The
Law of Large Numbers (OCCURRENCES)
The law of large numbers is a
principle of probability according to which the
frequencies of events with the same likelihood
of occurrence even out, given enough trials or
instances. As the number of experiments
increases, the actual ratio of outcomes will
converge on the theoretical, or expected, ratio
of outcomes. What this means is that if you have
a trading edge which has an expected win rate of
70% then its likely that as your sample size of
trades increases so does the expected
probability of 70%.
This means that if you did
the same type of trading set up 10 times, 100
times, 1000 times and 10,000 times that the
percentage win rate for each interval would
stabilize around 70% as you move from 10 trades
all the way up to 10,000 trades.
I have run simulations of 10
trades based on a probability win of 70% and
found that in some cases the overall probability
of wins after 10 trades was 80%, 40%, 70%, 60%,
%50. However when I ran a number of simulations
for a 1000 trades I found that the outliers at
40% and 90% disappeared and I was left with
multiple occurrences of 60%, 65%, 70%, 75% and
80%. If I had carried onto 10,000 I believe
probably the 80% and 60% would have disappeared
and I would have been left with results much
closer to the 70%.
This law of large numbers is
so profound that even if your edge was 51%,
meaning you had a slight edge, and your risk to
reward was 1R, if you ran millions of trades
there would be enough significant volume of
trades to ensure that you were profitable
despite the fact you may think that 51% to 49%
is too small of an edge to be worth much
difference.
In fact see below see a short
extract from an article about the one hedge fund
in the world which has never lost more than 1
day in over 4 years.

Virtu’s combination of microscopic
electronic surveillance,
lightningfast algorithms, and
rigorous risk management also
explains why this relatively small
company has become the most
consistently profitable market maker
in the history of electronic
trading. “The discipline they apply
to the wholesale marketmaking game
is different than other firms,” says
Rich Repetto, an analyst at Sandler
O’Neill & Partners.
The consistency with which Virtu
earns a profit is almost beyond
belief. From 2009 to 2014 it lost
money on only one day. (The
aberration happened when it missed a
special dividend payment for a
stock, throwing off its model and
causing a sevenfigure loss.) In
2014, the last year it disclosed its
daily winloss ratio, Virtu was in
the black every day, generating
revenue of $723.1 million and net
income of $190.1 million.
Rather than go
for big trades that could blow up
and lose a lot of money, Virtu
prides itself on making small
amounts—as in $10—millions of times
a day. Between the Watcher’s
sporadic warnings on that Friday in
May, the company was making markets
in gold exchangetraded funds and
futures. Over a series of
23 transactions in Chicago and New
York, it earned all of $36. “They’re
consistently profitable in an area
that’s not known for that,” Repetto
says. Profitability, however, hasn’t
prevented the company’s share price
from slumping since its initial
public offering in 2015.
What the company wants to do over
and over is to sell to buyers and
buy from sellers, not take positions
of its own, and either hedge its
risks at the end of the day or go
home with no risk at all. It does
this with stocks, currencies,
futures, and fixedincome
securities. All told, Virtu makes
markets in more than 12,000
financial assets. The ruthless
efficiency at its core is most
evident when you consider that it
trades on more markets (230plus
venues in 35 countries around the
world) than it has employees (148 at
the end of 2015).
Cifu (The CFO), 51, points to
probability theory and the millions
of times Virtu trades every day to
explain its winning ways. The law of
large numbers dictates that the
company, seeking only to earn the
spread on each transaction and not
bet on the direction of markets,
will make money close to 50 percent
of the time. (It loses a lot, too—on
the whole its strategies just happen
to win slightly more than they
lose.) “This firm is about making a
tick,” Cifu says. “We don’t hold on
to positions.” 
You can read the whole
article by seeing the link on the right hand
bottom of this page under "Articles".
Another hedge fund which has
an identical achievement to Virtu is the
Czech based RSJ. See below an extract
that appeared in the UK Times on January 10th
2017 regarding the firms incredible achievements
over the years using an algorithmic strategy
that is based on the law of large numbers.
Its an astonishing
achievement considering just like Virtu
the edge they have is just 51% yet it is the
ability to find and place millions of trades per
day which creates their positive expectancy. And
this is why hedge funds like these that believe
and align themselves with the law of large
numbers experienced growth rates of over 20% in
2016 when the majority of the hedge fund
industry lost 1.5% for 2016.
However retail traders like
you and me cannot possibly trade even hundreds
of times per day. Its simply impossible. So the
point is this.
If you trade very few times
and infrequently then you will be exposed to
more randomness and variance in the way that
probabilities play out. This will mean you will
find it takes a much longer period of time to
get to where you would have if you had traded
more times. However what is the ideal sweet spot
for trading set ups per day, per week, per
month? I would say taking into account that over
trading could actually end up doing far more
damage to a human who is finite and experiences
tiredness, emotional swings, and external
factors, the sweet spot for me is around 46 trades per day.
Its also important to add
here that I am talking about high probability
trades, not taking low probability or suspect
trades just to meet the quota of making your
maximum trade volume target. If I only have to
take 1 trade per day because there is simply a
lack of high probability trades then so be it.
However when I make 6 trades and know I could
take an extra 23 for the day then I will do it.
THE LAW OF STREAKS
The law of streaks more than
anything was what prepared me for embracing
uncertainty and the normal distribution of
probabalistic trading results. I realized that
just because a system may have a 60% win rate it
doesn't necessarily mean in the first 10 trades
you will get around 6 winners and 4 losers. It
is entirely possible the first 5 or even 6
trades could end up as losses. Does this mean
the 60% win rate is flawed?
Not many traders or trading
systems delve intensely on the subject of winning
and losing streaks but for me it was fully
engaging with this scientific phenomenon which
helped me to understand that uncertainty was an
unavoidable part of trading and it was better
to understand possible worst case scenarios of a
trading strategy so that you could be mentally
prepared when they strike.
In the image directly below
you are seeing the probabilistic results for a
trading strategy which wins 60% of the time over
a sample size of 60 trades. Accepting that anything above 1% could
happen you will see that within a given
distribution of 60 trades that it is possible
to hit a streak of 15 positive
consecutive trades (2.60%). It is also possible
within 60 trades to have a streak of 8
winners in a row up to 3 times (1.10%).
However the table below shows the probabilities
of the same strategy with a probability of
40% of a losing trade. Accepting that anything
above 1% could happen you can see that there
there is always the chance in 60 trades that one could have a losing streak of 8 losses in a row
(0.78%). One could also have a losing streak of 5
trades twice within a sample of 60 trades
(1.40%).
The contrast between the two
tables shows that the odds of sustained periods
of streaks clearly favours the first table
representing winning trades rather than the last
table showing losing trades. However by
accepting that with a 1% chance of probability
an event could happen it means that within a
sample of 60 trades should one experience a losing
streak of 78 trades you don't have to question
the
trading strategy because the streak behaviour
falls within the expected range of probability.
The final thing to bear in
mind is that because the winning trades are
twice the size of the losing trades it means even if
one has a sustained streak of losses (drawdowns)
that the biased probability towards winning
trades means that it doesn't take a long to
recover from drawdowns. Also its more
likely for an event to happen the greater the
size of the probability. So for instance an
outcome with a value of 50% or more is more
likely to occur than an event which is below
10%. So clearly the strength of the strategy is weighted towards not just risk
reward but also win rate. The best of both
worlds.