Lecture Presentation Software
to accompany
Investment Analysis and
Portfolio Management
Sixth Edition
by
Frank K. Reilly & Keith C. Brown
Chapter 8
Version 1.2
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copies of any part of the work should be mailed to:
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Chapter 8 - An Introduction to
Portfolio Management
Questions to be answered:
• What do we mean by risk aversion and what
evidence indicates that investors are generally
risk averse?
• What are the basic assumptions behind the
Markowitz portfolio theory?
• What is meant by risk and what are some of the
alternative measures of risk used in
investments?
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Chapter 8 - An Introduction to
Portfolio Management
• How do you compute the expected rate of
return for an individual risky asset or a
portfolio of assets?
• How do you compute the standard deviation of
rates of return for an individual risky asset?
• What is meant by the covariance between rates
of return and how do you compute covariance?
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Chapter 8 - An Introduction to
Portfolio Management
• What is the relationship between covariance
and correlation?
• What is the formula for the standard deviation
for a portfolio of risky assets and how does it
differ from the standard deviation of an
individual risky asset?
• Given the formula for the standard deviation of
a portfolio, how and why do you diversify a
portfolio?
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Chapter 8 - An Introduction to
Portfolio Management
• What happens to the standard deviation of a
portfolio when you change the correlation
between the assets in the portfolio?
• What is the risk-return efficient frontier?
• Is it reasonable for alternative investors to
select different portfolios from the portfolios on
the efficient frontier?
• What determines which portfolio on the
efficient frontier is selected by an individual
investor? Copyright © 2000 by Harcourt, Inc. All rights reserved.
Background Assumptions
• As an investor you want to maximize the
returns for a given level of risk.
• Your portfolio includes all of your assets
and liabilities
• The relationship between the returns for
assets in the portfolio is important.
• A good portfolio is not simply a collection
of individually good investments.
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Risk Aversion
Given a choice between two assets with
equal rates of return, risk averse
investors will select the asset with the
lower level of risk.
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Evidence That
Investors are Risk Averse
• Many investors purchase insurance for:
Life, Automobile, Health, and Disability
Income. The purchaser trades known costs
for unknown risk of loss
• Yield on bonds increases with risk
classifications from AAA to AA to A….
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Not all investors are risk averse
Risk preference may have to do with amount
of money involved - risking small amounts,
but insuring large losses
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Definition of Risk
1. Uncertainty of future outcomes
or
2. Probability of an adverse outcome
We will consider several measures of risk that
are used in developing portfolio theory
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Markowitz Portfolio Theory
• Quantifies risk
• Derives the expected rate of return for a
portfolio of assets and an expected risk measure
• Shows the variance of the rate of return is a
meaningful measure of portfolio risk
• Derives the formula for computing the variance
of a portfolio, showing how to effectively
diversify a portfolio
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Assumptions of
Markowitz Portfolio Theory
1. Investors consider each investment
alternative as being presented by a
probability distribution of expected returns
over some holding period.
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Assumptions of
Markowitz Portfolio Theory
2. Investors maximize one-period expected
utility, and their utility curves demonstrate
diminishing marginal utility of wealth.
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Assumptions of
Markowitz Portfolio Theory
3. Investors estimate the risk of the portfolio
on the basis of the variability of expected
returns.
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Assumptions of
Markowitz Portfolio Theory
4. Investors base decisions solely on expected
return and risk, so their utility curves are a
function of expected return and the
expected variance (or standard deviation) of
returns only.
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Assumptions of
Markowitz Portfolio Theory
5. For a given risk level, investors prefer
higher returns to lower returns. Similarly,
for a given level of expected returns,
investors prefer less risk to more risk.
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Assumptions of
Markowitz Portfolio Theory
Using these five assumptions, a single asset or
portfolio of assets is considered to be
efficient if no other asset or portfolio of
assets offers higher expected return with the
same (or lower) risk, or lower risk with the
same (or higher) expected return.
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Alternative Measures of Risk
• Variance or standard deviation of expected return
• Range of returns
• Returns below expectations
– Semivariance - measure expected returns below some
target
– Intended to minimize the damage
• Scale of returns (if stock returns follow a stable
distribution, in which case their variance would be
infinite)
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Expected Rates of Return
• Individual risky asset
– Sum of probability times possible rate of return
• Portfolio
– Weighted average of expected rates of return
for the individual investments in the portfolio
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Computation of Expected Return for an
Individual Risky Investment
Table 8.1
Probability
0.25
0.25
0.25
0.25
Possible Rate of
Return (Percent)
0.08
0.10
0.12
0.14
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Expected Return
(Percent)
0.0200
0.0250
0.0300
0.0350
E(R) = 0.1100
Computation of the Expected Return
for a Portfolio of Risky Assets
Weight (Wi )
(Percent of Portfolio)
Expected Security
Return (R i )
0.20
0.30
0.30
0.20
0.10
0.11
0.12
0.13
Expected Portfolio
Return (Wi X Ri )
0.0200
0.0330
0.0360
0.0260
E(Rpor i) = 0.1150
n
E(R por i )   Wi R i
i 1
Table 8.2
where :
Wi  the percent of the portfolio in asset i
E(R i )  the expected rate of return for asset i
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Variance (Standard Deviation) of
Returns for an Individual Investment
Standard deviation is the square root of the
variance
Variance is a measure of the variation of
possible rates of return Ri, from the
expected rate of return [E(Ri)]
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Variance (Standard Deviation) of
Returns for an Individual Investment
n
Variance ( )   [R i - E(R i )] Pi
2
2
i 1
where Pi is the probability of the possible rate
of return, Ri
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Variance (Standard Deviation) of
Returns for an Individual Investment
Standard Deviation
( ) 
n
 [R
i 1
2
i
- E(R i )] Pi
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Variance (Standard Deviation) of
Returns for an Individual Investment
Table 8.3
Possible Rate
of Return (R i )
Expected
Return E(R i )
Ri - E(Ri )
[Ri - E(Ri )]2
0.08
0.10
0.12
0.14
0.11
0.11
0.11
0.11
0.03
0.01
0.01
0.03
0.0009
0.0001
0.0001
0.0009
Variance (  2) = .0050
Standard Deviation ( ) = .02236
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Pi
0.25
0.25
0.25
0.25
[Ri - E(Ri )]2 Pi
0.000225
0.000025
0.000025
0.000225
0.000500
Variance (Standard Deviation) of
Returns for a Portfolio Table 8.4
Computation of Monthly Rates of Return
Coca - Cola
Date
Dec-97
Jan-98
Feb-98
Mar-98
Apr-98
May-98
Jun-98
Jul-98
Aug-98
Sep-98
Oct-98
Nov-98
Dec-98
Closing
Price
Dividend
66.688
0.14
64.750
68.625
77.438
0.15
75.875
78.375
85.500
0.15
80.500
65.125
57.625
0.15
67.563
70.063
0.15
67.000
E(RCoca-Cola)=
Exxon
Return (%)
-2.91%
5.98%
13.06%
-2.02%
3.29%
9.28%
-5.85%
-19.10%
-11.29%
17.25%
3.92%
-4.37%
0.61%
Closing
Price
61.188
59.313
63.750
67.625
73.063
70.500
71.375
70.250
65.438
70.625
71.625
75.000
73.125
Dividend Return (%)
0.41
-3.06%
0.41
8.17%
6.08%
8.04%
0.41
-2.95%
1.24%
-1.58%
0.41
-6.27%
7.93%
1.42%
0.41
5.28%
-2.50%
E(RExxon)= 1.82%
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Time Series Returns for
Coca-Cola: 1998
20.00%
15.00%
10.00%
5.00%
0.00%
-5.00%
-10.00%
-15.00%
-20.00%
-25.00%
J
F
M
A
M
J
J
A
Copyright © 2000 by Harcourt, Inc. All rights reserved.
S
O
Figure 8.1
N
D
Times Series Returns for
Exxon: 1998
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
-2.00%
-4.00%
-6.00%
-8.00%
J
F
M
A
M
J
J
A
Copyright © 2000 by Harcourt, Inc. All rights reserved.
S
O
Figure 8.2
N
D
Times Series Returns for
Coca-Cola and Exxon: 1998
15.00%
10.00%
5.00%
0.00%
-5.00%
J
F
M A M J
J
A S O N D
-10.00%
-15.00%
-20.00%
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Variance (Standard Deviation) of
Returns for a Portfolio
For two assets, i and j, the covariance of rates
of return is defined as:
Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}
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Computation of Covariance of Returns
for Coca-Cola and Exxon: 1998Table 6.5
Date
Jan-98
Feb-98
Mar-98
Apr-98
May-98
Jun-98
Jul-98
Aug-98
Sep-98
Oct-98
Nov-98
Dec-98
E(RCoca-Cola)=
Rate of
Return (%)
-2.91%
5.98%
13.06%
-2.02%
3.29%
9.28%
-5.85%
-19.10%
-11.29%
17.25%
3.92%
-4.37%
0.61%
E(RExxon)=
Covij =
Rate of
Return (%)
-0.0306
8.17%
6.08%
8.04%
-2.95%
1.24%
-1.58%
-6.27%
7.93%
1.42%
5.28%
-2.50%
1.82%
205.16
Coca-Cola
Ri - E(Ri)
-0.0352
0.0537
0.1245
-0.0263
0.0268
0.0867
-0.0646
-0.1971
-0.1190
0.1664
0.0331
-0.0498
Exxon
Rj - E(Rj)
Coca-Cola
[Ri - E(Ri)]
Exxon
X [Rj - E(Rj)]
-0.049
0.064
0.043
0.062
-0.048
-0.006
-0.034
-0.081
0.061
-0.004
0.035
-0.043
Sum =
/ 12 =
17.10
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17.18
34.10
53.04
-16.36
-12.78
-5.03
21.96
159.45
-72.71
-6.66
11.45
21.51
205.16
Scatter Plot of Monthly Returns
for Coca-Cola and Exxon: 1998
Figure 8.3
10.00%
Monthly Return for Exxon
8.00%
6.00%
4.00%
2.00%
0.00%
-8.00% -6.00% -4.00% -2.00% 0.00% 2.00%
-2.00%
4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00%
-4.00%
-6.00%
-8.00%
Monthly Returns for Coca-Cola
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Covariance and Correlation
Correlation coefficient varies from -1 to +1
Cov ij
rij 
 i j
where :
rij  the correlatio n coefficien t of returns
 i  the standard deviation of R it
 j  the standard deviation of R jt
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Computation of Standard Deviation of
Returns for Coca-Cola and Exxon: 1998
Jan-98
Feb-98
Mar-98
Apr-98
May-98
Jun-98
Jul-98
Aug-98
Sep-98
Oct-98
Nov-98
Dec-98
Variancei=
[Ri - E(R i )]2
Ri - E(R i )
Date
-3.63%
5.47%
12.54%
-2.72%
2.78%
8.77%
-6.53%
-19.62%
-11.80%
16.42%
3.41%
-5.09%
1166.37 / 12 =
Standard Deviationi =
97.20
1/2
=
[Rj - E(Rj)]2
Rj - E(Rj)
13.18
29.92
157.25
7.40
7.73
76.91
42.64
384.94
139.24
269.62
11.63
25.91
1,166.37
97.20
-5.27%
6.61%
3.84%
6.48%
-4.50%
-0.90%
-3.13%
-7.82%
5.70%
-0.14%
3.73%
-4.59%
Variancej=
9.86
Standard Deviationj =
27.77
43.69
14.75
41.99
20.25
0.81
9.80
61.15
32.49
0.02
13.91
21.07
287.70
287.70 / 12 = 23.98
23.98
1/2
=
These figures have not been rounded to two decimals at each step as was in the book
Copyright © 2000 by Harcourt, Inc. All rights reserved.
4.90
Table 8.6
Portfolio Standard Deviation
Formula
 port 
n
w 
i 1
2
i
n
2
i
n
  w i w j Cov ij
i 1 i 1
where :
 port  the standard deviation of the portfolio
Wi  the weights of the individual assets in the portfolio, where
weights are determined by the proportion of value in the portfolio
 i2  the variance of rates of return for asset i
Cov ij  the covariance between th e rates of return for assets i and j,
where Cov ij  rij i j
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Portfolio Standard Deviation
Calculation
• Any asset of a portfolio may be described
by two characteristics:
– The expected rate of return
– The expected standard deviations of returns
• The correlation, measured by covariance,
affects the portfolio standard deviation
• Low correlation reduces portfolio risk while
not affecting the expected return
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Combining Stocks with Different
Returns and Risk
Asset
1
2
Case
a
b
c
d
e
 2i
.10
Wi
.50
.0049
.07
.20
.50
.0100
.10
E(R i )
Correlation Coefficient
+1.00
+0.50
0.00
-0.50
-1.00
i
Covariance
.0070
.0035
.0000
-.0035
-.0070
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Combining Stocks with Different
Returns and Risk
• Assets may differ in expected rates of return
and individual standard deviations
• Negative correlation reduces portfolio risk
• Combining two assets with -1.0 correlation
reduces the portfolio standard deviation to
zero only when individual standard
deviations are equal
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Constant Correlation
with Changing Weights
Asset
E(R i )
1
.10
2
.20
rij = 0.00
Case
W1
W2
f
g
h
i
j
k
l
0.00
0.20
0.40
0.50
0.60
0.80
1.00
1.00
0.80
0.60
0.50
0.40
0.20
0.00
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E(Ri )
0.20
0.18
0.16
0.15
0.14
0.12
0.10
Constant Correlation
with Changing Weights
Case
W1
W2
E(Ri )
E(Fport)
f
g
h
i
j
k
l
0.00
0.20
0.40
0.50
0.60
0.80
1.00
1.00
0.80
0.60
0.50
0.40
0.20
0.00
0.20
0.18
0.16
0.15
0.14
0.12
0.10
0.1000
0.0812
0.0662
0.0610
0.0580
0.0595
0.0700
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Portfolio Risk-Return Plots for
Different Weights
E(R)
0.20
0.15
0.10
0.05
With two perfectly
correlated assets, it
is only possible to
create a two asset
portfolio with riskreturn along a line
between either
single asset
2
Rij = +1.00
1
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
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Portfolio Risk-Return Plots for
Different Weights
E(R)
0.20
0.15
0.10
f
g
2
With uncorrelated
h
assets it is possible
i
j
to create a two
Rij = +1.00
asset portfolio with
k
lower risk than
1
either single asset
Rij = 0.00
0.05
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
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Portfolio Risk-Return Plots for
Different Weights
E(R)
0.20
0.15
0.10
f
g
2
With correlated
h
assets it is possible
i
j
to create a two
Rij = +1.00
asset portfolio
k
Rij = +0.50
between the first
1
two curves
Rij = 0.00
0.05
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
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Portfolio Risk-Return Plots for
Different Weights
E(R) With
0.20 negatively
correlated
assets it is
0.15
possible to
create a two
0.10 asset portfolio
with much
0.05 lower risk than
either single
asset
Rij = -0.50
f
2
g
h
j
k
i
Rij = +1.00
Rij = +0.50
1
Rij = 0.00
-
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
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Portfolio Risk-Return Plots for
Figure 8.7
Different Weights
E(R)
0.20
Rij = -0.50
Rij = -1.00
f
2
g
h
0.15
0.10
0.05
-
j
k
i
Rij = +1.00
Rij = +0.50
1
Rij = 0.00
With perfectly negatively correlated
assets it is possible to create a two asset
portfolio with almost no risk
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
Standard Deviation of Return
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Estimation Issues
• Results of portfolio allocation depend on
accurate statistical inputs
• Estimates of
– Expected returns
– Standard deviation
– Correlation coefficient
• Among entire set of assets
• With 100 assets, 4,950 correlation estimates
• Estimation risk refers to potential errors
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Estimation Issues
• With assumption that stock returns can be
described by a single market model, the
number of correlations required reduces to
the number of assets
• Single index market model:
R i  a i  bi R m   i
bi = the slope coefficient that relates the returns for security i
to the returns for the aggregate stock market
Rm = the returns for the aggregate stock market
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Estimation Issues
If all the securities are similarly related to
the market and a bi derived for each one,
it can be shown that the correlation
coefficient between two securities i and j
is given as:
 m2
rij  b i b j
 i j
where  m2  the variance of returns for the
aggregate stock market
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The Efficient Frontier
• The efficient frontier represents that set of
portfolios with the maximum rate of return
for every given level of risk, or the
minimum risk for every level of return
• Frontier will be portfolios of investments
rather than individual securities
– Exceptions being the asset with the highest
return and the asset with the lowest risk
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Efficient Frontier
for Alternative Portfolios
E(R)
Efficient
Frontier
A
Figure 8.9
B
C
Standard Deviation of Return
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The Efficient Frontier
and Investor Utility
• An individual investor’s utility curve
specifies the trade-offs he is willing to make
between expected return and risk
• The slope of the efficient frontier curve
decreases steadily as you move upward
• These two interactions will determine the
particular portfolio selected by an individual
investor
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The Efficient Frontier
and Investor Utility
• The optimal portfolio has the highest utility
for a given investor
• It lies at the point of tangency between the
efficient frontier and the utility curve with
the highest possible utility
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Selecting an Optimal Risky
Portfolio
Figure 8.10
E(R port )
U3’
U2’
U1’
Y
U3
X
U2
U1
E( port )
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The Internet
Investments Online
www.pionlie.com
www.investmentnews.com
www.micropal.com
www.riskview.com
www.altivest.com
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End of Chapter 8
–An Introduction to Portfolio
Management
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Future topics
Chapter 9
•
•
•
•
•
Capital Market Theory
Capital Asset Pricing Model
Beta
Expected Return and Risk
Arbitrage Pricing Theory
Copyright © 2000 by Harcourt, Inc. All rights reserved.
Copyright © 2000 by Harcourt, Inc. All rights reserved.