Classical Portfolio Construction 235
In Markowitz’s work, risk was quantified for the first time. He described
risk as the variation in a portfolio’s returns, a definition many people have
challenged.
DEFINITION OF THE PROBLEM
For the moment we are dropping the entire idea of optimalf; it will catch
up with us later. It is easier to understand the derivation of the efficient
frontier if we begin from the assumption that we are discussing a portfolio
of stocks. These stocks are in a cash account and are paid for completely.
That is, they are not on margin.
Under such a circumstance, we derive the efficient frontier of portfo-
lios.^2 That is, for given stocks we want to find those with the lowest level
of expected risk for a given level of expected gain, the given levels being
determined by the particular investor’s aversion to risk. Hence, this basic
theory of Markowitz (aside from the general reference to it as Modern Port-
folio Theory) is often referred to asE–Vtheory (Expected return – Variance
of return). Note that the inputs are based on returns. That is, the inputs to
the derivation of the efficient frontier are the returns we would expect on a
given stock and the variance we would expect of those returns. Generally,
returns on stocks can be defined as the dividends expected over a given
period of time plus the capital appreciation (or minus depreciation) over
that period of time, expressed as a percentage gain (or loss).
Consider four potential investments, three of which are stocks and one a
savings account paying 8^1 / 2 % per year. Notice that we are defining the length
(^2) In this chapter, an important assumption is made regarding these techniques. The
assumption is that the generating distributions (the distribution of returns) have
finite variance. These techniques are effective only to the extent that the input data
used has finite variance. For more on this, see Fama, Eugene F., “Portfolio Analysis
in a Stable Paretian Market,”Management Science11, pp. 404–419, 1965. Fama has
demonstrated techniques for finding the efficient frontier parametrically for stably
distributed securities possessing the same characteristic exponent, A, when the re-
turns of the components all depend upon a single underlying market index. Readers
should be aware that other work has been done on determining the efficient frontier
when there is infinite variance in the returns of the components in the portfolio.
These techniques are not covered here other than to refer interested readers to per-
tinent articles. For more on the stable Paretian distribution, see Chapter 2. For a
discussion of infinite variance, see “The Student’s Distribution” in Chapter 2.