[R-SIG-Finance] Mathematical Expectation for a trading system

[Mark Knecht]

Hi Mark,

On Thu, Oct 15, 2009 at 1:28 AM, Mark Breman <breman.mark at gmail.com> wrote:
> Hhmmm, this is all very strange.
> The subject of "Mathematics of money management" is about optimizing trade
> size under reinvestment of profits (optimizing the Growth function G(f),
> i.e. finding the optimal f)

To be clear but somewhat off topic - Optimal f is only one of many
ways to size a position. Optimal f itself might not be appropriate for
your risk tolerances as it can cause large drawdowns.

> Ralph Vince warns the reader that the trade system to start with should have
> a positive Mathematical expectation to start with, because the optimal f can
> not turn a losing system into a winning system.

Independent of the calculations did your system make money
historically? If it did then it has a positive expectation.

> From the reactions here I conclude that it's impossible to calculate a
> reliable Mathematical expectation for the system to start with,

I do not understand this conclusion unless you are keying on the word
'reliable' and have some specific idea in mind about what that means.

> so what's
> the value of optimal f if you can never be confident that the system is
> profitable to start with?

The results of any sizing algorithm operating into the future is only
as valid as the idea that the system will continue to perform in a
similar manner. There is no guarantee trade by trade of anything, but
if the system's average return per trade was $1 over the past 1000
trades then we should expect that over the next 100 trades it will
also return $1 per trade. Further we should expect that we'll see
similar win/loss ratios and the largest winner and loser will
hopefully be smaller than the similar trades in the history of the
system. If that turns out to be true then we say the system is
continuing to operate as it did in the past.

However there are NO guarantees.


> Some reactions on this thread referred to the validity of historical data to
> future performance of the system.
> I think it's clear that there are no guarantees for the future whatsoever if
> we formulate expectations solely based on data from the past. I think
> this uncertainty is part of trading. But is it not possible to calculate a
> Mathematical expectation for a system based on historical results which only
> says something about the validity/performance of the system in the past?

I believe it is and that's what I do.

> Would the following approach be sensible/possible:
> Take the historical profits and loses from the system and look at the
> distribution of these results. If the distribution looks like a normal
> distribution (as expected for stock market returns, at least in theory), use
> the normal distribution to calculate the P (probability of winning or
> losing) and calculate the Mathematical expectation?

I don't know about this, but I'm not clear why it's needed unless you
have a requirement to trade systems that have 'normal' distributions
of trade-by-trade returns.

Hope this helps,
Mark

> -Mark-
> 2009/10/14 Mark Knecht <markknecht at gmail.com>
>>
>> On Wed, Oct 14, 2009 at 1:39 AM, Mark Breman <breman.mark at gmail.com>
>> wrote:
>> > Hello,
>> > In "The mathematics of money management" by Ralph Vince there is a
>> > formula
>> > for calculating the Mathematical Expectation of a game (in R pseudo
>> > code):
>> >
>> > ME  =  for(i in 1:N) { Pi * Ai}
>> >
>> > where
>> > P = Probability of winning or losing
>> > A = Amount won or lost
>> > N = Number of possible outcomes.
>> >
>> > Or in text: "Mathematical expectation is the amount you expect to make
>> > or
>> > lose, on average, each bet".
>> >
>> > Now suppose I want to know the Mathematical expectation of a trading
>> > system.
>> >
>> > I have a series of trade returns:
>> >
>> >> trades$PnL
>> >  [1]  -5.75  10.00  -1.25  96.00 -16.00 -35.00  29.00 -18.25  -2.25
>> > -10.25
>> > -21.75  -5.50   8.50 -20.50  -6.00  14.25  18.00
>> > [18]   3.75  -4.25  24.00  17.75  -9.50  11.25 -33.75   6.25 -28.00
>> > 1.00
>> >  36.75  14.00 -30.75  -0.50   6.75  19.25   5.25
>> > [35] -10.00 -23.25   9.25  11.00 -33.00 -19.00 -17.50  -5.50  -5.75
>> >  -8.50
>> > -24.50 -24.00   2.25  -1.00   0.75  -1.75  -2.25
>> > [52]   9.25  15.00  -2.25  -6.75   5.25  -4.75 -10.00  -2.00  63.50
>> > -18.00
>> > -18.00  58.00  -8.75   1.00 -36.75 -23.50 -64.00
>> > [69] -15.75 -10.00 -34.75  27.75 -57.00 204.75 -45.00 -71.00 133.75
>> >
>> > So I have A = trades$PnL and N=77, but how do I calculate P?
>> >
>> > -Mark-
>>
>> Hi Mark,
>>   The simple answer would be:
>>
>> 1) Look at all the data you have today. How many trades won? How many
>> trades total? P = Total wins/ total trades
>>
>> 2) Start trading. After some fixed number of trades - say 30 more
>> trades - how did the win/loss ratio compare?
>>
>>   Don't think only of the probability, but also how much does the
>> probability vary? I have systems that trade 4000 times in 6 months. I
>> constantly track win/loss ratios as a rolling calculation just to
>> watch how the system might be doing in a new market type. My system
>> might have a probability of winning 82% of the time over 4000 trades
>> but goes up and down by 5% when looking at any 100 consecutive trades.
>> (So 77%-87% wins) I consider that 'normal'. If it gets outside of 5% I
>> might stop trading it until it's back in the 'normal' range.
>>
>>   Note that I do this also for what you call 'A' since the product
>> represents the potential for making money if everything works out
>> 'normally'. ;-)
>>
>> HTH,
>> Mark
>
>