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A lot of people missed the original thread on this subject several years
ago and Chuck's coin flipping example is being taken somewhat out of
context. The question at the time was what are the pluses and minuses of
reducing size during a drawdown. The plus (and it's a big one) is you
reduce your risk of total ruin. The minus is it takes you longer to
recover from a drawdown and your total profit over time is lower. If you
have a really good system, that never experiences the mother of all
drawdowns, you will come out ahead if you increase size, according to
the fixed fractional model, but never decrease size. Of course, we can
never count on our systems remaining good into the future so most of us
will elect to reduce size during a drawdown. We should do this knowing
there is a financial penalty.
Several people asked me privately to respond to Aaron's quite correct
statistical analysis below. It's accurate as far as it goes but I
believe it misses Chuck's point. We aren't asking about a hypothetical
future where it's possible to have 100% winners. Rather, let's look
backward at a series of trades that have already occurred. The question
now is would our account be bigger if we had reduced size during
drawdowns or if we had not. If you do the sims for a good system, the
answer is your account is bigger if you do not reduce size. The
difference between the two values is the "insurance premium" you pay to
give you some protection against going broke. Most of us, including me,
gladly pay that premium because survival is more important than
maximizing profits. However, as in most things, there is no free lunch
and there is a financial cost for our increased security.
--
Dennis
Schindler Trading wrote:
>
> 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
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