CHAPTER 7
Optimal Risky Portfolios
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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
– 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
 wE r 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 wECovrD , rE 
 D2 = Variance of Security 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 wDCov(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
• How to solve for this weights?
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Three-Asset Portfolio
E (rp )  w1E (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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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 7.9 The Proportions of the Optimal
Overall Portfolio
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Markowitz Portfolio Selection Model
• Security Selection
– The first step is to determine the riskreturn opportunities available.
– All portfolios that lie on the minimumvariance frontier from the global
minimum-variance portfolio and upward
provide the best risk-return
combinations (i.e. this defines the
efficient set of mean-st.dev)
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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
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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.
• Under CAPM we known the answer
“without even looking”.
– 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:
1 2
n 1
   
Cov
n
n
2
P
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Table 7.4 Risk Reduction of Equally Weighted
Portfolios in Correlated and Uncorrelated Universes
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