Compound Interest Rate with Periodic Payment

Purpose

This calculator solves for the compound interest rate implied by a starting amount, an ending amount, and a series of regular payments in between — for example, an investment that starts with a lump sum, adds a fixed contribution every month, and reaches a target value after several years. It reports the rate both per payment period and as an equivalent annual rate.

Background

Solving for the rate with cash flows in and out

Unlike a simple lump-sum growth calculation, this scenario has money moving both at the start/end (Present Value and Future Value) and periodically in between (Periodic Payment). Each of these amounts can be either money going out of your pocket or money coming in, and the calculator needs to know which is which to solve correctly. It then searches numerically for the single compound rate per period that makes all these cash flows consistent with each other — the same approach used to solve for a loan's interest rate when you know the loan amount, the payment, and the payoff amount.

Inputs

Present Value

The amount at the start of the period.

Present Value Direction

Whether the Present Value is Incoming (received by you, such as a loan you take out) or Outgoing (paid by you, such as an initial investment).

Future Value

The amount at the end of the Duration.

Future Value Direction

Whether the Future Value is Incoming (received by you, such as an investment payout or savings goal) or Outgoing (paid by you, such as a loan balance you must pay off).

Duration

The total time from the Present Value to the Future Value (for example, 5 yr).

Periodic Payment

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

Periodic Payment Direction

Whether the Periodic Payment is Incoming (received by you) or Outgoing (paid by you, such as a regular contribution or loan installment).

Payment Interval

How often the Periodic Payment occurs (for example, every month). Together with Duration, this determines the total number of payments.

Payment When

Whether each Periodic Payment is made at the Period Start (an annuity due, common for contributions made at the beginning of a period) or the Period End (an ordinary annuity, common for loan payments made at the end of a period).

Starting Guess

An initial estimate of the annual interest rate, used to start the numerical search for the actual rate. The default is usually sufficient; adjust it only if the calculation fails to converge.

Tolerance

How precise the calculated rate must be before the numerical search stops. Smaller values give a more precise answer but may need more iterations.

Maximum Iteration

The maximum number of attempts the numerical search will make to find a rate precise enough to meet the Tolerance. If this limit is reached without converging, the result may be unreliable — try a different Starting Guess.

Results

Periodic Interest Rate

The compound interest rate per Payment Interval that reconciles the Present Value, Future Value, and Periodic Payment, given their directions and timing, expressed as a percentage per interval.

Annual Interest Rate

The same rate restated as an equivalent effective annual rate, useful for comparing against annual rates quoted elsewhere, regardless of how often payments and compounding actually occur.

Understanding the Calculation

The number of payment periods is the Duration divided by the Payment Interval. The calculator treats Present Value, Future Value, and Periodic Payment as signed cash flows — incoming amounts are positive, outgoing amounts are negative, based on each amount's direction setting — and numerically searches for the periodic rate $r$ that satisfies the standard time-value-of-money relationship linking a present value, a series of equal payments, and a future value, starting from the Starting Guess and refining the estimate until it is within Tolerance or Maximum Iteration is reached. The Annual Interest Rate is then the effective annual rate equivalent to compounding at $r$ every Payment Interval.

Example

Using the default inputs — a Present Value of 100,000 (Outgoing, an initial investment), a Future Value of 200,000 (Incoming, the payout after 5 years), a Periodic Payment of 1,000 (Outgoing, contributed every month at the start of each month) — the calculator finds a Periodic Interest Rate of about 0.451% per month, equivalent to an Annual Interest Rate of about 5.54% per year. This is the rate of return that reconciles putting in 100,000 up front and 1,000 every month, for it to grow to 200,000 after 5 years.

As a check, if the Periodic Payment is set to 0 and the Future Value is set equal to the Present Value (no growth needed and no payments made), the calculated rate comes out essentially 0% — confirming the solver correctly reduces to "no growth" when there is nothing to explain.

Important Assumptions and Interpretation

Getting the Present Value Direction, Future Value Direction, and Periodic Payment Direction settings right is essential — mixing up incoming and outgoing will produce a meaningless or unsolvable result.
The Periodic Payment amount and Payment Interval are assumed constant and evenly spaced throughout the Duration; irregular contributions are not supported.
Because the rate is found by numerical search, an unusual combination of inputs (or a poor Starting Guess) may fail to converge within Maximum Iteration — try adjusting the Starting Guess or Tolerance if the result looks implausible.
The result describes the rate implied by the cash flows given; it is not a forecast of future investment or loan performance.