Position Sizing: The Arithmetic of Survival

Three lessons have been about measurement: what a risk number says, where its distribution fails, and how leverage converts a loss into a forced exit. None of them changes an outcome.

The decision that does is size. Given a view, how much to put behind it.

This is more consequential than it sounds, because sizing is not a scaling of the outcome. A strategy with a genuine edge, sized too large, loses money. Not less money: actual losses, from a positive-expectation bet. The mechanism is compounding, and it is worth deriving rather than asserting.

That result reframes the job. Edge determines whether a strategy is worth trading at all. Sizing determines whether the edge is ever realised, and the two are separable problems that get solved by different people using different tools.

Take a favourable bet: a 60% chance of gaining the amount staked, 40% of losing it. Expected value is clearly positive.

Stake everything each time and ruin is certain, since one loss ends the sequence. That much is obvious. What is less obvious is that sizes well short of everything also fail.

The reason is that returns multiply rather than add. A 50% loss requires a 100% gain to recover. Gains and losses of equal percentage do not cancel: lose 50%, gain 50%, and you are at 75% of where you started.

So the quantity that grows wealth is not the average return but the average log return, which penalises large losses far more heavily than it rewards equally large gains. Increasing bet size raises the average return and, past a point, lowers the average log return.

Between those two points sits an optimum. Below it, growth is left on the table. Above it, growth falls, and beyond a further point it turns negative despite every individual bet being favourable.

J. L. Kelly Jr. derived the growth-optimal fraction in "A New Interpretation of Information Rate", Bell System Technical Journal, volume 35, 1956, pages 917 to 926. The paper is about communication channels: a gambler receiving information over a noisy channel can grow capital exponentially, at a rate equal to the channel's rate of information transmission.

For a bet paying odds b to 1 with win probability p, the growth-optimal fraction of capital to stake is:

f* = (p * b - (1 - p)) / b        which for even money is  2p - 1

import math

def growth_rate(f, p=0.6, b=1.0):
    """Expected log growth per bet at fraction f."""
    return p * math.log(1 + f * b) + (1 - p) * math.log(1 - f)

for f in (0.05, 0.20, 0.40, 0.60, 0.80):
    print(f, round(growth_rate(f), 5))
# 0.05  0.00876
# 0.2   0.02014     <- Kelly for p=0.6 is exactly 0.20
# 0.4   -0.00245    <- twice Kelly: growth has already crossed zero
# 0.6   -0.08451    <- still a 60% edge, now shrinking capital
# 0.8   -0.29110

At 60% and even money the optimum is 20% of capital. At 60% of capital, the same favourable bet destroys wealth at roughly 8% per bet.

The curve rises to a peak and then falls through zero, and the crossing sits at roughly twice the Kelly fraction. It is exactly twice in the continuous limit where returns are Gaussian; for the discrete bet above, growth at 0.40 is already slightly negative. Either way the landmark is the same and it is the single most useful fact in the subject.

It means a trader who correctly identifies an edge and then bets about twice the optimal amount earns nothing at all in the long run, and anything beyond that loses. No error in the view is required; the error is entirely in the sizing.

The asymmetry around the peak matters too. Betting half of Kelly gives about three quarters of the maximum growth rate. Betting double gives zero. Erring small is cheap and erring large is catastrophic, and the penalty is not symmetric in any respect.

That asymmetry is the whole argument for the practice in the next step.

Kelly is growth-optimal under one assumption that never holds: that p and b are known exactly.

In trading they are estimated, and estimated with considerable error. The previous lessons supply the reasons: distributions are fat-tailed, volatility shifts regime, and correlations move. An edge estimated from historical data is a number with a wide confidence interval, and the lesson Selection Bias and the Deflated Sharpe Ratio shows that the estimate is biased upward whenever the strategy was selected for looking good.

Now combine that with the shape of the curve. Overestimating the edge means overbetting, and the penalty for overbetting is severe and asymmetric.

So practitioners use fractional Kelly, commonly a half or a quarter. The trade is explicit and favourable: half Kelly gives roughly three quarters of the growth rate with substantially less volatility and far more tolerance for having overestimated the edge.

Quarter Kelly is defensible when the edge estimate is genuinely uncertain, which is most of the time.

The reason large losses are disproportionately costly is arithmetic, and the numbers are worth memorising because intuition gets them badly wrong.

def recovery_needed(drawdown):
    return drawdown / (1 - drawdown)

for dd in (0.10, 0.25, 0.50, 0.75, 0.90):
    print(f"{dd:.0%} down -> need {recovery_needed(dd):.0%} to get back")
# 10% down -> need 11% to get back
# 25% down -> need 33% to get back
# 50% down -> need 100% to get back
# 75% down -> need 300% to get back
# 90% down -> need 900% to get back

The relationship is convex. Losing 10% is nearly symmetric and easily recovered. Losing 50% requires doubling. Losing 90% requires a tenfold gain, which for most strategies means the capital is simply gone.

This is the quantitative justification for capping drawdown rather than optimising average return. A strategy is not a sequence of independent bets on a fresh account; it is one compounding path, and a deep enough hole cannot be climbed out of at any plausible rate of return.

The non-arithmetic version is worse: investors redeem and desks are closed at drawdowns far shallower than the ones that are mathematically unrecoverable.

Formal risk limits look bureaucratic and are better understood as a control system with a specific design goal.

A limit is not a prediction. Nobody believes the loss will stop at the limit. Its function is to force a decision at a predetermined point, chosen while calm, rather than in the middle of a loss when judgement is worst and the incentive to hold on is strongest.

A workable structure uses several layers that bind at different times. A position limit caps concentration in one instrument. A book-level risk limit caps aggregate exposure. A drawdown limit reduces size after cumulative losses, which is the layer that makes the recovery arithmetic tolerable. A stop-loss governs a single trade.

The drawdown layer is the one most often missing and the one that matters most, because it is the only one that responds to the compounding path rather than to a snapshot.

The design question for any limit is what happens when it binds. A limit with no defined action is a report. A limit that automatically reduces size is a control.

Most real books combine several, and the combination is usually the point: volatility targeting to normalise across instruments, fractional Kelly to set the overall scale, and a drawdown rule as the backstop.

The last row's weakness is real and worth stating plainly. Reducing size after losses means participating less in the recovery. That is a genuine cost, accepted deliberately, in exchange for guaranteeing that no single episode ends the enterprise. It is the same trade as buying insurance, and it is judged the same way: not by whether it paid off, but by whether the exposure it removed was one you could afford to keep.

Four lessons, and the arc runs from measurement to decision.

A risk number compresses a distribution and hides the tail, and the coherent replacement fixes one specific defect rather than the general problem. The distribution itself is estimated and wrong in known directions, all of which understate danger and all of which fail together. Leverage converts those errors from paper losses into forced exits by handing the exit decision to a lender. And sizing is the only one of these that is actually a decision.

The transferable idea is that survival and optimisation are different objectives that trade against each other, and that the trade is quantifiable rather than a matter of temperament. Twice Kelly earns zero. A 50% drawdown needs a double. Those are arithmetic facts, not attitudes to risk.

The practical residue is a habit: before sizing anything, compute what the position does in the case you are confident will not happen. Not because the estimate of its probability is right, but because the whole point of the preceding three lessons is that it is not.