investments | bodie, kane, marcus chapter seven optimal risky portfolios copyright © 2014...

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



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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



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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:


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


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



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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

investments | bodie, kane, marcus chapter seven optimal risky portfolios copyright © 2014...

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