Efficient Diversification I
Covariance and Portfolio Risk
Mean-variance Frontier
Efficient Portfolio Frontier
Some Empirical Evidence
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In 2000, 40% of stocks in Russell 3000
had returns of -20% or worse.
Meanwhile, less than 12% of U.S. stock
mutual funds had returns of -20% or
below.
Of the 2,397 U.S. stocks in existence
throughout 1990s, 22% had negative
returns.
In contrast, 0.4% of U.S. equity mutual
funds had negative returns.
Investments 9
2
Diversification and Portfolio Risk
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“Don’t put all your eggs in one basket”
Effect of portfolio diversification

Diversifiable risk, non-systematic risk,
firm-specific risk, idiosyncratic risk
Non-diversifiable risk, systematic risk,
market risk
5
10
15
20
# of securities in the portfolio
Investments 9
3
Covariance and Correlation
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Covariance and correlation
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Degree of co-movement of two stocks
Covariance: non-standardized measure
Cov[r1 , r2 ]  E[(r1  1 )(r2  2 )]  E[r1r2 ]  12
Correlation coefficient: standardized measure
Cov[r1 , r2 ]
12 
 Cov[r1 , r2 ]  12 1 2 and  1  12  1
 1 2
r2
r2
0<12 <1
Investments 9
r2
r1
r1
-1<12 <0
r1
12 =0
4
Covariance and Correlation
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Example: Two risky assets
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Calculating the covariance
Cov[r1 , r2 ]  E[(r1  1 )(r2  2 )]  E[r1r2 ]  12
E[r1r2 ]   p(s)r1 (s)r2 (s)
s
s
p
r1
r2
p*r1
boom 0.33 -7.00 17.00 -2.33
normal 0.33 12.00 7.00 4.00
bust
0.33 28.00 -3.00 9.33
11.00
p*r2 p*[r1-mu1]^2 p*[r2-mu2]^2 p*[r1*r2]-mu1*mu2
5.67
108.00
33.33
-65.33
2.33
0.33
0.00
2.33
-1.00
96.33
33.33
-53.67
7.00
204.67
66.67
-116.67
14.31
8.16
-1.00
Means
Investments 9
Std. Dev.
Cov.
Corr.
5
Diversification and Portfolio Risk
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A portfolio of two risky assets
portfolioreturn: rp  w1r1  w2r2 and w1  w2  1
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w1: % invested in bond
w2: % invested in stock
Expected return
 p  E[rp ]  w1E[r1 ]  w2 E[r2 ]  w11  w2 2
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Variance
 2p  Var[rp ]  E[(rp   p )(rp   p )]
 w12 12  w22 22  2w1w2 12 1 2
Investments 9
6
Diversification and Portfolio Risk
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Example: Portfolio of two risky securities
w in security 1, (1 – w) in security 2
1  0.10  1  0.15
12  0.2
2  0.14  2  0.20
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Expected return (Mean):
 p  0.10 w  0.14 (1  w)
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Variance
 2p  0.152 w2  0.202 (1 w)2  2  0.2  0.15 0.20 w(1  w)
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What happens when w changes?
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Expected return decreases with increasing w
How about variance ?..
Investments 9
7
Mean-Variance Frontier
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w from 0 to 1
GMVP:
Global Minimum Variance Portfolio
Expected return
0.150
w
1.00
0.80
0.60
0.40
0.20
0.00
1-w
0.00
0.20
0.40
0.60
0.80
1.00
E(rp)
0.100
0.108
0.116
0.124
0.132
0.140
Var(rp) std dev
0.02250
0.150
0.01792
0.134
0.01738
0.132
0.02088
0.144
0.02842
0.169
0.04000
0.200
Mean-variance frontier
0.140
0.130
0.120
Security 2
0.110
0.100
0.090
GMVP
0.080
0.100
Security 1
0.150
0.200
0.250
standard deviation
Investments 9
8
Mean-Variance Frontier
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Global Minimum Variance Port. (GMVP)
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A unique w
 22  12 1 2
.202  .2  .15  .20
w 2
 2
 .6733
2
2
 1   2  2 12 1 2 .15  .20  2  .20  .15  .20
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Associated characteristics
E[rGMVP ]  .6733  .10  .3267  .14  .1131
2
σ GMVP
 .67332  .152  .3267 2  .20 2
 2  .6733  .3267  .2  .15  .20
 .0171
σ GMVP  .0171  .1308
Investments 9
9
Efficient Portfolio Frontier
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67% in Security 1 and 33% in Security 2,
what’s so special?
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Efficient portfolio has < 67% in 1, and > 33% in 2
Expected return
0.150
w1=0
0.140
P
0.130
Efficient
Frontier
0.120
0.110
0.100
0.090
0.080
0.100
w1 =
.6733
GMVP
Inefficient
Frontier
w1=1
0.150
0.200
0.250
standard deviation
Investments 9
10
Efficient Portfolio Frontier
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Portfolio “P” dominates Security 1
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The same standard deviation
The higher expected return
How to find it?
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Since the portfolio has the same standard
deviation as Security 1
 2p  0.152 w2  0.202 (1  w) 2  2  0.2  0.15 0.20 w(1  w)
 0.152
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Solve the quadratic equation
w = 1 (Security 1) or w = .3465 (Portfolio P)
 p  0.10 0.3465 0.14 (1  0.3465)  0.1267 1  0.10
Investments 9
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Efficient Portfolio Frontier
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The effect of correlation
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Lower correlation means greater risk reduction
If  = +1.0, no risk reduction is possible
0.140
Expected Return
0.135
0.130
Rho=-1
0.125
Rho=-.5
0.120
Rho=0
0.115
Rho=.5
rho=1
0.110
0.105
0.100
0.00
0.10
0.20
0.30
Standard Deviation
Investments 9
12
Efficient Portfolio Frontier
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Efficient Portfolio of Many securities
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E[rp]: Weighted average of n securities
p2: Combination of all pair-wise covariance
measures
Construction of the efficient frontier is complicated
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Analytical solution without short-sale constraints
Numerical solution with short-sale constraints
General Features
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Optimal combination results in lowest risk for
given return
Efficient frontier describes optimal trade-off
Portfolios on efficient frontier are dominant
Investments 9
13
Efficient Frontier
E[r]
Efficient
frontier
Global
minimum
variance
portfolio
Individual
assets
Minimum
variance
frontier
St. Dev.
Investments 9
14
Wrap-up
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How to estimate portfolio return and
risk?
What is the mean-variance frontier?
What is the efficient portfolio frontier?
Why do portfolios on efficient frontier
dominate other combinations?
Investments 9
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