Shannon's Demon and how returns can be created out of thin air
richmondquant.com
richmondquant.com
> To illustrate this concept, you could consider a simple coin-flip game where you make a 50% return on your money if the coin lands on heads, or lose 33.3% of your money if the coin lands on tails.
Let's say you bet $100. Your expected outcome is 100 * (1/2 * 3/2 + 1/2 * 2/3) = 108.33. Greater than 100. That's all there is to it, the "magic" comes from the made up security which can generate such amazing returns.
The interesting thing, which is confused by the blog, is that "expectation" is not that same thing as "expectation of log". The "expectation of log" is a simple and effective way to adjust for risk, so trading off between these two can be useful.
In the simple betting case, you're multiplying your money by {e^k, e^-k}. In this case, k≈0.4, so you end up with
Y = P * {0.66, 1.50}
This has expectation of `1.08 * P`, and it has an expectation-of-log of `log P`.
In the rebalancing case, you're multiplying your money by (1 + {e^k, e^-k})/2. You end up with
Z = P * {0.83, 1.25}
This has an expectation of `1.04 * P`, and it has an expectation-of-log of `log P + 0.0204`.
So you actually have a smaller expectation, but a larger expectation-of-log.
This could be helpful, depending on your goals, because it is a simple way to describe how "risky" the strategy is. For example, after 5 rounds of this, the first strategy has a median value of +8.5%, while the second strategy has a median of +13%. On the other hand, on average the first strategy has grown +49% while the second has grown only +23%.
You can be assured that the possibilities of a portfolio balanced between a set of logarithmically random walking assets -- if one were to believe that individual stocks' behavior had such behavior -- is a well-contemplated concept in finance.
Often in finance, securities are assumed to follow something like a geometric Brownian pattern. This is the idea behind the Black-Scholes model, for instance. This means that you view your price ticker as a random Gaussian walk (e.g. Brownian motion) when your ticker is placed on a logarithmic scale.
One way to think of this is that, given some (sufficiently large) unit of time, you have an equal chance of your security either doubling or halving in value. Or being multiplied/divided by 1.5, or 1.1, or any other figure you like, given whatever unit of time you like.
It's really easy to see how this basic pattern has a positive expected value. Suppose you put $1 into a fair coin toss game, and if you win it doubles and if you lose it halves. You stand to win $1 or lose only $0.5, thus the expected value is $1.25.
Now suppose, to keep it simple, you have a ridiculously volatile security which doubles or halves in value every day. So the ideal trading strategy is to choose some betting amount, say $1000, and just put it in at the start of the day. Then the next day, just reset your position size to $1000, either taking your $1000 in profit or kicking in another $500 to cover losses. You will quickly become a trillionaire.
No, you will quickly become a thousandaire, because your strategy is very very far from optimal. You will gain an expected $250 per day, your earnings will not compound at all, and you will wonder why you are not ultra-rich at the end of the year.
Of course, this is better than getting all your money every day, which, in your example, will net you nothing on expectation: the expected change of the log of your assets every day is exactly zero.
If you actually use the Kelly criterion, you will indeed become very wealthy very quickly.
https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf
To “Shannon’s demon’s” credit, the first word of that paper is Shannon, but that’s about it. Kelly and Shannon were different people, and the paper makes no mention of demons.
This also has nothing to do with making returns out of thin air. It’s about how to avoid losing your returns by betting too much at once, and how to optimally choose the size of a bet.
Literally value machines. Hence their demand for exotic products with uncorrelated volatility characteristics for risk pairing. I superficially get it now. I can see how someone could get the bug for this.
Only for a definition of "value" that seems pretty highly questionable on a bunch of different levels.
And like, yes: there is a moment in the lecture where he explicitly sets up a scenario where the expectation value of your bets sums to zero and yet the rebalancing makes money.
It isn't a free lunch, of course: if the two assets both go down and stay down then you lose money, and you don't gain as much money as you'd want on the way up; but if they are both sputtering around you definitely make money.
If you were to somehow be able to rebalance infinitely fast it also doesn't make money; and yet, it only makes money in relationship to the volume you are able to rebalance, so there is another tradeoff.
At the end of the day you can thereby model it as making money by providing the service of "providing liquidity" to the market, and what you are effectively doing is taking a spread on that as a market maker.
You double your bet each time you loose, so you have a better chance of making back to your losses, however, you increase your chance of loosing everything to an unlikely event.
https://en.wikipedia.org/wiki/Martingale_(betting_system)
You can do it with any amount of money, but it still isn't long term profitable unless you have an infinite bankroll.
Finite money can't properly implement a martingale. It can implement this strategy.
The description is also flawed in the blog. If anyone offers you a 50/50 bet to win $50 or lose $33,take it.
Given that it's posted by financial advisors , The blog is probably intentionally misleading
A strategy like "always bet 10% of your bankroll on a coin flip" doesn't break. Whether it makes or loses money in the long run depends on the specific returns, but you can keep doing it indefinitely.
You can't keep doing a martingale indefinitely. Bad luck doesn't just lose you money, it makes the strategy itself explode. Once that happens you can't recover even if you have good luck or a house edge.
Martingale eventually stops because you don't have an infinite bankroll.
Proportional betting eventually stops if you can't gamble less than a penny.
Neither strategy can give long-term profit on a neutral EV bet
If the EV is exactly neutral, that's never going to happen in the real world. While martingale will fail moderately quickly in most setups.
And saying "ignore minimum bets" is far less egregious than "pretend you have infinite money".
Also neither strategy is specific to neutral EV.
I think the more interesting and testable question is about the claims of turning Neutral EV games profitable.
for example, the blogs claim that proportional betting can do just that, turn neutral EV coinflip profitable. It only holds up with a number of bogus and unstated conditions.
But it fails for any bet where you'd care about the money, unlike this strategy.
> for example, the blogs claim that proportional betting can do just that, turn neutral EV coinflip profitable. It only holds up with a number of bogus and unstated conditions.
There's different kinds of expectation. The EV for a single iteration is positive, but the EV for leaving money invested indefinitely is neutral.
This makes the title somewhat misleading, but it's still an interesting question. How can you make money in a compounding way when basic reinvestment cuts your returns to zero?
There are no unstated conditions that I can see. The model is trivial.
>To illustrate this concept, you could consider a simple coin-flip game where you make a 50% return on your money if the coin lands on heads, or lose 33.3% of your money if the coin lands on tails. This simple game has a expected long-term return of zero.
The slight of hand is that they confuse % change of portfolio or market and the % profit on the bet.
The game is positive EV and the only reason that figure 1 doesn't make money is they wager less every time they will win and more every time they will lose.
Having a reserve/bankroll to recapitalize is key to winning any +EV game.
This is what is demonstrated in their figures 3 and 4. If the bet was truly neutral, +/- 50% each time, there would be no gain.
It is obvious if you look at figure 3. Each $100 bet will make $50 or lose $33.
There is no strategy to make money on a neutral 50/50 bet, balance all you want.
The same is true for a stock with a 50/50 chance of going up or down equal amounts.
Otoh, if you knew it was oscillating around a mean, and not doing a random walk, you could always sell high and buy low
But that's how investing in a stock works. It's not a weird anti-strategy they came up with. If you put money in and let it ride, it bounces around and you gain nothing.
And you gain nothing in part because they do focus on that single asset, with no diversification.
If it was +50% and -40% you'd lose money if you tried to do a normal investment, but you'd gain money if you rebalanced properly.
So I still think they're discussing some useful properties and scenarios, even if they don't make it clear that there are different types of expected outcome.
> Having a reserve/bankroll to recapitalize is key to winning any +EV game.
Not really. If it was +10% and -5% you wouldn't need to recapitalize ever.
> Otoh, if you knew it was oscillating around a mean, and not doing a random walk, you could always sell high and buy low
Notably, the strategy in the post does work for a random walk.
I don't think this is true. Rebalancing doesn't work in the oscillating example provided if you change the rewards from +50%/-33% of bet to +/-50%.
I built their model in excel and tested it, and you can too.
The Kelly Criterion defines how much of your bankroll to invest in a asset with known upside and downside.
For the plus EV game of +50%/-33% you should bet 50% of your bankroll to optimize long term gains.
for the neutral EV game of +/-50%, the criterion yields a bet of 0% of bankroll because there is no edge.
The payout percents are a different issue. I wasn't saying it works with arbitrary payouts.
Note that the +50%/-50% model is not very realistic either when it comes to stocks or assets. The value over time rapidly spirals to zero.
It talks about rebalancing between two assets. For a gambler this could simply be the bet and the bankroll. (this is the example used in the blog).
(ignoring the fact that the blog is playing a +EV game) what was described is "proportional betting". The short term outcome is a normal distribution with a mean of zero. Like the martingale, If you run it to infinity, you will always go broke. for proportional betting, the failure mode is a string of bad bets taking the bankroll below minimum bet size (the long tail on the normal distribution).
When we talk about real world trading, the whole idea of "shannon's demon" is trivial. In the zero growth case, it assumes the mean value is known and constant so it just boils down to sell high, buy low.
And before the information age, the market likely did act like this.
But now adays, the dark pools, random subreddit s, etc, almost guarantee that a ticket will fluctuate a price incoherently.
So to find the mythic mean is probably impossible. But the demon could be generate via a novel method of analysis, which definitely wouldn't be presented here, as knowing the method would invalidate mythic mean.
The main difference I see, or innovation if it is one, is that the proportion by which you increase or reduce the bet in this system (1/2 the difference between your bankroll and your betting account) prevents you from outrunning your bankroll on any given bet.
If you have two assets that really do have zero expected return but vary independently, it's trivial to see that moving equity from the one that went up to the one that went down (or went up less) will net you a return.
The 'trick' if you will is that there aren't actually any uncorrelated assets.
The key is that the game (1.5 vs 0.66) has a positive arithmetic expectation and zero geometric expectation, so you should be able to profit from it somehow. Do it with a double or nothing game and you always lose because the geometric expectation is negative infinity and arithmetic is zero.
“can actually make it possible to generate positive returns at the portfolio level, even when all of the portfolio’s individual holdings are expected to produce zero, or even negative, returns over time.” -False
The illusion comes from the fact that, for small n you can flip heads and tails the same number of times and still have over half the money you started with - whereas with the invest all strategy you would have lost everything. For a fair coin, as n approaches infinity you still win nothing either way though. That graph is completely misleading. We can represent the amount won as the recursive sequence
X_(n+1) = X_n + (X_n)/2
def recursive_sequence(n):
""" Money won after winning n times in a row"""
result = 1 # money we start with
rebalance_ratio = .5
for i in range(0,n):
result += result*rebalance_ratio
return result
temp_profit = recursive_sequence(i)
net = temp_profit / 2**i # assuming you loose as many times as you won after
percent = abs((temp_profit-1))*100
print(f"savings after {i} heads:", recursive_sequence(i), f"\npercent profit: {percent}","\n....money after loosing n times after: ", net, "\n")
savings after 2 heads: 2.25 percent profit: 125.0 ....money after loosing n times after: 0.5625
savings after 3 heads: 3.375 percent profit: 237.5 ....money after loosing n times after: 0.421875
savings after 4 heads: 5.0625 percent profit: 406.25 ....money after loosing n times after: 0.31640625
savings after 5 heads: 7.59375 percent profit: 659.375 ....money after loosing n times after: 0.2373046875
savings after 6 heads: 11.390625 percent profit: 1039.0625 ....money after loosing n times after: 0.177978515625
savings after 7 heads: 17.0859375 percent profit: 1608.59375 ....money after loosing n times after: 0.13348388671875