Long-horizon expected returns at the holding level
Shrinkage, finite excess, and unfaded volatility in retirement projection.
Draft — 11 September 2026 · Empirical quantities from the locked house-blend record (9–10 September 2026) and the 1 September 2026 price-history cache. Not investment advice. Method is the customer write-up; this page is the evidence.
Modelling expected returns on individual common stocks over long horizons is notoriously difficult. A large literature supplies estimators for the next month, the next quarter, or the next few years — volatility clustering, short-horizon momentum, valuation ratios — and those tools are well understood on their own ground. Once the question becomes thirty, fifty, or seventy years, which is the horizon a retirement plan actually has to answer, consensus falls away, and treating the past decade’s compound growth as if it were that answer is a modelling choice rather than a measurement. Merton (1980) showed that the mean of a return series is far noisier than its volatility, and Damodaran and Morningstar, among others, have long argued that competition does not grant a firm a perpetual extra over the market. This paper therefore sets out a transparent holding-level model for that longer clock. Year-one drift shrinks each name’s chart toward the investor’s stated view of the market — six to ten percent — with a James–Stein weight whose scale is the standard error of the mean, after which any excess above that destination is assumed to decay linearly over twenty years until none remains. Twenty years is the long published window in which a wide-moat firm is more likely than not still to earn extra, and after which extra is taken to have ceased. Index trackers are placed under the same law as names rather than frozen at the market from the first day, and volatility is left at the name’s full-sample climate, because there is no literature that walks a surviving mega-cap’s long-run bounce down to index weather over a plan.
Monte Carlo is kept as independent monthly shocks around that faded centre. Replacing those shocks with a residual block bootstrap fattens the tails while leaving the middle where it was; replacing them with name-level mean reversion collapses the fan in a way Bessembinder (2018) will not allow for single stocks. The practical consequence is easiest to see on a concentrated growth illustration. Once extra is treated as finite, mean and median terminal wealth sit in the same order of magnitude even though the right tail remains wide, which is exactly what one should expect if the typical outcome of a hot name is not a thirty-percent career and a lucky high-volatility path can still print a Buffett-class rate. That is the shape of the evidence rather than a defect to be tidied out of the fan.
Modelling stock-price forecasts over long horizons is notoriously difficult. Several methodologies exist for shorter time frames — GARCH and implied volatility for the coming weeks (Poon and Granger 2003; Christoffersen and Diebold 2000), momentum for three to twelve months (Jegadeesh and Titman 1993), valuation ratios for the next few years (Campbell and Shiller) — and on those clocks they earn their keep. Few of them, however, were built to be compounded for ten or twenty years, and beyond that point there is no settled method at all. Retirement planning has no shorter question to fall back on. A person of thirty-five who intends to spend to ninety-five is asking what a holding is assumed to earn, both in expectation and in the tails, over a clock on which almost no firm’s extra has been shown to survive, and the planner cannot honestly answer that question by pasting last decade’s compound growth onto the next six.
Two further difficulties distinguish this setting from the textbook asset-allocation exercise. The first is that the object is often a name rather than a 60/40 sleeve, and Bessembinder (2018) documents how lopsided that object is: four in seven CRSP stocks never beat Treasury bills over their lives, while roughly four percent of firms created all net United States market wealth between 1926 and 2016. A model that reports only a middle, or only a right tail, is therefore answering half the question the investor actually asked. The second is that the mean which would justify compounding last decade’s chart is not known to anything like the precision of the bounce around it. Merton (1980) remains the classic statement of that fact: expected return is the parameter on which historical averages are least informative. Granger and Newbold (1974) add a second warning, which is that a line fitted to a wandering price inflates R², so that a high R² cannot be read as evidence that the line continues.
This paper contributes a complete, citable specification for that gap, as implemented in Loxoda’s holding-level engine. The investor states a view of the market — six percent ultra conservative through ten percent very bullish, with eight percent balanced as the house anchor, itself a haircut on the Dimson, Marsh and Staunton long-run United States record — and each growth holding’s chart is then shrunk toward that destination in proportion to how noisily its mean is known. Extra above the destination is assumed gone after twenty years, which is Morningstar’s long published wide-moat window and Damodaran’s treatment of high growth and of excess in terminal value; a weak chart is not walked up to that same destination over the same window. Trackers of the same index share one history and then follow the same shrink and fade as names, volatility stays the measured climate rather than being walked toward the index, and luck is independent monthly shocks around the faded centre — two thousand of them, with correlations stressed toward one in falling markets after Longin and Solnik (2001).
The rest of the paper proceeds as follows. Section 2 reviews the literature the specification is licensed by, and the literatures it will not over-read. Section 3 states the model. Section 4 describes the tapes. Section 5 reports year-one rates, then where extra goes, then what that choice does to wealth. Section 6 examines alternative luck engines. Section 7 concludes.
2. Related literature
2.1 The mean is the hard parameter
Merton (1980) showed that effort in empirical finance has not been uniform across parameters, and that this nonuniformity is not without reason: variance rates and betas can be estimated from time series with useful precision, whereas expected returns cannot. The sampling error of a mean return is σ / √T, so that at the volatilities of individual names even decades of history still leave a standard error of several percentage points. James and Stein (1961), Blume (1971), and Vasicek’s beta shrinkage supply the structure for pulling such a noisy sample toward a grand mean. What they do not supply, and what this paper will not invent, is a licence to treat R², or a fixed-percent mix weight, as if it were that structure.
2.2 Extra is finite
Damodaran’s notes on how long high growth lasts, and on excess returns in terminal value, are explicit on the point that matters here: competition kills extra, and in stable growth that extra is gone, with beta often walked toward one. Using cost of capital as a discount rate — a riskier claim is worth less today — is not the same operation as compounding wealth at that required return for fifty years, and the two should not be asked to do each other’s jobs. Morningstar’s equity-research methodology publishes the long window: a wide-moat firm is more likely than not to earn excess for twenty years, after which excess ceases and ROIC fades toward WACC. Damodaran’s typical high-growth window is five to ten years, so twenty years is the long published assumption rather than a cautious one. Frazzini and Pedersen (2014) further caution that high-beta names do not earn the full textbook premium, which is relevant if one were tempted to leave a CAPM extra sitting in the compounding rate forever.
2.3 Names are not the market
Bessembinder (2018) is the citation for the median and the left tail. Buffett’s realised 19.9 percent from 1965 to 2024, as tabulated in the Berkshire Hathaway 2024 letter, is a surviving, leveraged, multi-company path, and Frazzini, Kabiller and Pedersen (2018) show that much of that record is cheap, safe and quality with leverage rather than one-stock magic. It is a scale for the right tail of a high-volatility name, and it is not a target for the mean. Distress raises failure odds (Campbell, Hilscher and Szilagyi 2008) without underwriting a rebound for one beaten-up name, and De Bondt and Thaler (1985) is a basket result in any case. Cross-section evidence that young firms are wilder (Pástor and Veronesi 2003; Fink et al. 2010) is likewise not a clock on one survivor; Brown and Kapadia (2007) find that later listing cohorts stay persistently wilder, which is the opposite of a seasoning story one could apply to a mega-cap that has already survived.
2.4 Volatility as climate, not this week’s storm
Predictability of volatility dies in weeks (Christoffersen and Diebold 2000), and for a long forecast the long historical average outperforms short-window GARCH (Figlewski 1994/1997), while implied volatility remains the short-horizon tool (Poon and Granger 2003). Campbell and Viceira (2005) show that if the market mean-reverts, a long lump of the index can look calmer per year than monthly volatility suggests, but that result is about the market and not a name-climate glide, and Kitces and Pfau debate the same point for retirement Monte Carlo on indexes. Ibbotson and Morningstar planner practice uses all available relevant history on asset classes; Bessembinder bootstraps real months on names and does not season σ. There is, in short, a literature that looks like a fade if one reads it quickly — mean reversion on the index, GARCH dying in weeks, implied volatility as a tool for the next storm. None of that literature says that a single name’s long-run bounce, fifty-seven percent in NVIDIA’s case, should be walked down to the index’s fifteen percent over the length of a retirement plan, and this specification does not invent that walk.
3. The model
3.1 Objects
Cash earns a user-stated rate, defaulting to zero. Income holdings are a toggle rather than a bond sleeve: near-par names contribute coupon with capital held flat, while other Income names add capital from a split-adjusted path with dividends stripped. Crypto retains a separate CAPM-and-hazard machine, leftover 0.30, with a fifteen percent ceiling. All other growth holdings — a single company or a physical index tracker — share one equity law, and before that law is applied, trackers of the same index are assigned one canonical tape so that a shorter share class does not inherit a hotter chart.
3.2 Destination
Let d be the investor’s market view for the holding: 6, 7, 8, 9 or 10 percent according to stance. After twenty years, assumed extra above d is zero, while d itself does not fade. The destination is therefore neither a collapse of every name to a single eight percent nor a CAPM required return with beta held at its sample value forever.
3.3 Year one
Let μ̂ be the log-linear chart trend on monthly total-return closes, σ the full-sample annualised monthly log-return standard deviation, and T the length of history in years. Then
SE = σ / √Tw = τ² / (τ² + SE²)r₁ = (1 − w) d + w μ̂
with τ = 0.03 (three percentage points). That constant is the one invented scale in the specification: it says a name’s true long-run extra may be a few points rather than a different planet, and Merton’s observation that the market mean itself is known only to a couple of points even with a century of data is the licence for putting the scale there. If r₁ ≤ d, r₁ is left unchanged, so there is neither a promotion of a weak chart to the destination nor a lid ladder on a strong one.
3.4 Fade
Let extra e = max(r₁ − d, 0). At elapsed year t ≤ 20,
r(t) = d + e (1 − t / 20)
and r(t) = d thereafter. The path is linear in the assumed extra return, not in price. Exponential decay would kill extra faster and a cliff would keep it longer, so linear sits in the middle of those two. Leftover on equities is zero; crypto leftover 0.30 is a separate, signed choice on that machine and is not inherited here.
3.5 Wealth and luck
The deterministic path (“hero”) compounds each holding at r(t), buys contributions on the investor’s split, does not rebalance existing shares, and applies a 2.5 percent selling cost only to owned assets such as property. Extra annual platform or plan cost is a user input defaulting to zero; fund charges already sit in a tracker’s chart and are not subtracted again. The fan then applies two thousand independent monthly shocks to the book, sized by book volatility after pairwise correlations are pulled toward one, with displayed plan confidence capped at 99 percent. Hero and fan share the same faded mix.
4. Data
Tapes are split-adjusted monthly total-return closes, with volatility and trend taken from the full sample rather than a trailing window. The snapshot used for the illustrations below is the 1 September 2026 cache.
Holding
Trend CAGR
R²
Vol
History (yr)
β vs SPY
NVIDIA
30.0%
0.83
57.5%
27.8
2.12
QQQ
12.1%
0.82
23.5%
27.6
1.33
Apple
21.1%
0.91
44.0%
45.8
1.32
SPY
8.9%
0.92
14.8%
33.8
1.00
Table 1. Fitted statistics, cache 1 September 2026. Volatility is annualised monthly log-return σ, the name’s long-run average bounce, not last-year vol.
5. Results
5.1 Precision of the mean
Table 2 applies Merton’s standard error to Table 1. NVIDIA’s 30 percent chart cannot be distinguished from the S&P at conventional significance, which is a property of the tape rather than a modelling preference.
CAGR
Vol
Years
±1 SE
Rough 95% range
NVIDIA
30.0%
57.5%
27.8
±10.9 pp
~8% to 52%
Apple
21.1%
44.0%
45.8
±6.5 pp
~8% to 34%
QQQ
12.1%
23.5%
27.6
±4.5 pp
~3% to 21%
SPY
8.9%
14.8%
33.8
±2.5 pp
~4% to 14%
Table 2. SE = vol / √years. Source: same tapes as Table 1.
NVIDIA
±10.9 pp
Apple
±6.5 pp
QQQ
±4.5 pp
SPY
±2.5 pp
Figure 1. Sampling error on the mean return. At τ = 3 pp the implied trust weight on NVIDIA’s chart is about 7 percent; on SPY’s, about 58 percent. A rule that treated how smoothly the line fits as if it were knowledge of the mean would give NVIDIA roughly the same weight as SPY, even though NVIDIA’s sampling error is more than four times as large. Smoothness of a price path is not knowledge of the mean.
5.2 Year-one rates
Holding
Stance
Chart
Destination d
Year one r₁
NVIDIA
Very bullish
30%
10%
11.4%
NVIDIA
Balanced
30%
8%
9.5%
NVIDIA
Ultra conservative
30%
6%
7.7%
SPY / VOO
Very bullish
8.9%
10%
9.4%
SPY / VOO
Balanced
8.9%
8%
8.5% → 8%
SPY / VOO
Ultra conservative
8.9%
6%
7.7% → 6%
Table 3. Locked year-one rates at τ = 3 pp. SPY at very bullish is not walked up from 9.4% to 10%; a cooler chart is left at or below d, and fade never lifts.
5.3 Where extra goes after year one
Year one is only the start of the assumed path. What this specification does next is let any extra above the investor’s market view decay linearly over twenty years until none remains, so that NVIDIA on very bullish starts at 11.4 percent and sits at 10.0 percent from year twenty onward — ten percent being the destination the investor stated, not a second rate invented from beta.
The natural rival for that destination is the capital asset pricing model. CAPM says a security’s expected return is the risk-free rate plus beta times the market risk premium: a name that moves twice with the market is required to earn twice the premium, not because it is a better company, but because the holder must be compensated for systematic risk (Sharpe 1964; Lintner 1965). In valuation that required return is a discount rate — a riskier claim is worth less today. It is not a forecast of the rate at which the same claim will compound wealth for fifty years, and the two jobs should not be swapped. Leave NVIDIA’s sample beta of 2.12 stuck against a ten percent market and a risk-free rate in the neighbourhood of four percent, and the implied floor is about 16.7 percent in perpetuity: 4.8 points of chart extra can die and 6.7 points of beta premium remain. Damodaran’s own stable-state practice walks beta toward one, so even a CAPM-consistent terminal value does not keep that extra, and Frazzini and Pedersen (2014) find that high-beta names do not earn the textbook premium in the first place. Figure 2 therefore compares compounding at that stuck-beta floor with the path this paper uses. The difference is whether extra is a finite window that ends at the investor’s market view, or a second destination that never ends. Section 5.4 asks what a 6.7-point gap of that kind does to terminal wealth on a live book.
CAPM floor, β stuck at 2.12This paper: extra gone at 20yInvestor’s destination (10%)
Figure 2. Assumed drift through time, NVIDIA very bullish — not a simulated price path. Hero wealth is smooth because it is the no-luck path; Monte Carlo prices remain jagged. After year twenty the two specifications are 6.7 percentage points apart, and they stay that far apart for the rest of the plan.
5.4 What that difference does to wealth
Figure 2 is a path of assumed rates. Figure 3 is the same choice in money, on a live book: 25 NVIDIA (very bullish), 12 QQQ, 10 Apple and $1,800 cash — about $19,200 today — with $1,400 a month of contributions, a plan to age 95, and results stated in today’s money. If NVIDIA’s sample beta is left in the compounding rate after year twenty, the expected pot is about $500 million, with P50 at $76 million and P90 at about $3 billion: a mean that has left the median behind because the book becomes the high-beta name and never stops compounding that extra. Under the specification of this paper, NVIDIA’s drift after twenty years is 10.0 percent, and mean and median sit together.
Expected
$14.16m
P50
$14.55m
P10 (paths deplete)
$0
P90
$640m
Expected
P50
CAPM floor, β = 2.12This paper
Figure 3. Terminal wealth at age 95, today’s money, same live book and contributions ($ millions), under the two destinations in Figure 2. Hero and P50 under this paper are about 0.97× one another, with confidence 82%. At retirement age 60 the same book reads expected $2.08m, P10 $462k, P50 $2.59m and P90 $22.1m. P90 at 95 remains $640 million and P10 includes depletion, so the shape is not crushed, and τ is not retuned to tidy the fan.
5.5 Lump-sum fan at sixty years
Table 4 isolates one name: ten thousand dollars, no contributions, two thousand paths, leftover zero, today’s money, and NVIDIA volatility held at 57.5 percent.
Stance
Year 1
After 20y
Hero
P10
P50
P90
Ultra conservative
7.7%
6.0%
$88k
$351
$92k
$22m
Balanced
9.5%
8.0%
$266k
$1.1k
$279k
$67m
Very bullish
11.4%
10.0%
$786k
$3.1k
$827k
$198m
Table 4. P90/P10 remains on the order of 63,000×, which is tail over tail. Start-to-P90 very bullish is about 19,800× ($10k → $198m real); the corresponding nominal P90 is about $872 million, a compound rate near 21 percent.
Ultra
Balanced
Very bullish
Hero ($k, real)P50 ($k, real)
Figure 4. Mean and median remain aligned across stances. The compass still moves a tracker treated as a name: on SPY, very bullish versus ultra conservative at sixty years is 5.55×, which is neither a nine-times cartoon of a gap that never dies nor a decorative ~1.05×.
5.6 Long-horizon scale, not identity
Figure 5 places thirty-year nominal outcomes from $10,000 beside two well-documented careers. Berkshire Hathaway compounded at 19.9 percent from 1965 to 2024 versus 10.4 percent for the S&P with dividends (2024 letter). Druckenmiller’s Duquesne record is well-reported at about 30 percent over about 30 years, without a public year table, and is not buy-and-hold (Reuters, 18 August 2010; New York Times, 19 August 2010). Neither figure is a percentile of this model; both are rulers against which a tail can be read.
Druckenmiller 30% × 30y
$26.2m
NVIDIA P90
$11.54m
Buffett 19.9% × 30y
$2.32m
NVIDIA P50
$0.218m
S&P ~10.8% × 30y
$0.217m
Figure 5. NVIDIA P10 over the same clock is $3.8k and sits off the scale. Druckenmiller’s reported 30 × 30 sits above NVIDIA P90, while a sixty-year NVIDIA P90 (~21 percent, ~$872m nominal from $10k) sits in the same neighbourhood as Buffett’s realised path ($10k → ~$550m). Neighbourhood is not identity: Berkshire is many companies, float and leverage, and 1974 was −48.7 percent. CPI-U 1965–2024 is about 3.9 percent a year, so these bars are nominal rather than Loxoda’s 2.5 percent real deflator.
6. The luck engine
Volatility is left at the full-sample monthly climate in Table 1 — 57.5 percent for NVIDIA, 14.8 percent for SPY — rather than being walked toward the index over the plan. The production fan is then two thousand independent monthly shocks around the faded centre of Section 3. The rest of this section asks whether a more elaborate luck engine would change the sentence that fan is there to tell.
Table 5 reports one-shot experiments, subsequently discarded as production methods: $10,000 lump, nominal, NVIDIA very bullish fade 11.4 → 10 percent, two thousand paths. The Buffett column is the Berkshire 2024 letter path ($10k → $550.24 million at 19.9 percent), a realised history, not a percentile of this model.
Clock
Engine
Hero
P10
P50
P90
Share ≥ Buffett
30y
i.i.d. monthly
$198k
$4k
$221k
$11.8m
22%
30y
Residual blocks
$198k
$2k
$203k
$14.4m
23%
60y
i.i.d. monthly
$3.46m
$15k
$4.11m
$1.11bn
13.6%
60y
Residual blocks
$3.46m
$5k
$3.71m
$1.42bn
14.0%
60y
Berkshire path
—
—
$550m
—
—
Table 5. Residual block bootstrap (Künsch 1989; Politis and Romano 1994; filtered historical simulation in the spirit of Barone-Adesi et al. 1999) peels the sample mean, replays leftover months in chunks, and glues the faded house mean. At tape length it matches i.i.d. At sixty years it fattens P90 by about 28 percent and raises the Buffett-beat rate by 0.4 points. The middle does not catch Buffett. Ornstein–Uhlenbeck mean reversion with a three-year half-life produces P90 $10.4 million and a Buffett-beat rate of zero: a tidy fan that forbids staying rich or dead, contrary to Bessembinder on names.
i.i.d. 60y
Residual 60y
OU, 3y half-life
Berkshire path
Figure 6. Alternative engines for the same faded centre. Mean reversion is the specification that makes the tail look kind; residual blocks are the specification that looks more sophisticated and only fattens the tail. Production therefore remains independent monthly shocks around the faded mean.
7. Discussion and conclusion
Long-horizon expected return at the holding level cannot be read off a decade of prices, cannot be a perpetual beta premium, and cannot honestly borrow the index’s mean-reversion gift for a single name. The specification here is deliberately small: one shrinkage scale, a twenty-year linear extra, a destination the investor can state, and a climate taken from the whole tape. What it will not do is as important as what it asserts. It will not treat R² as a measure of trust in the mean, will not walk a weak name up to the destination, will not fade NVIDIA’s weather to 15 percent merely because the fan looks rude, and will not retune τ to shrink P90 once the middle is already in the same order of magnitude as the median.
Two limitations should be stated as such. Distress and firm failure are parked until they can be estimated rather than invented. Block-bootstrap and Student-t shocks remain available as robustness checks, but they are not the production generator, because on the experiments above they do not change the sentence the fan is there to tell. Balanced eight percent remains a house haircut on a lucky United States century (Dimson, Marsh and Staunton), not a claim that the next century will repeat including re-rating.
The usable conclusion is therefore a narrow one. A great decade is not a thirty-year assumption: extra is taken to die in twenty years, what remains is the market return the investor chose, and volatility stays what the name has been. The middle of a hot name is not Buffett, even though a lucky high-volatility path can be, and that is Merton, Morningstar, Damodaran and Bessembinder — together with the tapes in Tables 1–5 — rather than a cover-up of a wide fan.
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