risk declined at a decreasing rate until the

portfolio consisted of ( or so securities


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;i/ersification

!n the asis of studies such as agnerand "auDs:

i0 Aout 2=* of risk can e eli1inated through

a?/e di/ersificationii0 Risk can e di/ided intoiii0 ;i/ersifiale @ fir1 specific

i/0 on di/ersifiale @ due to factors that affect the

fortunes of all fir1s e0g0 econ gro6th, inflation


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;i/ersification

0

5 10 15

Number of Securities in the Portfolio

Portfo

lios

tandard

de

viation

;iagra11atic illustration of risk reduction through na?/edi/ersification:

Diversifiableris !or non"

s#stematic

ris$

%on"

Diversifiableris !or

s#stematic

ris$


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20(09 Risk & Ret: Modern Portfolio

Theory ;i/ersification to reduce risk e$posure

Marko6itE %(.2+ @ pro/ided analytical asis

Measure of interdependence of ret on assets through

co/ariance %C!V+ & correlation of rets %C!RR+

Co1ining securities into portfolios can reduceelo6 le/el otained fro1 a si1ple 6eighted

a/erage calculation if in/est1ents included in

the portfolio are not perfectly correlated

3correlation coefficient, F G 8(0'4

2 ( 9 Ri k & R t M d P tf li


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20(09 Risk & Ret: Modern Portfolio

Theory Correlation coefficient @ 9 possiilities:

i0 Perfect Positi/e Correlation 3F 7 8(0'4

Rets on 2 securities al6ays 1o/e in step

Risk of portfolio @ 6eighted a/g of risks of constituent securities

iii0 Perfect egati/e Correlation 3F 7 -(0'4

#ach de/iation for one security 6ill e 1atched y an eHual,

proportionate de/iation in the other security, ut 6ith the opp0 sign

i/0 Iero Correlation 3F 7 '4

Also descried as Jno correlationD

;e/iations fro1 e$p rets tend to ha/e sa1e sign in 'K of the

outco1es

Considerale scope for risk reduction

/0 1perfect positi/e correlation 3' L F L (0'4

Rets on 1ost securities

Rets influenced partly y co11on factors

Risk of portfolio 6ill e lo6er than the 6eighted a/g of risk of assets in

portfolio


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20(0 Risk & Ret: T6o Asset

Portfolios

ntro nitial focus: 2 asset portfolio "ater findings 6ill e e$panded to larger

portfolios %i0e0 1ore than 2 securities+

Modern portfolio theory @ co1putation of

portfolio return and portfolio risk, 6ith

e1phasis of ris0 reduction t#rou/#

diversification

2 ( Ri k & R t T A t


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Portfolios Portfolio Return: 2 asset portfolio 1ade up of

security A and security

E(RP) = WAE(RA) + WBE(RB)

here #%RP+ 7 e$pec0 ret on the portfolio

#%RA+ 7 e$pec0 ret on security A

#%R+ 7 e$pec0 ret on security

A 7 proportion of portfolio in/ested in security A 7 proportion of portfolio in/ested in security

And A 8 7 (0'



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Portfolios Portfolio Risk @ calculated in 2 6ays:

i0 !n the asis of dist of rets e$pec on

portfolio

Var%RP+ 7 NPi%RPi ) #%RP++2

7 #%RP ) #%RP++2

iii0 !n the asis of /ariance of rets assets in

portfolio, and relationship t6n these rets

Var(RP) = WA2Var(RA) + WB2Var(RB) +

2WAWBCov(RA,RB)

or P2 = WA2A2 + WB2B2 + 2WAWBAB



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Portfolios ote: ;eri/ation

Var%RP+7 #%RP ) #%RP++2

7 #3%ARA 8 R+ ) %A#%RA+ 8 #%R++42

7 #3%A%RA - #%RA++ 8 %%R - #%R++42

7 #3%A2%RA - #%RA++2 8 %2%R - #%R++2 8 2A%RA -

#%RA++%R - #%R++4

7 A2#%RA - #%RA++28 2#%R - #%R++2 8 2A#%RA -

#%RA++%R - #%R++

= WA2A2 + WB2B2 + 2WAWBAB

2 ( Ri k & Ret T o A et


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Portfolios Co/ariance and the Correlation Coefficient Co/ariance is the 6eighted a/g of the product of the de/iations of

2 /ariales fro1 their respecti/e e$pec /alues, the 6eights eing

gi/en y the proaility of the pairs of de/iations occuring

Cov(RA,RB) = AB = Pi(RAi E(RA))(RBi E(RB))

= E(RA ! E(RA))(RB ! E(RB))

A 1easure of relatedness dependent on unit of 1easure1ent @difficult to interpret

Co/ariance can e standardiEed y di/iding its /alue y the

product of std de/s @ results in a si1ple no0: corr coeffi

Corr Coeffi, "AB = #Cov(RA,RB)$%&'(RA)&'(RB)$

= AB%(AB)

ote: can be t#ou/#t of as a cov #ere all random variables #ave

been rescaled to #ave a variance of -



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Portfolios Portfolio Risk:

ack to Variance of portfolios:

P2 = WA2A2 + WB2B2 + 2WAWBAB

= WA2A2 + WB2B2 + 2WAWB"ABAB

Thus /ariance depends on: Variance of returns on assets in the portfolio

Correlation coefficient of the rets on the assetsincluded in the portfolio

2 ( Risk & Ret: T6o Asset


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Portfolios #$ercises

(0 The follo6ing infor1ation is pro/ided for 2 securitiesO

a0 Calculate the #%R+ for oth securities

0 Calculate the /ar and std de/ for oth securities

c0 Calculate the portfolio ret and risk if A 7 '09 and 7 '0

cf chap2 - risk and return (2012)

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