Rates and units
Lump-sum FV = PV × (1+r/m)^(m×t). Present value divides by the same factor. A nominal annual rate r is divided by m to obtain a compounding-period rate. An effective annual return is converted with exponential factors. Equivalent payment-period rate = (1+r/m)^(m/p) − 1. Daily compounding means 365 periods, and fractional terms use fractional exponents.
Payments and schedule order
Ordinary annuity FV = PMT × ((1+i)^n − 1)/i; due timing multiplies by 1+i. Present value divides the future-value factor by (1+i)^n. At zero rate, the annuity factor is n. Beginning payments enter before growth; end payments enter after growth and balance fees. Only complete payment periods receive deposits or withdrawals, including beginning timing. A partial final period earns growth without another payment.
Fees, withdrawals and reconciliation
Annual balance fees use retention (1−fee)^periodLength after growth. Annual deposit or spending increases begin after each completed year. Closing balance reconciles to principal + deposits + gross growth − fees − funded withdrawals. A withdrawal is capped at available funds. Unmet requested spending is tracked separately and no negative investment balance is created. Taxes are excluded.
Cash flows, NPV and bonds
For explicit equal periods, PV = Σ Cₜ/(1+i)^t and FV = Σ Cₜ(1+i)^(N−t). Row one is t=1. NPV subtracts its separate positive initial-outlay input once at t=0. Signed cash flows remain signed. Bond price discounts coupons and redemption on a coupon date; coupon and yield frequencies match. Accrued interest, irregular dates and clean/dirty settlement adjustments are not modeled.
Annualized return and purchasing power
CAGR = (ending/starting)^(1/years) − 1, assuming no intervening cash flows. Real terminal value = nominal terminal value/(1+inflation)^years. Exact real annual return = (1+nominal effective return)/(1+inflation) − 1. Historical-dollar conversion instead uses the ratio of two sourced monthly CPI observations.
Limits and numerical stability
Monetary inputs are bounded to one trillion, modeled balances to one quintillion, horizons to 200 years and schedules to 2,400 payment periods. Cash-flow input is limited to 120 rows. Individual form bounds can be narrower. Full internal precision is retained; UI and export presentation may round. Near-zero annuity rates use log1p/expm1. Non-finite or out-of-domain inputs produce errors.
Solver scope
The TVM equation uses signed financial cash flows. Rate solving searches −5% to 100% per period and rejects opposing PV/PMT signs to avoid claiming unique solutions for unsupported arrangements. Period solving is bounded to 0–2,400 with 200 bisection iterations and explicit failure. A fractional solved period is an algebraic result, not an invented payment schedule.
Source registry
The CPI source is the BLS historical CPI-U table: all items, U.S. city average, not seasonally adjusted, monthly observations January 2016–December 2024, checked September 11, 2026. Excel syntax follows Microsoft’s FV documentation. The financial-calculator guides identify their TI BA II Plus model and official manual. No live market rates or statutory contribution-limit tables are included.
Interpretation limits
Constant-rate projections do not assess market volatility, sequence risk, suitability, eligibility or complete tax outcomes. Display currencies change formatting only. The stock dividend model assumes annual distributions, fractional reinvestment and interpolated fractional final years. Maturity payouts use simple interest with a prorated tail period. These assumptions can differ from actual product contracts.