Investment Analysis & Portfolio Management
Lecture#0
9
PORTFOLIO THEORY
Measuring Risk:
Risk is often associated with the dispersion in the likely outcomes. Dispersion refers to
variability. Risk is assumed to arise out of variability', which is consistent with our
definition of risk as the chance that the actual outcome of an investment will differ
from the expected outcome. If an asset's return has no variability, in effect it has no
risk. Thus, a one-year treasury bill purchased to yield 10 percent and held to maturity
will, in fact, yield (a nominal) 10 percent No other outcome is possible, barring default
by the U.S. government, which is not considered a reasonable possibility.
Consider an investor analyzing a series of returns (TRs) for the major types of
financial asset over some period of years. Knowing the mean of .this series is not
enough; the investor also needs to know something about the variability in the returns.
Relative to the other assets, common stocks show, the largest variability (dispersion)
in returns, with small common stocks showing f ten greater variability. Corporate
bonds have a much smaller variability and therefore a more compact distribution of
returns. Of course, Treasury bills are the least risky. The "dispersion of annual returns
for bills is compact.
Standard Deviation:
The risk of distributions' can be measured with an absolute measure of dispersion, or
variability. The most commonly used measure of dispersion over some period of years
is the standard deviation, which measures the deviation of each observation from the
arithmetic mean of the observations and is a reliable measure of variability, because all
the information in a sample is used.
The standard deviation is a measure of the total risk of an asset or a portfolio. It
captures the total variability in the assets or portfolios return whatever the source of
that variability. The standard deviation can be calculated from the variance, which is
calculated as:
n
σ2 = ∑(X - X)
i=1
n-1
Where;
σ2 = the variance of a set of
values X = each value in the set
X = the mean of the observations
n = the number of returns in the
sample σ2 = (σ2) 1 / 2 = standard
deviation
Knowing the returns from the sample, we can calculate the standard deviation quite
easily.
Dealing with Uncertainty:
Realized returns are important for several reasons. For example, investors need to
know how their portfolios have performed. Realized returns, also can be particularly
important in helping investors to form expectations about future returns, because
investors must concern themselves with their best estimate of return over the next
year, or six months, or whatever.
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Investment Analysis & Portfolio Management
How do we go about estimating returns, which is what investors must actually do in
managing their portfolios?
The total return measure, TR, is applicable whether one is measuring realized returns;
or estimating, future (expected) returns. Because it includes everything the investor
can expect to receive over any specified future period, the TR is useful in
conceptualizing the estimated returns from securities.
Similarly, the variance, or its square root, the standard deviation, is an accepted
measure of variability for both realized returns and expected returns. We will calculate
both the variance and the standard deviation below and use them interchangeably as
the situation dictates. Sometimes it is preferable to use one and sometimes the other.
Using Probability Distributions:
The return an investor will earn from investing is not known; it must be estimated.
Future return is an expected return and may or may not actually be-realized. An
investor may expect the TR on a particular security to be 0.10 for the coming year, but
in truth this is only a "point estimate." Risk, or the chance that some unfavorable event
will occur, is involved when investment decisions are made. Investors are often overly
optimistic about expected returns.
Probability Distributions:
To deal with the uncertainty of returns, investors need to think explicitly about a:
security's distribution of probable TRs. ln other words, investors need to keep in mind
that, although they may expect a security to return 10 percent, for example, this is only
a one-point estimate of the entire range of possibilities. Given that investors must deal
with the uncertain future, a number of possible returns can, and will, occur.
In the case of a Treasury bond paying fixed rate of interest, the interest payment will
be made with l00 -percent certainty barring a financial collapse of the economy. The
probability of occurrence is 1.0; because no other outcome is possible.
With the possibility of two or more outcomes, which is the norm for common stocks,
each possible likely outcome must be considered and a probability of its occurrence
assessed. The probability for a particular outcome is simply the chance that the
specified outcome will occur. The result of considering these outcomes and their
probabilities together is a probability distribution consisting of the specification of the
likely outcomes that may occur and the probabilities associated with these likely
outcomes.
Probabilities represent the likelihood of various outcomes and are typically expressed
as a decimal. The sum of the probabilities of all possible outcomes must be 1.0,
because they must completely describe all the (perceived) likely occurrences.
How are these probabilities and associated outcomes obtained? In the final analysis,
investing for some future period involves uncertainty, and therefore subjective
estimates. Although past occurrences (frequencies) may be relied on heavily to
estimate the probabilities the past must be modified for any changes expected in the
future.
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Investment Analysis & Portfolio Management
Calculating Expected Return for a Security:
To describe the single most likely outcome from a particular probability distribution, it
is necessary to calculate its expected value. The expected value is the weighted
average of'all possible return outcomes, where each outcome is weighted by its
respective probability of occurrence. Since investors are interested in returns, we willcall this expected value the expected rate of return, or simply expected-return, and for
any security, it is calculated as;
m
E (R) = ∑Ri pri
i=1
Where;
E (R) = the expected return on a security'
Ri
= the ith possible return
pri
= the probability of the ith return Ri
m
= the number of possible returns
Calculating Risk for a Security:
Investors must be able to quantify and measure risk. To calculate the total risk
associated with the expected return, the variance or standard deviation is used, the
variance and, its square root, standard deviation, are measures of the spread or
dispersion in the probability distribution; that is, they measure the dispersion of a
random variable around its mean. The larger this dispersion, the larger the variance or
standard deviation.
To calculate the variance or standard deviation from the probability distribution, first
calculate the expected return of the distribution. Essentially, the same procedure used
to measure risk, but now the probabilities associated with the outcomes must be
included,
m
The variance of returns = σ2 = ∑- [Ri – E (R)]2pri
i=1
And
The standard deviation of returns = σ= (σ2)1/2
Portfolio Expected Return:
The expected return on any portfolio is easily calculated as a weighted average of the
individual securities expected returns. The percentages of a portfolio’s total value that
are invested in each portfolio asset are referred to as portfolio weights, which will
denote by w. The combined portfolio weights are assumed to sum to 100 percent of,
total investable funds, or 1.0, indicating that all portfolio funds are invested. That is,
n
w1 + w2 + … + wn = ∑wi = 1.0
i=1
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Investment Analysis & Portfolio Management
Portfolio Risk:
The remaining computation in investment analysis is that of the risk of the portfolio.
Risk is measured by the variance (or standard deviation) of the portfolio's return,
exactly as in the case of each individual security. Typically, portfolio risk is stated in
terms of standard deviation which is simply the square root of the variance.
It is at this point that the basis of modern portfolio theory emerges, which can be
stated as follows: Although the expected return of a portfolio is a weighted average of
its expected returns, portfolio risk (as measured by the variance or standard deviation)
is not a weighted average of the risk of the individual securities in the portfolio.
Symbolically,
n
E (Rp) = ∑wi E (Ri)
i= 1
But
n
σ2p ≠∑wi σ2i
i=1
Precisely, investors can reduce the risk of a portfolio beyond what it would be if risk
were, in fact, simply a weighted average of the individual securities' risk. In order to
see how this risk reduction can be accomplished, we must analyze portfolio risk in
detail.
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