Chapter 6
Efficient
Diversification
McGraw-Hill/Irwin
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 6
Efficient
Diversification
6-2
6.1 Diversification and Portfolio Risk
6.2 Asset Allocation With Two Risky
Assets
6-3
Combinations of risky assets
1-market
risk,systematic risk or
nondiversifiable risk.
2-unique risk, firmspecific risk,
nonsystematic risk, or
diversifiable risk.
6-4
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.
6-5
Correlation
– Highly correlated investments - moves
proportionally in the same direction~ do not
diversify away risk(+1)~Poor
– Negatively correlated investments : moves in
opposition to each other ~provide high degree
of risk reduction(-1)~Excellent
– Uncorrelated investments - no predictions
about the movement ~provide some overall
reduction in risk(0)~Good
The effects of correlation &
covariance on diversification
Asset A
Asset B
Portfolio AB
6-7
The effects of correlation &
covariance on diversification
Asset C
Asset C
Portfolio CD
6-8
Naïve diversification
The power of diversification
Most of the diversifiable risk
eliminated at 25 or so stocks
6-9
Covariance calculations
• 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
• If when r1 > E[r1], r2 > E[r2], and when
r1 < E[r1], r2 < E[r2], then COV will be postive
• 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?
6-10
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
6-11
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
6-12
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%
6-13
Two-Security Portfolio: Risk
sp2 = W12s12 + W22s22 + 2W1W2 Cov(r1r2)
s12 = Variance of Security 1
s22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
6-14
Problem 3
a.Subscript OP refers to the original portfolio, ABC to the new stock,
and NP to the new portfolio.
i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9  0.67) + (0.1  1.25) = 0.728%
0.40  .0237  .0295 = .00027966  0.00028
ii Cov =   sOP  sABC =
iii. sNP = [wOP2 sOP2 + wABC2 sABC2 + 2 wOP wABC (CovOP , ABC)]1/2
= [(0.92  .02372) + (0.12  .02952) + (2  0.9  0.1  .00028)]1/2
= 2.2673%  2.27%
6-15
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
6-16
Summary: Portfolio Risk/Return
Two Security Portfolio
• Amount of risk reduction depends critically
or covariances
on correlations
_________________________.
<1
• Adding securities with correlations _____
will result in risk reduction.
• To sum up the idea: The less correlated
the greater the risk reduction possible
through diversification
6-17
Minimum Variance
Combinations -1<  < +1
Choosing weights to minimize the portfolio variance
s 2 - Cov(r1r2)
2
W1 =
s 12 + s 22 - 2Cov(r1r2)
W2 = (1 - W1)
E(r)
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
13%
 = -1
=0
 = .3
=1
8%
12%
20%
St. Dev
6-18
Minimum Variance
Combinations -1<  < +1
Stk 1 E(r1) = .10
Stk 2 E(r2) = .14
s 22 - Cov(r1r2)
s 1 = .15
12 = .2
s 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)
s 12 + s 22 - 2Cov(r1r2)
W2 = (1 - W1)
Cov(r1r2) = 1,2s1s2
W1 = .6733
W1 = .6733
W2 = (1 - .6733) = .3267
W2 = (1 - .6733) = .3267
6-19
Minimum Variance: Return and
Risk with  = .2
s = .15
12 = .2
s = .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%
sp2 = W12s12 + W22s22 + 2W1W2 1,2s1s2
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
s p  0.01711 / 2  13.08%
6-20
Minimum Variance Combination
with  = -.3
Stk 1 E(r1) = .10
Stk 2 E(r2) = .14
s 2 - Cov(r1r2)
2
W1 =
s1 = .15
12 = .2-.3
s 2 = .20
2 - (.2)(.15)(-.3)
(.2)
(.2)2 - (-.3)(.15)(.2)
W1W=1 =
2
2
(.15)
-.3)
s 12 + s 22 - 2Cov(r1r2)
(.15)2++(.2)
(.2)2-- 2(.2)(.15)(
2(-.3)(.15)(.2)
W2 = (1 - W1)
W1 = .6087
Cov(r1r2) = 1,2s1s2
W2 = (1 - .6087) = .3913
6-21
Minimum Variance Combination
with  = -.3
s = .15
12 = .2-.3
s = .20
(.2)2 - (.2)(.15)(-.3)
Stk 1 E(r1) = .10
W1 =
(.15)2 + (.2)2 - 2(.2)(.15)(-.3) 1
Stk 2 E(r2) = .14
2
W1 = .6087
W2 = (1 - .6087) = .3913
E[rp] = 0.6087(.10) + 0.3913(.14) = .1157 = 11.57%
sp2 = W12s12 + W22s22 + 2W1W2 1,2s1s2
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) 


s 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%
sp = 13.08%
6-22
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
6-23
E(r)
The minimum-variance frontier of
risky assets
Efficient Frontier is the best diversified set of
investments with the highest returns
Efficient
frontier
Found by forming
portfolios of securities
with the lowest
covariances at a given
E(r) level.
Individual
assets
Minimum
variance
frontier
St. Dev.
6-24
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.
6-25
6.3 The Optimal Risky Portfolio With A
Risk-Free Asset
6.4 Efficient Diversification With Many
Risky Assets
6-26
Including Riskless Investments
• The optimal combination becomes linear
• A single combination of risky and riskless
assets will dominate
6-27
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
sP&F
sP sP&F
s
6-28
Dominant CAL with a Risk-Free
Investment (F)
• CAL(P) = Capital Market Line or CML dominates
other lines because it has the the largest slope
• Slope = (E(rp) - rf) / sp
(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
6-29
6.5 A Single Index Asset Market
6-30
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?
6-31
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.
6-32
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
6-33
Risk Premium Format
Let: Ri = (ri - rf)
Rm = (rm - rf)
Risk premium
format
The Model:
Ri = i + ßi(Rm) + ei
6-34
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
6-35
Components of Risk
• Market or systematic risk:
ßiM + ei
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
si2 = Systematic risk + Unsystematic Risk
6-36
Comparing Security
Characteristic Lines
Describe
• 
• 
• se
for each.
6-37
Measuring Components of Risk
si2
=
i2 sm2 + s2(ei)
where;
si2
= total variance
i2 sm2 = systematic variance
s2(ei) = unsystematic variance
6-38
Examining Percentage of
Variance
Total Risk =
Systematic Risk + Unsystematic Risk
Systematic Risk / Total Risk
2
=
ßi2 s m2 / si2 = 2
i2 sm2 / (i2 sm2 + s2(ei)) = 2
6-39
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) / sp
• You can also use the Sharpe ratio to evaluate an
individual stock if the investor does not diversify
6-40
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.
6-41