Why maximise log-wealth instead of expected wealth?

Maximising expected wealth tells you to bet everything on any favourable gamble, which guarantees eventual ruin. Log-wealth penalises large losses heavily, so maximising it gives the bet size with the highest long-run compound growth.

Is full Kelly too aggressive?

Usually. Full Kelly assumes you know your edge and odds exactly; in reality you estimate them, and overestimating makes full Kelly dangerously large. Half-Kelly keeps most of the growth with far less volatility, so it's the common practical choice.

Does Kelly apply to trading, not just betting?

Yes — sizing a position under an estimated edge is the same problem. Size with your edge, never bet the farm, and respect estimation error: the intuition carries directly to allocating capital across trades.

The Kelly Criterion, Explained

Knowing a bet is profitable is only half the problem. The other half — how much to stake — is where fortunes are made and lost. Bet too little and you leave growth on the table; bet too much and a single losing streak wipes you out. The Kelly criterion answers this precisely, which is why it turns up in trading interviews and on real desks.

What Kelly optimises

Kelly picks the bet size that maximises the long-run growth rate of your bankroll — equivalently, it maximises the expected logarithm of wealth, $\mathbb{E}[\log W]$ .

Key insight. Why log wealth and not wealth itself? Maximising expected wealth tells you to bet everything on any favourable gamble — which guarantees eventual ruin, since a single loss takes you to zero. The logarithm punishes large losses severely, so maximising it balances growth against survival. That trade-off is the whole idea.

The formula

For a bet that pays $b$ -to-1 odds, with probability $p$ of winning and $q = 1 - p$ of losing, the optimal fraction of your bankroll to wager is

$f^* = \frac{bp - q}{b}.$

A cleaner way to read it: f equals your edge divided by the odds. With no edge, $f^* = 0$ — you bet nothing.

A worked example

You are offered even (1-to-1) odds on a coin that lands heads 60% of the time, so $b = 1$ , $p = 0.6$ , and $q = 0.4$ :

$f^* = \frac{(1)(0.6) - 0.4}{1} = 0.20.$

Kelly says stake 20% of your bankroll. Bet more and your long-run growth actually falls, even though each bet is still +EV; bet your whole bankroll and ruin becomes certain.

Why traders use fractional Kelly

Full Kelly assumes you know $p$ and $b$ exactly. In reality you estimate them, and overestimating your edge makes full Kelly dangerously large — so practitioners bet a fraction of it.

Strategy	Long-run growth	Volatility and drawdowns
Half Kelly	~75% of the maximum	Much lower
Full Kelly	Maximum	High
Over-betting (> Kelly)	Falls, can turn negative	Severe; ruin likely

Half-Kelly keeps about three-quarters of the growth for a fraction of the swings — a trade most traders take happily.

Watch out. The growth-versus-bet-size curve is a hump: it rises to a peak at the Kelly fraction, then falls. Being a little under Kelly costs a sliver of growth; being over Kelly costs growth and piles on risk. When unsure, err low.

How to practise

The Kelly Betting game lets you size bets under variance and feel how over- and under-betting play out over many rounds. It builds on solid expected-value fundamentals, and connects to the St. Petersburg paradox, where maximising log-wealth — Kelly's exact objective — resolves a puzzle that pure expected value cannot.