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 · discounted to today
Present value of promised payments and a future amount
What a stream of periodic payments and a future amount are worth today, discounted at a periodic rate over a chosen number of periods, with payment timing respected.
Verified engine
What the engine returns
The present value returned is the size of loan that stream of promises supports — the amount a lender should hand over today in exchange for it. Its sign is opposite to the payments, because the lump moves one way and the instalments move back the other.
Inputs the result is built from
Number of periods
Periodic interest rate
Periodic payment
Future amount
Payment timing
How the result is derived
MethodThe payment stream is scaled by the annuity factor and the timing factor, the future amount is discounted by the compound factor across the full span, and the two are combined under the opposed-sign cash-flow convention.
StandardTime-value-of-money relation solved for the present value
GuardA periodic rate at or below minus one hundred percent is refused — discounting is undefined once a single period consumes the entire balance, and the pack declares this refusal among its vectors.
What money promised later is worth now
Discounting is compounding read in reverse. Where its mirror page pushes today’s money out to a horizon, this one pulls promised money back, shrinking each payment by every period it has yet to wait. The further away a payment sits, the less of it survives the trip to the present — which is why distant promises are cheap and near ones are not.
This is what a loan principal actually is. The amount a lender hands over is the present value of the payments the borrower promises back, discounted at the periodic rate. The signed pack demonstrates it on itself: its declared vectors take level payments solved elsewhere in the pack and price them back to exactly the amount originally lent.
Timing changes worth. A payment made at the beginning of its period arrives one period sooner, so it is worth more today; a stream on beginning-of-period timing therefore reaches the same present value with a slightly smaller payment than the end-of-period stream does. The pack’s vectors carry both timings against a single balance, and the gap between the two payments is the price of that one period.
The future amount is a separate promise from the payments and is discounted on its own: a lone lump due at the horizon, shrunk by the full journey. Signs follow the cash-flow convention throughout — money moving toward you and money moving away carry opposite signs, and the sign of the answer reports which way today’s equivalent flows.
The payment stream is scaled by the annuity factor and the timing factor, the future amount is discounted by the compound factor across the full span, and the two are combined under the opposed-sign cash-flow convention.
When this calculation is used
- Pricing a stream of promised payments — what a schedule of instalments is worth as a single amount today.
- Recovering the loan amount implied by a quoted payment, rate and term, to check a lender’s arithmetic.
- Weighing a lump sum now against a schedule of payments later on explicit, stated terms.
- Seeing what one period of timing — beginning versus end — does to the worth of an identical stream.
Worked example
Price a mortgage-shaped promise: a level payment every period for a term of many years at a small per-period rate, with nothing further owed after the final payment.
The present value returned is the size of loan that stream of promises supports — the amount a lender should hand over today in exchange for it. Its sign is opposite to the payments, because the lump moves one way and the instalments move back the other.
Switch the timing to beginning-of-period and reprice: the same balance is now reached by a slightly smaller payment, the one period of waiting it removes being worth exactly the difference. The signed pack’s declared vectors round-trip both timings back to the same lent amount. Every figure is computed by the verified engine when the page loads; the page stores none of them.
What each input represents
Number of periods
How many periods the promises span, in the same unit as the rate. A count of periods, not years: each payment is discounted by exactly the number of periods it makes the present wait.
Periodic interest rate
The discount rate for a single period, as a percentage. At a rate of zero, waiting costs nothing and the present value is simply the payments and the future amount summed with their signs.
Periodic payment
The level amount promised each period, its sign carrying its direction. Optional and defaulting to nothing, for the case where only a single future amount is being priced.
Future amount
A lump promised at the horizon, over and above the payment stream. Also optional: it is discounted across the full span on its own, so distance erodes it more than it erodes any payment.
Payment timing
Whether each payment falls at the beginning or the end of its period. Beginning-of-period payments wait one period less, so the same stream is worth more today under that convention.
Assumptions and limits
- The periodic rate is constant over the whole span; a promise discounted at a changing rate is a different problem.
- Payments are level and occur exactly once per period at the chosen timing.
- Rate and period count share one unit — the relation cannot notice an annual rate paired with a monthly count.
- Signs follow the cash-flow convention, and the sign of the result is part of the answer rather than decoration.
- The promises are treated as certain: no default, fees or taxes enter the pricing.
What the guards protect against
- A periodic rate at or below minus one hundred percent is refused — discounting is undefined once a single period consumes the entire balance, and the pack declares this refusal among its vectors.
- The period count must be greater than zero: a promise spanning no periods has nothing to discount.
- The payment and the future amount are each bounded to a realistic magnitude so the arithmetic stays meaningful.
Provenance
Time-value-of-money relation solved for the present value
The payment stream is scaled by the annuity factor and the timing factor, the future amount is discounted by the compound factor across the full span, and the two are combined under the opposed-sign cash-flow convention.
An educational reference on discounting — not financial advice, and not a valuation of any particular instrument. The signed pack states its own derivation provenance; the page reports the verification state of the release it mounted rather than asserting one.