Chapter 11
Diversification and Risky
Asset Allocation
11-1
Learning Objectives
1. How to calculate expected returns and
variances for a security.
2. How to calculate expected returns and
variances for a portfolio.
3. The importance of portfolio
diversification.
4. The efficient frontier and the
importance of asset allocation.
11-2
Expected Return of an Asset
Expected return = the “weighted average”
return on a risky asset expected in the future.
s
E(R )   Pri Ri
i 1
Prs = probability of a state
Rs = return if a state occurs
s = number of states
11-3
Expected Return & Risk Premium
State
S
1
2
3
4
5
Prob of
Return
State
in State
Pr
R
0.10
-5%
0.20
5%
0.40
15%
0.20
25%
0.10
35%
Expected Return
Pr * R
-0.5%
1.0%
6.0%
5.0%
3.5%
15.0%
Risk-free rate = 8%
Risk Premium = 15% - 8% = 7%
11-4
Measuring Variance and
Standard Deviation

s
   PrS  R s  E R 
2
i 1
 

2
2
11-5
Calculating Dispersion of Returns
State
S
1
2
3
4
5
Prob of
Return
State
in State
Pr
R
0.10
-5%
0.20
5%
0.40
15%
0.20
25%
0.10
35%
Expected Return
Pr * R
-0.5%
1.0%
6.0%
5.0%
3.5%
15.0%
Deviation Deviation
Deviation Squared Sq * Pr
R - E(R)
-20%
0.040
0.004
-10%
0.010
0.002
0%
0.000
0.000
10%
0.010
0.002
20%
0.040
0.004
Variance
Standard Deviation
0.012
10.95%
11-6
Calculating Dispersion of Returns
State
S
1
2
3
4
5
C=
1 =
D=
E=
A
B
Prob of
Return
State
in State
Pr
R
0.10
-5%
0.20
5%
0.40
15%
0.20
25%
0.10
35%
Expected Return
AXB
Sum of C
B - 1
= Deviation
D squared
C
Pr * R
-0.5%
1.0%
6.0%
5.0%
3.5%
15.0%
1
D
E
F
Deviation Deviation
Deviation Squared Sq * Pr
R - E(R)
-20%
0.040
0.004
-10%
0.010
0.002
0%
0.000
0.000
10%
0.010
0.002
20%
0.040
0.004
Variance
Standard Deviation
F=
2 =
3 =
0.012
10.95%
2
3
ExA
Sum of F
Square root of 2
11-7
Portfolios
• Portfolios = groups of assets, such as stocks and
bonds, that are held by an investor.
• Portfolio Description = list the proportion of the
total value of the portfolio that is invested into each
asset.
• Portfolio Weights = proportions
• Sometimes expressed in percentages.
• In calculations, make sure you use proportions
(i.e., decimals).
11-8
Portfolio Return
The rate of return on a portfolio is a weighted average
of the rates of return of each asset comprising the
portfolio, with the portfolio proportions as weights.
E(R P )  x 1  E(R 1 )  x 2  E(R 2 )  ...  x n  E(R n )
n
E(R P )   x i  E(R i )
i 1
xi = Proportion of funds in Security i
E(Ri) = Expected return on Security i
11-9
Portfolio Return
Prob of
State
State
S
Pr
1
0.10
2
0.20
3
0.40
4
0.20
5
0.10
Expected Return
Asset A
60%
Asset B
40%
Return
in State
R (A)
-5%
5%
15%
25%
35%
15%
Portfolio
Return Return in
in State
State
R (B)
25%
7.0%
15%
9.0%
10%
13.0%
5%
17.0%
-10%
17.0%
9.5%
Portfolio
0.7%
1.8%
5.2%
3.4%
1.7%
12.8%
7.0% = .60(-5%) + .40(25%)
11-10
Portfolio Return
Alternate Method – Step 1
Prob of
State
State
S
Pr
1
0.10
2
0.20
3
0.40
4
0.20
5
0.10
Expected Return
Asset A
Return
in State
R (A)
Pr * S
-5%
-0.5%
5%
1.0%
15%
6.0%
25%
5.0%
35%
3.5%
15.0%
Asset B
Return
in State
R (B)
Pr * S
25%
2.5%
15%
3.0%
10%
4.0%
5%
1.0%
-10%
-1.0%
9.5%
11-11
Portfolio Return
Alternate Method – Step 2
Assume 60% in Stock A; 40% in Stock B
Stocks
E(R)
X (wgt)
E(R)*X
A
15.0%
60%
9.00%
B
9.5%
40%
3.80%
12.80%
11-12
Portfolio Variance & Standard
Deviation

s
 
 P   PrS  R P,S  E R P
2
i 1
P  P
2
2
11-13
Portfolio Variance & Standard
Deviation
Portfolio
Portfolio
Return in Portfolio
Deviation
State
S
Pr
1
0.10
2
0.20
3
0.40
4
0.20
5
0.10
Expected Return
7.0%
9.0%
13.0%
17.0%
17.0%
12.8%
Variance =
Standard Deviation =
-5.8%
-3.8%
0.2%
4.2%
4.2%
Portfoiio
Deviation
Squared
0.003364
0.001444
4.00E-06
0.001764
0.001764
Deviation Sq
x Pr
0.0003364
0.0002888
0.0000016
0.0003528
0.0001764
0.0011560
0.0011560
3.4%
11-14
Risk-Return Comparison
E(R)
Variance
Std Dev
Stock Alone
Stock A
Stock B
15%
9.50%
0.012
0.007
10.95%
8.50%
Portfolio
60A-40B
12.80%
0.001156
3.40%
11-15
Stocks A & B vs. States
Returns by State
40%
A
Expected Return
30%
B
A
20%
B
A
B
10%
A
B
0%
A
-10%
-20%
B
1
2
3
4
5
A
-5%
5%
15%
25%
35%
B
25%
15%
10%
5%
-10%
State
11-16
Portfolio Returns: 60% A – 40% B
Returns by State
40%
A
Expected Return
30%
B
A
20%
B
A
B
10%
A
B
0%
A
-10%
-20%
B
1
2
3
4
5
A
-5%
5%
15%
25%
35%
B
25%
15%
10%
5%
-10%
P
7.00%
9.00%
13.00%
17.00%
17.00%
State
11-17
Diversification and Risk
11-18
Diversification and Risk
11-19
Why Diversification Works
• Correlation = The tendency of the returns on
two assets to move together.
• Positively correlated assets tend to move up
and down together.
• Negatively correlated assets tend to move in
opposite directions.
• Imperfect correlation, positive or negative, is
why diversification reduces portfolio risk.
11-20
Correlation Coefficient
•Correlation Coefficient = ρ (rho)
•Scales covariance to [-1,+1]
 r = -1.0  Two stocks can be combined to
form a riskless portfolio
 r = +1.0  No risk reduction at all
• In general, stocks have r ≈ 0.35 - 0.67
• Risk is lowered but not eliminated
r ab
 ab

 CORR(R A ,R B )
 a b
Will Not Be on a Test
11-21
Returns distribution for two perfectly
negatively correlated stocks (ρ = -1.0)
Stock W
Stock M
Portfolio WM
25
25
25
15
15
15
0
0
0
-10
-10
-10
11-22
Returns distribution for two perfectly positively
correlated stocks (ρ = 1.0)
Stock M’
Stock M
Portfolio MM’
25
25
25
15
15
15
0
0
0
-10
-10
-10
11-23
Why Diversification Works
ρ = +1
ρ = -1
ρ=0
11-24
Why Diversification Works
11-25
Why Diversification Works
11-26
Covariance of Returns
• Measures how much the returns on two
risky assets move together
Cov ( a , b )   ab
 ab   ra  rˆa rb  rˆb Pi
i
i
i
Will Not Be on a Test
11-27
Covariance vs. Variance of Returns
Cov(a , b)   ab
 ab   ra  rˆa rb  rˆb Pi
i
i
i
Var(a )   aa  
2
a
   ra  rˆa ra  rˆa Pi
2
a
i
i
i
Will Not Be on a Test
11-28
Covariance
Cov (a , b)   ab
 ab   ra  rˆa rb  rˆb Pi
i
i
i
Economy
Recession
Below Avg
Average
Above Avg
Boom
Prob
0.1
0.2
0.4
0.2
0.1
E(R )
A
-5%
5%
15%
25%
35%
15.0%
B
25%
15%
10%
5%
-10%
9.5%
A Dev
-20.0%
-10.0%
0.0%
10.0%
20.0%
B Dev
AxB
15.5%
-3.100%
5.5%
-0.550%
0.5%
0.000%
-4.5%
-0.450%
-19.5%
-3.900%
COV(A,B) =
x Prob
-0.0031
-0.0011
0.0000
-0.0009
-0.0039
-0.0090
Deviation = RA,S – E(RA)
Covariance (A:B) = -0.009
Will Not Be on a Test
11-29
Covariance
Cov (a , b)   ab
 ab   ra  rˆa rb  rˆb Pi
i
i
i
Economy
Recession
Below Avg
Average
Above Avg
Boom
1 =
2 =
D=
E=
A
B
C
Prob
0.1
0.2
0.4
0.2
0.1
E(R )
A
-5%
5%
15%
25%
35%
15.0%
1
B
25%
15%
10%
5%
-10%
9.5%
2
Sum of A x B
Sum of A x C
B- 1
C- 2
F=
G=
3 =
D
E
F
Deviations
A Dev x
A Dev
B Dev B Dev
-20.0%
15.5% -3.100%
-10.0%
5.5% -0.550%
0.0%
0.5%
0.000%
10.0%
-4.5% -0.450%
20.0%
-19.5% -3.900%
COV(A,B) =
G
x Prob
-0.0031
-0.0011
0.0000
-0.0009
-0.0039
-0.0090
3
DxE
FxA
Sum of G
Will Not Be on a Test
11-30
Correlation Coefficient
 ab
r ab 
 a b
Std Dev
Cov
Correlation
Coefficient
A
B
10.95% 8.50%
-0.0090
-0.967
Will Not Be on a Test
11-31
Calculating Portfolio Risk
• For a portfolio of two assets, A and B, the
variance of the return on the portfolio is:
σ p2  x A2 σ A2  x B2 σ B2  2x A x BCOV(A,B)
σ p2  x A2 σ A2  x B2 σ B2  2x A x B σ Aσ BCORR(RA RB )
Where: xA = portfolio weight of asset A
xB = portfolio weight of asset B
such that xA + xB = 1
11-32
Portfolio Risk Example
• Continuing our 2-stock example
E(R)
Variance
Std Dev
Pf Weight
Correlation
R
σ2
σ
X
Corr(RA,RB)
Stock Alone
Stock A
Stock B
15%
9.50%
Portfolio
60A-40B
12.80%
0.012
10.95%
60%
0.001156
3.40%
0.007
8.50%
40%
-0.97
 P2  x A2 A2  xB2 B2  2x A xB A BCorr(RA , RB )
11-33
 of n-Stock Portfolio
n
n
n
 p2   w i2 i2    w i w j i j r ij
i
i 1 j 1
i j
n
n
 ab
r ab 
 a b
n
 p2   w i2 i2    w i w j ij
i
i 1 j 1
i j




Subscripts denote stocks i and j
ri,j = Correlation between stocks i and j
σi and σj =Standard deviations of stocks i and j
σij = Covariance of stocks i and j
Will Not Be on a Test
11-34
Portfolio Risk-2 Risky Assets
n
n
n
 p2   w i2 i2    w i w j i j r ij
i 1 j 1
i
i j
W
60%
40%
A
B
i
1
2
1
2
j
1
2
2
1
Std Dev Variance
Cov
10.95%
0.0120
-0.0090
8.50%
0.0072
ρ
-0.97
for n=2
0.0043
0.0012
(0.0022)
(0.0022)
0.0012 Variance
3.40%
Std Dev
11-35
Diversification & the Minimum
Variance Portfolio
Assume the following statistics for two
portfolios, one of stocks and one of bonds:
Table 11.9
E(R)
Std Dev
Corr Coeff
Stocks
Stocks
Bonds
12%
6%
15%
10%
0.10
11-36
Correlation and Diversification
11-37
Correlation and Diversification
11-38
Portfolio Results
E(R)
Std Dev
12.00%
15.00%
11.70%
14.31%
11.40%
13.64%
11.10%
12.99%
10.80%
12.36%
10.50%
11.77%
10.20%
11.20%
9.90%
10.68%
9.60%
10.21%
9.30%
9.78%
9.00%
9.42%
8.70%
9.12%
8.40%
8.90%
8.10%
8.75%
7.80%
8.69%
7.50%
8.71%
7.20%
8.82%
6.90%
9.01%
6.60%
9.27%
6.30%
9.60%
6.00%
10.00%
Table 11.9
Stocks
Stocks
Bonds
E(R)
12%
6%
Std Dev
15%
10%
Corr Coeff
0.10
Risk-Return with Stocks & Bonds
Figure 11-4
100% Stocks
13%
12%
Expected Return
Portfolio Weights
Stocks
Bonds
1.00
0.00
0.95
0.05
0.90
0.10
0.85
0.15
0.80
0.20
0.75
0.25
0.70
0.30
0.65
0.35
0.60
0.40
0.55
0.45
0.50
0.50
0.45
0.55
0.40
0.60
0.35
0.65
0.30
0.70
0.25
0.75
0.20
0.80
0.15
0.85
0.10
0.90
0.05
0.95
0.00
1.00
The Minimum Variance
Portfolio
11%
10%
MVP
9%
8%
7%
100% Bonds
6%
5%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
Standard Deviation
11-39
The Minimum Variance Portfolio
Table 11.9
E(R)
Std Dev
Corr Coeff
Stocks
Stocks
Bonds
12%
6%
15%
10%
0.10
2

0.0085
*
B   A BCorr(RA , RB )
xA  2

 28.8136%
2
 A  B2 A BCorr(RA , RB ) 0.0295
XA* = 28.8136%
Putting 28.8136% in stocks and 71.1864% in bonds
yields an E(R) = 7.73% and a standard deviation of
8.69% as the minimum variance portfolio. (Ex 11.7 p.366)
11-40
Correlation and Diversification
• The various combinations of risk and return
available all fall on a smooth curve.
• This curve is called an investment opportunity set
,because it shows the possible combinations of risk
and return available from portfolios of these two
assets.
• A portfolio that offers the highest return for its level
of risk is said to be an efficient portfolio.
• The undesirable portfolios are said to be dominated
or inefficient.
11-41
The Markowitz Efficient Frontier
• The Markowitz Efficient frontier = the set of portfolios
with the maximum return for a given risk AND the
minimum risk given a return.
• For the plot, the upper left-hand boundary is the
Markowitz efficient frontier.
• All the other possible combinations are inefficient.
That is, investors would not hold these portfolios
because they could get either
• More return for a given level of risk, or
• Less risk for a given level of return.
11-42
Markowitz Efficient Frontier
Risk-Return with Stocks & Bonds
Figure 11-4
13%
Efficient Frontier
Expected Return
12%
11%
10%
9%
8%
7%
6%
Inefficient Frontier
5%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
Standard Deviation
11-43
Efficient Frontier
Markowitz Efficient Frontier
13%
Expected Return
12%
11%
10%
9%
8%
7%
6%
5%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
Standard Deviation
11-44
The Importance of Asset Allocation
• Suppose we invest in three mutual funds:
•
•
•
One that contains Foreign Stocks, F
One that contains U.S. Stocks, S
One that contains U.S. Bonds, B
Expected Return
Standard Deviation
Foreign Stocks, F
18%
35%
U.S. Stocks, S
12
22
U.S. Bonds, B
8
14
• Figure 11.6 shows the results of calculating various expected
returns and portfolio standard deviations with these three
assets.
11-45
Risk and Return with Multiple Assets
11-46
Useful Internet Sites
•
•
•
•
•
•
www.411stocks.com (to find expected earnings)
www.investopedia.com (for more on risk measures)
www.teachmefinance.com (also contains more on risk
measure)
www.morningstar.com (measure diversification using “instant xray”)
www.moneychimp.com (review modern portfolio theory)
www.efficentfrontier.com (check out the reading list)
11-47