The pitch is sound. The market is real, the margin works, the downside is bounded at the amount committed. You run the numbers and the expected value is comfortably positive. And the amount committed is forty per cent of what you have left.
Take it and you have made a good bet in a game you may not be in long enough to play again. Expected value is an average over worlds you will never visit. You live in one sequence, and that sequence has an absorbing state.
That is not a metaphor. It is a known defect in how expected value is applied. The ergodic hypothesis is what licenses treating the time average of a process and its expectation value as the same quantity, the conditions for its validity are restrictive, and they are more restrictive still for systems far from equilibrium, which is what growth is.4 Expected utility theory and its descendants assume it indiscriminately.4 A company compounding through time is precisely the case where the assumption fails.
Why these three models
The decision is how large to make a single commitment. The features that fire are deep uncertainty about the outcome, a sequence of similar decisions ahead, and a quantity that accumulates and drains without anyone drawing it.
Three lenses. Survival sizing produces a random-class answer about ruin probability over a sequence. Rule design produces an equilibrium answer about who sets the cap and when. Stock and flow produces a complex answer about the denominator that both of the others quietly assume. The three disagree about what the problem even is, which is the useful part: one says it is arithmetic, one says it is governance, one says it is measurement.
1. Survival sizing: the arithmetic that outranks the arithmetic
Take the cleanest version, from a chapter on money management written for traders and applying without modification to founders. Start with a bankroll and a sequence of bets, each risking a fixed fraction of what remains. Bet ten per cent of the bankroll and you need a net run of ten losses to be wiped out, and the chance of that happening somewhere within a thousand bets is close to certain. Bet one per cent and you need a net run of a hundred, and the chance of that within a thousand bets is under one per cent.1
Nothing changed about the quality of the bets. Only the fraction changed, and the fraction moved the outcome from near-certain ruin to near-certain survival.
The line that matters most in that chapter is not the arithmetic. It is the ordering: it is impossible to make millions if you are wiped out, so you have to make sure you can continue to play the game if you are ever going to win.1 That is a statement about precedence. Survival is not one objective among several to be traded off against return. It is the condition under which return exists at all.
Note what this model does not require. It does not require a behavioural finding, a replication, or a claim about how people misperceive risk. It follows from the structure of repeated exposure to a process with an absorbing barrier. That is why it survives scrutiny that most business advice does not.
Survival sizing, the sequence lens
- Assumes: decisions come in a sequence, losses are bounded per decision, and zero is absorbing.
- Fits because: you will make many capital decisions and only need to lose once terminally.
- Breaks when: exposures are correlated, in which case several decisions are secretly one decision and this model will not warn you.
- Evidence: grade A. Structural rather than empirical, so it does not depend on any contested finding.
- Counteracts: expected-value reasoning applied to a non-repeatable sequence.
- May reinforce: chronic under-sizing, and a portfolio too small to matter.
2. Rule design: who sets the cap, and when
The second lens moves the question from what the right size is to who decides it and at what moment.
Mechanism design starts from the equilibrium you want and engineers the rules so that self-interested behaviour produces it. Applied here, the desired equilibrium is that no single decision can end the company. The self-interest working against it is a founder in a room with a live opportunity, a closing window and a counterparty applying pressure.
The design answer is that the cap is not a judgement made at the moment of the opportunity. It is a rule set before the opportunity exists, by a person whose incentives at the time of setting are different from their incentives at the time of deciding. That is the same logic that makes a commitment device work, and it is why a policy beats an intention.
The practical form is a single sentence in your operating rules. No single commitment exceeds X per cent of unrestricted cash, and exceeding it requires a named second signature. The number matters less than the fact that it was chosen while nothing was at stake.
The condition that decides whether this works is enforcement, and it is where most founder implementations quietly fail. A rule the bound party can waive is not a rule, it is a preference with formatting. The test is simple and slightly unpleasant: name the person who would have to be overruled, and ask what overruling them would cost you. If the answer is a short conversation, you do not have a mechanism. If the answer involves a board minute, a co-founder’s written objection, or an investor notification, you do.
This is also why the cap should be expressed as a fraction rather than an amount. An amount ages badly. It was set against last year’s balance sheet, it is never revisited, and it becomes either absurdly restrictive or meaningless. A fraction re-prices itself every time the denominator moves, which means the rule keeps working without anyone maintaining it.
Rule design, the governance lens
- Assumes: you can bind your future self through a rule with an enforcer.
- Fits because: the failure happens under time pressure, not under analysis.
- Breaks when: the enforcer can be overruled by the person being bound, which is the default in a founder-controlled company.
- Evidence: grade A as theory. The enforcement condition is where real implementations fail.
- Counteracts: the belief that you will be disciplined in the moment.
- May reinforce: rule-following that blocks a genuinely correct exception.
3. The denominator: survival capacity is a stock
The third lens attacks the number the first two both depend on and neither examines.
And this is the lens founders are measurably worst at. Highly educated people are often unable to infer the behaviour of even simple stock and flow systems, and the failure is not explained away by the usual excuses: experiments show persistent poor performance is not attributable to an inability to interpret graphs, to contextual knowledge, to motivation, or to cognitive capacity.5 The authors name it stock-flow failure and describe it as a fundamental reasoning error rather than an artefact.5
Every ruin calculation is a fraction, and a fraction has a denominator. That denominator is survival capacity, and survival capacity is a stock. A stock changes only through its flows, which means you cannot set it, you can only move what fills it and what drains it.3 Founders routinely discuss it as though it were directly settable, which is where the arithmetic quietly detaches from the company.
It also means the denominator is not the number on the bank statement. A stock is conserved, in the sense that items entering remain until they flow out and what one stock loses another gains exactly.3 What leaves it has to arrive somewhere, and anything already committed has left even if it has not yet moved. Restricted grant money is in the account and is not yours. A receivable is an inflow with a delay and a default rate attached, not a balance. Committed spend is an outflow that has already happened in every sense except timing.
The correlated-exposure problem is the same point again. Three commitments at eight per cent each look like three modest positions. If they drain the same stock when the same event fires, because they depend on one customer, one currency or one approval, then they were never three flows. They were one outflow at twenty-four per cent, and nobody wrote that number down because nobody drew the stock.
Stock and flow, the denominator lens
- Assumes: survival capacity accumulates and drains through identifiable flows, and is conserved.
- Fits because: the cap is a fraction, and the fraction is only as good as what it divides by.
- Breaks when: the stock leaks somewhere unmodelled. The arithmetic stays right and the answer goes wrong, silently.
- Evidence: grade A. Structural and near-definitional, and confirmed by the experimental record on how badly stocks and flows are read.
- Counteracts: treating the bank balance as the denominator.
- May reinforce: false precision, because a carefully drawn stock still omits whatever you did not think to draw.
The levers, cheapest first
- Write the number down. Choose a maximum single-commitment fraction of unrestricted cash today, while nothing is pending. Cost: fifteen minutes. This is the whole intervention and everything else is elaboration.
- Group the correlated ones. List your live commitments and mark which fail together. Sum each group. Compare the group total, not the item, against the cap.
- Name a second signature. A rule without an enforcer is an intention. Pick a person who can say no and whose no is expensive to overrule.
- Separate the reversible from the irreversible. The cap applies hardest to what you cannot undo. A reversible commitment at the same size is a smaller problem, but it is still a problem, because reversibility is not a defence against magnitude.
- Kill the exception clause. A cap with a founder override is not a cap. If you need an exception path, make it slow rather than discretionary.
What to do this week
Do now, sized at one sitting. Set the fraction. Write it in the operating rules with a date and a name against it. Reversible, dominant across every scenario, and it costs nothing.
Hedge, where the premium is the whole loss. Take your single largest live commitment and split it into two tranches with a checkpoint between them. If the opportunity was as good as you thought, you have lost a little speed. That is the entire downside.
Defer and trigger, sized now. Do not renegotiate every existing commitment. Pre-commit instead: the next time a correlated group crosses the cap, the newest member of that group is the one that gets restructured, automatically. Fix the size of that restructuring now, because a size chosen at the moment of the trigger is a size chosen under pressure.
What history says
Run the break test first. Has a rule changed, has an actor entered or left, has a measurement become a target? A cap set against a cash balance that includes restricted grant money is a cap set against a number that is not yours, and if your funding mix changed, the historical base rate on your own resilience is describing a different company.
With that clear, the reference class is blunt. Companies do not usually fail because a series of decisions were wrong. They fail because one decision was large. The distribution of causes is not symmetric with the distribution of decisions, and that asymmetry is the entire argument for a cap.
Subtract the counterfactual before you conclude that your aggressive sizing worked. A founder who committed heavily and survived has one observation from a sequence where the alternative is unobserved. This is the same error that makes trading strategies look profitable in retrospect: a review of ninety-two studies of technical trading found fifty-eight reporting positive results, and concluded that most of those studies were compromised by data snooping and by selecting rules after the fact.2 The pattern generalises. Surviving a large bet is evidence about the world you are in, not about the decision you made.
What this ensemble cannot see
The three models tell you how to avoid ruin. None of them tells you what your survival capacity actually is.
The fraction is computed against a denominator, and the denominator is the hard part. Cash is not the same as unrestricted cash. Committed revenue is not the same as collected revenue. A receivable from a customer who is themselves close to their own limit is not an asset in the scenario where you need it. Every ruin calculation is only as good as the denominator, and the denominator is the number founders are most optimistic about.
There is a second limit. This apparatus is built for bounded, roughly independent exposures. Where a single event can take several positions at once, the model understates the risk by exactly the amount of the correlation, and it does so silently.
The one action that survives the ignorance: this week, compute your denominator twice. Once the way you normally would, and once counting only cash you could spend tomorrow without anyone’s permission. Set the cap against the second number. If the two are far apart, that gap is the real finding, and it is more important than the fraction you choose.
Who has to move
The person who needs this rule is the person least likely to want it, which is whoever holds the mandate to commit. If that is you, the rule needs an enforcer who is not you. The cheapest first test is to name the fraction out loud to your co-founder or your board chair this week and ask them to hold you to it. If saying the number out loud feels uncomfortable, the number you were about to choose was too high.
Sources and notes
- Courtney Smith, on money management, in Rick Bensignor (editor), New Thinking in Technical Analysis: Trading Models from the Masters, Bloomberg Press. The worked comparison between a ten per cent and a one per cent bet size over a thousand sequential bets, and the fixed fractional method, appear in the money management chapter. The ordering claim quoted here is the chapter’s own: that it is impossible to make millions if you are wiped out, and that you have to be able to continue playing if you are ever going to win.
- John D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World, McGraw-Hill. Stocks, flows and the conservation of material in stock and flow networks are developed in chapter 6, section 6.2.3, which also draws the distinction this section rests on: the contents of a stock and flow network are conserved, while information about a stock is not. The experimental record on how poorly stocks and flows are read, and the finding that these misperceptions are robust to experience, financial incentives and market institutions, is chapter 1, section 1.3 and Table 1-4.
- Cheol-Ho Park and Scott H. Irwin, The Profitability of Technical Analysis: A Review, AgMAS Project Research Report 2004-04, University of Illinois. https://farmdoc.illinois.edu/assets/marketing/agmas/AgMAS04_04.pdf. The abstract reports that among ninety-two modern studies, fifty-eight found positive results, twenty-four found negative results and ten were mixed, and concludes that most empirical studies are subject to problems in their testing procedures including data snooping, ex post selection of trading rules, and difficulties estimating risk and transaction costs.
- Ole Peters, The ergodicity problem in economics, Nature Physics 15, 2019, pages 1216 to 1221. https://www.nature.com/articles/s41567-019-0732-0. The abstract states that the ergodic hypothesis underlies the assumption that the time average and the expectation value of an observable are the same, that the conditions for validity are restrictive and more so for non-equilibrium systems, that economics typically deals with systems far from equilibrium and specifically with models of growth, and that the prevailing formulations of economic theory make an indiscriminate assumption of ergodicity.
- Matthew A. Cronin, Cleotilde Gonzalez and John D. Sterman, Why don’t well-educated adults understand accumulation? A challenge to researchers, educators, and citizens, Organizational Behavior and Human Decision Processes 108(1), 2009, pages 116 to 130. Author copy: https://www.mit.edu/~jsterman/CroninGonzalezSterman061210.pdf. The abstract states that highly educated people are often unable to infer the behaviour of simple stock-flow systems, and that in a series of experiments persistent poor performance is not attributable to an inability to interpret graphs, contextual knowledge, motivation, or cognitive capacity.
A note on citing a trading book for a founder audience. Two things in that volume survive contact with evidence, and money management is one of them. The pattern-recognition systems in the same book do not, which is why the second source here is the review that examines them. Using a book for its one durable idea while citing the evidence against the rest of it is the discipline, not an inconsistency.
Joshua Agonya Pi’Rwot, Founder.