Chapter 6
Efficient
Diversification
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
6.1 Diversification and
Portfolio Risk
6.2 Asset Allocation With
Two Risky Assets
6-2
Two-Security Portfolio: Return
E(rp )= W1 r1 + W2r2
W1 = Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2
n
E(r p ) 
W r ;
i i
i1
n  # securities in the portfolio
n

Wi = 1
i=1
6-3
Two-Security Portfolio Return
E(rp) = W1r1 + W2r2
W1 = 0.6
Wi = % of total money
W2 = 0.4
invested in security i
r1 = 9.28%
r2 = 11.97%
E(rp) = 0.6(9.28%) + 0.4(11.97%) = 10.36%
6-4
Combinations of risky assets
When we put stocks in a
portfolio, p < (Wii)
Why? Averaging principle
When Stock 1 has a return <
E[r1] it is likely that Stock 2 has
a return > E[r2] so that rp that
contains stocks 1 and 2 remains
close to E[rp]
n = # securities
in the portfolio
What statistics measure the
tendency for r1 to be above
expected when r2 is below
expected?
Covariance and Correlation
6-5
Portfolio Variance and
Standard Deviation
Q
2
σp 
Q
 [W W
I
J
Cov(r I , rJ )]
I1 J1
WI , WJ  Percentage of the total portfolio invested in stock I and J respective ly
Q  The total number of stocks in the portfolio
Cov(r I , rJ )  Covariance of the returns of Stock I and Stock J
If I  J then Cov (rI , rJ )  σ I 2 & Cov(r I , rJ )  Cov (rJ , rI )
Variance of a Two Stock Portfolio:
 p 2  W12 12  2W1W2 Cov (r1, r2 )  W2 2  2 2
6-6
Expost Covariance Calculations
N (r
 r 1 )  (r 2,T  r 2 )
n
1,T
Cov(r 1, r2 ) 
n  1 T 1
n

r 1  average or expected return for stock 1
r 2  average or expected return for stock 2
n  # of observatio ns
• If when r1 > E[r1], r2 > E[r2], and when
r1 < E[r1], r2 < E[r2], then COV will be
positive
_______.
• If when r1 > E[r1], r2 < E[r2], and when
r1 < E[r1], r2 > E[r2], then COV will be
negative
_______.
Which will “average away” more risk?
6-7
Covariance and Correlation
• The problem with covariance


Covariance does not tell us the intensity of the
comovement of the stock returns, only the
direction.
We can standardize the covariance however
and calculate the correlation coefficient which
will tell us not only the direction but provides a
scale to estimate the degree to which the
stocks move together.
6-8
Measuring the Correlation
Coefficient
• Standardized covariance is called the
correlation coefficient or 
_____________________
ρ (1,2) 
Cov(r 1, r2 )
σ1  σ 2
For Stock 1 and Stock 2
6-9
 and Diversification in a 2 Stock Portfolio
•  is always in the range __________
-1.0 to +1.0 inclusive.
• What does (1,2) = +1.0 imply?
The two are perfectly positively correlated. Means?
If (1,2) = +1, then (1,2) = W11 + W22
– What does (1,2) = -1.0 imply?
There are very large diversification benefits from combining 1 and 2.
Are there any diversification benefits from combining 1 and 2?
The two are perfectly negatively correlated. Means?
If (1,2) = -1, then (1,2) = ±(W11 – W22)
It is possible to choose W1 and W2 such that
(1,2) = 0.
6-10
 and Diversification in a 2 Stock Portfolio
• What does -1 < (1,2) < 1 imply?
– If -1 < (1,2) < 1 then
There are some diversification benefits from
combining stocks 1 and 2 into a portfolio.
p2 = W1212 + W2222 + 2W1W2 Cov(r1r2)
And since Cov(r1r2) = 1,212
p2 = W1212 + W2222 + 2W1W2 1,212
6-11
 and Diversification in a 2 Stock Portfolio
•
Typically  is greater than ____________________
zero and less than 1.0
(1,2) = (2,1) and the same is true for the COV
•
The covariance between any stock such as Stock 1 and
itself is simply the variance of Stock 1,
•
(1,1) = +1.0 by definition
•
We have no measure for how three or more stocks
move together.
6-12
The Effects of Correlation &
Covariance on Diversification
Asset A
Asset B
Portfolio AB
6-13
The Effects of Correlation &
Covariance on Diversification
Asset C
Asset C
Portfolio CD
6-14
Naïve Diversification
The power of diversification
Most of the diversifiable risk
eliminated at 25 or so stocks
6-15
Two-Security Portfolio: Risk
p2 = W1212 + W2222 + 2W1W2 Cov(r1r2)
12 = Variance of Security 1
22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
6-16
1
2
3
4
5
6
7
8
9
10
AAR
Returns
ABC
XYZ
0.2515
-0.2255
0.4322
0.3144
-0.2845
-0.0645
-0.1433
-0.5114
0.5534
0.3378
0.6843
0.3295
-0.1514
0.7019
0.2533
0.2763
-0.4432
-0.4879
-0.2245
0.5263
0.09278
0.11969
Squared deviations
from average
ABC
XYZ
0.025192 0.119156
0.115206 0.037912
0.14234 0.033926
0.055734 0.398275
0.212171 0.047572
0.349896 0.04402
0.059624 0.338968
0.025767 0.024527
0.287275 0.369166
0.100667 0.165332
Sum 1.37387 1.578853
Average 0.137387 0.157885
Calculating
Variance and
Covariance
Ex post
2ABC = 1.37387 / (10-1) = 0.15265
ABC = 39.07%
2XYZ = 1.57885 / (10-1) = 0.17543
XYZ = 41.88%
6-17
1
2
3
4
5
6
7
8
9
10
AAR
Returns
ABC
XYZ
0.2515
-0.2255
0.4322
0.3144
-0.2845
-0.0645
-0.1433
-0.5114
0.5534
0.3378
0.6843
0.3295
-0.1514
0.7019
0.2533
0.2763
-0.4432
-0.4879
-0.2245
0.5263
0.09278
0.11969
Deviation from
average
ABC
XYZ
0.15872 -0.34519
0.33942 0.19471
-0.37728 -0.18419
-0.23608 -0.63109
0.46062 0.21811
0.59152 0.20981
-0.24418 0.58221
0.16052 0.15661
-0.53598 -0.60759
-0.31728 0.40661
Product
of
deviations
-0.05479
0.066088
0.069491
0.148988
0.100466
0.124107
-0.14216
0.025139
0.325656
-0.12901
Sum 0.533973
Average 0.053397
COV(ABC,XYZ) = 0.533973 / (10-1) = 0.059330
ABC,XYZ = COV / (ABCXYZ) = 0.059330 / (0.3907 x 0.4188)
ABC,XYZ = 0.3626
ABC = 39.07%
N (r
 r 1 )  (r 2,T  r 2 )
n
1,T
Cov(r 1, r2 ) 
n  1 T 1
n

XYZ = 41.88%
6-18
Ex ante Covariance Calculation
• Using scenario analysis with probabilities
the covariance can be calculated with the
following formula:
S
Cov(rS , rB )   p (i )  rS (i )  rS   rB (i )  rB 
i 1
6-19
Two-Security Portfolio Risk
2ABC = 0.15265
σp
2

Q Q
  [WI WJ Cov(I, J)]
I 1J 1
2XYZ = 0.17543
COV(ABC,XYZ) = 0.05933
ABC,XYZ = 0.3626
p2 = W1212 + 2W1W2 Cov(r1r2) + W2222
Let W1 = 60% and W2 = 40% Stock 1 = ABC; Stock 2 = XYZ
p2 = 0.36(0.15265) + 2(.6)(.4)(0.05933) + 0.16(0.17543)
p2 = 0.1115019 = variance of the portfolio
p = 33.39%
p < W11 + W22
33.39% < [0.60(0.3907) + 0.40(0.4188)] = 40.20%
ABC = 39.07%
XYZ = 41.88%
6-20
Three-Security Portfolio n or Q = 3
σp
2

Q Q
  [WI WJ Cov(r I , rJ )]
I 1J 1
2p = W1212 + W2222 + W3232
For an n security
portfolio there would be
_
n variances and
n(n-1) covariance terms.
_____
The ___________
covariances are the
dominant effect on
 2p
+ 2W1W2 Cov(r1r2)
+ 2W1W3 Cov(r1r3)
+ 2W2W3 Cov(r2r3)
6-21
E(r)
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
WA = 0%
13%
WB = 100%
 = -1
=0
8%
WA = 100%
50%A
 = .3
50%B
 = +1
WB = 0%
12%
Stock A
20%
Stock B
St. Dev
6-22
Summary: Portfolio Risk/Return
Two Security Portfolio
• Amount of risk reduction depends critically
on _________________________.
correlations or covariances
<1
• Adding securities with correlations _____
will result in risk reduction.
• If risk is reduced by more than expected
return, what happens to the return per unit
of risk (the Sharpe ratio)?
6-23
Minimum Variance
Combinations -1<  < +1
Choosing weights to minimize the portfolio variance
 2 - Cov(r1r2)
2
W1 =
 12 +  22 - 2Cov(r1r2)
W2 = (1 - W1)
E(r)
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
13%
 = -1
=0
 = .3
=1
8%
12%
20%
St. Dev
6-24
Minimum Variance
Combinations -1<  < +1
Stk 1 E(r1) = .10
Stk 2 E(r2) = .14
 22 - Cov(r1r2)
 1 = .15
12 = .2
 2 = .20
(.2)2 -2(.2)(.15)(.2)
(.2) - (.2)(.15)(.2)
W1 = =
W1 (.15)2 + (.2)2 - 2(.2)(.15)(.2)
W1 =
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
 12 +  22 - 2Cov(r1r2)
W2 = (1 - W1)
Cov(r1r2) = 1,212
W1 = .6733
W1 = .6733
W2 = (1 - .6733) = .3267
W2 = (1 - .6733) = .3267
6-25
Minimum Variance: Return and
Risk with  = .2
 = .15
12 = .2
 = .20
(.2)2 - (.2)(.15)(.2)
1
E(r
)
=
.10
=
1
WStk
1
1
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
Stk 2 E(r2) = .14
2
W1 = .6733
W2 = (1 - .6733) = .3267
E[rp] = .6733(.10) + .3267(.14) = .1131 or 11.31%
p2 = W1212 + W2222 + 2W1W2 1,212
σ p  (0.6733 2 ) (0.15 2 )  (0.3267 2 ) (0.2 2 )  2 (0.6733) (0.3267) (0.2) (0.15) (0.2) 


1/2
 p  0.01711 / 2  13.08%
6-26
Minimum Variance Combination
with  = -.3
Stk 1 E(r1) = .10
Stk 2 E(r2) = .14
 2 - Cov(r1r2)
2
W1 =
1 = .15
12 = .2-.3
 2 = .20
2 - (.2)(.15)(-.3)
(.2)
(.2)2 - (-.3)(.15)(.2)
W1W=1 =
2
2
(.15)
-.3)
 12 +  22 - 2Cov(r1r2)
(.15)2++(.2)
(.2)2-- 2(.2)(.15)(
2(-.3)(.15)(.2)
W2 = (1 - W1)
W1 = .6087
Cov(r1r2) = 1,212
W2 = (1 - .6087) = .3913
6-27
Minimum Variance
Combination with  = -.3
= .15

W
1
12 = .2-.3
(.15) + (.2) - 2(.2)(.15)(-.3)
Stk 2 E(r2) = .14  2 = .20
2 - (.2)(.15)(-.3)
(.2)E(r
Stk
1
1) = .10
=
1
2
2
W1 = .6087
W2 = (1 - .6087) = .3913
E[rp] = 0.6087(.10) + 0.3913(.14) = .1157 = 11.57%
p2 = W1212 + W2222 + 2W1W2 1,212
σ p  (0.6087 2 ) (0.15 2 )  (0.3913 2 ) (0.2 2 )  2 (0.6087) (0.3913) (-0.3) (0.15) (0.2) 


 p  0.01021 / 2  10.09%
Notice lower portfolio
standard deviation but
higher expected return
with smaller 
1/2
12 = .2
E(rp) = 11.31%
p = 13.08%
6-28
Extending Concepts to All
Securities
•
•
•
Consider all possible combinations of securities,
with all possible different weightings and keep
track of combinations that provide more return
for less risk or the least risk for a given level of
return and graph the result.
The set of portfolios that provide the optimal
trade-offs are described as the efficient frontier.
The efficient frontier portfolios are dominant or
the best diversified possible combinations.
All investors should want a portfolio on the
efficient frontier. … Until we add the
riskless asset
6-29
E(r)
The minimum-variance frontier of
risky assets
Efficient
frontier
Global
minimum
variance
portfolio
Efficient Frontier is the best
diversified set of investments
with the highest returns
Found by forming
portfolios of securities
with the
lowest
Individual
covariances
assetsat a given
E(r) level.
Minimum
variance
frontier
St. Dev.
6-30
E(r)
The EF and asset allocation
EF including
international &
alternative
investments
80% Stocks
20% Bonds
60% Stocks
40% Bonds
40% Stocks
60% Bonds
100% Stocks
Efficient
frontier
20% Stocks
80% Bonds
100% Stocks
Ex-Post 20002002
St. Dev.
6-31
Efficient frontier for international
diversification Text Table 6.1
6-32
Efficient frontier for international
diversification Text Figure 6.11
6-33
6.3 The Optimal Risky
Portfolio With A RiskFree Asset
6.4 Efficient Diversification
With Many Risky Assets
6-34
Including Riskless Investments
•
•
The optimal combination becomes linear
A single combination of risky and riskless
assets will dominate
6-35
ALTERNATIVE CALS
CAL (P)
E(r)
Efficient
Frontier
E(rP&F)
P
E(rP)
E(rA)
CAL (A)
CAL (Global
minimum variance)
A
G
F
Risk Free
A P P&F

6-36
The Capital Market Line or CML
CAL (P) = CML
E(r)
Efficient
Frontier
E(rP&F)
P
E(rP)
o The optimal CAL is
called the Capital
Market Line or CML
o The CML dominates
the EF
E(rP&F)
F
Risk Free
P&F
P P&F

6-37
Dominant CAL with a RiskFree Investment (F)
•
•
CAL(P) = Capital Market Line or CML dominates
other lines because it has the the largest slope
Slope = (E(rp) - rf) / p
(CML maximizes the slope or the return per unit of risk
or it equivalently maximizes the Sharpe ratio)
Regardless of risk preferences some
combinations of P & F dominate
•
6-38
The Capital Market Line or CML
A=2
E(r)
Efficient
Frontier
E(rP&F)
P
E(rP)
E(rP&F)
CML
A=4
Both investors
choose the same well
diversified risky
portfolio P and the
risk free asset F, but
they choose different
proportions of each.
F
Risk Free
P&F
P P&F

6-39
Practical Implications
o The analyst or planner should identify what they
believe will be the best performing well
diversified portfolio, call it P.
P may include funds, stocks, bonds, international and
other alternative investments.
o This portfolio will serve as the starting point for all
their clients.
o The planner will then change the asset allocation
between the risky portfolio and “near cash”
investments according to risk tolerance of client.
o The risky portfolio P may have to be adjusted for
individual clients for tax and liquidity concerns if
relevant and for the client’s opinions.
6-40
6.5 A Single Index Asset
Market
6-41
Individual Securities
• We have learned that investors should diversify.
• Individual securities will be held in a portfolio.
Consequently, the relevant risk of an individual
security is the risk that remains when the security
is placed in a portfolio.
• What do we call the risk that cannot be diversified
away, i.e., the risk that remains when the stock is
put into a portfolio? Systematic risk
• How do we measure a stock’s systematic risk?
6-42
Systematic risk
• Systematic risk arises from events that effect the
entire economy such as a change in interest
rates or GDP or a financial crisis such as
occurred in 2007and 2008.
• If a well diversified portfolio has no unsystematic
risk then any risk that remains must be
systematic.
• That is, the variation in returns of a well
diversified portfolio must be due to changes in
systematic factors.
6-43
Individual Securities
How do we measure a stock’s systematic
risk?
Systematic Factors

Returns
Stock A
Returns
well
diversified
portfolio
Δ interest rates,
Δ GDP,
Δ consumer spending,
etc.
6-44
Single Factor Model
Ri = E(Ri) + ßiM + ei
Ri = Actual excess return = ri – rf
E(Ri) = expected excess return
Two sources of Uncertainty
M = some systematic factor or proxy; in this case
M is unanticipated movement in a well
diversified broad market index like the
S&P500
ßi
= sensitivity of a securities’ particular return to
the factor
ei = unanticipated firm specific events
6-45
Single Index Model
Parameter Estimation
r  r     r  r  e
i
f
i
i
m
f
i
Risk Prem
Market Risk Prem
or Index Risk Prem
αi = the stock’s expected excess return if the
market’s excess return is zero, i.e., (rm - rf) = 0
ßi(rm - rf) = the component of excess return due to
movements in the market index
ei = firm specific component of excess return that is not
due to market movements
6-46
Risk Premium Format
Let: Ri = (ri - rf)
Rm = (rm - rf)
Risk premium
format
The Model:
Ri = i + ßi(Rm) + ei
6-47
Estimating the Index Model
Scatter
Plot
Excess Returns (i)
. ..
. ..
.
.
.
.
.
. . ..
.. . .
Security
.
.
.
.
Characteristic
.
.
.
. .
Line
.
. .. . .
.
. . . Excess returns
. . . . on market index
.
.
.
.
.
.
. R =.  + ß R + e
i
i
i
m
i
Slope of SCL = beta
y-intercept = alpha
6-48
Estimating the Index Model
Scatter
Plot
Excess Returns (i)
Ri =  i + ßiRm + ei
. ..
. ..
.
.
.
.
.
. . ..
.. . .
Security
.
.
.
Characteristic
.
.
.
.
. .
Line
.
. .. . .
.
. . . Excess returns
on market index
.Variation
.
. in. R .explained
by the line is
.
.
.
systematic
risk
the. stock’s
_____________
.
.Variation.in R unrelated to the market
i
i
(the line) is unsystematic
________________
risk
6-49
Components of Risk
ßiM + ei
• Market or systematic risk:
risk related to the systematic or macro economic factor
in this case the market index
• Unsystematic or firm specific risk:
risk not related to the macro factor or market index
• Total risk =
Systematic + Unsystematic
i2 = Systematic risk + Unsystematic Risk
6-50
Comparing Security
Characteristic Lines
Describe
• 
• 
• e
for each.
6-51
Measuring Components of Risk
i2 =
i2 m2 + 2(ei)
where;
i2 = total variance
i2 m2 = systematic variance
2(ei) = unsystematic variance
6-52
Examining Percentage of
Variance
Total Risk = Systematic Risk + Unsystematic Risk
2

Systematic Risk / Total Risk =
ßi2  m2 / i2 = 2
i2 m2 / (i2 m2 + 2(ei)) = 2
6-53
Advantages of the Single
Index Model
•
Reduces the number of inputs needed to
account for diversification benefits
If you want to know the risk of a 25 stock
portfolio you would have to calculate 25
variances and (25x24) = 600 covariance terms
With the index model you need only 25 betas
•
Easy reference point for understanding stock risk.
βM = 1, so if βi > 1 what do we know?
If βi < 1?
6-54
Sharpe Ratios and Alphas
•
When ranking portfolios and security performance
we must consider both return & risk
“Well performing” diversified portfolios provide
high Sharpe ratios:
Sharpe = (rp – rf) / p
•
–
•
You can also use the Sharpe ratio to evaluate an
individual stock if the investor does not diversify
6-55
Sharpe Ratios and Alphas
•
“Well performing” individual stocks held in
diversified portfolios can be evaluated by the
stock’s alpha in relation to the stock’s
unsystematic risk.
Skip TreynorBlack Model
6-56
The Treynor-Black Model
• Suppose an investor holds a passive portfolio M but
believes that an individual security has a positive alpha.
– A positive alpha implies the security is undervalued.
Suppose it is Google.
• Adding Google moves the overall portfolio away from the
diversified optimum but it might be worth it to earn the
positive alpha.
• What is the optimal portfolio including Google?
• What is the resulting improvement in the Sharpe ratio?
6-57
The Treynor-Black Model
• Weight of Google in the optimal portfolio O:
WGO
*
*
W 
;
W

1

W
M
G
1  WGO (1  βG )
*
G
G  Google, M  Passive
• The improvement in the Sharpe ratio (S) over the Sharpe
of the passive portfolio M can be found as:
 α 
SO2  SM2   G 
 σ(e G ) 
2
 αG 

;
 σ(e G ) 
• Notice that the improvement in the Sharpe ratio is a
function of
This ratio is called the “information ratio”
6-58
The Treynor-Black Model
• For multiple stocks in the active portfolio:
i
1
2
n



...
i 2 (e ) 2 (e ) 2 (e ) 2 (e )
i
1
2
n
n
• The optimal weight of each security in the
active portfolio is found as:
i
 2 ( ei )
Wi 
i
i 2 (ei )
*
• A larger alpha increases the weight of
stock i and larger residual variance
reduces the weight of stock i.
6-59
The Treynor-Black Model
• If A stands for the “active portfolio,” the
active portfolio’s alpha, beta and residual
risk can be found from:
n
n
α A   WiA αiA ;  A   WiA iA
i
i
n
&
2
 (e A )   WiA2 iA
2
i
6-60
6-61
Treynor-Black Allocation
CAL
CML
E(r)
P
A
M
Rf

6-62
6.6 Risk of Long-Term
Investments
6-63
Are Stock Returns Less Risky in
the Long Run?
• Consider the variance of a 2-year investment
with serially independent returns r1 and r2:
Var (2-year total return) = Var ( r1  r2
)
 Var ( r1 )  Var (r2 )  2Cov (r1 , r2 )
  2  2  0
 2 2 and standard deviation of the return is 
2
• The variance of the 2-year return is double that
of the one-year return and σ is higher by a
multiple of the square root of 2
6-64
Are Stock Returns Less Risky in
the Long Run?
• Generalizing to an investment horizon
of n years and then annualizing:
Var(n year total return)  nσ 2
Standard deviation( n year total return)  σ n
One can show that for a portfolio of uncorrelat ed stocks with identical 
σp 
σ
n
• For a portfolio:
6-65
The Fly in the ‘Time
Diversification’ Ointment
• The annualized standard deviation is only
appropriate for short-term portfolios
• The variance grows linearly with the
n
number of years
• Standard deviation grows in proportion to
6-66
The Fly in the ‘Time
Diversification’ Ointment
• To compare investments in two different
time periods:
– Examine risk of the total rate of return rather
than average sub-period returns
– Must account for both magnitudes of total
returns and probabilities of such returns
occurring
6-67