Irwin/McGraw-Hill
10-1
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Single Index
and
Multifactor Models
Chapter 6
Irwin/McGraw-Hill
10-2
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
? Reduces the number of inputs for
diversification.
? Portfolio of 50 assets
? ---50 expected returns; 50 variances;
? 1225 covariance.
? ---too difficult a task.
Advantages of the Single Index Model
Irwin/McGraw-Hill
10-3
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
(r
i
-r
f
)=
i
+ ?
i
(r
m
-r
f
)+ e
i
α
Risk Premium
Market Risk Premium
or Index Risk Premium
i
= the stock’s expected return if the
market’s excess return is zero
?
i
(r
m
-r
f
) = the component of return due to
movements in the market index
E(r
m
-r
f
)= 0
e
i
= firm specific component, not due to market
movements, E(ei)=0
?α
Single Index Model
Irwin/McGraw-Hill
10-4
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Let: R
i
= (r
i
-r
f
)
R
m
= (r
m
-r
f
)
Risk premium
format
R
i
= α
i
+ ?
i
(R
m
)+ e
i
Risk Premium Format
Irwin/McGraw-Hill
10-5
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Security Characteristic Line
Excess Returns (i)
SCL
.
.
.
.
.
.
.
.
. .
.
. .
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
. .
.
. .
.
. .
. .
. .
Excess returns
on market index
R
i
= α
i
+ ?
i
R
m
+ e
i
Irwin/McGraw-Hill
10-6
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Jan.
Feb.
.
.
Dec
Mean
Std Dev
5.41
-3.44
.
.
2.43
-.60
4.97
7.24
.93
.
.
3.90
1.75
3.32
Excess
Mkt. Ret.
Excess
GM Ret.
Using the Text Example
from Table 10-1
Irwin/McGraw-Hill
10-7
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Estimated coefficient
Std error of estimate
Variance of residuals = 12.601
Std dev of residuals = 3.550
R-SQR = 0.575
?
-2.590
(1.547)
1.1357
(0.309)
r
GM
-r
f
= + ?(r
m
-r
f
)
α
α
Regression Results
Irwin/McGraw-Hill
10-8
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
? Market or systematic risk: risk related to the
macro economic factor or market index
? Unsystematic or firm specific risk: risk not
related to the macro factor or market index
? Total risk = Systematic + Unsystematic
Components of Risk
Irwin/McGraw-Hill
10-9
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
σ
i
2
= β
i
2
σ
m
2
+ σ
2
(e
i
)
where;
σ
i
2
= total variance
β
i
2
σ
m
2
= systematic variance
σ
2
(e
i
) = unsystematic variance
Measuring Components of Risk
Irwin/McGraw-Hill
10-10
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Total Risk = Systematic Risk + Unsystematic
Risk
Systematic Risk/Total Risk =
2
?
i
2
σ
m
2
/ σ
2
=
2
β
i
2
σ
m
2
/ (β
i
2
σ
m
2
+ σ
2
(e
i
) )=
2
Examining Percentage of Variance
R
R
R
Irwin/McGraw-Hill
10-11
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Index model and portfolio construction
? n assets
? n expected returns
? n betas
?
)(
2
i
en σ
2
M
σ
Irwin/McGraw-Hill
10-12
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Index Model and Diversification
)(
1
1
1
2222
1
1
1
PMP
P
N
i
iP
N
i
iP
N
i
iP
PMPPP
e
e
N
e
N
N
eRR
σσβσ
αα
ββ
βα
+=
=
=
=
++=
∑
∑
∑
=
=
=
Irwin/McGraw-Hill
10-13
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Risk Reduction with Diversification
Number of
Securities
variance
Market Risk
Unique Risk
σ
2
(e
P
)=σ
2
(e) / n
β
P
2
σ
M
2
Irwin/McGraw-Hill
10-14
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Industry Prediction of Beta
? Merrill Lynch Example
- Use returns not risk premiums
?α has a different interpretation
?α = α + r
f
(1-β)
? Forecasting beta as a function of past beta
? Forecasting beta as a function of firm size,
growth, leverage etc.
Irwin/McGraw-Hill
10-15
? The McGraw-Hill Companies, Inc., 1999
INVESTMENTS
Fourth Edition
Bodie Kane Marcus
Multifactor Models
? Use factors in addition to market return
- Examples include industrial production,
expected inflation etc.
- Estimate a beta for each factor using multiple
regression
? Fama and French
- Returns a function of size and book-to-market
value as well as market returns