1
Finance School of Management
Chapter 13,The Capital
Asset Pricing Model
Objective
?The Theory of the CAPM
?Use of CAPM in benchmarking
? Using CAPM to determine
correct rate for
discounting
2
Finance School of Management
Chapter 13 Contents
13.1 The Capital Asset Pricing Model in Brief
13.2 Determining the Risk Premium on the
Market Portfolio
13.3 Beta and Risk Premiums on Individual
Securities
13.4 Using the CAPM in Portfolio Selection
13.5 Valuation & Regulating Rates of Return
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Finance School of Management
Introduction
u CAPM is a theory about equilibrium prices
in the markets for risky assets
u It is important because it provides
– a justification for the widespread practice of
passive investing called indexing
– a way to estimate expected rates of return for
use in evaluating stocks and projects
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Finance School of Management
The Capital Asset Pricing Model
in Brief
u CAPM is an equilibrium theory based on the theory
of portfolio selection
u The basic question
What would risk premiums on securities be
in equilibrium if people had the same set of
forecasts of expected returns and risks and all
chose their portfolios optimally according to the
principles of efficient diversifications?
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Finance School of Management
Assumptions of CAPM
u Assumption 1 (homogeneous in information
processing)
Investors agree in their forecasts of expected
rates of return,standard deviation,and
correlations of the risky securities
u Assumption 2 (homogeneous in behavior)
Investors generally behave optimally
according the theory of portfolio selection
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Finance School of Management
Intuitive of CAPM
u All the investors will allocate their investments between
the riskless asset and the same tangent portfolio
u In equilibrium,the aggregate demand for each security is
equal to its supply
u The only way the asset market can clear is if the relative
proportions of risky assets in tangent portfolio are the
proportions in which they are valued in the market place,
i.e,the market portfolio
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Finance School of Management
The Capital Market Line (CML)
Standard Deviation s
Expected Return (%) CML
rf
M
E(rM)- rf
sM
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Finance School of Management
Efficient Risk-reward
s
s M
fM
f
rrE
rrE
])([
)(
?
??
u In equilibrium,any efficient portfolio should be a
combination of the market portfolio and the
riskless asset
u The best risk-reward depends on how much the
market-related a portfolio bears
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Finance School of Management
Determining the Risk Premium on the
Market Portfolio
u The equilibrium risk premium on the market
portfolio is the product of
– variance of the market,s2M
– weighted average of the degree of risk aversion
of holders of risk,A
2)(
MfM ArrE s??
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Finance School of Management
Example,To Determine ‘A’
0.2
20.0
06.014.0
)(
)(
,06.0,20.0,14.0)(
2
2
2
?
?
?
?
????
???
A
rrE
AArrE
rrE
M
fM
MfM
fMM
s
s
s
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Finance School of Management
Contribution of Security i to the
Market Risk
??
i
iiMM rExrE )()(
??
i
iMiMM x ss
2
iM
M
fM
fi
rrE
rrE s
s
])([
)(
?
??
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Finance School of Management
Security Market Line (SML)
MM
i
iMiMx ?? ??? 1
2MiMiM ss? ?
])([)( fMiMfi rrErrE ??? ?
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Finance School of Management
Security Market Line (SML)
Expected Return
Risk = ?
SML
rf
E(rM)
Market Portfolio
1
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Finance School of Management
A Simple Derivation of CAPM
21
pprUM a x s???
u Utility maximization
– risk tolerance,risk-adjusted expected return
?
?
???
n
i
fiifp rrwrr
1
)(
? ??
i j
ijjip ww ss
2
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Finance School of Management
A Simple Derivation of CAPM
? ??????
j
jijfi
i
wrrwU 02)( s?
? ??
j
fijij rrw )(2
?s
)(2t a n RrΣw 1 fr?? ??
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Finance School of Management
A Simple Derivation of CAPM
)(2)( fik
j
k
jij rrw ???
?s
M
M
j
fik
k
k
j
k
j
k
kij
w
rrIwI
?
?s
??
?? ?? ? )()(
2
1
)( )(
– For investor k
– Aggregation
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Finance School of Management
A Simple Derivation of CAPM
iM
M
fi rr s?
2??
22
M
M
fM rr s???
)( fMiMfi rrrr ??? ?
2MiMiM ss? ?
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Finance School of Management
Sec urity Price s
10
20
30
40
50
60
70
0, 0 0 0 0, 0 8 3 0, 1 6 7 0, 2 5 0 0, 3 3 3 0, 4 1 7 0, 5 0 0 0, 5 8 3 0, 6 6 7 0, 7 5 0 0, 8 3 3 0, 9 1 7 1, 0 0 0
Y e a rs
V
a
l
u
e
M a r k e t _ P r ic e S t o c k _ Z_P r ic e
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Finance School of Management
Table of Prices
m o n t h M kt _ P ri ce Z _ P ri ce h p r_ M kt h p r_ Z
0 5 0, 0 0 3 0, 0 0 h p r_ M kt h p r_ Z a n n u a l _ co n t _ m kta n n u a l _ co n t _ zre g _ l i n e
1 5 5, 8 4 3 3, 8 7 1 1, 6 8 % 1 2, 9 0 % 1 3 2, 5 5 % 1 4 5, 5 6 % 1 7 6, 8 2 %
2 5 2, 8 7 3 3, 6 5 -5, 3 2 % -0, 6 4 % -6 5, 5 9 % -7, 7 5 % -6 4, 5 4 %
3 5 8, 1 9 3 9, 1 9 1 0, 0 7 % 1 6, 4 7 % 1 1 5, 1 5 % 1 8 2, 9 8 % 1 5 5, 6 2 %
4 6 0, 3 3 4 1, 3 0 3, 6 6 % 5, 3 8 % 4 3, 1 9 % 6 2, 9 0 % 6 7, 9 7 %
5 5 6, 9 7 3 8, 9 3 -5, 5 7 % -5, 7 4 % -6 8, 7 1 % -7 0, 8 9 % -6 8, 3 5 %
6 5 1, 5 2 3 4, 2 0 -9, 5 6 % -1 2, 1 5 % -1 2 0, 5 6 % -1 5 5, 4 0 % -1 3 1, 5 0 %
7 5 2, 8 0 3 5, 8 8 2, 4 7 % 4, 9 1 % 2 9, 3 2 % 5 7, 5 4 % 5 1, 0 8 %
8 5 5, 0 4 3 8, 2 4 4, 2 4 % 6, 5 6 % 4 9, 8 3 % 7 6, 2 2 % 7 6, 0 6 %
9 5 5, 7 6 4 0, 6 4 1, 3 2 % 6, 2 8 % 1 5, 7 0 % 7 3, 0 8 % 3 4, 4 8 %
10 6 2, 2 0 4 6, 2 6 1 1, 5 5 % 1 3, 8 3 % 1 3 1, 1 2 % 1 5 5, 4 6 % 1 7 5, 0 9 %
11 5 6, 8 4 4 1, 0 1 -8, 6 2 % -1 1, 3 4 % -1 0 8, 2 3 % -1 4 4, 4 3 % -1 1 6, 4 9 %
12 5 5, 3 0 3 9, 5 4 -2, 7 1 % -3, 5 8 % -3 2, 9 3 % -4 3, 7 8 % -2 4, 7 6 %
a n _ a n _ f a ct 1, 1 0 5 9 3 4 1, 3 1 8 1 5 1 mu 1 0, 0 7 % 2 7, 6 2 %
a n _ co n t _ ra t e 0, 1 0 0 6 9 0, 2 7 6 2 3 s i g 0, 2 5 9 0 9 9 0, 3 2 5 7 9 6
rh o 0, 9 6 8 7 7 7
b e t a 1, 2 1 8 1 5 7
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Finance School of Management
R e g res s i o n o f R e turns o f Z o n Mark e t
-200%
-150%
-100%
-50%
0%
50%
100%
150%
200%
-150% -100% -50% 0% 50% 100% 150%
M a rk e t R e t ur n
R
e
t
u
r
n
o
n
Z
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Finance School of Management
Model and Measured Values of
Statistical Parameters
? m s m ? z s z ? ?
m od l 15% 20% 12% 25% 90% 1,1 3
M eas 10% 26% 28% 33% 97% 1,2 2
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Finance School of Management
S e c u ri ty Mark e t L i n e
M ark et
Portfo l i o
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
- 2, 0 - 1, 5 - 1, 0 - 0, 5 0, 0 0, 5 1, 0 1, 5 2, 0
B e t a ( R i s k )
E
x
p
e
c
t
e
d
R
i
s
k
P
r
e
m
i
u
m
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Finance School of Management
The Beta of a Portfolio
u When determining the risk of a portfolio
– using standard deviation results in a formula
that’s quite complex
– using beta,the formula is linear
??????
i iinnp
wwww ?????,,,2211
? ? ? ?
2
1
,1
2 2
??
?
?
??
?
?
?? ??
?? ji
ijiiji
ni
iip www ?ssss
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Finance School of Management
Return-Generating Process
? ? ifMiifi rrrr ??? ?????
Where,
? ? ? ? 0,0 ?? iiM ErE ??
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Finance School of Management
Mispriced
Return
,
rf
Market Portfolio
1
·
· A
Q
Risk = ?i
?
?
????
o v e r p ic e d
u n d e p ic e d
rr
i
ie
iii,0
,0
＜
＞
?
?
?
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Finance School of Management
Risk Decomposition
u Total risk for a security = Market risk + Unique risk
u Since the unique risk can be diversified out,the
market compensates only for the market-related
risk
s ? s s ?i i m i2 2 2 2? ?
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Finance School of Management
Using the CAPM in Portfolio Selection
u Passive portfolio management
– Proxy for market portfolio
– indexing
u Active portfolio management
– Positive ALPHA
– Beat the market
28
Finance School of Management
Valuation and Regulating Rates of
Return
u Suppose you are considering investing in a new
project in the same industry of Betaful Corp,
u The market rate is 15%,and the risk-free rate is
5%
u The beta of Betaful’s stock is 1.3
u The capital structure of Betaful,80% of equity,
20% of bond
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Finance School of Management
Valuation and Regulating Rates of
Return
u Compute the beta of Betaful’s operations
04.1
0*20.03.1*80.0
b o n d
?
??
??
c o m p a n y
c o m p a n y
bonde q u i t ye q u i t yc o m p a n y ww
?
?
???
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Finance School of Management
u Beta of Betaful’s operations is equal to the beta of
the new project
u To find the required return on the new project,
apply the CAPM
? ?
? ?
%4.15
05.015.004.105.0
?
????
??? fMf rrrk ?
Valuation and Regulating Rates of
Return
31
Finance School of Management
u Assume that you start up a company which is just a
vehicle for the new project,60% capital is financed
by issuing equity and 40% by issuing bond
u The beta of your unquoted equity is
73.1
0*40.0*60.004.1
b o n d
?
??
??
e q u i t y
e q u i t y
bonde q u i t ye q u i t yc o m p a n y ww
?
?
???
Valuation and Regulating Rates of
Return
32
Finance School of Management
u If your company is all-equity financed
u Your company has an expected dividend of $6
next year,and that it will grow annually at a rate
of 4% for ever,the value of a share is
63.52$
04.0154.0
61
0 ????? gk
Dp
Valuation and Regulating Rates of
Return