Stepan, can we please discuss this in some other thread? I really hope that we can stay on topic.
Risk (pain) vs Reward (gain)
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Stepan, can we please discuss this in some other thread? I really hope that we can stay on topic.
Sorry, but I think topic already exhausted, since you yourself already answered your original question very well.
Yes
In the single-proposition case, the convergence (of the expected growth-optimal point) is very fast. However, the inflection point and the reward/risk maximizing point (which do not appear until after a certain number of periods or trades have transpired) towards the peak of these points is not fast.
Other convergence take place as well. The convergence to an asymptotic probability of ruin or drawdown does not occur quickly. Nor does the actual amount you would "expect" to make converge with the traditionally-considered "mathematical expectation" necessarily converge quickly.
It looks like the convergence is very fast. After some 20 periods, it would be almost indistinguishable from the infinite horizon, correct?
I am including the figure from one of your papers below (hope you don't mind). In my case, the number of periods is around 300.
In the single-proposition case, the convergence (of the expected growth-optimal point) is very fast. However, the inflection point and the reward/risk maximizing point (which do not appear until after a certain number of periods or trades have transpired) towards the peak of these points is not fast.
Other convergence take place as well. The convergence to an asymptotic probability of ruin or drawdown does not occur quickly. Nor does the actual amount you would "expect" to make converge with the traditionally-considered "mathematical expectation" necessarily converge quickly.
My apologies if going way off-topic, but I think we should really be careful with any ideas that occur in the limit. A lot of what all of us are dealing with in these markets is a little theoretical to begin with, but when we start looking at ideas in the limit, we go through the looking glass often and aren;t aware of it, aren't aware of what I refer to as these "fallacies in the limit."
The fundamental notion of expectation (the probability-weighted mean outcome), foundational to so much in game theory, trading, etc. is sheer fallacy (what one "expects" is the median of the sorted, cumulative outcomes at the horizon, which is therefore a function of the horizon).
To see this, consider a not-so-fair coin that pays 1:-1 but falls in your favor with a probability of .51 The classical expectation is .02 per play, and after N plays, .5N is what you would expect to make or lose for player and house, as the math of this fallacious approach - and I say fallacious as it does not comport to real-life. That is, if I play it on million times, sequentially, I expect to make 20,000 and if a million guys play it against a house, simultaneously, (2% in the house's favor) the house expect to make 20,000
And I refer to the former as horizontal ergodicity (I go play it N times), the latter as vertical ergodicity (N guys come play it one time each). But in real-life, these are NOT equivalent, given the necessarily finite nature of all games, all participants, all opportunities.
To see this, let is return to our coin toss game, but inject a third possible outcome -- the coin lands on it;s side with a probability of one-in-one-million and an outcome which costs us one million. Now the classical thinking person would never play such a game, the mathematical expectation (in classical terms) being:
.51 x 1 + .489999 x -1 + .000001 x - 1,000,000 = -.979999 per play.
A very negative game indeed. Yet, for the player whose horizon is 1 play, he expects to make 1 unit on that one play (if I rank all three possible outcomes at one play, and take the median, it i a gain of one unit. Similarly, if I rank all 9 possible outcomes after 2 plays, the player, by my calculations should expect to make a net gain of .0592146863 after 2 plays of this three-possible-outcome coin toss versus the classical expectation net loss of -2.939997 (A wager I would have gladly challenged Messrs. Pascal and Huygens with). To see this, consider the 9 possible outcomes of two plays of this game:
outcome
0.51 0.51 1.02
0.51 -0.489999 0.020001
0.51 -1000000 -999999.49
-0.489999 0.51 0.020001
-0.489999 -0.489999 -0.979998
-0.489999 -1000000 -1000000.489999
-1000000 0.51 -999999.49
-1000000 -0.489999 -1000000.489999
-1000000 -1000000 -2000000
The outcomes are additive. Consider the corresponding probabilities for each branch:
product
0.51 0.51 0.260100000000
0.51 0.489999 0.249899490000
0.51 0.000001 0.000000510000
0.489999 0.51 0.249899490000
0.489999 0.489999 0.240099020001
0.489999 0.000001 0.000000489999
0.000001 0.51 0.000000510000
0.000001 0.489999 0.000000489999
0.000001 0.000001 0.000000000001
The product at each branch is multiplicative. Combining the 9 outcomes, and their probabilities and sorting them, we have:
outcome probability cumulative prob
1.02 0.260100000000 1.000000000000
0.020001 0.249899490000 0.739900000000
0.020001 0.249899490000 0.490000510000
-0.979998 0.240099020001 0.240101020000
-999999.49 0.000000510000 0.000001999999
-999999.49 0.000000510000 0.000001489999
-1000000.489999 0.000000489999 0.000000979999
-1000000.489999 0.000000489999 0.000000490000
-2000000 0.000000000001 0.000000000001
And so we see the median, the cumulative probability of .5 (where half of the event space is above, half below -- what we "expect") as of .020001 after two plays in this three-possible-outcome coin toss. This is the amount wherein half of the sample space is better, half is worse. This is what the individual, experiencing horizontal ergodicity to a (necessarily) finite horizon (2 plays in this example) expects to experience, the expectation of "the house" not withstanding.
And this is an example of "Fallacies of the Limit," regarding expectations, but capital market calculations are rife with these fallacies. Whether considering Mean-Variance, Markowitz-style portfolio allocations or Value at Risk, VAR calculations, both of which are single-event calculations extrapolated out for many, or infinite plays or periods (erroneously) and similarly in expected growth-optimal strategies which do not take the finite requirement of real-life into account.
Consider, say, the earlier mentioned, two-outcome case coin toss that pays 1:-1 with p = .51. Typical expected growth allocations would call for an expected growth-optimal wager of 2p-1, or 2 x .51 - 1 = .02, or to risk 2% of our capital on such an opportunity so as to be expected growth optimal. But this is never the correct amount -- it is only correct in the limit as the number of plays, N - > infinity. In fact, at a horizon of one play our expected growth-optimal allocation in this instance is to risk 100%.
Finally, consider our three-outcome coin toss where it can land on its side. The Kelly Criterion for determining that fraction of our capital to allocate in expected growth-optimal maximization (which, according to Kelly, to risk that amount which maximizes the probability-weighted outcome) would be to risk 0% (since the probability-weighted outcome is negative in this opportunity).
However, we correctly us the outcomes and probabilities that occur along the path to the outcome illustrated in our example of a horizon of two plays of this three-outcome opportunity, (whic is .51, -.489999) which is (expected-growth) maximized at risking 100% of our capital, not 0% as the Kelly Criterion would mistakenly suggest to us.
I should point out two things on expectation here which may not be obvious. First, in the limit, as the number of trials increases N approaches infinity, the classical expectation and my horizon-specific expectation converge (I.e. the classical expectation is asymptotic).
Secondly, it is the horizon specific expectation which living organisms on earth innately operate by, as evidence by their actions.
The fundamental notion of expectation (the probability-weighted mean outcome), foundational to so much in game theory, trading, etc. is sheer fallacy (what one "expects" is the median of the sorted, cumulative outcomes at the horizon, which is therefore a function of the horizon).
To see this, consider a not-so-fair coin that pays 1:-1 but falls in your favor with a probability of .51 The classical expectation is .02 per play, and after N plays, .5N is what you would expect to make or lose for player and house, as the math of this fallacious approach - and I say fallacious as it does not comport to real-life. That is, if I play it on million times, sequentially, I expect to make 20,000 and if a million guys play it against a house, simultaneously, (2% in the house's favor) the house expect to make 20,000
And I refer to the former as horizontal ergodicity (I go play it N times), the latter as vertical ergodicity (N guys come play it one time each). But in real-life, these are NOT equivalent, given the necessarily finite nature of all games, all participants, all opportunities.
To see this, let is return to our coin toss game, but inject a third possible outcome -- the coin lands on it;s side with a probability of one-in-one-million and an outcome which costs us one million. Now the classical thinking person would never play such a game, the mathematical expectation (in classical terms) being:
.51 x 1 + .489999 x -1 + .000001 x - 1,000,000 = -.979999 per play.
A very negative game indeed. Yet, for the player whose horizon is 1 play, he expects to make 1 unit on that one play (if I rank all three possible outcomes at one play, and take the median, it i a gain of one unit. Similarly, if I rank all 9 possible outcomes after 2 plays, the player, by my calculations should expect to make a net gain of .0592146863 after 2 plays of this three-possible-outcome coin toss versus the classical expectation net loss of -2.939997 (A wager I would have gladly challenged Messrs. Pascal and Huygens with). To see this, consider the 9 possible outcomes of two plays of this game:
outcome
0.51 0.51 1.02
0.51 -0.489999 0.020001
0.51 -1000000 -999999.49
-0.489999 0.51 0.020001
-0.489999 -0.489999 -0.979998
-0.489999 -1000000 -1000000.489999
-1000000 0.51 -999999.49
-1000000 -0.489999 -1000000.489999
-1000000 -1000000 -2000000
The outcomes are additive. Consider the corresponding probabilities for each branch:
product
0.51 0.51 0.260100000000
0.51 0.489999 0.249899490000
0.51 0.000001 0.000000510000
0.489999 0.51 0.249899490000
0.489999 0.489999 0.240099020001
0.489999 0.000001 0.000000489999
0.000001 0.51 0.000000510000
0.000001 0.489999 0.000000489999
0.000001 0.000001 0.000000000001
The product at each branch is multiplicative. Combining the 9 outcomes, and their probabilities and sorting them, we have:
outcome probability cumulative prob
1.02 0.260100000000 1.000000000000
0.020001 0.249899490000 0.739900000000
0.020001 0.249899490000 0.490000510000
-0.979998 0.240099020001 0.240101020000
-999999.49 0.000000510000 0.000001999999
-999999.49 0.000000510000 0.000001489999
-1000000.489999 0.000000489999 0.000000979999
-1000000.489999 0.000000489999 0.000000490000
-2000000 0.000000000001 0.000000000001
And so we see the median, the cumulative probability of .5 (where half of the event space is above, half below -- what we "expect") as of .020001 after two plays in this three-possible-outcome coin toss. This is the amount wherein half of the sample space is better, half is worse. This is what the individual, experiencing horizontal ergodicity to a (necessarily) finite horizon (2 plays in this example) expects to experience, the expectation of "the house" not withstanding.
And this is an example of "Fallacies of the Limit," regarding expectations, but capital market calculations are rife with these fallacies. Whether considering Mean-Variance, Markowitz-style portfolio allocations or Value at Risk, VAR calculations, both of which are single-event calculations extrapolated out for many, or infinite plays or periods (erroneously) and similarly in expected growth-optimal strategies which do not take the finite requirement of real-life into account.
Consider, say, the earlier mentioned, two-outcome case coin toss that pays 1:-1 with p = .51. Typical expected growth allocations would call for an expected growth-optimal wager of 2p-1, or 2 x .51 - 1 = .02, or to risk 2% of our capital on such an opportunity so as to be expected growth optimal. But this is never the correct amount -- it is only correct in the limit as the number of plays, N - > infinity. In fact, at a horizon of one play our expected growth-optimal allocation in this instance is to risk 100%.
Finally, consider our three-outcome coin toss where it can land on its side. The Kelly Criterion for determining that fraction of our capital to allocate in expected growth-optimal maximization (which, according to Kelly, to risk that amount which maximizes the probability-weighted outcome) would be to risk 0% (since the probability-weighted outcome is negative in this opportunity).
However, we correctly us the outcomes and probabilities that occur along the path to the outcome illustrated in our example of a horizon of two plays of this three-outcome opportunity, (whic is .51, -.489999) which is (expected-growth) maximized at risking 100% of our capital, not 0% as the Kelly Criterion would mistakenly suggest to us.
I should point out two things on expectation here which may not be obvious. First, in the limit, as the number of trials increases N approaches infinity, the classical expectation and my horizon-specific expectation converge (I.e. the classical expectation is asymptotic).
Secondly, it is the horizon specific expectation which living organisms on earth innately operate by, as evidence by their actions.
In the world's richest 1%? Almost all of them, undoubtedly. Maybe even in the world's richest 0.1%.
Understood. The novelty of your determination of the optimal leverage is that it's a function of the time horizon (and not just the function of the returns distribution). Your optimal leverage is equal to full Kelly when time horizon is infinite, and is less than full Kelly when time horizon is finite. The shorter the time horizon, the further away (to the left of full Kelly) we should go. Correct? Perhaps an easier way to understand (and to explain) this concept is to think about it in terms of the certainty of the total outcome, which is proportional to the square root of time. By going "less than full Kelly", you effectively compensate for uncertainty of the aggregate outcome over a finite stretch of time.
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Forgive me a possible silly question, but has the optimal point changed now (please see attachments)?
If this is a pure/approximate log-function, why would you expect an optimum based on that graph alone.
Reading the in-depth explanation above, most of which fly way over my head, are these concerns possible related. I do believe you need to adapt to the realities of your real trades, however you see fit to model/define your requirements for trading.
If this is a pure/approximate log-function, why would you expect an optimum based on that graph alone.
Reading the in-depth explanation above, most of which fly way over my head, are these concerns possible related. I do believe you need to adapt to the realities of your real trades, however you see fit to model/define your requirements for trading.
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Ralph and I have a different definition of the "optimal point", so I am not sure which one you are questioning.
Please watch the scales on the two attachments in my previous post and what they do to your green optimum. It seems the method/solution is not scale-invariant. If it's based on a log-function, I think you will be hard pressed to find a perfect optimum without using some other criteria. Time horizon could be one, but there could be other criterias as well.
Try scaling your graph in different ways to see what I mean. Basically, you can make the graph slope look quite different just by re-scaling the graph.
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