Ralph Vince - Portfolio Mathematics

200 THE HANDBOOK OF PORTFOLIO MATHEMATICS

This function is true only when the return (i.e., the HPR) is constant, which
is not the case in trading.
The real growth function in trading (or any event where the HPR is not
constant) is the multiplicative product of the HPRs. Assume we are trading
coffee, and our optimalfis one contract for every $21,000 in equity, and
we have two trades, a loss of $210 and a gain of $210, for HPRs of .99 and
1.01, respectively. In this example, our TWR would be:

TWR= 1. (^01) *. 99
=. 9999
An insight can be gained by using the estimated geometric mean (EGM),
which very closely approximates the geometric mean:
G=

√

A^2 −S^2

or:

G=

√

A^2 −V

where: G=geometric mean HPR

A=arithmetic mean HPR

S=standard deviation in HPRs

V=variance in HPRs

Now we take Equations (4.18) and (3.04) to the power ofnto estimate
the TWR. This will very closely approximate themultiplicativegrowth
function, the actual TWR, of Equation (4.17):

TWR=(

√

A^2 −S^2 )T (5.10)

where: T=Number of periods.

A=Arithmetic mean HPR.

S=Population standard deviation in HPRs.

The insight gained is that we can see, mathematically, the trade-off
between an increase in the arithmetic average trade (the HPR) versus the
dispersion in the HPRs (the standard deviations or the variance), hence the
reason that the 70% one-to-one system did better than the 10% twenty-eight-
to-one system.
Our goal should be to maximize the coefficient of this function, to max-
imize Equation (3.04): Expressed literally, to maximizethe square root of
the quantity HPR squared minus the variance in HPRs.