# Investment Analysis and Portfolio Management

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Investment Analysis & Portfolio Management (FIN630)
VU
Lesson # 33
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
2
σ = (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
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Investment Analysis & Portfolio Management (FIN630)
VU
themselves with their best estimate of return over the next year, or six months, or whatever.
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 (FIN630)
VU
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 will-call 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 (FIN630)
VU
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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