Analytical by default. Monte Carlo when you ask.

Returns are lognormal in both engines. The closed-form path uses the Kan & Zhou methodology to compute analytical percentiles directly — fully deterministic, with no randomness to seed. The Monte Carlo path draws monthly return factors from a lognormal distribution using a seeded pseudorandom generator, so a given seed reproduces the same set of simulated paths exactly. It runs 1,000 paths by default and reports the 10th / 25th / 50th / 75th / 90th percentiles; selection is nearest-path, so every reported value is an actually-simulated outcome, not an interpolation.

One parameter picks the engine: closed-form (the default), a single deterministic path, Monte Carlo, or auto — which falls back to a stepping engine when something path-dependent, such as a balance-based fee, can't be expressed analytically. The response always reports which method actually ran.

For plans with ongoing contributions, the closed-form engine's lower percentiles sit below the Monte Carlo result: over 30 years at 15% volatility, its 10th percentile is about 12% lower and its median about 3% lower. At 30% volatility over 25 years the gaps widen to about 24% and 9%. The error runs in the cautious direction, and a fix is decided but not yet shipped — when the downside band matters, ask for Monte Carlo.

Inflation defaults to 2.5%, and every result is reported in both nominal (face-value future dollars) and real (constant purchasing power) terms. Real values are deflated with the Fisher relation, and the same inflation assumption is applied consistently across returns, contributions, and fees — so a real projection is internally coherent, not a nominal number with a rough haircut.

Projections can be converted to after-tax spendable values by account type: a pre-tax 401(k)/IRA is reduced by your marginal ordinary rate; Roth, HSA, and 529 balances pass through untaxed; a taxable brokerage is reduced by long-term capital-gains tax on its gain only — and because cost basis is fixed while the balance grows, the taxed gain fraction rises over time, exactly as it does in reality.

These conversions use the marginal ordinary and capital-gains rates you supply, not a full bracket-by-bracket return — for that, the tax tools compute the whole return directly.

A full plan projects each account under its own allocation — a 401(k) and a cash savings account grow differently — and feeds the household's net monthly surplus (income minus expenses) back in as contributions. You can pin a contribution to a specific account and let the rest of the surplus fall where it will. Employer 401(k) match is routed to each current-employer plan on top of the surplus, and goals are scored with a success probability at their target date.

Accounts are aggregated by summing matching percentiles: a perfectly-correlated (comonotonic) assumption that is deliberately conservative on the downside, and is stated in every response rather than buried.

Age-based income and expense rules (a start or end age) are not evaluated yet: age-ruled income is left out and age-bounded expenses run for the whole horizon. Employer match holds the IRS compensation limit at its current-year value, does not enforce the employee deferral limit, and does not grow the pay it matches against. Required minimum distributions are computed by the RMD tools, not forced out of accounts during a whole-plan projection. Goals gated on another goal's progress are listed as skipped rather than projected. The response says so whenever one of these applies.