Ch 6

Report
CHAPTER 6
Efficient Diversification
The Goals of Chapter 6
Introduce the market risk and the firm-specific
risk
Portfolio Theory:
– Diversification in the case of two risky assets
without or with the risk-free asset
– Extension to the multiple risky-asset case
Introduce the single-factor model
– It is a statistical model
– To identify the components of firm-specific and
market risks
– The Treynor-Black model to construct portfolios
with higher Sharpe ratios
– To examine the CAPM (discussed in Ch7)
6-2
6.1 DIVERSIFICATION AND
PORTFOLIO RISK
6-3
Diversification and Portfolio Risk
Firm-specific risk (公司特定風險)
– Risk factors that affect an individual firm without
noticeably affecting other firms, like the results of
R&D, the management style, etc.
– Due to the possible offset of the firm-specific risks
from different firms in a portfolio, this risk can be
eliminated through diversification
– Also called unique, idiosyncratic, diversifiable, or
nonsystematic risk
Market risk (市場風險)
– Risk factors common to the whole economy,
possibly from business cycles, inflation rates,
interest rates, exchange rates, etc.
– The risk cannot be eliminated through diversification
– Also called nondiversifiable or systematic risk
6-4
Portfolio Risk as a Function of Number of
Securities on NYSE
Firm-specific risk
Market risk
※ For one single stock, the average total risk can be measured as the standard
deviation of 50%. The market and firm-specific risks represent about 40% and
60% of the total risk, respectively
※ The extremely diversified portfolio will be the whole market portfolio traded on
NYSE
※ International diversification may further reduce the portfolio risk, but global risk
factors affecting all countries will limit the extent of risk reduction
6-5
6.2 ASSET ALLOCATION WITH
TWO RISKY ASSETS
6-6
Diversification and Portfolio Theory
The portfolio theory was first introduced by
Harry Markowitz in 1952, who is a Nobel
Prize laureate in 1990
Sections 6.2 to 6.4 will introduce the results
of Markowitz’s work of showing how to make
the most of the power of diversification
In this section, the effect of diversification is
illustrated by a case of two risky assets
6-7
Two Asset Portfolio Return
– Stock and Bond Funds
Here we assumed that the risky portfolio
comprised a stock and a bond fund, and
investors need to decide the weight of each
fund in their portfolios
r p  w B rB  w S rS (portfolio return)
w B : w eight of the bond fund
rB : return of the bond fund
w S : w eight of the stock fund
rS : return of the stock fund
6-8
Scenario Analysis-Mean and Variance
Mean (Expected Return):
s
E (ri )   p(k )ri (k ), for i = S and B
k 1
Variance:
s
var(ri )     p(k )  ri (k )  E (ri )  , for i = S and B
2
i
2
k 1
※ A property for variance
s
var( wi ri )   p(k )  wi ri (k )  wi E (ri ) 
2
k 1
w
2
i
s
 p(k )  ri (k )  E (ri )  wi2 var(ri )
k 1
2
6-9
Scenario Analysis-Covariance
Covariance between rS and rB: a measure of
co-varying behavior of returns on two assets
s
cov(rS , rB )   p(k )  rS (k )  E (rS )  rB (k )  E (rB ) 
k 1
※ Three properties for covariance
cov(rS , rB )  cov(rB , rS )
s
cov(rS , rS )   p(k )  rS (k )  E (rS )   var(rS )
2
k 1
s
cov( wS rS , wB rB )   p(k )  wS rS (k )  wS E (rS )  wB rB (k )  wB E (rB ) 
k 1
s
 wS wB  p (k )  rS (k )  E (rS )  rB (k )  E (rB ) 
k 1
 wS wB cov(rS , rB )
6-10
Scenario Analysis-Covariance
※ Another expression for covariance
s
cov ( rS , rB ) 

p ( k )  rS ( k )  E ( rS )  rB ( k )  E ( rB ) 
k 1
s


k 1
s
p ( k ) rS ( k ) rB ( k )   p ( k ) rS ( k ) E ( rB ) 
k 1
s

s
p ( k ) rB ( k ) E ( rS ) 
k 1

p ( k ) E ( rS ) E ( rB )
k 1
 E ( rS rB )  E ( rS ) E ( rB )  E ( rB ) E ( rS )  E ( rS ) E ( rB )
 E ( rS rB )  E ( rS ) E ( rB )
6-11
Scenario Analysis-Correlation Coefficient
Correlation Coefficient: standardize the
covariance with two standard deviations
 SB 
cov(rS , rB )
 S B
 cov(rS , rB )   SB S B
Range of possible values for ρij
–1.0 < ρij < 1.0
※ ρij = 1 (perfectly positively correlated): the strongest tendency
for two returns to vary in the same direction
※ ρij = 0: the returns on two assets are unrelated to each other
※ ρij = –1 (perfectly negatively correlated): the strongest
tendency for two returns to vary inversely
6-12
Sample Covariance
For two series of returns in the same period
(with n pairs of observations), the covariance is
computed as follows
cov ( r1 , r2 ) 
1

n 1
t
 r1,t  r1   r2 ,t  r2 , w here ri 
1
r

n
i ,t
t
※ To correct the underestimation of the covariance when using sample
average to estimate the true mean, the sum of the product of deviations
are divided with (n–1) instead of n
※ Note that the function “COVAR” in Excel is not appropriate to compute
the covariance between two series of returns, because the sum of the
product of deviations is divided by n
※ In Excel 2010, the function “COVARIANCE.S” takes the (n–1) adjustment
into account and thus can be applied to computing the covariance given
historical series of observations of two returns
6-13
An Simple Example for Diversification
Expected Return (%):
Variance (%2) and Standard Deviation (%):
6-14
An Simple Example for Diversification
Covariance (%2) and Correlation:
Effect of Diversification (reducing risk (s.d.)):
※ Portfolio risk heavily depends on the covariance or correlation between
the returns of the assets in the portfolio
6-15
An Simple Example for Diversification
Summary:
Expected
Return (E(ri))
Standard
Deviation (σi)
Sharpe Ratio
(rf = 1%)
Stock fund (i = S)
10%
18.63%
0.483
Bond fund (i = B)
5%
8.27%
0.484
Combined Portfolio (i = P)
7%
6.65%
0.902
※ Since E(rP) > E(rB) and σP < σB, the combined portfolio is strictly
better than the bond fund according to the mean-variance
analysis
※ Because the stock fund is with higher return and higher standard
deviation than those of the portfolio, it is difficult to choose
between the stock fund and the combined portfolio
※ With the help of the Sharpe ratio, we can identify that the
combined portfolio is the best investment target
6-16
Three Rules of Two-Risky-Asset Portfolios
Rate of return on the portfolio:
rP  wB rB  wS rS
Expected rate of return on the portfolio:
E ( rP )  w B E ( rB )  w S E ( rS )
Variance of the rate of return on the
portfolio (proved on Slide 5-43):
 P = w B  B  2 w B w S  SB  B  S  w S  S
2
2
2
2
2
6-17
Standard Deviation of the Portfolio of
Two Assets
※ Alternative way to derive the formula of the portfolio variance:
Calculating the sum of the covariances of different combinations
of the terms in rP
(In this two-asset case, the four combinations of wBrB and wSrS are
considered)
 P  portfolio variance  var( rP )  var ( w B rB  w S rS )
2
 cov( w B rB , w B rB )  cov( w B rB , w S rS )  cov( w S rS , w B rB )  cov( w S rS , w S rS )
 w B var( rB )  2 w B w S cov ( rS , rB )  w S var( rS )
2
2
 w B  B  2 w B w S  SB  B  S  w S  S
2
2
2
2
6-18
Standard Deviation of the Portfolio of
Two Assets
※ Verify the above formula numerically by the above two-asset
example
 P = (0.4) (18.63)  2(0.4)(0.6)(  74.8)  (0.6) (8.27 )
2
2
2
2
2
= (0.4) (18.63)  2(0.4)(0.6)(  0.49)(18.6 3)(8.27 )  (0.6) (8.27 )
2
2
2
2
= 43.93
 P = 6.65
※ With the above formula, it is not necessary to calculate rp in
different scenarios. We can derive the portfolio variance
directly if we know the weights for each asset, the variances
of the return of each asset, and the covariance between the
returns of these two assets
6-19
Three Rules for an n-Security Portfolio:
G iven rP  w1 r1  w 2 r2 
 w n rn (w eighted average of the returns of n securities)
n
E ( rP )  w1 E ( r1 )  w 2 E ( r2 ) 
 w n E ( rn ) 
 w E (r )
i
i
i 1
 P  portfolio variance
2
 sum of n pair-w ise cov( w i ri , w j r j )
2
 w1 w1 cov ( r1 , r1 )  w1 w 2 cov ( r1 , r2 ) 
+ w 2 w1 cov ( r2 , r1 )  w 2 w 2 cov ( r2 , r2 ) 
 w 2 w n cov ( r2 , rn )
+ w n w1 cov ( rn , r1 )  w n w 2 cov ( rn , r2 ) 
 w n w n cov ( rn , rn )
n

 w1 w n cov ( r1 , rn )
n
ww
i
i 1
j
cov ( ri , r j )
j 1
6-20
Numerical Example: Portfolio Return and
S.D. of Bond and Stock Funds
Returns
Bond fund E(rB) = 5% Stock fund E(rS) = 10%
Standard deviations
Bond fund σB = 8%
Stock fund σS = 19%
Weights
Bond fund WB = 0.6
Stock fund WS = 0.4
Correlation coefficient between returns of the
bond fund and stock fund = 0.2
6-21
Numerical Example: Portfolio Return and
S.D. of Bond and Stock Funds
Portfolio return
0.6(5%) + 0.4(10%) = 7%
Portfolio standard deviation
[(0.6)2(8%)2 + (0.4)2(19%)2 + 2(0.6)(0.4)
(0.2)(8%)(19%)]½ = 9.76%
6-22
Numerical Example: Portfolio Return and
S.D. Given Different Correlation Coefficients
Different values of the correlation coefficient
(given wB = 0.6 and wS = 0.4)
ρSB
E(rP)
σP
–1
7%
2.80%
–0.5
7%
6.66%
0
0.5
1
7%
7%
7%
8.99% 10.83% 12.40%
※ If the correlation between the component securities (or
portfolios) is small or negative, there is a greater tendency
for the variability in the returns on the two assets to offset
each other, and thus the portfolio is with a smaller σP
6-23
Numerical Example: Portfolio Return and
S.D. Given Different Weights
Different weights on the bond and stock funds
minimumvariance portfolio
6-24
Numerical Example: Portfolio Return of
Bond and Stock Funds (Page 155)
The weights for the minimum-variance portfolio
Find w S to m inim ize  P = w B  B  w S  S  2 w B w S  SB  B  S
2
2
2
2
2
= (1  w S )  B  w S  S  2(1  w S ) w S  SB  B  S
2
2
2
2
First order condition (FO C )  0 w ith respec t to w S
 2(1  w S ) B (  1)+ 2 w S  S  (2  4 w S )  SB  B  S  0
2
2
 B   B  S  SB
2
 wS 
 B   S  2 B  S  SB
2
2
(0.08)  (0.08)(0.19)(0.2)
2

(0.08)  (0.19)  2(0.08)(0.19)(0.2)
2
2
 0.0932
 m inim al  P  0.006090242
2
 P  0.07804  7.804%
6-25
Investment Opportunity Set (投資機會集合)
for the Stock and Bond Funds
※ By varying different weights on the bond and stock funds, we can
construct the investment opportunity set, which is a set of all
available portfolio risk-return combinations
※ The blue curve is the investment opportunity set for these two
risky portfolios with the correlation coefficient to be 0.2
6-26
Mean-Variance Criterion
Investors prefer portfolios with higher expected
return and lower volatility
Portfolio A is said to dominate (宰制) Portfolio B
if E(rA)  E(rB) and σA  σB
(Thus, the stock fund dominates the portfolio Z in Figure 6.3)
An important feature of the portfolios in the
investment opportunity set:
– Under the same expected return, the portfolio in the
investment opportunity set dominates all portfolios
to its right due to the smaller s.d.
– This feature will be employed to define the
investment opportunity set in n risky-asset case in
6-27
Section 6.4
Mean-Variance Criterion
Efficient vs. Inefficient Portfolios (效率與非效
率投資組合)
– Any portfolios that lies below the minimumvariance portfolio (MVP) can therefore be viewed
as inefficient portfolios because it must be
dominated by a counterpart portfolio above the
MVP with the same volatility but with higher
expected return
For the investment opportunity set above the
MVP, because higher expected return is
accompanied with greater risk, the best
choice depends on the investor’s willingness
to trade off risk against expected return
6-28
Investment Opportunity Sets for the Stock and
Bond Funds with Various Correlations
※ For more negative ρSB, it tends to generate investment opportunity
sets with smaller s.d. (Amount of risk reduction depends critically on
correlation or covariance)
※ For ρSB = –1, since the movements of rS and rB are always in different
directions, it is possible to construct a portfolio with a positive return
and a zero s.d.
6-29
6.3 THE OPTIMAL RISKY PORTFOLIO
WITH A RISK-FREE ASSET
6-30
Extension to Include the Risk-Free Asset
Combinations of any risky portfolio P and the
risk-free asset are in a linear relation on the
E(r)-σ plane
P ortfolio P  invests w f in r f (  r f  0)
w P in rP ( var( rP )   P )
2
E ( rP  )  w f r f  w P E ( rP )  (1  w P ) r f  w P E ( rP )  r f  w P [ E ( rP )  r f ]
 P   w f  r  w P  P  2 w f w P cov ( r f , rP )
2
2
2
2
2
f
 w f 0  w P P  2 w f w P 0  w P P
2
2
2
2
2
2
  P   w P  P (suppose w P is positive)  w P   P  /  P
R eplacing w P in the equation of E ( rP  )
 E ( rP  )  r f   P 
 E ( rP )  r f 

 ( E ( rP  ) and  P  are in a linear relation)
P


6-31
Extension to Include the Risk-Free Asset
Combinations with different weights form a
capital allocation line (CAL)
– According the derivation on the previous slide, the
reward-to-volatility ratio of any combined portfolio is
the slope of the CAL, i.e.,
slope of CAL P 
E (rP )  rf
P
(reward-to-volatility ratio of P)
– Note that the geometric representation of the above
result is illustrated on Slide 5-46
6-32
Investment Opportunity Set Using Portfolio
MVP or A and the Risk-Free Asset
Consider the combination of any efficient
portfolio above the MVP and rf = 3%
– It is obvious that the reward-to-volatility ratio of
portfolio A is higher than that of MVP
– Mean-variance criterion also suggests that portfolios
on CALA is more preferred than those on CALMIN
6-33
Figure 6.6 Dominant CAL associated with
the Risk-Free rate
We can continue to choose the CAL upward
until it reaches the ultimate point of tangency
with the investment opportunity set, i.e., finding
the tangent portfolio O as follows
6-34
Dominant CAL with the Risk-Free Asset
CALO dominates other CALs and all
portfolios in the investment opportunity set: it
has the best risk/return or the largest slope
E (rO )  rf

E (rP )  rf
O
P
– Note that the mean-variance criterion also
suggests the same conclusion
The tangent portfolio O is the optimal risky
portfolio associated with the risk-free asset
– Given a different risk-free rate, we can find a
different portfolio O such that the combinations of
the risk-free asset and the portfolio O are the
most efficient portfolios
6-35
Dominant CAL with the Risk-free Asset
Since portfolios on the CALO are with the
same reward-to-volatility ratio, investors will
choose their preferred complete portfolios
along the CALO
– Among different combinations of the portfolio O
and the risk-free asset, more risk-averse (risktolerant) investors prefer low-risk, lower-return
(higher-risk, higher-return) portfolios near rf (near
rO or to the right of rO)
– Recall that on Slide 5-48, the optimal weight on
Portfolio O can be derived as
y
P rice of risk of the portfolio O
Invester's coefficient of risk aversion
6-36
The Complete Portfolio
※ For point O, the investor allocates 100% of asset in portfolio O
(with expected return of 7.16% and the s.d. of 10.15%)
※ For point C, the investor allocates 55% of his asset in portfolio
O and 45% of his asset in the risk-free asset
E (rC )  3%  0.55  (7.16%  3%)  5.29%
 C  0.55 10.15%  5.58%
6-37
Dominant CAL with the Risk-free Asset
In this two-asset case, the weights in the stock
and bond funds of the optimal tangent portfolio
O can be derived through Eq. (6.10)
wB* 
[ E (rB )  rf ] S2  [ E (rS )  rf ] B S  BS
[ E (rB )  rf ] S2  [ E (rS )  rf ] B2  [ E (rB )  rf  E (rS )  rf ] B S  BS
wS*  1  wB*
The expected return and the standard
deviation of the portfolio O thus can be derived
through
E ( rO )  w B E ( rB )  w S E ( rS )
*
*
 O2 = (wB* )2  B2  ( wS* )2  S2  2wB* wS* SB B S
6-38
6.4 EFFICIENT DIVERSIFICATION WITH
MANY RISKY ASSETS
6-39
Extension to All Securities
For n-risky assets, the investment opportunity
set consists of portfolios with optimal weights
(could be negative) on assets to minimize
variance given the expected portfolio return, i.e.,
n
m in 
wi
2
P

n
ww
i
i 1
n
j
cov( ri , r j ) 
j 1
s.t. w1 E ( r1 )  w 2 E ( r2 ) 
w1  w 2 
n
ww
i
i 1
j
 i , j  i
j
j 1
 w n E ( rn )  E ( r p )
 wn  1
In other words, for given expected portfolio
returns E(rp), the portfolios on the investment
opportunity set dominate other portfolios, i.e.,
with less risk given the same expected return
6-40
Extension to All Securities
The curve for the investment opportunity set
is also called the portfolio frontier (投資組合
前緣)
The upper half of the portfolio frontier is
called the efficient frontier (效率前緣), which
represents a set of efficient portfolios that
offer higher expected return at each level of
portfolio risk (comparing to the lower half of
the portfolio frontier) (see the next slide)
6-41
The Efficient Frontier of Risky Assets and
Individual Assets
※ For rational investors, they prefer the portfolio of risky assets on
the efficient frontier due to the mean-variance criterion
6-42
Extension to All Securities
Using the current risk-free rate, we can search for
the CAL with the highest reward-to-volatility ratio
and thus find the optimal risky portfolio O (see
Slides 6-33 and 6-34)
Finally, investors choose the appropriate mix
between the optimal risky portfolio O and the riskfree asset (see Slide 6-37)
The separation property: the process to choose
portfolio combinations can be separated into two
independent tasks
1. Determination of the optimal risky portfolio O, which is
a purely technical problem
2. The personal choice of the mix of the risky portfolio
and the risk-free asset depending on his preference 6-43
Extension to All Securities
An illustrative example: Construct a global
portfolio using six stock market indices
※ To take the forbiddance of short sales into account, one need an
additional constraint that 1 , 2 ,…,  ≥ 0 in the minimization problem
specified on Slide 6-40
6-44
6.5 SINGLE-INDEX MODEL
6-45
Index models
The index model (指數模型) is a statistical model
to measure or identify the components of firmspecific and systematic risks for a particular
security or portfolio
William Sharpe (1963), who is a Nobel Prize
laureate in 1990, introduced the single-index
model (單一指數模型) to explain the benefits of
diversification
– To separate the systematic and nonsystematic risks in
the single-index model, it is intuitive to use the rate of
return on a broad portfolio of securities, such as the
S&P 500 index, as a proxy for the common macro
factor (market risk factor)
– The single-index model is also called the market model6-46
Specification of a Single-Index Model of
Security Returns
Excess return of security i can be stated as:
Ri  i  i RM  ei
– Ri (= ri – rf) denotes the excess return on security i
– RM (= rM – rf) denotes the excess return on the market index
– ei denotes the unexpected risk relevant only to this security
and E(ei) = 0, var(ei) = σ2(ei), cov(RM, ei) = 0, and cov(ei, ej)
=0
– αi is the security’s expected excess return if RM is zero
– βi measures the sensitivity of the excess return Ri with
respect to the market excess return RM
– This model specifies two sources of risks for securities:
1. Common macro factor (RM) (or the market risk factor): represented
by the fluctuation of the market index return
2. Firm-specific components (ei): representing the part of uncertainty
specific to individual firms but independent of the market risk factor6-47
Scatter Diagram (點散圖) for Ri and RM
※ The linear relation is drawn so as to minimize the sum of all the
squared errors (which is measured as the vertical distance between
each node and the examined straight line) around it. Hence, we say
the regression line “best fits” the data in the scatter diagram
※ The line is called the security characteristic line (SCL)
6-48
Scatter Diagram (點散圖) for Ri and RM
Geometric interpretation of αi and βi
– The regression intercept is αi, which is
measured from the origin to the intersection of
the regression line with the vertical axis (see
the previous slide)
– The regression coefficient βi is measured as
the slope of the regression line
The larger the beta of a security  the greater
sensitivity of the security price in response to the
market index  the higher the security’s systematic
risk
6-49
Deriving αi and βi using Historical Data
The slope and intercept of the best-fit
regression line can be derived as follows
A
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
B
Week
1
2
3
4
5
6
7
8
9
10
C
D
E
Annualized Rates of Return (%)
ABC
XYZ
Mkt. Index Risk Free
65.13
-22.55
64.40
5.23
51.84
31.44
24.00
4.76
-30.82
-6.45
9.15
6.22
-15.13
-51.14
-35.57
3.78
70.63
33.78
11.59
4.43
107.82
32.95
23.13
3.78
-25.16
70.19
8.54
3.87
50.48
27.63
25.87
4.15
-36.41
-48.79
-13.15
3.99
-42.20
52.63
20.21
4.01
Average:
F
G
H
I
Excess Returns (%)
ABC
XYZ
Market
59.90
-27.78
59.17
47.08
26.68
19.24
-37.04
-12.67
2.93
-18.91
-54.92
-39.35
66.20
29.35
7.16
104.04
29.17
19.35
-29.03
66.32
4.67
46.33
23.48
21.72
-40.40
-52.78
-17.14
-46.21
48.62
16.20
15.20
7.55
9.40
COVARIANCE MATRIX
ABC
XYZ
Market
ABC
3020.933
XYZ
442.114 1766.923
Market
773.306 396.789 669.010
SUMMARY OUTPUT OF EXCEL REGRESSION
Regression Statistics
Multiple R
0.544
R-square
0.296
Adj. R-square
0.208
Standard
28
48.918
Error
29 Observations
10.000
30
31
32
Coefficients Std. Error
33 Intercept
4.336
16.564
34 Market Return
1.156
0.630
35
 ABC 
cov( R A B C , R M )
var( R M )

773.31
 1.156
669.01
E ( RABC )   ABC   ABC E ( R M )
  ABC  E ( RABC )   ABC E ( R M )
 R ABC   ABC  R ABC
 15.20%  1.156  9.40%
 4.33%
※ The SCL of ABC is given by
RABC = 4.33% + 1.156 RMarket
t-stat
p-value
0.262
0.800
1.834
0.104
※ The regression results in the left
table is generated by the Data
Analysis tool in Excel
6-50
Components of Risk
Because the firm-specific components of the
firm’s return is uncorrelated with the market
return, we have the following equation
T otal risk  var( R i )  var( i   i R M  e i )
 var(  i R M  e i )
 var(  i R M )  var( e i )
  i  M   ( ei )
2
2
2
 S ystem atic risk + Firm -specific risk
– The systematic risk of each security depends on both
the volatility in RM (that is, σM) and the sensitivity of the
security to fluctuations in RM (that is, βi)
– The firm-specific risk is the variance in the part of the
stock’s return that is independent of market return (that
is, σ2(ei))
6-51
Components of Risk
One method to measure the relative importance of
systematic risk is to calculate the ratio of systematic
variance to total variance
S ystem atic variance
T otal variance

 
2
i
i
2
2
M
2
2
 cov ( R i , R M )   M2
cov ( R i , R M )
2
2





R

iM
2
2
2
2




M
i
M
i


– A larger correlation coefficient (in absolute value terms)
indicates that the systematic risk represents a larger
portion of the total risk and is more important
– At the extreme, when the correlation is either 1 or –1, the
return of the individual stock is perfectly positive or
negative correlated with the market return. Thus the
security return can be fully explained by the market
return and there are no firm-specific effects
※This ratio is also called R-squared, which measures how
well the market return can explain the individual return 6-52
Interpretation of Regression Lines and Scatter
Diagrams
※ For R1 to R6, β > 0, and for R7 and R8, β < 0
※ For R1, R2, and R6, β’s are larger because the regression lines are steeper
※ For R2, R3, and R7, the degree of deviations from the regression line are smaller
and relatively stable, which implies a smaller σ2(ei) or a higher R-squared value
※ For R1, R4, and R8, α > 0, which is a preferred feature because for stocks with the
same β, a higher α means a higher expected excess return for that stock
6-53
Diversification in a Single-Factor Security Market
The systematic component of each security
return βiRM is perfectly correlated with the
systematic part of any other security’s return
Thus there are no diversification effects on
systematic risk no matter how many securities
are involved
The beta of a portfolio is the weighted average
of the individual security betas in that portfolio
n
If rP   wi ri , then  P 
i 1
n
w 
i 1
i
i
(This result will be
verified numerically in
Ch. 7)
This is why the systematic risk is also called
the nondiversifiable risk
6-54
Diversification in a Single-Factor Security Market
Since the firm-specific risks (ei) are
independent of each other, their effects could
be offset or almost eliminated through
diversification, i.e., through investing in many
securities
Consider a equally weighted portfolio
T otal firm -specific risk
n
 var( w1 e1  w 2 e 2 
 w n en ) 
ww
i
i 1

var( e1 )  var( e 2 ) 
n
2
n
 var( e n )

n
j
cov( e i , e j ) 
j 1
n
n
2
2

w
2
i
var( e i )
i 1

2
n 
 0
n
So the firm-specific risk is also called the
diversifiable risk
6-55
The Treynor-Black (1973) Model
To improve the Sharpe ratio of the market
portfolio based on stocks with nonzero α’s in
the single-index model
– Construct a new portfolio based on the market
portfolio M and these non-zero alpha stocks
– First, construct the active portfolio A by optimally
combining non-zero alpha stocks with the weights
 i /  ( ei )
2
wi 

/  ( ei )
2
i
 A 
 w
i
i
i
,A 
w
i
i
,  (e A ) 
2
i
w
2
i
 ( ei )
2
i
i
Positive (negative) αi  positive (negative) wi 
purchase (short) the stock i  wi αi always has positive
contribution to αA
Smaller σ2(ei)  Higher R2 based on the market model
 higher confidence level of the accuracy of αi  higher
weight wi on the stock i
6-56
The Treynor-Black (1973) Model
– Second, construct the investment opportunity set
using the portfolios M and A, and find the optimal
tangent portfolio O given the risk-free rate rf
 E ( rA )   A  r f   A [ E ( rM )  r f ]
rA  r f   A   A ( rM  r f )  e A  
2
2
2
2
(e A )






M
A
A

S ubstitute the ( E ( rA ),  A ), ( E ( rM ),  M ), and
2
2
 M A   A M /  A into the form ulae on S lide
6.38, and the optim al w eights on the p ortfolios
A and M can be derived as follow s.
0
w 
*
A
wA
and w M  1  w A ,
*
1  w (1   A )
0
A
 A /  (e A )
*
2
w here w 
0
A
RM / 
2
M
( R M  E ( rM )  r f )
– The improvement of the Sharpe ratio
2
SRO2  SRM2   A /  (eA )   (information ratio of A)2  0
6-57
The Treynor-Black (1973) Model
wGoogle and wDell in
the active portfolio
A
A
 2 (e A )
wA0 (derived based on
 A ,  2 (eA ), RM , and  M2 )
wM* and w*A
※ Note that the performance of the Treynor-Black model critically depends on
the accuracy of the prediction of αi‘s and βi‘s in a future period of time
6-58
6.6 RISK OF LONG-TERM INVESTMENTS
6-59
Risk of Long-Term Investments
Vast majority of financial advisers believe that
stocks are less risky if held for the long run
– Risk premium for the T-year investment is RT (= R+
R +…+ R)
– Variance for the T-year investment is σ2T (Under the
assumption that excess returns are serially
uncorrelated, var(R + R +…+ R) = var(R) + var(R)
+…+ var(R) = Tvar(R) = σ2T
– Standard deviation for the T-year investment is σ 
– As a result, the Sharpe ratio becomes RT / σ  =
R /σ
When T is large, R  / σ is higher than R / σ, the 1-year
Sharpe ratio
6-60
Risk of Long-Term Investments
Time diversification effect:
– The overperforming and underperforming effects
could offset for each other and therefore reduce the
variance of the T-year investment
However, the time diversification effect is widely
rejected in practice primarily due to
– The serial correlation for successive returns cannot
be ignored, so var(R + R +…+ R) ≠ var(R) + var(R)
+…+ var(R)
6-61

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