Download Portfolio Theory

Portfolio Management-Learning
Objective
Basic assumptions of Markowitz portfolio theory?
What is meant by risk, and alternative measures of
risk?
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
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
Exhibit 7.2
i 1
where :
Wi  the percent of the portfolio in asset i
E(R i )  the expected rate of return for asset i
11
Variance (Standard Deviation) of
Returns for an Individual
Investment
n
Variance ( )   [R i - E(R i )] Pi
2
2
i 1
Standard Deviation
( ) 
n
 [R
i 1
2
i
- E(R i )] Pi
12
Returns for an Individual
Investment
Exhibit 7.3
Possible Rate
Expected
of Return (R i )
Return E(R i )
Ri - E(Ri )
[Ri - E(Ri )]
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
2
Pi
0.25
0.25
0.25
0.25
2
[Ri - E(Ri )] Pi
0.000225
0.000025
0.000025
0.000225
0.000500
Variance (  2) = .0050
Standard Deviation ( ) = .02236
13
Variance (Standard Deviation) of
Exhibit 7.4
Returns for a Portfolio
Computation of Monthly Rates of Return
Date
Dec.00
Jan.01
Feb.01
Mar.01
Apr.01
May.01
Jun.01
Jul.01
Aug.01
Sep.01
Oct.01
Nov.01
Dec.01
Closing
Price
Dividend
Return (%)
60.938
58.000
-4.82%
53.030
-8.57%
45.160
0.18
-14.50%
46.190
2.28%
47.400
2.62%
45.000
0.18
-4.68%
44.600
-0.89%
48.670
9.13%
46.850
0.18
-3.37%
47.880
2.20%
46.960
0.18
-1.55%
47.150
0.40%
E(RCoca-Cola)= -1.81%
Closing
Price
Dividend Return (%)
45.688
48.200
5.50%
42.500
-11.83%
43.100
0.04
1.51%
47.100
9.28%
49.290
4.65%
47.240
0.04
-4.08%
50.370
6.63%
45.950
0.04
-8.70%
38.370
-16.50%
38.230
-0.36%
46.650
0.05
22.16%
51.010
9.35%
E(Rhome
E(RExxon)=
Depot)= 1.47%
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:
Covij = E{[Ri - E(Ri)][Rj - E(Rj)]}
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
rij 
Cov ij
 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
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 
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
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
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
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
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
E(Ri )
0.20
0.18
0.16
0.15
0.14
0.12
0.10
23
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
24
Constant Correlation
with Changing Weights
Asset
1
E(R i )
.10
rij = 0.00
with changing weig
2 Constant correlation
.20
Case
W1
W2
f
g
h
i
j
k
0.00
0.20
0.40
0.50
0.60
0.80
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
Constant Correlation
with Changing Weights
Case
W1
W2
E(Ri )
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
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
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
27
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
28
Portfolio Risk-Return Plots for
Exhibit 7.13
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
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  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
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
Exhibit 7.15
E(R)
Efficient
Frontier
A
B
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
Exhibit 7.16
E(R port )
U3’
U2’
U1’
Y
U3
X
U2
U1
E( port )
36