The Leverage Space Portfolio Model in the Real World 409
With the second technique, to be presented now, we can extrapolate out
that line and hence seek its horizontal asymptote. Fortunately, lines derived
from the Equations (12.05), (12.05a), and (12.05b) do possess an asymptote
and are of the form:
RX′(b,q)=asymptote-variableA * EXP(−variableB∗q) (12.11)RX′(b,q) will be the surrogate point, the value along theyaxis for a
givenqalong thexaxis in the Cartesian plane.
We can use equation (12.11) as a surrogate for the actual calculations
in (12.05), (12.05a), or (12.05b) whenqgets too computationally expensive.
To do this, we need only know three values: the asymptote, variableA,
and variableB.
We can find these values by any method of mathematical minimization
whereby we minimize the squares of the differences between the observed
values and the values given by (12.11). Those values with the minimum sum
of the differences squared are those values that best fit this line, this proxy
of actualRX(b,q) values whenqis too computationally expensive.
The process is relatively simple. We take those values we were able to
calculate forRX(b,q). For each of these values, we compare corresponding
points derived from (12.11) and square the differences between the two. We
then sum the squares.
Thus, we have a sum of the squared differences of our points to (12.11)
for a given (asymptote, variableA, variableB). Proceeding with a mathe-
matical minimization routine (Powell’s, Downhill Simplex, even the genetic
algorithm, though this will be far from the most efficient means— for a list
and detailed explanation of these methods, see “Numerical Recipes,”^9 Press
et al.) we arrive at that set of variable values that minimizes the sum of the
differences squared between the observed points and their corresponding
points as given by (12.11).
Returning, for example, to our two-to-one coin toss, we had calculated
by equation (12.05) thoseRR(.6) values, and these were given in Table 4.2.
Here, using Microsoft Excel’s Solver function, we can calculate the param-
eters in (12.11) that yield the best fit:
asymptote 0.48406
variableA 0.37418
variableB 0.137892(^9) Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; and Vetterling, William
T. ,Numerical Recipes: The Art of Scientific Computing, (New York: Cambridge
University Press, 1986).