Kelly Criterion for Optimal Bet Sizing

Optimization · Medium · Free problem

You face a sequence of independent, identically distributed bets. Each bet wins with probability $p$ and loses with probability $q = 1 - p$. When you bet a fraction $f$ of your current wealth: - On a win, your wealth is multiplied by $(1 + bf)$, where $b > 0$ is the net odds (you gain $b$ times your bet). - On a loss, your wealth is multiplied by $(1 - f)$ (you lose your bet).

  1. Derive the Kelly-optimal fraction $f^{*}$ that maximizes the expected logarithmic growth rate of wealth.
  1. What condition on $p$ and $b$ is required for $f^{*} > 0$ (i.e., for it to be worth betting at all)?
  1. What happens if you bet more than $f^{*}$? What if you bet less?

Hints

  1. Maximize the expected log-growth rate per round, not the expected arithmetic return. Why does log-wealth matter for long-run compounding?
  2. Write the growth rate as $G(f) = p \ln(1 + bf) + q \ln(1 - f)$ and differentiate with respect to $f$. The concavity of $\ln$ guarantees a unique maximum.
  3. Set $G'(f) = 0$ and solve: $\frac{pb}{1+bf} = \frac{q}{1-f}$. Cross-multiply and use $p + q = 1$.

Worked Solution

How to Think About It: This is the foundational problem of bankroll management. You want to grow your wealth as fast as possible over many bets, but betting too aggressively risks ruin. The insight is that maximizing expected wealth (arithmetic mean) leads to insane bets, but maximizing expected log-wealth (geometric mean) automatically balances growth against risk. The Kelly fraction is the sweet spot: it maximizes the long-run compound growth rate.

Quick Estimate: For a coin flip with 2:1 odds ($p = 0.5$, $b = 2$), the edge is $pb - q = 1.0 - 0.5 = 0.5$ and Kelly says bet $f^{*} = 0.5/2 = 25\%$ of your wealth each round. That feels right -- you have a nice edge but still lose half the time, so you should not bet too heavily.

Approach: Maximize the expected log-growth rate per bet.

Formal Solution:

After one bet at fraction $f$, the wealth changes from $W$ to: - $W(1 + bf)$ with probability $p$ (win) - $W(1 - f)$ with probability $q = 1-p$ (loss)

The expected log-growth rate per round is: $G(f) = p \ln(1 + bf) + q \ln(1 - f)$

To maximize, take the derivative and set it to zero: $G'(f) = \frac{pb}{1 + bf} - \frac{q}{1 - f} = 0$

Solving: $pb(1 - f) = q(1 + bf)$ $pb - pbf = q + qbf$ $pb - q = pbf + qbf = bf(p + q) = bf$ $f^{*} = \frac{pb - q}{b}$

Since $q = 1 - p$, this can also be written as: $f^{*} = \frac{pb - (1 - p)}{b} = p - \frac{1 - p}{b} = p - \frac{q}{b}$

Verification that this is a maximum: $G''(f) = -\frac{pb^2}{(1+bf)^2} - \frac{q}{(1-f)^2} < 0$ for all valid $f \in [0, 1)$. So $G$ is strictly concave and $f^{*}$ is the unique global maximum.

Part 2: When is it worth betting?

$f^{*} > 0$ requires $pb - q > 0$, i.e., $pb > q = 1 - p$, which gives: $p > \frac{1}{1 + b}$

This is the positive edge condition: the expected gain per dollar bet ($pb - q$) must be positive. If $pb \leq q$, the Kelly fraction is zero or negative -- you should not bet (or if allowed, bet against the outcome).

Part 3: Over-betting and under-betting.

  • **Betting $f < f^{*}$ (under-betting):** You grow slower than optimal but with lower volatility. The growth rate $G(f)$ is still positive (as long as $f > 0$ and you have edge), just not maximal. This is the "fractional Kelly" approach -- many practitioners bet $f = cf^{*}$ with $c \in (0.3, 0.7)$ to reduce variance.
  • **Betting $f > f^{*}$ (over-betting):** You grow slower AND have higher volatility. This is strictly worse than Kelly on both dimensions. In the extreme, betting $f = 1$ (all-in every round) gives $G(1) = p \ln(1+b) + q \ln(0) = -\infty$ -- guaranteed ruin after the first loss.
  • **Betting $f = 2f^{*}$ (double Kelly):** The expected growth rate equals zero. You neither grow nor shrink on average (in log terms), but the path is extremely volatile. Any bet beyond
f^{*}$ actually shrinks wealth on average.

The growth rate as a function of $f$ is a concave curve: it rises from $G(0) = 0$, peaks at $f^{*}$, crosses zero again at f^{*}$ (approximately), and goes to $-\infty$ as $f \to 1$.

Answer: The Kelly-optimal fraction is $f^{*} = \frac{pb - q}{b}$, where $pb > q$ is required for a positive bet. Over-betting is strictly worse than under-betting: it decreases growth while increasing volatility. The Kelly fraction maximizes long-run compound growth rate $G(f) = p \ln(1 + bf) + q \ln(1 - f)$.

Intuition

The Kelly Criterion is one of the most important results in applied probability. The core insight is that repeated multiplicative gambles require you to think in terms of geometric means, not arithmetic means. A sequence of $+50\%$ and $-50\%$ returns does not average to 0 -- it averages to a loss of 5\%$ per round (

.5 \times 0.5 = 0.75$). Maximizing the arithmetic expected return leads to overbetting and eventual ruin; maximizing the log (geometric) growth rate automatically accounts for the "volatility drag" that erodes compounded returns.

In practice, almost nobody bets full Kelly. The growth rate curve is flat near the top, so you give up little growth by betting $0.5f^{*}$ (half Kelly), but you dramatically reduce the drawdowns and the variance of outcomes over any finite horizon. This is why real trading desks use fractional Kelly or equivalently impose position limits well below what the edge would justify -- the theoretical optimal is too volatile for finite-horizon survival.

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