Future Value of an Annuity (and Present Value)
The forward time-value-of-money primitive the finance set (loan-amortization, upgrade-roi) builds on but never states.
Example
You enter
- Payment per period ($) 500
- Interest rate PER PERIOD (%) 0.5
- Number of periods 120
- Payment timing ordinary
You get
- Future value $81939.7
- Present value $45036.7
- Interest earned 21939.7
Details, formula, and sources
For a level payment PMT at a periodic rate i over n periods: FV = PMT [((1+i)^n - 1)/i] is what equal deposits grow to (a sinking fund for equipment replacement or a savings goal), and PV = PMT [(1-(1+i)^-n)/i] is the lump sum today worth that stream. An annuity-due (payments at the start of each period) multiplies both by (1+i). Depositing $500/month at 0.5%/month (6%/yr) for 120 months grows to $81,940 -- $60,000 contributed, $21,940 interest. The rate must be PER PERIOD (divide an annual rate by periods per year); to find the deposit for a target, PMT = FV i/((1+i)^n - 1). A bookkeeping aid; the account terms and a CPA govern.
Ordinary annuity FV = PMT [((1+i)^n - 1)/i]; PV = PMT [(1-(1+i)^-n)/i]; annuity-due multiplies each by (1+i). At i = 0 both reduce to PMT n. i is the periodic rate, n the number of payments; sinking-fund deposit PMT = FV i/((1+i)^n - 1).
Standard time-value-of-money identities (public-domain finance); the TILA/Reg Z (12 CFR 1030) compounding convention, by name.
The annuity FV/PV identities are universal public formulas; a finance text or the SEC investor.gov compound-interest material covers them.
Estimate. AHJ and licensed professional govern.
Field names used by the API: payment, rate_pct, periods, timing, future_value, present_value, interest_earned
- Annuity FV/PV FV = PMT [((1+i)^n - 1)/i], PV = PMT [(1-(1+i)^-n)/i]; due x (1+i)time-value-of-money identities
- Per-period rate i is the rate per payment period (annual rate / periods per year); n is the number of paymentsTVM convention