I would like to pose a question here for those of you familiar with the mathematics of random numbers. I will pose it in terms of coin flipping, but you could easily ask the question in terms of buying and selling equities: If you flip an unweighted coin 100 times, it should come up heads about 50 times out of the 100. Suppose you do this experiment and you find that it comes up heads 60 times and tails 40 times. What is your expectation for heads and tails in the next 100 flips? The answer should be 50 heads and 50 tails for the next 100 flips. What concerns me however is that the expectation for the total of 200 flips is 100 heads and 100 tails, so one might expect that for the second 100 flips, the expectation is 40 heads and 60 tails, not 50 heads and 50 tails. Can anyone enlighten me on this subject. In particular what is the error in the expectation value and how does it relate to the number of flips. (I am thinking that it is something like 1/square root of the number of flips or about 10% for 100 flips)
Check out: Beyond Greed and Fear: Finance and the Psychology of Investing by Hersh Shefrin Most people tend to have that fallacy (the book gives some example of financial experts making similar adjustments in expectations). The law of large numbers doesn't quite work in small samples. If someone is baseball is "due" to have a homerun b/c of his long time average and lack of homeruns in the near past - that's another example of people making the same mistake
If I remember my stats correctly, one over root n is the rate at which the sample estimate converges to the population parameter.
The assumption of a 50% probability for, say, head will work out in the long run only. How long is long? In a series of 4040 flips you may expect a deviation of around 0.5%. (I did not compute that (it can be done) but took it from a book.) In absolute numbers: of 4040 flips there might be 2048 heads and 1992 tails. For trading simulations watch that absolute difference of 56 "Trades", they might bias a simulation to one side or other more than you expect. I recommend to set up a spreadsheet in Excel and do some simulations. In a series of 1000 flips repeated 200 times you will be surprised how often you will get "head" in a row: 8 heads in a row per 1000 flips is almost guaranteed, there is a chance of getting 15 heads in a row once. I attached a little Excel spreadsheet where I simulated 1000 flips 200 times. The first column shows the number of heads in a row, the last column the probability (frequency) of getting that number in this simulation. Ignore the other columns. The same logic applies for trading simulations where you do not have such a simple probability distribution as in the coin flip. You should not look at probabilities here: you are interested in expectancy that is (prob x profit). In terms of probability only, a losing series might last longer than you remain solvent and still be in accordance with your tests. Regards Bernd Kuerbs
the original question was about expectation. lots of people confuse probability with expectation. probability of winning a coin toss = 50% over a long series expectation, is a different story. expectation is what you would expect to get if you bet on each coin flip. That of course depends on how much you bet each time ( position sizing ), and what the payoffs are ( risk vs. reward ), which don't have to be symmetrical.
BKuerbs- In a series of 4040 flips you may expect a deviation of around 0.5% That's what I would like to know. Is there a formula for computing the expected deviation for a small sample such as 100 flips.
I don't believe your explanation is accurate, with all due respect, unless I misunderstood something. If the probability of an upward move is equal to that of a downward move - say 50/50 if we ignore the case of no move at all (and if there is no reason to believe the moves in one direction will be consistently larger than those in the other - in other words, if the move magnitudes are random and independent of the direction), and you keep betting randomly, random amounts (hence the position sizing is irrelevant), in the long run, ignoring transactions costs, the payoff will be zero. As for probability vs. expectation, isn't expectation merely a probability-weighted average???