The Geometry of Mean Variance Portfolios 271
Since we know that when N=1, G will be less than A, we can rephrase
the question to “At how many N will G equal A?” Mathematically this is:
GHPRN= 1 +N*(AHPR−1) (8.08a)which can be written as:
1 +N*(AHPR−1)−GHPRN= 0 (8.08b)or
1 +N*AHPR−N−GHPRN= 0 (8.08c)or
N=(GHPR*N−1)/(AHPR−1) (8.08d)The N that solves (8.08a) through (8.08d) is the N that is required for the
geometric HPR to equal the arithmetic. All three equations are equivalent.
The solution must be arrived at by iteration. Taking our geometric optimal
portfolio of a GHPR of 1.01542 and a corresponding AHPR of 1.031, if we
were to solve for any of Equations (8.10a) through (8.10d), we would find
the solution to these equations at N=83.49894. That is, at 83.49894 elapsed
trades, the geometric TWR will overtake the arithmetic TWR for those TWRs
corresponding to a variance coordinate of the geometric optimal portfolio.
Just as the AHPR has a CML line, so too does the GHPR. Figure 8.5
shows both the AHPR and the GHPR with a CML line for both calculated
from the same risk-free rate.
The CML for the GHPR is calculated from the CML for the AHPR by the
following equation:
CMLG=
√
CMLA^2 −VT*P (8.09)
where: CMLG=The E coordinate (vertical) to the CML line to the
GHPR for a given V coordinate corresponding to P.
CMLA=The E coordinate (vertical) to the CML line to the
AHPR for a given V coordinate corresponding to P.
P=The percentage in the tangent portfolio, figured from
(8.02).
VT=The variance coordinate of the tangent portfolio.You should know that, for any given risk-free rate, the tangent portfolio
and the geometric optimal portfolio are not necessarily (and usually are