Portfolio Management-Learning
Objective
Basic assumptions of Markowitz portfolio theory?
What is meant by risk?
risk, and alternative measures of
Expected rate of return for an individual risky asset and a portfolio of assets?
Standard deviation of return for an individual risky asset and portfolio of assets?
Covariance and correlation between rates of return.
1

Portfolio Management-Learning
Objective
Portfolio diversification.
What is the risk-return efficient frontier?
What determines which portfolio on the efficient frontier is selected by an individual investor?
Asset Allocation Models.
2

Markowitz Portfolio Theory
Quantifies risk
Derives the expected rate of return for a portfolio of assets and an expected risk measure
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
3

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.
2. Investors minimize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.
4

Assumptions of
Markowitz Portfolio Theory
3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.
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.
5

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.
6

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.
7

Alternative Measures of Risk
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
8

Expected Rates of Return
For an individual asset - sum of the potential returns multiplied with the corresponding probability of the returns
For a portfolio of assets - weighted average of the expected rates of return for the individual investments in the portfolio
9

Computation of Expected Return for an Individual Risky Investment
Exhibit 7.1
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
10

Computation of the Expected
Return for a Portfolio of Risky
Assets
Weight (W i
)
(Percent of Portfolio)
Expected Security
Return (R i
)
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
Exhibit 7.2
where
W i
E(R
 i i

1
) the
 percent
the of the expected portfolio rate of in return asset for i asset i
11

Variance (Standard Deviation) of
Returns for an Individual
Investment
Variance (

2
)

Standard Deviation n  i

1
[ R i
E(R i
)]
2
P i
(

)
 n  i

1
[ R i
E(R i
)]
2
P i
12

Variance (Standard Deviation) of
Returns for an Individual
Investment
Exhibit 7.3
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
[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

13

Variance (Standard Deviation) of
Returns for a Portfolio
Exhibit 7.4
Computation of Monthly Rates of Return
Closing Closing
Date Price Dividend Return (%) 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%
14

Covariance of Returns
A measure of the degree to which two variables “move together” relative to their individual mean values over time
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
)]}
15

Covariance and Correlation
The correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations
16

Covariance and Correlation
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
17

Correlation Coefficient
It can vary only in the range +1 to -1. A value of +1 would indicate perfect positive correlation. This means that returns for the two assets move together in a completely linear manner. A value of –1 would indicate perfect correlation. This means that the returns for two assets have the same percentage movement, but in opposite directions
18

Portfolio Standard Deviation
Formula
 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
 of the portfolio the weights of the individual assets in the portfolio, where weights are determined by the proportion of value in the portfolio
 i
2
Cov
 the variance of rates of return for asset i ij
 the covariance between th e rates of return for assets i and j, where Cov ij
 r ij
 i
 j
19

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
20

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
21

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
22

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
23

Case j k h i l f g
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
24

Constant Correlation with Changing Weights
Asset E(R ) i
1 .10 r ij
= 0.00
Case j i k f g h
W
1
0.00
0.20
0.40
0.50
0.60
0.80
W
2
1.00
0.80
0.60
0.50
0.40
0.20
E(R i
)
0.20
0.18
0.16
0.15
0.14
0.12
25

Case h i f g j k l
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
26

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
27

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
28

Portfolio Risk-Return Plots for
Different Weights
Exhibit 7.13
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
29

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
30

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
 b i
R m
  i b i
= the slope coefficient that relates the returns for security to the returns for the aggregate stock market i
R m
= the returns for the aggregate stock market
31

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
32

Efficient Frontier for Alternative Portfolios
E(R)
Efficient
Frontier
B
Exhibit 7.15
A C
Standard Deviation of Return
33

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
34

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
35

Selecting an Optimal Risky
Portfolio
E(R port
)
U
3’
U
2’
U
1’
Exhibit 7.16
Y
U
3
U
2
U
1
X
E(
 port
)
36