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.