Setting the equity share: what is arithmetic, what is preference, and what neither can reach

Question. The repository has declared net geometric growth as its objective and has refused to set the equity/bond split. Given both, what can actually be said about setting it?

Decision it informs. What a disciplined answer to the largest decision in the portfolio looks like, and which parts an application may compute rather than ask. It does not set the split, and the refusal in the recommendation §1.1 stands.

Out of scope. A forecast of any market. A recommended number. Glide paths as products. Annuities, which change the problem rather than the parameter.

as of 2026-08-17. Every measured figure in §§1–7 regenerates from studies/equity_share.py and is pinned in research/tests/unit/test_studies_equity_share.py; conclusion 8’s figures come from studies/fixed_income_shelf.py and are set out in the alternative sleeves audit.

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Conclusion

1. The objective does not choose the split. The constraint does. Growth-optimal sizing contains no risk-aversion parameter at all, and under the zero-leverage rule the growth objective alone returns a corner solution — 100% equity, for any equity premium over bonds above about 2.2%/yr arithmetic. Every bond in variants B and C is there because of the constraint, and the constraint is a single number nobody has supplied.
2. Because the constraint binds, the only available error is underbetting, and underbetting is cheap. Growth at exposure f times the growth-optimal exposure retains 1 − (1 − f)**2 of the peak excess growth — a parameter-free expression. Half the growth-optimal exposure keeps 75%. Twice it keeps 0%. With leverage at zero and the unconstrained optimum above 1.0, every admissible portfolio sits on the gentle left branch of that parabola.
3. The equity share is worth about as much per year as the entire measured edge budget, and takes about twenty-five years to prove rather than twelve months. Moving 60/40 to 90/10 was worth +127 bp/yr against 485 bp/yr of tracking error on 1963-07…2025-12 — a 0.924 chance of being ahead after thirty years, 90% confidence at 24 years. The contractual budget is ~109 bp against an assumed 46 bp. Same order of magnitude, wholly different certainty class, and they may never be added.
4. The case for fractional Kelly is variance, not bias, and it is weaker than usually made. The expected annual growth given up by using an estimated optimum is exactly 1/(2T) — 0.80%/yr on a 62-year sample, free of every other parameter. The growth-maximising shrinkage is f* = S**2 T / (S**2 T + 1), which on this repository’s own US equity sample is 0.931, not 0.5. Half Kelly on this asset is the claim that 62 years of record are worth 4.7 years of stationary information. That may be a good claim — but it is a claim about non-stationarity and should be argued as one.
5. Sequence risk has a sign. Across 20,000 reorderings of one fixed 360-month record, a lump sum’s terminal wealth was identical to floating-point precision. A contributor’s spanned 2.18× and a withdrawer’s 1.77×, with correlations to first-decade returns of −0.775 and +0.775 — the same number with opposite signs.
6. Variant C bundles two situations with opposite answers. In real terms over a 30-year retirement, a 20% equity portfolio drawing 4%/yr real failed in 6.82% of reorderings against 2.43% at 60% equity; at a 5% draw, failure fell monotonically to 90% equity.
7. The bond side is a risk brake whose braking is regime-dependent. On this repository’s own data the equity/bond beta ran +0.129 (1974–1999), −0.055 (2000–2022Q3), +0.116 (2022Q4–2024Q2) and −0.109 since. The all-bond portfolio drew down −25.1%, deeper than the 30/70 mix’s −17.9%. Bonds are not a floor.
8. Added 2026-08-17: TIPS do not fix that, and they are not a second asset. The obvious answer to point 7 is an inflation-indexed bond, on the argument that it responds to a different state variable and so should hold its correlation where the nominal bond does not. Measured, it is the reverse. On the 275 months where both exist, TIPS’ correlation to US equity is +0.131 against the nominal ten-year’s −0.076 — a gap of 3.5 standard errors — and their five-year-block dispersion is 0.200 against 0.114. TIPS correlate +0.76 to +0.85 with the nominal bond funds they would sit beside, so holding both is the same fake breadth as holding credit beside Treasuries. And the era that would settle it cannot be reached: no TIPS return exists before 2003, which is entirely inside the period when the nominal bond’s correlation had already flipped negative. Full working in alternative sleeves audit; the practical reading is that point 7’s regime dependence has no fixed-income remedy on any evidence held here.
9. Status. §§1–3 are arithmetic — closed forms, exact given their inputs, and not evidence about any market. §§5–7 are exploratory at best: one sample, one country, one modelled bond series, and in the retirement case a permutation null that deliberately destroys serial dependence. Point 8’s TIPS figures are exploratory too and rest on a modelled real-yield series over 275 months. Nothing here is promoted.

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1. The part that is arithmetic

1.1 The growth parabola, and why being wrong is cheap on one side

Written about its vertex, g(L) = r + 0.5 sigma**2 [(L*)**2 − (L − L*)**2] with L* = (mu − r)/sigma**2. Divide the excess of g over cash by its peak and every parameter cancels. At exposure L = f L*, growth retained is 1 − (1 − f)**2:

The parabola is symmetric in L, so the asymmetry people mean is multiplicative. Half the growth-optimal exposure and twice it are both a factor of two away; one costs a quarter of the peak, the other costs all of it and carries four times the variance while doing so. That is why MacLean, Thorp and Ziemba (2010) write that “it never pays to bet more than the Kelly strategy.”

The consequence here is specific. Leverage is zero and §1.2 shows the unconstrained optimum sits well above 1.0, so the whole feasible range is on the left branch. With leverage at zero you cannot overbet the equity/bond decision.

1.2 Where the optimum is, and why the constraint binds

For a fully invested long-only two-asset mix rebalanced continuously,

w* = ( mu_e − mu_b + sigma_b**2 − rho sigma_e sigma_b )
     / ( sigma_e**2 + sigma_b**2 − 2 rho sigma_e sigma_b )

Both mu terms are forecasts. So are all three second moments. This step cannot be run without a forecast, and the repository does not make one.

The honest direction to read it in is backwards. Invert it, and a chosen equity share becomes the forecast it always was. At the sample second moments of this repository’s own US series (sigma_e 15.40%, sigma_b 6.73% for the modelled ten-year bond), the arithmetic equity-over-bond premium at which each share is growth-optimal:

Read the last column: a 100% equity portfolio is growth-optimal for any expected equity-over-bond premium above about 2.1 to 2.7%/yr. Read the middle: choosing 60/40 asserts that equities will beat bonds by about 1.2%/yr, and no more. Anyone who would not write that forecast down should notice that holding 60/40 writes it down for them.

For scale, and as an illustration rather than a forecast: over 1963-07…2025-12 the realised arithmetic premium of US equity over the modelled bond was 5.51%/yr, at which the unconstrained w* is 2.28.

This is the finding that most needs stating plainly. Growth-optimal sizing, under the zero-leverage rule, does not produce a balanced portfolio. It produces a corner. Every bond in variants B and C is bought by something the objective does not contain.

1.3 What one step up the ladder is worth

Same window, constant mix rebalanced monthly, log tracking error against the lower rung:

The information ratio is nearly constant up the ladder, because edge and tracking error both scale with the weight difference — so the size of the step barely changes how long the decision takes to prove. About twenty-five years either way.

The comparison with the rest of the repository. The contractual budget is ~109 bp against an assumed 46 bp, 99% settled in about twelve months; the 60/40 → 90/10 move is +127 bp against 485. Those are the same order of magnitude in expected return and more than twenty times apart in how fast you find out. The often-repeated claim that the equity share dwarfs everything else is right about the risk — 15 points of maximum drawdown against an edge budget with no drawdown term at all — and roughly wrong about the return. Both are historical, and the second must never be added to the first: they carry different benchmarks and the study code raises on the attempt.

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2. Why full Kelly is not the answer even on its own terms

Breiman’s theorem is asymptotic, and its own statement says so. Asymptotically is not a horizon, and Samuelson’s objection at finite horizons stands.

Three arguments get made for cutting the fraction. They are not equally good.

2.1 The estimation-error argument, done exactly

With sigma known and muhat ~ N(mu, sigma**2/T) over T years:

SE(Lhat*) = 1 / (sigma sqrt(T))
E[g(Lhat*)] = g(L*) − 0.5 sigma**2 Var(Lhat*) = g(L*) − 1/(2T)

The second line is exact and is the sharpest thing here. The expected annual growth given up by estimating the growth-optimal exposure is 1/(2T), free of mu, sigma and r — a pure consequence of the objective being quadratic in the exposure error. Verified against a 400,000-draw seeded simulation.

Shrink the plug-in by f and minimise the same expected shortfall, and the optimum is f* = S**2 T / (S**2 T + 1), depending on the data only through S**2 T, which is the sample’s whole information content about the mean. On this repository’s US market series, S = 0.4631:

The cost of not knowing the mean is large — 0.80%/yr over the longest sample anyone has is about three-quarters of the whole contractual budget, and at twenty years it is 2.50%/yr, more than twice it. This is the quantitative form of Merton (1980): the precision of a mean estimate improves with the calendar span of the sample and not with sampling more finely inside it, so it improves slowly and there is no way to buy your way out. It is also why Chopra and Ziemba find errors in means about eleven times as damaging as errors in variances and twenty-two times as damaging as errors in covariances — the widely quoted 20:2:1 ratio, and worse still as risk aversion falls, a log investor’s being about as low as it gets. (Their table is reproduced in MacLean, Thorp and Ziemba; the original is paywalled and was not read here.)

And the honest shrinkage is 0.93, not 0.5. This is where the usual telling goes wrong. With sigma known the plug-in optimum is not biased upward — it is unbiased and noisy, and what the noise damages is the achieved growth, not the estimate. Estimating sigma too does bias it upward, by (n−1)/(n−3), which on 750 monthly observations is 1.00268 — a rounding error. Selection across many candidate assets biases the winner’s estimate properly, but there is no selection in an equity/bond split: there are two assets and both are held.

So the estimation-error argument, run correctly, supports a fraction around 0.9. Inverting the formula makes the folk rule legible: T = f / ((1 − f) S**2), so half Kelly on a 0.4631-Sharpe asset is the assertion that the entire 62-year record is worth 4.66 years of stationary information. That is defensible — regimes change, and §6’s bond-stock sign flip is direct evidence for it — but it is a claim about non-stationarity and should be defended as one rather than smuggled in as a statistical correction.

2.2 The other two arguments

Variance of the growth rate. Full Kelly’s wealth path is violent: “the Kelly criterion can be very risky in the short term.” MacLean, Thorp and Ziemba read Buffett as behaving “similar to a fully Kelly bettor (subject to the constraint of no borrowing)” and Keynes as an 80% Kelly bettor. Note the parenthesis: it is the same constraint imposed here, and under it full Kelly on equities is simply 100% equity.

Risk-constrained Kelly. Framework open decision 8 notes a ~34% growth advantage at matched drawdown risk on a finite-outcome case, and that the advantage vanished on that paper’s own fat-tailed mixture. Untested here and correctly deferred: it sizes an edge, and there is no edge.

What the argument actually supports is a fraction between about 0.5 and 0.9, the low end justified by non-stationarity and the high end by the arithmetic. On §1.2’s numbers that maps to an equity share of roughly 1.1 to 2.1 times a fully invested portfolio — so the no-leverage constraint still binds across the whole of it. Fractional Kelly does not produce a bond allocation either.

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3. Sequence risk, verified and given a sign

Terminal wealth without flows is W0 · prod(1 + r_t), a product, and multiplication commutes. With flows it is sum_t C_t · prod_{s>t}(1 + r_s), which does not.

Measured rather than asserted: 20,000 random reorderings of one fixed 360-month record (the US market, 1996-01…2025-12), seed 20260812, 100% equity throughout, so the multiset of returns is identical in every draw and ordering is the only thing that varies.

The lump-sum row is the identity, confirmed to about 1 part in 10¹⁵. The other two are the finding, and the correlations are the same magnitude with opposite sign, because level contributions and level withdrawals are algebraically dual. A bad first decade is good for someone buying through it and bad for someone selling through it.

The consequence is that horizon is the wrong input. A 30-year accumulator and a 30-year retiree have the same horizon and mirror-image problems. The input that matters is the schedule of cash flows: sign, size relative to the portfolio, and when they start.

One limitation, stated because it cuts against the result. Permuting imposes an iid null. It holds the multiset fixed, which is what makes it the right test of the identity, but it destroys serial dependence, so it neither confirms nor denies mean reversion. If real returns mean-revert, real sequence risk for a withdrawer is smaller than this; if volatility clusters, it is larger.

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4. Human capital, honestly

The standard argument: a young investor’s future earnings are a large, bond-like asset, so to hold a given fraction of total wealth in equities the financial portfolio must be equity-heavy and should de-risk as human capital is spent down. That is the intellectual basis of every target-date glide path on the market. It is a real result with a real derivation, and four objections are usually left out.

1. Labour income is not bond-like for everyone. Benzoni, Collin-Dufresne and Goldstein (2007) model labour income and dividends as cointegrated — a long-run tie that a short-run correlation of roughly zero hides completely. Under cointegration the young agent’s human capital is stock-like, because there is time for the tie to bind, and only the older agent’s is bond-like. Their model implies young investors should short equities and produces hump-shaped lifetime holdings. The mechanism the standard argument uses can run the other way, and in that paper it does.
2. Occupation and employer stock decide it, not age. A tenured public employee and a commission-paid salesperson at a cyclical firm do not hold the same asset, and anyone holding employer stock in a qualified plan holds a position correlated with their own income at the moment they most need the money.
3. Human capital is illiquid and cannot be pledged. Treating it as a bond holding in a mix that is then rebalanced quarterly treats an unsellable asset as a tradeable one.
4. The argument as usually deployed implies leverage this repository forbids. If human capital is 80% of a 25-year-old’s total wealth and the target total-wealth equity share is 60%, the implied financial portfolio is 300% equity. The literature knows this and says so; retail practice quietly caps it at 100% and keeps the conclusion. And the capped version is just “hold 100% equity while young”, which §1.2 already gets from the growth objective without any human-capital argument at all.

The one piece that survives all four cleanly is Bodie, Merton and Samuelson (1992): the ability to vary work effort ex post — to work longer, save more, or retire later — induces more risk-taking ex ante. That is a statement about flexibility, not about age, and it is the version an application can actually ask about. “How old are you” is a proxy for it, and a poor one.

Nothing in this section is measured here. It is published theory with published objections, and it is not tested. It should inform the questions an application asks and must not be turned into a formula that outputs a weight.

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5. The drawdown anchor, which is the operational form of the answer

Constant mix, rebalanced monthly, 1963-07…2025-12, 750 months. Equity is Ken French’s US market total return — the same series that produced the 10.80% / 15.40% / −50.3% / 72-month line in Experiment 007, reproduced here through a different code path as a check. Two safe assets, because the difference between them is itself a finding: cash is French’s RF, measured; the ten-year bond is modelled from FRED GS10.

This table is the answer in the only form that can be handed to a person. Pick the row whose drawdown you would have held through — not the one you would tolerate in the abstract, the one you would have held through for the months under water in the same row — and read the equity share off the left.

Three warnings that belong beside it and not in a footnote. Drawdown deepens mechanically with sample length, so no number here may be compared against a drawdown from a different window. And bonds shortened the drawdown but did not remove it: the 90/10 rung is −45.5%, 4.8 points better than all-equity.

The third limitation is measured, and it is worse than a warning about one country would suggest. Jordà–Schularick–Taylor R6 supplies annual real total returns for sixteen countries, 1870–2020:

In the same 1963-onward window this ladder is built from, every one of the other fifteen countries did worse, and fourteen of fifteen did worse than −50%. France fell −97.7% from its 1942 peak and had not regained it 78 years later; Japan’s −93.0% (1937→1945) is a floor rather than a measurement, because 1946–47 are missing from the source and inflation in those years ran +91% and +125%.

Three qualifications travel with that table and none of them rescue the anchor. Portugal’s −98.4% leans on source-flagged interpolations for 1975–77; dropping them leaves −80.1%, and the cleanest fully-measured near-total loss is France’s. Germany’s −97.9% is contaminated by hyperinflation arithmetic and the 1948 currency reform. And these are annual and real against this page’s monthly and nominal −50.3%, so the like-for-like US comparator is −47.2% — which is the number that ranks last of sixteen.

Read the ladder accordingly. −50.3% is not a bound and not a typical case. It is close to the most fortunate outcome the developed world produced, and an equity share chosen against it is chosen against the best draw rather than the median one.

5.1 Withdrawals invert part of the table

The drawdown ladder is monotone; the failure ladder is not. In real terms — CPI deflated, level real withdrawal, 30-year horizon, 20,000 reorderings, 748 months (real equity 6.66%/yr, real bond 1.98%/yr) — the probability of running out:

The minimum walks right as the withdrawal rate rises. At 3% real the safest portfolio is genuinely the safe one; at 4% the minimum is at 60% equity and a 20% equity portfolio is nearly three times as likely to fail; at 5% and 6% failure falls almost all the way to 100% equity. Above about a 4% real draw, holding too few equities is the larger risk. The mechanism is not subtle: the withdrawal outruns the return, and no ordering of a 1.98%/yr real bond return supports a 5% real draw for thirty years.

This is exploratory — one country, one modelled bond, an iid permutation null, and a real bond return no reader should assume forward. It is enough to show that variant C conflates two cases.

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6. What the bond side is actually for

The recommendation books bonds as “a different benchmark, not an edge”, sized by risk capacity, citing Campbell, Pflueger and Viceira (2025) for the sign flip. Two corrections to how they have been quoted here, and one measurement.

The era boundaries usually quoted are not the ones the authors state. In their own February 2026 summary the negative era runs to 2022Q3, not 2022, and the positive era that follows runs 2022Q3 to 2024Q2. They also name three earlier sub-periods: negative 1964Q1–1967Q3, positive 1967Q4–1971Q3, and no significant beta 1971Q4–1974Q2. The picture is not two eras. It is at least six.

The “US, UK and Eurozone” attribution could not be verified. The NBER landing page, the abstract and the authors’ summary are US-only in what they state; the working paper PDF returned HTTP 403 from two hosts. The multi-country claim may well be in the paper. It is not supported by anything read here.

Measured on this repository’s own data, with the modelled bond and their sub-period boundaries translated to months:

The three long eras reproduce the published sign pattern independently. The three short ones do not and should not be expected to — 18 to 45 observations carry no power, and the two earliest disagree with the published sign outright. The final row is the one worth sitting with: on the eighteen months since the published sample ends, this repository’s own data says the sign has flipped back to negative again.

What follows for the split: bonds are held as a risk brake whose diversification benefit is regime-dependent — the sign changed three times in the published record and, on this data, a fourth time since it ended. That is weaker than “bonds diversify equities”. It is strong enough to keep them, since §5 shows the brake working at every rung, but not strong enough to size them from a covariance estimate, and it is direct evidence for the non-stationarity §2.1 says is the real argument for a fractional exposure.

And the row that should end any conversation about bonds as a floor: the all-bond portfolio drew down −25.1% and spent 65 months under water, deeper and longer than the 30/70 mix’s −17.9% over 41 months. Its worst stretch was 2020-08 to 2023-10, inside the era when the beta had turned positive again.

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7. The decision structure

This is the part an application renders. It is a structure, not a number, and it fails loudly at the step where a forecast is required rather than substituting one.

Step 1 — inputs the reader must supply. None can be inferred.

Step 2 — arithmetic that runs on them, with no forecast.

Step 3 — the range where being wrong is cheap. Under zero leverage the whole admissible range sits on the left branch, so the only available error is underbetting. Against a growth-optimal exposure of 2.28 (§1.2, at the sample premium, an illustration and not a forecast): 40% equity retains 0.32 of peak excess growth, 60% retains 0.45, 80% 0.58, 100% 0.68. Read that as a warning about the premise, not a recommendation — at a forecast premium of 1.2%/yr instead, w* is 0.6 and the table inverts. The gradient is entirely a function of the forecast you are unwilling to make, which is the whole reason the split is not set here.

Step 4 — the drawdown you are signing up for. §5’s table, filtered to the chosen rung, with its as of date, its window, its one-country limitation, and the note that the bond column is modelled.

What the application must not do. Present §1.2’s w* without the premium forecast that produced it. Add the §1.3 ladder edge to the contractual budget — different benchmarks, and the study code raises. Output an equity share from age; §4 is the reason. Describe any of §§5–7 as anything but one historical sample.

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Verified, assumed, open

Verified here. The 1 − (1 − f)**2 identity, cross-checked against an independently written growth_rate_vertex to 1e-15. The two-asset w* and its inverse, against a 20,001-point grid search at twelve parameter combinations. 1/(2T) and f*, against a 400,000-draw seeded simulation. (n−1)/(n−3) likewise. Permutation invariance of terminal wealth without flows, to floating-point precision. The 1963-07…2025-12 market line reproduced from Experiment 007 through a different code path. The Campbell–Pflueger–Viceira sign pattern in its three long eras.

Assumed on this page, and nowhere else.

1. The ten-year bond total return on this page is modelled from GS10, not measured, and as of 2026-08-17 that is a choice rather than a necessity. Goyal–Welch ltr carries 1,200 months of measured long-Treasury total return from 1926-01 and eighteen bond and TIPS ETFs carry investable Item B.5 returns from 2019-09 (evidence base). This page has not been re-run on either, and the two are not interchangeable with the proxy: ltr is a roughly twenty-year exposure against GS10’s ten-year point, they correlate +0.663 over the 750 months both cover, and the proxy is 3.4 pp/yr less volatile and 0.8 pp/yr lower in excess return. Rebuilding §5’s ladder on ltr would make the bond column a different question, not a better answer to the same one — so it is left as it is, labelled, and the substitution is named as open below. The proxy still carries no on-the-run premium, no bid/ask, no tax and no index roll rules. Every bond figure on this page inherits it.
2. Monthly rebalancing in every constant mix. Experiment 003 priced the policy difference at 0.3–1.2 bp/yr in cost and nothing in return, so this is small — but it is one.
3. sigma known and returns Gaussian and independent in §2.1. Both fail in practice, and both failures push f* down, which is the direction §2.1 already argues.
4. An iid permutation null in §3 and §5.1 — deliberate, and it destroys serial dependence.
5. A 30-year horizon in §5.1, CPI-U as the deflator.
6. Nominal, US, pre-tax throughout §§1, 3, 5 and 6. §5.1 alone is real.

Open. What drawdown constraint the objective is subject to — the binding input, and nothing here can supply it. Which estimation window and regime-conditioning scheme the covariance matrix should use, on which §6 is now direct evidence that the question is live. Whether risk-constrained Kelly beats fractional Kelly on real returns. Whether the multi-country bond-stock claim holds. How labour income actually covaries with equities for a given reader.

One item left this list on 2026-08-16. “What a non-US drawdown ladder looks like” is answered in §5 above: sixteen countries are now loaded, and the US ranks 15th of 16 on the full sample and 16th of 16 from 1963. What remains open is the constant-mix ladder on those countries — this page’s rungs are still computed on US data alone, and rebuilding them per country is a study nobody has run.

One item joined it on 2026-08-17. §5’s ladder should be rebuilt on the measured ltr bond leg beside the modelled GS10 one, so that the drawdown a reader is asked to choose from is a drawdown someone could have taken. It is a study, not an acquisition: the series is held. What it will not change is the corner solution, which is set by the equity premium over the safe leg and the zero-leverage rule — and ltr’s realised excess return over 1991-2025 is higher than the proxy’s by 1.76 pp/yr at a higher volatility, so the bonds still come from the constraint.

Reproducibility. cd research && uv run python -m portfolio_edge.studies.equity_share. Equity: Ken French Mkt-RF + RF, 1963-07…2025-12, 750 observations. Cash: the RF column. Bond: FRED GS10, modelled by pricing a semiannual par bond struck at last month’s yield and repricing at this month’s. Inflation: CPIAUCSL, ending two months before the equity series, so §5.1 runs on 748 months. Seed 20260812, 20,000 permutation draws per cell. No experiment was run and no ledger entry was written: this is a study, not an experiment, and it decides nothing a frozen specification would have to adjudicate.

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Consequence for this repository

1. §1.1 of the recommendation keeps its refusal and gains the reason. Not “risk capacity” as an undefined faculty, but a drawdown constraint the objective is explicitly subject to and that nobody has supplied.
2. Variant C should be split. “Under 10 years” and “withdrawals have begun” have opposite answers above a 4% real draw.
3. The bond–stock citation needs tightening. The era boundaries and the three-currency-area attribution are not supported by anything reachable; §6 states what is. What §6 asserts is now also measured here: on Goyal–Welch ltr against Ken French’s market, the correlation to equity spans 0.802 across twelve non-overlapping five-year blocks, seven positive then five negative — compactly, +0.352 to 1998-06 and −0.206 after, on a break date chosen by eye and reported as descriptive. The sign change is real and it is not a citation problem.
4. An application may render §7 and must render it whole. Every input in step 1, the stop-here marker at step 2’s last row, and the drawdown table with its limitations. A calculator that outputs an equity share from an age or a risk questionnaire is the thing this page exists to prevent.
5. Nothing here reopens the zero-leverage rule. The observation that the growth objective would like more than 100% equity is an observation about the objective, not a case for borrowing.
6. studies/equity_share.py is the executable record. Any change to a number here changes a test, by construction.