CHAPTER 7
Optimal Risky Portfolios
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McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
7-2
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
– Systematic or nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic
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Figure 7.1 Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
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Figure 7.2 Portfolio Diversification
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Covariance and Correlation
• Portfolio risk 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 vary
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Two-Security Portfolio: Return
rp

rP
 Portfolio Return
wr
D
D
 wEr E
wD  Bond Weight
rD
 Bond Return
wE  Equity Weight
rE
 Equity Return
E (rp )  wD E (rD )  wE E (rE )
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Two-Security Portfolio: Risk
 p2  wD2  D2  wE2 E2  2wD wE CovrD , rE 
 = Variance of Security D
2
D

2
E
= Variance of Security E
CovrD , rE  = Covariance of returns for
Security D and Security E
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Two-Security Portfolio: Risk
• Another way to express variance of the
portfolio:
 P2  wD wD Cov (rD , rD )  wE wE Cov (rE , rE )  2wD wE Cov (rD , rE )
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Covariance
Cov(rD,rE) = DEDE
D,E = Correlation coefficient of
returns
D = Standard deviation of
returns for Security D
E = Standard deviation of
returns for Security E
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Correlation Coefficients: Possible Values
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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Correlation Coefficients
• When ρDE = 1, there is no diversification
 P  wE E  wD D
• When ρDE = -1, a perfect hedge is possible
wE 
D
 D  E
 1  wD
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Table 7.2 Computation of Portfolio
Variance From the Covariance Matrix
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Three-Asset Portfolio
E (rp )  w1 E (r1 )  w2 E (r2 )  w3 E (r3 )
 p2  w1212  w22 22  w32 32
 2w1w2 1, 2  2w1w3 1,3  2w2 w3 2,3
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Example 7.1
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Table 7.3 Portfolio Standard Deviation for
a given correlation
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Figure 7.3 Portfolio Expected Return as a
Function of Investment Proportions
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Figure 7.4 Portfolio Standard Deviation as
a Function of Investment Proportions
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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.
• When correlation is
less than +1, the
portfolio standard
deviation may be
smaller than that of
either of the individual
component assets.
• When correlation is 1, the standard
deviation of the
minimum variance
portfolio is zero.
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Figure 7.5 Portfolio Expected Return as a
Function of Standard Deviation
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Correlation Effects
• The amount of possible risk reduction
through diversification depends on the
correlation.
• The risk reduction potential increases as
the correlation approaches -1.
– 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 The Opportunity Set of the Debt and Equity Funds
with the Optimal CAL and the Optimal Risky Portfolio
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Figure 7.8 Determination of the Optimal
Overall 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
Overall Portfolio
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The steps to arrive at the complete portfolio
The steps we followed to arrive at the complete portfolio.
1. Specify the return characteristics of all securities (expected returns,
variances, covariances).
2. Establish the risky portfolio (asset allocation):
a. Calculate the optimal risky portfolio, P (Equation 7.13).
b. Calculate the properties of portfolio P using the weights determined in
step (a) and Equations 7.2 and 7.3.
3. Allocate funds between the risky portfolio and the risk-free asset
(capital allocation):
a. Calculate the fraction of the complete portfolio allocated to portfolio P
(the risky portfolio) and to T-bills (the risk-free asset) (Equation 7.14).
b. Calculate the share of the complete portfolio invested in each asset
and in T-bills
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Markowitz Portfolio Selection Model
• Security Selection
– The first step is to determine the riskreturn opportunities available
(summarized by the minimum-variance
frontier of risky assets).
– All portfolios that lie on the minimumvariance 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 Selection Model
• We now search for the CAL with the
highest reward-to-variability ratio
• The CAL generated by the optimal
portfolio, P, is the one tangent to the
efficient frontier.
• This CAL dominates all alternative
feasible lines (the broken lines that are
drawn through the frontier).
• Portfolio P, therefore, is the optimal risky
portfolio
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Figure 7.11 The Efficient Frontier of Risky
Assets with the Optimal CAL
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Markowitz Portfolio Selection Model
• 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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Capital Allocation and the
Separation Property
• The separation property tells us that the
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 Tbills 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
n
• Remember:  2 

P
i 1
n
 w w Cov(r , r )
j 1
i
j
i
j
• If we define the average variance and average
covariance of the securities as:
1 n 2
   i
n i 1
2
n
1
Cov 

n(n  1) j 1
j i
n
 Cov(r , r )
i 1
i
j
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The Power of Diversification
• We can then express portfolio variance as
(for equally weighted portfolio):
1 2
n 1
   
Cov
n
n
When the average covariance among security returns is zero, as it
2
P
•
would be if all risk were firm-specific, portfolio variance can be
driven to zero
• If Cov is positive and n increases, portfolio variance remain positive,
but firm-specific risk  0. Thus, the irreducible risk of a diversified
portfolio depends on the covariance of the returns of the component
securities, which in turn is a function of the importance of systematic
factors in the economy
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Table 7.4 Risk Reduction of Equally Weighted
Portfolios in Correlated and Uncorrelated Universes
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