344 THE HANDBOOK OF PORTFOLIO MATHEMATICS
where: T=The variableTderived from Equation (10.05).
FRAC=The initial active portion of funds in an account.
Geometric Mean=The raw geometric mean HPR; there is no
adjustment performed on it as there is in
Equation (5.07b).
Example:
Initial Active Equity Percentage=5% (i.e.,.05)
Geomean HPR per period= 1. 004171
T= 316We know at 316 periods, on average, the dynamic will begin to out-
perform the corresponding staticffor the same value off, per Equation
(10.05). This is the same as saying that, starting at an initial active equity of
5%, when the account is up by 13.63% (.05 * 1.004171^316 −.05), the dynamic
will begin to outperform the corresponding staticffor the same value off.
So, we can see that there is a minimum number of holding periods
which must elapse in order for the dynamic fractionalfto overtake its static
counterpart (prior to which, reallocation is harmful if implementing the
dynamic fractionalf, and, after which, it is harmful to trade the static
fractionalf), which can also be converted from a horizontal point to a
vertical one. That is, rather than a minimum number of holding periods, a
minimum profit objective can be used.
Reallocating when the equity equals or exceeds this target of active eq-
uity will generally result in a much smoother equity curve than reallocating
based onT, the horizontal axis. That is, most money managers will find it
advantageous to reallocate based on upside progress rather than elapsed
holding periods.
What is most interesting here is that for a given level of initial active
equity, the upside target will always be the same, regardless of what values
you are using for the geometric mean HPR orT! Thus, a 5% initial active
equity level will always see the dynamic overtake the static at a 13.63%
profit on the account!
Since we have an optimal upside target, we can state that there is, as
well, an optimal delta on the portfolio on the upside. So, what is the formula
for the optimal upside delta? This can be discerned by Equations (10.06a)
and (10.06b), where FRAC equals that fraction of active equity which would
be seen by satisfying Equation (10.09). This is given as:
FRAC=
(Initial Active Equity+Upside Target)
1 +Upside Target