270 THE HANDBOOK OF PORTFOLIO MATHEMATICS
FIGURE 8.4 The efficient frontier with/without reinvestment
We let the AHPR at the same V as our geometric optimal portfolio be
our goal and use the geometric mean of our geometric optimal portfolio in
the denominator of the equation just mentioned. We can now discern how
many trades are required to make our geometric optimal portfolio match
one trade in the corresponding arithmetic portfolio. Thus:
T=ln(1.031)/ln(1.01542) (8.06)
=. 035294 /. 0153023
= 1. 995075We would thus expect 1.995075, or roughly 2, trades for the optimal
GHPR to be as high up as the corresponding (same V) AHPR after one
trade.
The problem is that the ATWR needs to reflect the fact that two trades
have elapsed. In other words, as the GTWR approaches the ATWR, the ATWR
is also moving upward, albeit at a constant rate (compared to the GTWR,
which is accelerating). We can relate this problem to Equations (8.07) and
(8.06), the geometric and arithmetic TWRs respectively, and express it math-
ematically: