Why identical returns produce different investors
Three interactive experiments on one objection to Episode 4, and what answering it costs.
Episode 4 made its case with a deck of cards. Take every daily return the S&P has produced over forty years, shuffle them, and deal them again. Every statistic a risk model can see survives the shuffle untouched. The worst loss moves fifty points.
That argument is complete, and we stand behind it. But it leaves one objection standing, and it is the objection a careful reader raises first.
If every ordering finishes in the same place, why does the order matter?
It is a fair question, and it deserves an answer built from evidence rather than rhetoric. So this addendum runs three experiments, and the first of them concedes the point completely.
Experiment one shows how many different lives one set of risk statistics can describe.
Experiment two shows the decisions you take inside one of those lives, without hindsight, on the only evidence anyone ever holds.
Experiment three shows what you can build that does not depend on which life you got.
Each one runs in the page. You can change the settings, re-deal the data and watch every figure quoted here recompute in front of you.
Experiment one: the same deck, ten ways
Below are ten portfolios. Every one of them is thirty years of S&P 500 monthly returns, from August 1996 to July 2026, dealt in a different order. Nothing has been added, nothing removed, and no number altered.
The risk report is identical across all ten. Not similar. Identical, to fifteen decimal places, on compound return, standard deviation, Sharpe, Sortino, downside deviation, mean absolute deviation, skew, kurtosis, Value at Risk at both thresholds, expected shortfall, best month, worst month and the share of positive months. The largest relative difference between any two portfolios on any of those measures is three parts in a thousand million million, which is the limit of the arithmetic rather than a real difference.
They also begin at the same value and close at the same value, to the cent. Multiplication does not care what order you multiply in, so a shuffled deck of returns compounds to precisely the same number every time.
That deserves stating precisely, because it is the one place this argument can be misread. Compounding is not order-blind. Its entire path is determined by order and by nothing else: every drawdown, every stretch underwater, every value along the way. What is order-blind is the endpoint alone, and only for as long as nothing responds to the path while it unfolds. The moment a dollar moves, a position is sized, or a decision is taken, the endpoint stops being order-blind as well.
Experiment one holds the endpoint fixed so that the path can be seen on its own. Experiments two and three let go of it.
Before you run it, the panel asks you to commit. Ten identical tearsheets still have to be ranked by somebody, and the only numbers that differ are the recent ones, so the tie-break panel shows what a committee would actually look at. Pick the portfolio you think will do most damage, then press play.
Three things come out of it.
The worst loss ranges from around twenty-two percent to around forty-six percent across a single deal of ten, and from eighteen to sixty-nine percent across ten thousand of them. The longest stretch underwater ranges from under two years to over seventeen. These are histories that every risk model on earth regards as the same asset.
The ordering the report supports is worse than useless. Across twenty thousand simulated committees choosing on the trailing three-year Sharpe ratio, which Episode 2 established is the horizon on which capital decisions actually get made, the first choice lands in the safer half of its ten just under forty percent of the time. A coin lands there half the time. The diligence is not neutral. It points slightly the wrong way, because a strong recent run means the difficult months had to sit somewhere earlier, clustered, which is what a deep drawdown is made of.
And the rolling panel above the chart never stops moving. The report says the Sharpe ratio is 0.6128, once, for all ten, for thirty years. Measured over rolling three-year windows the same asset produces readings from about minus one to about plus two and a half, roughly a tenth of them below zero, with a gap of nearly three between the best and worst of the ten at a single moment.
This also sharpens the correction Episode 4 published rather than withdrawing it. Standard deviation explains about three quarters of the variation in worst drawdown between markets. It explains none of the variation within one market’s own possible orderings, because it does not vary there at all. Between markets and within a market are different questions, and volatility answers only the first.
One report described ten histories, the worst of which lost twice what the best did, and nothing on that report could have told you which one you were about to live through.
Why that is not the end of it
Now the objection. All ten of those portfolios closed at 11,703.60. If you were going to arrive at the same place regardless, the path was scenery.
The endpoints coincide for one reason: nobody touched the portfolio for thirty years. That is the only condition under which the objection holds, and it is a condition that has never been met by any investor, any fund, or any allocator in the history of the industry.
Money goes in. Money comes out. Fees are charged. Tax is paid in the good years and carried forward in the bad. Investors watch, then act. None of it commutes. The moment a dollar crosses the path, the order decides the outcome.
And here is the part that matters most, because it is the part nobody can design around.
The decision is never taken on the thirty-year statistic. It cannot be. That number does not exist yet. The sequence ahead is unwritten, and everything anyone has ever had to work with is the sequence behind them, which is a short and recent sample of a long and unfinished path. So the measurement that drives the decision is a property of where you happen to be standing rather than a property of the asset.
The path produces the measurement. The measurement produces the decision. The decision alters the path. The statistics the experiment began with never enter the chain at any point.
Experiment two: what actually happens to the money
Same ten portfolios. Same returns, same orderings, same identical report. This time you can switch on the things that happen to real capital, and the panel splits into three so the chain is visible in one screen: what the tearsheet reports, what the decision is taken on, and what ends up in the account.
Start with Left alone to confirm the ten still close together. Then try Drawing an income, which is the cleanest case in the whole addendum, and then Sacking the manager.
Drawing an income takes four point eight percent a year out of the portfolio, which is a retirement drawdown, an endowment spend, or a fee load. The withdrawals are identical to the dollar in every one of the ten orderings, so there is no free parameter and nothing to tune. Closing capital ranges from 941 to 6,735. One investor finishes with seven times another, taking the same money out of the same asset, and the tearsheet reports 8.55% a year to both of them.
Sacking the manager is the one that should give an allocator pause. A single rule, nothing else switched on: redeem in full the first month the trailing three-year Sharpe ratio goes below zero. Nobody would call that reckless. It fires in ten orderings out of ten, at months ranging from 38 to 268, and closing capital ranges from 768 to 8,367. The money-weighted return runs from minus 0.88 percent to plus 7.34 percent. The same rule, the same asset, and the outcome settled entirely by when the measurement happened to dip.
That gap is worth naming properly, because the profession already has the vocabulary. Time-weighted return is what a tearsheet reports and what performance tables rank, and it deliberately strips out the effect of when money arrived. Money-weighted return is what the investor earned per dollar at risk. In every scenario here the time-weighted return is 8.55 percent for all ten portfolios. The money-weighted return does whatever the sequence tells it to.
The returns were identical and the report was identical. Once money crossed the path, the order decided how much of it you kept.
Nobody decides over thirty years
Both experiments above run the full sample, and that is a courtesy no allocator has ever been extended. The thirty-year number is the one figure in this entire piece that nobody could have acted on, because it only exists once the thirty years are over.
So take the same series and ask what a real evaluation window looks like. Every possible start month, held for a fixed period, on the actual history rather than a shuffle.
Nearly one investor in eight who held this asset for a full decade lost money on it, in a sample that compounded at 8.55 percent a year. Not because the asset failed. Because of where they came in.
This is the whole difficulty compressed into one table. The strategy was profitable across the entire sequence. Almost nobody experiences the entire sequence. What they experience is a segment, chosen for them by when they happened to have capital, and the segments disagree with each other violently. Managers are dismissed on those segments. Investors move between strategies on those segments. Allocators build and unwind positions on those segments.
And the arrangement is not accidental. It is manufactured by the measurement regime itself. A committee that ranks on volatility, Sharpe and Value at Risk is using instruments that are silent about order, so the only remaining discriminator is the recent segment, which is nothing but order. The industry has built a selection process that is blind to path in principle and driven by path in practice.
Experiment three: two ways to calm a path
The objection is answered. What follows asks the harder question: once the sequence is identified as the source of the problem, what kind of intervention can alter it?
Shortly after Episode 4 appeared, David Dredge, Chief Investment Officer of Convex Strategies, made an observation that captures the distinction this addendum has been circling.
"We see far too much of fiduciaries hedging their risk methodology, as opposed to fiduciaries hedging their actual risk. Uncertainty, ambiguity and intractability are generally not conducive to capture in most traditional risk metrics. Sharpe World risk metrics are small world tools. Convexity is the solution to large world realities."
Two ideas in that are worth separating before the experiment runs. The first is the distinction between hedging a methodology and hedging a risk, which names the failure precisely. The second is small world tools, which is Leonard Savage’s distinction: a small world is one where the possible outcomes and their probabilities are known in advance, and a large world is the one everybody actually invests in, where they are not.
None of that is evidence for anything that follows, and it is not offered as any. It is an experienced allocator arriving at the same distinction from an entirely different direction, which is worth more than agreement because it is independent of the method. The experiment that follows asks whether those two approaches can be told apart empirically.
Practitioners who have seen this demonstration usually reach for the obvious remedy. If the trouble is that the realised path swings too far from the long-run statistic, smooth the path. Target a volatility. Damp the swings. Bring the short-window experience closer to the thirty-year number.
It works, and it costs more than it is worth. The experiment below runs two interventions against buy and hold, tuned to arrive at roughly the same realised volatility and roughly the same average exposure, so that the only thing being tested is the shape of the intervention rather than how much of it there is.
Smooth it scales exposure to hold a constant volatility. It is symmetric, and it does not know which kind of volatility it is removing. Shape it holds full exposure while the index sits above where it was some months earlier, and none otherwise. It is asymmetric, and it chooses a side. Both are applied to the same buy and hold position, and both use only information available before the month they act on.
Neither is offered as a strategy. They are the two available ways of altering the shape of a compounded path, reduced to their simplest form.
How each one is built
Smooth it. Each month, measure the volatility of the previous twelve months and set exposure to the target divided by it, capped at one and a half times. The target is ten percent, adjustable from six to eighteen. Exposure ranges from 0.33 to 1.50, averages 0.82, and changes in 329 of the 359 months.
Shape it. Each month, compare the index with where it stood ten months earlier. Higher, hold the position at full size. Not higher, hold nothing. The lookback is adjustable from four to twenty-four months. Exposure is zero or one, never anything between, averaging 0.78 across eleven round trips.
Five things are true of both, and they are what makes the comparison fair.
Neither has a stop, a target, a forecast or a view. Neither is optimised: both settings are on sliders so that no single choice is doing the work. No costs are modelled, and at the turnover shown neither result would move materially, with what little cost there is falling on the shaped version.
The last row is the one that matters. Smoothing cannot function without an estimate of volatility, and that estimate is necessarily backward-looking, which is why it arrives late. Shaping estimates nothing at all. It asks a single question about a price that has already printed and acts on the answer.
Three things, in order. One: commit to a prediction, then run the actual history. Two: press Remove the order, which leaves a market with the same returns and no memory, and watch which approach stops working. Three: move the two sliders and confirm that nothing depends on the settings we chose. Everything else on the panel can wait.
On the actual history, at matched volatility and matched exposure:
The two betas in that table are the heart of it. Up beta is the share of the market’s move the approach captured in months when the market rose; down beta is the share it took in months when the market fell. Buy and hold is 1.00 on both by definition, since it is the market. Anything that consistently takes less of the fall than it sacrifices of the rise shows the positive asymmetry we mean here by convexity. That is a statement about conditional payoff, not about the strict mathematical sense of the word.
Smoothing gives up 1.87 points of compound return and still leaves a forty percent loss. Look at its two betas. It is fractionally more exposed on the way down than on the way up, because volatility only rises once the damage has started, so it deleverages late and re-levers late. Measured against the worst and best deciles of months, it held an average exposure of 0.68 during the worst and 0.62 during the best. It is, on average, more invested in the months you would least want to be there.
What is left is a tepid, low-volatility holding with its tails clipped at both ends. The upside volatility was the part that was paying for everything, and smoothing hands it back in the same proportion as the downside, because it cannot tell them apart. It is a worse asset wearing a better tearsheet.
Shaping arrives at the same volatility and the same exposure by a different route. Down beta of 0.32 against up beta of 0.49 means it keeps roughly half the good side while taking about a third of the bad. That gap is what convexity is, and it is visible directly in the scatter panel as a kink where the smoothed version shows a straight line. The worst loss more than halves, the compound return goes up rather than down, and the twelve-month skew turns positive.
What the scramble is actually testing
Then remove the order. Every return is still present, in the same quantity. What has gone is every trend, every cluster and every stretch that persisted, which is to say every property that lives in the sequence rather than in the set.
This is a control experiment, not a stress test. It cannot tell you whether an approach would survive a different future. What it can tell you, precisely, is which approach was extracting something from order in the first place.
Buy and hold’s return does not move, because the product of a fixed set of returns is the same in any order. Nothing was removed from it, because it was never using order to begin with. Its worst loss does move, from minus 52.6 to minus 42.2, which is experiment one restated.
Smoothing is a more interesting case, and it is worth being careful here, because a volatility rule obviously is sequence-dependent. Its exposure each month is computed from the previous twelve returns, so removing the order changes every one of those windows. It does. Its average exposure falls from 0.82 to 0.70, which is a substantial change to how the rule behaved.
Its compound return moves by 0.31 of a point.
So smoothing responds to the sequence, at some length, and gets almost nothing from it. Volatility targeting is sold as a reaction to how markets behave, and in a market that behaves in no particular way at all it reacts differently and arrives in the same place. What it is extracting is not the directional persistence the scramble removes. It is mostly an average reduction in exposure, and that is available without reading the path at all.
Shaping is the only one of the three that stops working, and it stops working completely. Four points of annual return, and a worst loss that doubles.
Buy and hold never used the sequence. Smoothing responded to it and lost almost nothing when the directional memory went. Shaping was different in kind: its entire advantage disappeared, because order was the whole of what it was working on.
It is worth being clear that the scrambled market does not exist. Real returns cluster: the autocorrelation of absolute monthly returns in this series is plus 0.195, which sits outside the entire range produced by two thousand shuffles. Nobody has ever invested in a market with no memory. The point of visiting one is not to ask what would happen there. It is to find out what each approach was living on here.
And it sharpens what the risk report is capable of registering. Every statistic on a tearsheet is a property of the distribution, and removing the order leaves that distribution untouched to fifteen decimal places. So this is a procedure that strips out exactly what the report cannot see and preserves exactly what it can. Buy and hold survives it in terminal return by construction. Smoothing changes only modestly. Shaping loses the entire feature that distinguished it.
That is a statement about differential sensitivity, not about completeness: one scramble cannot prove that a tearsheet describes anything in full. But it does locate where the advantage lived, and in shaping’s case the advantage lived in precisely the dimension no statistic on the page is measured along.
A committee comparing these three would see the smoothed version as the disciplined one and the shaped version as an unexplained anomaly. They would be right that it is unexplained. The explanation is not available in the language they are using.
The only intervention that repairs the path is the one whose entire benefit lives inside the path. It is therefore the one intervention that no additive statistic can see, recommend, or defend. The metrics did not merely fail to warn you about the problem. They cannot describe the solution either.
Both interventions calmed the path. Only one did it by choosing a side rather than shrinking both, and only that one is invisible to the report.
The report is looking at the wrong frequency
There is a second reason the statistics cannot see this, and it has nothing to do with order at all. It is the interval they sample at.
To show it we need a longer run than experiment three uses, so the figures below come from Shiller’s monthly S&P series back to 1871. Every number in every risk report is computed on periodic returns, monthly or daily. But a shaped position does not experience the market in calendar buckets. It experiences it in positions, each with a beginning, a middle and an end, and chopping those positions into months destroys the thing that makes the approach work.
Read monthly, the shaped path has worse skew than buy and hold, which over the same 152 years reports plus 0.45. Read as positions, it has a win rate near seventy percent, an average winner around ten times the average loser, a largest gain of plus 230 percent against a largest loss of minus 14 percent, and the top decile of positions producing 44 percent of all gains. Those are two descriptions of one identical set of returns.
The monthly figure is not wrong. It is measuring something else. Sitting out a collapse removes the violent rebound months along with the decline, so monthly sampling penalises the exact behaviour that protected the capital, while the occasional whipsaw loss stays in the sample.
The convexity was always there. The report could not see it because it was sampling in calendar months, and the mechanism does not live in calendar months. Choosing a reporting frequency is a modelling decision, and nobody treats it as one.
What survives a century
Everything above this line was measured on thirty years, which is the exact horizon this addendum has spent three experiments warning you about. So we took the architecture to a window it had never seen: Shiller’s monthly S&P series from 1871, holding out the 1,499 months before 1996.
A rule with nothing to estimate matched buy and hold’s compound return on seventy percent of the exposure and cut the worst loss almost in half, across a century and a half that included 1929.
Essentially the same compound return as buy and hold for seventy percent of the exposure, with the worst loss cut from minus eighty-two to minus forty-six, across the 1929 collapse and everything either side of it. Smoothing is again slightly concave in a period it had never seen. Shaping is the only one of the three with positive asymmetry in both windows.
The asymmetry runs from plus 0.03 over the century to plus 0.17 in the modern thirty years. Both are the real number for their window. The drawdown benefit is large and stable across both; the convexity benefit varies with the period, which is what you would expect of something built out of the sequence rather than the distribution.
The result does not hang on a setting. The pattern holds at every lookback we tested, from six months to twenty-four, and the slider is there so you can check that yourself. The rule forecasts nothing and estimates nothing. It asks one question about a price that has already printed, and it does that better across a century and a half than a method built on estimating volatility manages over thirty years.
Hedging the methodology
Return to the distinction that opened experiment three: hedging a methodology rather than hedging a risk.
The smoothing result is that distinction with numbers attached, and it is worth setting out in full because it is the most damning table in this addendum.
Every measure a committee would look at improves. Volatility falls by four points. Sharpe and Sortino both rise. Value at Risk improves by more than three points and the worst month by five and a half. A manager presenting this would be presenting an unambiguously better risk profile, and would be telling the truth.
The investor ends with 4,755 less per thousand dollars committed.
That is not a manager doing a bad job of protecting capital. It is a manager doing an excellent job of protecting the report. The methodology has been hedged. The risk has not. And the mechanism is visible in the exposure figures given earlier: the smoothed version held an average 0.68 of full exposure through the worst decile of months and 0.62 through the best, because volatility only rises once the damage has begun. The instrument that is supposed to be reducing risk is systematically more invested in the months that do the harm.
This is why the industry keeps arriving at smoothing. It is the only intervention that improves every number a fiduciary is asked about. Convexity does the opposite. It costs volatility, it costs premium or whipsaw, and on a short window it will often look worse on the very metrics used to judge it. The correct solution is the one that reports badly.
Think like an engineer
Traditionally accepted risk statistics, applied over the short windows on which decisions actually get taken, are making bad decisions about terminal wealth. That is not a claim about their mathematics. Every number in every report in this addendum is correctly calculated. It is a claim about their silence. They describe a distribution, decisions are taken on a sequence, and no amount of care in computing the first will tell you anything about the second.
The answer is not to abandon them. They are useful descriptions and Episode 4 conceded, and still concedes, that volatility explains about three quarters of the variation in drawdown between markets. The answer is to demote them. They are descriptions, not decision inputs. Nobody navigates by a thermometer.
What replaces them is an engineering question rather than a statistical one, and it is a large-world question rather than a small-world one. You do not measure your way out of path dependency. You build for it, the same way a bridge is built for loads nobody has yet measured, by importing structure that changes the shape of the outcome rather than the estimate of it. That structure is convexity, and there are three ways to get it. They are not interchangeable and this page can only speak to one.
Experiment three is the replicated kind, and the scramble is what that means made visible. A bought option is convex whether the sequence cooperates or not, because the payoff is written into the contract. A rule is convex only while the sequence cooperates. Whipsaw is the price of replication, and unlike a premium it is charged unpredictably rather than agreed in advance. That is not an argument against it, since it survived a century and a half. It is an argument for knowing which kind you are holding, because the two fail differently and only one of them prices itself up front.
The third route is the one a single index cannot demonstrate at all. Convexity built from breadth comes from holding many weakly related exposures, so that a large number of small losses fund a small number of large gains. That is a property of the cross-section rather than of the time series, and one instrument has no cross-section. The experiments here can diagnose the problem with some confidence. They cannot prescribe, and we are not going to pretend otherwise on one index.
What this establishes, and what it does not
It establishes four things, and none of them is marginal.
The entire additive toolkit is silent about order, and that silence has a measurable price the moment cash moves across the path. In this experiment a short-window measurement, which is the only kind anyone will ever hold, carried no useful information about the damage ahead and pointed slightly the wrong way. A governance rule that any committee would approve, applied to an identical asset, produces outcomes that differ by an order of magnitude. And the intervention that improves every number on the report is the one that costs the investor forty percent of their money, while the intervention that repairs the path is invisible to every number on it.
What it does not establish is worth naming, because the argument is stronger for being bounded. The withdrawal rate, the allocation rule and the redemption threshold are behavioural assumptions rather than measured facts, which is why each of them sits on a slider. The direction of the result holds wherever you set them. The magnitude does not, and you should find that out by moving them rather than by taking our word for it.
On convexity, one index is a thin base for a general claim, however clean the result on it. And the effect is genuinely horizon-dependent: monthly skew stays negative at minus 0.39 while the twelve-month figure is plus 0.37 and the position-level figure is plus 1.32. The convexity is built by avoiding runs of losses rather than individual bad months, which is why it does not appear in a monthly statistic at all. Anyone reaching for one number to check this will reach for the wrong one.
One further caveat belongs to us rather than to a critic. Shuffling destroys volatility clustering, which is real in this series: the autocorrelation of absolute monthly returns is +0.195 in the actual history against effectively zero across two thousand shuffles, placing the real path outside the entire shuffled distribution. That does not weaken the first two experiments, which are about what the statistics can see rather than about what the market does. It matters here, and it cuts in the same direction. The structure a shuffle destroys is precisely the structure that any path engineering has to exploit.
And it is one index. The three experiments run on thirty years of it and the holdout on a century and a half, but it remains a single market, measured monthly. The five episodes rest on 68 markets and 647,629 market-days. This is a demonstration rather than a survey, and it is offered as one.
What it adds to Episode 4 is the mechanism. The episode showed that ruin is a property of a sequence. These experiments show how an ordinary, unlevered, prudently governed investor gets there, without a crash, without a fat tail, and without anyone making a decision they could not defend in writing at the time.
And they point at where the answer has to come from. Not from a better estimate, because the estimate was never the problem. From building something whose shape survives a sequence you will not get to see in advance.
The order was never scenery. It was the thing making the decisions.
NEXT, IN EPISODE FIVE
What Works and What We Got Wrong
This addendum ends on the one route to convexity a single index cannot demonstrate. Convexity built from breadth is a property of the cross-section, and one instrument has no cross-section. So that is where Episode Five starts.
What to do instead, and the five things we got wrong. Plus the experiment that isolates why diversification fails at the tail, the mechanism the whole industry relies on and almost nobody has measured. We can prove why breadth does not buy you the protection it promises.
Method. S&P 500 monthly returns computed from index closes, August 1996 to July 2026, 360 observations. These are price returns and exclude dividends, so every level quoted is lower than a total return series would show; the comparison between approaches is unaffected, since all of them are applied to the same series. Ten orderings generated by uniform random permutation, verifiable in the browser with the deal button. All statistics computed in the page from the same underlying series, so every figure quoted above can be reproduced by running the experiments. Money-weighted return solved by bisection on the monthly internal rate of return and annualised. Redemption proceeds are assumed to sit in cash earning nothing for the remainder of the run. Holding-period figures use every available start month on the actual history. Volatility targeting uses a trailing twelve-month realised volatility scaled to a ten percent target and capped at 1.5 times exposure; shaping holds full exposure while the index sits above its level ten months earlier and none otherwise. Both use only information available before the month they are applied to. The out-of-sample section uses Shiller’s monthly S&P total return series from 1871, which differs slightly from the index series used elsewhere, so figures across the two are close rather than identical.
Richard Brennan writes on systematic trading, complex adaptive markets, and the philosophical foundations of trend following at atstradingsolutions.com. His books include The Fractals of Finance, Complex Adaptive Markets, Carved by Impossibility and The Aussie Turtles Trend Following Guide.
Want to explore why structure exists at all?
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