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# The Arcsine Law: Why Traders Get Stuck Either in the Red or in the Green

This result explains why 50% of the people consistently lose money, while 50% consistently win. Let's compare stock trading to coin flipping (tails = loss, heads = gain). Then
• The probability that the number of heads exceeds the number of tails in a sequence of coin-flips by some amount can be estimated with the Central Limit Theorem and the probability gets close to 1 as the number of tosses grows large.
• The law of long leads, more properly known as the arcsine law, says that in a coin-tossing games, a surprisingly large fraction of sample paths leave one player in the lead almost all the time, and in very few cases will the lead change sides and fluctuate in the manner that is naively expected of a well-behaved coin.
• Interpreted geometrically in terms of random walks, the path crosses the x-axis rarely, and with increasing duration of the walk, the frequency of crossings decreases, and the lengths of the “waves” on one side of the axis increase in length.

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### Replies to This Discussion

This is Fascinating.

It is amazing that the probability of throwing a "head" with a fair coin is "1/2" and yet if we throw the coin 100 times we are not likely to throw 50 heads and 50 tails. The 50-50 split might happen only 20% of the time.

You have tapped into a amazing and important subject.
I don't see how the three bullets you specify lead to your initial conclusion: 50% consistently lose money. In fact, if I understand you bullets properly, shouldn't it be likely that the percentage of consistent losers or winners would be anything BUT 50%?
If we are dealing with pure arbitrage (no insider trading etc.) on a neutral market, mathematical laws pertaining to Markov processes dictate that while 50% of the trades lose money, and 50% win, the cumulative activity (over time) results in traders being split in two categories: those who have been in the green (black) for a long time (that is, despite recent losses, they are still above water), and those almost permanently in the red - possibly because of one single trade where they lost 80% of their principal, despite otherwise having profitable trades all the times.

Good post, Sun.

Are you aware of the St Petersburg Paradox? Another unintuitive mathematical result based on the infinite coin-flipping exercise.

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