Ch # 07-An Introduction to Portfolio Managemen.ppt

An Introduction to Portfolio
Management
Copyright © 2000 by Harcourt, Inc. All rights reserved.

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?

• 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?

• 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?

• 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?

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.

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.

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….

Not all investors are risk averse
Risk preference may have to do with amount
of money involved - risking small amounts,
but insuring large losses

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

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

Assumptions of
1. Investors consider each investment
alternative as being presented by a
probability distribution of expected returns
over some holding period.

Assumptions of
2. Investors maximize one-period expected
utility, and their utility curves demonstrate
diminishing marginal utility of wealth.
 Diminishing marginal utility refers
to the phenomenon that each additional unit
of gain leads to an ever-smaller increase in
subjective value

Assumptions of
3. Investors estimate the risk of the portfolio
on the basis of the variability of expected
returns.

Assumptions of
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.

Assumptions of
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.

Assumptions of
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.

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)

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

Computation of Expected Return for an
Individual Risky Investment
0.25 0.08 0.0200
0.25 0.10 0.0250
0.25 0.12 0.0300
0.25 0.14 0.0350
E(R) = 0.1100
Expected Return
(Percent)
Probability
Possible Rate of
Return (Percent)
Table 8.1

Computation of the Expected Return
for a Portfolio of Risky Assets
0.20 0.10 0.0200
0.30 0.11 0.0330
0.30 0.12 0.0360
0.20 0.13 0.0260
E(Rpor i) = 0.1150
Expected Portfolio
Return (Wi X Ri)
(Percent of Portfolio)
Expected Security
Return (Ri)
Weight (Wi)
Table 8.2
i
asset
for
return
of
rate
expected
the
)
E(R
i
asset
in
portfolio
the
of
percent
the
W
:
where
R
W
)
E(R
i
i
1
i
por


 

n
i
i
i

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)]




n
i 1
i
2
i
i
2
P
)]
E(R
-
R
[
)
(
Variance 
where Pi is the probability of the possible rate
of return, Ri




n
i 1
i
2
i
i P
)]
E(R
-
R
[
)
(
Standard Deviation

Possible Rate Expected
of Return (Ri) Return E(Ri) Ri - E(Ri) [Ri - E(Ri)]2
Pi [Ri - E(Ri)]2
Pi
0.08 0.11 0.03 0.0009 0.25 0.000225
0.10 0.11 0.01 0.0001 0.25 0.000025
0.12 0.11 0.01 0.0001 0.25 0.000025
0.14 0.11 0.03 0.0009 0.25 0.000225
0.000500
Table 8.3
Variance ( 2) = .0050
Standard Deviation ( ) = .02236



Returns for a Portfolio
Computation of Monthly Rates of Return
Table 8.4
Coca - Cola Exxon
Closing Closing
Date Price Dividend Return (%) Price Dividend Return (%)
Dec-97 66.688 0.14 61.188 0.41
Jan-98 64.750 -2.91% 59.313 -3.06%
Feb-98 68.625 5.98% 63.750 0.41 8.17%
Mar-98 77.438 0.15 13.06% 67.625 6.08%
Apr-98 75.875 -2.02% 73.063 8.04%
May-98 78.375 3.29% 70.500 0.41 -2.95%
Jun-98 85.500 0.15 9.28% 71.375 1.24%
Jul-98 80.500 -5.85% 70.250 -1.58%
Aug-98 65.125 -19.10% 65.438 0.41 -6.27%
Sep-98 57.625 0.15 -11.29% 70.625 7.93%
Oct-98 67.563 17.25% 71.625 1.42%
Nov-98 70.063 0.15 3.92% 75.000 0.41 5.28%
Dec-98 67.000 -4.37% 73.125 -2.50%
E(RCoca-Cola)= 0.61% E(RExxon)= 1.82%

Time Series Returns for
Coca-Cola: 1998
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
20.00%
J F M A M J J A S O N D
Figure 8.1

Times Series Returns for
Exxon: 1998
-8.00%
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
Figure 8.2

Times Series Returns for
Coca-Cola and Exxon: 1998
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%

• Covariance is a measure to indicate the
extent to which two random variables
change in tandem.
• Correlation is a measure used to represent
how strongly two random variables are
related to each other.
• Covariance is nothing but a measure of
correlation.

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)]}

Computation of Covariance of Returns
for Coca-Cola and Exxon: 1998
Rate of Rate of Coca-Cola Exxon Coca-Cola Exxon
Date Return (%) Return (%) Ri - E(Ri) Rj - E(Rj) [Ri - E(Ri)] X [Rj - E(Rj)]
Jan-98 -2.91% -0.0306 -0.0352 -0.049 17.18
Feb-98 5.98% 8.17% 0.0537 0.064 34.10
Mar-98 13.06% 6.08% 0.1245 0.043 53.04
Apr-98 -2.02% 8.04% -0.0263 0.062 -16.36
May-98 3.29% -2.95% 0.0268 -0.048 -12.78
Jun-98 9.28% 1.24% 0.0867 -0.006 -5.03
Jul-98 -5.85% -1.58% -0.0646 -0.034 21.96
Aug-98 -19.10% -6.27% -0.1971 -0.081 159.45
Sep-98 -11.29% 7.93% -0.1190 0.061 -72.71
Oct-98 17.25% 1.42% 0.1664 -0.004 -6.66
Nov-98 3.92% 5.28% 0.0331 0.035 11.45
Dec-98 -4.37% -2.50% -0.0498 -0.043 21.51
E(RCoca-Cola)= 0.61% E(RExxon)= 1.82% Sum = 205.16
Covij = 205.16 / 12 = 17.10
Table 6.5

Scatter Plot of Monthly Returns
for Coca-Cola and Exxon: 1998
-8.00%
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00%
Monthly Returns for Coca-Cola
Monthly
Return
for
Exxon
Figure 8.3

Covariance and Correlation
Correlation coefficient varies from -1 to +1
jt
it
i
ij
R
of
deviation
standard
the
R
of
deviation
standard
the
returns
of
t
coefficien
n
correlatio
the
r
:
where
Cov
r




j
j
i
ij
ij





Computation of Standard Deviation of
Returns for Coca-Cola and Exxon: 1998
Date Ri - E(Ri) [Ri - E(Ri)]2
Rj - E(Rj) [Rj - E(Rj)]2
Jan-98 -3.63% 13.18 -5.27% 27.77
Feb-98 5.47% 29.92 6.61% 43.69
Mar-98 12.54% 157.25 3.84% 14.75
Apr-98 -2.72% 7.40 6.48% 41.99
May-98 2.78% 7.73 -4.50% 20.25
Jun-98 8.77% 76.91 -0.90% 0.81
Jul-98 -6.53% 42.64 -3.13% 9.80
Aug-98 -19.62% 384.94 -7.82% 61.15
Sep-98 -11.80% 139.24 5.70% 32.49
Oct-98 16.42% 269.62 -0.14% 0.02
Nov-98 3.41% 11.63 3.73% 13.91
Dec-98 -5.09% 25.91 -4.59% 21.07
1,166.37 287.70
Variancei= 1166.37 / 12 = 97.20 Variancej= 287.70 / 12 = 23.98
Standard Deviationi = 97.20
1/2
= 9.86 Standard Deviationj = 23.98
1/2
= 4.90
These figures have not been rounded to two decimals at each step as was in the book
Table 8.6

Portfolio Standard Deviation
Formula
j
i





ij
ij
ij
2
i
i
port
n
1
i
n
1
i
ij
j
n
1
i
i
2
i
2
i
port
r
Cov
where
j,
and
i
assets
for
return
of
rates
e
between th
covariance
the
Cov
i
asset
for
return
of
rates
of
variance
the
portfolio
in the
value
of
proportion
by the
determined
are
weights
where
portfolio,
in the
assets
individual
the
of
weights
the
W
portfolio
the
of
deviation
standard
the
:
where
Cov
w
w
w






  
  

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

Combining Stocks with Different
Returns and Risk
Case Correlation Coefficient Covariance
a +1.00 .0070
b +0.50 .0035
c 0.00 .0000
d -0.50 -.0035
e -1.00 -.0070
W
)
E(R
Asset i
i
2
i
i 

1 .10 .50 .0049 .07
2 .20 .50 .0100 .10

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

Constant Correlation
with Changing Weights
Case W1 W2
E(Ri)
f 0.00 1.00 0.20
g 0.20 0.80 0.18
h 0.40 0.60 0.16
i 0.50 0.50 0.15
j 0.60 0.40 0.14
k 0.80 0.20 0.12
l 1.00 0.00 0.10
)
E(R
Asset i
1 .10 rij = 0.00
2 .20

Constant Correlation
with Changing Weights
Case W1 W2 E(Ri) E(F
port)
f 0.00 1.00 0.20 0.1000
g 0.20 0.80 0.18 0.0812
h 0.40 0.60 0.16 0.0662
i 0.50 0.50 0.15 0.0610
j 0.60 0.40 0.14 0.0580
k 0.80 0.20 0.12 0.0595
l 1.00 0.00 0.10 0.0700

Portfolio Risk-Return Plots for
Different Weights
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = +1.00
1
2
With two perfectly
correlated assets, it
is only possible to
create a two asset
portfolio with risk-
return along a line
between either
single asset

Different Weights
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
f
g
h
i
j
k
1
2
With uncorrelated
assets it is possible
to create a two
asset portfolio with
lower risk than
either single asset

Different Weights
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
Rij = +0.50
f
g
h
i
j
k
1
2
With correlated
assets it is possible
to create a two
asset portfolio
between the first
two curves

Different Weights
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
Rij = -0.50
Rij = +0.50
f
g
h
i
j
k
1
2
With
negatively
correlated
assets it is
possible to
create a two
asset portfolio
with much
lower risk than
either single
asset

Different Weights
-
0.05
0.10
0.15
0.20
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
E(R)
Rij = 0.00
Rij = +1.00
Rij = -1.00
Rij = +0.50
f
g
h
i
j
k
1
2
With perfectly negatively correlated
assets it is possible to create a two asset
portfolio with almost no risk
Rij = -0.50
Figure 8.7

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

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:
i
m
i
i
i R
b
a
R 



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

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:
market
stock
aggregate
for the
returns
of
variance
the
where
b
b
r
2
m
i
2
m
j
i
ij






j

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

Efficient Frontier
for Alternative Portfolios
Efficient
Frontier
A
B
C
Figure 8.9
E(R)

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

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

Selecting an Optimal Risky
Portfolio
)
E( port

)
E(Rport
X
Y
U3
U2
U1
U3’
U2’
U1’
Figure 8.10

Ch # 07-An Introduction to Portfolio Managemen.ppt

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