The Shape of Growth

Growth is a curve, and the curve has a peak

Everything so far has treated the risk fraction as something to keep small. This lesson says how small, and the answer comes from a different question: not "how much can I survive?" but "what size makes the balance grow fastest?" Growth is a compounding process, so the quantity to maximise is the average of the logarithms of the outcomes — and logarithms are concave, which is why the answer is a peak rather than "as much as possible". For a bet that wins w times the stake with probability p and loses l times the stake with probability q, the growth rate of risking a fraction f of equity is p·ln(1 + f·w) + q·ln(1 − f·l). Maximising it gives the **Kelly fraction**: f* = (p·w − q·l) ÷ (w·l). For the house strategy — 45% wins, 1.5R winners, 1R losses — that is (0.675 − 0.55) ÷ 1.5 = **8.33%**. At that size the growth rate is **0.516% a trade**, which compounds into a large number over hundreds of trades, and it is the fastest it can go. Now the part that matters for discipline. Double the size to 16.67% and growth collapses to **0.014% a trade** — a third of a basis point, statistically indistinguishable from standing still — while the drawdowns become far larger. At 25% the growth rate is **negative**, about −1.5% a trade, even though the strategy wins 45% of the time at 1.5R on every single trade. That is what the simulation in R7 was measuring from the other end when it found a 65% ruin rate at 20% risk with a positive edge. Betting more than the peak does not buy a better version of the same return; it buys a slower one and a more violent path to it. The same edge at five sizes — 2% per trade: 0.219% growth a trade · 8.33% — the Kelly fraction: 0.516% — the peak ← · 16.67% — twice Kelly: 0.014% · 25% per trade: negative, about −1.49% ← Every field in the workbench is computed from these two formulas rather than from a simulation, so the curve is a fact about the arithmetic and not about a sample.

Why the peak is a ceiling and not a target

Kelly maximises long-run growth and says nothing about the path. The drawdowns at the peak are severe — for an edge this modest, the fraction that maximises growth is a fraction almost nobody could hold, because the variance around the median is larger than most households tolerate, and the estimate of the edge itself is uncertain. That last point is the practical one: the formula takes p, w and l as known, and in markets they are estimated from a sample, which means the computed optimum is itself an estimate with error. The standard answer is to bet a fraction of Kelly. Half Kelly gives **0.388%** a trade against the full 0.516% — about three quarters of the growth — for a substantial reduction in variance and drawdown. Quarter Kelly gives roughly half the growth for a much quieter ride. This is not timidity: it is the recognition that the optimum is a knife-edge computed from uncertain inputs, and that the penalty for overshooting is asymmetric. Being 20% below the peak costs a little growth; being 50% above it costs the growth rate entirely and then some, as the 25% row shows. There is also a structural reason to stay well under the peak: the input that is most wrong is usually the edge itself. A strategy believed to earn +0.125R a trade that actually earns +0.05R has a much smaller Kelly fraction, and sizing at the estimated optimum means sizing at the true over-optimum. The safe direction of error is obvious once stated — size as though the edge were smaller than you believe, because that error costs growth while the opposite one costs the account. Fractions of Kelly — Full Kelly — 8.33%: 0.516% growth a trade · Half Kelly — 4.17%: 0.388%, about three quarters of the growth ← · Twice Kelly — 16.67%: 0.014%, a fraction of the growth and far worse drawdowns · What the estimate of the edge does: moves the whole curve, which is why the safe error is downwards The Kelly fraction assumes you know p, w and l. In markets they are estimates from a sample, so the computed optimum is over-optimistic by construction — and the cost of being above the peak is far larger than the cost of being below it.

Kelly assumes you know the odds

The optimal fraction is derived from the win probability and the payoff, both treated as known. In a casino they nearly are. In markets they are estimates drawn from a finite sample, and that difference changes the recommendation. If your estimate of the edge is too high, the Kelly fraction oversizes relative to the truth, and the growth penalty for oversizing is much steeper than the penalty for undersizing — the curve is asymmetric, so an error in the optimistic direction costs more than the same error in the cautious one. Estimates of edge are also unstable. A system with a genuine 55% win rate can produce a sample that looks like 60%, and a Kelly fraction computed from that sample is a bet on a number that does not exist. This is the core of the argument for **fractional Kelly**: betting a fixed fraction of the theoretically optimal size keeps most of the growth while building in a margin for the possibility that the edge estimate is wrong. In practice, half-Kelly or less is common in systematic trading, and the choice is explicitly a hedge against model error rather than a claim about the true optimum. There is a second caveat in the multi-asset case. Independent Kelly bets can be sized one at a time; correlated bets cannot, because their worst outcomes arrive together, and a portfolio of individually optimal positions can be collectively over-bet. When positions share a common exposure — the same factor, the same market beta — the fraction should be set at the *portfolio* level, which is why correlation is a sizing input and not just a diversification nicety. Kelly tells you the best bet for a *known* edge. With an estimated edge, the honest answer is a fraction of it, plus a rule that shrinks further when the estimate is uncertain.

Half Kelly, and the version with many bets at once

The previous read said practitioners bet a fraction of the optimum and that the ride at the peak is one almost nobody holds. The arithmetic behind that remark is unusually friendly, and it is the reason half Kelly is the standard rather than a compromise. The growth curve is **flat near its peak** — it is a parabola in the fraction to a first approximation, because it comes from maximising a logarithm — so moving from the optimum to half of it gives up only about a quarter of the maximum growth rate while cutting the variance of the outcome by roughly three quarters. Worked on this lesson’s numbers: the peak at 8.33% produces 0.516% a trade, and at 4.17% the growth rate is still around 0.39% a trade, while the drawdowns the holder has to sit through are roughly halved. The ratio is the argument — you keep three quarters of the growth for a quarter of the variance — and it is why the estimation error in the edge, which biases the computed optimum upward, costs so little when you bet a fraction of it. Betting half of an overestimated peak lands close to the true peak; betting all of it lands well past. The single-bet formula also has a form that generalises, and the generalisation is where most of the practical difficulty lives. For a continuous distribution of returns rather than a binary win-or-lose bet, the growth-optimal fraction of capital is approximately the **expected excess return divided by the variance** — which is the same shape as the mean-variance optimum and explains why the two ideas keep converging in these lessons. With several simultaneous bets the calculation becomes a matrix problem: the optimal vector of positions is the inverse of the covariance matrix multiplied by the vector of expected excess returns, and the result has two properties that catch people. The first is that it produces large positions in low-volatility assets and reduces positions in assets that are highly correlated with the rest of the book, which often looks nothing like the intuition a trader has about their best ideas. The second is its **violence to error**: inverting a covariance matrix computed from a sample amplifies the estimation noise, so small differences in estimated correlations produce large differences in the recommended positions. That is why the practitioner’s version is Kelly scaled by a fraction and then constrained — caps per position, caps per driver, and a stated maximum heat — with the constraints doing the work and the formula setting the direction. Two boundary conditions keep the whole apparatus honest. Kelly assumes you can **re-bet continuously** and that the edge does not change; a strategy whose edge decays as size grows — which is most of them, because the returns come from a finite opportunity — has a growth-optimal size lower than the formula’s, and the formula does not know that. And Kelly maximises the growth of *capital*, not the utility of its owner: an individual with obligations, a horizon and a tolerance for drawdown is not a growth-maximising machine, which is why the fraction is chosen against the drawdown a person can hold (R5) and not against the peak of a curve. The synthesis is the one the subject keeps making: the formula tells you where the ceiling is, the constraints keep you inside it, and the fraction converts a mathematical optimum into a plan someone will actually follow. The curve is flat at the top — Full Kelly — 8.33%: Growth 0.516% a trade, with the drawdowns nobody holds · Half Kelly — 4.17%: About 0.39% a trade — three quarters of the growth, near a quarter of the variance ← · Twice Kelly — 16.67%: Growth collapses to 0.014% a trade on the same edge ← · Many bets at once: Inverse covariance matrix × expected excess returns, then scaled and capped The link to the correlation lesson is the reason the constrained version wins: the matrix formula sizes positions by their contribution to portfolio variance, which means it is using the same effective-bet arithmetic. Kelly with correlated bets is not more aggressive than Kelly with independent ones — it is more concentrated in the few drivers it can identify, which is the honest answer and the one most people find uncomfortable.

Sources

• The Kelly criterion: maximising the expected logarithm of wealthJ. L. Kelly, "A New Interpretation of Information Rate" (1956)
• Fractional Kelly and the trade-off between growth and drawdownMacLean, Thorp & Ziemba, "The Kelly Capital Growth Investment Criterion"
• Estimation error and why full Kelly is fragile in practiceThorp, "The Kelly Criterion in Blackjack, Sports Betting and the Stock Market"
• Geometric growth and the negative growth rate above twice KellyStandard log-growth (geometric mean) arithmetic