Workspace
The calculator's own fields, action and results arrive with the verified pack when you load it. Nothing is computed in this page.
Time value of money · read at the horizon
Future value of a present amount and periodic payment
The closing balance a present amount and a standing periodic payment reconcile to after a chosen number of periods at a periodic rate, with payment timing respected.
Verified engine
What the engine returns
The closing balance comes back with the opposite sign to the payments — cash out became value in — and it exceeds the plain sum of the payments. That excess is what the periodic rate added between each payment and the horizon, and it belongs mostly to the earliest payments.
Inputs the result is built from
Number of periods
Periodic interest rate
Present value
Periodic payment
Payment timing
How the result is derived
MethodThe present value and the payment stream — the latter scaled by the annuity factor and the timing factor — are carried to the horizon by the compound factor, under the opposed-sign cash-flow convention.
StandardTime-value-of-money relation solved for the future value
GuardA periodic rate at or below minus one hundred percent is refused — beyond that point a single period wipes out more than the whole balance and compounding stops meaning anything. The pack declares this refusal as one of its vectors.
What the ledger reads at the horizon
Think of this as the closing line of a ledger. A present amount, a payment that repeats every period, and a periodic rate are all carried forward together, and the future value is what they reconcile to on the last day. Its mirror page discounts the same quantities back to today; the two are one relation with the reader standing at opposite ends of it.
The result carries the opposite sign to the money you put in. Cash flowing in and cash flowing out have opposite signs throughout, so a stream of outgoing payments produces a positive closing balance and vice versa. A sign that surprises you is the relation reporting the direction of the flow, not a mistake to be corrected by hand.
A future value of exactly nothing is one of the most useful readings this page can give. Feed it a borrowed amount together with the level payment that amortises it, and the horizon balance comes back at zero — the loan viewed from its final day. The signed pack proves this on itself: its declared vectors take payments solved elsewhere in the pack and confirm they close the ledger.
The rate here is a periodic rate and the count is a count of periods — the same discipline as every solver in this family. Timing is a genuine input as well: a payment at the start of its period spends one extra period in the balance, so beginning-of-period timing reaches a higher horizon figure from identical payments.
The present value and the payment stream — the latter scaled by the annuity factor and the timing factor — are carried to the horizon by the compound factor, under the opposed-sign cash-flow convention.
When this calculation is used
- Projecting what an existing balance plus a standing periodic payment reconciles to at a chosen horizon.
- Verifying an amortisation: confirming that a quoted payment really drives a borrowed amount to nothing over the stated term.
- Reading the horizon-end difference between beginning-of-period and end-of-period timing on the same terms.
- Answering the savings direction of the family, where the ending balance is the unknown rather than the payment or the term.
Worked example
Start the ledger empty: no opening balance, a steady outgoing payment every period at a small per-period rate, kept up for many periods with ordinary end-of-period timing.
The closing balance comes back with the opposite sign to the payments — cash out became value in — and it exceeds the plain sum of the payments. That excess is what the periodic rate added between each payment and the horizon, and it belongs mostly to the earliest payments.
Then run the mirror check: enter a borrowed amount as the present value together with the exact level payment that amortises it, and the future value reads nothing — at either timing setting. The signed pack’s declared vectors perform precisely this round trip. Every figure on the page is computed by the verified engine at load; nothing here is a stored answer.
What each input represents
Number of periods
How many periods lie between now and the horizon, in the same unit as the rate. It is a count of periods, never a count of years — a monthly schedule over several years is a much larger number than the years alone.
Periodic interest rate
The rate for a single period, as a percentage. A rate of zero is allowed and meaningful: with nothing compounding, the horizon figure is simply the balance plus the payments, signed and summed.
Present value
The balance the ledger opens with. Optional and defaulting to nothing, for the pure payment-stream case. Its sign declares the direction of the opening flow relative to the payments.
Periodic payment
The amount that moves every period, with its sign carrying its direction. Also optional: left at nothing, the page reduces to carrying a lone balance forward to the horizon.
Payment timing
Whether each payment falls at the beginning or the end of its period. Beginning-of-period timing — the annuity-due convention — gives every payment one extra period in the balance before the horizon.
Assumptions and limits
- The periodic rate is constant across every period between now and the horizon.
- The payment is level — identical every period — and occurs exactly once per period at the chosen timing.
- Rate and period count share one unit; the relation cannot detect an annual rate paired with a monthly count.
- Signs follow the cash-flow convention: inflows and outflows are opposite, and the result’s sign is part of the answer.
- No fees, taxes or charges enter — this is the bare reconciliation, not a total cost.
What the guards protect against
- A periodic rate at or below minus one hundred percent is refused — beyond that point a single period wipes out more than the whole balance and compounding stops meaning anything. The pack declares this refusal as one of its vectors.
- The period count must be greater than zero: there is no horizon reading over no elapsed periods.
- The opening balance and the payment are each bounded to a realistic magnitude, so the arithmetic stays meaningful rather than merely finite.
Provenance
Time-value-of-money relation solved for the future value
The present value and the payment stream — the latter scaled by the annuity factor and the timing factor — are carried to the horizon by the compound factor, under the opposed-sign cash-flow convention.
An educational reference on the mechanics of compounding forward — not financial advice. The signed pack states its own derivation provenance, and the page reports the verification state of the release it mounted rather than asserting one.