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Fixed fractional sizing does not have a negative expectation.
Chuck LeBeau has done a disservice to his readers. A system model that gets
exactly 5 wins and 5 losses and for every 10 trades and always gets exactly
5 wins and 5 losses is a poor model of a system. If you have ever traded a
strategy you'll recognize that a strategy can sometimes have a good run and
do 6 or 7 wins in a 10 trade set. And sometimes it'll have a bad set and
see only 3 or 4 wins in a 10 trade set.
A better model of a strategy is the binomial distribution. This is the
distribution of the number of times heads shows up when you flip a coin. We
assume each trade and each toss of the coin are independent. Chuck LeBeau
would have us believe that if we have tossed the coin 9 times (made 9
trades) and have gotten 5 heads (winners) and 4 tails (losers) then we will
automatically have a tail (loser) on the next flip (trade) -- might as well
skip the trade! That is not true. The next flip (trade) still has a 50/50
chance of turning up heads or tails (winner or loser for a strategy with a
50% chance of having a winner).
We won't necessarily have exactly 5 wins and 5 losses for every 10 trades.
If we flip a coin 10 times the most likely result is 5 heads and 5 tails,
but there is a chance of having anywhere from 0 heads and 10 tails all the
way up to 10 heads and 0 tails.
Now if we take Dennis' email where we bet 25% of the available capital each
trade, and start with $100, he showed that with 5 wins and 5 losses you
would end up with $72.42. But the full range of possibilities, with the
ending capital, and the chance of that possibility happening are:
0 wins, $5.63, 0.1%
1 win, $9.39, 1.0%
2 wins, $15.64, 4.4%
3 wins, $26.07, 11.7%
4 wins, $43.45, 20.5%
5 wins, $72.42, 24.6% (Dennis' example)
6 wins, $120.70, 20.5%
7 wins, $201.17, 11.7%
8 wins, $335.28, 4.4%
9 wins, $558.79, 1.0%
10 wins, $931.79, 0.1%
To get the expectation for this model of a strategy, we need to multply the
ending capital for each possibility by the chance of that possibility
occurring and then add across all possibilities. If you do this you'll find
that the strategy has a $100 expectation -- exactly what we started with!
Fixed fractional trading does not have a negative expectation. Nor does it
turn good systems into losing systems.
You might think intuitively -- "well my strategy has ups and downs and I'll
always be playing the largest size on the losers and then after the losers
I'll be playing smaller size when I have winners, so it makes sense that I
would lose money on a 50/50 system." I reply that the human brain is not
very good at intuiting probabilities. Each trade is independent. Whether
we previously had a winner or a loser and whether we just upped or reduced
the size, doesn't affect whether the next trade will be a winner or a loser.
Maybe it would be easier to think of it this way if you want to be
intuitive... Having 0 wins is just as likely as having 10 wins. With 0
wins we lose about $94. But with 10 wins we make a whopping $832! Average
these out and you are way ahead. Keep doing this with opposing pairs... 1
win loses less money than 9 wins gains. Etc. The positive expectation of
the five opposing pairs offsets the expected loss from exactly 5 wins.
I think backtesting and the proper use of the statistics gained in
backtesting are the most important thing in being a profitable trader and it
pains me to see Chuck LeBeau misleading people.
Regards,
Aaron Schindler, CFA
Schindler Trading
1243 Yorkshire Lane
Barrington, IL 60010
telephone: 847-719-2846
fax: 847-719-2846
email: aaron@xxxxxxxxxxxxxxxxxxxx
www.schindlertrading.com
----- Original Message -----
From: "John Lynch" <kiwi_trader@xxxxxxxxxxxx>
To: "Schindler Trading" <schindlertrading@xxxxxxxxxxx>;
<omega-list@xxxxxxxxxx>
Sent: Sunday, August 25, 2002 7:25 PM
Subject: Re: Geometric Capital Growth / Optimal-f
> Aaron,
>
> No references needed. Its a simple demonstration which I've
> borrowed from one of Chuck LeBeau's Traders Club Bulletins:
>
> "Here are the numbers: Risk is always 5% of current capital.
> (I'm going to round the numbers to two decimals.)
>
> Capital $ Risk W/L Account balance
> 100.0 5.00 L 95.00
> 95.00 4.75 L 90.25
> 90.25 4.51 L 85.74
> 85.74 4.29 L 81.45
> 81.45 4.07 L 77.38
>
> OK we are already tired of losing. Let's have five winners
> in a row and see if we can get our money back.
>
> Capital $ Risk W/L Account balance
> 77.38 3.87 W 81.25
> 81.25 4.06 W 85.31
> 85.31 4.27 W 89.58
> 89.58 4.48 W 94.06
> 94.06 4.70 W 98.76
>
> As you can see we had an equal number of winners and losers
> yet somehow we lost money. Perhaps it is because we had bad
> luck and got started in the wrong direction. Lets reverse
> the sequence of trades so that we start out on a winning
> streak instead of losing. Maybe that will help.
>
> Capital $ Risk W/L Account balance
> 100.00 5.00 W 105.00
> 105.00 5.25 W 110.25
> 110.25 5.51 W 115.76
> 115.76 5.79 W 121.55
> 121.55 6.08 W 127.63
>
> Looks good so far. Starting off with winners looks much
> better than starting with losses. But now we have five
> losers coming up.
>
> Capital $ Risk W/L Account balance
> 127.63 6.38 L 121.25
> 121.25 6.06 L 115.19
> 115.19 5.76 L 109.43
> 109.43 5.47 L 103.96
> 103.96 5.20 L 98.76
>
> Hmmm. It doesn't seem to matter if we start out with a
> string of winners or a string of losses. Somehow we wound up
> losing the same amount of money either way."
>
> The full bulletin is at:
>
> http://traderclub.com/discus/messages/107/681.html?SundayApr
> il3020001039pm
>
> I recommend Chuck's group for some interesting discussions
> although some of the members are a bit puerile. His
> bulletins are excellent material.
>
> So the problem is that if your system has marginal
> expectancy then fixed fractional will reduce it further.
> The other side of it makes up for this. On the positive
> side, because you reduce bet size in a losing streak you can
> start with a larger percentage bet for a given maximum
> drawdown. As you can see from my previous posting this not
> only gives you a higher return for a given maxDD but you
> also get a smaller maxDD at 2 Standard Deviations indicating
> a reduced risk of ruin.
>
> Regards, John
>
>
> ----- Original Message -----
> From: "Schindler Trading" <schindlertrading@xxxxxxxxxxx>
> To: "John Lynch" <kiwi_trader@xxxxxxxxxxxx>;
> <omega-list@xxxxxxxxxx>
> Sent: Sunday, August 25, 2002 11:55 AM
> Subject: Re: Geometric Capital Growth / Optimal-f
>
>
>
> Mr. Lynch:
>
> Fixed fractional has a negative expectancy? What do you
> mean? Do you have
> any references? I don't believe this is true, but I've got
> an open mind.
>
> At Schindler Trading we use fixed fractional position sizing
> and believe it
> is the wisest choice -- as I believe you also concluded
> after your Monte
> Carlo testing.
>
>
> Regards,
> Aaron Schindler, CFA
>
> Schindler Trading
> 1243 Yorkshire Lane
> Barrington, IL 60010
> telephone: 847-719-2846
> fax: 847-719-2846
> email: aaron@xxxxxxxxxxxxxxxxxxxx
> www.schindlertrading.com
>
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