238 THE HANDBOOK OF PORTFOLIO MATHEMATICS
of the securities under consideration and calculate the returns and their
variances over the specified holding periods. Again the termreturnsmeans
not only the dividends in the underlying security, but any gains in the value
of the security as well. This is then specified as a percentage.Varianceis the
statistical variance of the percentage returns. A user of this approach would
often perform a linear regression on the past returns to determine the return
(the expected return) in the next holding period. The variance portion of the
input would then be determined by calculating the variance of each past data
point from what would have been predicted for that past data point (and
not from the regression line calculated to predict the next expected return).
Rather than gathering these figures empirically, the investor can also simply
estimate what he or she believes will be the future returns and variances
in those returns. Perhaps the best way to arrive at these parameters is to
use a combination of the two. The investor should gather the information
empirically, then, if need be, interject his or her beliefs about the future of
those expected returns and their variances.
The next parameters the investor must gather in order to use this tech-
nique are the linear correlation coefficients of the returns. Again, these
figures can be arrived at empirically, by estimation, or by a combination of
the two.
In determining the correlation coefficients, it is important to use data
points of the same time frame as was used to determine the expected returns
and variance in returns. In other words, if you are using yearly data to
determine the expected returns and variance in returns (on a yearly basis),
then you should use yearly data in determining the correlation coefficients.
If you are using daily data to determine the expected returns and variance
in returns (on a daily basis), then you should use daily data in determining
the correlation coefficients.
It is also very important to realize that we are determining the correla-
tion coefficients ofreturns(gains in the stock price plus dividends), not of
the underlying price of the stocks in question.
Consider our example of four alternative investments—Toxico, In-
cubeast Corp., LA Carb, and a savings account. We designate these with
the symbols T, I, L, and S, respectively. Next, we construct a grid of the
linear correlation coefficients as follows:
ILST −.15 .05 0
I .25 0
L0