266 THE HANDBOOK OF PORTFOLIO MATHEMATICS
Thus, the CML line at the standard deviation coordinate .08296, the last
entry in the table, is divided by the standard deviation coordinate of the
tangent portfolio, .02986, yielding 2.7782, or 277.82%.
The last column in the table, the CML line AHPR, is the AHPR of the
CML line at the given standard deviation coordinate. This is figured as:
ACML=(AT*P)+((1+RFR)*(1−P)) (8.03)where: ACML=The AHPR of the CML line at a given risk coordinate,
or a corresponding percentage figured from (8.02).
AT=The AHPR at the tangent point, figured from (8.01a).
P=The percentage in the tangent portfolio, figured from
(8.02).
RFR=The risk-free rate.On occasion you may want to know the standard deviation of a certain
point on the CML line for a given AHPR. This linear relationship can be
obtained as:
SD=P*ST (8.04)where: SD=The standard deviation at a given point on the CML line
corresponding to a certain percentage, P, corresponding
to a certain AHPR.
P=The percentage in the tangent portfolio, figured from
(8.02).
ST=The standard deviation coordinate of the tangent
portfolio.The Geometric Efficient Frontier
The problem with Figure 8.1 is that it shows the arithmetic average HPR.
When we are reinvesting profits back into the program we must look at the
geometric average HPR for the vertical axis of the efficient frontier. This
changes things considerably. The formula to convert a point on the efficient
frontier from an arithmetic HPR to a geometric is:
GHPR=
√
AHPR^2 −V)
where: GHPR=The geometric average HPR.
AHPR=The arithmetic average HPR.
V=The variance coordinate. (This is equal to the
standard deviation coordinate squared.)