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Yes, there is a formula, but it is difficult to describe via e-mail.
The main thing to remember is that since you have so
many "observations", i.e., hypothetical trades, the profit numbers
for all these trades follow a normal distribution law (as you
observed).
The key parameters that determine the normal distribution are the
mean and the standard deviation. You already know the mean, namely
$144 profit per trade. The standard deviation is the square root of
the average, squared deviation from the mean.
To calculate the standard deviation here is what you do. For each
trade, calculate the profit on that trade, then subtract 144 (the
answer here can be a negative number!) and then square the result.
Having done this for each trade, add up all the resulting answers and
divide the sum by the number of trades. Finally, take the square
root of this last number. This is the standard deviation.
The significance of the standard deviation is this. 66% of the
trades will show a profit within 1 standard deviation (either way) of
the average profit ($144). 90% of the trades will show a profit
within 2 standard deviations of the average profit.
Now what happens if you look at groups of trades instead of
individual trades?
To take an example, suppose you look at groups of 100 trades because
that is the average number of trades your system makes in a month. As
it happens, the average profit on groups of 100 trades will also
follow a normal distribution. In this example the mean of this
distribution will be 100 x $144 = $14,400. What about the standard
deviation?
Well, suppose that D was the standard deviation you calculated above
for the distribution of individual trades.
Then the standard deviation on groups of 100 trades would be the
square root of 100 times D,i.e., 10 x D. Notice that the standard
deviation goes up much more slowly than does the mean as you increase
the number of trades in a group. This means that as the number of
trades in a group goes up, the probability that there will be a
profit on the group of trades taken as a whole also goes up.
The way you calculate these probabilities is to use the normal
distribution tables found in statistical reference books. These
tables are tabulated for normal distributions with mean 0 and
standard deviation 1. To use them you have to convert your total
profit on a group of trades into a number that has a normal
distribution with mean 0 and standard deviation 1.
So let us say that P symbolizes the profit on a group of N trades and
that the standard deviation on individual trades (which we calculated
above ) is D.
Then [(P - (N x $144)) / (square root of N)x D ] will be a quantity
than follows a normal distribution with mean 0 and standard deviation
1. Let's call this quantity A.
Then the profit P on a group of N trades will be greater than 0 if
and only if A is greater than the number (-144 N)/ (D x square root
of N). So if you look up this last number in the statistical tables
for the normal distribution with mean 0 and standard deviation 1 you
will find next to it the probability that a group of N trades in your
system will show a net profit.
Carl
--- In realtraders@xxxx, "Erika Toth Fluke" <erika@xxxx> wrote:
> Hi Carl,
>
>
> Thanks for the explanation.
>
> Is the square root relationship changing if look at the monthly or
quarterly
> numbers (125, 355 trades)?
>
> Can this relationship be expressed in a formula?
>
> Thanks again,
> ***** Erika ***** :-)
>
>
> -----Original Message-----
> From: topos8 [mailto:topos8@x...]
> Sent: Monday, August 26, 2002 5:53 PM
> To: realtraders@xxxx
> Subject: [RT] Re: Help needed in mathematics
>
>
> The answer to your question is a standard exercise in probabilty
> theory.
>
> The reason that the probability of a profitable week is higher
that
> the probability of a profitable trade is simple. There are 28
trades
> per week and each trade on average makes $144. So if you divide
the
> total weekly profit by the number of trades that week you will
get a
> number whose average is again $144 but whose variablity is much
lower
> (by a factor proportional to the square root of 28) than the
> variability of the profit on a single trade. So you are much more
> likely to make money in a typical week of many trades than you
are to
> make money on a single trade.
>
> The same phenomenon explains why owning a casino is the sure road
to
> wealth: On any give roll of the dice or deal of the cards the
casino
> has only a small advantage over the customer. But when you look
at
> the casino's results over millions of dice rolls etc, it is a sure
> winner.
>
> Carl
>
>
> --- In realtraders@xxxx, "Erika Toth Fluke" <erika@xxxx> wrote:
> > HI,
> >
> > I'm trying to prove that a 422 sample win/loss distribution's
> profitability
> > will increase if I look at the weekly, monthly etc. data.
> >
> > The original dataset gives 45% win/loss on the trade by trade
basis
> and
> > 71.6% on the weekly basis (about 28 trades/week).
> >
> > The data has the characteristics of the normal distribution,
where
> the
> > average trade wins: $144
> > standard deviation is: $1266.34
> >
> > Can somebody point it out why the weekly profitability
increases so
> > significantly or show me a formula that can be used to
calculate the
> > increase in profitability depending on the time frame?
> >
> > The result is there but, somehow I'm just not getting it.
> >
> > Thanks for your help.
> >
> > ***** Erika ***** :-)
>
>
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