Expectation is a statistical fiction, like having 2.5 children. A gambler’s actual wealth varies wildly.
The code that Morse devised for his telegraph was relatively good because the most common letter, E, is represented with the shortest code, a single dot. Uncommon letters, like Z, have longer codes with multiple dots and dashes. This makes most messages more concise than they were in some of the early telegraphic codes. This principle, and many more subtle ones, figures in today’s codes for compressing digital pictures, audio, and video.
Assuming you wanted your spouse to bring home Shamu, you wouldn’t just say, “Pick up Shamu!” You would need a good explanation. The more improbable the message, the less “compressible” it is, and the more bandwidth it requires. This is Shannon’s point: the essence of a message is its improbability.
Collectively, the world’s investors own 100% of all the world’s stock. That means that the average return of all the world’s investors has to be identical to the average return of the stock market as a whole. It can’t be otherwise.
Even more clearly, the average return of passive investors is equal to the average stock market return.
Subtract the return of passive investors from the whole. This leaves the return of the active investors … Collectively, active investors must do no better or worse than the passive investors.
Active investing is therefore a zero-sum game. The only way for one active investor to do better than average is for another active investor to do worse than average. You can’t squirm out of this conclusion by imagining that the active investors’ profits come at the expense of those wimpy passive investors who settle for the average return. The average return of the passive investors is exactly the same as that of the active investors, for the reason just outlined.
Take Shannon’s pipe dream of turning a dollar into $2,048. You buy a stock for $1. It doubles every year for eleven years (100 percent annual return!) and then you sell it for $2,048. That triggers capital gains tax on the $2,047 profit. At a 20 percent tax rate, you’d owe the government $409. This leaves you $1,639. That is the same as getting a 96 percent return, tax-free, for eleven years. The tax knocks only 4 percentage points off the pretax compound return rate.
Suppose instead that you run the same dollar into $2,048 through a lot of trading. You realize profit each year, so you have to pay capital taxes each year. The first year, you go from $1 to $2 and owe tax on the $1 profit. For simplicity, pretend that the short-term tax rate is also 20 percent (it’s generally higher). Then you pay the government 20 cents and end the first year with $1.80 rather than $2.00.
This means that you are not doubling your money but increasing it by a factor of 1.8 – after taxes. At the end of eleven years you will have not 2^11 but 1.8^11. That comes to about $683. That’s less than half what the buy-and-hold investor is left with after taxes.
When people invest based on useless advice, there may be an opportunity to profit.
If people’s valuation of money were in direct proportion to their wealth, the chart would be a straight line. With Bernoulli’s rule of thumb, the line curves. This reflects the fact that it takes a large dollar gain to make the same difference to a rich person as does a smaller dollar gain would to a poor person. The shape of this curve describes a logarithmic function.
People don’t reason this way. Both you and your neighbor would be nuts to agree to this wager. You have far more to lose by forfeiting everything you have than to gain by doubling your net worth.
Look at the geometric mean. You compute it by multiplying the two equally possible outcomes together—$200,000 times $0— and taking the square root. Since zero times anything is zero, the geometric mean is zero. Accept that as the true value of the wager, and you’ll prefer to stick with your $100,000 net worth.
The geometric mean is almost always less than the arithmetic mean. (The exception is when all the averaged values are identical. Then the two kinds of mean are the same.) This means that the geometric mean is a more conservative way of valuing risky propositions. Bernoulli believed that this conservatism better reflects people’s distaste for risk.
Because the geometric mean is always less than the arithmetic mean in a risky venture, “fair” wagers are in fact unfavorable. This, says Bernoulli, is “Nature’s admonition to avoid the dice altogether.” (Bernoulli does not allow for any enjoyment people may get from gambling.)
Kelly’s prescription can be restated as this simple rule: When faced with a choice of wagers, choose the one with the highest geometric mean of outcomes.
One portfolio is better when it offers higher mean return for a given level of volatility; or a lower volatility for a given level of return.
The geometric mean can be estimated from the standard (arithmetic) mean and variance. The geometric mean is approximately the arithmetic mean minus one-half the variance.
Suppose you got an MP3 of an unreleased song. It’s the only copy. If you erase it, it’s gone forever. To erase is to destroy a small part of history. Erasing increases uncertainty about the past state of the world. Uncertainty is entropy.
The bankroll fluctuations in Kelly betting obey a simple rule. In an infinite series of serial Kelly bets, the chance of your bankroll ever dipping down to half its original size is 1/2.This is exactly correct for an idealized game in which the betting is continuous. It is close to correct for the more usual case of discrete bets (blackjack, horse racing, etc.). A similar rule holds for any fraction of 1/n. The chance of ever dipping to 1/3 your original bankroll is 1/3. The chance of being reduced to 1% of your original bankroll is 1%.The good news is that the chance of ever begin reduced to zero is zero. Because you never go broke, you can always recover from losses.The bad news is that no matter how rich you get, you run the risk of serious dips. The 1/n rule applies at any stage of betting. If you’ve run up your bankroll to a million dollars, it’s as if you are starting over with a $1 million bankroll. You run a 50% chance of losing half that million at some point in the future. This loss is “temporary.” Any way you slice it, the Kelly bettor spends a lot of time being less wealthy than he was.
Like adultery, insider trading is not a sing that can be committed alone.
A man who risks his entire fortune acts like a simpleton, however great may be the possible gain.