Financial Analysis, Planning and
Forecasting
Theory and Application
Chapter 8
Risk and Return Trade-off Analysis
By
Alice C. Lee
San Francisco State University
John C. Lee
J.P. Morgan Chase
Cheng F. Lee
Rutgers University
1
Outline

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




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8.1 Introduction
8.2 Capital Market Line, Efficient-market hypothesis, and
capital asset pricing model
8.3 The market model and beta estimation
8.4 Empirical evidence for the risk-return relationship
8.5 Why beta is important in financial management
8.6 Systematic risk determination
8.7 Some applications and implications of the capital asset
pricing model
8.8 Arbitrage pricing theory
8.9 Summary
Appendix 8A. Mathematical derivation of the capital asset
pricing model
Appendix 8B. Arbitrage Pricing Model
2
8.2 Capital Market Line, Efficient-market hypothesis, and
capital asset pricing model
 Lending,
borrowing, and the market portfolio
 The capital market line
 The efficient-market hypothesis
- Weak-form efficient-market hypothesis
- Semistrong-form efficient-market hypothesis
- Strong-form efficient-market hypothesis
 The capital asset pricing model
3
8.2 Capital Market Line, Efficient-market hypothesis, and
capital asset pricing model
From Figure 8.1, we can
derive the capital market line
as follows.
Step 1. Since
PRf E  M p Rf D
therefore,
E ( Rp )  R f  p

E( Rm )  R f  m
Figure 8.1 The Capital Market Line
then we can obtain Eq. 8.2.
Eq(8.2)
P
E ( RP )  R f  [ E ( Rm )  R f ]
m
4
8.2 Capital Market Line, Efficient-market hypothesis, and
capital asset pricing model
market value of an individual asset
x
market value of all asset
(8.3)
Figure 8.2 The Capital Asset Pricing Model (SML)
E(Ri )  Rf  i [E(Rm )  Rf ]
(8.4)
Derivation of Equation (8.4) can
be found in Appendix 8A
5
8.2 Capital Market Line, Efficient-market hypothesis, and
capital asset pricing model
COV ( Ri , Rm ) i ,m i m


VAR ( Rm )
 m2
i ,m i

m
i
E ( Ri )  R f 
[ E ( Rm )  R f ]
m
(8.5)
(8.6)
6
8.3 The market model and beta estimation
Ri, t = αi + i Rm, t + εi, t
(8.7)
Ri, t – Rf, t = αi + i (Rm, t – Rf, t) + εi, t
(8.8)
ˆi  (Ri  Rf )  i (Rm  Rf )
(8.9)
7
8.3 The market model and beta estimation
 Example
8-1
i 
 JNJ 
cov( Ri , Rm )
 m2
cov( RJNJ , Rm )
 m2
 MRK 
 PG 

.000407
 .3246
.00126
cov( RMRK , Rm )
 m2
cov( RPG , Rm )
 m2

.000974
 .7763
.00126
.000244

 .1943
.00126
8
8.3 The market model and beta estimation
Table 8-1
Average Return ( Ri )
Variance (2i)
Covariance [COV (Ri, Rm)]
JNJ
.0042
.00162
.000407
MRK
.0021
.00610
.000974
PG
.0105
.00124
.000244
Market
.0043
.00126
ˆ JNJ  (.0042 .004)  (0.3255)(.0043 .004)  .0001
ˆ MRK  (.0021 .004)  (0.7790)(.0043 .004)  .0021
ˆ PG  (.0105 .004)  (0.1941)(.0043 .004)  .0064
9
8.3 The market model and beta estimation
Figure 8-3 Mean-Variance Comparison
Mean-Variance Comparison
0.007
MRK
Mean Return
0.006
0.005
0.004
0.003
JNJ
0.002
Market
PG
0.001
0
0
0.002
0.004
0.006
0.008
0.01
0.012
Variance
10
8.3 The market model and beta estimation
Figure 8-4 SML Comparison
SML Comparison
Mean Return
0.012
0.01
PG
0.008
0.006
JNJ
0.004
Market
MRK
0.002
0
0
0.2
0.4
0.6
0.8
1
1.2
Beta
11
8.4 Empirical evidence for the risk-return relationship
E(Ri) – Rf = [E(Rm) – Rf] i
(8.10)
 im i ,m i
i  2 
m
m
(8.5)
Rp  Rf  r0  r1 p  ep
(8.11)
12
8.5
Why beta is important in financial management
The use of beta is of great importance to the financial manager because it
is a key component in the estimation of the cost of capital. A firm’s
expected cost of capital can be derived from E(Ri), where E(Ri) = Rf +
βi [E (Rm) – Rf]. This assumes that the manager has access to the
other parameters, such as the risk-free rate and the market rate of
return. (This is explored in more detail in chapter 11.)
Ri, t = αi + βi Rm, t + єi, t
(8.7)
Beta is also an important variable because of its usefulness in security
analysis. In this type of analysis, beta is used to measure a security’s
response to a change in the market and so can be used to structure
portfolios that have certain risk characteristics.
13
8.6 Systematic risk determination
 Business
risk
 Financial risk
 Other financial variables
 Capital labor ratio
 Fixed costs and variable costs
 Market-based versus accounting-based beta
forecasting
14
8.6 Systematic risk determination
 Business
risk and financial risk
risk of financial leverage   ROE   ROA
 Other
Financial Variables
Salest  Salest 1
gs 
100%
Salest 1
gTA
total assetst  total assetst 1

100%
total assetst 1
15
8.6 Systematic risk determination
EPSi,t  a0  a1EPSm,t  Ei,t
(8.12)
N
 2 ( EBIT ) 
2
(
EBIT

EBIT
)

t
t 1
N 1
Capital labor ratio
Q=f(K, L)
  a0  a1 X1  a2 X 2  a3 X 3  ...  an X n
(8.13)
(8.14)
16
8.7 Some applications and implications of the
capital asset pricing model
Table 8-2 Regression Results for the Accounting Beta
  a0  a1 X1  a2 X 2  ...  an X n
Coefficients
t Values
Intercept ( a0 )
.911
11.92
Financial leverage ( a1 )
.704
5.31
Dividend payout ( a2 )
-.175
-3.50
Sales beta( a3)
.03
3.02
Operating income beta ( a4 )
.011
2.18
17
8.7 Some applications and implications of the
capital asset pricing model
E ( P1 )  P0
E ( Ri ) 
P0
E ( P1 )  P0
 R f  [ E ( Rm )  R f ]i
P0
E ( P1 )
P0 
1  R f  [ E ( Rm )  R f ]( i )
(8.15)
(8.16)
(8.17)
18
8.7 Some applications and implications of the
capital asset pricing model
FIGURE 8-5
Capital Budgeting and Business Strategy Matrix
19
8.8 Liquidity and Capital Asset Pricing Model
Acharya and Pedersen (2005) proposed the following model for liquidity
considering the impact of liquidity risk on security pricing
[E(Rit )]  a  bj1 (factor 1)  bj 2 (factor 2)  ...  bjk (factor k )
(8.18)
The overall risk of a security account for three kinds of liquidity risk
defined as follows
 L1 
Cov(Ci , CM )
Var ( RM  CM )
L2 
Cov( Ri , CM )
Var ( RM  CM )
 L3 
Cov(Ci , RM )
Var ( RM  CM )
First,  L1 measures the sensitivity of security illiquidity to market illiquidity. In general, investors
demand higher premium for holding an illiquid security when the overall market liquidity is low.
Further, L 2 measures the sensitivity of security’s return to market illiquidity. Investors are willing
to accept a lower average return on security that will provide higher returns when market
 L3
illiquidity is higher. Finally
measures
the sensitivity of security illiquidity to the market rate of
return. In general, investors are willing to accept a lower average return on security that can be20
sold more easily.
8.9 Arbitrage Pricing Theory
Ross (1976, 1977) has derived a generalized capital asset pricing relationship
called arbitrage pricing theory (APT). To derive the APT, Ross assumed the
expected rate of return on ith security be explained by k independent
influences (or factors) as
[E(Rit )]  a  bj1 (factor 1)  bj 2 (factor 2)  ...  bjk (factor k )
(8.19)
Using Equation 8-18, Ross has shown that the risk premium of jth security can
be defined as
E(Ri )  Rf  [E(Fi )  Rf ]bi1  ...  [E(Fk )  Rf ]bik
(8.20)
By comparing Equation 8-19 with the CAPM equation, we can conclude that
the APT is a generalized capital asset pricing model. Therefore, it is one of
the important models for students of finance to understand. This model is
generally discussed in upper-level financial management or investment
courses.
21
8.10 Intertemporal CAPM
Merton (1973,1992) derived an Intertemporal CAPM which
allows the change of investment opportunity set as defined
in Eq. (8.21). This Intertemporal CAPM will reduce to the
one-period CAPM defined in Eq. (8.4).
i  r 
 i iM  in nM 
2
 M (1  nM
)
( M  r ) 
 i in  iM nM 
2
 M (1   Mn
)
( n  r )
(8.21)
22
8.8 Summary
In Chapter 8 we have discussed the basic concepts of risk
and diversification and how they pertain to the CAPM, as
well as the procedure for deriving the CAPM itself. It was
shown that the CAPM is an extension of the capital market
line theory. The possible uses of the CAPM in financial
analysis and planning were also indicated.
The concept of beta and its importance to the financial
manager were introduced. Beta represents the firm’s
systematic risk and is a comparative measure between a
firm’s security or portfolio risk in comparison with the
market risk. Systematic risk was further discussed by
investigating the relationship between the beta coefficient
and other important financial variables.
23
Appendix 8A. Mathematical derivation of the
capital asset pricing model
E( RP )  wi E( Ri )  (1  wi ) E( Rm )
 P  [w   (1  wi )   2(1  wi )wi im ]
2
i
2
i
2
2
m
E ( RP )
 E ( Ri )  E ( Rm )
wi
(8A-1)
1
2
(8A-2)
(8A-3)
1

( P )
 1 2[wi2 i2  (1  wi )2  m2  2wi (1  wi )] 2 (2wi i2  2 m2  2wi m2  2 im  4wi im )
wi
24
Appendix 8A. Mathematical derivation of
the capital asset pricing model
When wi  0
E ( RP ) E ( Ri )  E ( Rm )
wi
 im   m2
E ( RP )



(

)
 ( P )
m
P
wi
CML Slop:
E ( Rm )  R f
m
(8A-4)
(8A-5)
 im
E ( Ri )  R f  [ E ( Rm )  R f ] 2
m
(8A-6)
E( Ri )  Rf  [ E( Rm )  R f ]i
(8A-7)
25
Appendix 8A. Mathematical derivation of
the capital asset pricing model
FIGURE 8-A1 The Opportunity Set Provided by Combinations of
Risky Asset / and Market Portfolio, M
26
Appendix 8B. Arbitrage Pricing Model
Rj  E(Rj )  b1 j1  b2 j 2 
 bkj X j  0
 bkj k   j
(8B-1)
(8B-2)
n
 Xj 0
j 1
 X j Rj  0
(8B-3)
(8B-4)
27
Appendix 8B. Arbitrage Pricing Model
E(Rj )  0  1bj1  2bj 2 
 k bjk
(8B-5)
i  Ei  E0
E(Rj )  E0  (E1  E0 )bj1 
Rjt  a0  a1R1t 
 (Ek  E0 )bjk
(8B-6)
 ai 1Ri 1t  ai 1Ri 1t  ak Rk  it (8B-7)
28