Risk, ruin, and expected value

A coin pays you $2 when it lands heads and costs you $1 when it lands tails. Played once, it is a good bet. Played with your whole net worth on every flip, it will eventually take everything you have — not because the odds turned against you, but because a single tails, encountered at full stakes, ends the game. This is the most important and least taught idea in all of finance, and the rest of this curriculum silently assumes you already hold it: average return and risk of ruin are different questions, and the second one is the one that kills.

Daniel Bernoulli noticed the gap in 1738, wrestling with a St. Petersburg lottery that had infinite expected value yet no sane person would pay much to play. The resolution — that we value the next dollar less the more we already have, and that we are right to fear the ruinous path — is the seed of risk aversion, insurance (Lesson 104), and the whole architecture of portfolio theory. John Kelly turned it into a formula in 1956: there is a bet size that maximizes long-run growth, and betting bigger than it makes you poorer, not richer, even when every individual wager is in your favor.

Why “it’s positive expected value” is not enough

The casino does not beat you because any single hand is hopeless; it beats you because it plays a million hands at a survivable fraction of its bankroll while you play a few hands at a reckless fraction of yours. The same asymmetry runs through every lottery ticket, every all-in options trade, every leveraged position that “can’t lose.” Ole Peters’ work on ergodicity makes the point sharp: the average outcome across a crowd of gamblers can look healthy while the outcome for almost every individual through time trends to zero. You do not live the average. You live one path.