The book Fortune’s Formula describes a simple investing scheme invented by Claude Shannon, referred to as “Shannon’s Demon”, that’s specifically designed to make money in markets described by log random walks. I found a blog post describing the scheme here. (Some previous discussion.) I’d expect this kind of volatility harvesting scheme to work better for Bitcoins than for other assets because Bitcoins are more volatile.
However, I’m not convinced that the market for Bitcoins is efficient… for example, there are going to be 84 million Litecoins to Bitcoins’ 21 million, but typical investors don’t know that, so 4 Litecoins for $100 feels like more of a steal than 1 Bitcoin for $100 (even Silicon Valley software engineers commonly forget to account for this basic division operation). There was talk on /r/bitcoin about how once the price got to the $1000 range, people seemed reluctant to invest since it seemed so expensive and how things should be reframed as “mBTC”. And I’d expect that quant firms are reluctant to trade bitcoins due to factors like institutional regulation and it not being serious-seeming enough for themselves or their investors.
I’m doing this (Shannon’s Demon). So far it’s profitable, although I think I’ve taken on more risk premium than investing 50% BTC 50% USD and not balancing.
However, I’m not convinced that the market for Bitcoins is efficient… for example, there are going to be 84 million Litecoins to Bitcoins’ 21 million, but typical investors don’t know that, so 4 Litecoins for $100 feels like more of a steal than 1 Bitcoin for $100
There no reason at all to believe that the total value of Litecoins should have an easy relationship to Bitcoins. Bitcoin has much more infrastructure for real world usage behind it.
I agree, I’m just arguing that typical investors are not valuing either currency rationally and “failure to account for the denominator” is an argument in favor of this position.
For a player who gambles his entire bankroll each round, it appears to be a wash. No matter the order of returns, if there are an equal number of heads and tails, the player ends up having exactly as much as he did at the start.
If the sequence of coin flips has an equal number of heads and tails, you wouldn’t need any complicated scheme to win—you could just bet $0 on everything except the last flip, and you would know with 100% certainty what the result of the last flip would have to be to produce equal numbers, so you’d bet everything on it. This would even work if the win and loss payoffs are equal numerically instead of equal in percent.
I don’t see why anyone would postulate that the sequence of coin flips contains an equal number of heads and tails unless they are confusing “as you flip a lot of coins, it gets closer to 50% heads and 50% tails” (true) with “as you flip a lot of coins, the number of heads gets closer to the number of tails” (not true).
This doesn’t give me high confidence for the rest of that link. (My first suspicion is that the whole thing actually amounts to a proof that this type of random walk is nonexistent.)
If the sequence of coin flips has an equal number of heads and tails
It only does in expectation; the underlying process is a martingale. They’re using an illustrative example to show you that investing everything in that random walk leads to a modal expectation of having the same at the end as you do at the beginning.
But that’s an expected value of 0 in log terms; the expected value in linear terms of course follows 1.25^n, where n is the number of flips. Shannon’s Demon reduces the variance in return at the price of reducing the mean return. If you’re only half in the market, your expected value grows at 1.125^n.
But if you have a log utility function, the decreased variance is helpful because then your expected utility grows each period rather than staying flat. (With 100% exposure, your EV of one period is .5*log(2)+.5*log(.5), which is obviously 0, but with 50% exposure your EV of one period is .5*log(1.5)+.5*log(.75), which is positive.) If you have a log utility function, 50% exposure happens to maximize your growth in expected return.
(I do agree with you that the link saying that the offer is a “wash” without bringing in the log utility function, or the tradeoff between variance and expected return, is bad, but those are somewhat subtle issues that they might not want to introduce along with the game.)
They’re using an illustrative example to show you that investing everything in that random walk leads to a modal expectation of having the same at the end as you do at the beginning.
That illustrative example highly depends on the number of heads being exactly equal. If the number of heads and the number of tails differed even slightly, the result would not be the same amount that you started with, and the fact that the ratio of heads to tails was close to 50% would not affect that. If you had 100 heads and 101 tails, you’d end up with half as much as you started with, and if you had 10000 heads and 10001 tails, you’d still end up with half as much as you started with.
And if the number of heads and the number of tails was exactly equal, I could guarantee doubling my money simply by waiting until the last flip to bet anything.
Everything else you’re saying is correct, but the example is bad. And I still suspect that this just proves it’s impossible for a real life stock to actually have equal chances of doubling and halving.
And I still suspect that this just proves it’s impossible for a real life stock to actually have equal chances of doubling and halving.
Well, real life models generally operate on much smaller timescales, with much smaller step sizes. A model where you increase or decrease by .01 on a log scale (roughly 1% increase and 1% decrease) each step seems much more reasonable, but again the same strategy (of 50% exposure, rebalanced continuously) is optimal for a log utility function.
I have no doubt that a real-life stock can change in a manner similar to a log scale, but if it changed in a manner exactly like a log scale, the company could never fail (sending the value to 0) and it could grow larger than the size of the entire economy.
Given that “it can only grow to a certain size before you exceed the real-life limit” transforms the St. Petersburg paradox from infinite expected value to a small expected value, I would expect to see anyone proposing this model show that such real-life limits don’t destroy this model in the same way.
The optimization that I’ve been linking to- take the derivative with respect to exposure, set it equal to 0- is a 1-step optimization problem. That is, the strategy I’m describing as optimal (Shannon’s Demon) is optimal even if there’s only one coin flip, and because of the nature of the setup and the log utility function what’s optimal for one coin flip is optimal for an arbitrary number of coin flips.
The book Fortune’s Formula describes a simple investing scheme invented by Claude Shannon, referred to as “Shannon’s Demon”, that’s specifically designed to make money in markets described by log random walks. I found a blog post describing the scheme here. (Some previous discussion.) I’d expect this kind of volatility harvesting scheme to work better for Bitcoins than for other assets because Bitcoins are more volatile.
However, I’m not convinced that the market for Bitcoins is efficient… for example, there are going to be 84 million Litecoins to Bitcoins’ 21 million, but typical investors don’t know that, so 4 Litecoins for $100 feels like more of a steal than 1 Bitcoin for $100 (even Silicon Valley software engineers commonly forget to account for this basic division operation). There was talk on /r/bitcoin about how once the price got to the $1000 range, people seemed reluctant to invest since it seemed so expensive and how things should be reframed as “mBTC”. And I’d expect that quant firms are reluctant to trade bitcoins due to factors like institutional regulation and it not being serious-seeming enough for themselves or their investors.
I think it’s worth mentioning the phrase “Kelly criterion,” because it is so much more popular than “Shannon’s Demon” (eg, it has a wikipedia entry).
I’m doing this (Shannon’s Demon). So far it’s profitable, although I think I’ve taken on more risk premium than investing 50% BTC 50% USD and not balancing.
There no reason at all to believe that the total value of Litecoins should have an easy relationship to Bitcoins. Bitcoin has much more infrastructure for real world usage behind it.
I agree, I’m just arguing that typical investors are not valuing either currency rationally and “failure to account for the denominator” is an argument in favor of this position.
Thanks for the link to Stable Investing. The Permanent Portfolio looks awesome.
That blog post describing the scheme starts out
If the sequence of coin flips has an equal number of heads and tails, you wouldn’t need any complicated scheme to win—you could just bet $0 on everything except the last flip, and you would know with 100% certainty what the result of the last flip would have to be to produce equal numbers, so you’d bet everything on it. This would even work if the win and loss payoffs are equal numerically instead of equal in percent.
I don’t see why anyone would postulate that the sequence of coin flips contains an equal number of heads and tails unless they are confusing “as you flip a lot of coins, it gets closer to 50% heads and 50% tails” (true) with “as you flip a lot of coins, the number of heads gets closer to the number of tails” (not true).
This doesn’t give me high confidence for the rest of that link. (My first suspicion is that the whole thing actually amounts to a proof that this type of random walk is nonexistent.)
It only does in expectation; the underlying process is a martingale. They’re using an illustrative example to show you that investing everything in that random walk leads to a modal expectation of having the same at the end as you do at the beginning.
But that’s an expected value of 0 in log terms; the expected value in linear terms of course follows 1.25^n, where n is the number of flips. Shannon’s Demon reduces the variance in return at the price of reducing the mean return. If you’re only half in the market, your expected value grows at 1.125^n.
But if you have a log utility function, the decreased variance is helpful because then your expected utility grows each period rather than staying flat. (With 100% exposure, your EV of one period is .5*log(2)+.5*log(.5), which is obviously 0, but with 50% exposure your EV of one period is .5*log(1.5)+.5*log(.75), which is positive.) If you have a log utility function, 50% exposure happens to maximize your growth in expected return.
(I do agree with you that the link saying that the offer is a “wash” without bringing in the log utility function, or the tradeoff between variance and expected return, is bad, but those are somewhat subtle issues that they might not want to introduce along with the game.)
That illustrative example highly depends on the number of heads being exactly equal. If the number of heads and the number of tails differed even slightly, the result would not be the same amount that you started with, and the fact that the ratio of heads to tails was close to 50% would not affect that. If you had 100 heads and 101 tails, you’d end up with half as much as you started with, and if you had 10000 heads and 10001 tails, you’d still end up with half as much as you started with.
And if the number of heads and the number of tails was exactly equal, I could guarantee doubling my money simply by waiting until the last flip to bet anything.
Everything else you’re saying is correct, but the example is bad. And I still suspect that this just proves it’s impossible for a real life stock to actually have equal chances of doubling and halving.
Well, real life models generally operate on much smaller timescales, with much smaller step sizes. A model where you increase or decrease by .01 on a log scale (roughly 1% increase and 1% decrease) each step seems much more reasonable, but again the same strategy (of 50% exposure, rebalanced continuously) is optimal for a log utility function.
I have no doubt that a real-life stock can change in a manner similar to a log scale, but if it changed in a manner exactly like a log scale, the company could never fail (sending the value to 0) and it could grow larger than the size of the entire economy.
Given that “it can only grow to a certain size before you exceed the real-life limit” transforms the St. Petersburg paradox from infinite expected value to a small expected value, I would expect to see anyone proposing this model show that such real-life limits don’t destroy this model in the same way.
The optimization that I’ve been linking to- take the derivative with respect to exposure, set it equal to 0- is a 1-step optimization problem. That is, the strategy I’m describing as optimal (Shannon’s Demon) is optimal even if there’s only one coin flip, and because of the nature of the setup and the log utility function what’s optimal for one coin flip is optimal for an arbitrary number of coin flips.