An Introduction to Geometrical Implications in Money Management
Editors note: Ralph Vince is a widely-acknowledged expert in the field of
trading and money management. This article on money management from Ralph
reminds us that deciding how much capital to risk per trade is every bit as
important as is the decision of when to trade (i.e., “when to risk”) and what
instrument to trade (i.e., “how to risk”). Although for advanced traders,
Vince’s discussion on money management is worthwhile reading for traders of all
types who want to understand- and safeguard against- the risks inherent in
trading.
In a discussion of money management, and the notion of Optimal f, it becomes
evident how the question of quantity to assume for a given position is every bit
as crucial to profit or loss as the timing of the position itself or the
instrument chosen to be traded.
By way of an example, suppose we have a simple 2 to 1 coin toss game, whereby if
the coin comes up heads, we win $2, and if tails, we lose $1. Betting different
percentages (f values) on each play, results in a different expected multiple on
our starting stake as the following graph illustrates:
The graph shows the multiple of our stake we bet on each play as the horizontal
axis. The vertical axis is the multiple we can expect to make on our stake. The
graph depicts this expectation after 40 plays of this 2 to 1 coin toss game.
Although the peak of expectation is at .25, there are many other critical points
of interest on this Optimal f graph. One of these points is where as the
function goes off to the right, the multiple comes down to 1.0 and then crosses
below 1.0. This occurs at the horizontal point of .5, or at an f of .5.
In other words, even in this very favorable game, where you win $2 and lose $1,
each with a .5 probability, if you bet more than .5 your stake on each play, you
will lose money, and as you continue to play you will go broke with a
probability that approaches certainty.
Every single market that can be traded, and every means of trading a market, has
an optimal f curve of similar shape. The peak and other points of interest may
change with each market approach, but the shape is the same and each has a point
where the function crosses below 1.0 as the function moves to the right.
So we can say that every single market and every single technique can be a loser
if you bet the wrong quantity!
As for what to buy or sell and the timing of such, we have so many tools- systems, chart reading, fundamental analysis, etc. There is no shortage of tools
for selection and timing. Yet, for the equally critical decision of quantity,
there is next to nothing. We have been operating in the dark on this evidently
critical element of trading.
Optimal f, and its evolution into the Leverage Space Portfolio model create a
paradigm for examining money management decisions where one had heretofore not
existed.
As someone begins to study these facets, a beautiful geometry begins to emerge.
There are considerable geometrical implications as by-products in determining
Optimal f and The Leverage Space Model. Often, these can dictate or illuminate
changes in our behavior.
For example, with each scenario we have a probability of its occurrence and an
outcome. For a given scenario set (i.e. component of a portfolio) we have an
assigned f value (between 0 and 1). From this f value, we can convert a
scenario’s outcome into a “Holding Period Return,” or “HPR.” That would be the
percent return we would see on our account equity, should that scenario
manifest, trading at that level of f.
In essence, a holding period can be either a trade, or a unit of time (you can
use days, weeks, months, or whatever you like). Using time is preferable to
using trades as with time you can “align” things up if you are trading more than
one market.
The “Holding Period Return,” or HPR, is simply what you would have made or lost
on one unit of a given market traded in a certain manner over that holding
period, expressed as a multiple.
One unit can be anything you like- it can be one futures contract, or 100. It
can 100 shares of stock- or a $1 risk in our 2:1 coin toss game. The point is
that it must be consistent across time.
So to figure our HPR, to express it as a multiple, we simply express the one
unit gain or loss as a percentage and add 1 to it.
“As a percentage of what?” you ask. Well, as a percentage of the biggest loss
divided by the f value that you are using.
So we can convert the f we are using to a variable we might call f$ as:
f$ = -Biggest Loss / f
then use this to calculate our HPR
HPR = 1 + GainOrLoss / f$
For example, in our 2:1 coin toss game, if we look at an f value of .25, we are
thus making 1 bet for every $4 in account equity (f$ is $4). Therefore, if we
win $2 on heads, the heads scenario can be said to have returned 50% on our
equity. Adding 1.0 to this value gives us the HPR. So for heads, with an f of
.25, our HPR is 1.5.
For tails, losing $1, we have a 25% loss, or an HPR for tails of .75.
Remember, we are looking to maximize the geometric growth of an account, since
what we have to trade with today is a function of what we made or lost
yesterday. Thus, we are interested in the geometric average of our HPRs. That
is, what we really end up with after a sequence of holding periods is the
multiplicative product of our HPRs. It is this multiplicative product that
represents the multiple made on our stake at the end of those holding periods.
And if we take the multiplicative product to the root of however many periods
there are, we get the Geometric Mean:
(HPR1 *…* HPRn) ^ 1/n
The Geometric Mean represents what we expect to make on our stake, each play, on
average, as a multiple.
The Arithmetic Mean, what most people call “the average,” expressed as:
(HPR1 *…* HPRn) / n
has essentially no meaning to us as traders, since what we have to trade with
today is a function of what we made or lost up to this point (yet, people still
misguidedly focus on “average trade” and “average return,” etc.).
We can estimate the Geometric Mean by the following formula:
As you can now see, this shows that the two are related per the Pythagorean
Theorem* as demonstrated in the following figure for our 2:1 coin toss at f=
.25:
From this figure, we can see that any reduction in variance (variance = standard
deviation squared = SD2) in HPRs is equivalent to an increase in the arithmetic
average HPR in terms of its effect on our geometric average HPR! In other words,
reducing the dispersion in returns is equivalent to increasing the average
return (and sometimes may be easier to accomplish).
This is why I say this (Optimal f and its extensive by-product geometry) is not
just merely a superior portfolio model, but is even more so a framework- a
paradigm for examining our actions.
There are numerous cataloged geometric relationships regarding this paradigm and
without a doubt many to still be discovered. Ultimately it presents a new
starting point for further study and analysis in an essentially undiscovered
field which can be as vast and interesting, and every bit as important,
as the study of timing and selection in the markets.
*The geometric mean HPR is very closely approximated from the arithmetic mean
HPR and variance in those HPRs.
Ralph Vince will be offering a 2 day intensive workshop on portfolio management for European professionals in Paris in late May or early June. The workshop will focus on how to use the money management techniques he has written about to satisfy the demands of professional money managers, and software to perform these tasks will be included. Interested persons should email events@ggrip.com .