Present Value (PV) of a Future Cash and Periodic Payment

Purpose

This calculator answers the reverse of a future-value question: what is a future lump sum, plus a series of regular payments, worth today? It is useful for evaluating a loan, an annuity, or any arrangement where you know a target future amount and a stream of periodic payments, and want to know their combined value in today's terms at a given interest rate.

Background

Discounting future money back to today

Money received or paid in the future is worth less today than the same amount in hand now, because money today can earn interest. This calculator discounts both the Future Value and every Periodic Payment back to the present using the given Interest Rate, and adds them together. A payment due sooner is discounted less (worth closer to its face amount) than one due later.

Direction of cash flows

Both Future Value and Periodic Payment are marked as either Incoming (money you receive) or Outgoing (money you pay), using the Future Value Direction and Periodic Payment Direction settings. The calculated Present Value is always shown as a positive amount, with a separate Cash Flow result telling you whether that present-day amount is money you would pay (Outgoing) or receive (Incoming), based on the combined directions of what you entered.

Inputs

Future Value

The lump sum due at the end of the Duration.

Future Value Direction

Whether the Future Value is Incoming (money you will receive) or Outgoing (money you will owe or pay).

Interest Rate

The annual nominal interest rate used to discount future amounts back to today (for example, 6 pct/yr). It is converted internally to a rate per Payment Interval.

Duration

The total length of time from today until the Future Value is due (for example, 5 yr). Together with Payment Interval, this determines the number of discounting/payment periods.

Periodic Payment

The fixed amount paid or received at every Payment Interval throughout the Duration.

Periodic Payment Direction

Whether the Periodic Payment is Incoming (money you receive each period) or Outgoing (money you pay each period).

Payment Interval

How often the Periodic Payment occurs and discounting is applied (for example, every month).

Payment When

Whether each Periodic Payment is made at the Period Start (an annuity due) or the Period End (an ordinary annuity). This affects how much each payment is discounted, since a payment at the start of a period is worth slightly more today than one at the end of the same period.

Results

Present Value

The value today of the Future Value and all Periodic Payment amounts combined, each discounted back from when it occurs. Always shown as a positive amount.

Cash Flow

States whether the Present Value represents money Outgoing (you would need to pay it today) or Incoming (you would receive it today), based on the combined direction of the Future Value and Periodic Payment you entered.

Understanding the Calculation

The annual Interest Rate is converted to an equivalent rate per Payment Interval. The number of periods is Duration divided by Payment Interval. The Present Value is then:

$$PV = \dfrac{FV}{(1+r)^{n}} + PMT \times (1 + r \times w) \times \dfrac{1-(1+r)^{-n}}{r}$$

where $FV$ and $PMT$ are the signed Future Value and Periodic Payment (positive if Incoming, negative if Outgoing), $r$ is the rate per period, $n$ is the number of periods, and $w$ is 1 for payments at Period Start or 0 for payments at Period End.

Example

Using the default inputs — a Future Value of 100,000 (Incoming), an Interest Rate of 6% per year, a Duration of 5 years, and a Periodic Payment of 1,000 (Outgoing) made monthly at the start of each month — the calculator returns a Present Value of approximately 22,549.25, with Cash Flow shown as Outgoing.

For comparison, making the same monthly payments at the end of each period instead (Payment When = Period End) slightly increases the result to about 22,802.00, since payments made later are discounted a little more, reducing how much they offset the incoming future value's present worth. Removing the periodic payments entirely (Periodic Payment = 0) leaves only the discounted future value, giving a Present Value of about 74,725.82.

Important Assumptions and Interpretation

• Periodic Payment is assumed to be a fixed, evenly spaced amount for every period; irregular payments are not supported.
• The Interest Rate is assumed constant for the entire Duration.
• Getting Future Value Direction and Periodic Payment Direction right matters for interpreting Cash Flow correctly, even though Present Value itself is always reported as a positive number.
• The result is a present-day valuation based on the assumed constant interest rate; it is not a market price and does not account for risk, fees, or taxes.