Kelly Criterion for Optimal Bet Sizing
You have a trading opportunity with probability $p = 0.6$ of making $W$ dollars and probability $q = 1 - p = 0.4$ of losing $L$ dollars per unit bet. Your current bankroll is $B$.
- Using the Kelly criterion, derive the optimal fraction $f^{*}$ of your bankroll to bet.
- For the even-money case ($W = L = 1$), what is $f^{*}$?
- What is the long-run growth rate at the optimal bet size?
Hints
- The Kelly criterion maximizes $E[\ln(\text{wealth})]$, not $E[\text{wealth}]$. Write the expected log growth as a function of the bet fraction $f$.
- For even money, the growth rate is $g(f) = p \ln(1+f) + q \ln(1-f)$. Differentiate with respect to $f$ and set to zero.
- You should get $f^{*} = p - q$ for even money. Verify: $g''(f) < 0$ everywhere, so this is a maximum. Plug back in to get the growth rate.
Worked Solution
How to Think About It: The Kelly criterion answers the most fundamental question in bankroll management: how much of your capital should you risk on a single bet? Bet too little and you leave money on the table. Bet too much and a string of losses wipes you out. The Kelly fraction maximizes the long-run geometric growth rate of your wealth -- equivalently, it maximizes $E[\ln(\text{wealth})]$. For an even-money bet with a 60% edge, your gut should say "bet some meaningful fraction but not all of it" -- and the answer turns out to be 20%.
Quick Estimate: For even money ($W = L$), the answer is just $f^{*} = p - q = 0.6 - 0.4 = 0.2$. Bet 20% of your bankroll. The growth rate is $g = 0.6 \ln(1.2) + 0.4 \ln(0.8) \approx 0.109 - 0.089 = 0.020$, or about 2% per bet.
Approach: Maximize the expected logarithm of wealth after one bet.
Formal Solution:
Part 1: General Kelly formula.
If you bet fraction $f$ of your bankroll $B$ on an even-money bet, your wealth after one round is:
- $B(1 + f)$ with probability $p$ (win)
- $B(1 - f)$ with probability $q$ (lose)
For a general bet paying $W$-to-$L$ odds (risk $fB \cdot L$ to win $fB \cdot W$, or equivalently, bet fraction $f$ and gain $fW$ on a win, lose $fL$ on a loss):
- Wealth $= B(1 + fW)$ with probability $p$
- Wealth $= B(1 - fL)$ with probability $q$
The expected log growth rate is:
$g(f) = p \ln(1 + fW) + q \ln(1 - fL)$
Differentiate and set to zero:
$g'(f) = \frac{pW}{1 + fW} - \frac{qL}{1 - fL} = 0$
$pW(1 - fL) = qL(1 + fW)$
$pW - pWfL = qL + qLfW$
$f(pWL + qLW) = pW - qL$
$f \cdot WL(p + q) = pW - qL$
Since $p + q = 1$:
$f^{*} = \frac{pW - qL}{WL} = \frac{p}{L} - \frac{q}{W}$
For odds $b = W/L$ (win $b$ dollars per
$f^{*} = \frac{bp - q}{b}$
Part 2: Even-money case ($W = L = 1$).
$f^{*} = \frac{p \cdot 1 - q \cdot 1}{1 \cdot 1} = p - q = 0.6 - 0.4 = 0.2$
Bet 20% of your bankroll each time.
Part 3: Long-run growth rate.
The growth rate at $f^{*} = 0.2$:
$g(f^{*}) = 0.6 \ln(1.2) + 0.4 \ln(0.8)$
$= 0.6 \times 0.1823 + 0.4 \times (-0.2231)$
$= 0.1094 - 0.0893 = 0.0201$
So the bankroll grows at approximately
Answer:
- General Kelly fraction: $f^{*} = \frac{p}{L} - \frac{q}{W}$
- Even money ($W = L = 1$): $f^{*} = p - q = 0.2$ (bet 20%)
- Long-run growth rate: $g(f^{*}) \approx 0.0201$ per bet
Intuition
The Kelly criterion is one of the most important results in quantitative finance and gambling theory. The core insight is that maximizing expected wealth (arithmetic mean) is not the right objective for repeated bets -- it leads to over-betting and eventual ruin. Instead, you maximize the expected log of wealth (geometric mean), which accounts for the compounding nature of sequential bets. The logarithm naturally penalizes large losses more than it rewards large gains, producing a conservative but growth-optimal strategy.
In practice, most traders bet a fraction of Kelly ("half-Kelly" is common) because full Kelly produces large drawdowns and the edge estimate $p$ is itself uncertain. If you overestimate your edge and bet full Kelly, you are actually over-betting relative to your true Kelly fraction, which can reduce growth or even produce negative expected growth. Half-Kelly sacrifices about 25% of the growth rate but reduces variance by 50%, which is usually a good trade-off when your edge is estimated rather than known.