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Methodology · Finance
Finance Methodology
Every finance calculator here is a model of one of five things: a cash flow through time, a claim priced against a rate, a portfolio’s risk, a rule set by statute, or a household’s plan. This page states the conventions those models share, the primary sources each one descends from, and the point at which the model stops being valid.
§ 1
The discounting identity
One identity underlies most of the catalog: a cash flow C at time t is worth C/(1+r)t today. The annuity factor is that series summed, duration is its time-weighted average, and an IRR is the rate at which the sum crosses zero.
The five forms below are the same statement rearranged. Every loan, bond, DCF and retirement page is one of them with different labels on the variables — which is why the conventions in §2 matter more than the algebra: they decide what r and t actually mean in the period you are looking at.
| Form | Relationship | Where it is used |
|---|---|---|
| Present value | PV = Σ Cₜ · (1+r)⁻ᵗ | DCF, bond pricing, NPV |
| Future value | FV = PV · (1+r)ⁿ | Compound interest, savings, 401(k) |
| Annuity factor | a(n,r) = (1 − (1+r)⁻ⁿ) / r | Loan sizing, affordability, pension |
| Level payment | PMT = P · r / (1 − (1+r)⁻ⁿ) | Amortisation, auto loan, mortgage |
| Internal rate | Σ Cₜ · (1+IRR)⁻ᵗ = 0 | IRR, YTM, break-even financing |
§ 2
Conventions
These 7 settings are shared by the lending, valuation and risk models in the catalog. Each one changes a number, not just a label — the third column is what breaks if you assume otherwise.
| Convention | Setting | What it changes |
|---|---|---|
| Rate basis | Nominal annual rate, monthly compounding (APR ÷ 12) unless the page states otherwise | A 6.00% APR is a 6.17% effective annual rate — the gap compounds with the term |
| Payment timing | Annuity-immediate: every payment lands at the end of its period | The first payment accrues one full period of interest before it is made |
| Working precision | 5 pages use Decimal.js, 2 route currency arithmetic through Big.js in a worker, the rest use doubles | Ledger pages reconcile to the cent; projection pages carry float noise instead |
| Rounding | Only in the formatter — never inside a schedule | A 60-month amortisation closes on exactly 0.00 rather than a few cents short |
| Payment factor | PMT = P · (r + r / expm1(n · log1p(r))) | Identical to P·r/(1−(1+r)⁻ⁿ), but stable where the textbook form cancels to noise |
| Randomness | 11 pages seed mulberry32 from an input; normals via Box–Muller | The same inputs and seed reproduce the same paths, percentile and chart |
| Sign | Outflows negative, inflows positive in every NPV / IRR page | A sign flip turns an IRR into a rate you cannot earn |
§ 3
Citation index
Built from the reference each calculator already publishes: a model appears under a work only because its own page names it. All 179 finance calculators resolve to a source here.
US Internal Revenue CodeIRC / IRS publications
401(k)529 College SavingsAlternative Minimum TaxAsset LocationCapital Gains TaxCharitable TrustDepreciationDynasty TrustEstate TaxFederal Income TaxGift TaxGRAT and IDGT SaleGST and Charitable EstateHome Sale ProceedsHousehold Financial PlanHSAMarriage PenaltyNet Investment Income TaxPaycheckQBI Deduction (199A)Quarterly Estimated TaxRetirement Tax & WithdrawalRMDRoth IRASelf-Employment TaxState Estate TaxTax Return Analyzer (1040)Tax-Loss HarvestingTraditional IRA
29
BrovermanMathematics of Investment and Credit
AmortizationAnnuityAverage ReturnBoat LoanCash Back vs Low InterestCompound InterestConstruction LoanCredit Card PayoffDebt ConsolidationDebt SnowballFuture ValueHome Equity LoanHouse AffordabilityInterest RateMortgageMortgage PayoffMortgage RecastPersonal LoanPresent ValueRefinanceRetirementRetirement Plan Monte CarloSavingsSavings GoalSimple InterestUK Mortgage
26
Reg Z / Reg DD / Reg M / Reg TConsumer, deposit, lease and margin disclosure rules
12
Brealey, Myers & AllenPrinciples of Corporate Finance
Corporate DebtDCF ValuationDebt SculptingIRRLBOM&A Accretion / DilutionNPVPayback PeriodUnit Economics
9
Basel Committee (BCBS)IRB · FRTB · IRRBB standards
CDO TrancheCredit Risk CapitalCVA Hedge and CapitalExpected LossExpected ShortfallFRTB GIRRIRRBBValue at Risk (VaR)
8
Brigham & EhrhardtFinancial Management: Theory & Practice
Burn Rate and RunwayCovenant HeadroomHousehold Cash Flow PlanNet WorthROIThree-Statement ModelWorking Capital and Cash Cycle
7
HullOptions, Futures and Other Derivatives
6
34 CFR Part 685Federal student aid regulations
4
Appraisal InstituteThe Appraisal of Real Estate
4
National Association of Insurance CommissionersLife Insurance Buyer's Guide · Shopper's Guide to Long-Term Care Insurance
2
US Small Business AdministrationSOP 50 10 — Lender and Development Company Loan Programs
2
Cited once each: 31 CFR Part 360 · Albanese & Andersen · Almgren & Chriss · Bengen · Kitces · Black & Litterman · Bollerslev · Bouzoubaa & Osseiran · Brace, Gatarek & Musiela · Centers for Medicare & Medicaid Services · CFA Institute · CME Group · Cox, Ingersoll & Ross · Du, Tepper & Verdelhan · Dupire · Breeden & Litzenberger · European Central Bank · Geltner & Miller · Government of Canada · Hagan et al. · Hamilton · Heath, Jarrow & Morton · Heston · HM Revenue & Customs · IAAO · IAS 19 · IAS 26 · IRS · J.P. Morgan · Kelly / Thorp · Kyle · Longstaff & Schwartz · McNeil, Frey & Embrechts · National Center for Education Statistics · O'Kane & Turnbull · Schwartz · Streamlined Sales Tax Governing Board · Tax Foundation · UK Government · US Bureau of Economic Analysis · US Securities and Exchange Commission · US Social Security Administration · US Treasury · Vasicek · Li · Warren & Tyagi.
§ 4
The register
All 179 finance calculators, with the relationship each one implements and the reference it is implemented against. The typeset formula and a worked example are on each calculator’s own page.
401(k) Calculator401k-calculator
e_t = min(s_t · c, limit), m_t = μ · min(e_t, s_t · κ), B_(t+1) = (B_t + e_t + m_t)(1+r)
IRC §402(g) elective-deferral limit; typical safe-harbour match design (50% of the first 6%).
529 College Savings Calculator529-college-savings-calculator
B_n = B_0(1+r)^n + c ((1+r)^n-1)/(r); penalty = E_(nq) (t + 0.10)
IRC §529 for qualified tuition programmes; the 10% additional tax is in §529(c)(6). State deductions follow each state's own statute.
Alternative Minimum Tax Calculatoramt-calculator
AMT = max(0, T_(min) - T_(reg))
IRS Rev. Proc. 2025-32 §3.10 (2026 exemption, phase-out and 28% threshold).
American Option Calculatorlongstaff-schwartz-american-option-engine
V_0 = E[ e^(-rτ) h_τ(S_τ) ], C_t(S_t) ≈ Σ_(j=0)^(3) β_j L_j((S_t)/(K)), exercise at t if h_t(S_t) > C_t(S_t)
Longstaff & Schwartz (2001), "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies 14(1); Cox, Ross & Rubinstein (1979) binomial benchmark; Hull, Options, Futures, and Other Derivatives.
Amortization Calculatoramortization-calculator
M = P (r(1+r)^(n))/((1+r)^(n)-1), r = (APR)/(1200), n = months
Standard annuity-immediate amortization formula (Broverman, Mathematics of Investment and Credit), evaluated with expm1/log1p for numerical stability. The APR is treated as a nominal annual rate compounded monthly — the usual US consumer-loan payment convention. Reg Z Appendix J defines the actuarial APR itself, which includes finance charges and is not simply the note rate divided by twelve.
Annuity Calculatorannuity-calculator
PV = PMT·(1-(1+i)^(-n))/(i); FV = PMT·((1+i)^(n)-1)/(i)
Broverman, Mathematics of Investment and Credit, ch. 2; the same identity underlies the Reg Z amortisation convention.
APR Calculatorapr-calculator
P - F = M · (1 - (1+a)^(-n))/(a), M = (P i)/(1-(1+i)^(-n))
Regulation Z, 12 CFR §1026.22 (APR); standard amortising payment formula.
APY Calculatorapy-calculator
APY = (1 + (r)/(n))^(n) - 1, APY_(∞) = e^(r) - 1, i = (r)/(n)
Truth in Savings Act (Regulation DD) definition of annual percentage yield; standard effective-rate algebra.
ARM Calculatorarm-calculator
r_(reset) = min(i + m, r_0 + cap_(periodic), r_0 + cap_(life))
Regulation Z (12 CFR §1026.20) adjustable-rate reset: index plus margin, periodic and lifetime caps.
Asset Location Calculatorasset-location-calculator
After-tax future value in a taxable versus a tax-deferred account, applying the IRC §1 and §408 treatment of contributions and growth.
Auto Lease Calculatorauto-lease-calculator
M = (C - R)/(n) + (C + R)· MF; APR = 2400· MF
Standard lease algebra as used by the Federal Reserve's leasing disclosure (Reg M) and by ALG residual guides.
Auto Loan Calculatorauto-loan-calculator
F = Price - Down - Trade + Tax + Fees, PMT = (F · i)/(1-(1+i)^(-n)), i = (APR)/(12)
Standard vehicle-retail financing under Regulation Z (12 CFR §1026) APR disclosure; state motor-vehicle sales-tax treatment of trade-ins.
Autocallable Note Calculatorautocallable-note-calculator
S_(t+Δ) = S_texp[(r-q- 12σ^2)Δ + σ√(Δ) Z]
Standard Monte Carlo structured-product practice; the payoff convention follows the Phoenix autocallable described in Bouzoubaa & Osseiran, Exotic Options and Hybrids.
Average Return Calculatoraverage-return-calculator
CAGR is the geometric mean and the arithmetic average is linear (Broverman, Mathematics of Investment and Credit).
Black-Litterman Portfolio Calculatorblack-litterman-portfolio-equilibrium-engine
= δ Σ w_(mkt), E[R] = [(τΣ)^(-1) + P^( )Ω^(-1)P]^(-1)[(τΣ)^(-1) + P^( )Ω^(-1)Q], w^(*) = (δΣ)^(-1)E[R]
Black & Litterman (1992), "Global Portfolio Optimization," Financial Analysts Journal; He & Litterman (1999) on calibrating the view-uncertainty matrix; Idzorek (2007) on confidence-based Omega.
Black-Scholes Options Calculatorblack-scholes-options-pricing-core
C = S e^(-qT) N(d_1) - K e^(-rT) N(d_2), d_1 = (ln(S/K) + (r - q + (1)/(2)σ^2)T)/(σ√(T)), d_2 = d_1 - σ√(T)
Black & Scholes (1973), "The Pricing of Options and Corporate Liabilities"; Merton (1973) extension to continuous dividend yields; Hull, Options, Futures, and Other Derivatives (closed-form Greeks and put-call parity).
Boat Loan Calculatorboat-loan-calculator
Level payment amortisation at a fixed rate (Broverman, Mathematics of Investment and Credit).
Bond Duration Calculatorbond-duration-calculator
P = Σ_(t=1)^(n)(c)/((1+y/f)^t) + (F)/((1+y/f)^n), D = (Σ (t/f)PV_t)/(P), D^(*) = (D)/(1+y/f)
Fabozzi, "Bond Markets, Analysis and Strategies", ch. 4–5.
Bond Yield Calculatorfixed-income-bond-yield-matrix
P(y) = Σ_(k=1)^(N) (CF_k)/((1 + y/m)^k), D_(Mac) = (1)/(P)Σ_(k=1)^(N) t_k PV_k, D_(mod) = (D_(Mac))/(1 + y/m)
Frank J. Fabozzi, Bond Markets, Analysis, and Strategies — present-value pricing of bonds and the measurement of duration and convexity.
Break-Even Calculatorbreak-even-calculator
CM = P - V, Q^(*) = (F)/(P-V), MoS = (Q - Q^(*))/(Q), π = Q(P-V) - F
Horngren, Datar & Rajan, "Cost Accounting", ch. 3 (cost-volume-profit analysis).
Budget Calculatorbudget-calculator
needs≤ 50%, wants≤ 30%, savings≥ 20%, Σ shares=100%
Warren & Tyagi, All Your Worth (2005) — the 50/30/20 heuristic, not a regulatory standard.
Burn Rate and Runwayburn-rate-runway
cash_m = cash_(m-1) + R_m - E_m
Monthly cash roll-forward with compounding revenue and expenses (Brigham & Ehrhardt, Financial Management, cash budgeting).
Business Loan Calculatorbusiness-loan-calculator
Level amortisation plus the DSCR coverage test; SBA SOP 50 10 sets the 1.25× minimum.
CAGR Calculatorcagr-calculator
CAGR = ((V_n)/(V_0))^(1/n) - 1, g_(real) = (1+CAGR)/(1+π) - 1, t_2 = (ln 2)/(ln(1+CAGR))
Damodaran, "Investment Valuation", ch. 4; Fisher (1930) on real versus nominal rates.
Canadian Mortgage Calculatorcanadian-mortgage-calculator
Interest Act (R.S.C. 1985, c. I-15) s. 6: nominal rate compounded semi-annually, paid monthly.
Cap Rate Calculatorcap-rate-calculator
cap = (NOI)/(P), DSCR = (NOI)/(DS), CoC = (NOI - DS)/(E)
Appraisal Institute, "The Appraisal of Real Estate", 15th ed.
Cap Table Calculatorcap-table-calculator
p = (pre-money)/(S_(existing) + P), P = (q)/(1-q)(S_(existing) + n)
Standard priced-round arithmetic with a pre-money pool, following the NVCA model legal documents.
Capital Allocation Calculatorcapital-allocation-matrix
w_T ∝ Σ^(-1)(μ - r_f 1), E[R_c] = r_f + y (E[R_T] - r_f), y^(*) = (E[R_T] - r_f)/(A σ_T^(2))
Markowitz (1952) mean-variance portfolio selection; Tobin (1958) separation theorem; two-asset tangency weights from the inverse covariance matrix; quadratic (mean-variance) utility.
Capital Gains Tax Calculatorcapital-gains-tax-calculator
T = Σ_i τ_i · G_i + 0.038 · min(NII, MAGI - threshold)
IRC §1(h) (2025 long-term rates, Rev. Proc. 2024-40); IRC §1411 (net investment income tax).
Caplet Bootstrap Calculatorcaplet-bootstrap-calculator
Cap_n = Σ_(i=1)^(n) Caplet_i; Caplet_n = Cap_n - Cap_(n-1)
Brace, Gatarek & Musiela (1997) for the LIBOR market model; the caplet identity is standard Black (1976) cap pricing.
CAPM Calculatorcapm-asset-equilibrium-core
E[R_i] = R_f + β_i(E[R_m] - R_f), β_i = (Cov(R_i, R_m))/(Var(R_m)) = (ρ_(iM) σ_i)/(σ_m)
Sharpe (1964), "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk"; Lintner (1965); Mossin (1966); textbook treatment in Hull, Options, Futures, and Other Derivatives.
Cash Back vs Low Interestcash-back-vs-low-interest-calculator
Both offers compared over the same term and price as present values (Broverman, Mathematics of Investment and Credit).
CD Calculatorcd-calculator
APY = (1 + (r)/(n))^n - 1, M = P(1 + (r)/(n))^(nt)
Truth in Savings Act, 12 CFR §1030.4 (APY definition); standard compound-interest formula.
CDO Tranche Calculatorcdo-tranche-calculator
p(M) = [( ^(-1)(PD) - √(ρ) M)/(√(1-ρ))]; L = LGD· p(M)
Vasicek (1987) loan portfolio value; Li (2000) for the Gaussian copula applied to CDOs; BCBS (2005) for the regulatory use.
Charitable Trust Calculatorcharitable-trust-calculator
CRUT: V(1-p)^n; CRAT: V(1-p a(n,r)); CLAT: V - A a(n,r)
IRC §664 (charitable remainder trusts), §7520 (valuation of annuities and split interests), §170(f)(2) (charitable lead trusts).
CIR Interest Rate Calculatorcox-ingersoll-ross-stochastic-term-engine
dr_t = κ(θ - r_t) dt + σ√(r_t) dW_t, P(t,T) = A(t,T) e^(-B(t,T)r_t), γ = √(κ^2 + 2σ^2)
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985), "A Theory of the Term Structure of Interest Rates," Econometrica 53(2), 385-407 (closed-form affine bond price).
Closing Costs Calculatorclosing-costs-calculator
points = L× p; prepaid = (L r d)/(365); cash to close = D + Σ C
The Loan Estimate and Closing Disclosure forms required by the TILA-RESPA integrated disclosure rule (12 CFR §1026.19) define the line items.
College Cost Calculatorcollege-cost-calculator
Cost inflation compounding to the first year of study, on NCES/IPEDS tuition data.
Commission Calculatorcommission-calculator
Marginal tiering: each rate applies only to the slice above its threshold (Horngren et al., Cost Accounting).
Commodity Mean-Reversion Calculatorcommodity-mean-reversion-schwartz-engine
dX_t = κ(a - X_t) dt + σ dZ_t, X_t=ln S_t, F(S,T) = exp (e^(-κ T)ln S + (1-e^(-κ T))a^(*) + (σ^2)/(4κ)(1-e^(-2κ T)) )
Schwartz, E. S. (1997), "The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance 52(3), 923-973 (Model 1).
Compound Interest Calculatorcompound-interest-calculator
A = P(1 + (r)/(n))^(nt), APY = (1 + (r)/(n))^(n) - 1
Standard compound-interest future value and effective annual yield (Broverman, Mathematics of Investment and Credit; any algebra-of-finance text). Continuous mode uses A = Pe^{rt}, evaluated in daily steps. Contributions are added monthly before that month's compounding.
Construction Loan Calculatorconstruction-loan-calculator
I = L · r · (m)/(12) · d, d = 0.5
Interest capitalised on the average outstanding balance, the standard construction draw convention (Broverman, Mathematics of Investment and Credit).
Corporate Debt Calculatorcorporate-debt-financing-node
P = Σ_(t=1)^(n)(C)/((1+y)^(t)) + (F)/((1+y)^(n)), V_L = V_U + tD, WACC = (E)/(V)k_e + (D)/(V)k_d(1-t)
Modigliani & Miller (1963), corporate taxes and the value of the firm; Sharpe (1964) CAPM; standard bond-pricing and APV conventions (Brealey, Myers & Allen, Principles of Corporate Finance).
Covenant Headroomcovenant-headroom
leverage = (net debt)/(EBITDA), cover = (EBITDA)/(interest)
Lender definitions: leverage as a maximum, cover and DSCR as minimums (Brigham & Ehrhardt, Financial Management, ratio analysis).
Credit Card Payoff Calculatorcredit-card-payoff-calculator
r = (APR)/(1200), B_(k+1) = B_k (1+r) - pay_k, surplus paid to the priority debt
Revolving-credit amortisation with monthly interest = balance × APR ÷ 12, minimums paid first, simulated month by month (Broverman, Mathematics of Investment and Credit).
Credit Default Swap (CDS) Calculatorcredit-default-swap-hazard-rate-solver
S(t) = exp (-∫_0^t λ(u) du), s Σ_i α_i D(t_i) S(t_i) = (1-R)Σ_i D(t_i) [S(t_(i-1)) - S(t_i) ]
O'Kane, D. & Turnbull, S. (2003), "Valuation of Credit Default Swaps," Lehman Brothers Fixed Income Quantitative Research; ISDA Standard CDS Model (reduced-form hazard-rate pricing).
Credit Risk Capital Calculatorcredit-risk-capital-calculator
CVA ≈ LGD · EPE · PD(t) · DF, RWA = E · RW
Basel standardised approach; flat-exposure CVA approximation.
Currency Convertercurrency-converter
A_(to) = A_(from) × (X_(to))/(X_(from)) × (1 - f)
European Central Bank euro foreign exchange reference rates (daily fixing, 16:00 CET).
CVA Hedge and Capital Calculatorcva-capital-calculator
CVA ≈ LGD ·(1-e^(-λ T))· DF; K = [LGD N(·) - PD LGD] MA
Basel II/III IRB corporate formula (BCBS 128, ¶272); CVA as the risk-neutral expected loss on expected positive exposure.
DCF Valuation Calculatordiscounted-cash-flow-valuation-core
FCFF = EBIT (1-T) + D&A - CapEx - Δ NWC, EV = Σ_(t=1)^(n) (FCFF_t)/((1+WACC)^t) + (TV)/((1+WACC)^n), TV = (FCFF_(n+1))/(WACC - g)
Standard corporate-finance treatment as in Brealey, Myers, and Allen, Principles of Corporate Finance (valuing a business by discounted cash flow); Gordon-growth terminal value after Gordon and Shapiro (1956).
Debt Consolidation Calculatordebt-consolidation-calculator
APR_w = (Σ b_i r_i)/(Σ b_i); I_(saved) = I_(current) - I_(loan)
Level-payment amortisation with the weighted average taken by balance, the convention lenders quote (Broverman, Mathematics of Investment and Credit).
Debt Sculptingdebt-sculpting
Debt capacity as the present value of the maximum annual service (Brealey, Myers & Allen, Principles of Corporate Finance).
Debt Snowball Calculatordebt-snowball-calculator
B_i arrow B_i(1 + (r_i)/(12)) - min_i, pool = extra + Σ_(cleared) min_i, snowball: argmin B_i, avalanche: argmax r_i
Consumer debt-rollover simulation with minimum-payment conventions for revolving credit (Broverman, Mathematics of Investment and Credit).
Debt-to-Income Calculatordebt-to-income-calculator
DTI_(back) = (housing + other debt)/(gross monthly income)
Consumer Financial Protection Bureau qualified-mortgage standards (43% back-end); Fannie Mae and FHA underwriting guidelines.
Depreciation Calculatordepreciation-calculator
SL: D = (C-S)/(n), DDB: D_t = BV_(t-1)·(2)/(n), SYD: D_t = (C-S)(n-t+1)/(n(n+1)/2)
IAS 16 (property, plant and equipment) and ASC 360 for financial-reporting depreciation; standard accounting texts for the SL, DDB-with-switch and SYD schedules. These are book methods — IRS Publication 946 MACRS tables use different conventions (half-year, no salvage) and are not implemented here.
Development Feasibilitydevelopment-feasibility
Residual land value after build cost, fees and profit (Appraisal Institute, The Appraisal of Real Estate).
Discount Calculatordiscount-calculator
P_1 = P(1-d_1), P_2 = P_1(1-d_2), effective = 1-(1-d_1)(1-d_2)
Standard retail markdown arithmetic (Horngren et al., Cost Accounting).
Distribution Waterfalldistribution-waterfall
catch-up = pref × (c)/(1 - c)
European (whole-fund) waterfall with a simple non-compounding preference, following the ILPA waterfall and reporting standards.
Dividend Reinvestment Calculatordividend-reinvestment-calculator
n_(t+1) = n_t(1 + (d_0(1+g)^t)/(P_t)); yield on cost = (D_t)/(cost)
Standard dividend-reinvestment arithmetic; the yield-on-cost convention follows the CFA Institute's equity income treatment.
Down Payment Calculatordown-payment-calculator
S_n = S_0(1+i)^n + c ((1+i)^n - 1)/(i)
Homeowners Protection Act, 12 USC §4902, for the 78% automatic termination and 80% request thresholds.
DSCR Calculatordscr-calculator
DSCR = (NOI)/(annual debt service)
Commercial underwriting covenant, typically 1.20×–1.35×; SBA SOP 50 10 requires 1.25× for 7(a) loans.
Dual-Curve Swap Calculatordual-curve-swap-calculator
NPV = NΣ_i τ_i (f_i - K) DF_i^(OIS)
Post-crisis swap pricing convention (OIS discounting); see Hull & White, 'The Valuation of Collateralized Derivatives'.
Dynasty Trust Calculatordynasty-trust-calculator
inclusion = 1 - (X_(allocated))/(V); GST = 0.40 × V × inclusion
IRC §2601–2664 (generation-skipping transfer tax), §2632 (allocation of exemption); state estate and trust taxation.
Emergency Fund Calculatoremergency-fund-calculator
target = S × m, B_n = B_(n-1)(1 + (apy)/(12)) + c
Standard sinking-fund recursion with monthly compounding; target follows the conventional months-of-essential-spend rule used by CFPB and FDIC consumer guidance.
Estate Tax Calculatorestate-tax-calculator
T = 0.40 × max(0, E - D - X)
IRC §2001 (estate tax), §2010(c) (basic exclusion, portability), §2503(b) (annual gift exclusion).
Expected Loss Calculatorexpected-loss-calculator
EL = PD × LGD × EAD
Basel/IFRS 9/CECL expected-loss framework.
Expected Shortfall Calculatorexpected-shortfall-calculator
ES_α = μ + σ(φ(z_α))/(1-α)
Basel Committee, Minimum capital requirements for market risk (d457, 2019), §3 for the 97.5% ES; McNeil, Frey & Embrechts, Quantitative Risk Management, ch. 2 for the closed forms.
Federal Income Tax Calculatorfederal-income-tax-calculator
T = Σ_i τ_i · (min(I, U_i) - L_i), I = max(0, G - D_(pre) - D_(std))
IRS Rev. Proc. 2024-40 (2025 brackets and standard deduction), 26 U.S.C. §1.
FHA Loan Calculatorfha-loan-calculator
MIP_(annual) = rate(LTV, loan) × (B + 0.0175B)
HUD Handbook 4000.1 II.A.4.d for the premium schedule; the 2025 FHA floor is $524,225 for a one-unit property.
Forex & Currency Calculatorcross-border-forex-liquidity-node
F = S e^((r_d - r_f - b)T), b = r_d - r_f - (1)/(T)ln (F)/(S), r_f^(hedged) = r_f + (1)/(T)ln (F)/(S)
Covered interest rate parity (continuous-compounding form); Du, Tepper & Verdelhan (2018), "Deviations from Covered Interest Rate Parity," Journal of Finance 73(3), 915-957; BIS Quarterly Review on the cross-currency basis.
FRTB GIRR Calculatorfrtb-sa-calculator
WS_k = RW_k s_k; ρ_(kl) = max(e^(-θ|T_k-T_l|/min(T_k,T_l)), 0.40); K = Σ WS_k^2 + Σ_(k≠ l)ρ_(kl) WS_k WS_l
BCBS MAR21 (FRTB standardised approach), general interest rate risk delta charge, intra-bucket aggregation.
Fund Metrics Calculatorfund-metrics-tvpi-dpi
TVPI = (D + NAV)/(P), DPI = (D)/(P), Σ (CF_t)/((1+r)^t) = 0
TVPI, DPI and IRR by bisection on the dated flows, per the ILPA reporting template.
Future Value Calculatorfuture-value-calculator
Compound growth with monthly contributions at the monthly-equivalent rate (Broverman, Mathematics of Investment and Credit).
GARCH Volatility Calculatorgarch-volatility-forecasting-solver
σ_t^2 = ω + α ε_(t-1)^2 + β σ_(t-1)^2, E_t[σ_(t+h)^2] = σ^2 + (α + β)^(h-1)(σ_(t+1)^2 - σ^2), σ^2 = (ω)/(1-α-β)
Engle (1982) for ARCH; Bollerslev (1986), "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Econometrics 31, for GARCH; forecast algebra as in standard time-series texts (e.g. Tsay, Analysis of Financial Time Series).
GDP & Macroeconomics Calculatormacroeconomic-gdp-output-engine
GDP = C + I + G + X - M, Real GDP = (Nominal GDP)/(Deflator/100), g_(ann) = (1+g_(QoQ))^(4) - 1
US Bureau of Economic Analysis (BEA) expenditure approach and implicit GDP price deflator; World Bank production (value-added) approach; standard annualization of quarterly growth.
Gift Tax Calculatorgift-tax-calculator
G_(taxable) = G - D_(marital) - D_(charity) - 19,000 × n
IRC §2503(b) annual exclusion and §2010(c) basic exclusion, at the 2025 amounts from Rev. Proc. 2024-40; rate schedule at §2001(c).
GRAT and IDGT Sale Calculatorgrat-idgt-calculator
GRAT gift = V - A a(n,r); remainder = V(1+g)^n - A((1+g)^n-1)/(g)
IRC §2702 (retained interests), §7520 (valuation); the AFR is published monthly under §1274.
GST and Charitable Estate Calculatorgst-charitable-estate-calculator
GST = 0.40 × max(0, T - X_(allocated))
IRC §2601 (GST tax), §2055 (charitable deduction), §2632 (allocation).
HELOC Calculatorheloc-calculator
M_(draw) = P · i, M_(repay) = (P i)/(1-(1+i)^(-n))
HELOC structure — interest-only draw, then fully amortising repayment — under Regulation Z open-end credit rules (12 CFR §1026.40).
Heston Volatility Calculatorheston-stochastic-volatility-surface-engine
C = S e^(-qT) P_1 - K e^(-rT) P_2, P_j = (1)/(2) + (1)/(π)∫_0^(∞) Re [(e^(-iφ ln K) φ_j(φ))/(iφ)] dφ, dv_t = κ(θ - v_t) dt + ξ√(v_t) dW_t^(v), d W^S, W^v _t = ρ dt
Heston (1993), "A Closed-Form Solution for Options with Stochastic Volatility and Applications to Bond and Currency Options", Review of Financial Studies 6(2), 327-343; Gatheral, The Volatility Surface (2006), for the P1/P2 decomposition and numerical integration.
HJM Interest Rate Calculatorheath-jarrow-morton-interest-rate-framework
df(t,T) = α(t,T) dt + σ(t,T) dW_t, α(t,T) = σ(t,T)∫_t^T σ(t,u) du, P(t,T) = exp (-∫_t^T f(t,u) du)
Heath, D., Jarrow, R., & Morton, A. (1992), "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica 60(1), 77-105.
Home Equity Loan Calculatorhome-equity-loan-calculator
CLTV = (first + second)/(value)
Lender CLTV policy, typically 85%, with level-payment amortisation for the second mortgage (Broverman, Mathematics of Investment and Credit).
Home Sale Proceeds Calculatorhome-sale-proceeds-calculator
gain = P - C - (P_0 + I); taxable = max(0, gain - exclusion)
IRC §121 for the primary-residence exclusion; IRC §1001 and §1016 for amount realised and adjusted basis.
House Affordability Calculatorhouse-affordability-calculator
L = PI·((1+r)^(n)-1)/(r(1+r)^(n)), Price = (L)/(1-d), PI = min(Gf, Gb - D) - T - I - H
Inverted annuity-immediate loan sizing (Broverman, Mathematics of Investment and Credit); front- and back-end DTI conventions from US mortgage underwriting (28/36 conventional, 31/43 FHA, 41 VA). Computed with Big.js decimal arithmetic.
Household Cash Flow Planhousehold-cash-flow-plan
Net_t = (income_t + W_t + sale_t) - (spending_t + T_t)
Cash-flow view of the deterministic household projection, net of sale costs and mortgage (Brigham & Ehrhardt, Financial Management, cash budgeting).
Household Financial Planhousehold-financial-plan
B_(t+1) = B_t(1+r) + S_t - W_t, W_t = spending_t + T_t
Deterministic multi-year projection; tax from the 2025 federal brackets, drawdown order from the retirement tax engine.
HSA Calculatorhsa-calculator
B_(m+1) = B_m(1+(r)/(12)) + (c)/(12); saved = c (t + FICA)
IRC §223 and IRS Rev. Proc. 2024-25 for the 2025 limits; the payroll exclusion follows IRC §125 cafeteria-plan treatment.
Inflation Calculatorinflation-calculator
V_(real) = (V_(nom))/((1+π)^n), (1+r_(real)) = (1+r_(nom))/(1+π), r_(nom)^(*) = (π)/(1-t)
Fisher (1930), "The Theory of Interest"; standard CPI deflation practice.
Interest Rate Calculatorinterest-rate-solver
P(1+i)^n - M((1+i)^n - 1)/(i) = 0
Amortising loan balance equation solved for i by bisection (Broverman, Mathematics of Investment and Credit).
Interest Rate Swap Calculatorinterest-rate-swap-calculator
Net = N(S - L), NPV = Net · (1-(1+r)^(-n))/(r)
Single-curve swap valuation; break-even at the par swap rate.
IRMAA and Medicare Premium Calculatorirmaa-medicare-calculator
C_(yr) = 12 (P_B + S_D + P_D)
CMS, 2026 Medicare Parts A & B Premiums and Deductibles fact sheet (14 November 2025).
IRR Calculatorinternal-rate-of-return-solver-core
NPV(r) = Σ_(t=0)^(N) (CF_t)/((1+r)^t) = 0, MIRR = ((FV_(+))/(-PV_(-)))^(1/N) - 1
Brealey, Myers, and Allen, Principles of Corporate Finance — capital budgeting with NPV and IRR; modified IRR as presented in Chan S. Park, Contemporary Engineering Economics.
IRRBB Calculatorirrbb-calculator
EVE = Σ (A_t)/((1+r_a)^t) - Σ (L_t)/((1+r_l)^t)
BCBS (2016), Interest rate risk in the banking book, standardised framework and supervisory outlier test.
Jump-Diffusion Option Calculatorjump-diffusion-merton-levy-process-engine
C = Σ_(n=0)^(∞) (e^(-λ' T)(λ' T)^(n))/(n!) C_(BS) (S,K,r_(n),σ_(n),T,q), λ' = λ(1+κ), κ = e^μ_J + (1)/(2)δ^(2) - 1, σ_(n)^(2) = σ^(2) + (nδ^(2))/(T), r_(n) = r - λκ + (nln(1+κ))/(T)
Merton, R. C. (1976), "Option pricing when underlying stock returns are discontinuous", Journal of Financial Economics 3 (1–2), 125–144.
Kelly Criterion Calculatorkelly-criterion-calculator
f^(*) = (p(b+1)-1)/(b), g(f) = pln(1+fb) + qln(1-f)
Kelly (1956); Thorp (2006) on the Kelly criterion in markets.
LBO Calculatorlbo-calculator
MOIC = (equity_(exit))/(equity_(entry)), IRR = MOIC^(1/t) - 1
Single entry, single exit IRR with no interim distributions or fund fees (Brealey, Myers & Allen, Principles of Corporate Finance).
Life Insurance Needsinsurance-needs-calculator
DIME needs analysis — debt, income, mortgage, education, less existing assets — the needs-based approach set out in the NAIC Life Insurance Buyer’s Guide.
Loan Payment Calculatorloan-payment-calculator
PMT = (P · i)/(1 - (1+i)^(-n)), i = (r)/(12), n = 12 · years, B_t = P(1+i)^t - PMT((1+i)^t - 1)/(i)
Standard annuity-immediate amortisation formula; the APR is taken as a nominal annual rate compounded monthly (US consumer-loan payment convention). Reg Z Appendix J defines the disclosed actuarial APR, which includes finance charges.
Local Volatility Calculatorlocal-volatility-calculator
σ^2(K,T) = (∂_T C + (r-q)K ∂_K C + qC)/( 12 K^2 ∂_(KK) C)
Dupire (1994), Pricing with a Smile; Breeden & Litzenberger (1978) for the density identity.
Long-Term Care Costltci-needs-calculator
Inflated care costs against available assets and income, following the NAIC Shopper’s Guide to Long-Term Care Insurance.
M&A Accretion / Dilutionmerger-accretion-dilution
Pro forma EPS after synergies and financing cost (Brealey, Myers & Allen, Principles of Corporate Finance).
Margin & Leverage Calculatormargin-liquidation-leverage-core
P_(liq)^(long) = P_0 (1-m_0)/(1-m), P_(liq)^(short) = P_0 (1+m_0)/(1+m), L = (Q P_0)/(Q P_0 m_0)
US Federal Reserve Regulation T (initial margin) and FINRA Rule 4210 (maintenance margin); standard broker margin-account accounting.
Market Microstructure Calculatorhigh-frequency-microstructure-signal-core
P^(μ) = (P^(a)V^(b) + P^(b)V^(a))/(V^(b) + V^(a)) = M + (S)/(2) I, M = (P^(a)+P^(b))/(2), S = P^(a)-P^(b), I = (V^(b)-V^(a))/(V^(b)+V^(a)), Δ p_t = λ q_t + ε_t, S_(Roll) = 2 max(-Cov(Δ p_t, Δ p_(t-1)), 0), RV = Σ_i r_i^2, BV = μ_1^(-2)Σ_i |r_i||r_(i-1)|
Kyle (1985), "Continuous Auctions and Insider Trading", Econometrica 53(6); Glosten & Milgrom (1985); Roll (1984), "A Simple Implicit Measure of the Effective Bid-Ask Spread"; Barndorff-Nielsen & Shephard (2004) for realized and bipower variation; Ait-Sahalia, Mykland & Zhang (2005) for the signature plot.
Markup & Margin Calculatormarkup-margin-calculator
P = C(1+m), g = (P-C)/(P) = (m)/(1+m), P_(target) = (C)/(1-g)
Standard retail pricing identity; margin-on-price convention used in gross-margin reporting (Horngren et al., Cost Accounting).
Marriage Penalty Calculatormarriage-penalty-calculator
P = T_(MFJ)(a+b-D_(joint)) - [T(a-D_a) + T(b-D_b)]
IRC §1(j) 2025 rate schedules and §164(b)(6) SALT cap; the standard deduction is §63(c).
MBS Prepayment Calculatormbs-prepayment-calculator
CPR(t) = 6%×min(t/30,1); SMM = 1-(1-CPR)^(1/12)
Public Securities Association prepayment model; the WAL and passthrough conventions follow Fabozzi, Fixed Income Analysis.
Monte Carlo Simulation Calculatormonte-carlo-stochastic-path-engine
S_(t+Δ t) = S_t exp((μ - (1)/(2)σ^2)Δ t + σ√(Δ t) Z), ln S_T ~ N(ln S_0 + (μ - (1)/(2)σ^2)T, σ^2 T)
Hull, Options, Futures, and Other Derivatives (Wiener processes and Ito’s lemma); Glasserman, Monte Carlo Methods in Financial Engineering (2003) for path simulation and antithetic variates.
Mortgage Calculatormortgage-principal-calculus-engine
M = P (r(1+r)^(n))/((1+r)^(n)-1), r = (APR)/(1200), n = months
Standard annuity-immediate amortization formula (Broverman, Mathematics of Investment and Credit); the APR is taken as a nominal annual rate compounded monthly, the usual US consumer-loan payment convention. Reg Z Appendix J defines the disclosed actuarial APR.
Mortgage Payoff Calculatormortgage-payoff-calculator
B_(k+1) = B_(k)(1+r) - (M + E), M = P (r(1+r)^(n))/((1+r)^(n)-1), r = (APR)/(1200)
Annuity-immediate payment (Broverman, Mathematics of Investment and Credit); the accelerated path is simulated month by month with 20-significant-digit decimal arithmetic (Decimal.js), capping the final payment at the remaining balance.
Mortgage Recast Calculatormortgage-recast-calculator
Re-amortising the remaining balance over the original term (Broverman, Mathematics of Investment and Credit).
Mutual Fund Calculatormutual-fund-calculator
Net return = gross return − expense ratio, compounded annually, per the SEC Form N-1A fee disclosure.
Net Investment Income Taxniit-calculator
IRC §1411; thresholds are statutory and not indexed for inflation.
Net Worth Calculatornet-worth-calculator
net worth = Σ A - Σ L, liquid = (A_(cash) + A_(invest)) - Σ L
Standard personal financial statement — assets at fair market value less liabilities (Brigham & Ehrhardt, Financial Management).
NPV Calculatornpv-calculator
NPV = -C_0 + Σ_(t=1)^(T) (CF_t)/((1+r)^t), 0 = -C_0 + Σ_(t=1)^(T) (CF_t)/((1+IRR)^t), PI = (Σ PV(inflows))/(C_0)
Brealey, Myers & Allen, "Principles of Corporate Finance", ch. 5–6; Graham & Harvey (2001) survey of CFO hurdle-rate practice.
Option Pool Dilutionoption-pool-dilution
The pool is carved out before the round, per the NVCA model legal documents.
Payback Period Calculatorpayback-period-calculator
PB = t^* + (|CumCF_(t^*)|)/(CF_(t^*+1)), CumCF_t = -C_0 + Σ_(i=1)^(t) CF_i
Brealey, Myers & Allen, "Principles of Corporate Finance", ch. 5 (payback as a screening rule).
Paycheck Calculatorpaycheck-calculator
net = G - G d - T (max(0, G - Gd - D)) - FICA - s(G - Gd) - 12p
IRC §1 (2026 brackets, IRS Rev. Proc. 2025-32); IRC §3101 (FICA); IRS Publication 15 (2026), social-security wage base $184,500.
Pension Calculatordefined-benefit-pension-projection-core
S_t = S_0(1+g)^(t), PBO = (a S_T (T+1) e^(-μ T))/((1+d)^(T)), Interest cost = d × PBO
IAS 19 Employee Benefits and IAS 26 Accounting and Reporting by Retirement Benefit Plans; projected unit credit method (actuarial present value of accrued benefits).
Personal Loan Calculatorpersonal-loan-calculator
Level payment amortisation M = P·i/(1 − (1+i)^−n) (Broverman, Mathematics of Investment and Credit).
PMI Calculatorpmi-calculator
PMI_(annual) = L · r_(pmi), LTV = (L)/(V)
Homeowners Protection Act: automatic termination at 78% LTV, cancellation on request at 80%.
Portfolio Optimization Calculatormarkowitz-efficient-frontier-optimization-matrix
σ_p^2(μ_p) = (Aμ_p^2 - 2Bμ_p + C)/(D), A = 1^( )Σ^(-1)1, B = 1^( )Σ^(-1)μ, C = μ^( )Σ^(-1)μ, D = AC - B^2
Markowitz, H. (1952), "Portfolio Selection", Journal of Finance 7(1); closed-form frontier constants following Merton (1972), "An Analytic Derivation of the Efficient Portfolio Frontier".
Present Value Calculatorpresent-value-calculator
Discounted cash flow with the discount factor 1/(1 + r/n)^(nt) (Broverman, Mathematics of Investment and Credit).
Property Pro Formaproperty-proforma
NOI = EGI - opex, cap = (NOI)/(price)
Standard real estate underwriting — NOI before capex, TI and leasing commissions (Appraisal Institute, The Appraisal of Real Estate).
Property Tax Calculatorproperty-tax-calculator
tax = max(0, V · r - E) · (m)/(1000)
Local assessment statutes; one mill equals $1 of tax per $1,000 of taxable value (IAAO, Property Appraisal and Assessment Administration).
PSLF Calculatorpslf-calculator
eligible = Direct employer full time plan certified
34 CFR 685.219, Public Service Loan Forgiveness Program.
QBI Deduction (199A) Calculatorqbi-deduction-calculator
D = min(0.2 QBI, 0.2 TI, W)
IRC §199A; IRS Rev. Proc. 2025-32 §3.26 for the 2026 thresholds.
Quarterly Estimated Tax Calculatorquarterly-estimated-tax-calculator
T = Σ slice × rate; harbour = min(0.9 T, 1.0 or 1.1 T_(-1))
IRC §6654(d) for the safe harbours; brackets and the standard deduction from Rev. Proc. 2024-40; self-employment tax under §1401.
Real Estate Investment Calculatorreal-estate-asset-yield-engine
R_0 = (NOI_1)/(V_0), PV = Σ_(t=1)^(n)(CF_t)/((1+Y)^(t)) + (TV_n)/((1+Y)^(n)), TV_n = (NOI_(n+1))/(R_(exit))
Appraisal Institute yield-capitalization and DCF conventions (The Appraisal of Real Estate); NCREIF definitions of NOI and cap rate; Gordon-growth alternative for terminal value.
Refinance Calculatordebt-refinancing-arbitrage-matrix
Break-even = (Closing costs)/(M_(old) - M_(new)), M = P (r(1+r)^(n))/((1+r)^(n)-1)
Annuity-immediate payment formula (Broverman, Mathematics of Investment and Credit) with the APR treated as a nominal annual rate compounded monthly; break-even is fixed cost divided by periodic saving. Reg Z Appendix J governs the disclosed APR, which this model takes as an input rather than recomputing.
Regime-Switching Model Calculatorregime-switching-hamilton-markov-engine
ξ_(t|t-1) = P ξ_(t-1|t-1), ξ_(t|t) = (ξ_(t|t-1) η_t)/(1^( )(ξ_(t|t-1) η_t)), η_(t,j) = (1)/(√(2π)σ_j)exp (-((y_t-μ_j)^2)/(2σ_j^2)), log L = Σ_(t=1)^(T)log [1^( )(ξ_(t|t-1) η_t)], E[duration_j] = (1)/(1-p_(jj))
Hamilton (1989), "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle", Econometrica 57(2), 357-384; Kim (1994) smoothing recursion; Hamilton, Time Series Analysis (1994), ch. 22.
Rent Affordability Calculatorrent-affordability-calculator
rent = min(0.30 G_(m), G_(a)/40, N(1-s)-E-D)
HUD defines housing above 30% of gross income as a cost burden and above 50% as severe; the 40× rule is the standard landlord screen.
Rent vs Buy Calculatorrent-vs-buy-calculator
C_(buy)(t) = D + K + Σ_(s=1)^(t)(M_s + carry_s) - benefit_t - E_t; C_(rent)(t) = Σ_(s=1)^(t) R_s - (D+K)[(1+i)^t-1]
Net-cost comparison as set out in the NYT rent-versus-buy model and in Geltner & Miller, Commercial Real Estate Analysis; equity is measured net of selling costs.
Retirement Calculatorretirement-calculator
B_(k+1) = (B_k + C_k)(1 + r_(pre)), B_(k+1) = (B_k - G_k)(1 + r_(post)), G_k = max (0, (I - S)(1+i)^(k))
Deterministic year-by-year accumulation with user-set returns (Broverman, Mathematics of Investment and Credit); Monte Carlo mode draws log-normal returns r = exp(mu − sigma²/2 + sigma·z) − 1 with z by Box-Muller.
Retirement Plan Monte Carloretirement-plan-monte-carlo
r_t = μ + σ z_t, success = Pr[balance ≥ 0 t]
Monte Carlo simulation with independent normal annual returns, replayed by the household projection engine on the accumulation identity (Broverman, Mathematics of Investment and Credit).
Retirement Tax & Withdrawalretirement-tax-withdrawal
RMD = (D)/(divisor(age)), T = Σ_i band_i × τ_i, W = max(0, S + T(W) - I)
IRS Rev. Proc. 2024-40 and Rev. Proc. 2025-32 (2025 brackets and deductions); IRS Pub. 915 Worksheet 1 (taxable Social Security); Uniform Lifetime Table, Treas. Reg. §1.401(a)(9)-9; SECURE 2.0 RMD start age of 73.
RMD Calculatorrmd-calculator
RMD_t = (B_(t-1))/(D(age_t)); B_(t) = (B_(t-1) - RMD_t)(1+g)
IRC §401(a)(9) and Treas. Reg. §1.401(a)(9)-9, Uniform Lifetime Table; applicable age from SECURE 2.0 §107.
ROI Calculatorroi-calculator
ROI = ((V_(exit) + Income) - (Cost_0 + Costs))/(Cost_0 + Costs), r_(annual) = (1+ROI)^(1/n) - 1
Brigham & Ehrhardt, "Financial Management: Theory & Practice", ch. 11 (profitability measures).
Roth IRA Calculatorroth-ira-calculator
FV = C((1+r)^n - 1)/(r), Roth: FV_R = FV(C), Traditional: FV_T = FV ((C)/(1-t_0))(1-t_1)
IRC §408A (Roth IRA); standard tax-deferred equivalence argument.
Rough Volatility Calculatorrough-volatility-fractional-brownian-core
v_t = ξ_0 exp (η B^(H)_t - (1)/(2)η^2 t^(2H)), Cov(B^(H)_t, B^(H)_s) = (1)/(2)(t^(2H) + s^(2H) - |t-s|^(2H)), E[v_t] = ξ_0, Var[v_T] = ξ_0^2(e^η^2 T^(2H) - 1)
Gatheral, Jaisson & Rosenbaum (2018), "Volatility is Rough", Quantitative Finance 18(6), 933-949; Bayer, Friz & Gatheral (2016), "Pricing under Rough Volatility", for the rough Bergomi variance specification.
RSU Calculatorequity-compensation-rsu
Ordinary income at vest; flat supplemental withholding rate.
SABR Swaption Calculatorstochastic-alpha-beta-rho-swaption-matrix
σ_B(K,F) = (α)/((FK)^((1-β)/2)[1 + ((1-β)^2)/(24)ln^2 (F)/(K) + ((1-β)^4)/(1920)ln^4 (F)/(K)])·(z)/(x(z))·1 + [((1-β)^2)/(24)(α^2)/((FK)^(1-β)) + (ρβνα)/(4(FK)^((1-β)/2)) + (2-3ρ^2)/(24)ν^2]T, z = (ν)/(α)(FK)^((1-β)/2)ln(F)/(K), x(z) = ln(√(1-2ρ z + z^2) + z - ρ)/(1-ρ)
Hagan, Kumar, Lesniewski & Woodward (2002), "Managing Smile Risk", Wilmott Magazine, eqs. (2.17a-2.17c) for the implied-volatility smile and (2.18) for the at-the-money limit.
Salary to Hourly Calculatorsalary-to-hourly-calculator
r_(pay) = (S - bonus)/(h_w · w); r_(eff) = (S_(total))/(h_w · w - h_(leave))
FLSA regular-rate definition (29 CFR §778.109) for the payslip rate; the effective rate is the same identity with hours actually worked as the denominator.
Sales Tax Calculatorsales-tax-calculator
c = state + local, G = N(1+c), N = (G)/(1+c)
State revenue codes; Streamlined Sales Tax sourcing conventions.
Savings Calculatorsavings-calculator
B_(k+1) = B_k(1+r_m) + D, r_m = (1+(r_(net))/(m))^(m/12)-1, r_(net) = APY (1-t)
Monthly-compounded future value with level deposits (compound interest combined with the future value of an annuity; Broverman, Mathematics of Investment and Credit). Doubling time uses the Rule-of-72 approximation 72 ÷ rate; the real value deflates the balance by (1 + inflation)^(years).
Savings Goal Calculatorsavings-goal-calculator
PMT = ([FV - PV(1+i)^n] i)/((1+i)^n - 1), i = (r)/(12), n = 12t
Standard sinking-fund formula (Broverman, Mathematics of Investment and Credit).
Scenario DCF Valuation Calculatormulti-scenario-dcf-valuation-matrix
FCFF_t = NOPAT_t - CapEx_t - _t, EV = Σ_(t=1)^(n)(FCFF_t)/((1+WACC)^(t)) + (TV)/((1+WACC)^(n)), TV = (FCFF_(n)(1+g))/(WACC-g)
Standard two-stage FCFF discounted-cash-flow valuation; Gordon (1959) growth model for terminal value; Damodaran, Investment Valuation, and Koller, Goedhart & Wessels, Valuation.
Scenario Provision Calculatorscenario-provision-calculator
EL = Σ_i w_i · PD_i · LGD_i · EAD
IFRS 9 / CECL multi-scenario expected credit loss.
Self-Employment Tax Calculatorself-employment-tax-calculator
SECA = 12.4%·min(0.9235P, 176,100) + 2.9%·0.9235P
IRC §1401 (SECA), 2025 Social Security wage base $176,100, IRS Schedule SE.
Sequence-of-Returns Backtestsequence-of-returns-backtest
B_t = B_(t-1)(1 + w r_(e,t) + (1-w) r_b - f) - W(1+π)^t
Bengen (1994), “Determining Withdrawal Rates Using Historical Data”; Kitces on sequence-of-returns risk. Equity data: S&P 500 total return, 1926–2025.
Series I Bond Calculatorseries-i-bond-calculator
c = f + 2π + fπ; V_m = P(1 + (c)/(2))^(m/6)
Treasury formula under 31 CFR Part 360; rates published by TreasuryDirect each May and November.
Sharpe Ratio Calculatorsharpe-ratio-calculator
Sharpe = (R_p - R_f)/(σ_p), Sortino = (R_p - R_f)/(σ_(down)), Treynor = (R_p - R_f)/(β), α = R_p - [R_f + β(R_m - R_f)]
Sharpe (1966, 1994); Sortino & Price (1994); Treynor (1965); Jensen (1968).
Simple Interest Calculatorsimple-interest-calculator
I = P · r · t, A = P + I
Simple interest with months converted to years (Broverman, Mathematics of Investment and Credit).
Social Security Claiming Calculatorsocial-security-claiming-calculator
Early benefit=PIA[1-min(m,36)(5)/(900)-max(m-36,0)(5)/(1200)], Delayed benefit=PIA(1+dc)
SSA, Early or Late Retirement (https://www.ssa.gov/OACT/quickcalc/early_late.html); SSA, Annual Statistical Supplement 2025, Appendix C (https://www.ssa.gov/policy/docs/statcomps/supplement/2025/apnc.html); SSA, OASDI Program Reference, Tables 9–10 (https://www.ssa.gov/policy/about/oasdi-reference.html).
State Estate Tax Calculatorstate-estate-tax-calculator
T = 0.40max(0, E - D - X_f) + r_smax(0, E - D - X_s)
IRC §2001 (federal); state estate tax statutes differ and should be checked directly.
State Income Tax Calculatorstate-income-tax-calculator
T = Σ_i (I_i × r_i)
Tax Foundation, 2026 State Individual Income Tax Rates and Brackets, compiled from state statutes.
Student Loan Calculatorstudent-loan-calculator
PMT = (B · i)/(1-(1+i)^(-n)), I_(total) = n · PMT - B, DTI = (12 · PMT)/(income)
Federal Direct Loan standard repayment mechanics (34 CFR 685.208).
Student Loan IDR Calculatorstudent-loan-idr-calculator
M = (s (AGI - m FPL))/(12)
34 CFR 685.209 (PAYE, ICR, SAVE) and 685.221 (IBR), with the 2025 HHS poverty guidelines.
Student Loan Refinance Calculatorstudent-loan-refinance-calculator
net = (I_(current) - I_(new)) - F_(forfeited)
Level-payment amortisation; the federal protections forfeited are 34 CFR 685 (IDR) and 685.219 (PSLF).
Swap Curve Calculatorswap-curve-calculator
DF_i = (1 - s_i Σ_(j<i) DF_j)/(1 + s_i), NPV = N(SΣ DF - (1 - DF_n))
Standard par-rate bootstrapping; single-curve swap valuation.
Tax Return Analyzer (1040)tax-return-analyzer
T = Σ_i (B_i r_i) + T_(LTCG) + T_(NIIT) - C
IRS Rev. Proc. 2025-32 (2026 brackets and standard deduction).
Tax-Loss Harvestingtax-loss-harvesting-calculator
Immediate tax saving and the time value of the deferral, subject to the IRC §1091 wash-sale rule (30 days).
Three-Statement Modelthree-statement-model
Income statement, cash flow and balance sheet from one set of drivers (Brigham & Ehrhardt, Financial Management).
Tip Calculatortip-calculator
T = base× r, total = base + tax + T, each = (total)/(n)
US service-tipping convention; state and local sales-tax schedules. The definition of a tip follows the FLSA tipped-employee rule at 29 CFR §531.
Trade Execution Calculatoroptimal-execution-almgren-chriss-protocol
x(t) = X (sinh (κ (T - t) ))/(sinh(κ T)), κ = (λ σ^(2))/(η), min_(x) E(x) + λ V(x)
Almgren, R. & Chriss, N. (2000/2001), "Optimal Execution of Portfolio Transactions," Journal of Risk 3(2), 5-39; the continuous-time hyperbolic-sine liquidation trajectory with linear temporary and permanent impact.
Traditional IRA Calculatortraditional-ira-calculator
D = C × [1 - (MAGI - a)/(b - a)]
IRC §219(b)(5) contribution limits and §219(g) deductibility phase-out, at the 2025 amounts.
Treasury Bill Calculatortreasury-bill-calculator
P = F(1 - d(n)/(360)), y_(inv) = (F-P)/(P)·(365)/(n)
US Treasury bill auction conventions: bank-discount basis, actual/360.
Treasury Futures Hedge Calculatortreasury-futures-hedge-calculator
N = (V · D_p)/(P · 1000 · D_f / CF)
Duration-based hedge ratio for Treasury futures; conversion factor adjusts for the cheapest-to-deliver.
UK Income Tax Calculatoruk-income-tax-calculator
T = Σ_i r_i × min(income - PA, band_i)
HMRC income tax rates and allowances 2025/26; NI Class 1 employee contributions; Pensions Act 2008 automatic enrolment.
UK Mortgage Calculatoruk-mortgage-calculator
Repayment mortgage with a product fee (Broverman, Mathematics of Investment and Credit).
UK VAT Calculatoruk-vat-calculator
VAT = net × r; net = (gross)/(1 + r)
VAT Act 1994: s.2(1) standard rate, Schedule 7A reduced, Schedule 4 zero-rated.
Unit Economics Calculatorstartup-unit-economics
LTV = (ARPA · GM)/(churn), payback = (CAC)/(ARPA · GM)
Gross-profit LTV with constant churn, as the present value of a decaying perpetuity (Brealey, Myers & Allen, Principles of Corporate Finance).
Unit Price Calculatorunit-price-calculator
u = (p)/(q), saving = (u_(dear) - u_(cheap))/(u_(dear)), waste = n/q q - n
Unit-pricing disclosure rules (EU 98/6/EC; US state unit-pricing statutes).
VA IRRRL Calculatorva-irrrl-calculator
recoupment = (closing costs + 0.005 × B)/(P_(old) - P_(new))
VA Pamphlet 26-7 Chapter 6; funding fee under 38 CFR 36.4313.
VA Loan Calculatorva-loan-calculator
F = r_(fee) × (P - D)
VA Lender’s Handbook (VA Pamphlet 26-7), Chapter 8 for the funding fee; 2025 FHFA conforming limit of $806,500.
Value at Risk (VaR) Calculatorvalue-at-risk-volatility-core
VaR_c = V_0 max(0, -(μ h + σ√(h) z_(1-c))), ES_c = V_0(-μ h + σ√(h) (φ(z_(1-c)))/(1-c)), σ_t^2 = λσ_(t-1)^2 + (1-λ) r_(t-1)^2
RiskMetrics Technical Document (J.P. Morgan, 1996) for the EWMA recursion with lambda = 0.94 on daily data; Basel Committee on Banking Supervision market-risk framework for VaR at 99% over 10 days; Hull, Options, Futures, and Other Derivatives (VaR and expected shortfall chapters).
VAT Calculatorvat-calculator
Add: G = N(1+r), VAT = Nr Remove: N = (G)/(1+r), VAT = G - N
EU VAT Directive 2006/112/EC, Art. 73 (taxable amount inclusive of tax); national GST legislation follows the same algebra.
WACC Calculatorwacc-calculator
K_e = r_f + β · ERP, WACC = (E)/(V)K_e + (D)/(V)K_d(1-T), V = E + D
Modigliani & Miller (1958, 1963) — the after-tax WACC follows the 1963 tax correction; Damodaran, "Applied Corporate Finance", ch. 7–8.
Working Capital and Cash Cycleworking-capital-cash-cycle
CCC = DIO + DSO - DPO
DIO and DPO against COGS, DSO against revenue (Brigham & Ehrhardt, Financial Management, ratio analysis).
XVA Calculatorxva-capital-calculator
FVA = s_f Σ EPE(t) DF(t) Δ t; KVA = k_e Σ K(t) DF(t) Δ t
XVA framework: FVA/FBA (Albanese & Andersen), MVA and KVA as the cost of funding margin and capital.
§ 5
Assumptions & limits
Every page states its own assumptions. These 6 recur across the catalog — if a decision depends on one of them, the model is not the last word.
No frictions. No transaction costs, bid–ask spread, slippage or market impact unless a page models them explicitly (only the execution and microstructure pages do).
One rate. Reinvestment at the same rate, and a flat term structure unless the page builds a curve. Real curves slope; that is what the bond and rate-model pages are for.
Thin tails. Monte Carlo pages draw normal innovations, so tails are thinner than observed markets. Value at Risk therefore reports parametric and historical estimates beside the simulated one.
Statutory constants. Tax brackets, contribution limits and RMD divisors carry the tax year they belong to and cite the notice they came from — they are data, not model.
Deterministic projections. Retirement and pension pages are year-by-year recursions, not optimisations: they show what a fixed plan does, not the plan that maximises something.
Single currency. No page converts between currencies mid-model except the FX page; mixing currencies is a decision the model cannot make for you.
§ 6
Verification
A finance number is only useful if it can be reproduced. Three checks run against every model before its page ships.
01
Identity check
Every schedule is reconciled against the annuity identity: the final balance must land on zero within one cent, or the page fails its own test.
02
Second implementation
Worked examples are generated from an independent implementation of the same formula, so a transcription error cannot hide behind a plausible number.
03
Seeded reproduction
Simulation pages re-run the same seed in CI and compare the output byte for byte; a change in the path set is a regression, not noise.
Where a model cannot be checked against a closed form — a stochastic volatility surface, a regime-switching filter — the page publishes the residual, the iteration count or the seed instead, so the answer can be audited rather than believed.
179 finance calculators · reviewed 2026-10-23Engineering methods: methodology/engineering · How they compute: the calculation system