Edge and Bet Sizing

Every trade or bet asks you two separate questions, and beginners answer only the first. Question one: do I have an edge?, is the expected value in my favor, as computed in Expected-Value Games. Question two, which decides whether the edge ever turns into money: how much do I bet? You can have a real, positive edge and still go broke by betting too much, and you can waste a fine edge by betting too little. Sizing is not a footnote to the edge; it's half the game.

Your edge is your expected profit per dollar staked. For a bet that wins with probability $p$ at $b$-to-1 odds and loses your stake otherwise, the edge is

That's just the expected value of a one-dollar bet: probability of winning times what you win, minus probability of losing times what you lose. A positive edge means "over many repetitions this makes money." It says nothing yet about how much to put at risk.

Sizing: bet edge over odds

The sizing rule that maximizes long-run growth is the Kelly criterion: bet the fraction

of your bankroll. In words, bet your edge divided by the odds. Bigger edge, bet more; longer odds (bigger $b$), bet less of your roll because each loss is a bigger fraction of what you risked. For a simple even-money bet ($b = 1$) this collapses to just betting your edge, $f^\star = 2p - 1$.

Worked example: the 55% coin

You can bet any fraction of your $1,000 bankroll on a coin that lands your way 55% of the time, at even money. The edge is $2(0.55) - 1 = 0.10$, a healthy 10 cents of expected profit per dollar. Kelly says bet a fraction $f^\star = 0.10$, i.e. $100 on the first flip, then 10% of whatever your bankroll becomes after each result.

The temptation is to think "I've got a big edge, let me press it." Watch what over-betting does to the same favorable coin:

The edge never changed, the coin is favorable in every row. What changed is that betting too much lets variance compound against you faster than the edge compounds for you. Past twice Kelly, a favorable bet becomes a wealth destroyer.

Why real traders bet a fraction of the "right" size

Kelly assumes you know your edge exactly. You never do. Your win probability is an estimate, and estimates are optimistic exactly when it matters, the bets you found most attractive are the ones where you most likely overestimated the edge, a cousin of the winner's curse. Because over-betting is punished so much harder than under-betting, the sane response to uncertainty is to shrink. Most practitioners bet half Kelly or less, which keeps roughly three-quarters of the growth for far less than half the variance. The same logic drives real Position Sizing on trading desks: size to a risk budget, not to a fragile point estimate of the edge.

The interview version

The canonical prompt is exactly the biased coin: "a coin lands heads 60% of the time, you can bet any fraction of your bankroll at even odds, what fraction maximizes long-run growth?" The edge is $2(0.6) - 1 = 0.20$, so full Kelly is 20% and the slick answer notes you'd more likely run half that in practice. The trap the interviewer is watching for is the person who says "bet it all", that maximizes expected wealth but guarantees eventual ruin, the whole reason sizing exists as a separate discipline from finding the edge.