Continuous & Fractional Kelly

Last lesson, Kelly lived in a casino. You knew the exact odds — win probability $p$, payout $b$ to 1 — and out popped a tidy fraction to bet: $f^* = p - q/b$. Crisp, finite, almost smug. Markets are not casinos. A stock doesn’t pay you “2-to-1 with probability 0.55”; it dribbles out a continuous stream of returns with a mean and a wobble, and there’s no fixed payout to plug in. So we need a Kelly for the continuous world — and remarkably, it’s even prettier than the discrete one. Then comes the punchline every quant and pro gambler already knows: even when you can compute full Kelly exactly, you shouldn’t bet it. This lesson is about that gap.

From coins to markets

Analogy. A casino bet is a vending machine: insert stake, the odds are stamped on the front, the payout is fixed. A stock is more like a tide — it carries you up on average, but it sloshes, and there’s no sign telling you the exact odds of the next wave. You can’t write down “win probability” and “payout ratio” for owning the market for a year; what you can write down is its average return and its variability.

Definition. For a continuously-traded asset we summarise the return stream by two numbers: the excess return $\mu$ (the average return above the risk-free rate — the genuine reward for taking risk) and the volatility $\sigma$ (the standard deviation of returns, i.e. how violently it sloshes). Its variance is $\sigma^2$. The Kelly question becomes: what fraction of my wealth $f$ should I hold in this asset — where $f$ above 1 means borrowing to lever up, and $f$ below 1 means keeping some in cash — to grow my money fastest over the long run?

Notice the binary bet’s two knobs ($p$ and $b$) have collapsed into two new knobs ($\mu$ and $\sigma$). The spirit is identical — reward versus risk — but now risk is a continuous spread, not a coin landing one of two ways. Everything you learned about geometric growth and volatility drag carries straight over; we just feed it continuous returns.

The continuous Kelly formula

Analogy. Picture growth as a hill you’re climbing by choosing how much leverage $f$ to carry. Reward ($\mu$) is the tailwind pushing you uphill — it grows your expected return in proportion to how much you bet. But volatility drag is a headwind that grows with the square of how much you bet, because variance scales as $f^2$. A little leverage: lots of tailwind, trivial headwind. Too much leverage: the squared headwind overpowers everything and shoves you back down. The optimal $f$ sits exactly where the next gust of tailwind equals the next gust of headwind.

Definition. Hold a fraction $f$ of an asset with excess return $\mu$ and volatility $\sigma$. A position of size $f$ has mean return $f\mu$ and variance $f^2\sigma^2$. Plugging those into the volatility-drag (geometric-growth) formula from earlier in this topic, the long-run growth rate is

$g(f) = f\mu - \tfrac{1}{2}f^2\sigma^2.$

The first term is the reward, linear in $f$; the second is the drag, quadratic in $f$ — a downward parabola in $f$. To find the peak, differentiate and set to zero:

$\frac{dg}{df} = \mu - f\sigma^2 = 0 \quad\Longrightarrow\quad f^* = \frac{\mu}{\sigma^2}.$

That’s it — the continuous Kelly fraction, the most quotable result in position sizing:

$\boxed{\,f^* = \dfrac{\mu}{\sigma^2}\,}$

Bet more when reward is high, much less when volatility is high (note the square — doubling volatility quarters your Kelly bet). Now substitute $f^*$ back to find the growth you actually achieve at the optimum:

$g(f^*) = \frac{\mu}{\sigma^2}\mu - \frac{1}{2}\left(\frac{\mu}{\sigma^2}\right)^2\sigma^2 = \frac{\mu^2}{\sigma^2} - \frac{1}{2}\frac{\mu^2}{\sigma^2} = \frac{\mu^2}{2\sigma^2}.$

And here’s the gorgeous part. Since the Sharpe ratio is $S = \mu/\sigma$, that maximum growth is

$g(f^*) = \frac{\mu^2}{2\sigma^2} = \frac{1}{2}\left(\frac{\mu}{\sigma}\right)^2 = \tfrac{1}{2}S^2.$

The best achievable growth rate depends only on the Sharpe ratio. Not on $\mu$ alone, not on $\sigma$ alone — only on their ratio. Two strategies with the same Sharpe can be Kelly-optimised to the same long-run growth, however different their raw returns look. Sharpe isn’t just a risk-adjusted return number you report to investors; under Kelly it is your growth engine.

Worked example — leverage falls out fast

Let’s put real numbers on it. Take an asset (or a whole strategy) with excess return $\mu = 8\%$ and volatility $\sigma = 16\%$. The Kelly fraction is

$f^* = \frac{\mu}{\sigma^2} = \frac{0.08}{0.16^2} = \frac{0.08}{0.0256} \approx 3.125.$

Full Kelly says hold about 3.1× your wealth in this asset — i.e. borrow to run roughly 3:1 leverage. The growth you’d earn:

$g(f^*) = \frac{\mu^2}{2\sigma^2} = \frac{0.08^2}{2 \times 0.0256} = \frac{0.0064}{0.0512} = 0.125 = 12.5\%.$

A 12.5% long-run compound growth rate — and notice it equals $\tfrac{1}{2}S^2 = \tfrac{1}{2}(0.08/0.16)^2 = \tfrac{1}{2}(0.5)^2 = 0.125$, exactly as promised.

Now keep $\mu = 8\%$ but make the asset choppier: $\sigma = 20\%$.

$f^* = \frac{0.08}{0.20^2} = \frac{0.08}{0.04} = 2.0.$

Same reward, more volatility, and full Kelly drops from 3.1× to 2.0×. Crank volatility higher still and the recommended bet keeps shrinking, because that $\sigma^2$ sits in the denominator. The lesson: for any asset with a decent Sharpe, full Kelly routinely tells you to use serious leverage — which is the first clue that betting it literally is a wild idea.

Why full Kelly is too wild

Analogy. Full Kelly is the engine redlined. Yes, it’s the configuration that maximises long-run growth in theory — but it runs the motor at the absolute edge of its tolerance, and at that edge the slightest miscalibration or rough patch sends it into violent shaking. Nobody drives their daily commute at the redline. The reason isn’t that the redline is slow; it’s that it’s unsurvivable in practice.

The drawdowns. Full Kelly maximises growth, but it does so with brutal swings. A classic result: under full Kelly, the probability that your bankroll at some point dips to half its starting value is roughly 50%. Deep, frequent drawdowns aren’t a tail event at full Kelly — they’re the everyday weather. Most humans (and most investors with redemptions, margin desks, and a pulse) cannot stomach a coin-flip chance of watching half their money evaporate, however “optimal” the math calls it.

The estimation problem — and it’s the killer. The formula $f^* = \mu/\sigma^2$ assumes you know $\mu$. You don’t. Expected return is the single hardest quantity in finance to measure — volatility you can estimate reasonably well from data, but $\mu$ is buried in noise and needs decades of data to pin down. And $f^*$ is linear in $\mu$: overestimate your edge by 30% and you over-bet by 30%. Recall the over-betting cliff from the previous lesson — growth falls off a cliff once you bet past optimal, and far enough past it growth goes negative and you’re compounding your way to ruin. Since your $\mu$ estimate is always optimistically uncertain, your “full Kelly” is very probably an over-bet in disguise.

Fractional Kelly

Analogy. Think of full Kelly as the peak of a hill that’s flat on top but drops away in a sheer cliff on the over-betting side. If you stand a little back from the edge (bet a fraction of Kelly), you’re barely any lower — but you’re a comfortable margin from the cliff. The view (growth) is almost as good; the footing (safety) is vastly better.

Definition. Fractional Kelly means betting $k \cdot f^*$ for some $k$ in $(0, 1]$ — a constant fraction of the full Kelly amount. The magic is in how growth and risk respond differently to $k$:

• Volatility scales linearly: betting $k f^*$ gives $k$ times the volatility of full Kelly. Bet a third, get a third of the swings.
• Growth scales as a parabola: the growth you keep, as a fraction of the maximum, is

$\frac{g(k f^*)}{g(f^*)} = 2k - k^2 = k(2 - k).$

(That comes straight from $g(f) = f\mu - \tfrac12 f^2\sigma^2$ evaluated at $f = kf^*$ — the linear term scales by $k$, the quadratic by $k^2$.) Because that’s a downward parabola peaking at $k=1$, it’s flat near the top — so pulling $k$ back from 1 costs you almost nothing at first.

The headline case: half-Kelly, $k = 0.5$.

$\text{growth kept} = k(2-k) = 0.5 \times 1.5 = 0.75, \qquad \text{volatility} = 0.5.$

Three-quarters of the growth for half the volatility. You give up a quarter of your growth rate and cut your swings in half. That trade is so good it’s practically folklore. Quarter-Kelly is even more conservative: $k(2-k) = 0.25 \times 1.75 = 0.4375$, so ~44% of the growth at just 25% of the volatility.

That last row is the kicker — we’ll return to it. Drag the slider below and watch the two bars diverge: growth (the parabola) barely moves as you pull back from full Kelly, while volatility (the straight line) drops in lockstep with $k$.

The drawdown-vs-growth trade-off

Analogy. Imagine a thermostat where comfort (growth) flattens out near the ideal temperature but the energy bill (risk) keeps climbing in a straight line. Nudging the dial a touch below “perfect comfort” you’ll barely feel the difference — but the bill drops noticeably. Only a fool pays the linearly-rising bill to chase the last sliver of a flattening benefit.

The principle. This is the single deepest idea in the lesson, and it falls straight out of the math. Growth is a parabola in the bet size, so near its peak it is flat — the derivative is zero at the top, meaning the first bit of bet you shave off costs essentially zero growth. Risk (volatility) is a straight line in the bet size, so every bit you shave off buys a proportional, immediate reduction in swings. Flat benefit versus linear cost: shaving the bet is nearly free in growth and richly rewarded in safety. That asymmetry — and only that asymmetry — is why every serious practitioner bets a fraction of Kelly rather than the full amount.

Worked feel. Going from full Kelly ($k=1$) to three-quarter Kelly ($k=0.75$) drops growth from 100% to $0.75 \times 1.25 = 93.75\%$ — a loss of barely 6 percentage points — while cutting volatility by a full 25%. You bought a quarter less risk for a sixteenth less growth. Near the peak, safety is on sale.

Over-betting is strictly dominated. Now the other direction. Push past full Kelly to $k = 1.5$: growth is $1.5 \times 0.5 = 0.75$ — back down to 75%, the same as half-Kelly — but volatility is now 150%, three times that of half-Kelly. You’ve matched half-Kelly’s growth while tripling its risk. Over-betting (any $k$ above 1) is never a trade-off; it’s strictly worse on both axes: less growth and more risk than betting less. There is simply no reason to ever sit on the over-betting side of the peak.

Pitfalls

Estimation error makes your “full Kelly” an over-bet. Since $f^* = \mu/\sigma^2$ is linear in the one quantity you can’t measure well, plugging in an optimistic $\mu$ silently lands you past the peak — on the dominated, growth-destroying side. Fractional Kelly is partly insurance against being wrong about your own edge: betting half-Kelly leaves room for your $\mu$ estimate to be too high without tipping you over the cliff. A common rule of thumb: if you think you’ve estimated full Kelly, you’ve probably already over-bet, so halve it.

Kelly maximises growth, which may not be your goal. The whole formula assumes you want to maximise long-run geometric growth (equivalently, log-utility) and that you have an effectively infinite horizon to let it compound. A real investor has a finite horizon, redemption dates, a boss, and a stomach. If a 30% drawdown gets you fired or makes you capitulate at the bottom, the “optimal” growth rate is irrelevant — you won’t be around to collect it. Rationally betting even less than half-Kelly can be the right call for a risk-averse investor.

$f^*$ can exceed 1 — and the basic formula ignores the cost of that. When $f^*$ exceeds 1 (as in both worked examples), Kelly is telling you to borrow. But $g(f) = f\mu - \tfrac12 f^2\sigma^2$ quietly assumes borrowing is free and unlimited. In reality leverage costs interest (which eats into $\mu$), and margin calls can force you to sell at the worst possible moment — a path-dependent risk the smooth formula never sees. Lever up on the say-so of the bare formula and a sharp drawdown can liquidate you before the long run ever arrives.

When it matters

This is not academic. Wherever you size a position from a return-and-risk estimate, continuous and fractional Kelly are the governing logic:

• Sizing a trading strategy from its Sharpe. A strategy’s backtested Sharpe immediately tells you its Kelly-optimal growth ($\tfrac12 S^2$) and, with its volatility, the leverage full Kelly implies. Quants routinely compute it — then deliberately run a fraction of it.
• Leverage decisions. Whether to run a portfolio at 1×, 2×, or 3× is a Kelly question in disguise. The formula warns you how fast the over-betting cliff arrives, and the borrowing-cost pitfall warns you the smooth math understates the danger.
• Why “half-Kelly” is folklore. Across professional gambling, sports betting, and quant trading, “bet half-Kelly” is the near-universal default — not because anyone is timid, but because the 75%-growth-for-50%-vol trade, plus insurance against a wrong $\mu$, is mathematically close to a free lunch. When practitioners disagree, it’s usually about betting an even smaller fraction, never a larger one.

Putting it together

Markets don’t post their odds, so Kelly goes continuous: with excess return $\mu$ and volatility $\sigma$, the growth-optimal fraction is $f^* = \mu/\sigma^2$, and the best growth it buys is $\tfrac12 S^2$ — a function of the Sharpe ratio alone. But full Kelly is the redline: it implies aggressive leverage, carries a ~50% chance of halving your bankroll, and — because it’s linear in the un-knowable $\mu$ — is almost certainly an over-bet in disguise. So practitioners bet a fraction $k$ of it. The arithmetic is decisive: growth scales as the parabola $k(2-k)$ while volatility scales linearly as $k$, so half-Kelly keeps ~75% of the growth for 50% of the volatility, and over-betting (any $k$ above 1) is strictly dominated — less growth and more risk. Flat benefit, linear cost: shaving the bet is the closest thing position-sizing has to a free lunch.

You now have Kelly for the real world: a continuous formula, the Sharpe-only growth ceiling it implies, and — most importantly — the disciplined humility to bet a fraction of what it recommends. The thread running through this whole topic has been geometric growth: how volatility drags on compounding, and how to size bets to grow fastest without blowing up. Next, we tie it together and put it to the test.