Bet sizing, Kelly and risk of ruin

MM · Chapter 612 min readAsked at SIG, Jane Street, Optiver, Akuna

After this lesson you should be able to

  • Size a bet from its edge and its odds rather than from how confident you feel.
  • Explain why over-betting a positive-edge game still loses money.
  • Say when you would take a lower expected value in exchange for lower variance.

Having an edge tells you to bet. It does not tell you how much. Bet too little and the edge never compounds; bet too much and a positive-edge game takes your whole stack — which is the answer interviewers are usually fishing for when they ask "how much would you put on that?".

Equation 6.1

The Kelly criterion

The fraction of your bankroll that maximises the long-run growth rate of a repeated bet.

f∗=bp−qb=edgeoddsf^{*} = \frac{bp - q}{b} = \frac{\text{edge}}{\text{odds}}
pp
Probability of winning; q=1−pq = 1 - p.
bb
Net odds received on a win — a bet paying even money has b=1b = 1.
f∗f^{*}
Fraction of the bankroll to stake.

Example 6.2

A coin lands heads with probability 0.60.6 and you are paid even money on heads. What fraction of your bankroll should you stake on each flip?

Show the worked solutionHide the worked solution

Worked solution

  1. Formula
    f∗=bp−qbf^{*} = \frac{bp - q}{b}
  2. Substitute
    =1(0.6)−0.41= \frac{1(0.6) - 0.4}{1}
  3. Solve
    =0.2= 0.2
  4. Answer
    f∗=20%f^{*} = 20\%

Sanity check. With even money, Kelly reduces to p−qp - q, which is just your edge. A 60/40 coin gives a 20% edge and a 20% stake.

Why over-betting ruins a winning game. Growth is multiplicative, so a loss of xx per cent needs more than xx per cent to recover: lose half and you need to double. Staking above Kelly raises your expected wealth but lowers your median wealth, and past twice Kelly the long-run growth rate turns negative — you are near-certain to go broke playing a game you have an edge in. This is the concrete reason "bet it all" is the wrong answer to a positive-edge question.

StakeLong-run growthWhat happens
5%5\% — quarter KellyPositive, modestSlow, very smooth
20%20\% — full KellyMaximisedOptimal growth, uncomfortable swings
40%40\% — double KellyZeroAll that variance for nothing
60%60\% — treble KellyNegativeRuin, despite a genuine edge
Table 6.3 · Staking the 60/40 coin. Growth is flat near the peak and falls away sharply past it, which is why practitioners use a fraction of Kelly: you give up very little growth and a great deal of variance.

Proposition 6.4

Why nobody bets full Kelly

Full Kelly assumes you know pp exactly. You do not — you estimated it. Over-estimating your edge pushes you past the peak, where the penalty is severe, while under-betting costs you almost nothing. Half Kelly gives three quarters of the growth with half the volatility, which is why half or quarter Kelly is the practical answer.

Holds when

  • Kelly assumes the bet repeats and that you can re-size each time.
  • It assumes ruin is possible and permanent: with a bankroll you can top up, the calculus changes.

Proposition 6.5

When to take less expected value

Expected value is the right objective only when you can repeat the bet enough times for the average to arrive. When a single outcome can end the game — a position larger than your limit, a trade that breaches a risk budget — reducing variance is worth paying for, and saying so is not timidity.

Holds when

  • Sizing down, hedging, or crossing the spread to get flat all cost expected value and buy survival.
  • Interviewers ask this as "would you rather reduce the mean or the standard deviation?" — the answer depends on which constraint binds.

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Bet sizing, Kelly and risk of ruin · QuantMax