The Kelly criterion answers the question trading interviewers love most: given an edge, how much do you bet? It is the bet size that maximizes the long-run growth rate of your bankroll, and it shows up everywhere from market-making games to "I flip a biased coin, what fraction do you wager?" prompts.
The formula
For a bet that pays $b$-to-1 with win probability $p$ (lose probability $q = 1-p$), the optimal fraction of bankroll is:
$$f^* = \frac{bp - q}{b}$$
For even-money bets ($b=1$) this collapses to the form worth memorizing: $f^* = p - q = 2p - 1$ — your edge, bet directly.
Worked example
A coin lands heads with probability 0.6 and pays even money. Kelly says bet $f^* = 0.6 - 0.4 = 0.2$ — 20% of bankroll per flip. Why not more? Betting 100% goes bankrupt on the first tails, and even 50% per flip shrinks wealth long-run despite the positive edge: growth rate is $g(f) = p\ln(1+f) + q\ln(1-f)$, which peaks at $f^*$ and goes negative past roughly $2f^*$. That non-obvious fact — overbetting a winning game loses money — is exactly what the interviewer is checking you understand.
What interviewers actually probe
- The log-growth derivation. Can you set up $g(f)$ and differentiate, not just quote the formula?
- Fractional Kelly. Real desks bet half-Kelly or less: growth is nearly flat near the peak but variance is not, and your estimate of $p$ carries error. Half-Kelly gives up ~25% of growth for ~50% less variance.
- When Kelly breaks. Estimated (not known) edges, correlated simultaneous bets, and finite horizons all argue for smaller fractions.
- Game contexts. Market-making games often hide a Kelly question inside: your quote width and size decision is a bet-sizing decision.
Practice with solutions
- Half-Kelly betting with equal wins and losses — the fractional-Kelly trade-off, worked.
- Kelly sizing for a mean-reversion signal — Kelly applied to an actual trading signal.
- Monte Carlo backtest of Kelly vs fractional Kelly — see the variance argument in simulation.
- The betting game — practice sizing live against a clock, the way trading finals actually test it.
More topic guides
- Bayes' Theorem in Quant Interviews
- Coin Flip Questions in Quant Interviews
- Dice Questions in Quant Interviews
- Gambler's Ruin in Quant Interviews
- Markov Chains in Quant Interviews
- Martingales in Quant Interviews
- The Monty Hall Problem — and the Variants Interviews Actually Ask
- Optimal Stopping in Quant Interviews
- Random Walks in Quant Interviews
- All guides & explainers
Frequently asked questions
What is the Kelly criterion in simple terms?
It is the bet size that maximizes long-run compound growth: bet the fraction f* = (bp − q)/b of your bankroll, where p is your win probability, q = 1−p, and b is the payout ratio. For even-money bets it reduces to betting your edge, 2p − 1.
Why do trading firms ask Kelly criterion questions?
Bet sizing IS trading: the same logic sets position sizes and quote widths. The question tests whether you understand that overbetting a positive-edge game destroys wealth, and whether you can derive the log-growth argument rather than recite a formula.
What is half-Kelly and why use it?
Betting half the Kelly fraction. Growth is nearly flat near the optimum but variance is not — half-Kelly keeps about 75% of the growth with roughly half the variance, and it protects against overestimating your edge, which in practice you always risk doing.
What happens if you bet more than Kelly?
Growth falls, and past roughly twice the Kelly fraction the long-run growth rate turns negative — you lose money on a winning game. This is the counterintuitive core of most interview follow-ups.
Practice the real thing
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