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Peter,
Thanks for the reference. My only caution to readers of this List would
be to go back and re-read that section very, very carefully *before*
tossing out any high-optimal-f systems based solely upon Vince's
statement that, "...the lower the optimal f, the better the system".
Here's why:
Vince states the following: "So, there are two basic measures for
comparing systems, the geometric means at the optimal f's, with the
higher geometric mean being the superior system, and the optimal f's
themselves, with the lower optimal f being the superior system. Thus,
rather than having a single, one-dimensional measure of system
performance, we see that performance must be measured on a
two-dimensional plane, one axis being the geometric mean, the other
being the value for f itself. The higher the geometric mean at the
optimal f, the better the system. Also, the lower the optimal f, the
better the system" [this last sentence being the original sentence which
motivated these posts].
Vince next goes on to state that "Geometric mean does not imply anything
regarding drawdown. That is, a higher geometric mean does not mean a
higher (or lower) drawdown. The geometric mean only pertains to
return. The optimal f is the measure of minimum expected historical
drawdown as a percentage of equity retracement. A higher optimal f does
not mean a higher (or lower) return. We can also use these benchmarks
to compare a given system at a fractional f value and another given
system at its full optimal f value. Therefore, when looking at systems,
you should look at them in terms of how high their geometric means are
and what their optmal f's are".
He next goes on to cite the following example:
Geometric Mean Optimal f
System A 1.050 0.80
System B 1.025 0.40
He then states, "System A at half f level will have the same minimum
historical worst-case equity retracement (drawdown) of 40%, just as
System B's at full f, but System A's geometric mean at half f will still
be higher than System B's at the full amount. Therefore, System A is
superior to System B" [note: in spite of the fact that System B has the
lower optimal f value].
So what's wrong with this picture? In one sense, absolutely nothing
(given Vince's statement that "System A's geometric mean at half f will
still be higher than System B's at the full amount" is true). By
equating the two optimal f's, we can then directly compare the two
geometric means and choose the system with the higher of the two
geometric means to be the "superior" system. This is precisely what
Vince does.
However, go back and re-read Vince's original quote: "...the lower the
optimal f, the better the system" [the quote which motivated these
posts]. Taking this quote out of context, one would be led to compare
System A's optimal f at 0.80 with System B's optimal f at 0.40 and draw
the (incorrect) conclusion that the "lower optimal f" System B is
superior to System A...a clear contradiction to Vince's just arrived at
conclusion that "System A is superior to System B."
It is for this reason that I raised the Caution Flag about not
automatically throwing out high optimal f systems "based solely upon
Vince's statement that, '...the lower the optimal f, the better the
system'".
As far as what it would take to reconcile the apparent contradiction,
let's try for a statement that any rational person should agree upon.
First, in English:
Given two systems with equivalent risks, the system with the higher
return is the superior system. Conversely, given two systems with
equivalent returns, the system with the lower risk is the superior
system.
Translating from English to Vince-speak:
Given two systems with equivalent optimal f's, the system with the
higher geometric mean is the superior system. Conversely, given two
systems with equivalent geometric means, the system with the lower
optimal f is the superior system.
Clearly this not only makes intuitive sense but is also in line with
Vince's example above where he first equivalenced the optimal f's of the
two systems and then compared their geometric means to arrive at his
conclusion that "System A [even though it's the higher optimal f system]
is superior to System B".
Although Vince did not make an example using the converse of his example
(e.g. equivalencing the geometric means of the two systems and choosing
the system with the lowest optimal f), the following "thought
experiment" should justify our being able to do so:
First, we know (according to Vince) that when we cut System A's optimal
f value in half from 0.8 to 0.4, that "System A's geometric mean at half
f will still be higher than System B's at the full amount". This
implies that we must therefore cut System A's optimal f value by *more*
than half (and therefore to some value less than 0.40) in order to
equivalence the geometric means of the two systems. This then leads us
directly to the conclusion that, at equivalent geometric means, System A
(with the lower optimal f value) is the Superior system...a conclusion
which maintains the logic of our proposed statement.
Comparing one system to another, therefore, is not merely a matter of
comparing either optimal f's or geometric means...they must *both* be
taken into consideration in order to effect a valid comparison [a
statement in 100% agreement with Vince's statement that "...rather than
having a single, one-dimensional measure of system performance, we see
that performance must be measured on a two-dimensional plane, one axis
being the geometric mean, the other being the value for f itself"].
Dave
> Hey Dave
>
> Check on pg 81 about comparing trading systems in the Mathematics of
> Money Management book.
>
> Have a look at the example that follows on pg 81. You want to maximise
> geometric mean (return) whilst at the same time minimise optimal f
> (risk). And yes of course these are contradictory!!!
>
> In practice I found that systems with higher gmeans also tend to have
> higher optfs - I think this is what Vince is alluding to in your quote
> below. It is very difficult to design high return low risk systems -
> shame huh?
>
> So that leaves us with a tradeoff, the question is how to choose between
> different combinations of gmean/optf?
> I don't have a thoroughly researched answer to that question. Frankly I
> am more concerned with first satisfying the underlying assumptions, like
> the system having a positive mathematical expectation (not
> historical!!!). Crystal ball please...
>
> The rest comes easy-er . . .
>
> Later
>
> Peter
>
> >
> > Peter,
> >
> > Your quote of Vince's statement that, "...the lower the optimal f, the
> > better the system" takes me by suprise.
> >
> > I have read all three of Vince's books and do not recall him making such
> > a statement. As a matter of fact, on page 42 of his latest book, The
> > New Money Management, he states, "One quickly realizes that the better a
> > trading approach appears when traded on a one-unit basis, the higher the
> > optimal f is". This statement appears to directly contradict the
> > statement which you had quoted.
> >
> > Would you do me the favor of citing the book and page number containing
> > the statement you were referencing? It's quite obviously a passage I
> > need to go back to and re-read.
> >
> > Thanks.
> >
> > Dave
> >
> > > Subject:
> > > Re: optimal f values
> > > Date:
> > > Fri, 11 Sep 1998 14:52:04 +0800
> > > From:
> > > HeyPeter <heypeter@xxxxxxxxxxxxxx>
> > > To:
> > > omegalist <omega-list@xxxxxxxxxx>
> > >
> > >
> > > Oops . . took my own advice and read Vince's books again.
> > > I was incorrect in stating the higher the f the better the system.
> > > Someone was correct in stating that TWR was what you wanted to maximise
> > > ie geometric mean. To quote Vince:
> > >
> > > "The higher the geometric mean at the optimal f, the better the system.
> > > Also, the lower the optimal f , the better the system."
> > >
> > >
> > > Apologies
> > >
> > > Peter
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