Part 3 of the Inefficient Markets sequence.

[I am not your financial advisor! Examples are illustrations of principles for educational purposes only, to help you make your own financial decisions with due diligence. Read part 1 first.]

Suppose I offer you a deal: You will flip a fair coin. If it's heads, I will pay you a dollar. If it's tails, you will pay me a dollar.

Does this game strike you as unfair? Does it seem dangerous? Your expected value is zero. If we iterate this game, you'll experience some gains and some losses, but they will tend to even out over time. So if anything, it's pointless. Entertaining maybe, but dangerous? No, not really.

Now suppose I let you bet any number of dollars X. If you win, you will pay me X dollars. If I win, I will pay you X dollars.

Is this game unfair? Is it dangerous? How much should you bet each game? What is your expected value?

It's the same deal, right?

Not quite.

If you bet all of your money, you could double your money, or you could lose all your money. If we iterate, and I have enough dollars to pay off your winning streak, you are almost certain to eventually flip a tail and lose. Now I have all of your dollars, and I keep them, because you can't afford to play.

The Second Law of Averting Ruin

Don't Bet the Farm

I surreptitiously gave myself a massive advantage in the second game. I said "I have enough dollars to pay off your winning streak". The player with more money has a massive advantage. But if you are betting against "the market", you had best assume that "the market" has a lot more dollars than you.

Kelly's Optimal Strategy

I will not derive the formula here, but the optimal betting strategy is to bet a certain fraction of your bankroll.

For an even-money bet, the formula is simply

$fraction=P(win)−P(loss)$

The fraction is only 1 if the probability of winning is also 1.

The Kelly formula becomes more complex if it's not an even-money bet. It must be computed from the distribution of payoffs (including negative payoffs).

It's important that you understand this formula exists for the insights it provides, that the optimal strategy is to bet an optimal fraction, that underbetting will underperform, overbetting will underperform, that double the fraction has an expected growth rate of zero, and for anything bigger than that it will be negative, so Betting the Farm is a sure path to ruin, but we can't actually use it, be...