The Growth Curve: One Peak, Then Ruin

Last lesson we derived the Kelly fraction $f^{*}=p-q/b$ — the single stake that makes a compounding bankroll grow as fast as it can. But a formula that spits out one magic number is dangerous if you don’t understand the landscape around it. What happens if you’re a little over? A little under? Twice over? The Kelly fraction lives on a curve, and the shape of that curve — where it peaks, where it crosses zero, how steeply it falls on each side — is where the real wisdom of Kelly betting hides.

This lesson maps that curve, the growth function $g(f)$. We’ll see it is a lopsided hill: it starts at zero, climbs to a single summit at $f^{*}$, comes back down to zero at roughly twice $f^{*}$, and then plunges into negative territory, dragging a positive-edge gambler all the way to ruin. And we’ll extract the most practical lesson in all of Kelly betting: when in doubt, bet too little, never too much.

The shape of the curve: a lopsided hill

Plot the growth rate $g(f)=p\ln(1+fb)+q\ln(1-f)$ against the fraction $f$ and you get a distinctive hump with four landmarks worth memorising:

• At $f=0$: you bet nothing, so nothing compounds. $g(0)=0$. Flat ground.
• Climbing to $f^{*}$: every extra bit of stake adds growth, at a diminishing rate, up to the single summit at the Kelly fraction $f^{*}$, where the slope is exactly zero.
• Back to zero near $2f^{*}$: past the peak the curve descends, and it returns to $g=0$ at approximately twice the Kelly fraction. Here you carry maximum risk for zero reward.
• Negative beyond, diving to $-\infty$: stake more than $2f^{*}$ and growth goes negative — you decay to zero over time. As $f\to1$ (betting everything) a single loss zeroes you, so $g\to-\infty$.

The analogy. Picture hiking a ridge in fog. From the trailhead ($f=0$) the path climbs to a summit ($f^{*}$). Keep walking the same direction and you don’t stay high — you descend the far slope back to trailhead elevation (at $2f^{*}$) and then off a cliff into a ravine ($g<0$). The summit is the only good place to stand. The tragedy is that both the far slope and the cliff feel like “betting more aggressively on a winning game” — the map gives no warning that you’ve crossed the top.

Why it’s a near-parabola — the quick derivation

Near the peak the hump is almost exactly a downward parabola, which makes its symmetry precise. Expand $\ln(1+fX)$ for small $f$ (where $X=+b$ on a win, $-1$ on a loss) using $\ln(1+u)\approx u-\tfrac12u^{2}$:

$g(f)=E[\ln(1+fX)] \approx f\,E[X] - \tfrac12 f^{2}\,E[X^{2}] = \mu f - \tfrac12\sigma^{2}f^{2},$

where $\mu=E[X]=pb-q$ is your edge (the mean per-unit return) and $\sigma^{2}=\operatorname{Var}(X)\approx E[X^{2}]$ is its variance. This parabola makes both landmarks exact:

$\text{peak: } g'(f)=\mu-\sigma^{2}f=0 \;\Rightarrow\; f^{*}=\frac{\mu}{\sigma^{2}}, \qquad \text{zero: } \mu f-\tfrac12\sigma^{2}f^{2}=0 \;\Rightarrow\; f=\frac{2\mu}{\sigma^{2}}=2f^{*}.$

So in the continuous (diffusion) approximation, growth is a clean parabola that peaks at $f^{*}=\mu/\sigma^{2}$ and hits exactly zero at $2f^{*}$. For discrete bets it’s slightly off — recall from Example A that double-Kelly $f=0.40$ gave $g\approx-0.0025$, a hair below zero rather than dead on — but “twice Kelly ≈ zero growth” is close enough to run your life by.

The symmetry trap: equal growth cost, unequal danger

Here is the subtle part almost everyone gets wrong. Because the hump is nearly a symmetric parabola about its peak, overbetting by some amount and underbetting by the same amount cost about the same in growth. Land at $0.5f^{*}$ or at $1.5f^{*}$ and you sacrifice a similar slice of growth. If growth were the only thing that mattered, over and under would be equally forgivable errors. They are not — because they carry wildly different risk.

Let’s tabulate the 60/40 even-money coin ($f^{*}=20\%$) at four fractions, tracking both growth and a rough measure of round-to-round risk. (Volatility of log-wealth scales roughly linearly with $f$, so it doubles from $0.5f^{*}$ to $f^{*}$ and doubles again by $2f^{*}$ — while growth barely moves near the top.)

Stare at the middle two data rows: $10\%$ and $30\%$ deliver almost identical growth (~1.5%/round, ~three-quarters of the peak). Symmetric in growth. But the $30\%$ bettor endures roughly three times the volatility of the $10\%$ bettor for that same return — and sits perilously close to the $40\%$ cliff where growth vanishes. The costs are symmetric; the consequences are not.

Watch the over-bet crash

Numbers on a curve are one thing; watching bankrolls die is another. In the simulator below, the chosen fraction is pre-set to 40% — exactly double Kelly for the 60/40 coin, right on the zero-growth crossing. Run it and compare the three staking lines: half-Kelly crawls steadily upward, full Kelly climbs fastest, and the 40% over-bet spikes early then collapses, typically ending near or below where it started despite the identical winning edge. Then flip the chosen line to the mean view: the average appears to rocket up — because a handful of absurdly lucky paths drag it skyward — while the median player (and almost everyone else) has been wiped out. That gap between the soaring mean and the sinking median is the ergodicity gap, and it is exactly what overbetting manufactures.

Under-betting: safe but slow

None of this makes under-betting free. Sit at half-Kelly forever and you genuinely leave growth on the table — about a quarter of it, in the 60/40 case (1.5% vs 2.0% per round), which compounds into a real difference over thousands of rounds. Under-betting is the tortoise: it will never blow up, never hit the ruin cliff, never suffer a heart-stopping drawdown — but it ambles. The trade-off is honest and worth stating plainly:

• Under-betting ($f<f^{*}$): slower growth, gentler ride, no ruin risk, forgiving of a mis-estimated edge. The tortoise.
• Full Kelly ($f=f^{*}$): fastest growth, but violent volatility and only exactly correct if you truly know $p$ and $b$. The knife’s edge.
• Over-betting ($f>f^{*}$): less growth than full Kelly, more volatility, and past $2f^{*}$ negative growth — fast, then fatal. The hare that dies.

Because the summit is flat (its slope is zero), a modest step to the low side costs almost nothing in growth while buying a lot of safety — which is the entire rationale for the fractional Kelly strategy the next lesson is built around. The asymmetry means the smart default isn’t the razor-edge peak; it’s a deliberate step down the safe slope.