154 THE HANDBOOK OF PORTFOLIO MATHEMATICS
We have already obtained a new geometric mean by equalizing the past
data. Thef$ variable, which is constant when we do not equalize the past
data, now changes continuously, as it is a function of the current underlying
price. Hence, our geometric average trade changes continuously as the price
of the underlying instrument changes.
Our threshold to the geometric also must be changed to reflect the
equalized data.
T=AAT/GATBiggest Loss/−f (4.13)
where: T=The threshold to the geometric.
AAT=The arithmetic average trade.
GAT=The geometric average trade.
f=The optimalf(0 to 1).
This equation can also be rewritten as:
T=AAT/GATf$ (4.13a)
Now, not only do the AAT and GAT variables change continuously as
the price of the underlying changes, so too does thef$ variable.
Finally, when putting together a portfolio of market systems we must
figure daily HPRs. These too are a function off$:
Daily HPR=D$/f$+ 1 (4.14)
where: D$=The dollar gain or loss on 1 unit from the previous day.
This is equal to (Tonight’s Close – Last Night’s Close)*
Dollars per Point.
f$=The current optimalfin dollars, calculated from
Equation (4.12). Here, however, the current price variable
is last night’s close.For example, suppose a stock tonight closed at $99 per share. Last night it
was $102 per share. Our biggest percentage loss is−15. If ourfis .09, then
ourf$ is:
f$=−. (^15) (^102) 1/−.09
=−15.3/−.09
= 170
Since we are dealing with only 1 share, our dollars per point value is $1. We
can now determine our daily HPR for today as:
Daily HPR =(99−102)* 1 / 170 + 1
=− 3 / 170 + 1
=−. 01764705882 + 1
=. 9823529412