Start with the arithmetic, because it settles more arguments than any forecast does. A position that goes to zero costs you its full weight, and the rest of the portfolio has to make that back. At 2% of the portfolio, the survivors need to gain 2.04%. At 5%, they need 5.26%. At 10%, 11.1%. At 25%, a third.
None of those numbers depend on what the asset is. They don't need a probability, a valuation model or a view. They only need one assumption: that the thing can plausibly reach zero and stay there. That is the whole test, and it is the cheapest piece of analysis available to anyone holding something speculative.
Total loss is not a tail case for single assets
The base rate here is worse 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 a position-sizing decision the 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 select more carefully. The evidence on that is discouraging. 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 the characteristics of those companies at their peak share prices, before anything went 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. If profitability, leverage, valuation and consensus all failed to flag the losers in advance, then the screen you're planning to run has already been run by people with more data. 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 determines 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. His set-up was a gambler receiving results over a noisy wire. 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 comment on that strategy is blunt: the gambler "would probably be broke" for large N, "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. Kelly was careful about why. The log has nothing to do with how much the gambler values money, he wrote. It is 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. Kelly notes that if the gambler's wife let him bet one dollar a week but not reinvest his winnings, he should maximise expectation instead, and bet 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 never repeat.
Kelly's 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 is 222% of wealth in the index, or 2.2 times leverage. The reward for that is 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.
Nobody sensible runs that. Thorp doesn't either, and 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 call this the criterion's main disadvantage in plain terms: 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. Thorp's figures for a favourable game: 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.
| Fraction of the Kelly bet | Chance of doubling before halving | Relative growth rate |
|---|---|---|
| 0.5 (half Kelly) | 0.89 | 0.75 |
| 1.0 (full Kelly) | 0.67 | 1.00 |
| 1.5 | 0.56 | 0.75 |
| 2.0 (double Kelly) | 0.50 | 0.00 |
Those figures come from a blackjack game with a 2% advantage, where the probabilities really are known. Note the symmetry in the growth column and the lack of it everywhere else. Half Kelly and 1.5 times Kelly deliver 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 on, the safe side is obvious.
Why the answer is always a fraction of the answer
Growth-optimal sizing is exquisitely sensitive to the expected return, and the expected return is the 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. If 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 is 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: overbetting is punished far more severely than underbetting, so to the extent future probabilities are uncertain, a long-term compounder limits the fraction further.
This is what "sizing dominates selection" means mechanically. Two investors buy the same asset, one at 3% of the portfolio and one at 30%. If the asset multiplies tenfold, the first is meaningfully richer and the second is transformed. If it goes to zero, the first is annoyed and the second has lost a decade. The spread of outcomes across the sizing choice is wider than the spread across most plausible selection choices, which is also why the gap between volatility and drawdown as risk measures shows up so clearly in concentrated positions.
The counter-argument: nobody has the distribution
Here is 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 could not open, so I am not characterising its argument beyond its title. The 1971 objection stands on its own: growth-optimality is a criterion someone chose, not a law.
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 the plan. That is 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 need to know the optimal fraction to know that being well below it costs you a quarter of the growth while being well above it costs you the portfolio. When the input is uncertain, that asymmetry is an argument for a small number, and the more uncertain the input, the smaller.
Third, one of Kelly's best properties quietly fails for the assets people are worried 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 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 matters more than sizing. The whole 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. A diversified index fund, a government bond in the issuer's own currency, a property with a tenant — these 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 holding cash instead.
The Kelly machinery in particular assumes you compound the position: same process, repeated, with 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 is a deliberate purchase of 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 is a preference, not a finding.
What doesn't change is the order of operations. Decide the weight at which a total loss leaves the plan intact, write it down before the position exists, and let selection compete for what's 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 this framework is short. You can't size a speculative position correctly, because correct requires a distribution you don't have. You can size it survivably, and the two are not the same claim.