Position Sizing When an Asset Can Go to Zero

Key takeaways

• A position wiped out at 5% of a portfolio needs a 5.26% gain from everything else to break even; at 25% weight it needs 33%.
• Of 25,967 US stocks from 1926 to 2016, only 42.6% beat Treasury bills over their lifetimes, and 4.3% created all the net wealth.
• Half Kelly keeps 75% of the growth rate while cutting the chance of ever losing half your capital from 1 in 2 to 1 in 8.
• Betting exactly twice the Kelly fraction drops the long-run growth rate to the risk-free rate; errors in means matter roughly 20 times more than covariances.
• Among J.P. Morgan's archive of catastrophic decliners, 54% were profitable at their peak price and 63% carried net debt below 2 times EBITDA.

Two people buy the same speculative asset on the same morning. One of them is you, and you put 2% of your portfolio in. The other puts 25% in, because they've done the reading and they're confident.

Eighteen months later it's worth nothing. A fraud, a failed protocol, a bankruptcy filing — the cause barely matters now. What matters is what each of you has to do to get back to level.

You'd need everything else to gain 2.04%. Call that a decent year. They'd need 33% — a third, from every remaining asset, just to draw even. Had you gone in at 5%, your number would have been 5.26%. At 10%, 11.1%.

Same asset. Same ending. Same conviction, even. The only thing that differed was a number each of you picked before buying anything.

Now notice what that arithmetic didn't require. No probability, no valuation model, no view on the technology. It needs one assumption and one only: that the thing can plausibly reach zero and stay there. That's the whole test, and it's the cheapest piece of analysis available to anyone holding something speculative.

Total loss is not a tail risk case for single assets

So how often does that assumption bite? More often than most people expect.

Hendrik Bessembinder studied 25,967 US stocks in the CRSP database between July 1926 and December 2016. Just 42.6% of them beat one-month Treasury bills over their listed lifetimes. More than half delivered negative lifetime returns. The median stock was listed for seven and a half years.

The distribution is the point. Rounded to the nearest 5%, the most frequent decade-horizon outcome for an individual stock was a loss of 100%. Wealth creation went the other way: 1,092 firms, 4.3% of the sample, accounted for the entire net gain of the US market, and 90 firms accounted for half of it.

One caveat belongs next to that headline, and Bessembinder puts it in a footnote himself. Only 375 stocks in the data had a delisting return of exactly -100%. CRSP records a final price from another exchange or the over-the-counter market, and in an involuntary delisting that price is often small but not literally zero. So the mode is "near-total loss", not "provably total loss". For your sizing decision, that difference is cosmetic.

The follow-up study widened the lens. Bessembinder, Te-Feng Chen, Goeun Choi and K.C. John Wei looked at more than 64,000 global stocks from January 1990 to December 2020. They found 55.2% of US stocks and 57.4% of non-US stocks underperformed one-month Treasury bills in compound terms. The top 2.4% of firms accounted for all of the $75.7 trillion in net global wealth created. In 21 of 43 markets, the median stock had a negative buy-and-hold return over the full period.

That paper carries its own caveat, also disclosed by the authors. Outside the US there are no reliable delisting returns, so they follow the literature and impute a -30% return when a non-US firm exits the data. Near-total losses therefore look more common in the US figures than the non-US ones partly by construction.

Picking better doesn't fix it

The natural response is to screen harder. If most stocks are duds, avoid the duds. So how well does that work?

J.P. Morgan's Michael Cembalest has kept an archive of what he calls catastrophic declines — a stock that falls 70% from its peak and does not recover. His October 2024 update, the fourth in the series, looked at what those companies were like at their peak share prices, before anything had gone wrong. He doesn't state the start year of the sample in that edition, so treat the counts as covering the archive rather than a stated window.

Fifty-four percent of them were profitable at the peak. Sixty-three percent carried net debt of two times EBITDA or less. Most traded on forward price/earnings multiples between 10 and 50, which is to say they weren't obviously bubbles. Wall Street consensus at those same peaks was heavily tilted toward "strong buy", and around half of the stocks still had projected upside to analysts' price targets. The sector with the most catastrophic decliners was healthcare and biotech.

Read that as a statement about information, not about analysts. Profitability, leverage, valuation and consensus all failed to flag the losers in advance. The screen you're planning to run has already been run, by people with better data than you and more time to stare at it.

Which leaves sizing as the control that still works. It's the same reason fund overlap concentrating five funds into ten companies matters more than any single manager's stock picks: what you hold in size decides what a bad outcome does to you.

What the growth-optimal literature actually says

The formal treatment starts with John Kelly at Bell Labs in 1956. Picture his set-up: a gambler receiving results over a noisy wire, slightly ahead of everybody else. Betting the whole bankroll each time maximises the expected value of capital — the expectation grows as (2q) to the power of N.

Kelly's own verdict on that strategy is blunt. The gambler "would probably be broke" for large N, he wrote, "and, in fact, would be broke with probability one if he continued indefinitely."

His alternative was to bet a fixed fraction of capital, chosen to maximise the expected logarithm of wealth. In the simple two-outcome case that fraction is the edge divided by the odds. He was careful about why. The log has nothing to do with how much the gambler values money, he wrote. It's simply the function that adds up across repeated bets, and to which the law of large numbers applies.

That caveat matters more than the formula does. Kelly notes that if the gambler's wife let him bet one dollar a week but never reinvest his winnings, he should maximise expectation instead and put everything on the highest-expected-value event. Growth-optimal sizing is a rule for capital you compound. It says nothing useful about a one-off wager you'll never repeat.

The Kelly criterion answer for real assets is absurd

Edward Thorp worked the theory through for equities. Using rough historical estimates for the S&P 500 — an 11% mean, 15% volatility and a 6% risk-free rate — the full Kelly fraction comes out at 2.22. That's 222% of your wealth in the index. Borrowing to hold 2.2 times your money in shares, in other words.

And what do you get for it? A long-run growth rate of 11.5% against 9.9% unlevered, with the standard deviation of log wealth rising from 0.15 to 0.33. Would you take that trade?

Nobody sensible does, and Thorp doesn't either. The reason is structural rather than squeamish. In continuous time, betting exactly twice the Kelly fraction drives the growth rate down to the risk-free rate. Past that, growth turns negative and keeps falling. MacLean, Thorp and Ziemba put the criterion's main disadvantage plainly: the suggested wagers may be very large, so Kelly can be very risky in the short term.

The asymmetry around the optimum is the useful part. At half Kelly you keep three quarters of the growth rate, the probability of doubling before halving rises from 2/3 to 8/9, and the chance of ever losing half your starting capital falls from 1/2 to 1/8. You give up a quarter of the growth and cut the odds of a severe loss by a factor of four.

Those figures come from a blackjack game with a 2% advantage, where the probabilities really are known. Look at the growth column, then at the one beside it. Half Kelly and 1.5 times Kelly hand you the same growth rate. One of them halves your risk of a deep drawdown; the other raises it. If you can't tell which side of the optimum you're standing on, the safe side is obvious.

Why position sizing always ends at a fraction of the answer

Growth-optimal sizing is exquisitely sensitive to the expected return, and the expected return is the one input nobody knows.

Chopra and Ziemba measured this for portfolio choice generally. As reported by MacLean, Thorp and Ziemba, errors in means, variances and covariances matter in roughly a 20:2:1 ratio, measured by the cash-equivalent loss in final wealth. At a risk tolerance of 50, an error in the mean cost about 22.5 times what the same error in a covariance cost.

Thorp puts a number on what happens when the error runs the wrong way. Imagine your estimated mean is twice the true mean, and you size at 1.5 times your estimated Kelly fraction. The growth rate goes to -0.75 in his units. That's ruin. Size at exactly your estimated fraction with the same bad mean and growth is zero, which produces increasingly wild oscillations around your starting capital.

His conclusion is worth carrying around: overbetting is punished far more severely than underbetting, so to the extent future probabilities are uncertain, a long-term compounder limits the fraction further.

The same position sizing decision, in money

Say your portfolio is £100. It's an illustration — scale it to whatever you actually hold and nothing in the arithmetic changes.

Version one. You put £5 into the speculative asset and leave the rest where it was. The asset goes to zero. To get back to £100, everything else has to gain 5.26%. That's a decent year, or two quiet ones.

Version two. You put £25 in, because the case looked strong and the price had been rising for months. Same asset, same zero. Now the rest has to gain 33% just to draw level. That isn't a good year. That's a bull market you have to sit and wait for, holding less capital than you started with.

Now run it the other way, because zero isn't the only ending. Suppose the asset multiplies tenfold instead. The £5 version leaves you meaningfully richer. The £25 version transforms the portfolio, and the person who chose it looks brilliant for the rest of their life.

Both futures are real, which is exactly what makes this hard. Two investors buy the same asset, one at 3% of the portfolio and one at 30%, and the spread of outcomes across that single sizing choice is wider than the spread across almost any selection choice either of them could make. It's also why the gap between volatility and drawdown as risk measures shows up so sharply in concentrated positions.

So which version would you rather be holding on the morning you turn out to be wrong? That's the question the arithmetic is really asking, and it has almost nothing to do with the asset.

The counter-argument: nobody has the distribution

Here's the strongest objection, and it's a good one. Kelly sizing needs p, the odds and a stable process. For a novel asset there is no p. There's a short price history, a story about adoption, and a spread of expert opinion far too wide to pin a mean on. Applying a formula built for blackjack to that is false precision dressed up as rigour.

The authors agree with the objection. MacLean, Thorp and Ziemba write that the theory is straightforward when the underlying distributions are fairly accurately known, and that in investment applications this is usually not the case. They add that realised future equity returns may be very different from what history suggests. Thorp says the same thing in his conclusion. Nobody in this literature claims the inputs are knowable.

Paul Samuelson pressed a separate objection for decades. In his 1971 PNAS paper he argued that maximising the geometric mean is asymptotically valid but not optimal for any finite horizon, and not even a good approximation, for an investor whose utility isn't logarithmic. He returned to it in 1979 in a paper written almost entirely in words of one syllable. That second paper I couldn't open, so I'm not characterising its argument beyond its title. The 1971 objection stands on its own: growth-optimality is a criterion someone chose, not a law.

So does the whole framework fall over? Three things survive both objections.

First, the recovery arithmetic needs no distribution at all. It needs one scenario — the asset is worth nothing — and an honest answer about what that does to your plan. That's a question about your capacity to absorb a loss rather than your tolerance for one, and capacity is the half that can be measured.

Second, the asymmetry does the work the point estimate can't. You don't have to know the optimal fraction to know that sitting well below it costs you a quarter of the growth, while sitting well above it costs you the portfolio. When the input is uncertain, that asymmetry argues for a small number, and the more uncertain the input, the smaller.

Third, one of Kelly's best properties quietly fails for exactly the assets people worry about. The log-optimal bettor never risks ruin, but that result assumes continuously divisible bets and continuous rebalancing. An asset that gaps to zero overnight — a fraud, an exchange failure, a bankruptcy filing — breaks the assumption. The theory's own safety guarantee doesn't apply, which argues for a smaller fraction, not a larger one.

What would change the conclusion

The test stops binding when a total loss stops mattering. If the largest weight you'd plausibly hold is 1% of the portfolio, a wipeout costs 1.01% in recovery terms, and at that size selection genuinely does matter more than sizing. The framework only earns its keep in the range where a zero would change your plan.

It also fails if the asset can't reach zero. Think of a diversified index fund, a government bond in the issuer's own currency, a property with a tenant in it. Those have floors that a single company or a single protocol does not. Applying total-loss logic to them overstates the risk and understates the cost of sitting in cash instead. That cost is the whole case for an invested emergency fund, and the case against it is that the fall arrives with the job loss.

The Kelly machinery in particular assumes you compound the position: same process, repeated, winnings reinvested. Kelly's own example of the weekly one-dollar bet shows the rule flipping when that isn't true. A single, terminal, unrepeatable wager is an expected-value problem, not a growth-rate problem.

And if the edge really is knowable — a card counter with a measured 2% advantage, a market maker with a fill-rate distribution — then fractional sizing buys comfort rather than protection against error. Thorp is explicit that most people prefer half Kelly for psychological reasons, in exchange for giving up a quarter of the growth rate. That's a preference, not a finding.

What doesn't change is the order of operations. Decide the weight at which a total loss leaves your plan intact. Write it down before the position exists. Then let selection compete for whatever is left. Pre-commitment is the part that survives contact with a rising price, which is the argument for keeping the number in a written investment policy statement rather than in your head. LedgerTouch reports position weights against a target, which is the mechanical half of the same job.

The intellectually honest version of all this is short. You can't size a speculative position correctly, because correct requires a distribution you don't have. You can size it survivably. Those aren't the same claim, and only one of them is actually available to you.