The Kelly Criterion in Quant Interviews

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.