Position Sizing: The Arithmetic

Scope — this article carries the refusal in its own body, and it is the sharpest refusal in the pillar. MarketClue publishes no maximum position size, no sizing rule or formula, no concentration limit, and does not tell any reader how much of anything to hold. This article sets out what size does arithmetically — the consequences, not the choice. All figures use the illustrative teaching parameters fixed on this pillar's hub and are not forecasts. The mechanics of borrowing to increase exposure belong to Margin and Leverage in Pillar 29; this article deals only with what exposure does to a portfolio's arithmetic.

A correct view held in a size that cannot be sustained produces a loss; an ordinary view held in a modest size produces an ordinary result. Size converts an opinion into a consequence.

The first-order arithmetic, which is simple

A position's effect on a portfolio is its weight multiplied by its own movement. That is trivially true and worth tabulating because the numbers are frequently smaller than people expect at one end and larger at the other.

The asymmetry from Investing and Speculation: Where the Line Is Drawn then applies to the portfolio figure: a 10% portfolio loss requires +11.11% to recover, a 20% loss requires +25.00%, and a 40% loss requires +66.67%.

The second-order arithmetic, which is not simple

A position's share of a portfolio's risk is not the same as its share of the money, and the two diverge sharply as the position grows.

Taking twenty holdings, each with a 30.0% standard deviation and an average correlation of 0.20 with the others, and varying only how much sits in the largest one.

Worked example — a position's share of the risk outruns its share of the money. At equal weight the two match exactly: a 5% position supplies 5% of the variance. Everywhere else they come apart. A holding at 20% of the money supplies 28% of the risk; at 40% of the money it supplies 61%; at 60% it supplies 82%. The relationship is quadratic rather than linear, because a position contributes both its own variability and its co-movement with everything else, and both terms scale with weight. The consequence is that a portfolio can look diversified by count and be dominated by one holding. Twenty holdings with one at 40% is, in risk terms, closer to a single-holding portfolio with a hedge than to a twenty-holding portfolio. This portal sets no maximum and draws no line — the arithmetic states what a given size does, and the choice belongs to the reader.

Why more exposure eventually reduces growth

The most counterintuitive result in sizing is that increasing exposure raises expected return and, past a point, lowers expected compound growth. The two are different quantities and the gap between them widens with variability.

On the hub parameters — a 4.0% risk-free rate and equities at a 9.0% expected return with 16.0% variability — varying total exposure to equities:

Worked example — a positive expected return with a negative expected growth rate. At five times exposure the expected arithmetic return is +29.00% a year and the expected compound growth is −3.00%. Both figures are correct and they describe different things: the arithmetic mean is what one would expect to earn in a single period, while compound growth is what actually accrues to a sum of money held across many periods, and the gap between them grows with the square of variability. On these parameters growth peaks at 1.95× exposure and reaches zero at 3.91×. Beyond the peak, taking more risk raises the expected return and lowers the money. This is the arithmetic behind why size cannot be set by conviction alone — a holder certain of a positive expected return can still size the position into a negative outcome, and the certainty makes no difference to the calculation. MarketClue recommends no level of exposure, and the numbers above depend entirely on the assumed parameters. (Compound growth here is the continuous-time approximation, arithmetic return less half the variance; the peak at (μ − r) ÷ σ² and the zero at twice that follow from it.)

The formal answer, and why it is not usable

There is a mathematically correct sizing rule. The criterion associated with Kelly identifies the position size maximising the long-run compound growth rate, and it is genuinely optimal for that objective.

It requires as an input the probability distribution of the position's returns. Not an estimate of expected return — the distribution. Nobody has it.

The consequences of that gap are asymmetric, which is what makes the rule dangerous rather than merely unusable. Sizing below the criterion costs growth gradually. Sizing above it costs growth quickly, and the penalty accelerates — which is the shape of the table above, where the decline past the peak is far steeper than the climb toward it. Practitioners who use the framework generally apply a fraction of what it prescribes, and that convention exists precisely because the inputs are unknown. This portal names the rule so a reader encountering it knows what it is, and supplies neither it nor any fraction of it.