The Geometry of Mean Variance Portfolios 269
FIGURE 8.3 The efficient frontier with/without reinvestment
Thus, after 50 trades, an arithmetic average HPR of 1.03 would havemade 1+ (^50) (1. 03 −1)= 1 + (^50) . 03 = 1 + 1. 5 = 2 .5 times our starting
stake. Note that this shows what happens when we do not reinvest our
winnings back into the trading program. Equation (3.06) is the TWR you
can expect when constant-contract trading.
Just as Figure 8.2 shows the TWRs, both arithmetic and geometric,
for one trade, Figure 8.3 shows them for a few trades later. Notice that
the GTWR line is approaching the ATWR line. At some point for N, the
geometric TWR will overtake the arithmetic TWR. Figure 8.4 shows the
arithmetic and geometric TWRs after more trades have elapsed. Notice that
the geometric has overtaken the arithmetic. If we were to continue with
more and more trades, the geometric TWR would continue to outpace the
arithmetic. Eventually, the geometric TWR becomes infinitely greater than
the arithmetic.
The logical question is, “How many trades must elapse until the geo-
metric TWR surpasses the arithmetic?” See the following equation, which
tells us the number of trades required to reach a specific goal:
T=ln(Goal)/ln(Geometric Mean)
where: T=The expected number of trades to reach a specific goal.
Goal=The goal in terms of a multiple on our starting stake, a
TWR.
ln( )=The natural logarithm function.