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№2 in the series

A Wide & Deep Pond

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 assigned 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 the distribution of outcomes. We combine long-run measurement with a deliberately concentration-friendly model to show why broad diversification is a durable default, then estimate the cost of retaining a small speculative allocation.

Median outcome, the pond
$362,909
today’s dollars
≈ 6.0 years of your spending
Median outcome, one stock
$25,550
today’s dollars
≈ 0.4 years of your spending
Odds one stock beats the pond
12.5%
a coin flip is the ceiling
Fee-equivalent of annual rebalancing
1.33%
per year, a 5% sleeve at the current assumptions

A note on posture. This paper does not assume that skill or mispricing can never exist. It asks a narrower question: whether a concentrated, active, or speculative alternative has evidence strong enough to displace a broad, low-cost default after implementation costs and uncertainty. Foundational theory and the measured record set that burden of proof.

№ I — MEASURED

The shape of the pond

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.

Fig. 1

Most stocks are not the market. A handful of extraordinary companies carry the whole pond; everything else, on net, could have been T-bills.

The wealth field. Each dot is 0.1% of all US-listed firms since 1926; 43 account for all net wealth creation6, 574 trailed T-bills over their lifetime. The market’s return is a tail event. MEASURED
Beat one-month T-bills over their lifetime642.6%
Median lifetime buy-and-hold return6−2.29%
Median listing life690 mo (7.5 yr)
Suffered an effectively complete loss611.83%
Reduced aggregate shareholder wealth (1926–2022)758.6%
Firms for half of all net wealth (2016 → 2022 → 2025)6,7,890 → 72 → 46
Top wealth creators, 1926–2022 ($B)9,7: Apple 2.7T · Microsoft 2.1T · ExxonMobil 1.2T · Alphabet 1.0T · Amazon 764B · Berkshire Hathaway 704B · Johnson & Johnson 661B · Walmart 629B · Chevron 583B · Procter & Gamble 581B
Anything that is huge, profitable, famous, or influential is the result of a tail event.
— Morgan Housel, “Tails, You Win,” The Psychology of Money (2020)48

№ II — MODELED

Width

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.

Assumptions console — governs every modeled figure
Market world

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.

Fig. 2

Ten stocks feel diversified. The wobble says otherwise.

The price of narrowness. Volatility diversifies fast — the next figure states why that is not enough. MODELED
Fig. 3

Adding stocks gets you back to a coin flip against the market — never past it.

Odds of trailing the pond. Width buys you back the coin flip; nothing buys you past it. MODELED
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.
— Harry Markowitz, “Portfolio Selection,” Journal of Finance (1952)2

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.
— attributed to Merton Miller by his colleagues Eugene Fama and David Booth — attributed2

№ III — MODELED

Depth

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.

Fig. 4

Both bets promise the same average. Only one of them usually keeps the promise.

Two ponds, one expected return. Same mean path, diverging medians — pond $0 vs one stock $0 to the median holder. MODELED
Expected (mean) terminal — both
$532,803
identical for pond and stock, by construction
≈ 8.9 years of your spending
Median terminal, the pond
$362,909
what the typical diversified holder gets
≈ 6.0 years of your spending
Median terminal, one stock
$25,550
what the typical concentrated holder gets
≈ 0.4 years of your spending
Fig. 5

Under the calibrated model, time lowers the diversified portfolio’s chance of trailing cash while concentration retains a wider range of outcomes.

What time does to risk. Probability of ending below the cash (3%) floor — the Bessembinder benchmark when set to cash. Time heals wide risk and deepens narrow risk. MODELED
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.
— Warren Buffett, 1993 Berkshire Hathaway letter — his rendering of Benjamin Graham’s 1934 voting-machine passage41,42

№ IV — MEASURED

What the professional record adds

SPIVA measures active fund populations, not the odds that any particular stock will beat a diversified portfolio. It addresses a related implementation question: how often professionally selected portfolios overcome their costs, and whether strong relative rankings persist. The scorecards below provide that narrower evidence.

Fig. 6

The people paid most to beat the market mostly don't — and the longer the race, the fewer survive.

The professionals’ record. Share of active large-cap funds underperforming the S&P 500 by horizon, SPIVA Mid-Year 202513. The gray rule is the coin flip. MEASURED

Persistence was weak in this sample: of large-cap funds in the top quartile as of December 2021, 0% remained top-quartile through December 202515. This does not prove that skill never exists; it shows that a recent top-quartile rank was not a reliable selection rule over the measured period.

The bet — index fund vs five funds-of-hedge-funds, 2008–201717
+125.8%
the index fund (8.5%/yr)
+36.3%
the five funds, average (21.7% · 42.3% · 87.7% · 2.8% · 27%)

“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.
— William F. Sharpe, “The Arithmetic of Active Management,” Financial Analysts Journal (1991)4
[A] major industry appears to be built largely on an illusion of skill.
— Daniel Kahneman, Thinking, Fast and Slow (2011), ch. 2045

№ V — MEASURED + MODELED

A speculative sleeve

Some investors will retain a concentrated or speculative allocation for interest, conviction, or entertainment. The relevant questions are its expected return, range of outcomes, recurring funding policy, and maximum portfolio weight. The section below makes those choices explicit and estimates their cost.

The gameSum after costsThe measured resultIf you insist
The pond (broad index)Positive — you are paid to waitthe equity premium6this is the savings, not the sleeve
One ordinary stockPositive expectation, lottery-shapedmost lifetime-trail T-bills; ~4 in 10 suffer a catastrophic, unrecovered decline6,10same expectation in the model, wider outcomes; cap the allocation
Active fundsPositive minus ~1%/yrmost trail the pond at every horizon; winners don’t persist13,15you already met this arithmetic at /fees
Retail optionsZero-sum minus spreads, fees, and the counterparty knowing moremeasured retail losses at every study24,28the default sleeve preset above is deliberately conservative
CryptoNo cash flows; a monetary-premium wagermost retail buyers entered and lost; drawdowns >75% are routine30,31if held at all: lottery-ticket sizing, one ticket
Prediction marketsZero-sum minus the fee — but priced by informationinformation value is real; retail P&L unstudied — absence of evidence, noted35,37worked fees at /vig — house rule 5
ARKK −81.0%11GameStop −91.6%11Bitcoin −77.0%31

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.
— John Maynard Keynes, The General Theory (1936), ch. 1243
The sleeve — price the policy
Game
Policy
Fig. 7

A small speculative allocation can have a bounded cost. Rebalancing it repeatedly can make that cost compound like a fee.

The price of the sleeve. An annually rebalanced 5% allocation at −20%/yr expected return has a 1.33%/yr fee-equivalent under this closed-form comparison. MODELED
Fee-equivalent
1.33%
per year, annually rebalanced — closed form
Median cost of the sleeve
$0
vs 100% pond, today’s dollars
Mean cost of the sleeve
$173,994
expected shortfall — closed form

House rules for a sleeve

  1. 1Size it once: at most 5% of investable assets, in a separate account39.
  2. 2Choose the funding rule: a one-ticket sleeve is funded once and allowed to drift. A refilled sleeve is rebalanced back to its target weight each year, which repeatedly commits capital to losses as well as gains. Use the latter only when the expected-return case can withstand review.
  3. 3Value above the cap returns to the diversified portfolio rather than increasing the speculative weight.
  4. 4Keep score against the pond, in writing and after all costs. Review whether any measured advantage survives the comparison.
  5. 5Prediction markets only: your edge must exceed the fee plus your own overconfidence; this is the one counter where domain knowledge plausibly matters, but the market remains zero-sum before fees. Treat the allocation as a bounded discretionary expense rather than core savings. The posted fees are worked at The House Percentage.
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.
— John C. Bogle, MarketWatch interview (2014)39

Readers of the companion survey argued at length over a fund charging 0.66%. The annually rebalanced sleeve above is that argument at a much larger scale. → 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.
— Fred Schwed, Where Are the Customers’ Yachts? (1940)44

№ VI — MEASURED · MODELED · AS OF JULY 2026

High water

This section is dated, on purpose. It was written at midsummer 2026, with the pond at high water: the index has roughly doubled since the start of 2023, a fourth consecutive double-digit year is in progress, and a century of records contains only a handful of stretches like it. Readings this high reliably produce two urges — to celebrate, and to leave. The record supports neither. It supports an audit.

A major bank enters mid-2026 forecasting a fourth consecutive double-digit year for the index and reads a diversified 2025 as newly rewarded54,56. The observation is the occasion for this section; the arithmetic below is computed from a century of primary data and does not depend on it.

Fig. 8

Number went up — more than in almost any four-year stretch since 1928. That is a fact about the past, not an instruction about the future.

A century of four-year stretches. Every rolling four-calendar-year total-return multiple, 1928–2025 (95 windows), dividends included; vintage: Damodaran, January 2026 data (fetched 2026-07-08)65. The ~95 windows overlap — only about two dozen are independent — so percentile talk here is descriptive, not inferential. MEASURED
Last complete window
1.5×
2022–25 · 45% of windows since 1928 were smaller
Through mid-2026
2.0×
provisional 3.5-year multiple — the pond roughly doubled
Live ≥10% streak
3
consecutive double-digit years, 2023–25, a fourth in progress
≥4-year streaks ever
3
1942–45, 1949–52, 1995–99 — only a handful like this

The gauge is not a tide table. After record highs, the next years’ returns have averaged roughly what they average after ordinary days; after the best four-year stretches, the spread of outcomes widens in both directions while the middle barely moves. The reader who stepped out after 1998 missed a fifth double-digit year; the reader who stayed concentrated in what had just won found 2000 waiting. The instrument that distinguishes those futures in advance has not been invented, and §IV documented what happened to the best-funded attempts.

Far more money has been lost by investors preparing for corrections, or trying to anticipate corrections, than has been lost in corrections themselves.
— Peter Lynch, “Fear of Crashing,” Worth (September 1995)67
Fig. 9

After the best stretches, the next few years were sometimes wonderful and sometimes ugly. The spread widens; the middle barely moves; there is no exit signal in here.

The next three years’ annualized return, after every four-year window and after only the top-decile four-year windows. Overlapping windows again: the conditional strip is a handful of episodes, not a sample — read it as a picture, not a probability65. MEASURED
After a record high
≈13.7% after highs

since 1926, the average 12-month S&P 500 return after a month-end all-time high (≈13.7%) is comparable to the return after any other month; 3- and 5-year results are essentially indistinguishable57

Highs as floors
~7% of days; ⅓ held

since 1950, about 7% of trading days set an all-time high; roughly a third of those highs were never revisited to the downside — new highs became floors as often as ceilings58

What valuation foretold
CAPE ≈ 37 → +17.8%

the Shiller CAPE stood near 37 entering 2025, a level many read as a poor-returns omen; the S&P 500 then returned +17.8% — valuations forecast decades, not years59

Both tails are real
+20.89% · then -9.0 / -11.8 / -22.0

Leaving after 1998 cost a fifth double-digit year (20.9%, 1999); staying concentrated found three straight losing years waiting (2000–02)65.

The forecast, twice
3.0% → 7.0%

the same bank’s 10-year annualized S&P 500 forecast moved from 3.0% (October 2024) to 7.0% (July 2026) — 21 months apart, a swing larger than most decades deliver55

In October 2024, a major bank’s strategists put the coming decade at three percent a year. Twenty-one months later, the same desk said seven. This is not an indictment — it is the honest volatility of forecasting, printed on bank letterhead — but a plan that changes when the forecast changes is a forecast, not a plan.

The audit — width

If the last four years made your portfolio, they also made it lopsided. A run this concentrated in one country’s largest companies quietly converts a pond into a puddle with excellent recent reviews: five American firms now outweigh entire national markets, and the case that the next decade’s leadership is already priced sits in the same forward-return tables that were wrong last time. The audit question is not whether to own stocks; it is which pond you are actually holding. The academic reading of a century of many countries’ records puts the answer near one-third home, two-thirds everywhere else — and 2025, when the rest of the world returned nearly twice the home market, was a one-year reminder of why the pendulum has two sides. Rebalancing is the only respectable way to sell high: it takes no view, requires no forecast, and is finished by lunchtime.

The pendulum, 2025
+32.4% vs +17.1%

2025 total returns: global ex-US (VXUS) +32.35% vs US (VTI) +17.1%; the dollar index fell ~9.4%, its worst year since 2017 — the pendulum swung abroad60

Concentration
five > any nation

the combined value of the five largest S&P 500 companies exceeds the total market capitalization of any national equity market outside the US62

Where the record points
≈33% home / 67% abroad

a century of many countries’ records implies a lifetime-optimal allocation near one-third domestic, two-thirds international equities at all ages — the diversification lever is international, not merely “more tickers”63

What the models expect
US 4–5% / ex-US 5–7%

Vanguard’s model puts 10-year annualized US equities near 4–5% and non-US near 5–7% — a point-in-time projection, cited as such61

The audit — depth

For the reader within a decade of spending this money, high water is not a reason to leave the pond; it is the cheapest moment there will ever be to build the jetty. The danger with a name — sequence risk — is a bad run in the first years of withdrawals, and its defense is structural, not predictive: enough safe assets, matched to when the money is needed, that no market can force a sale at the bottom64. The literature argues about how much and for how long; it does not argue about whether. The drill below prices your version of the question.

The undulation drill

A retiree spending $60,000 a year, real, over the global horizon of 30 years, starting from a corpus of 25× that spending. The equity sleeve rides the market; the safe bucket funds withdrawals whenever equity is below its peak and is topped back up only when equity makes a new high. The question is how many years of groceries to keep out of the storm.

Forced sales vs the size of the jetty

You cannot schedule the storm. You can decide, today, how many years of groceries ride outside it.

Probability that a bad sequence forces a sale of equity while it is ≥20% below its peak, against the years of spending held safe. Seed 42, 6,000 antithetic paths. MODELED
Forced trough sale
84%
at 3 years held safe
Ran out by year 30
19%
paths that exhausted the corpus
Median terminal (real)
$1,271,870
what the median path left behind
≈ 21 years of your spending

What this section does not claim

The windows overlap

The ~95 rolling four-year windows contain only about two dozen independent ones; the percentile talk is descriptive, not inferential.

The other tail is real

Nothing here proves the market won’t fall; high readings were sometimes followed by 1973, 2000, 2008. The claim is only that the gauge cannot tell you which, and the base rate after highs is unremarkable57,58.

The academics disagree about bonds

Cederburg’s optimal portfolio held essentially none; Kitces’s tent holds many at the retirement boundary63,64. Safe assets are a ruin-tolerance decision, not a return maximizer.

The liturgy in the next section does not change at high water. That is the point of a liturgy.

№ VII

Practical conclusion

The default can be stated without claiming that every exception is impossible.

  1. 1Own everything — a global, capitalization-weighted index at a few basis points.
  2. 2Hold enough safe assets, matched to when you need the money, that no market can force you to sell.
  3. 3Feed the pond on a schedule; when you have won, stop playing.

What this survey does not claim

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: for a saver without a demonstrated edge or a binding constraint, a broad, low-cost portfolio is the most defensible starting point. Active competition helps make market prices informative; an index investor benefits from that price discovery without needing to identify its winners in advance.

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.
— Warren Buffett, 2013 Berkshire Hathaway letter (the instruction in his own will)40
When you’ve won the game, why keep playing it?
— William Bernstein, Money interview (2012)47

The companion survey prices the other tax on wealth — fees. → The Arithmetic of Fees

▸Notes on method
The model

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.

Calibration at your settings

m_m = 0.067659 · A_m = 8.38% · m_s = -0.020791 · σ_i = 42.1%

Closed-form cross-checks (live)
  • Mean multiple, both assets: 11.2×
  • Median multiple — pond 7.6×, one stock 0.54×
  • P(one stock beats the pond): 12.5%
  • Sleeve mean — refilled 7.5×, one ticket 10.6×
  • Fee-equivalent of the refilled sleeve: 1.33% per year

“Refilled” means rebalanced back to the selected sleeve weight annually. “One ticket” means funded at inception and then allowed to drift without new capital.

The Monte Carlo figures reproduce these within sampling error; if a simulated value drifts from the closed form, the closed form is right.

The random numbers

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.

seed 42
§VI — High water (dated on purpose)

This section pins to a data vintage and will read as a period piece later — that is intended. Its exhibits are computed, not quoted: the S&P 500 total-return series is vendored from Damodaran’s January 2026 file (dividends included), and every rolling four-year multiple, percentile, and streak is derived from it in-app. The ~95 rolling windows overlap — only about two dozen are independent — so percentile talk is descriptive, never inferential. The undulation drill is modeled: a retiree holds an equity sleeve (the market lognormal, real) and a safe bucket earning cash; withdrawals come from the bucket whenever equity is below its running peak, and the bucket is refilled toward its Y-year target only when equity makes a new high — that refill-at-peak rule is load-bearing. The starting corpus is 25× annual spending. No market data is fetched; the pond stays static and self-contained.

What the model is not

It prices composition, not accumulation — contributions, decumulation, and taxes are out of scope (contributions are the companion /fees paper’s subject). And it is intentionally favorable to concentration: the stock is assigned the market’s full expected return, while the measured record contains additional company-specific risks. The resulting comparison should not be read as a worst-case estimate.

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In the series

Series index
← Previous · No.1The Arithmetic of FeesHow recurring costs, inflation, and allocation assumptions affect long-horizon wealth.Next · No.3 →The Yield IllusionHow dividends, option premiums, and leveraged products relate to total return after friction.

Solved Problems in Personal Finance

№1 The Arithmetic of Fees·№2 A Wide & Deep Pond·№3 The Yield Illusion·№4 The Myth Ledger·№5 Retirement Money Secrets: An Evidence Review·№6 The Employer Stock Transition Guide·№7 The House Percentage: A Note for Michael Batnick·№8 What Money Is For·№9 The Bearer Asset: Bitcoin, Crypto, and the Price of Control

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