Optimal Bet Sizing via the Kelly Criterion

Worked Solution

How to Think About It: This is the Kelly Criterion -- the single most important result in bankroll management. The key tension: bet too small and you leave edge on the table; bet too large and the geometric drag from losses destroys you. Arithmetic expectation says "bet it all" (your edge is positive), but compounding says otherwise. The right objective is not maximizing expected wealth (which is linear in $f$) but maximizing expected *log*-wealth (which captures the multiplicative nature of reinvestment). A senior trader would frame it this way: "What fraction makes my median outcome grow fastest?"

Quick Estimate: With $p = 0.53$ on an even-money bet, Kelly says $f^{*} = 2p - 1 = 0.06$. That feels small, but the edge ($p - q = 0.06$) is small. The optimal growth rate should be on the order of $f^{*2}/4 \approx 0.0009$ per round -- tiny but positive. Betting $f = 0.50$ is massively over-betting: the log-growth would be $0.53 \log 1.5 + 0.47 \log 0.5 \approx 0.53(0.405) + 0.47(-0.693) = 0.215 - 0.326 = -0.111$. That is a *negative* growth rate -- you go broke with certainty despite having a positive edge. This is the whole point of Kelly.

Approach: Maximize the expected log-growth rate, which is a concave function of $f$, by taking the first derivative and setting it to zero.

Formal Solution:

The one-round log-growth rate is:

$g(f) = E[\log W_{t+1} - \log W_t] = p \log(1 + f) + q \log(1 - f)$

Differentiate with respect to $f$:

$g'(f) = \frac{p}{1 + f} - \frac{q}{1 - f}$

Set $g'(f) = 0$:

$\frac{p}{1 + f} = \frac{q}{1 - f}$

$p(1 - f) = q(1 + f)$

$p - pf = q + qf$

$p - q = f(p + q) = f$

Since $p + q = 1$, we get:

$f^{*} = p - q = 2p - 1$

With $p = 0.53$: $f^{*} = 0.06$, i.e., bet 6% of your bankroll each round.

To confirm this is a maximum, check the second derivative:

$g''(f) = -\frac{p}{(1+f)^2} - \frac{q}{(1-f)^2} < 0$

This is always negative, so $g(f)$ is strictly concave and $f^{*}$ is indeed a global maximum.

The optimal growth rate is:

$g(f^{*}) = p \log(1 + (p - q)) + q \log(1 - (p - q))$

$= p \log(2p) + q \log(2q)$

$= p \log p + q \log q + \log 2$

$= \log 2 - H(p)$

where $H(p) = -p \log p - q \log q$ is the binary entropy of the bet. In natural logs with $p = 0.53$:

$g(f^{*}) = \ln 2 - (-0.53 \ln 0.53 - 0.47 \ln 0.47)$

$= 0.6931 - (0.53 \times 0.6349 + 0.47 \times 0.7550)$

$= 0.6931 - (0.3365 + 0.3549) = 0.6931 - 0.6914 = 0.0018$

So you grow at about 0.18% per round at the optimum.

For Part 3, at $f = 0.50$:

$g(0.50) = 0.53 \ln 1.5 + 0.47 \ln 0.5 = 0.53(0.4055) + 0.47(-0.6931) = 0.2149 - 0.3258 = -0.1109$

Negative growth rate: your bankroll decays geometrically. Despite a positive expected return per round, you are almost certain to go broke in the long run because the geometric mean of $(1.5)^{0.53} \times (0.5)^{0.47}$ is less than 1.

Answer: The optimal Kelly fraction is $f^{*} = p - q = 2p - 1 = 0.06$. The optimal growth rate is $g(f^{*}) = \log 2 - H(p)$, which equals approximately $0.0018$ per round. Over-betting at $f = 0.50$ produces a negative growth rate of about $-0.11$ per round, guaranteeing eventual ruin.