The retirement projection
1,000 simulations, drawn from a lognormal return distribution, with federal income tax on every retirement withdrawal solved year by year. Here is exactly how it works; the simplifications it still makes are listed at the end.
What it does
HelloMula runs 1,000 independent simulations of your portfolio from today to the end of your planning horizon. Each simulation applies a different randomly drawn sequence of annual returns, which is why the result is a probability rather than a single figure.
Returns are drawn from a lognormal distribution parameterised by the
expected return and volatility you set. Random normal variates are generated
with the Box–Muller transform, and the drift term is adjusted
(μ = ln(1 + r) − ½σ²) so that your stated
expected return is the arithmetic mean of the simulated distribution rather
than drifting high.
Each simulated year runs in two phases:
- Before retirement: the portfolio grows at the drawn return and your annual contributions are added.
- After retirement: your expenses are inflated forward — general expenses at your inflation rate, healthcare at a separate and usually higher healthcare inflation rate — and the portfolio is debited by the withdrawal needed to cover the shortfall between those expenses and your Social Security income after federal income tax.
How tax is handled
A retirement plan that debits the portfolio in pre-tax dollars overstates its own odds. Ours does not. If you need $80,000 of spending and the money sits in a traditional 401(k), the withdrawal has to be large enough that what survives the tax bill is $80,000. We solve for that gross figure in every retirement year:
- The tax-deferred share of each withdrawal is ordinary income. That share defaults to the split between your 401(k) and taxable contributions, and you can override it directly if your balances say otherwise.
- Tax is computed bracket by bracket against the same IRS data that powers the Federal Tax Estimator, less the standard deduction for your filing status.
- Social Security is taxed under the provisional-income rule of IRC §86 — zero, 50% or up to 85% of the benefit becomes taxable depending on your other income.
- Brackets and the standard deduction are indexed forward each year at your assumed inflation rate, so nominal growth alone does not invent bracket creep across a thirty-year projection. The §86 thresholds are not indexed in current law, and we hold them nominal to match — which correctly makes more of your benefit taxable as the years pass.
The projection reports the federal tax you are expected to pay across the whole of retirement, and your effective rate on it.
One consequence worth stating plainly: a model that ignores this returns a higher probability of success than ours does. That difference is not accuracy — it is a tax bill that still arrives, quietly left out of the arithmetic. If you have compared HelloMula against a tool that shows you a friendlier number, this is usually why.
Two honest caveats. Current tax law is held constant for the life of the projection, and a future Congress may change it. And the tax-deferred share of your portfolio at retirement is projected once at your expected return rather than separately in each of the 1,000 scenarios — the ratio moves far less across scenarios than the balance does, because both account types earn the same return in any given year.
We report the share of the 1,000 runs in which the portfolio was never exhausted, a percentile fan from the 10th to the 90th, a year-by-year table of the median path, and the average age at depletion across the runs that did fail. Up to five one-time future expenses — a college year, a roof, a wedding — can be placed in specific years.
