The math behind the calculators.
Every calculator on this site does math you can verify. This page documents the formula each calculator uses, the source it cites, and the assumptions it makes — so you can audit the result, copy the math into your own spreadsheet, or spot something we got wrong.
Calculators that show their working.
A calculator that gives you a number without showing how it got there isn’t a tool — it’s a black box. For a one-off curiosity that might be fine. For a six-figure decision — a mortgage, a retirement number, a tax-conversion strategy — it isn’t.
Every calculator on this site is built around three commitments:
- The formula is published. You’ll find it on this page, in plain notation. Copy it into Excel or Google Sheets and you’ll get the same answer the calculator gave you.
- The source is cited. Where federal math applies — payroll withholding, contribution limits, tax brackets — we cite the specific IRS publication, Federal Reserve release, or state revenue document.
- The assumptions are stated. A calculator that doesn’t tell you what it’s assuming is hiding something. Every assumption — constant rate, no fees, a particular filing status, an inflation guess — is named explicitly.
Compound growth, loans, retirement, debt.
These calculators apply universal financial mathematics — the formulas don’t change with the tax year or the state you live in. They do change based on the inputs you provide.
Calculates the future value of an initial principal plus regular contributions, compounding at a fixed rate over time.
A = P × (1 + r/n)^(nt) + PMT × [((1 + r/n)^(nt) − 1) / (r/n)]Afuture valuePinitial principalPMTcontribution per periodrannual interest ratencompounding periods per yearttime in yearsLong-term portfolio projection with a real-return option (inflation-adjusted). Built on the same compounding formula above, with an inflation toggle.
Real return = ((1 + nominal) / (1 + inflation)) − 1Monthly payment, total interest paid, and full amortization schedule for a fixed-rate auto loan.
M = P × [r(1 + r)^n] / [(1 + r)^n − 1]Mmonthly paymentPloan principalrmonthly interest rate (annual ÷ 12)ntotal payments (years × 12)Estimates the portfolio size needed to support a given annual spending level for a given retirement horizon, plus the years to reach it at a given savings rate.
Required portfolio = Annual spending ÷ Safe withdrawal rateProjects tax-free growth on Roth contributions over time, plus the after-tax value comparison against a traditional pre-tax IRA at a given expected retirement tax rate.
Roth value = Σ(contribution × (1 + r)^(years to retirement))
Traditional after-tax = Σ(contribution × (1 + r)^(years)) × (1 − tax rate at withdrawal)Compares two repayment strategies side by side using your actual balances and APRs. Snowball applies extra to the smallest balance first. Avalanche applies extra to the highest-APR balance first. Output: total interest paid and time to debt-free under each strategy.
For each debt, monthly interest = balance × (APR ÷ 12)
Apply minimum to each debt, then apply extra payment per the strategy
Iterate month-by-month until all balances reach zeroFederal & state withholding, income tax, capital gains.
These calculators depend on federal and state tax tables that change annually. Every figure is tied to the publishing agency’s current document — and updated within 30 days of any agency announcement.
Estimates take-home pay from gross wages — federal income tax withholding + FICA (Social Security and Medicare) + Additional Medicare for high earners.
Net = Gross − Federal income tax − FICA SS − FICA Medicare − Additional Medicare − State withholding − Pre-tax deductions- Federal income tax withholding: calculated using the percentage method from IRS Publication 15-T (Federal Income Tax Withholding Methods), based on filing status and the W-4 information you provide.
- FICA Social Security: 6.2% on wages up to the IRS-published annual wage base.
- FICA Medicare: 1.45% on all wages.
- Additional Medicare: 0.9% on wages above $200,000 (single filer threshold for employer-withholding purposes).
A state-specific paycheck calculator for every US state plus DC. Each uses the federal methodology above, layered with the state’s own withholding tables.
- Flat-rate states (Pennsylvania, Indiana, Michigan, Illinois, Colorado, North Carolina, Utah, Kentucky, Massachusetts, etc.) — a single percentage rate applied to taxable wages.
- Progressive states (California, New York, Oregon, Hawaii, Minnesota, etc.) — bracketed rates from the state’s withholding tables.
- No-income-tax states (Alaska, Florida, Nevada, New Hampshire, South Dakota, Tennessee, Texas, Washington, Wyoming) — no state withholding component.
- Local-tax states (some Pennsylvania municipalities, Ohio, New York City, etc.) — local income taxes added where applicable.
Estimates total federal income tax liability for the year given gross income, filing status, deductions and credits.
Taxable income = Gross income − Adjustments (above-line) − Standard or itemized deduction
Tax = Apply progressive bracket schedule to taxable income
Final liability = Tax − CreditsA state income tax calculator for every state that has an income tax. Uses federal AGI as the starting point, then applies state-specific adjustments, deductions, brackets, and credits.
Estimates federal (and where applicable, state) tax on realized capital gains, distinguishing short-term from long-term and accounting for the 0%/15%/20% long-term brackets plus the Net Investment Income Tax (NIIT).
- Short-term gains (held ≤1 year): taxed at ordinary income rates.
- Long-term gains (held >1 year): 0%, 15%, or 20% bracket based on taxable income.
- NIIT: additional 3.8% on net investment income above MAGI thresholds.
Mortgage, refinance, rent vs buy.
These calculators handle the math of major housing decisions. Mortgage variants for specific state markets are layered on top of the universal formula — adding state-specific recording fees, transfer taxes, or homestead exemptions where relevant.
Calculates the full monthly mortgage payment — principal & interest, plus PMI, property tax, and homeowners insurance.
P&I = Principal × [r(1 + r)^n] / [(1 + r)^n − 1]
Monthly payment = P&I + (PMI ÷ 12) + (Property tax ÷ 12) + (Insurance ÷ 12)rmonthly interest rate (annual ÷ 12)ntotal payments (years × 12)Computes the break-even point — how long you need to stay in a home for buying to beat renting financially. Accounts for mortgage interest, principal payoff, home appreciation, opportunity cost of the down payment, closing costs, maintenance, and tax effects.
Net cost of buying = Down payment + Closing costs + Σ(monthly costs) − Equity built − Home appreciation − Tax savings
Net cost of renting = Σ(rent) − Investment growth on down payment kept invested
Break-even = year where buying cost ≤ renting costCD ladder, savings goals, cost of living.
Builds a CD ladder of N rungs at staggered maturities, calculates total interest earned across the ladder, and shows reinvestment timing.
Solves for the monthly contribution needed to reach a savings target by a given date, or the date you’ll reach a target at a given contribution rate.
PMT = (FV × r) / [(1 + r)^n − 1]Compares the cost of living between two US metro areas, adjusting for housing, food, transportation, healthcare and taxes. Useful for relocation decisions.
What every calculator assumes — and what it can’t do.
Two things to keep in mind across every calculator on this site:
1. Calculators estimate, not predict. When a calculator uses a future return, an inflation rate, or an interest rate, those are inputs — not forecasts. The math is exact given the inputs. The inputs are guesses about the future, and the future doesn’t take instructions.
2. Calculators are not personalized advice. A calculator can tell you how much you’d save under strategy A vs strategy B, given your inputs. It cannot tell you which strategy is right for you — that depends on your specific situation, goals, and constraints. For decisions that turn on those, talk to a licensed fiduciary.
Specific universal assumptions across all calculators:
- Tax year: the current tax year unless you specify otherwise. Where a calculator uses tax tables, the year is shown explicitly on the calculator page.
- Rounding: dollar figures rounded to two decimals. Percentage figures rounded to four decimals internally, two for display.
- Currency: US dollars. Calculators do not convert other currencies.
- Inflation defaults: 3.0% long-term (CPI 1913–present average), adjustable on every calculator that uses it.
- Stock return defaults: 7% real / 10% nominal (S&P 500 ~1928–present, before taxes and fees), adjustable.
- Bond return defaults: 2.5% real, adjustable.
Specific limits worth knowing:
- State tax reciprocity agreements (e.g., NJ–PA, MD–VA–DC, IL–IN–IA–KY–MI–WI) are not automatically applied. Use the calculator for the state of employment, not residence, unless the state-specific page notes otherwise.
- Alternative Minimum Tax (AMT) is not modeled in the standard federal income tax calculator. A separate AMT calculator is at /calculators/amt/.
- Self-employment tax is handled in a separate self-employment calculator, not in the standard paycheck or income-tax calculators.
- International / expat scenarios are out of scope. Calculators assume US tax residency.
Update triggers and turnaround.
Universal formulas (compound interest, amortization, future-value) don’t change — they’re fixed financial mathematics. Tax-related calculators do change, and on a defined schedule:
- IRS annual Revenue Procedure (typically October–November, for the following tax year) — bracket adjustments, contribution limits, standard deduction. Calculators updated within 30 days of publication.
- IRS Publication 15-T updates — federal withholding methods. Calculators updated within 30 days.
- State revenue department announcements — withholding tables, bracket changes. Calculators updated within 30 days of state publication.
- FICA wage base announcement (typically October, by the SSA) — paycheck calculators updated within 30 days.
- Federal Reserve rate changes — interest-rate-dependent calculators (mortgage assumptions, HYSA tables) reflect the change within 7 days.
When a calculator’s methodology changes meaningfully — not just a tax-table refresh, but a formula or assumption change — the change is noted in the calculator’s update history on its own page, and summarized in our public corrections log at /corrections/.
How to flag a problem with a calculator.
If a calculator gives you a result you can’t reproduce using the formula on this page — or if you believe the formula itself is wrong — we want to know.
The fastest way is the contact form. Useful information to include:
- The calculator URL (e.g.,
/calculators/mortgage/california/) - The inputs you used
- The result the calculator gave
- The result you expected, and how you derived it
- The source you’re citing if you believe a tax-table or rate value is wrong
We investigate within 5 business days. If the calculator is wrong, it gets fixed, and the fix is logged on our public corrections log. If the calculator is right but the methodology page is unclear, the methodology page gets clarified.
The full corrections process is in our editorial policy.
You don’t have to trust the result.
You can verify it. Every calculator’s formula is on this page. Every source is cited. Every assumption is named. Copy the math into your own spreadsheet and you should get the same answer — and if you don’t, we’d rather hear about it than not.