Part III · Models · Chapter 3D

Position size and trading frequency

How much to trade, and how often

Even with a perfect model, oversizing a single trade can ruin the whole strategy. The intuition is brutal but simple: bet small, bet often. The best quant strategies often win 51% to 53% of their trades. If you oversize at that win rate, the variance of any single trade dominates the expected value while the probability of ruin climbs fast.

1 · The Kelly criterion

The canonical answer to the sizing question comes from Edward Thorp's gambling research and from Kelly (1956). Take a binary bet that, per dollar staked, wins $G$ dollars with probability $p$ or loses $L$ dollars with probability $1 - p$. If you stake a fraction $f$ of your wealth, each bet multiplies the account by $1 + fG$ or by $1 - fL$. By Chapter 1C, the account's long-run growth rate is the expected log return per bet:

The Kelly criterion stakes the fraction that maximizes $g$:

For the special case of symmetric payoffs scaled to 1, $G = L = 1$, it becomes

So a strategy with a 55% win rate at symmetric payoffs has a Kelly fraction of 0.10, which means betting 10% of wealth per trial. A strategy with a 60% win rate has a Kelly fraction of 0.20. A strategy with exactly a 50% win rate at symmetric payoffs has a Kelly fraction of zero: do not bet.

Where the Kelly fraction comes from

Step 1: the slope of a logarithm. The derivative of $\ln u$ is $1/u$ times the derivative of $u$, just as the square in Chapter 3C gave $2u$ times it. So the derivative of $\ln(1 + fG)$ with respect to $f$ is $G/(1 + fG)$, while that of $\ln(1 - fL)$ is $-L/(1 - fL)$.

Step 2: set the slope of (3D.1) to zero.

$$\frac{pG}{1 + fG} = \frac{(1-p)L}{1 - fL}$$ (3D.4)

Step 3: cross-multiply. Multiplying out gives $pG - pGLf = (1-p)L + (1-p)GLf$, in which collecting the terms in $f$ leaves $pG - (1-p)L = GLf$.

Step 4: divide by $GL$. The result is $f = p/L - (1-p)/G$, which is (3D.2).

Continuous returns

For continuous returns rather than binary outcomes, with $\mu$ the expected excess return and $\sigma^2$ the variance, the Kelly fraction is

Equivalently, the Kelly fraction equals the strategy's Sharpe ratio divided by its volatility, $(\mu/\sigma)/\sigma$, which makes it the leverage that maximizes expected compound growth. The derivation is short: holding $f$ dollars of the strategy per dollar of capital earns an excess return of $f\mu$ with a variance of $f^2\sigma^2$. By the variance drag of Chapter 1A, the excess growth rate is then $f\mu - f^2\sigma^2/2$, whose slope $\mu - f\sigma^2$ is zero at (3D.5).

Take $\mu = 10\%$ and $\sigma = 20\%$, a Sharpe ratio of 0.5, for which the Kelly fraction is $0.10/0.04 = 2.5$: hold $2.50 of the strategy per dollar of capital, which is 2.5 times leverage. Its volatility is then $2.5 \times 20\% = 50\%$, the level at which Chapter 1D found growth peaks, with an excess growth rate of $0.25 - 0.125 = 12.5\%$ a year.

The figure below draws (3D.1) against the fraction staked, starting at 55% and even odds. Move the win probability and the payoffs, and watch half and quarter Kelly keep about 75% and 44% of the peak while the curve past the peak falls faster.

win probability p

55%

win per dollar G

1.00

loss per dollar L

1.00

preset 55%, even odds the retail trade

f* = p/L − (1 − p)/G = 0.100

growth is negative past f = 0.199

Stake	f	g(f) a bet	Share of peak
quarter Kelly	0.025	0.0022	44%
half Kelly	0.050	0.0038	75%
full Kelly	0.100	0.0050	100%
twice Kelly	0.200	−0.0001	−3%

2 · Why Kelly is a ceiling, not a target

The Kelly criterion maximizes expected compound growth at the cost of enormous variance in outcomes. A strategy sized at full Kelly has roughly a 50% chance of seeing a drawdown of 50% or more at some point. The math is unforgiving: optimal compound growth requires accepting wild swings in path.

Chapter 1D's ruin formula shows where that 50% comes from, for a fall measured from the starting value. Stake a multiple $c$ of the Kelly fraction, so that the levered drift is $c f^* \mu$ and the levered variance $c^2 f^{*2}\sigma^2$. The exponent in the ruin formula, twice the drift over the variance minus one, becomes $2/c - 1$. So the chance of ever falling to half the starting value is

At full Kelly, $c = 1$, that is $0.5^1 = 50\%$, while growth has an equally simple form: substituting $f = c f^*$ into the growth rate gives the maximum growth times $2c - c^2$.

Most professional traders therefore use fractional Kelly, typically half-Kelly ($f = 0.5 \times f^*$) or quarter-Kelly. Half-Kelly gives up about 25% of expected compound growth but reduces the probability of drawdowns dramatically. Quarter-Kelly keeps about 44% of expected compound growth, giving up about 56%, in exchange for a further cut in the probability of drawdowns. The optimal fraction depends on the trader's tolerance for path-dependent risk and on the reliability of the estimated edge.

These rest on returns that follow the geometric Brownian motion of Chapter 1A, with the edge known exactly. Fat tails (Chapter 1B) make every number in the last column worse.

Why half-Kelly is the convention

The reason half-Kelly is the dominant convention is that estimates of edge are noisy. Suppose your true Kelly fraction is $f^*$ but your estimate has a 30% standard error. Betting at the point estimate means you are often betting at $1.3 \times f^*$, which has worse expected compound growth than betting at $f^*$: $2(1.3) - 1.3^2 = 91\%$ of the maximum.

Under-betting by the same 30% costs the same growth, because $2(0.7) - 0.7^2$ is also 91%. The two errors differ in risk: by (3D.6), the chance of ever halving is 69% at $1.3 \times f^*$ and 28% at $0.7 \times f^*$. Over-betting also hurts more than under-betting by the same factor: at twice Kelly, growth falls to zero, while at half Kelly it keeps 75%. Half-Kelly accommodates the estimation error by buffering against over-betting.

How much to shade a noisy estimate

Step 1: the error. Suppose your estimate is $f^*(1 + e)$, where the relative error $e$ has mean zero and standard deviation $s$. You bet a multiple $c$ of the estimate, which is $c(1 + e)$ times the true Kelly fraction.

Step 2: the expected growth. Using the growth factor $2x - x^2$ with $x = c(1 + e)$, together with $\mathbb{E}[(1 + e)^2] = 1 + s^2$, gives an expected share of maximum growth of $2c - c^2(1 + s^2)$.

Step 3: the best multiple. Its slope, $2 - 2c(1 + s^2)$, is zero at $c = 1/(1 + s^2)$. So a 30% standard error gives $1/1.09 = 0.92$, while an error as large as the estimate itself, $s = 1$, gives $1/2$, which is half-Kelly.

A standard error as large as the estimate is common. By Chapter 1D, a Sharpe ratio measured over four years has a standard error of about $1/\sqrt{4} = 0.5$, so an estimated Sharpe ratio of 0.5 is exactly that case. Selection among many backtests (Chapter 1D) biases the estimate upward as well, while fat tails (Chapter 1B) add risk the formula does not see. Together they push the sensible multiple from a half toward a quarter.

3 · A retail-scale worked example

A formula and a buffering principle are useful but abstract. Walking one full chain at a retail account size brings the operational shape into focus, including the seam between this framework and the loss-budget framework that Chapter 13C builds on top of it.

Setup: an account of $250,000 runs a swing strategy with an estimated 60% win rate, a $4 average winner per share and a $3 average loser per share. It trades an underlying at $50 a share with a $2 stop distance.

The two frameworks disagree by an order of magnitude. The 1% loss budget buys 1,250 shares, a fifth of quarter-Kelly and a twentieth of full Kelly. For this specific edge, Kelly is far more aggressive than the loss-budget rule. The edge explains why: $0.6 \times \$4 - 0.4 \times \$3 = \$1.20$ a share, 2.4% of the price on every trade, is a very large edge, and Kelly sizes large edges hard.

This disagreement is structural, not a bug in either rule. Kelly's "bet" is the full notional at stake, under a binary-outcome geometry in which the entire position rises or falls by $G$ or $L$ per unit. The loss budget's "size" is the position scaled to a fixed-stop maximum loss, under a stop-distance geometry. To compare them, put both in one unit, the share of capital lost on a losing trade. Full Kelly here risks 30% of the account on an average loser, against the loss budget's 1% at the stop.

In that unit, the two converge only when the edge is mild. With symmetric payoffs the Kelly fraction of capital at risk is $2p - 1$, whatever the stop distance. At a 52% win rate that is 4%, a quarter of which is the loss budget's 1%. At 53% it is 6%, a quarter of which is 1.5%. So quarter-Kelly and the 1% budget meet around a 52% to 53% win rate with near-symmetric payoffs.

The divergence answers the question of whether the loss-budget rule is sufficient on its own. For an edge as strong as this one, the loss budget is the binding limit by a wide margin, so it sizes each trade well inside the Kelly ceiling. It stops being sufficient only if your true edge is much weaker than your estimate. Below a 52% win rate at even payoffs, 1% at risk is more than quarter-Kelly, while below 50.5% it is more than full Kelly. Most retail strategies do not have an edge like the one above. Chapter 13C reconciles the two frameworks explicitly: the loss budget for individual-trade sizing, and Kelly as a sanity check on the aggregate.

One more lesson hides in the setup: the average loser is $3 against a $2 stop, so at 1,250 shares an average loss is $3,750, which is 1.5% of the account rather than 1%. A stop sets the planned loss, while gaps and slippage set the real one.

4 · Sizing functions

How big should the trade be when the model produces a prediction of strength $s$, a signed signal between −1 and +1? Three sizing functions are common, each mapping $s$ to a position with a cap $c$, the two scaled ones through a slope setting $k$:

The hyperbolic tangent, $\tanh x = (e^{2x} - 1)/(e^{2x} + 1)$, rises smoothly from −1 to +1 with a slope of 1 at zero.

Constant sizing takes the same position whatever the prediction strength, trading just the sign of the prediction. It is the cleanest baseline for backtesting because the interpretation is unambiguous. Every trade is the same size, so the profit and loss (P&L) is a pure function of directional accuracy.

Piecewise-linear (hard tanh) sizing scales linearly with prediction strength up to a cap. Stronger predictions get larger trades, while the cap ensures that a runaway prediction, from a single extreme outlier in the input data, cannot blow up the size.

Smooth nonlinear (tanh) sizing is like the hard version but smooth, and differentiable everywhere. That is useful if you are optimizing the sizing parameters with gradient-based methods (Chapter 3C) alongside the model. The smooth transition between zero prediction (zero size) and strong prediction (capped size) also avoids the corner where the hard version abruptly stops responding. With the corner goes the "I just barely crossed the threshold, should I scale up?" decision.

In practice, most production quant strategies use either piecewise-linear or smooth-tanh sizing, calibrated so that the cap corresponds to half-Kelly at the strongest prediction signal the model produces.

5 · Frequency scaling

If your model has a tiny edge per prediction, say a 51% win rate or a per-trade Sharpe ratio of 0.05, you do not want to bet once but many times. The reason follows directly from the frequency-Sharpe relationship of Chapter 1D:

A tiny per-bet Sharpe ratio becomes a robust strategy Sharpe ratio when $N$, the number of independent bets, is large. Suppose a model predicts the 12-hour-ahead return with a Sharpe ratio of 0.05. Split it into many smaller bets at finer temporal resolution, betting on the 1-hour-ahead return 24 times a day, each time at a smaller size. Suppose each hourly bet carries the same per-bet Sharpe ratio of 0.05 and the 24 bets are roughly independent. Then the daily Sharpe ratio of the high-frequency version is $\sqrt{24} \approx 4.9$ times that of a single 12-hour bet a day.

That condition matters, because chopping one 12-hour bet into twelve hourly slices of the same drift adds nothing: each slice carries a twelfth of the mean and $1/\sqrt{12}$ of the noise. The gain comes only when every short bet brings a fresh, independent edge.

This is the structural reason high-frequency strategies dominate the modern quant landscape. They are not smarter about direction: they trade the same edge many more times. The compounding logic is identical to the Sharpe argument of Chapter 1D: more independent bets with positive expected value (EV) leave the same terminal drift with lower variance. That means a higher Sharpe ratio, which makes leverage safer, which lets returns scale.

The retail implication is sobering: most retail strategies make on the order of 10 to 100 trades a year, so even with substantial per-trade edge the resulting annual Sharpe ratio is modest. To compete with institutional high-frequency strategies you would need one of two things. The first is a per-trade edge an order of magnitude larger than what high-frequency firms exploit, which means operating in a different market regime, probably in lower-liquidity products. The second is the operational infrastructure to trade at high frequency, which retail traders do not have.

The honest path for retail is to accept that you cannot compete on frequency. Focus instead on capturing risk premia that do not require high frequency, such as the variance, equity and trend premia of Part X. Size conservatively too, because your effective Sharpe ratio is going to be modest.

6 · Frequency against cost

There is a fundamental tradeoff between frequency and transaction costs. Each trade costs the spread plus commission plus slippage, costs that eat the edge at very high frequency. The optimal frequency for any strategy is the one at which the marginal Sharpe gain from one more trade a day equals the marginal cost in basis points.

For high-frequency trading (HFT) firms, with co-located servers, near-zero latency and maker strategies that earn rebates, the optimal frequency can be thousands of trades a second. For retail traders paying 1 cent a share in commission plus 5 bps of slippage per round trip, the optimal frequency is dramatically lower, perhaps 10 to 50 trades a year depending on the per-trade edge.

The same arithmetic explains why the "free trades" era, after 2019, when most US brokerages dropped commissions for retail equity orders, did not transform retail performance. Commissions were already small, while the dominant retail cost is slippage, which retail traders neither see directly nor can negotiate down. The dominant retail edge-destroyer is behavioral: overtrading, sizing inconsistencies and emotional exits. Zero commissions fixed none of these.

7 · Path against terminal wealth

A final point is worth emphasizing: Kelly maximizes the expected log of terminal wealth over a long horizon. It does not minimize the probability of drawdown, does not bound the worst case and does not account for behavioral capitulation during drawdowns.

For a real investor with finite emotional bandwidth, drawdown discipline is more important than Kelly optimization. A strategy sized at quarter-Kelly with a 30% maximum drawdown discipline (a close-and-reset rule) outperforms a strategy sized at full Kelly with no drawdown discipline, in any realistic behavioral model. The full-Kelly strategy gets abandoned during its first 50% drawdown, and the quarter-Kelly strategy does not. By the ruin formula, the chance of ever falling 30% below the start is $0.7^7 = 8\%$ at quarter-Kelly, against $0.7^1 = 70\%$ at full Kelly.

Chapter 13C returns to this with explicit kill rules at three levels. For now the principle is: size for survivability first, then for compound growth. Kelly is the ceiling, half-Kelly is the working compromise and quarter-Kelly is what you actually use if you want to sleep through drawdowns.

8 · Check yourself