1
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Chapter 7
Efficient
Diversification
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
2
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Two-Security Portfolio: Return
rp =
W1 =
W2 =
r1 =
r2 =
n
S
Wi = 1
i=1
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
3
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Two-Security Portfolio: Risk
sp2 =
s12 = Variance of Security 1
s22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
4
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Covariance
Cov(r1r2) =
r1,2 = Correlation coefficient of
returns
s1 = Standard deviation of
returns for Security 1
s2 = Standard deviation of
returns for Security 2
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
5
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Correlation Coefficients: Possible
Values
Range of values for r 1,2
If r = 1.0,
If r = - 1.0,
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
6
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Three-Security Portfolio
rp =
s2p =
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
7
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
In General, For an n-Security
Portfolio:
rp = Weighted average of the
n securities
sp2 = (Consider all pair-wise
covariance measures)
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
8
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Two-Security Portfolio
E(rp) =
sp2 =
sp =
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
Bodie • Kane • Marcus
9
E(r)
Essentials of Investments
Fourth
Edition
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
13%
r = -1
r=0
8%
r = -1
r=1
12%
Irwin / McGraw-Hill
r = .3
20%
St. Dev
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
10
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Portfolio Risk/Return Two Securities:
Correlation Effects
• Relationship depends on correlation
coefficient
• -1.0 < r < +1.0
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
11
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Minimum Variance Combination
Sec 1 E(r1) = .10
Sec 2 E(r2) = .14
s 1 = .15
r12 = .2
s 2 = .20
W1 =
W2 = (1 - W1)
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
12
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Minimum Variance
Combination: r = .2
(.2)2 - (.2)(.15)(.2)
W1 =
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
W1 =
W2 =
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
13
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Minimum Variance: Return and Risk
with r = .2
rp =
sp=
s p=
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
14
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Minimum Variance
Combination: r = -.3
(.2)2 - (-.3)(.15)(.2)
W1 =
(.15)2 + (.2)2 - 2(.2)(.15)(-.3)
W1 =
W2 =
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
15
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Minimum Variance: Return and Risk
with r = -.3
rp =
sp =
s p=
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
16
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Extending Concepts to All Securities
• The optimal combinations result in
lowest level of risk for a given return
• The optimal trade-off is described as the
efficient frontier
• These portfolios are dominant
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
17
Bodie • Kane • Marcus
E(r)
Essentials of Investments
Fourth
Edition
The minimum-variance frontier of
risky assets
Efficient
frontier
Global
minimum
variance
portfolio
Individual
assets
Minimum
variance
frontier
St. Dev.
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
18
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Extending to Include Riskless Asset
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
Bodie • Kane • Marcus
19
E(r)
Essentials of Investments
ALTERNATIVE CALS CAL (P)
Fourth
Edition
CAL (A)
M
M
P
P
A
CAL (Global
minimum variance)
A
G
F
P
Irwin / McGraw-Hill
P&F
M
A&F
s
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
20
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Dominant CAL with a Risk-Free
Investment (F)
CAL(P) dominates other lines -- it has the
best risk/return or the largest slope
Slope =
[ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA]
Regardless of risk preferences
combinations of P & F dominate
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
21
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Single Factor Model
ri =
ßi = index of a securities’ particular return
to the factor
F= some macro factor; in this case F is
unanticipated movement; F is commonly
related to security returns
Assumption: a broad market index like the
S&P500 is the common factor
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
Bodie • Kane • Marcus
22
Essentials of Investments
Fourth
Edition
Single Index Model
r  r  a   r  r  e
i
f
Risk Prem
a
i
i
i
m
f
i
Market Risk Prem
or Index Risk Prem
= the stock’s expected return if the
market’s excess return is zero (rm - rf) = 0
ßi(rm - rf) = the component of return due to
movements in the market index
ei = firm specific component, not due to market
movements
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
23
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Risk Premium Format
Let:
Risk premium
format
Ri = ai + ßi(Rm) + ei
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
24
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Estimating the Index Model
Excess Returns (i)
. ..
. ..
.
.
.
.
.
. . ..
.. . .
Irwin / McGraw-Hill
Security
.
.
.
.
.
.
Characteristic
. . .
Line
.
. .. . .
.
. . . Excess returns
. . . . on market index
.
.
.
.
.
.
.
.
Ri = a i + ßiRm + ei
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
25
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Components of Risk
• 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 =
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
26
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Measuring Components of Risk
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
27
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Examining Percentage of Variance
Total Risk = Systematic Risk +
Unsystematic Risk
Systematic Risk/Total Risk = r2
ßi2 s m2 / s2 = r2
i2 sm2 / i2 sm2 + s2(ei) = r2
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.
28
Bodie • Kane • Marcus
Essentials of Investments
Fourth
Edition
Advantages of the Single Index Model
• Reduces the number of inputs for
diversification
• Easier for security analysts to specialize
Irwin / McGraw-Hill
© 2001 The McGraw-Hill Companies, Inc. All rights reserved.