mula for calculating the TWR, to gain a little better understanding of how the
entire process works.
Just as one can calculate an arithmetic average for any data series, so too can
we calculate an average value representing all the HPRs making up the TWR.
Because we are working with multiplication and compounded returns, however,
this average must be a geometric average. For example, to calculate the average
arithmetic return from three profitable trades of 3, 5, and 7 percent, respectively,
simply add them up and divide by the number of trades. In this case, the arithmetic
average equals 5 percent [(3 5 7) / 3]. Without compounding the returns, the
total return can be calculated either as (3 5 7 15), or as (3 * 5 15). With
an initial account equity of, for example, $100,000, the resulting balance is
$115,000 [100,000 * (1 15 / 100)].
Using compounding, we need to use multiplication instead of addition.
Therefore, the three different returns need to be expressed as 1.03 (1 3 / 100),
1.05, and 1.07. Had one of the returns been negative because of a losing trade, the
plus sign would have been replaced by a minus sign, so that, for example a 3 per-
cent losing trade would have been stated as 0.97 (1 3 / 100). Multiplying these
values would result in a compounded return of 1.1572 (1.03 * 1.05 * 1.07), which
would result in a compounded resulting balance over the three trades of $115,720
(100,000 * 1.1572). Now, to calculate the average return over the three trades, we
need to calculate the geometric mean(GM), which, because we’re dealing with
multiplications will be 1.0499 (1.1572(1/3)), or 4.99 percent [(1.0499 1) / 100].
A more general relationship between TWR, HPR, and GM can be expressed as:
TWR GMN HPR1 * HPR2 * HPR3 * ... * HPRi
So that:
GM (HPR1 * HPR2 * HPR3 * ... * HPRi)(1/N)TWR(1/N)
Where:
GM The geometric mean
N Number of trades
The formulas might look a little advanced, but don’t forget that the entire
process is akin to calculating the regular arithmetic average profit per trade,
expressed in percentages. Now, the next step also might look a little advanced, but
it really isn’t. Let’s just settle for the fact that there is an alternative way to calcu-
late GM that incorporates the arithmetic average profit, which we calculated at 5
percent in the previous example. This formula looks like this:
GM = (APM^2 – SD^2 )(1/2)
Where:
CHAPTER 25 Fixed Fractional Trading 303