Chapter 7
An Introduction to Portfolio
Management
Why Should Capital Markets
Be Efficient?
The premises of an efficient market



A large number of competing profit-maximizing
participants analyze and value securities, each
independently of the others
New information regarding securities comes to the
market in a random fashion
Profit-maximizing investors adjust security prices rapidly
to reflect the effect of new information
Conclusion: the expected returns implicit in the
current price of a security should reflect its risk
Alternative
Efficient Market Hypotheses



Weak-form efficient market hypothesis
Semistrong-form EMH
Strong-form EMH
Efficient Capital Markets

Joint hypothesis problem




market efficiency must be tested at same time as
test’ of asset pricing model being used to generate
expected returns
no one APM shown to be “the true model” that
represents how returns are generated
for this reason, some believe market efficiency is
not truly able to be tested
Fama explains that we may never be able to say
for sure whether markets are efficient or not,
results of tests are still worthwhile
Implications of
Efficient Capital Markets


Overall results indicate the capital markets
are efficient as related to numerous sets
of information
There are substantial instances where the
market fails to rapidly adjust to public
information


So, what techniques will or won’t work?
What do you do if you can’t beat the market?
Efficient Markets
and Portfolio Management



Management depends on analysts
With superior analysts, follow them and look for
opportunities in neglected stock
Without superior analysts, passive management
may outperform active management by


Build a globally diversified portfolio with a risk level
matching client preferences
Minimize transaction costs (taxes, trading turnover,
liquidity costs)
The Rationale and
Use of Index Funds


Efficient capital markets and a lack of superior
analysts imply that many portfolios should be
managed passively (so their performance
matches the aggregate market, minimizes the
costs of research and trading)
Institutions created market (index) funds which
duplicate the composition and performance of a
selected index series
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, most
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
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
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
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 investments - weighted
average of the expected rates of return for
the individual investments in the portfolio
Expected Return for an
Individual Risky Investment
Exhibit 7.1
Probability
0.35
0.30
0.20
0.15
Possible Rate of
Return (Percent)
0.08
0.10
0.12
0.14
Expected Return
(Percent)
0.0280
0.0300
0.0240
0.0210
E(R) = 0.1030
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(Rport) 0.1150
n
E(R port)   Wi R i
i 1
Exhibit 7.2
where :
Wi  the percent of the portfolio in asset i
E(R i )  the expected rate of return for asset i
Variance of Returns for an
Individual Investment
Variance is a measure of the variation of
possible rates of return Ri, from the
expected rate of return [E(Ri)]
Standard deviation is the square root of
the variance
Variance 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
Standard Deviation of Returns
for an Individual Investment
Standard Deviation
( ) 
n
 [R
i 1
2
i
- E(R i )] Pi
Standard Deviation of Returns
for an Individual Investment
Exhibit 7.3
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
0.103
0.103
0.103
0.103
-0.023
-0.003
0.017
0.037
0.0005
0.0000
0.0003
0.0014
Variance (  2) = .000451
Standard Deviation ( ) = .021237
Pi
0.35
0.30
0.20
0.15
[Ri - E(Ri )]2 Pi
0.000185
0.000003
0.000058
0.000205
0.000451
Covariance of Returns

A measure of the degree to which two
variables “move together” relative to their
individual mean values over time
Covariance of Returns
For two assets, i and j, the covariance of
rates of return is defined as:
Covij = sum{[Ri - E(Ri)] [Rj - E(Rj)]pi}
 Correlation coefficient varies from -1 to +1

rij 
Cov ij
 i j
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
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
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
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
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
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
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
Portfolio Risk-Return Plots
Exhibit 7.13
for 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
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:
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
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
E(R)
Efficient
Frontier
A
Figure 8.9
B
C
Standard Deviation of Return
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
Selecting an Optimal Risky
Portfolio
Figure 8.10
E(R port )
U3’
U2’
U1’
Y
U3
X
U2
U1
E( port )