Page 70 - DCOM504_SECURITY_ANALYSIS_AND_PORTFOLIO_MANAGEMENT

Unit 2: Risk and Return

estimated. An investor may expect the TR (total return) on a particular security to be 10% for the Notes
coming year, but in truth this is only a "point estimate."

Probability Distributions

To deal with the uncertainty of returns, investors need to think explicitly about a security's
distribution of probable TRs In other words, investors need to keep in mind that, although they
may expect a security to return 10%, 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 a fixed rate of interest, the interest payment will be made
with 100 per cent 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 result of considering these outcomes and their
probabilities together is a probability distribution consisting of the specification of the likely
returns that may occur and the probabilities associated with these likely returns.

Probabilities represent the likelihood of various outcomes and are typically expressed as a
decimal (sometimes fractions are used). 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. Probability distributions can be either
discrete or continuous. With a discrete probability distribution, a probability is assigned to each
possible outcome. With a continuous probability distribution, an infinite number of possible
outcomes exists. The most familiar continuous distribution is the normal distribution depicted
by the well-known bell-shaped curve often used in statistics. It is a two-parameter distribution
in that the mean and the variance fully describe it.
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 average of all possible return
outcomes, where each outcome is weighted by its respective probability of occurrence. For
investors, this can be described as the expected return.
We have mentioned that it's important for investors to be able to quantify and measure risk.
To calculate the total risk associated with the expected return, the variance or standard deviation
is used. This is a measure of the spread or dispersion in the probability distribution; that is, a
measurement of the dispersion of a random variable around its mean. Without going into
further details, just be aware that the larger this dispersion, the larger the variance or standard
deviation. Since variance, volatility and risk can, in this context, be used synonymously, remember
that the larger the standard deviation, the more uncertain the outcome.
Calculating a standard deviation using probability distributions involves making subjective
estimates of the probabilities and the likely returns. However, we cannot avoid such estimates
because future returns are uncertain. The prices of securities are based on investors' expectations
about the future. The relevant standard deviation in this situation is the ex ante standard deviation
and not the ex post based on realized returns.

Although standard deviations based on realized returns are often used as proxies for ex ante
standard deviations, investors should be careful to remember that the past cannot always be
extrapolated into the future without modifications. Ex post standard deviations may be convenient,
but they are subject to errors. One important point about the estimation of standard deviation is

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