QuantCalc Methodology

Technical Documentation v1.7 · Last updated: September 2026

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1. Introduction

QuantCalc uses Monte Carlo simulation to model the uncertainty inherent in retirement planning. Unlike deterministic calculators that assume fixed returns, Monte Carlo methods generate thousands of possible future scenarios based on the statistical properties of financial markets.

The fundamental question we answer is: Given your savings, contributions, spending needs, and asset allocation, what is the probability that your portfolio will sustain your retirement?

Real vs nominal dollars

QuantCalc separates three distinct dollar concepts so that the math behind every figure is unambiguous:

• What you enter is in today's dollars. Monthly contribution, annual spending, Social Security, life-event amounts — all interpreted as the purchasing power those amounts have right now. You don't need to project them forward yourself. In Tax-Aware Mode annual spending is what you spend after income tax; otherwise the engine does not model tax and the spending figure is the total withdrawal.
• What the engine simulates is nominal. The published Capital Market Expectations (CME) used for asset returns are nominal (before inflation). Internally, the engine inflates your cash flows forward at your inflation rate so that nominal flows meet nominal returns — the arithmetically consistent way to model purchasing-power growth.
• What you see is deflated back to today's dollars. All percentile bands, projection-chart values, sample paths, and yearly statistics are divided by the cumulative inflation factor at each point so the chart reads in today's purchasing power. If you turn off "adjust for inflation," the engine treats your inputs as fixed nominal amounts and skips the deflation step.

Worked example: $100,000 starting balance, 7% nominal return, 3% inflation, no cash flows, 30-year horizon. Internally the engine grows the portfolio to $100,000 × 1.07³⁰ ≈ $761,225 in nominal dollars. Before display, it divides by (1 + 0.03/12)¹²·³⁰ ≈ 2.457 to give ≈ $309,800 in today's purchasing power — the figure you'll see on the chart. The 4% real return implied by the inputs (7% nominal − 3% inflation, approximately) is what's actually visible.

For stochastic-inflation models (AR(1), regime-switching, AR(1) multi-category), each simulation path has its own inflation trajectory, and each path is deflated by its own per-path cumulative CPI factor before percentiles are computed. The deflation is correct path-by-path, not just on average.

Cash-flow timing convention

The simulation runs in monthly steps. Cash flows you enter as annual amounts are divided by 12 and applied at the end of each month, after that month's growth. So a $12,000/year contribution becomes 12 separate $1,000 cash flows: the first lands at the end of month 1 and earns 11 more months of growth before "year 1" on the chart; the last lands at the end of month 12 and earns no further growth in year 1. The effective center of mass of each year's cash flow is roughly mid-year.

This is the institutional convention — it matches how paychecks, bills, and retirement withdrawals actually move through real life. It is not the convention used by typical textbook PMT/FV/PV formulas in Excel or Google Sheets, which assume one annual lump sum either at year-start (annuity due) or year-end (ordinary annuity). When you compare a QuantCalc result against a hand-built annual spreadsheet, the gap is real and predictable:

If you build a verification spreadsheet to sanity-check QuantCalc, you'll see these gaps. They are not arithmetic errors in either model — both are internally consistent. They reflect the difference between annual-lump and monthly-uniform cash-flow timing. To match a QuantCalc number with an annual spreadsheet exactly, your spreadsheet would need to compound monthly and place 1/12 of the annual cash flow at the end of every month.

2. Monte Carlo Simulation Overview

Monte Carlo simulation is a computational technique that uses repeated random sampling to obtain numerical results. For retirement planning, we simulate many possible "paths" that a portfolio might take over time.

The Basic Algorithm

For each simulation s = 1, 2, ..., N:
    Initialize portfolio value V₀
    For each time period t = 1, 2, ..., T:
        Generate random asset returns
        Update portfolio value based on returns and cash flows
        Record portfolio value Vₜ
    Record final outcome (success or failure)

Calculate statistics across all N simulations

With N = 1,000 to 10,000 simulations, we obtain a distribution of outcomes that reflects the range of possibilities given market uncertainty.

3. Asset Return Modeling

Log-Normal Returns

Financial returns are commonly modeled as log-normally distributed. If $P_t$ is the price of an asset at time $t$, the continuously compounded return over one period is:

We assume these log-returns follow a normal distribution:

where $\mu$ is the expected (mean) return per period and $\sigma$ is the volatility (standard deviation) per period.

Converting Annual to Monthly Parameters

Capital Market Expectations are typically quoted as annual figures. For monthly simulation:

The division by $\sqrt{12}$ for volatility follows from the property that variance scales linearly with time for independent returns.

Growth Factor

Given a monthly return $r_t$, the portfolio growth factor is:

4. Generating Correlated Returns

The Challenge

Real asset classes don't move independently. Stocks tend to move together, bonds often move inversely to stocks during crises, etc. We must generate returns that respect these correlations.

Correlation Matrix

For $n$ asset classes, the correlation matrix $\mathbf{R}$ is an $n \times n$ symmetric matrix where $R_{ij}$ is the correlation between assets $i$ and $j$:

Covariance Matrix

The covariance matrix $\mathbf{\Sigma}$ combines volatilities and correlations:

Or in matrix form: $\mathbf{\Sigma} = \mathbf{D} \cdot \mathbf{R} \cdot \mathbf{D}$, where $\mathbf{D} = \text{diag}(\sigma_1, \sigma_2, \ldots, \sigma_n)$.

Cholesky Decomposition

To generate correlated random variables, we use the Cholesky decomposition. Any positive-definite covariance matrix can be factored as:

where $\mathbf{L}$ is a lower triangular matrix.

5. Capital Market Expectations

QuantCalc references return assumptions derived from publicly available publications by major asset managers. QuantCalc is not affiliated with any of these firms. The data is sourced from publicly available reports, press releases, and financial media coverage.

Capital Market Expectations (CME) are forward-looking estimates of expected returns, volatilities, and correlations for each asset class. Major asset managers publish CMEs annually, and these forecasts are widely reported in public financial media.

QuantCalc's Asset Classes

Source mapping: Each firm publishes forecasts at varying levels of granularity. Where a source provides separate forecasts for US Large Cap, Mid Cap, and Small Cap equities, QuantCalc uses the Large Cap figure for "US Stocks" as it most closely represents the S&P 500 universe. Similarly, "Bonds" uses investment-grade aggregate or core bond forecasts, and "International Stocks" uses developed-market ex-US figures. For exact sub-category details, consult each firm's published report linked below.

Volatility estimates: Most firms do not publicly publish volatility estimates for exactly QuantCalc's 5 asset classes. Where a source publishes volatility data (e.g., J.P. Morgan's LTCMA includes volatility estimates in its full report, checked 2026-09-10), those figures inform our estimates. For other sources, volatilities are representative estimates based on long-term historical realized volatility for each asset class. Typical ranges: US equities 15-17%, international equities 17.5-20%, bonds 5-6.5%, real estate 13-16%, cash ~1%.

Correlation estimates: Published correlation matrices vary by source and methodology. J.P. Morgan's LTCMA includes a comprehensive correlation matrix in their full report. For sources that do not publish complete correlation matrices for these specific asset classes, QuantCalc uses representative estimates consistent with long-term empirical relationships (e.g., US/Intl equity correlation of 0.75-0.85, equity/bond correlation of 0.05-0.15, as documented in Dimson, Marsh & Staunton and other academic research). All correlation matrices are verified to be positive semi-definite for valid Cholesky decomposition.

Real vs. nominal returns: Most sources publish nominal returns (before adjusting for inflation). GMO is a notable exception — it publishes real returns (after inflation); as of 2025-12-31 its US large-cap forecast is −6.0% real (gmo.com, checked 2026-09-10). QuantCalc converts GMO's real returns to nominal by adding an assumed 2.5% long-term inflation rate, as noted in the GMO source description below. This enables consistent comparison across all sources.

Arithmetic vs. geometric return conventions: Published expected returns come in two conventions. Forward-looking firm forecasts (BlackRock, J.P. Morgan, Vanguard, GMO, Schwab, Invesco — each firm's own publication is linked in the source list below, and the set is restated in Morningstar's 2026 roundup, checked 2026-09-10) are geometric (compound annual) expectations — the long-run growth rate a buy-and-hold investor would experience. Long-run historical averages, by contrast, are conventionally quoted as arithmetic means of annual returns, which for a volatile asset overstate the achievable compound growth rate by approximately σ²/2 (about 1.9 percentage points for an equity series with 19.5% volatility). QuantCalc tags every source with its convention and converts arithmetic inputs to geometric (μ_geometric ≈ μ_arithmetic − σ²/2) before they enter the lognormal return-generation step, so the Historical source no longer compounds an arithmetic mean as if it were a growth rate — a subtle convention mismatch that roughly doubled long-horizon Historical wealth projections before being corrected in May 2026. All sources therefore simulate on a consistent geometric basis, and Historical-source results are directly comparable with the forward-looking forecasts.

Independent Source Selection

QuantCalc allows you to select separate sources for each of the three CME components:

• Return Expectations: The expected annual return for each asset class. All 8 sources are available since every source publishes return estimates.
• Correlation Matrix: The pairwise correlations between asset classes. Only sources that publish verified correlation data are available (currently JPMorgan LTCMA, Historical, and Equilibrium).
• Volatility: The expected annual standard deviation of returns. Only sources that publish volatility estimates are available (currently JPMorgan, Vanguard, Historical, and Equilibrium).

This separation is important because different firms have different strengths. For example, you might prefer GMO's bearish return outlook combined with JPMorgan's well-documented correlation structure and historical volatility estimates. The backend automatically recomputes the Cholesky decomposition of the correlation matrix whenever sources are mixed, ensuring mathematically consistent correlated return generation.

By default, QuantCalc uses JPMorgan for returns and correlations (their LTCMA is the most comprehensive publicly available dataset) and Historical for volatility (long-term realized volatility is a robust baseline).

Near-Term and Long-Term Assumptions PRO

A single CME set applies one view of the world to the entire horizon, which is not how the published sources are written. GMO's headline numbers are a 7-year forecast; BlackRock's and Vanguard's are 10-year; JPMorgan's and Schwab's are explicitly long-term. Applying a seven-year forecast unchanged across thirty years is arguably the misuse, not the alternative.

QuantCalc therefore accepts an optional second set: a near-term source governs the first N years and the long-term set governs the remainder. Both slots take real published assumptions — no near-term view is interpolated or synthesised, because QuantCalc does not produce its own forecasts and manufacturing one here would break that. Only sources whose own published horizon is short are offered in the near-term slot, and the default length follows that source's stated horizon (GMO defaults to 7 years, BlackRock and Vanguard to 10), so a forecast is applied over the period it was written for. Each set keeps its own returns, volatilities and correlation matrix, and the engine switches between two fully-prepared sets at the boundary month.

The split is refused — visibly, with the reason reported in the results panel — in four cases: when regime switching is active, because its volatility correction is computed from a single set and the near-term months would otherwise silently carry long-term volatilities; when the return model is the bootstrap or historical sequences, which read no capital market expectations at all; when the named source is not one we carry; and when the requested length falls outside 1–50 years. A near-term period longer than the plan itself is treated as a single set rather than pretending there is a second phase.

Example CME Data (Illustrative — BlackRock, as of Sep 2025)

CME Sources

QuantCalc references Capital Market Expectations from six publicly available sources, each with distinct methodological approaches:

Historical Bootstrap PRO

As an alternative to the parametric CME framework above, PRO users can switch to a historical bootstrap return model that resamples actual monthly history instead of drawing from a fitted log-normal distribution. Block lengths follow a geometric distribution averaging 24 months (a Politis–Romano stationary block bootstrap), and all five asset classes are resampled jointly — the same historical month drives every asset at once — so real correlation, volatility clustering, and regime sequencing carry through directly rather than being approximated.

The dataset spans July 1990 through May 2026 (431 months): US and developed ex-US market returns and the listed real-estate industry portfolio (REIT-dominated in the modern era) from the Ken French Data Library, par-bond returns derived from FRED's 10-year constant-maturity yield, and cash returns from FRED's 3-month T-bill yield. Two notes on what those series are. The real-estate series is an industry portfolio, so it includes real-estate operating companies as well as REITs and runs at about 26% annualized volatility against roughly 19–20% for a REIT index — real-estate risk is overstated relative to a pure REIT allocation. And returns are nominal: they are deflated by a CPI path drawn from QuantCalc's inflation model, not by the inflation that actually accompanied each resampled month (see the coupling note below).

Window bias — read bootstrap success rates as an optimistic bound. The 1990 start excludes the 1970s stagflation era, and the direction of that omission is one-sided. Within this sample a 60/40 US-equity/bond mix compounded about 8.7%/yr nominal, US equity 11.16%/yr and bonds 5.04%/yr — and the bond figure is the average of two very different halves, 7.67% annualized in 1990–2007 against 2.69% in 2008–2026, so roughly a third of the window is the tail of a bond bull market that cannot repeat from today's yields. Every forward-looking capital market expectation in the parametric model is materially lower. Switching a plan from parametric to Historical Bootstrap therefore tends to raise its success rate for reasons that are a property of the window, not evidence about the future. A related caveat on precision: 431 months at a 24-month mean block length is only about 18 effectively independent segments, so the Wilson interval on the success rate (Section 26) covers sampling error within the resampled distribution and captures none of that window uncertainty — which dominates here.

Fat tails and regime clustering are already embedded in the resampled history, so those parametric add-ons don't apply here. Return–inflation coupling is a different case: because the resampled series is nominal and the CPI path is drawn independently, the coupling is absent under this model rather than embedded — the historical link between a month's returns and its inflation is not carried through. The portfolio optimizer and the source-comparison view are parametric-only and reject this model rather than substituting one silently: optimizing an allocation against a resample of a single 36-year window maximizes performance inside that window, and comparing capital market expectations is meaningless for a model that does not read them. Sobol scrambling is likewise inert here — the resampler draws historical months directly, so there is no low-discrepancy sequence to rotate, and the run uses plain pseudo-random draws (the results panel reports which sampler actually ran).

Historical Sequences PRO

Where the bootstrap above resamples history in blocks, this model replays it. Simulation k is the actual run of months beginning at start month k and continuing in order, so sequence-of-returns risk is whatever it really was: 1929–32, 1966–82 and 2008 arrive in their true order, at their true length, with their true recoveries, rather than being reassembled from geometric blocks.

That requires a longer dataset than the bootstrap uses. The bootstrap panel begins in July 1990 because Ken French's developed ex-US series starts there and the listed real-estate series only reaches 1972 — fine for resampling short runs, useless for contiguous 30-year sequences: over 1990–2026 a 30-year horizon admits only about 83 heavily overlapping start months, roughly 7 distinct start years, and excludes every period that makes historical simulation informative in the first place. This model therefore runs on a separate panel that trades breadth for length: 1,178 contiguous months, August 1926 through September 2024, giving 819 distinct start months at a 30-year horizon and 699 at 40 years.

The price of that length is asset coverage. The panel carries three series — US equity and cash from the Ken French Data Library (market return including the risk-free rate, and the one-month Treasury bill respectively), and a par-bond total return derived from Shiller's monthly long-term yield by the same approximation used for the bootstrap panel's bond series. International equity and real estate are absent, not proxied: no public monthly series for either reaches 1926, and substituting an invented stand-in would fabricate the very data this model exists to respect. A five-asset allocation is folded onto the three — international and real estate join US equity, because both are equity-risk assets and dropping their weight would silently run a smaller portfolio than the client holds — and the results panel states that the folding happened.

Read these success rates against forecasts, not as forecasts. This model scores higher than the parametric one, and not because the plan is safer. Realized US equity compounded at 10.21%/yr nominal across this window; every forward-looking capital market expectation on offer here sits well below that — the J.P. Morgan Long-Term Capital Market Assumptions figure carried in this app for US equity is 6.7%. That gap is roughly three and a half points a year, which over a thirty-year horizon is the whole difference. The results panel reports the realized figure next to the number for exactly this reason. The same one-sided caution that applies to the bootstrap window applies here with a longer lever.

A second precision caveat: the 819 start months of a 30-year horizon overlap almost entirely, so the effective independent sample is one or two non-overlapping periods, not 819. The panel therefore reports the count of distinct start months rather than a path count — quoting “10,000 paths” would imply 10,000 independent draws from a distribution when nothing is being drawn at all. For the same reason the Wilson interval of Section 26 is not meaningful under this model, and Sobol sampling is inert: there is no low-discrepancy sequence to rotate, because nothing is sampled.

As with the bootstrap, the parametric overlays are disabled here — real history already contains its own regimes, fat tails and crashes, so layering synthetic ones on top would double-count them. An inline custom CME forces a downgrade to the parametric engine (the panel is a fixed three-asset set with no correspondence to an arbitrary roster), and the response echoes the model that actually ran rather than the one requested.

6. Portfolio Return Calculation

Weighted Portfolio Return

Given asset allocation weights $w_1, w_2, \ldots, w_n$ (summing to 1) and individual asset returns $r_1, r_2, \ldots, r_n$, the portfolio return is:

Portfolio Statistics

The expected portfolio return: $\mathbb{E}[r_p] = \sum_{i=1}^{n} w_i \cdot \mu_i$

The portfolio variance:

7. Cash Flow Modeling

Time Periods

QuantCalc simulates on a monthly basis from current age to end of plan. Total months:

Cash Flow Components

At each month $t$, the net cash flow $C_t$ consists of:

Pre-Retirement Phase

Before retirement (month $t < t_{\text{retire}}$): $C_t = M_{\text{contribution}} \times I_t$

Post-Retirement Phase

After retirement (month $t \geq t_{\text{retire}}$):

Income tax does not appear in $C_t$: the standard engine does not model it, so $M_{\text{withdrawal}}$ is a total withdrawal. Tax-Aware Mode (Part III) replaces this identity with $W_t - \text{tax}_t = S_t$.

Social Security and Pension

Social Security begins at the specified claiming age:

Supplemental income streams are opt-in. Social Security and pension income are each zero unless you enter an amount — the engine does not assume a benefit on your behalf. A simulation with no Social Security and no pension entered draws spending entirely from the portfolio, which is the conservative baseline for an early-retirement or FIRE plan. To include a benefit, enter its annual amount (today's dollars) and the age it begins; it is then inflated forward like every other cash flow.

Inflation Adjustment

Cash flows are adjusted for inflation to maintain purchasing power:

8. Simulation Execution

Portfolio Evolution

The portfolio value evolves according to:

where $V_t$ is portfolio value, $G_t = 1 + r_{p,t}$ is the growth factor, and $C_t$ is net cash flow.

Handling Ruin

If the portfolio value goes negative, it is floored at zero:

Once $V_t = 0$ during retirement, the simulation is marked as a "failure" (ruin).

9. Success Rate and Statistics

Success Rate

A simulation is "successful" if the portfolio never hits zero during retirement:

Percentiles

• 10th percentile: "Pessimistic" outcome (90% did better)
• 50th percentile (median): "Typical" outcome
• 90th percentile: "Optimistic" outcome

Confidence Interval

The standard error of the success rate estimate is:

For $p = 0.95$ and $N = 1000$: $SE \approx 0.007$, so the 95% CI is approximately $\pm 1.4\%$.

10. Quasi-Monte Carlo Methods

Sobol Sequences

Standard Monte Carlo uses pseudo-random numbers which can exhibit clustering. Quasi-Monte Carlo (QMC) methods use deterministic low-discrepancy sequences designed to fill the sample space more uniformly.

QuantCalc uses Sobol sequences, which provide:

• Deterministic, reproducible results
• Better space coverage than pseudo-random
• Faster convergence: $O(1/N)$ vs $O(1/\sqrt{N})$

Transformation to Normal

Sobol points $u \in [0,1]$ are transformed to standard normal via the inverse CDF:

11. Optimization Problem Formulation

QuantCalc's portfolio optimizer finds the asset allocation that maximizes retirement success probability (or other objectives) subject to real-world constraints. Unlike simple mean-variance optimization, our approach directly optimizes the Monte Carlo simulation output—accounting for sequence-of-returns risk, cash flows, and the full distribution of outcomes.

General Form

Decision Variables

Single-period: $\mathbf{w} = (w_{\text{US}}, w_{\text{Intl}}, w_{\text{Bonds}}, w_{\text{RE}}, w_{\text{Cash}})^T$ — 5 variables

Multi-period (glide path): For $k$ periods, we optimize $k \times n$ weights:

12. Objective Functions

Maximize Success Rate

Maximize Median Final Value

13. Constraints

Equality: For each period $p$: $\sum_{i=1}^{n} w_{p,i} = 1$

Bounds: $0 \leq w_{p,i} \leq 1$ for all weights (no short selling)

14. Sequential Quadratic Programming (SLSQP)

We use SLSQP because it handles exactly the constraint structure we need: equality constraints (weights sum to 1), bound constraints (0–100% per asset), and a smooth but simulation-derived objective.

The Quadratic Subproblem

At each iteration $k$:

Gradient Estimation

We estimate gradients numerically using finite differences:

A fixed random seed is used for all objective evaluations to eliminate stochastic noise.

BFGS Hessian Update

15. Multi-Period Optimization

For $k$ periods, we flatten the weight matrix into a single decision vector:

16. Initial Point Selection

Strategy 1: User's current allocation extended to all periods

Strategy 2: Balanced portfolio: $w_{p,i}^{(0)} = 1/n$ (20% each)

Strategy 3: Age-based heuristic: $w_{\text{stocks}}^{(0)} = 1 - \text{age}/100$

17. Convergence and Termination

The optimization terminates when any condition is met:

• Gradient tolerance: $\|\nabla f\| < 10^{-4}$
• Step tolerance: $\|\mathbf{w}^{(k+1)} - \mathbf{w}^{(k)}\| < 10^{-6}$
• Function tolerance: $|f^{(k+1)} - f^{(k)}| < 10^{-6}$
• Maximum iterations: $k > 100$

18. Practical Considerations

Simulation Noise

A fixed random seed is used for all objective evaluations within the optimization loop. The final optimized allocation is then validated with a fresh simulation run.

Computational Cost

Flat Regions

When success rate is very high (>95%) or very low (<5%), the objective surface becomes flat. Switching to median value objective can help.

19. Tax-Aware Withdrawal Optimization

QuantCalc's Tax-Aware Mode chooses the withdrawal policy across traditional, Roth and taxable accounts that minimizes lifetime tax cost. It simulates a set of whole-retirement candidates on the same market paths — traditional first, taxable first, filling the standard deduction or the top of the 10%, 12%, 22% or 24% bracket (or the first Medicare surcharge tier) from traditional each year before touching taxable and Roth, the same bracket and tier fills completed with a Roth conversion for any room the year's withdrawals leave (from retirement until the year before required distributions begin), and a "convert to the 10% or 12% top, then spend traditional first" pair — and recommends the one with the lowest present value of lifetime tax: every year's federal, state, Medicare-surcharge and penalty cost discounted at the portfolio's own return on that path, plus any spending the portfolio could no longer fund on paths where it runs out, plus the tax still owed on any traditional balance left at the end, valued as an heir drawing it evenly over ten years as a single filer with no other income in the plan's state. A single-year grid search that minimizes only that year’s bill is not among them: it was removed from the search in September 2026 because it lost to drawing traditional first on every realistic long plan we tested. It survives only as a forced comparison in the withdrawal-order study, which reports how often it would have helped or backfired. Spending is after tax: in every retirement year the engine withdraws enough to fund the year's spending plus that year's whole tax bill, and pays the bill from the accounts it drew — so the success rate, the ending balances and the recommendation all depend on your state and your withdrawal order. Because drawing traditional first is itself one of the candidates, the recommended policy can only tie it or beat it; the reported saving is the median, over paths, of the per-path present-value difference. A winner whose median saving is under $1,000 is treated as not a real recommendation and traditional-first is used instead, so the reported saving is then exactly $0; a bracket- or tier-fill candidate that recommends a Roth conversion is held to the same $1,000 bar against its own non-converting twin (the same fill with no conversion), so a household is never shown a multi-year conversion schedule that only edges out doing nothing by a few dollars. If you enter a Roth-conversion plan of your own, the conversion candidates are not used: every candidate order is simulated around your plan and the results say so. The pre-Medicare Roth Conversion Planner is a separate joint optimization of conversions and income that also prices ACA premium subsidies, healthcare costs and the 400% FPL cliff; Tax-Aware Mode's conversion candidates model no ACA subsidy.

How the advantage is explained. The two lifetime-tax figures the app shows are each the median of one order's own per-path tax paid, in today's dollars; the saving is the median of the per-path present-value difference. A median of a difference is not the difference of two medians, and the two are in different bases, so the app does not explain the saving from those figures. It takes the single simulated market history whose present-value advantage equals the reported median and reports that history's own parts — the present value of tax paid along the way, of tax still owed on any traditional balance at the end, and of any spending the portfolio could not fund — which sum exactly to the reported saving, alongside the same tax paid in today's dollars. When that history's tax is larger in today's dollars but smaller in present value, its tax fell later, and the app says so; when the advantage comes from less tax owed at the end, it says that instead. If that history's difference in tax paid lies outside the middle 80% of histories, the app says so as well.

The Tax Problem in Retirement

Retirees typically hold assets across three account types, each with different tax treatment:

Choosing which accounts to withdraw from—and in what amounts—directly impacts:

• Federal income tax (10% to 37% marginal rates)
• Capital gains tax (0%, 15%, or 20% depending on income)
• IRMAA surcharges (Medicare Part B/D premium increases based on MAGI)
• ACA subsidy eligibility (for early retirees under age 65)
• Early withdrawal penalties (10% penalty if withdrawing before age 59½ from traditional accounts)

No single fixed sequence is best for every household: which accounts to draw first depends on the horizon, the size of the traditional balance relative to required distributions, Social Security timing and the state. The candidate orders above span the choices a planner would actually consider; the engine measures each on the household's own paths rather than assuming one.

Account Structure

The optimizer tracks three account balances:

For the taxable account, we also track the cost basis ratio:

This determines what fraction of a taxable withdrawal is principal (not taxed) versus capital gains (taxed at preferential rates). The single cost-basis ratio is the default model, and is described first because it is what most plans use. The engine internally represents it as a single tax lot; the lot-level model below is a strict generalization that reduces to it exactly when there is one lot.

Taxable Account: Per-Lot Cost-Basis Accounting

Rather than carry a single blended cost-basis ratio for the whole taxable account, the engine tracks the taxable account as a ledger of individual tax lots, each with its own dollar value and cost basis. This matters because a real taxable account is not homogeneous: shares bought at different times have different embedded gains, and which shares you sell changes the tax bill.

The ledger evolves over each simulated path:

• Growth applies to the whole ledger through a single shared growth index, so a lot's current value is its units times that index. Growth never changes the ordering of lots by cost-basis-per-dollar, which keeps the per-month update inexpensive.
• Simulation-created purchases become new lots. After-tax money that flows into the taxable account during the run — reinvested required-minimum-distribution excess, or a positive life-event windfall — is added as a fresh lot whose basis equals the full amount deposited (new money is not phantom gain), tagged with the month it was purchased.
• Sales use specific identification, highest-basis-first (HIFO). When the optimizer draws from the taxable account, it sells the highest-cost-basis lots first, which realizes the smallest gain (or harvests the largest loss) for a given dollar of proceeds — the standard tax-aware lot-selection order.

Short-term vs. long-term character

Each sale carries the correct holding-period character. A lot sold less than 12 months after it was purchased realizes a short-term gain; a lot held 12 months or longer realizes a long-term gain. The distinction is not cosmetic:

• Long-term gains are taxed at the preferential 0% / 15% / 20% capital-gains rates (stacked on top of ordinary income) and flow into the Net Investment Income Tax base.
• Short-term gains are taxed as ordinary income at your marginal bracket, and are likewise included in net investment income for the 3.8% surtax.

Because the initial holdings and user-supplied lots are treated as long-term, short-term character arises only on money the simulation itself deposited and then sold within a year — an uncommon but correctly-priced case.

Optional custom lots PRO

By default the initial taxable balance becomes a single long-term lot whose basis is taxableBalance × costBasisRatio — the same information the single-ratio model used, expressed as one lot. PRO users who know their actual holdings can instead supply up to 8 custom lots, each a {current value, cost basis} pair. The supplied values must sum to the taxable balance (checked to within about 0.1% of the balance, with a $1 floor; otherwise the request is rejected with a clear message); when custom lots are provided the single cost-basis ratio is ignored. All custom lots are treated as long-term. The response reports which mode was used and how many lots are tracked, so the assumptions are visible rather than hidden.

Withdrawal Decision Variables

In each year, the optimizer chooses amounts to withdraw from each account:

where $W_t$ is the withdrawal that funds year $t$'s spending and its tax: the smallest amount whose own split under the policy satisfies $W_t - C(\mathbf{w}_t) = S_t$, with $S_t$ the year's spending net of Social Security and pension (a bracketed solve to within $1; when a required minimum distribution exceeds it, the after-tax excess is deposited in the taxable account).

Tax Cost Function

The total tax cost for a given withdrawal mix is:

Every term is evaluated for each candidate mix, and the optimizer selects the mix with the lowest total for that year. In addition to the federal income, capital-gains and penalty terms below, the objective includes the state income tax for the selected state (Section 25), the Net Investment Income Tax (the 3.8% surtax under §1411, levied on long- and short-term gains once modified AGI clears its threshold), and the IRMAA Medicare surcharge. Short-term gains realized from the taxable lot ledger are added to ordinary income before the income-tax term is computed.

Conversion candidates

Six of the whole-retirement candidates pair a bracket or Medicare-surcharge-tier fill with a Roth conversion. Within a candidate's year, the withdrawal is sized first, with no conversion income; the conversion is then sized on the full year so that withdrawal income plus the conversion lands exactly on the fill target, capped so the withdrawal itself stays fundable from what remains in the traditional account. Conversions never include RMD dollars — the required distribution is taken first and is not itself convertible. The window runs from retirement (or today, if already retired) through the year before the plan's required-distribution age begins; once RMDs start, the required dollars already occupy the low brackets, so later conversions were not evaluated for this branch.

The tax on a recommended conversion is paid the same way a real conversion is funded: from the taxable account first, where the sale itself realizes and pays tax on its own gain, and otherwise withheld from the converted dollars, in which case the withheld amount is treated as an early withdrawal and can carry the 10% additional tax under age 59½.

Every dollar of tax is paid from the accounts in the year it is owed. A Roth conversion pays its own marginal tax from the taxable account first (the funding sale's gain is itself taxed) and otherwise from the converted dollars, with the 10% additional tax under age 59½ withheld as well; the rest of the year's bill is drawn through the policy's own withdrawal order — "traditional to the top of the 12% bracket, then taxable, then Roth" pays its tax bill from taxable once traditional has reached the bracket top. Because nothing is left undebited, the present-value comparison between candidates is exact on solvent paths, and no allowance for an unmodelled funding account is needed.

Income Tax Component

Ordinary income includes traditional IRA withdrawals, Social Security (if applicable), pensions, and interest/dividends from the taxable account:

Federal income tax is computed using the progressive bracket structure. For married filing jointly in 2026 (indexed for inflation):

Capital Gains Component

When withdrawing from a taxable account, only the appreciated portion is subject to capital gains tax:

Long-term capital gains rates (for assets held >1 year) are preferential:

Notably, the 0% capital gains bracket creates an opportunity: if ordinary income is low enough, retirees can sell appreciated assets and pay zero tax on the gains.

IRMAA Surcharges

Medicare Part B and Part D premiums increase for high-income beneficiaries based on Modified Adjusted Gross Income (MAGI) from two years prior. For 2026 (based on 2024 MAGI):

IRMAA creates income cliffs: earning $1 over a threshold can cost thousands in additional premiums.

Early Withdrawal Penalty

Traditional IRA/401k withdrawals before age 59½ incur a 10% penalty (with certain exceptions like Rule 72(t) SEPP):

Required Minimum Distributions (RMDs)

Starting at age 73 (for those born 1951-1959), the IRS requires minimum annual withdrawals from traditional accounts based on the Uniform Lifetime Table:

The optimizer enforces: $w_{\text{trad}} \geq \text{RMD}_t$ when RMDs apply. Failing to take RMDs results in a 25% penalty on the shortfall (recently reduced from 50%). The RMD start age is birth-year dependent under SECURE 2.0: age 73 for those born 1951–1959 and age 75 for those born 1960 or later, derived from the plan's calendar base year rather than hard-coded, so conversion and IRMAA timing line up with the correct first-RMD year.

Standard Deduction and Senior Deductions

Ordinary taxable income is computed after the standard deduction for the filing status, plus two age-based additions once a filer reaches 65:

• Additional standard deduction for the aged (IRC §63(f)). A set add-on to the standard deduction for each filer 65 or older ($1,650 per qualifying married filer, $2,050 unmarried, 2026 amounts). On a joint return it is granted per spouse who is 65+, so a couple where both are 65+ receives it twice. It has no income phase-out.
• OBBBA senior deduction (§70103). A separate $6,000-per-qualifying-individual deduction for filers 65+, available for tax years 2025 through 2028 only, and unavailable to married-filing-separately filers. It phases out at 6% of modified AGI above $75,000 (single / head of household) or $150,000 (joint), applied to each qualifying individual's amount — so a joint return with both spouses 65+ loses $0.12 of deduction per dollar over the threshold and reaches zero at $250,000 of joint MAGI. Because the phase-out depends on MAGI, the deduction is applied after the year's income is known.

Yearly Cash-Flow Sequencing

Within each simulated year the engine resolves the interacting tax events in a set order so that later steps see the correct income base:

1. Required minimum distribution first. The RMD floor is computed and, where a Qualified Charitable Distribution is specified, the QCD is applied against that floor (a QCD satisfies the RMD without adding to taxable income). Any RMD beyond the QCD is a taxable distribution.
2. Roth conversion. A conversion is ordinary income in the year it occurs. Conversions run only inside their configured age window, which typically ends before RMDs begin.
3. Withdrawal under the policy's order funds the remaining spending need plus the year's tax (the gross-up above).
4. Conversion tax funded from real dollars. The additional tax caused by a Roth conversion is paid with actual money drawn from the taxable account, not conjured for free. The engine attributes this cost marginally — the tax with the conversion minus the tax without it — and debits the taxable ledger accordingly (a HIFO sale), so the conversion's true after-tax drag is reflected.
5. Reinvestment of RMD excess. When a required distribution forces more out of the traditional account than spending and tax require, the after-tax excess is not discarded — it is deposited into the taxable account as a new full-basis lot.

IRMAA Two-Year MAGI Lookback

Medicare's income-related surcharge for a given premium year is set from modified AGI two years earlier. The simulation carries this lag explicitly: each year's surcharge is driven by the MAGI recorded two years prior (a short ring buffer of recent MAGI values), rather than the current year's income. In the first years of Medicare, before two years of simulated history exist, the current-year MAGI is used. Married-filing-separately filers use the compressed IRMAA schedule (a steeper ladder with two surcharge steps rather than the five surcharge steps on joint or single returns), and on a joint return where both spouses are Medicare-age the surcharge is levied on each spouse at the couple's joint MAGI — effectively doubling it.

Tax-Exempt (Municipal) Interest

Tax-exempt interest is optional annual income that is deliberately treated the way the tax code treats it — it counts in exactly the places it legally counts, and nowhere else. It is added to the provisional-income base that determines how much Social Security is taxable, and to the modified AGI used for IRMAA, but it is not ordinary taxable income and is not part of the Net Investment Income Tax base. Like pension and Social Security amounts, it is entered in today's dollars and inflated forward.

Social Security Taxation (Provisional Income)

The taxable portion of Social Security is computed with the provisional-income tiers of IRS Publication 915: up to 85% of benefits become taxable as provisional income (AGI excluding Social Security, plus tax-exempt interest, plus half of benefits) rises through the tier thresholds. Married-filing-separately filers who lived apart from their spouse for the entire year use the single thresholds ($25,000 / $34,000); otherwise the separate-filer thresholds are $0 / $0, so their benefits are taxable from the first dollar. This lived-apart distinction is an optional input.

The Retired Single-Year Grid Search (comparison only, never recommended)

Until September 2026 QuantCalc chose each year’s mix with a brute-force grid search over that year’s withdrawal splits. It is described here because the withdrawal-order study still scores it as a comparison, and because results published before that date were produced by it. It is no longer a candidate in any projection. The search space was discretized as:

• Divide total withdrawal $W_t$ into increments of 10%
• Enumerate all combinations: $(w_{\text{trad}}, w_{\text{roth}}, w_{\text{tax}})$ where each is a multiple of $0.1 \times W_t$
• Subject to: $w_{\text{trad}} + w_{\text{roth}} + w_{\text{tax}} = W_t$ and $w_{\text{trad}} \geq \text{RMD}_t$

This produces 66 unique combinations per year. For each combination, we compute total tax cost $C(\mathbf{w}_t)$ and select the minimum.

Multi-Year Simulation

Each Monte Carlo simulation path is run once per candidate policy, all on the same market draws, and the candidate with the lowest present value of lifetime tax is re-run as the primary so every chart and table reflects the recommendation:

For each candidate order k:
    For each simulation path s = 1..N (identical paths for every k):
        Initialize accounts (B_trad, B_roth, B_tax), the tax-lot ledger, the MAGI ring
        For each year t = 1..T:
            Apply returns; apply contributions and life events
            Resolve spending (fixed real, or the guardrail / VPW rule on this candidate's own balances)
            QCD -> RMD floor -> Roth conversion -> conversion-tax funding -> withdrawal under order k grossed up so that W - tax = spending -> surplus deposit / shortfall record
            PV_k[s] += (tax_t + IRMAA_t + unfunded_t) / G_t          (G_t = portfolio growth from today to year t on path s)
        PV_k[s] += heir tax on ending traditional / G_T
    score_k = median_s ( PV_baseline[s] - PV_k[s] )       (baseline = traditional first; score_baseline = 0)
Recommend k* = argmax_k score_k; re-run k* as the primary strategy for every chart and table

Metrics Reported

Tax-Aware Mode adds these outputs:

• Lifetime tax paid: Median tax paid under the recommended order, today's dollars
• Traditional-first lifetime tax paid: The same, for the traditional-first baseline
• Tax still owed at the end: The heir valuation (Section on Conversion candidates' heir assumptions), for both the recommended order and traditional-first
• Lifetime tax saved: Present value, paired per path against traditional-first, never negative
• Recommended order: The candidate policy chosen, named
• Effective tax rate: Total taxes / Total withdrawals (withdrawals include the tax they fund)
• Account depletion sequence: Which accounts run out first
• Unfunded spending: Median spending and tax the portfolio could not fund, today's dollars — zero when the median path stays solvent

Practical Impact

How much the order is worth depends on the household. Measured results for nine households and a horizon sweep, with the recommended order for each, are published in When Withdrawal Order Helps, and When It Backfires (2026).

• Staying in lower tax brackets by mixing Roth withdrawals
• Harvesting capital gains at 0% when income allows
• Avoiding IRMAA cliffs
• Strategic Roth conversions in low-income years
• Drawing traditional first while brackets are low when a large traditional balance would otherwise be forced out later at higher rates and with Medicare surcharges

20. ACA Subsidy Cliff Management

For early retirees (ages 55-64, before Medicare eligibility), the Affordable Care Act (ACA) subsidy cliff represents one of the most significant financial risks in retirement planning. The cliff creates a scenario where earning just $1 of additional income can eliminate $15,000 to $30,000 in annual health insurance subsidies—an effective marginal tax rate exceeding 1,000%.

QuantCalc's Tax-Aware Mode includes ACA cliff detection and optimization to help early retirees navigate this challenge.

What is the ACA Subsidy Cliff?

The ACA provides premium tax credits (subsidies) to reduce the cost of marketplace health insurance for individuals and families below certain income thresholds. Prior to 2026, enhanced subsidies phased out gradually. Starting in 2026, the subsidies revert to the original ACA structure with a hard cutoff at 400% of the Federal Poverty Level (FPL).

For 2026, the cliff occurs at:

If Modified Adjusted Gross Income (MAGI) exceeds the threshold by even $1, the entire subsidy disappears. There is no gradual phase-out.

Subsidy Calculation

ACA subsidies are calculated as the difference between the "benchmark plan" premium (second-lowest-cost Silver plan in your area) and a capped percentage of your income:

where $r_{\text{cap}}$ is the income-based required-contribution percentage. With the ARPA/IRA enhanced subsidies expired after 2025 (and not extended), 2026 reverts to the indexed §36B applicable-percentage table of IRS Rev. Proc. 2025-25 — running from 2.10% up to 9.96%, with the hard 400% FPL cliff restored. The percentage is piecewise-linear within each band:

Example: The $1 Cliff

Consider a 62-year-old married couple in Denver, Colorado:

Scenario A: MAGI = $84,000 (below the $84,600 couple cliff)

• Benchmark Silver plan premium: $24,000/year
• Required contribution (9.96% of MAGI): $8,366
• ACA subsidy: $24,000 - $8,366 = $15,634
• Out-of-pocket: $8,366/year

Scenario B: MAGI = $85,000 (above the cliff by $400)

• Benchmark Silver plan premium: $24,000/year
• ACA subsidy: $0 (cliff)
• Out-of-pocket: $24,000/year

Result: Earning an extra $1,000 cost this couple $15,634 in lost subsidies—a 1,563% marginal tax rate.

MAGI for ACA Purposes

ACA subsidies are based on Modified Adjusted Gross Income (MAGI), which is AGI plus certain add-backs. Critically, not all retirement income counts toward MAGI.

Income that COUNTS toward ACA MAGI:

• Traditional IRA/401k withdrawals
• Wages and self-employment income
• Taxable interest and dividends
• Capital gains (including from portfolio rebalancing)
• Rental income (net of expenses)
• Pension income
• Taxable Social Security benefits (if applicable)

Income that DOES NOT count toward ACA MAGI:

• Roth IRA withdrawals (contributions or earnings)
• Municipal bond interest
• HSA withdrawals for qualified medical expenses
• Qualified Charitable Distributions (QCDs) from IRAs
• Return of principal from non-qualified annuities
• Borrowed money (home equity lines of credit, margin loans)

QuantCalc's ACA Cliff Detection

When Tax-Aware Mode is enabled and the user's age is under 65, QuantCalc checks for ACA cliff risk:

1. Determine household size from inputs
2. Calculate 400% FPL threshold for the applicable year
3. Monitor projected MAGI each year during simulation
4. Flag cliff proximity: if MAGI is within 5% of the threshold, mark as "cliff risk"

Optimization Strategy

When the ACA cliff is detected, the withdrawal optimizer adjusts its strategy:

Priority 1: Maximize Roth Withdrawals

Since Roth withdrawals don't count toward MAGI, the optimizer prioritizes Roth over traditional accounts when ACA subsidies are at stake:

Priority 2: Harvest Taxable Gains at 0%

If MAGI is low enough to stay in the 0% long-term capital gains bracket (under ~$94,000 for married couples), the optimizer may recommend withdrawing from taxable accounts to "harvest" gains tax-free while staying under the ACA cliff.

Priority 3: Avoid Traditional Withdrawals

Traditional IRA withdrawals add dollar-for-dollar to MAGI. The optimizer minimizes traditional withdrawals except when:

• RMDs require them (age 73+)
• Roth and taxable accounts are depleted
• The cliff is unavoidable (e.g., large RMDs push MAGI over regardless)

Priority 4: Reduce Spending if Necessary

If reducing traditional withdrawals would keep MAGI under the cliff and the subsidy value exceeds the foregone spending, the optimizer may recommend reducing spending (and, in Tax-Aware Mode, the withdrawal that funds it plus its tax) in that year.

Example: If crossing the cliff costs $18,000 in subsidies, it's better to spend $15,000 less that year (living on Roth or reducing discretionary expenses) than to lose $18,000 in subsidies.

Multi-Year Considerations

The ACA cliff is a multi-year optimization problem. Spending down Roth balances aggressively in early retirement (ages 55-64) preserves ACA subsidies but leaves less tax-free money for later years. The optimizer balances:

• Near-term subsidy preservation (Roth-heavy withdrawals during ACA years)
• Long-term tax efficiency (preserving Roth for high-tax years in late retirement)

Cliff Avoidance Tactics

QuantCalc models several advanced tactics automatically or via user configuration:

1. Roth Conversion Ladder (Pre-Retirement)

Convert traditional IRA funds to Roth IRA in low-income years (ages 50-54, or immediately post-retirement). This builds a Roth "cushion" for the ACA years.

2. Tax-Loss Harvesting (Taxable Accounts)

Sell losing positions to offset capital gains from winners, keeping net realized gains (and thus MAGI) low.

3. Bunching Income into Non-ACA Years

If a large income event is unavoidable (stock options vesting, real estate sale), time it for:

• Before retirement (still on employer insurance)
• After age 65 (on Medicare, ACA no longer applies)

4. Spending Flexibility

Model variable spending: reduce discretionary expenses during ACA years to stay under the cliff, then increase spending after Medicare eligibility.

Metrics Reported

When ACA cliff logic is active, QuantCalc reports:

• ACA cliff exposure: Number of years projected MAGI exceeds 400% FPL
• Subsidy value at risk: Total dollar value of subsidies lost due to cliff crossings
• Optimal withdrawal sequence: Recommended mix of Roth, traditional, and taxable withdrawals each year
• MAGI trajectory: Year-by-year projected MAGI relative to the cliff threshold
• Break-even analysis: Trade-off between spending less vs. losing subsidies

Limitations and Assumptions

QuantCalc's ACA cliff modeling assumes:

• Cliff remains at 400% FPL: If Congress restores enhanced subsidies, the cliff may be eliminated or softened
• Benchmark premium estimates: Actual premiums vary by location, age, and plan design
• No mid-year income changes: ACA subsidies are reconciled annually; mid-year income spikes require estimated tax payments

References and Further Reading

For detailed guidance on ACA subsidy optimization, see QuantCalc's ACA Subsidy Cliff 2026 blog post and the ACA Subsidy Calculator.

Part IV documents the PRO-gated modeling extensions. Each feature is off by default and preserves bit-for-bit output when disabled — the baseline Monte Carlo engine described in Parts I–III is a strict subset.

21. Gompertz Gender-Specific Mortality

By default, simulations run a fixed horizon from current_age to life_expectancy. Under the mortality model, we draw per-path terminal ages from a Gompertz distribution with gender-specific parameters, so outcomes reflect the joint distribution of portfolio performance and longevity.

where $M$ is the modal age at death and $b$ is the dispersion parameter. Calibration follows Milevsky (2020) using SSA period life tables:

For joint-life scenarios (joint filers with a spouse), each simulation path draws two independent Gompertz death ages. Supplying the spouse's current age enables stochastic widowhood: each path draws a spouse-death year from the gender-specific Gompertz table, so the timing of the first death varies across the distribution rather than being held constant. (A single deterministic spouse-death age can be specified instead when a constant assumption is preferred.)

The first death is a material tax event, and the engine models its consequences. At the start of the year following the first death the filing status flips from joint to single: the brackets compress, the standard deduction and its age-based additions halve, and the IRMAA thresholds halve. Conversely, while both spouses are alive and 65 or older, the couple receives the age-based deductions per spouse and pays the Medicare surcharge on each spouse — effects driven by the same spouse-age input. Social Security is reduced in the year following the first death to reflect the survivor benefit.

22. Fat-Tail Returns & Regime Switching

The two overlays in this section — fat tails and regime switching — add empirical realism to the parametric return model. They are the parametric route to the same properties that the Historical Bootstrap return model (Section 5) captures non-parametrically by resampling actual history. The two are alternatives: when the bootstrap return model is active these parametric overlays (and inflation coupling, Section 24) are disabled. Fat tails and volatility clustering are genuinely present in the resampled data, so the overlays would double-count them. Return–inflation coupling is not: the resampled series carries nominal returns only and the simulated CPI path is drawn from a separate process, so under the bootstrap model a month's returns and its inflation are independent — less coupling than the parametric path it replaces, not more.

Student-t Variance Mixture (Fat Tails)

The Gaussian return model understates the empirical kurtosis of equity returns. When Fat-Tail Returns is enabled, each monthly shock is drawn from a Student-t variance mixture:

The result is Student-t distributed with $\nu$ degrees of freedom. Variance is rescaled by $\sqrt{(\nu-2)/\nu}$ so the marginal variance still matches CME inputs. Recommended: $\nu=5$ reproduces empirical equity kurtosis $\approx 7$.

2-State Markov Regime Switching (Bull / Bear)

A Markov chain with transition matrix $P$ governs monthly switching between a bull and bear regime. In the bear state, the vol of the joint shock is multiplied by regime_bear_vol_mult and the drift shifted by regime_bear_drift_shift:

Defaults ($p_{bb}=0.01$, $p_{\bar b b}=0.15$, vol multiplier $1.8\times$) imply mean bear duration $\approx 6.7$ months and long-run bear probability $\approx 6.3\%$ — consistent with post-war US equity data.

Variance compensation. Under lognormal monthly growth $\exp(\mu_a + \sigma_a z)$, scaling the shock by $m$ in stressed months would, by Jensen's inequality, multiply that month's arithmetic expected return by $e^{(m^2-1)\sigma_a^2/2}$ — a hidden upward drift the user never asked for. The engine cancels it exactly: each asset's log-return in a stressed month receives the Itô correction $-(m^2-1)\,\sigma_a^2/2$. Stressed months therefore have higher variance but unchanged unconditional expected arithmetic return, so enabling regime switching widens the distribution without quietly changing its mean.

For a fully data-driven companion to this model — a Gaussian hidden-Markov fit to 150+ years of monthly real S&P total returns, with a current filtered regime reading refreshed monthly — see the Market Regime Monitor.

Fitted (1871–2026) Preset

The PRO regime panel offers a preset that transfers the fitted two-state structure into the engine: monthly transition probabilities $p_{bb}=0.0304$, $p_{\bar b b}=0.1488$, a stressed/calm volatility ratio of $2.55\times$, and the full fitted bull–bear drift spread of $-0.0277$/month (log). These four numbers are derived from the same monthly refit that drives the Market Regime Monitor rather than being chosen by hand, and the shipped preset is checked against the published fit before release, so the two cannot drift apart. Because forward-looking capital market expectations are unconditional — they already average over historical bear markets — applying the spread as a raw bear-only drag would double-count bears and depress the unconditional mean by $\pi_{bear}\times$ spread $\approx 5.6\%$/yr. The preset therefore re-centers the shift under the chain's stationary distribution: every month receives $-\pi_{bear}\,\Delta$, leaving the calm state at a small positive offset and the stressed state carrying the full spread.

Initial regime. Because the shift is de-meaned at the stationary stressed share $\pi_{bear}$, the chain must also start at that share or the re-centering over-corrects early. Each path therefore draws its month-0 regime from the stationary distribution, $s_0 \sim \text{Bernoulli}(\pi_{bear})$, instead of always opening in the calm state. Two consequences: the unconditional expected return matches your CME inputs from the first month rather than from about year two, and the first year of the plan — the window that dominates sequence-of-returns risk — carries the model's own long-run share of stressed months instead of a systematically calmer one. Opening every path in the calm state understated the stressed share for roughly 24 months and left about $+2\%$ of unintended cumulative drift on terminal values, almost all of it inside the first 12 months.

What the preset changes is where losses concentrate — contiguous stressed episodes — which is exactly the sequence-of-returns channel. The spread is estimated on real returns and applied to nominal paths; the difference is second-order because inflation is common to both regimes. Calm-state volatility equals your CME volatility, so unconditional volatility under the preset is $\sqrt{1-\pi_{bear}+\pi_{bear}m^2} \approx 1.39\times$ the CME volatility input (regime switching is deliberately a vol-increase overlay, as above).

23. Stochastic Inflation

The default engine compounds a constant CPI rate. The stochastic-inflation module replaces this with one of three user-selectable calibrated processes. Cash flows are stored in real (2025-dollar) terms and multiplied by a per-path cumulative inflation factor inside the simulation loop; this preserves bit-identical output when the model is deterministic.

Method 1 — AR(1) on CPI

Calibrated on FRED CPIAUCSL monthly log-differences, 1947–present. Anchored fit (post-1985 sample): $\bar\pi = 2.53\%$/yr, $\varphi = 0.4495$, $\sigma = 0.80\%$/yr.

Method 3 — Multi-Category Correlated AR(1)

Retirement spending is not single-index CPI. Medical, education, and housing categories compound at different rates with imperfect correlations. The multi-category model runs four parallel AR(1) processes on:

Shocks are cross-correlated via a Cholesky factor $L$ of a user-overridable correlation matrix. Each life event has a inflationCategory tag that selects which factor compounds that line item — medical events ride the medical-CPI path, tuition rides education, etc.

Method 4 — 2-State Regime (Anchored / Unanchored)

The 1970s and 2021–23 episodes are poorly described by a single Gaussian AR(1). We fit a 2-state hidden Markov model (Hamilton 1989) via the EM algorithm on the full postwar sample:

Stationary distribution gives $P(\text{unanchored}) \approx 17\%$, matching the empirical unanchored months in 1965–1982 + 2021–2023. Calibration is in backend/calibration/fit_inflation_regime.py.

Method 5 — Stationary Block Bootstrap (internal validation only)

Politis-Romano (1994) stationary block bootstrap on historical CPI returns. Blocks are drawn with Geometric lengths (default mean 30 months) and resampled with wraparound. Used as a non-parametric yardstick against which methods 1/3/4 are validated; not user-exposed.

24. Per-Asset Inflation Coupling (β)

Real assets don't respond neutrally to inflation surprises. When coupling is enabled, each asset's monthly log-return is shifted by its inflation beta times the current surprise:

Literature defaults (Ang & Piazzesi 2003; Fama & Gibbons 1984; Bekaert & Wang 2010):

25. State Tax, Roth Conversion, QCD & IRMAA Refinements

State Tax Engine (51 jurisdictions)

Federal-only tax mode still accounts for most of QuantCalc's audience, but single-state filers see meaningful divergence (FL/TX 0% vs CA 13.3% top). The state engine carries the real 2026 statutory law for all 50 US states plus the District of Columbia. It is not a single blended rate per state: each jurisdiction has its own bracket schedule per filing status, standard deduction, retirement-income treatment, Social Security rule, and capital-gains treatment. The 2025 schedules are checked against each state's Department of Revenue filings and projected forward inside the simulation loop by scaling every dollar-denominated threshold, cap and bracket edge by cumulative inflation (constant statutory thresholds, such as a $1,000,000 surtax floor that does not index, are left unscaled).

Rate schedules

Both progressive (multi-bracket) and flat-rate states are represented, with the correct schedule selected for single, married-filing-jointly, married-filing-separately, and head-of-household filers. Separate filers are handled per each state's actual rule — some states compress the joint brackets by half, others apply one shared schedule across statuses. Distinct head-of-household schedules are carried where a state publishes them.

Retirement-income exclusions

Retirement income is not taxed uniformly across states, and the engine encodes each state's actual rule rather than a flat inclusion:

• Full exclusions — a handful of states (for example Illinois, Pennsylvania, Mississippi) exclude qualified pension and IRA distributions entirely, with no dollar cap; in these states a traditional-to-Roth conversion is likewise not taxed.
• Capped exclusions — many states exclude the first $N of retirement distributions (for example a $20,000 private-pension exclusion, or larger age-tiered caps), sometimes with a lower cap for a younger age band.
• Age gates — exclusions frequently apply only at or above a threshold age.
• AGI cliffs and ramps — some states remove the exclusion entirely above an income limit, others phase it out ratably over an income range.
• Public vs. private pensions — certain states fully exempt government or employer-funded pensions while capping private pensions and IRAs. The optional pensionType input (public or private) routes the pension stream to the correct treatment; public/employer-funded pensions receive the full exemption where a state grants it.

Social Security

As of the 2026 tax year, eight states still apply some tax to Social Security benefits (Colorado, Connecticut, Minnesota, Montana, New Mexico, Rhode Island, Utah, Vermont), each under its own statutory formula — a full exemption below an AGI threshold with a hard cliff above, a fixed partial inclusion, a ratable phase-out of the exemption, a phase-out credit, an age-plus-income test, or full federal conformity. West Virginia completes its phase-out to full exemption for 2026. The remaining 42 states and the District of Columbia do not tax Social Security. The engine owns this exemption: callers pass the federally-taxable benefit and the filer's age, and each state's real formula decides how much (if any) enters the state base.

Capital gains

Most states tax long-term capital gains at their ordinary rates. The engine also captures the exceptions:

• Flat-rate states (for example Pennsylvania) tax gains at the single statutory rate.
• A few states exempt investment income entirely (for example New Hampshire), so the ordinary/gain split changes the result.
• Washington levies a standalone capital-gains excise — 7% on long-term gains above a large standard deduction, plus an additional 2.9% above $1,000,000 of gain — despite having no income tax.
• Maryland adds a 2% capital-gains surcharge above a federal-AGI threshold; Massachusetts and California apply high-income surtaxes on top of the bracket tax (one inflation-indexed, one set by statute at a constant threshold).
• Montana taxes net long-term gains on its own two-rate schedule; Hawaii caps the tax on gains at an alternative rate; Vermont and New Mexico exclude a flat first slice of gain.

Federal-coupled bases and additional taxes

States that begin from federal taxable income are computed federal-first: the engine calculates the federal result, then hands the post-deduction federal taxable ordinary income and gains (and, where relevant, the year's federal tax liability) to the state engine, so conformity states inherit the real federal deduction stack rather than re-deriving a proxy. Two states allow a deduction for federal income tax paid (Alabama in full including the surtax, Oregon capped and phased out with income), and county/local income taxes that ride on the state taxable base (for example Indiana and Maryland) are supported through an optional localTaxRate input, clamped to a reasonable range.

Roth Conversion Optimizer

Two modes are supported for a user-entered conversion plan:

• FIXED — convert a constant dollar amount each year in [rothConvStartAge, rothConvEndAge).
• BRACKET_FILL — convert up to the marginal federal bracket ceiling (e.g. the top of 12% or 22%). Amount varies annually with other income.

Conversions add to ordinary income in the year they occur, raising tax and potentially IRMAA tier. The simulator captures this interaction by re-running the full federal+state+Medicare pipeline per year.

When no conversion plan is entered, Tax-Aware Mode's whole-retirement policy search (Section 19) also considers six candidates that pair a withdrawal order with a recommended Roth-conversion rule of its own — the same FIXED/BRACKET_FILL-style sizing described above, chosen automatically per household rather than typed in. The two are mutually exclusive per run: entering a plan here disables the recommended-conversion candidates and the search instead recommends only among withdrawal orders, honoring the plan exactly as configured.

Qualified Charitable Distributions (QCD)

For filers aged $\geq 70\tfrac{1}{2}$, up to $111,000 (2026 cap, inflation-indexed — up from $108,000 in 2025) of IRA distributions can be directed to charity and excluded from gross income. QCD reduces AGI, lowering both federal tax and the IRMAA premium tier. The simulator applies the QCD against the required minimum distribution first — a QCD satisfies the RMD without generating taxable income — then treats any remaining distribution normally.

Medicare-Specific Inflation (IRMAA)

Medicare premiums and IRMAA thresholds have historically outpaced core CPI by 1–3 percentage points. Users can decouple the Medicare inflation rate from the headline rate (medicareInflationRate). Defaults use the Medicare Board of Trustees projections; empirical calibration lives in backend/data/medicare/.

IRMAA Two-Year Lookback Projector (/irmaa-projector/)

The IRMAA bracket lookup is a step function in MAGI with a fixed two-year lag between the tax year and the Medicare premium year. The projector accepts a year-by-year MAGI schedule from the user, identifies the lookback MAGI for each premium year (premium year t uses MAGI from year t−2), and applies the 2026 CMS bracket table to produce the projected per-month surcharge plus annual totals. Both single and MFJ filing statuses are supported; for MFJ households where both spouses are Medicare-enrolled, the per-person surcharge is doubled. The Form SSA-44 life-changing-event flow is documented inline so users can identify qualifying situations.

The math is a deterministic table lookup with the 2-year offset and the optional "both spouses on Medicare" multiplier; there's no optimization here, only projection. The free tier offers a 3-year horizon; the PRO tier extends to 10 years and integrates with the bridge optimizer (below) for conversion-window timing.

ACA Marriage Penalty Calculator (/aca-marriage-penalty/)

The marriage penalty under the OBBBA-restored 400% FPL cliff arises because the household-of-two FPL ($84,600 in 2026) is roughly 1.35× the household-of-one threshold ($62,600), not 2×. A two-earner couple with $60k each crosses the cliff jointly but stays under it filing separately. The calculator enumerates 4–6 filing+coverage scenarios (MFJ both ACA, MFJ A-employer-B-ACA, unmarried both ACA, unmarried A-employer-B-ACA, MFJ A-Medicare-B-ACA, MFJ both Medicare) and ranks them by total annual cost (premium + federal tax + IRMAA where applicable).

State-specific Silver-benchmark premiums use the published KFF 2026 marketplace data anchored at age 40 (kff.org, 2026 plan premiums, checked 2026-09-10), scaled by the CMS 45 CFR 147.102 age-rating curve (3:1 from age 21 to 64). Medicaid expansion status is encoded for all 50 states + DC; non-expansion states trigger a coverage-gap detection when MAGI < 100% FPL. The scenario engine handles the constraint that Medicare-age (≥65) household members cannot be on ACA marketplace plans.

ACA Bridge Optimizer (/aca-bridge-optimizer/)

The bridge optimizer solves the joint multi-year decision problem: given a household's portfolio split (traditional / Roth / taxable), required annual cash-out for living expenses, chronic-condition healthcare cost band, target FPL ceiling, and state of residence, find the year-by-year Roth conversion amounts and withdrawal sequence that minimize lifetime cost (federal + state tax + healthcare net of subsidy + IRMAA from the 2-year lookback) over the bridge from current age to Medicare at 65.

Problem class. The decision space is up to 4 variables per year × 25 bridge years (100 decision variables max). The objective is non-convex, non-smooth, and includes cliff discontinuities at 400% FPL (full subsidy forfeiture), 138% FPL (Medicaid eligibility edge), and the six IRMAA bracket boundaries. Federal and state tax are piecewise-linear with kinks at bracket boundaries. The cash-flow constraint couples decisions across buckets within each year; the IRMAA constraint couples decisions across years via the 2-year lookback.

Solver architecture. Two tiers:

• Free tier — per-year greedy heuristic. Each year, the algorithm computes a fixed-point iteration over (taxable withdrawal, Roth withdrawal, traditional withdrawal, Roth conversion) until cash-flow, MAGI, healthcare cost, and tax all converge mutually (typically <8 iterations, contractive because higher conversion → higher tax → larger cash need → larger withdrawals → larger LTCG MAGI → shrunk headroom → smaller conversion). The greedy captures roughly 95% of the lifetime benefit of full joint optimization per FIRE-community backtests.
• PRO tier — joint nlopt ISRES + SBPLX refinement. Warm-started from the greedy result, the optimizer uses nlopt's ISRES (Improved Stochastic Ranking Evolution Strategy — gradient-free, handles non-smooth constraints) at high population, followed by a deterministic SBPLX local-refinement pass to tighten the result. Cliff discontinuities are handled via a smoothed quadratic penalty added to the objective. The final plan is guaranteed to be at least as good as the greedy warm-start (the optimizer falls back to greedy if it can't improve).

Healthcare model. Chronic-condition out-of-pocket bands are QuantCalc's own model parameters (~$3k/yr for one mild condition, ~$8k/yr for 2–3 conditions, ~$18k/yr for 3+ severe), calibrated against the out-of-pocket spending distribution the Peterson-KFF Health System Tracker publishes (healthsystemtracker.org, checked 2026-09-10). KFF does not publish these bands. Premium subsidy follows the 2026 statutory applicable-percentage curve as indexed by IRS Rev. Proc. 2025-25 §3.01 (2.10% of income at ≤133% FPL, scaling to a flat 9.96% across 300–400% FPL, cliff above). The ARPA/IRA enhanced schedule (0%–8.5%, no cliff) expired after 2025 and is not modeled as current law. Medicaid (≤138% FPL in expansion states) and coverage-gap (≤100% FPL in non-expansion states) are detected and modeled separately.

State tax. Uses the same 51-jurisdiction engine as Section 25, with the user's MAGI split between ordinary income (other income + traditional withdrawal + conversion) and LTCG (taxable-account withdrawal × gain fraction). Most states tax LTCG at ordinary rates; a handful (e.g., NH) exempt investment income and the split matters.

Verification. The C implementation carries 35 unit tests covering cliff respect across many scenarios, infeasibility handling, money conservation (no money created or destroyed by the simulation), stochastic-run stability, naive-baseline domination, and state-tax variance. The frontend JS module carries an additional 56 tests on the greedy math. Both run as part of the deploy gate.

26. Randomized QMC & Wilson CI

Cranley-Patterson Scrambling

Unscrambled Sobol sequences are deterministic — two runs with the same inputs produce identical output and give no Monte Carlo standard-error estimate. With Sobol Scrambling enabled, each dimension is rotated by a random offset $r_d \sim U[0,1)$:

This preserves the low-discrepancy property while randomizing the realization (Cranley & Patterson 1976). Running $K$ independent scrambled replicates gives an unbiased estimate of the QMC error.

When Sobol is not in use. Three settings bypass the Sobol sequence entirely, and scrambling has nothing to act on in any of them: Fat-Tail Returns and regime switching both replace the draws with pseudo-random ones (the regime volatility multiplier destroys the low-discrepancy property, so keeping Sobol would be worse than not), and the Historical Bootstrap return model draws historical months directly and never touches the normal array at all. Under any of the three, $K$ replicates return $K$ identical answers. The scrambling toggle is disabled while the bootstrap preset is active, and the “Assumptions used” panel reports the sampler the run actually used rather than the one requested.

Wilson 95% Confidence Interval on Success Rate

With $N$ simulations and $k$ successes, the normal-approximation CI $\hat p \pm 1.96 \sqrt{\hat p(1-\hat p)/N}$ degrades badly when $\hat p$ is near 0 or 1. The Wilson score interval stays valid across the full range:

where $z = 1.96$. QuantCalc displays this CI next to every success-rate metric.

Pre-Computed Sobol Cache

At container-build time we generate a 10,000-sample × 6,480-dim Sobol cache using QuantLib's Joe-Kuo D7 direction numbers. Dimensions: 9 per month × 720 months (5 asset-return dimensions + 4 inflation-category dimensions), covering the widest model configuration.

27. Withdrawal Rules

Every simulation applies one withdrawal rule to the planned spending. The rule decides how much is taken from the portfolio each year; the tax-aware order (Part III) decides from which accounts. In Tax-Aware Mode the rule's amount is the year's after-tax spending; the engine withdraws that amount plus the year's taxes. One consequence for VPW: it amortises the balance as after-tax spending while taxes are drawn on top, so in Tax-Aware Mode a VPW plan can reach zero before the horizon, which the standard engine's VPW cannot. A second consequence, for Guyton-Klinger: the guardrail's withdrawal-rate test (current spending divided by the current portfolio, compared with the initial rate) keeps running on the after-tax spending amount, not on the grossed-up withdrawal that actually leaves the portfolio — so in Tax-Aware Mode the measured rate reads lower than a gross-withdrawal-over-portfolio reading would, and a guardrail cut or raise fires later than that gross reading would suggest. Guardrails and VPW run in Tax-Aware Mode (the rule sets each year's spending target; the tax-aware order chooses which accounts fund it); the bucket rule is fixed-path only, because it chooses which assets are sold — the decision the tax-aware order makes for itself. Dynamic rules are ignored when a multi-period glide path is set.

Fixed real (the 4% rule family)

The first year's spending is taken in dollars and indexed to the simulated inflation path every year thereafter, regardless of portfolio performance. This is the default and the baseline every other rule is compared against in the withdrawal-strategies study.

Guyton-Klinger guardrails

The initial withdrawal rate is recorded at retirement. Each year the current withdrawal rate (planned spending divided by the current portfolio) is compared with it. When the current rate rises above 120% of the initial rate, spending is cut by 10%; when it falls below 80% of the initial rate, spending is raised by 10%; otherwise spending is indexed to inflation. The cut is suppressed inside the final 15 years of the plan (the capital-preservation rule). Because the rule is closed-form, the two guardrail portfolio levels on page 1 of the client report are computed exactly: a cut fires below the retirement portfolio divided by 1.20, a raise above the retirement portfolio divided by 0.80. The report's survival rate under this rule is the rule's survival; the realised-spending distribution shows what the rule actually paid out.

Variable percentage withdrawal (VPW)

Each year's spending is the amortised payment of the remaining portfolio over the remaining horizon at the plan's assumed real return, so spending rises and falls with markets and cannot exhaust the portfolio before the horizon by construction. Survival is not the relevant measure for VPW; the realised-spending distribution is.

Bucket (cash reserve)

At retirement a chosen number of years of planned spending (one to ten, default three) is set aside in the plan's lowest-volatility asset and drawn on first; the rest stays invested at the plan's allocation and is rebalanced annually. The reserve is refilled to its target at year end either after any non-losing year or only when it has fallen below half its target. Spending itself follows the fixed-real rule, so the survival rate compares directly with fixed real. The apples-to-apples comparison against a total-return portfolio holds the same starting cash in both arms; see the bucket-strategy study.

Constants and formulas mirror js/guardrails.js and backend/src/monte_carlo.h; the three-way comparison of tent, bucket and guardrails on identical paths is in the sequence-risk defences study.

28. Implementation Notes

For $N$ simulations over $T$ months with $n$ assets, random numbers needed: $N \times T \times n$. For 10,000 sims × 720 months × 5 assets = 36 million random numbers. Stochastic inflation adds up to $4 \times 720 \times N$ more when method #3 is active.

Optimizations: Pre-generate random numbers in batch, vectorized operations, pre-compute Cholesky factors (asset returns and inflation categories), fixed seed during optimization, OpenMP parallelization of the per-simulation loop.

29. References

1. Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer.
2. Jorion, P. (2006). Value at Risk. McGraw-Hill.
3. Nocedal, J. & Wright, S. (2006). Numerical Optimization. Springer.
4. Kraft, D. (1988). "A software package for sequential quadratic programming." DFVLR-FB 88-28.
5. Sobol, I.M. (1967). "On the distribution of points in a cube." USSR Comp. Math. & Math. Phys.
6. Joe, S. & Kuo, F.Y. (2008). "Constructing Sobol sequences with better two-dimensional projections." SIAM J. Sci. Comput.
7. Cranley, R. & Patterson, T.N.L. (1976). "Randomization of number theoretic methods for multiple integration." SIAM J. Numer. Anal.
8. Hamilton, J.D. (1989). "A new approach to the economic analysis of nonstationary time series and the business cycle." Econometrica.
9. Politis, D.N. & Romano, J.P. (1994). "The stationary bootstrap." J. American Statistical Association.
10. Ang, A. & Piazzesi, M. (2003). "A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables." J. Monetary Economics.
11. Fama, E.F. & Gibbons, M.R. (1984). "A comparison of inflation forecasts." J. Monetary Economics.
12. Bekaert, G. & Wang, X. (2010). "Inflation risk and the inflation risk premium." Economic Policy.
13. Milevsky, M.A. (2020). Retirement Income Recipes in R. Springer.
14. Wilson, E.B. (1927). "Probable inference, the law of succession, and statistical inference." JASA.
15. US Bureau of Labor Statistics — CPI series CPIAUCSL, CPIMEDSL, CUSR0000SAE, CUSR0000SAH (FRED).

A. Editorial Standards

QuantCalc's articles and calculators are written and maintained by the QuantCalc Research team. The same engineering team that maintains the simulation backend also writes the public-facing research — there is no separate editorial layer paraphrasing technical content for a general audience.

Verification process

1. Primary-source-first: every numerical fact (tax brackets, FPL thresholds, IRMAA tiers, RMD divisors, contribution limits) is verified against the originating government publication, not against a secondary source aggregator. Each article's "Primary sources cited" block links directly to the IRS, SSA, CMS, HHS, or congressional bill text it relies on.
2. Computed against the backend: any worked example or scenario in an article is computed using the same C backend the calculators run on. If the math in the article disagrees with the calculator, the article gets corrected.
3. Forecast inputs are publicly cited: capital-market expectations (return, volatility, correlation) are sourced from six named investment firms (BlackRock, JPMorgan, Vanguard, GMO, Schwab, Invesco) whose 2026 publications are linked in §5 above. We do not modify these except for the documented GMO real→nominal conversion.
4. Update on rule change: when a tax-law or regulatory change is signed (e.g., the One Big Beautiful Bill Act of July 2025, the SECURE 2.0 Act of December 2022), affected articles are updated within 30 days. The "Last reviewed" date on each post reflects the most recent verification pass.

Review cadence

• Annual review — every article is reviewed at least once per calendar year for accuracy against the new tax-year brackets, FPL guidelines, and Medicare premium adjustments. The "Last reviewed" date stamp on each article shows the most recent pass.
• Event-triggered review — when a primary source releases new data (IRS Rev. Proc., CMS annual notice, HHS poverty guidelines, OBRA/OBBBA-class legislation), all affected articles are re-verified within 30 days.
• User-flagged corrections — readers can email [email protected] to flag a specific number; we investigate within 7 days and either correct or document why the existing figure stands.

What you will not find here

• Personalized financial advice. QuantCalc is an educational research tool. We do not know your tax situation, employer plan details, or risk tolerance. A scenario shown by the calculator is one of 10,000 possible paths, not a recommendation.
• Pay-to-play product reviews. When an article mentions a brokerage, fund family, or competing planning tool, we have no commercial relationship with them. There are no affiliate links in our content; the only revenue path is a one-time PRO unlock for our own calculator.
• Backdated edits hidden from readers. Material corrections are dated; the dateModified field in each article's JSON-LD reflects the actual file mtime, and "Last reviewed" dates appear in the post-meta strip.

Author

QuantCalc Research is the engineering and research team behind QuantCalc. The team's published methodology covers Monte Carlo simulation with quasi-Monte Carlo sampling (Sobol sequences), Cholesky-decomposed correlated returns, log-normal asset return modeling, regime-switching volatility, stochastic inflation with category-specific drift, IRS-compliant federal/state tax computation, and SECURE 2.0-aware RMD/IRMAA modeling. See sections 2-15 above for the full technical specification.

For questions, corrections, or research collaboration: [email protected].