Chapter Seven
Optimal Risky Portfolios
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Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
Chapter Overview
• The investment decision:
• Capital allocation (risky vs. risk-free)
• Asset allocation (construction of the risky
portfolio)
• Security selection
• Optimal risky portfolio
• The Markowitz portfolio optimization model
• Long- vs. short-term investing
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The Investment Decision
•
Top-down process with 3 steps:
1. Capital allocation between the risky portfolio and
risk-free asset
2. Asset allocation across broad asset classes
3. Security selection of individual assets within each
asset class
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Diversification and Portfolio Risk
• Market risk
• Risk attributable to marketwide risk sources and
remains even after extensive diversification
• Also call systematic or nondiversifiable
• Firm-specific risk
• Risk that can be eliminated by diversification
• Also called diversifiable or nonsystematic
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Figure 7.1 Portfolio Risk and
the Number of Stocks in the Portfolio
Panel A: All risk is firm specific. Panel B: Some risk is systematic or marketwide.
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Figure 7.2 Portfolio Diversification
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Portfolios of Two Risky Assets
• Portfolio risk (variance) depends on the
correlation between the returns of the
assets in the portfolio
• Covariance and the correlation coefficient
provide a measure of the way returns of
two assets move together (covary)
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Portfolios of Two Risky Assets:
Return
• Portfolio return: rp = wDrD + wErE
•
•
•
•
wD = Bond weight
rD = Bond return
wE = Equity weight
rE = Equity return
E(rp) = wD E(rD) + wEE(rE)
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Portfolios of Two Risky Assets:
Risk
• Portfolio variance:
 p2  wD2  D2  wE2 E2  2wD wE Cov  rD , rE 
•
 D2 = Bond variance
•

2
E
= Equity variance
• Cov  rD , rE  = Covariance of returns for bond
and equity
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Portfolios of Two Risky Assets:
Covariance
• Covariance of returns on bond and equity:
Cov(rD,rE) = DEDE
• D,E = Correlation coefficient of returns
• D = Standard deviation of bond returns
• E = Standard deviation of equity returns
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Portfolios of Two Risky Assets:
Correlation Coefficients
• Range of values for 1,2
- 1.0 >  > +1.0
• If  = 1.0, the securities are perfectly positively
correlated
• If  = - 1.0, the securities are perfectly negatively
correlated
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Portfolios of Two Risky Assets:
Correlation Coefficients
• When ρDE = 1, there is no diversification
 P  wE E  wD D
• When ρDE = -1, a perfect hedge is possible
wE 
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D
 D  E
 1  wD
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Table 7.2 Computation of Portfolio Variance
From the Covariance Matrix
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Figure 7.3 Portfolio Expected Return
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Figure 7.4 Portfolio Standard Deviation
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Figure 7.5 Portfolio Expected Return as a
Function of Standard Deviation
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The Minimum Variance Portfolio
• The minimum variance portfolio is the portfolio
composed of the risky assets that has the smallest
standard deviation; the portfolio with least risk
• The amount of possible risk reduction through
diversification depends on the correlation:
• If  = +1.0, no risk reduction is possible
• If  = 0, σP may be less than the standard deviation of
either component asset
• If  = -1.0, a riskless hedge is possible
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Figure 7.6 The Opportunity Set of the Debt
and Equity Funds and Two Feasible CALs
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The Sharpe Ratio
• Maximize the slope of the CAL for any possible
portfolio, P
• The objective function is the slope:
Sp 
E rp   rf
p
• The slope is also the Sharpe ratio
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Figure 7.7 Debt and Equity Funds with the
Optimal Risky Portfolio
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Figure 7.8 Determination of the Optimal
Overall Portfolio
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Figure 7.9 The Proportions of the Optimal
Complete Portfolio
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Markowitz Portfolio Optimization
Model
• Security selection
• The first step is to determine the risk-return
opportunities available
• All portfolios that lie on the minimum-variance
frontier from the global minimum-variance
portfolio and upward provide the best risk-return
combinations
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Figure 7.10 The Minimum-Variance
Frontier of Risky Assets
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Markowitz Portfolio Optimization
Model
• Search for the CAL with the highest reward-tovariability ratio
• Everyone invests in P, regardless of their
degree of risk aversion
• More risk averse investors put more in the riskfree asset
• Less risk averse investors put more in P
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Figure 7.11 The Efficient Frontier of
Risky Assets with the Optimal CAL
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Markowitz Portfolio Optimization
Model
• Capital Allocation and the Separation Property
• Portfolio choice problem may be separated into
two independent tasks
• Determination of the optimal risky portfolio is
purely technical
• Allocation of the complete portfolio to risk-free
versus the risky portfolio depends on personal
preference
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Figure 7.13 Capital Allocation Lines with
Various Portfolios from the Efficient Set
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Markowitz Portfolio Optimization
Model
• The Power of Diversification
• Remember:
   wi w j Cov  ri , rj 
n
2
p
n
i 1 j 1
• If we define the average variance and average
covariance of the securities as:
1 n 2
   i
n i 1
2
n
n
1
Cov 
Cov  ri , rj 

n  n  1 j 1 i 1
j i
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Markowitz Portfolio Optimization
Model
• The Power of Diversification
• We can then express portfolio variance as
1 2 n 1
   
Cov
n
n
2
p
• Portfolio variance can be driven to zero if the
average covariance is zero (only firm specific risk)
• The irreducible risk of a diversified portfolio
depends on the covariance of the returns, which is
a function of the systematic factors in the
economy
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Table 7.4 Risk Reduction of
Equally Weighted Portfolios
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Markowitz Portfolio Optimization
Model
• Optimal Portfolios and Nonnormal Returns
• Fat-tailed distributions can result in extreme
values of VaR and ES and encourage smaller
allocations to the risky portfolio
• If other portfolios provide sufficiently better VaR
and ES values than the mean-variance efficient
portfolio, we may prefer these when faced with
fat-tailed distributions
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Risk Pooling and
the Insurance Principle
• Risk pooling
• Merging uncorrelated, risky projects as a means to
reduce risk
• It increases the scale of the risky investment by
adding additional uncorrelated assets
• The insurance principle
• Risk increases less than proportionally to the
number of policies when the policies are
uncorrelated
• Sharpe ratio increases
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Risk Sharing
• As risky assets are added to the portfolio, a
portion of the pool is sold to maintain a risky
portfolio of fixed size
• Risk sharing combined with risk pooling is the
key to the insurance industry
• True diversification means spreading a
portfolio of fixed size across many assets, not
merely adding more risky bets to an evergrowing risky portfolio
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Investment for the Long Run
Long-Term Strategy
Short-Term Strategy
• Invest in the risky
portfolio for 2 years
• Invest in the risky
portfolio for 1 year and
in the risk-free asset for
the second year
• Long-term strategy is
riskier
• Risk can be reduced by
selling some of the
risky assets in year 2
• “Time diversification”
is not true
diversification
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