№2 in the series
a field survey of diversification · drewbreyer.com
The mathematical expectation of the speculator is zero.
L’espérance mathématique du spéculateur est nulle.
Louis Bachelier, Théorie de la Spéculation (1900), trans. Boness1
Abstract
We consider an investor placing $100,000 for 30 years in a diversified market portfolio returning 7.0% with 16% volatility, against a single ordinary stock granted — charitably — the same expected return at 45% volatility. The two bets have identical expected terminal wealth: $532,803 in today’s dollars. They are not the same bet. The pond delivers $362,909 to its median holder; the single stock delivers $25,550 to its median holder and beats the pond in only 12.5% of futures. Diversification does not shrink the prize; it changes who receives it — from a lucky few to nearly everyone. We survey a century of measurement showing that this arithmetic is the solved core of retirement investing, and we price what a gambling habit costs if, knowing this, you keep one anyway.
A note on posture. In an investment club, the contrarian position is not a hotter stock pick. Everyone in the room has one of those. The contrarian position is that the game was solved between 1900 and 1974 — priced by Bachelier, proportioned by Markowitz, indexed by Bogle — and that everything since has been measurement confirming it. What follows is that measurement. The author’s zeal is spent not on finding the winning stock but on refusing, with citations, to pretend anyone reliably can.
№ I — MEASURED
Begin with what the pond actually contains. The market’s long-run return is not a property of its typical fish; it is carried by a thin tail of extraordinary ones. Miss the tail and you did not fish a smaller pond — you fished a different one, with different odds.
Most stocks are not the market. A handful of extraordinary companies carry the whole pond; everything else, on net, could have been T-bills.
| Beat one-month T-bills over their lifetime6 | 42.6% |
| Median lifetime buy-and-hold return6 | −2.29% |
| Median listing life6 | 90 mo (7.5 yr) |
| Suffered an effectively complete loss6 | 11.83% |
| Reduced aggregate shareholder wealth (1926–2022)7 | 58.6% |
| Firms for half of all net wealth (2016 → 2022 → 2025)6,7,8 | 90 → 72 → 46 |
Anything that is huge, profitable, famous, or influential is the result of a tail event.
№ II — MODELED
The model below concedes everything a stock picker could ask: the stock is average, β = 1, and it is paid the market’s full expected return. Nothing is taken away except the other stocks. What remains is the arithmetic of what you gave up.
The calibration concession. The modeled stock is granted the market’s full arithmetic mean — 8.4% at these settings. Idiosyncratic risk is taken but not paid (CAPM). This is charity, not cynicism: the measured record in §I says the typical stock does worse than this model assumes.
Ten stocks feel diversified. The wobble says otherwise.
Adding stocks gets you back to a coin flip against the market — never past it.
Diversification is both observed and sensible; a rule of behavior which does not imply the superiority of diversification must be rejected both as a hypothesis and as a maxim.
The celebrated aphorism — that diversification is “the only free lunch in investing” — is attributed to Markowitz everywhere and sourced nowhere; we checked. The 1952 sentence above is what he actually wrote, and it is better.
Diversification is your buddy.
№ III — MODELED
Time is usually sold as the retail investor’s edge. It is — but only in wide water. Concentrated risk does not wash out with the years; it compounds in. The two curves below are the same investor, the same horizon, and the same expected return.
Both bets promise the same average. Only one of them usually keeps the promise.
Given time, a wide portfolio almost never loses to cash. A single stock usually does.
In the short-run, the market is a voting machine — reflecting a voter-registration test that requires only money, not intelligence or emotional stability — but in the long-run, the market is a weighing machine.
№ IV — MEASURED
Perhaps skill can beat the arithmetic. The most-resourced investors on earth have run that experiment continuously since 1976, in public, with scorekeeping. The scores follow.
The people paid most to beat the market mostly don't — and the longer the race, the fewer survive.
Persistence is worse than chance: of large-cap funds in the top quartile as of December 2021, 0% remained top-quartile through December 202515. Skill, properly measured, rarely clears its own costs.
“The five funds-of-funds got off to a fast start, each beating the index fund in 2008. Then the roof fell in.”
Properly measured, the average actively managed dollar must underperform the average passively managed dollar, net of costs.
[A] major industry appears to be built largely on an illusion of skill.
№ V — MEASURED + MODELED
None of this abolishes the itch. A fraction of readers — the author declines to estimate it from the room — will speculate regardless, and a survey that pretends otherwise is decoration. So we price the itch instead. Below: each game, its sum after costs, its measured retail result, and the rules under which a small sleeve does bounded damage.
| The game | Sum after costs | The measured result | If you insist |
|---|---|---|---|
| The pond (broad index) | Positive — you are paid to wait | the equity premium6 | this is the savings, not the sleeve |
| One ordinary stock | Positive expectation, lottery-shaped | most lifetime-trail T-bills; ~4 in 10 suffer a catastrophic, unrecovered decline6,10 | the cheapest itch to scratch — same expectation, wilder ride; sleeve it |
| Active funds | Positive minus ~1%/yr | most trail the pond at every horizon; winners don’t persist13,15 | you already met this arithmetic at /fees |
| Retail options | Zero-sum minus spreads, fees, and the counterparty knowing more | measured retail losses at every study24,28 | the default sleeve preset above, charitably |
| Crypto | No cash flows; a monetary-premium wager | most retail buyers entered and lost; drawdowns >75% are routine30,31 | if held at all: lottery-ticket sizing, one ticket |
| Prediction markets | Zero-sum minus the fee — but priced by information | information value is real; retail P&L unstudied — absence of evidence, noted35,37 | house rule 5 |
Myth check, both directions: roughly a third of option contracts expire worthless — not the 90% of lore29. The retail losses above come from spreads and prices paid, not from expiry trivia. This survey’s case does not need the myth. And on the other side: 75% of concentrated stockholders would have benefited from some diversification10.
When the capital development of a country becomes a by-product of the activities of a casino, the job is likely to be ill-done.
A small, capped gambling budget costs surprisingly little. A refilled habit compounds like a fee.
I recommend you have a funny money account of no more than 5% of your portfolio … I’d be astonished if at least 95% of those funny money accounts don’t do worse.
Readers of the companion survey argued at length over a fund charging 0.66%. The refilled sleeve above is that argument, twenty times over. → The Arithmetic of Fees
Speculation is an effort, probably unsuccessful, to turn a little money into a lot. Investment is an effort, which should be successful, to prevent a lot of money from becoming a little.
№ VI
Acceptance, written out, is three sentences long.
Not that markets are perfectly efficient — anomalies exist, factor premia exist, and a handful of investors have beaten the arithmetic for decades. Not that prices are magic: they are informative precisely because zealots spend careers competing to correct them (Grossman & Stiglitz’s paradox5). The claim is narrower and stronger: for a saver, net of costs and taxes, over decades, the diversified default has dominated every implementable alternative — and the reader who indexes free-rides on the very competition they decline to join. It is also, the author concedes, more fun to be right about this at a party than profitable to argue it further.
My advice to the trustee could not be more simple: Put 10% of the cash in short-term government bonds and 90% in a very low-cost S&P 500 index fund. (I suggest Vanguard’s.) I believe the trust’s long-term results from this policy will be superior to those attained by most investors — whether pension funds, institutions or individuals — who employ high-fee managers.
When you’ve won the game, why keep playing it?
The companion survey prices the other tax on wealth — fees. → The Arithmetic of Fees
Lump sum, annual steps. Every asset is lognormal in annual log-returns r = m + s·z, gross factor e^r. Geometric inputs (market, cash) map as m = ln(1+g); arithmetic inputs (sleeve games) as m = ln(1+E) − s²/2. The single stock is granted the market’s arithmetic mean (β = 1): its idiosyncratic risk is taken but not paid. The market shock z_mkt is shared between pond and stock — the common-random-numbers identity that makes every comparison ceteris paribus.
m_m = 0.067659 · A_m = 8.38% · m_s = -0.020791 · σ_i = 42.1%
The Monte Carlo figures reproduce these within sampling error; if a simulated value drifts from the closed form, the closed form is right.
A mulberry32 stream feeds a Box–Muller pair with caching (cosine first, sine held for the next draw). Each base path draws its full shock set in a fixed order, then runs forward and sign-flipped — antithetic variates. The width engine uses 8000 effective paths; comparisons share the market shock, so differences are pure composition.
It prices composition, not accumulation — contributions, decumulation, and taxes are out of scope (contributions are the companion /fees paper’s subject). And it is kinder to concentration than the measured record: the stock is paid the market’s full expected return, which §I shows the typical stock does not earn. Where the model and reality disagree, reality is worse for the concentrated investor.
Continue the series →
№3 The Yield Illusion
Dividends, covered-call income, and leveraged ETFs: total return relabeled, minus friction.
Solved Problems in Personal Finance
№1 The Arithmetic of Fees·№2 A Wide & Deep Pond·№3 The Yield Illusion
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