# Investment Analysis and Portfolio Management

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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
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 randomly--the 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
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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 'and--1.0, 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 (FIN630)
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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)
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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 n-SECURITY CASE:
The two-security case can be generalized to the n-security 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
ij
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
∑ ∑
together (i.e., all possible pairs of values for i and j)
i =1 j =1
It states exactly the same messages for the two-stock 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 matrix--two variances
and two covariances. For the case of four securities, there are n2, or 16 total terms in the
matrix--four 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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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 portfolio-risk 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 risk-return opportunities available to an
investor from a given set of securities. A large number of possible portfolios exist when we
realize that varying-percentages 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,
risk-averse 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 earlier--expected returns, variances, and covariances---we can
calculate the portfolio with the smallest variance, or risk, for a given level of expected
return based on these inputs. Given the minimum-portfolios, we can plot the minimum-
variance frontier. Point A represents, the global minimum-variance portfolio, because no
other minimum-variance 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 minimum-variance frontier above the global minimum variance
portfolio, AB, offers the best risk-return 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 dominance--portfolio 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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