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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
!" #$ternal %ntake 2'(2 ) 2'(*+
*'.2 Corporate inance
8/12/2019 CF Chap2 - Risk and Return (2012)
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T!PC Co/erage
20(: Portfolio Analysis
202: Mean-Variance Analysis
20*: The CAPM
8/12/2019 CF Chap2 - Risk and Return (2012)
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2.1 Portfolio Theory
Topic 2 Part (
8/12/2019 CF Chap2 - Risk and Return (2012)
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20(0( ntroduction: Risk and Return
Risk and Return
Risk A/erse Assu1p: #$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
8/12/2019 CF Chap2 - Risk and Return (2012)
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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
8/12/2019 CF Chap2 - Risk and Return (2012)
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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)
8/12/2019 CF Chap2 - Risk and Return (2012)
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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
8/12/2019 CF Chap2 - Risk and Return (2012)
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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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20(02 Measuring Risk & Return: 5ingle
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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20(02 Measuring Risk & Return: 5ingle
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
8/12/2019 CF Chap2 - Risk and Return (2012)
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20(02 Measuring Risk & Return: 5ingle
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$0 y a continuous cur/e
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20(02 Measuring Risk & Return: 5ingle
5ecurityExample: Discrete Dist.
Standard Deviation VS. Expected Return
Investment A
0
2
4
6
8
10
12
14
16
18
20
50 0 50
%probability
% return
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20(02 Measuring Risk & Return: 5ingle
5ecurityExample: Discrete Dist.
Standard Deviation VS. Expected ReturnInvestment B
0
2
4
6
8
10
12
14
16
18
20
50 0 50
%probability
% return
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20(02 Measuring Risk & Return: 5ingle
5ecurity Continuous dists often pro/ide a etter appro$ 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
8/12/2019 CF Chap2 - Risk and Return (2012)
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20(0* Risk & Ret: Portfolios & a?/e
;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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20(0* Risk & Ret: Portfolios & a?/e
;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$panded to larger
portfolios %i0e0 1ore than 2 securities+
Modern portfolio theory @ co1putation of
portfolio return and portfolio risk, 6ith
e1phasis of ris0 reduction t#rou/#
diversification
2 ( Ri k & R t T A t
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20(0 Risk & Ret: T6o Asset
Portfolios Portfolio Return: 2 asset portfolio 1ade up of
security A and security
E(RP) = WAE(RA) + WBE(RB)
here #%RP+ 7 e$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'
2 ( Ri k & R t T A t
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20(0 Risk & Ret: T6o Asset
Portfolios Portfolio Risk @ calculated in 2 6ays:
i0 !n the asis of dist of rets e$pec on
portfolio
Var%RP+ 7 NPi%RPi ) #%RP++2
7 #%RP ) #%RP++2
iii0 !n the asis of /ariance of rets assets in
portfolio, and relationship t6n these rets
Var(RP) = WA2Var(RA) + WB2Var(RB) +
2WAWBCov(RA,RB)
or P2 = WA2A2 + WB2B2 + 2WAWBAB
2 ( Ri k & R t T A t
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20(0 Risk & Ret: T6o Asset
Portfolios ote: ;eri/ation
Var%RP+7 #%RP ) #%RP++2
7 #3%ARA 8 R+ ) %A#%RA+ 8 #%R++42
7 #3%A%RA - #%RA++ 8 %%R - #%R++42
7 #3%A2%RA - #%RA++2 8 %2%R - #%R++2 8 2A%RA -
#%RA++%R - #%R++4
7 A2#%RA - #%RA++28 2#%R - #%R++2 8 2A#%RA -
#%RA++%R - #%R++
= WA2A2 + WB2B2 + 2WAWBAB
2 ( Ri k & Ret T o A et
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20(0 Risk & Ret: T6o Asset
Portfolios Co/ariance and the Correlation Coefficient Co/ariance is the 6eighted a/g of the product of the de/iations of
2 /ariales fro1 their respecti/e e$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 -
2 ( Ri k & R t T A t
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20(0 Risk & Ret: T6o Asset
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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20(0 Risk & Ret: T6o Asset
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