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INVESTMENTS | BODIE, KANE, MARCUS
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin
CHAPTER 7
Optimal Risky Portfolios
INVESTMENTS | BODIE, KANE, MARCUS
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
7-2
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INVESTMENTS | BODIE, KANE, MARCUS
Diversification and Portfolio Risk
• Market risk
– Risk attributable to market-wide risk
sources, and remains even after extensive
diversification
– aka Systemic or non-diversifiable
• Firm-specific risk
– Risk that can be eliminated by diversification
– aka diversifiable or non-systemic
7-3
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INVESTMENTS | BODIE, KANE, MARCUS
Fig. 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio
1-4
Panel A: All risk is firm specific. Panel B: Some risk is systematic or marketwide.
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INVESTMENTS | BODIE, KANE, MARCUS
Figure 7.2 Portfolio Diversification
1-5
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INVESTMENTS | BODIE, KANE, MARCUS
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
7-6
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INVESTMENTS | BODIE, KANE, MARCUS
A Two-Security Portfolio: Return
7-7
( ) ( ) ( )p D D E EE r w E r w E r
𝑤𝐷 = Bond weight
𝑤𝐸 = Equity weight
𝑟𝐷 = Bond return
𝑟𝐸 = Equity return
𝑟𝑃 = Portfolio return
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INVESTMENTS | BODIE, KANE, MARCUS
A Two-Security Portfolio: Risk
7-8
EDEDEEDD rrCovwwww ,222222
p
= Variance of Security D
= Variance of Security E
= Covariance of returns for
Security D and Security E
2
E
2
D
ED rrCov ,
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INVESTMENTS | BODIE, KANE, MARCUS
Two-Security Portfolio: Risk
• Another way to express variance of the portfolio is to think of Covariances:
7-9
EDED
EEEE
DDDD
rrCovww
rrCovww
rrCovww
,2
,
,2
p
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INVESTMENTS | BODIE, KANE, MARCUS
Covariance
7-10
𝜌𝐷,𝐸 = Correlation coefficient of returns
𝜎𝐷 = Standard deviation of returns
for Security D
𝜎𝐸 = Standard deviation of returns
for Security E
EDDEED rrCov ,
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INVESTMENTS | BODIE, KANE, MARCUS
Table 7.2 Computation of Portfolio Variance From the Covariance Matrix
1-11
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INVESTMENTS | BODIE, KANE, MARCUS
A portfolio of 3 Assets
1-12
1 1 2 2 3 3( ) ( ) ( ) ( )pE r w E r w E r w E r
• You have three assets with weights
w1, w2, w3
• The portfolio return is simply the linear
combination of the returns with same
coefficients:
Q. is the portfolio’s variance also the
linear combination of the 3 variances?
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INVESTMENTS | BODIE, KANE, MARCUS
Bordered Matrix for 3 Assets
w1 w2 w3
w1 Cov(1,1) Cov(1,2) Cov(1,3)
w2 Cov(2,1) Cov(2,2) Cov(2,3)
w3 Cov(3,1) Cov(3,2) Cov(3,3)
7-13
Step 1: write the covariance matrix and its weights
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INVESTMENTS | BODIE, KANE, MARCUS
Bordered Matrix for 3 Assets
w1 w2 w3
w1
w2
w3
7-14
Step 2: Symmetry! baabCovbaCov ,,,
2
1 2,1 3,1
2
2
2
3
3,2
3,1 3,2
2,1
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INVESTMENTS | BODIE, KANE, MARCUS
w1 w2 w3
w1
w2
w3
Bordered Matrix for 3 Assets
7-15
Step 3: multiply by the weights around the border
2
1w
2
2w
2
3w
21ww
21ww
31ww
31ww
32ww
32ww
2
1 2,1 3,1
2
2
2
3
3,2
3,1 3,2
2,1
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INVESTMENTS | BODIE, KANE, MARCUS
Bordered Matrix for 3 Assets
1-16
3,2323,1312,121
2
3
2
3
2
2
2
2
2
1
2
1
2
222
wwwwww
wwwp
Covariance terms
Step 4: add-up all the pieces
Remember ababbabaabba ,,,,
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INVESTMENTS | BODIE, KANE, MARCUS
Bordered Matrix for 3 Assets
w1 w2 w3
w1
w2
w3
7-17
All in one step, using correlations this time
2
1 212,1 313,1
2
2
2
3
323,2
313,1 323,2
212,1
2
1w
2
2w
2
3w
31ww
33ww
21ww
33ww31ww
21ww
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INVESTMENTS | BODIE, KANE, MARCUS
Bordered Matrix for 3 Assets
1-18
323,232
313,131
212,121
2
3
2
3
2
2
2
2
2
1
2
1
2
2
2
2
ww
ww
ww
wwwp
Add-up all the pieces
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INVESTMENTS | BODIE, KANE, MARCUS
Correlation: Possible Values
7-19
Range of values for correlation
−𝟏 ≤ 𝝆 ≤ +1
If = 1.0, the securities are perfectly
positively correlated
If = - 1.0, the securities are perfectly
negatively correlated
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INVESTMENTS | BODIE, KANE, MARCUS
Two-Security Portfolio: Variance
• Remember the variance of a two-asset portfolio
7-20
EDDEED
EEDD
ww
ww
2
22222
p
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INVESTMENTS | BODIE, KANE, MARCUS
Correlation Coefficients
• When 𝜌𝐷𝐸 = 1, there is no diversification
7-21
DDEEP ww
D
ED
DE
ED
ED www
1 and
0222222
p EDEDEEDD wwww
• When 𝜌𝐷𝐸 = −1, a perfect hedge is when:
the solution (which also makes 𝑤𝐷 + 𝑤𝐸 = 1) is:
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.3 Portfolio Expected Return as a Function of Investment Proportions
1-22
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.4 Portfolio Standard Deviation as a Function of Investment Proportions
1-23
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INVESTMENTS | BODIE, KANE, MARCUS
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.
• If correlation < +1
the portfolio standard
deviation may be
smaller than that of
either of the individual
component assets.
• If correlation = -1
the standard deviation
of the minimum
variance portfolio is
zero. 7-24
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.5 Portfolio Expected Return as a Function of Standard Deviation
1-25
Portfolio
opportunity
set for given
correlation
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INVESTMENTS | BODIE, KANE, MARCUS
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 𝜌 = +1.0, no risk reduction is possible.
– If 𝜌 = 0, 𝜎𝑃 may be less than the standard
deviation of either component asset.
– If 𝜌 = -1.0, a riskless hedge is possible.
7-26
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs
1-27
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INVESTMENTS | BODIE, KANE, MARCUS
The Sharpe Ratio
• Maximize the slope of the CAL for any possible portfolio, P.
• The objective function is the slope:
• The slope is also the Sharpe ratio.
7-28
P
fP
P
rrES
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INVESTMENTS | BODIE, KANE, MARCUS 7-29
Fig 7.7 The Opportunity Set of Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.8 Determination of the Optimal Overall Portfolio
1-30
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INVESTMENTS | BODIE, KANE, MARCUS
Markowitz Portfolio Selection 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
7-31
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.10 The Minimum-Variance Frontier of Risky Assets
1-32
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INVESTMENTS | BODIE, KANE, MARCUS
Markowitz Portfolio Selection Model
• We now search for the CAL with the highest reward-to-variability ratio
Q. How do we measure that?
• That means to find that optimal line that stems from the risk-free point and is tangent to the efficient frontier
7-33
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL
1-34
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INVESTMENTS | BODIE, KANE, MARCUS
Markowitz Portfolio Selection Model
• Everyone invests in P, regardless of their degree of risk aversion. However:
– More risk averse investors put more in the
risk-free asset.
– Less risk averse investors put more in P.
7-35
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INVESTMENTS | BODIE, KANE, MARCUS
Capital Allocation, Separation Property
• The separation property tells us that the portfolio choice problem may be separated into two independent tasks:
1. Determination of the optimal risky portfolio
(purely technical / mathematical)
2. Allocation of the complete portfolio to risk
free asset versus the risky portfolio
(depends on personal preference)
7-36
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INVESTMENTS | BODIE, KANE, MARCUS
Fig 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set
1-37
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INVESTMENTS | BODIE, KANE, MARCUS
The Power of Diversification
1-38
• Remember:
• Consider an equally weighted portfolio:
n
i
n
j
jijiP rrCovww1 1
2 ,
nwi
1
• Look at covariance the matrix structure:
2
P
n
i
in1
2
2
1
n
i
n
ijj
ji rrCovnn1 1
,11
• Rewrite covariance as:
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INVESTMENTS | BODIE, KANE, MARCUS
The Power of Diversification
1-39
• Rearrange:
• Define avg variance and avg covariance as:
n
i
in 1
22 1
terms1
1 1
,1
1
nn
n
i
n
ijj
ji rrCovnn
Cov
n
i
n
ijj
ji
n
i
iP rrCovnnnn 1 11
22 ,1111
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INVESTMENTS | BODIE, KANE, MARCUS
The Power of Diversification
7-40
• Then we can rewrite portfolio variance:
terms1
1 11
22 ,1111
2
nn
n
i
n
ijj
ji
n
i
iP rrCovnnnn
Covn
n
nP
11 22
as:
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INVESTMENTS | BODIE, KANE, MARCUS
The Power of Diversification
7-41
Study case where all assets have same
standard deviation and one correlation for all
Q. What happens for very large n?
222 11
n
n
nP
Q. What happens when correlation = 0?
22
P
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INVESTMENTS | BODIE, KANE, MARCUS
Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes
1-42
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INVESTMENTS | BODIE, KANE, MARCUS
Optimal Portfolios and Non-normal Returns
• The optimal portfolio approach we just studied assumes normal returns.
• Fat-tailed distributions can result in extreme values of Value-at-Risk (VaR) and Expected Shortfall (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.
7-43
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INVESTMENTS | BODIE, KANE, MARCUS
Risk Pooling - Insurance Principle
• Risk pooling: merging (adding) uncorrelated, risky projects as a means to reduce risk. – 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 insured when the policies are uncorrelated
– Sharpe ratio increases
7-44
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INVESTMENTS | BODIE, KANE, MARCUS
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 ever-growing risky portfolio.
7-45
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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”
7-46
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