328 THE HANDBOOK OF PORTFOLIO MATHEMATICS
(number of days, since the geometric means are daily) required to double
the static fractionalfis given by Equation (5.07) as:
ln(2)
ln(1.005)= 138. 9751
To double the dynamic fractionalfrequires setting the goal to 6. This
is because, if you initially have 20% of the equity at work, and you start out
with a $100,000 account, then you initially have $20,000 at work. The goal is
to make the active equity equal $120,000. Since the inactive equity remains
at $80,000, you will have a total of $200,000 on your account that started
at $100,000. Thus, to make a $20,000 account grow to $120,000 means you
need to achieve a TWR of 6. Therefore, the goal is 6 in order to double a .2
dynamic fractionalf:
ln(6)
ln(1.01933)= 93. 58634
Notice how it took 93 days for the dynamic fractionalfversus 138 days for
the static fractionalf.
Now let’s look at the .1 fraction. The number of days expected in order
for the static technique to double is expected as:
ln(2)
ln(1.002577)= 269. 3404
If we compare this to doubling a dynamic fractionalfthat is initially set
to .1 active, you need to achieve a TWR of 11. Hence, the number of days
required for the comparative dynamic fractionalfstrategy is:
ln(11)
ln(1.01933)= 125. 2458
To double the account equity, at the .1 level of fractionalfis, therefore,
269 days for our static example, compared to 125 days for the dynamic. The
lower the fraction forf, the faster the dynamic will outperform the static
technique.
Let’s take a look at tripling the .2 fractionalf.The number of days
expected by static technique to triple is:
ln(3)
ln(1.005)= 220. 2704
This compares to its dynamic counterpart, which requires:
ln(11)
ln(1.01933)