Optimalf 153
Thus, we would optimally buy one share for every $166.67 in account
equity. If we used 100 shares as our unit size, the only variable affected would
have been the number of dollars per full point, which would have been 100.
The resultingf$ would have been $16,666.67 in equity for every 100 shares.
Suppose now that the stock went down to $3 per share. Ourf$ equation
would be exactly the same except for the current price variable, which
would now be 3. Thus, the amount to finance one share by becomes:
f$=−. (^15) (^3) 1/−.09
=−.45/−.09
= 5
We optimally would buy one share for every $5 we had in account equity.
Notice that the optimal fdoes not change with the current price of
the stock. It remains at .09. However, thef$ changes continuously as the
price of the stock changes. This doesn’t mean that you must alter a position
you are already in on a daily basis, but it does make it more likely to be
beneficial that you do so. As an example, if you are long a given stock and it
declines, the dollars that you should allocate to one unit (100 shares in this
case) of this stock will decline as well, with the optimalfdetermined off
of equalized data. If your optimalfis determined off of the raw trade P&L
data, it will not decline. In both cases, your daily equity is declining. Using
the equalized optimal fmakes it more likely that adjusting your position
size daily will be beneficial.
Equalizing the data for your optimalfnecessitates changes in the by-
products. We have already seen that both the optimalfand the geometric
mean (and hence the TWR) change. The arithmetic average trade changes
because now it, too, must be based on the idea that all trades in the past
must be adjusted as if they had occurred from the current price. Thus, in
our hypothetical example of outcomes on one share of+2,−3,+10, and
−5, we have an average trade of $1. When we take our percentage gains and
losses of+.1,−15,+.2, and−.1, we have an average trade (in percent) of
+.5. At $100 per share, this translates into an average trade of 100.05 or
$5 per trade. At $3 per share, the average trade becomes $.15(3.05).
The geometric average trade changes as well.
GAT=G(Biggest Loss/−f)
where: G=Geometric mean−1.
f=Optimal fixed fraction.
(and, of course, our biggest loss is always a negative number). This equation
is the equivalent of:
GAT=(geometric mean−1)f$