Future Value (FV) of a Principal and Periodic Payment

Purpose

This calculator projects how much a starting amount and a series of regular contributions will grow to by a future date, given a compound interest rate. It is useful for planning savings goals, retirement contributions, or any scenario where you invest a lump sum and add to it periodically.

Background

Two sources of growth

The future value combines two things: the starting Present Value compounding on its own at the given Interest Rate, plus a series of equal Periodic Payment contributions, each of which also compounds from the time it is made until the end of the Duration. A payment made earlier has more time to grow than one made later, which is why when during each period the payment happens (Payment When) affects the result slightly.

Direction of cash flows

Both Present Value and Periodic Payment are marked as either Outgoing (money you put in) or Incoming (money you receive), using the Present Value Direction and Periodic Payment Direction settings. The calculated Future Value is shown as a positive amount with a direction as well, with a separate Cash Flow result telling you whether that final amount is money you would receive (Incoming) or a balance you would owe (Outgoing), based on what you had put in or received.

Inputs

Present Value

The lump sum invested (or borrowed) at the start.

Present Value Direction

Whether the Present Value is Outgoing (money you invest or deposit) or Incoming (money you receive, such as loan proceeds).

Interest Rate

The annual nominal interest rate at which the balance compounds (for example, 6 pct/yr). It is converted internally to a rate per Payment Interval.

Duration

The total length of time the investment grows for (for example, 5 yr). Together with Payment Interval, this determines the number of compounding/payment periods.

Periodic Payment

The fixed amount contributed (or withdrawn) at every Payment Interval throughout the Duration.

Periodic Payment Direction

Whether the Periodic Payment is Outgoing (money you contribute) or Incoming (money you receive).

Payment Interval

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

Payment When

Whether each Periodic Payment is made at the Period Start (an annuity due — the payment has a little extra time to earn interest within that period) or the Period End (an ordinary annuity).

Results

Future Value

The projected total value at the end of the Duration: the Present Value compounded over the full period, plus all Periodic Payment contributions compounded from when each was made. Always shown as a positive amount.

Cash Flow

States whether the Future Value represents money Incoming (you would receive it) or Outgoing (you would owe it), based on the combined direction of the Present 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 Future Value is then:

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

where PV and PMT are the signed Present Value and Periodic Payment (negative if Outgoing, positive if Incoming), 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 Present Value of 100,000 (Outgoing), an Interest Rate of 6% per year, a Duration of 5 years, a Periodic Payment of 1,000 (Outgoing) made monthly at the start of each month — the calculator returns a Future Value of approximately 203,646.57, with Cash Flow shown as Incoming (since both the initial deposit and the monthly contributions were outgoing, the resulting balance is money you would receive).

For comparison, making the same monthly payments at the end of each period instead (Payment When = Period End) slightly reduces the result to about 203,308.34, since each contribution then has slightly less time to compound. Removing the periodic payments entirely (Periodic Payment = 0) leaves only the initial deposit compounding on its own, reducing the Future Value to about 133,822.56.

Important Assumptions and Interpretation

• Periodic Payment is assumed to be a fixed, evenly spaced amount for every period; irregular contributions are not supported.
• The Interest Rate is assumed constant for the entire Duration.
• Getting Present Value Direction and Periodic Payment Direction right matters for interpreting Cash Flow correctly, even though Future Value itself is always reported as a positive number.

Quick Cross-Check with NPV/NFV

To match NFV from the same cashflow table, use FV with:

• Present Value = output of NPV
• Periodic Payment = 0
• same rate and interval as NPV/NFV
• Duration equal to the normalized last period distance (for periods 1..N, duration is N-1 intervals)

Then the values should satisfy:

$$FV = NPV \times (1+r)^T = NFV$$