

Investment
Analysis & Portfolio Management
(FIN630)
VU
Lesson
# 34
PORTFOLIO
THEORY Contd...
ANALYZING
PORTFOLIO RISK:
Risk
Reduction: The Insurance
Principle:
To
begin our analysis of how a
portfolio of assets can reduce risk,
assume that all
risk
sources
in a portfolio of securities are
independent. As we add securities to
this portfolio,
the
exposure to any particular source of
risk becomes small.
According to the Law of
Large
Numbers,
the larger the sample size,
the more likely it is that
the sample mean will be close
to
the population expected value.
Risk reduction in the case
of independent risk
sources
can be thought of as the insurance
principle, named for the
idea that an
insurance
company
reduces its risk by writing
many policies against many
independent sources of
risk.
We
are assuming here that rates
of return on individual securities
are statistically
independent
such that any one security's
rate of return is unaffected by another's
rate of:
return.
In this situation, the standard
deviation of the portfolio is
given by,
σp
= σi
/ n 1/2
Diversification:
The
insurance principle illustrates
the concept of attempting to
diversify the risk involved
in
a
portfolio of assets (or
liabilities). In fact, diversification is
the key to the management
of
portfolio
risk, because it allows
investors;
significantly to
lower portfolio risk
without
adversely
affecting return.
Random
Diversification:
Random
or naive diversification refers to
the act of randomly diversifying
without regard to
relevant
investment characteristics such as
expected return and industry
classification. An
investor
simply selects a relatively
large number of securities
randomlythe proverbial
"throwing
a dart at the Wall Street
Journal page showing stock
quotes. For simplicity, we
assume
equal dollar amounts are
invested in each
stock.
Markowitz
Portfolio Theory:
Before
Markowitz, investors dealt
loosely with the concepts of
return and risk.
Investors
have
known intuitively for many
years that it is smart to diversify;
that is, not to "put
all of
your
eggs in one basket? Markowitz
however, was the first .to
develop the concept
of
portfolio
diversification in a formal way he
quantified the concept of
diversification. He
showed
quantitatively why and how portfolio
diversification works to reduce the
risk of a
portfolio
to an investor.
Markowitz
sought to organize the
existing thoughts and practices into, a
more formal
framework
and to answer a basic question.
Does the risk of a portfolio
equal to the sum
of
the risks of the individual
securities comprising it?
Markowitz was the first to
develop
a
specific measure of portfolio
risk and to derive the
Expected return and risk for
a portfolio
based
on covariance relationships. We consider
covariances in detail in the
discussion
193
Investment
Analysis & Portfolio Management
(FIN630)
VU
below.
Portfolio
risk is not simply a
weighted average of the
individual security risks.
Rather, as
Markowitz
first showed, we must
account for the
interrelationships among, security
returns
in
order to calculate portfolio
risk, and in order to reduce portfolio
risk to its minimum
level
for
any given level of return.
The reason we need to
consider these, interrelationships,
or
comovements,
among security
return.
In
order to remove the
inequality sign from
Equation and develop that will
calculate the risk
of a
portfolio as measured by the
variance or standard deviation, we must
account for two
factors;
1.
Weighted individual security
risks (i.e. the variance of
each individual
security,
weighted
by the percentage of investable
funds placed in each individual
security.)
2.
Weighted comovements between
securities returns (i.e.,
the coyariance between,
the
securities
returns, again weighted by the
percentage of investable funds placed
in
each
security).
Measuring
Comovements in Security
Returns:
Covariance
is an absolute measure of the
comovements between security
returns used in the
calculation
of portfolio risk. We need
the actual covariance
between securities in a
portfolio
in
order to calculate portfolio
variance or standard deviation. Before
considering covariance,
however,
we can easily illustrate how
security returns move
together by considering
the
correlation
coefficient, a relative measure' of
association learned in statistics.
Correlation
Coefficient:
As
used in portfolio theory,
the correlation coefficient ρij (pronounced "rho")
is a statistical
measure
of the relative comovernents
between security returns. It
measures the extent
to
which
the returns on any two
securities are related,
however, it denotes only association,
not
causation.
It is a relative measure of association
that is bounded by +1.0
'and1.0, with;
ρij =
+1.0
=
perfect positive
correlation
ρij
=
1.0
=
perfect negative (inverse)
correlation
ρij
=
0.0
=
zero correlation
Covariance:
Given
the significant amount of
correlation among security
returns, we must measure
the
actual
amount of comovement and incorporate it
into any measure of
portfolio risk,
because
such
comovements affect the
portfolio's variance (or standard
deviation). The
Covariance
measure
does this.
The
covariance is an absolute measure of
the degree of association between
the returns for a
pair
of securities. Covariance is defined as
the extent to which two
random variables
covary
(move
together) over time. As is
true throughout our
discussion, the variables in
question
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Investment
Analysis & Portfolio Management
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are
the returns (TRs) on two
securities. As in the case of
the correlation coefficient,
the
covariance
can be:
1.
Positive, indicating that
the returns on the two
securities tend to move in
the same
direction
at the same time; when one
increases (decreases), the
other tends to do the
same.
When the covariance is
positive, the correlation
coefficient will also be
positive.
2.
Negative, indicating that
the returns on the two
securities tend to move
inversely;
when
one increases (decreases), the
other tends to decrease (increase),
When the
covariance
is negative, the correlation
coefficient will also be negative.
3.
Zero, indicating that the
returns on two securities
are independent and have
no
tendency
to move in the same or opposite
directions together.
The
formula for calculating
covariance on an expected basis
is;
m
σAB
= ∑
[RA,i E ( RA)] [RB,i
E(RB)] pri
i=1
Where;
σAB
= the
covariance between securities A and
B
RA
= one possible
return on 'security A
E ( RA) = the expected value of the
return on security A ,
m
= the
number of likely outcomes
for a security for the
period
Covariance
is the expected value of the
product of deviations from
the mean. The size of
the
covariance
measure depends upon the
units of the variables
involved and usually
changes
when
these units are changed.
Therefore, the, covariance
primarily provides
information
about
whether the association between
variables is positive, negative, or
zero because
simply
observing the number itself
is not very useful.
Relating
the Correlation Coefficient and the
Covariance:
The
covariance and the correlation
coefficient can be related in the
following manner:
ρAB
= σ
AB
/ σA
σB
This
equation shows that the
correlation coefficient is simply
the covariance standardized
by
dividing by the product of
the two standard deviations of
returns.
Given
this definition of the
correlation coefficient, the
covariance can be written
as;
σ
AB
= ρAB
σA
σB
Therefore,
knowing the correlation
coefficient, we can calculate the
covariance because
the
standard
deviations of the assets
rates of return will already be
available. Knowing
the
covariance,
we can easily calculate the
correlation coefficient.
Calculating
Portfolio Risk:
Co
variances account for the
comovements in security returns; we
are ready to
calculate
portfolio
risk. First, we will consider
the simplest possible case,
two securities, in order
to
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Investment
Analysis & Portfolio Management
(FIN630)
VU
see
what is happening in the
portfolio risk equation. We will
then consider the case of
many
securities,
where the calculations soon
become too large and complex
lo analyze with any
means
other than a
computer.
THE
nSECURITY CASE:
The
twosecurity case can be generalized to
the nsecurity case.
Portfolio risk can be
reduced by
combining assets with less
than perfect positive
correlation. Furthermore,
the
smaller
the positive correlation,
the better.
Portfolio
risk is a function of each
individual security's risk and
the covariances between
the
returns
on the individual securities.
Stated in terms of variances
portfolio risk is;
n
n
n
σ2p
= ∑
wi2 σi2
+ ∑ ∑
wi wj σij
i=1
i=1
j=1
i≠j
Where
σ2p
= the
variance of the return on
the portfolio
σi2
= the
variance of return for
security
σij
= the
covariance between the
returns for securities i and
j
.
wi
= the
portfolio weights or percentage of
investable funds invested in
security i
n
n
= a
double summation sign
indicating that n2 numbers are to be
added
∑ ∑
together
(i.e., all possible pairs of values
for i and j)
i =1 j
=1
It
states exactly the same
messages for the twostock
portfolio. This message is
portfolio
risk
is a function of;
·
The
weighted risk of each
individual security (as
measured by its
variance)
·
The
weighted covariance among
all pairs of securities
Note
that three variables
actually determine portfolio
risk: variances, covariances,
and
weights.
Simplifying
the Markowitz Calculations:
In
the case of two securities,
there are, two covariances,
and we multiply the
weighted
covariance
term by two,' since the
covariance of A with B is the
same as the covariance
of
B
with A. In the case of three
securities, there are six
covariances; with four
securities, 12
covariances;
and so forth, based on the
fact that the total
number of covariances in
the
Markowitz
model is calculated as n (n  1),
where n is the number of
securities.
For
the case of two securities,
there are n2, or
four,: total terms in
the matrixtwo
variances
and
two covariances. For the
case of four securities,
there are n2, or 16
total terms in the
matrixfour
variances and 12 covariances. The
variance terms are on the
diagonal of the
matrix;
in effect represent the covariance of a
security with itself.
Efficient
Portfolios:
Markowitz's
approach to portfolio selection is that
an investor should evaluate
portfolios on
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Investment
Analysis & Portfolio Management
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VU
the
basis of their expected returns and
risk as measured by the standard
deviation. He was
the
first to derive die concept of an
efficient portfolio, which is
defined as one that has
the
smallest
portfolio risk for a given
level of expected return or the
largest expected return
for
a
given level of risk.
Investors can identify efficient
portfolios by specifying an
expected
portfolio
return and minimizing the
portfoliorisk at this level of
return. Alternatively,
they
can
specify a portfolio risk
level they are willing to
assume and maximize the
expected
return
on the portfolio for this
level of risk. Rational
investors will seek efficient
portfolios,
because
these portfolios are
optimized on the two
dimensions of most importance
to
investors,
expected return and risk.
To
begin our analysis, we must
first determine the
riskreturn opportunities available to
an
investor
from a given set of
securities. A large number of possible
portfolios exist when
we
realize
that varyingpercentages of an investor's
wealth can be invested in each of
the assets
under
consideration investors should be
interested in only that
subset of the
available
portfolios
known as the efficient
set.
The
assets generate the
attainable set of portfolios, or
the opportunity set. The
attainable set
is
the entire set of all
portfolios that could be
found from a group of n
securities. However,
riskaverse
investors should be interested
only in those portfolios with
the lowest possible
risk
for any given level of
return. All other portfolios in
the attainable set are
dominated.
Using
the inputs described
earlierexpected returns, variances, and
covarianceswe can
calculate
the portfolio with the
smallest variance, or risk,
for a given level of
expected
return
based on these inputs. Given
the minimumportfolios, we can plot
the minimum
variance
frontier. Point A represents, the
global minimumvariance portfolio,
because no
other
minimumvariance portfolio has a
smaller risk. The bottom
segment of tile minimum
variance
frontier, AC, is dominated by
portfolios on the upper segment,
AB. For example,
since
portfolio X has a larger
return than portfolio Y for
the same level of risk,
investors
would
not want to own portfolio
Y.
The
segment of the minimumvariance frontier
above the global minimum
variance
portfolio,
AB, offers the best
riskreturn combinations available to
investors from this
particular
set of inputs, this segment is
referred to as the efficient
set of portfolios.
This
efficient
set is determined by the
principle of dominanceportfolio X
dominates portfolio
Y if it
has the same level of
risk but a larger expected
return or the same expected
return but
a
lower risk.
The
solution to the Markowitz
model revolves around the
portfolio weights, or
percentages
of
investable funds to be invested in
each security. Because the
expected returns,
standard
deviations,
and correlation coefficients for
the securities being considered
are inputs in,
the
Markowitz
analysis, the portfolio
weights are the only
variable that can be manipulated
to
solve
the portfolio problem of
determining efficient
portfolios.
Think
of efficient portfolios as being
derived in the following
manner. The inputs
are
obtained
and a level of desired expected return
for a portfolio is specified;
for example, 10
percent.
Then all combinations of
securities that can be combined to
form a portfolio with
an expected
return of I0 percent are
determined, and the one with
the smallest variance
of
return
is selected as the efficient
portfolio. Next, a new level
of portfolio expected return is
specified
for example; 11 percent  and
the process is repeated.
This continues until
the
feasible
range of expected returns is processed. Of
.course, the problem could
be solved by
specifying
levels of portfolio risk and
choosing that portfolio with
the largest expected
return
for the specified level of
risk.
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