You Do The Math

Over at Money Skater's blog, in the comment section (link here), a blogger posted his trading record. I have cut and repasted the record below.

Total trades.......123
Wins...............123
Losses.............0
% Return on Capital 26.2%
From 20 July 2007

This record provoked doubt in the minds of the blog readers. The probability of such a string of trades was questioned. The blogger responded, "Mathmatically [sic.] impossible you say, not at all..."

Since the mathematical likelihood of such a string of trades was briefly addressed and promptly dismissed, I decided that it might be illuminating to examine the probability of making, in a row, 123 winning trades.

First, we must assign a probability for a trade being a winner or a loser. My own trading record shows that 4 out of 10 of my trades are winners (40%). A record of 70% winners would make an outstanding track record for the majority of traders (assuming the 30% losers were not greater than the winners). For the purposes of illustrating the mathematical improbability of 123 winning trades with no losers, I will give this blogger/trader a generous likelihood of success by assigning a 95% probability that every trade he makes is a winning trade.

Be sure to understand what this means. I am saying that with every trade he makes, there is a 95% chance that he will win. This is clearly improbable, especially when considering that we are going to let him make 123 trades in 21 days.

Now to the mathematics: Below I will list the probability of a string of 123 winners with no losers, based on decreasing winning percentages.

Winners _________Probability of 123 in a row
95% _________________.173%
90% ________________0.0002118708%
80% ________________0.0000000001%

As the number of winners is decreased, we quickly see the improbability of making 123 winning trades in a row. For the purposes of argument, lets assume this trader's system generates 99% winners. Still, the likelihood of him having 123 winners in a row is only 28.76%.

Still, I feel there is more to this story. This blogger did post the results of his paper trades, on his own blog, in May. The following is what he posted:

Total Trades…………………………..11
Wins…………………………………….8
Losses………………………………….3
Win%…………………………………..62.5%

Lets take his winning percentage of 62.5% and calculate the probability of a string of winning trades with 62.5% winners.

Winners In A Row__________Probability
5 _____________________5.96%
10_____________________0.568%
20_____________________0.0052%
30_____________________0.000047%
40_____________________0.00000043%

Obviously, the results show that it is absurd to go any further with these calculations.

Another element of this record, not before questioned, is why was it not posted on this trader's own blog? It seems to me that if one experiences such a phenomenal success, it would be posted on his own domain, rather than in the comments section on another blog. Furthermore, the fact that this blogger has never divulged a strategy, or even posted a single actual trade, in my opinion, casts doubt on this record.

What can we take away from this discussion? The most important take-away is that every trader can calculate the probability of experiencing a string of losers. For example, since I have 60% losers, there is a 1% probability that I will have 8 losers in a row. If I'm betting large on every trade, I need to know what my likelihood is for a series of losers. However, it is also important that traders understand the Gambler's Fallacy when seeking to use probabilities to increase their profits and decrease the likelihood of a blow-up.

Update: Our blogger/trader has updated the comments section to state that his practice of arbitrage is 100% risk free. Certainly, true arb is risk free. I'll leave it up to the intelligence of the readers to determine whether or not they believe that 123 trades, even with 99% win rate, can be executed without a single loss. Probability at 99% win rate is 28.76%.