Arcsinus law: distinguishing trend from persistency of chance

The demonstration has been done by Levy: the probability P(Kn/n < a)
is the sum of combinatorial terms where Kn is the number
of times when the player's gain is positive and it tends towards 1/Pi * Arcsinus ,
the density is just derivation and it is because this density is infinite
when x = 0 or 1 (since the denominator is x(x-1)) that 1/2 of the time
is not the most probable but rather 0 (none of the time) or 1 (all of the time).

Quote from Mr Subliminal:

The density function of the arcsine distribution (as described in the link provided by OddTrader) depicts the results I obtained. This is a visual affirmation only.
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Now there is something related to this arsinus law that the quants and fondamentalists didn't take into account when they compare Stock market to casinos but hold on I'm tired for now I will come back another day :p .

Quote from harrytrader:

The conclusion - of Berstein but it is not "his" conclusion it is statistical implication - is that judging performance of traders on equity curve is very difficult since equity curve is a cumulated sum that will deviate larger and larger from the zero line even if it is positive how can you judge by this sole criteria that it is not due to chance since chance can produce the same kind of results ? So an equity curve that is trending positively is a necessary condition (if it is not positive it isn't even worth of course :D) but not sufficient condition. The problem is worsened if market doesn't follow a random walk because the jumps are rare events and that rare event are even more unpredictable stochastically. Random walk is the first kind of risk which is assimilated to usual noise, the jumps are a second kind of risk some calls uncertainty that is "more unpredictable" than the usual noise. That's what the so-called quants and fundamentalists base their reasoning to affirm that speculators who are winners are survival bias because as Berstein quoted Buffet in his book "if 215 millions of monkeys were playing the same game" the result would be the same - there will be some big winners and many losers of course but we don't just hear about the losers. It doesn't mean that these winners doesn't have any talent, for example by using money management (which is optimisation techniques and in fact just noble term for martingale term used by gamblers) but this money management cannot circumvent the edge for the global population. Of course you will find tremendous sucessful gamblers but for sure they are survival for nearly all of them since everybody knows that casinos are very carefull to give them no true edge. But the casinos know that they must let some players make big wins because it plays a role in their marketing so as to attract all the crowd. As for stock market, as long as it is not demonstrated academically that it is not so unpredictable, the market, for most traders, should be like the casinos, so that the big winners that have access to no special information are like the big winners in casinos: they would be survival bias - once again even if they have applied the optimal strategy it doesn't change the fact that the edge of the system was as it was: null or even negative so by pure logic any winner, in this hypothesis, has benefited from so called chance: chance is ONE realisation of a system that has no edge but that has deviated from the theorical value only due to sampling fluctuation.

I just exposed above the quants and fundamentalists point of view which cannot be challenged easily on scientific basis by the others schools - once again even if market doesn't follow a normal random walk. I can give you one of the big reason : there could be no strict frontier between deterministic process and stochastic process ... so that trying to prove that something is not stochastic whereas such thing imitates stochastic process perfectly - by using traditional stochastic criterias like autocorrelations or others (like spectral analysis) for example one of known deterministic process called tent map has been proved to behave like a random process.
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Quote from harrytrader:

The conclusion - of Berstein but it is not "his" conclusion it is statistical implication - is that judging performance of traders on equity curve is very difficult since equity curve is a cumulated sum that will deviate larger and larger from the zero line even if it is positive how can you judge by this sole criteria that it is not due to chance since chance can produce the same kind of results ? So an equity curve that is trending positively is a necessary condition (if it is not positive it isn't even worth of course :D) but not sufficient condition. The problem is worsened if market doesn't follow a random walk because the jumps are rare events and that rare event are even more unpredictable stochastically. Random walk is the first kind of risk which is assimilated to usual noise, the jumps are a second kind of risk some calls uncertainty that is "more unpredictable" than the usual noise. That's what the so-called quants and fundamentalists base their reasoning to affirm that speculators who are winners are survival bias because as Berstein quoted Buffet in his book "if 215 millions of monkeys were playing the same game" the result would be the same - there will be some big winners and many losers of course but we don't just hear about the losers. It doesn't mean that these winners doesn't have any talent, for example by using money management (which is optimisation techniques and in fact just noble term for martingale term used by gamblers) but this money management cannot circumvent the edge for the global population. Of course you will find tremendous sucessful gamblers but for sure they are survival for nearly all of them since everybody knows that casinos are very carefull to give them no true edge. But the casinos know that they must let some players make big wins because it plays a role in their marketing so as to attract all the crowd. As for stock market, as long as it is not demonstrated academically that it is not so unpredictable, the market, for most traders, should be like the casinos, so that the big winners that have access to no special information are like the big winners in casinos: they would be survival bias - once again even if they have applied the optimal strategy it doesn't change the fact that the edge of the system was as it was: null or even negative so by pure logic any winner, in this hypothesis, has benefited from so called chance: chance is ONE realisation of a system that has no edge but that has deviated from the theorical value only due to sampling fluctuation.

I just exposed above the quants and fundamentalists point of view which cannot be challenged easily on scientific basis by the others schools - once again even if market doesn't follow a normal random walk. I can give you one of the big reason : there could be no strict frontier between deterministic process and stochastic process ... so that trying to prove that something is not stochastic whereas such thing imitates stochastic process perfectly - by using traditional stochastic criterias like autocorrelations or others (like spectral analysis) for example one of known deterministic process called tent map has been proved to behave like a random process.
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Thanks Harry, excellent post. I appreciate such a down-to-earth explanation this time. :)
 
Quote from harrytrader:

great I will use the second one for illustrating my prob faqs :)
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Your site (from my last visit) looks great! :cool:

Perhaps more down-to-earth applications would be even better! :)
 
Good points Harry. I was just thinking, if as they say, only 10% of traders are successful, one has to wonder what percentage of that top 10% have been successful by pure chance. The real % of "good" traders could be much lower than 10%. :eek:
 
It's rather that at long term there are only a few percentage of "good traders" but on short term (this is relative of course to period and style of trading this means dozens of years for portfolio managers for example to a few months/years for short term traders) there can be even a majority of winners ... and after that a majority of loosers: that's what persistency means, it's easy to see it during bull phase where all the public and traders believe into a "new era" because it seems to last long time enough (remember arcsinus law is about time not about the distribution of mean which is another law - the central limit theorem) and that everybody seems to be winner :D. Since at each cycle there are newcomers the same illusion can begin again and again, this on multiple scales.

Quote from Bolts:

Good points Harry. I was just thinking, if as they say, only 10% of traders are successful, one has to wonder what percentage of that top 10% have been successful by pure chance. The real % of "good" traders could be much lower than 10%. :eek:
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What if Irving Fisher the most famous economist of his time before 1929 had known the persistency law (Levy discovered and demonstrated the law in 1939 only) ? Would he had declared that "stock prices were not overinflated but, rather, had achieved a new, permanent plateau." :D

http://cepa.newschool.edu/het/profiles/fisher.htm

"This Yale economist was an eccentric and colorful figure. When Irving Fisher wrote his 1892 dissertation, he constructed a remarkable machine equipped with pumps, wheels, levers and pipes in order to illustrate his price theory - see here for pictures of his draft and his first and second prototypes. Socially, he was an avid advocate of eugenics and health food diets. He made a fortune with his visible index card system - known today as the rolodex - and advocated the establishment of an 100% reserve requirement banking system His fortune was lost and his reputation was severely marred by the 1929 Wall Street Crash, when just days before the crash, he was reassuring investors that stock prices were not overinflated but, rather, had achieved a new, permanent plateau."
 
Kenneth Galbraith like to refer to Fisher's prediction as his "immortal estimate" . It's all the more funny that Fisher has created with 2 others the so-called Cowles Commission (see its history http://cowles.econ.yale.edu/reports/20yr/his_1.htm) which is at the very origin of the RMH (Random Market Hypothesis). In 1933 they published an article untitled "Can Stock Market Forecasters forecast ?" their conclusion is that it is doubtful. Notably the Cowles commission has attacked Hamilton father of so-called Dow Theory - Hamilton who has formalised Dow's idea - saying that his forecast results was no more or less good than if it was due to chance .

Quote from harrytrader:

What if Irving Fisher the most famous economist of his time before 1929 had known the persistency law (Levy discovered and demonstrated the law in 1939 only) ? Would he had declared that "stock prices were not overinflated but, rather, had achieved a new, permanent plateau." :D

http://cepa.newschool.edu/het/profiles/fisher.htm

"This Yale economist was an eccentric and colorful figure. When Irving Fisher wrote his 1892 dissertation, he constructed a remarkable machine equipped with pumps, wheels, levers and pipes in order to illustrate his price theory - see here for pictures of his draft and his first and second prototypes. Socially, he was an avid advocate of eugenics and health food diets. He made a fortune with his visible index card system - known today as the rolodex - and advocated the establishment of an 100% reserve requirement banking system His fortune was lost and his reputation was severely marred by the 1929 Wall Street Crash, when just days before the crash, he was reassuring investors that stock prices were not overinflated but, rather, had achieved a new, permanent plateau."
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