As an investor you want to maximize the returns for a given level of risk.
Your portfolio includes all of your assets and liabilities
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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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Given a choice between two assets with equal rates of return, most investors will select the asset with the lower level of risk.
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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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1. Uncertainty of future outcomes or
2. Probability of an adverse outcome
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Quantifies risk
Derives the expected rate of return and expected risk for a portfolio of assets.
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Shows that 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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1. Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.
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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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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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Variance or standard deviation of expected return
Range of returns
Returns below expectations
Semivariance – a measure that only considers deviations below the mean
These measures of risk implicitly assume that investors want to minimize the damage from returns less than some target rate
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Expected Return for an Individual
Risky Investment
Exhibit 6.1
Probability
0.25
0.25
0.25
0.25
Possible Rate of
Return (Percent)
0.08
0.10
0.12
0.14
Expected Return
(Percent)
0.0200
0.0250
0.0300
0.0350
E(R) = 0.1100
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Expected Return for a
Portfolio of Risky Assets
Weight (W i
)
(Percent of Portfolio)
Expected Security
Return (R i
)
Exhibit 6.2
Expected Portfolio
Return (W i
X R i
)
0.20
0.30
0.30
0.20
0.10
0.11
0.12
0.13
0.0200
0.0330
0.0360
0.0260
E(R por i
) = 0.1150
E(R por i
:
)
 n 
W i
R i where
W i
E(R
 i i

1
) the
 percent
the of the expected portfolio rate of in return asset for i asset i
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Variance (Standard Deviation) of
Returns for an Individual Investment
Variance (

2
)
 n  i

1
[ R i
E(R i
)]
2
P i where P i is the probability of the possible rate of return, R i
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Variance (Standard Deviation) of
Returns for an Individual Investment
Possible Rate Expected of Return (R i
) Return E(R i
) R i
- E(R i
) [R i
- E(R i
)]
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
P i
Exhibit 6.3
[R i
- E(R i
)]
2
P i
0.25
0.000225
0.25
0.000025
0.25
0.000025
0.25
0.000225
0.000500

2 ) = .0050
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Variance (Standard Deviation) of Returns for a Portfolio
Exhibit 6.4
Computation of Monthly Rates of Return
Closing
Date Price Dividend Return (%)
Closing
Price Dividend Return (%)
Dec.00
Jan.01
60.938
58.000
Feb.01
Mar.01
Apr.01
53.030
45.160
46.190
May.01
47.400
Jun.01
Jul.01
Aug.01
Sep.01
Oct.01
Nov.01
Dec.01
0.18
-4.82%
-8.57%
-14.50%
2.28%
2.62%
45.000
44.600
48.670
46.850
47.880
46.960
0.18
0.18
-4.68%
-0.89%
9.13%
-3.37%
2.20%
-1.55% 0.18
47.150
0.40%
E(RCoca-Cola)= -1.81%
45.688
48.200
42.500
43.100
47.100
49.290
47.240
50.370
45.950
38.370
38.230
46.650
51.010
0.04
0.04
0.04
0.05
5.50%
-11.83%
1.51%
9.28%
4.65%
-4.08%
6.63%
-8.70%
-16.50%
-0.36%
22.16%
9.35%
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A measure of the degree to which two variables “ move together ” relative to their individual mean values over time
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For two assets, i and j, the covariance of rates of return is defined as:
Cov ij
= E{[R i
- E(R i
)][R j
- E(R j
)]}
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Computation of Covariance of Returns for Coca cola and Home Depot: 2001
Exhibit 6.7
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Correlation coefficient varies from -1 to
+ 1 r ij

Cov ij
 i
 j where : r ij
 i
 j


 the correlatio n coefficien the standard deviation the standard deviation t of returns of R it of R jt
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 port
 i n 

1 w i
2
 i
2  i n n 

1 i

1 w i w j
Cov ij where
 port

: the standard deviation
W i
 the weights of the portfolio of the individual assets in the portfolio, where weights are determined by the proportion of value in the portfolio
 i
2  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
 r ij
 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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Time Patterns of Returns for Two Assets with
Perfect Negative Correlation
Exhibit 6.10
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Risk-Return Plot for Portfolios with Equal Returns and Standard
Deviations but Different Correlations (page 182-183)
Correlation affects portfolio risk
Exhibit 6.11
A: correlation=1
B: correlation=0.5
C: correlation=0
D: correlation=-0.5
E: correlation=-1
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Combining Stocks with Different
Returns and Risk
Asset E(R i
) W i

2 i
 i
1 .10 .50 .0049 .07
2 .20 .50 .0100 .10
Case Correlation Coefficient Covariance a +1.00 .0070
b +0.50 .0035
c 0.00 .0000
d -0.50 -.0035
e -1.00 -.0070
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Combining Stocks with Different
Returns and Risk
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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Risk-Return Plot for Portfolios with Different
Returns, Standard Deviations, and Correlations
Exhibit 6.12
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Constant Correlation with Changing Weights j k l h i f g
Asset E(R i
)
1 .10 r ij
= 0.00
Case
2 .20
W
1
W
2
E(R i
)
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
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Case k l j i f g h
Constant Correlation with Changing Weights
W
1
0.00
0.20
0.40
0.50
0.60
0.80
1.00
W
2
1.00
0.80
0.60
0.50
0.40
0.20
0.00
E(R i
)
0.20
0.18
0.16
0.15
0.14
0.12
0.10
E(
F port
)
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
0.20
E(R)
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
1
2
R ij
= +1.00
-
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
Different Weights
0.20
E(R)
0.15
0.10
With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset j k i
0.05
h g f
2
R ij
= +1.00
R ij
1
= 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
Different Weights
0.20
E(R)
0.15
0.10
With correlated assets it is possible to create a two asset portfolio between the first two curves j k i
0.05
h g f
2
R ij
= +1.00
R ij
R ij
1
= 0.00
= +0.50
-
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
0.20
E(R) With 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
-
R ij j k
= -0.50
i h
R ij
1 g
R ij
R
= 0.00
ij f
2
= +1.00
= +0.50
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 Exhibit 6.13
0.20
0.15
0.10
0.05
E(R)
R ij
= -1.00
R ij
= -0.50
g f
2 h i j
R ij
= +1.00
k
R ij
R ij
1
= 0.00
= +0.50
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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Results of portfolio allocation depend on accurate statistical inputs
Estimates of
Expected returns
Standard deviation
Correlation coefficient n(n-1)/2 correlation estimates: with 100 assets,
4,950 correlation estimates
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Single index market model :
R i
 a i
 b i
R m
  i b i
= the slope coefficient that relates the returns for security i to the returns for the aggregate stock market
R m
= the returns for the aggregate stock market
40/46

The correlation coefficient between two securities i and j is given as: r ij
 b i b j 
 m
 i
2 j where

2 m
 the variance of returns for the aggregate stock market
41/46

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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E(R)
Efficient Frontier for Alternative Portfolios
Exhibit 6.15
Efficient
Frontier
B
A C
Standard Deviation of Return
43/46

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 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
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Selecting an Optimal Risky Portfolio
E(R port
) Exhibit 6.16
U
3’
U
2’
U
1’
Y
U
3
U
2
U
1
X
E(
 port
)
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