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INVESTMENTS | BODIE, KANE, MARCUSINVESTMENTS | BODIE, KANE, MARCUS
Chapter Seven
Optimal Risky Portfolios
Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.
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• 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
Chapter Overview
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• 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
The Investment Decision
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• 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
Diversification and Portfolio Risk
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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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• 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)
Portfolios of Two Risky Assets
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• Portfolio return: rp = wDrD + wErE
• wD = Bond weight
• rD = Bond return
• wE = Equity weight
• rE = Equity return
E(rp) = wD E(rD) + wEE(rE)
Portfolios of Two Risky Assets: Return
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• Portfolio variance:
• = Bond variance
• = Equity variance
• = Covariance of returns for bond and equity
Portfolios of Two Risky Assets: Risk
2 2 2 2 2 2 Cov ,p D D E E D E D Ew w w w r r
2E
2D
Cov ,D Er r
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• Covariance of returns on bond and equity: Cov(rD,rE) = DEDE
• D,E = Correlation coefficient of returns
• D = Standard deviation of bond returns
• E = Standard deviation of equity returns
Portfolios of Two Risky Assets: Covariance
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• Range of values for 1,2
- 1.0 > r > +1.0
• If r = 1.0, the securities are perfectly positively correlated
• If r = - 1.0, the securities are perfectly negatively correlated
Portfolios of Two Risky Assets: Correlation Coefficients
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• When ρDE = 1, there is no diversification
• When ρDE = -1, a perfect hedge is possible
Portfolios of Two Risky Assets: Correlation Coefficients
DDEEP ww
D
ED
DE ww
1
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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 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 r = +1.0, no risk reduction is possible
• If r = 0, σP may be less than the standard deviation of either component asset
• If r = -1.0, a riskless hedge is possible
The Minimum Variance Portfolio
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Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two
Feasible CALs
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• Maximize the slope of the CAL for any possible portfolio, P
• The objective function is the slope:
• The slope is also the Sharpe ratio
The Sharpe Ratio
p
fpp
rrES
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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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• 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
Markowitz Portfolio Optimization Model
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Figure 7.10 The Minimum-Variance
Frontier of Risky Assets
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• Search for the CAL with the highest reward-to-variability ratio
• Everyone invests in P, regardless of their degree of risk aversion• More risk averse investors put more in the risk-
free asset• Less risk averse investors put more in P
Markowitz Portfolio Optimization Model
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Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL
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• 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
Markowitz Portfolio Optimization Model
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Figure 7.13 Capital Allocation Lines with Various Portfolios from the
Efficient Set
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• The Power of Diversification• Remember:
• If we define the average variance and average covariance of the securities as:
Markowitz Portfolio Optimization Model
2
1 1
Cov ,n n
p i j i ji j
ww r r
2 2
1
1 1
1
1Cov Cov ,
1
n
ii
n n
i jj ij i
n
r rn n
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• The Power of Diversification• We can then express portfolio variance as
• 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
Markowitz Portfolio Optimization Model
2 21 1Covp
n
n n
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Table 7.4 Risk Reduction of Equally Weighted Portfolios
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• 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
Markowitz Portfolio Optimization Model
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• 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
Risk Pooling and the Insurance Principle
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• 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 ever-growing risky portfolio
Risk Sharing
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Investment for the Long Run
Long-Term Strategy• Invest in the risky
portfolio for 2 years• 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
Short-Term Strategy• Invest in the risky
portfolio for 1 year and in the risk-free asset for the second year
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