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Compound Interest Rate without Periodic Payment
Purpose
This calculator works backward from a known starting amount and a known ending amount to find the compound interest rate that explains the growth between them, with no additional deposits or withdrawals along the way. It is useful for figuring out what rate of return an investment actually earned, or what rate you would need to grow a present amount into a target future amount over a chosen period.
Background
Solving for the rate instead of the future value
Most compound-interest calculations start with a rate and compute a future value. This calculator does the reverse: given the Present Value, the Future Value, and the time between them, it solves for the constant compounding rate that would turn one into the other, assuming no payments are made or received in between. It reports that rate both per compounding period and as an equivalent annual rate, so you can compare it with rates quoted on other terms.
Inputs
Present Value
The starting amount.
Future Value
The ending amount after the stated Duration.
Duration
The total time between the Present Value and the Future Value (for
example, 6 yr).
Interest Interval
How often interest is assumed to compound over that duration (for example, every month or every year). Together with Duration, this determines the number of compounding periods used to solve for the rate.
Results
Periodic Interest Rate
The constant compound growth rate per Interest Interval that would grow the Present Value into the Future Value over the calculated number of periods, expressed as a percentage per interval (for example, percent per month).
Annual Interest Rate
The same growth rate restated as an equivalent effective annual rate, regardless of the Interest Interval used. This makes it easy to compare the result with an annual rate quoted elsewhere, even if the underlying compounding happened monthly, quarterly, or on some other interval.
Understanding the Calculation
The number of compounding periods is the Duration divided by the Interest Interval (for example, 6 years at a monthly interval gives 72 periods). The Periodic Interest Rate per period is then the rate that satisfies:
solved for :
The Annual Interest Rate is the effective annual rate equivalent to compounding at rate every Interest Interval throughout a full year — it will equal the Periodic Interest Rate only when Interest Interval is already a year.
Example
Using the default inputs — a Present Value of 100, a Future Value of 200, a Duration of 6 years, and interest compounding monthly — the calculator finds a Periodic Interest Rate of about 0.967% per month, equivalent to an Annual Interest Rate of about 12.25% per year. This means an investment compounding at roughly 0.967% every month would double in value over 6 years.
As a sanity check, doubling an investment in exactly 1 year with monthly compounding gives a Periodic Interest Rate of about 5.95% per month and an Annual Interest Rate of exactly 100% per year — as expected, since doubling in one year is a 100% annual return regardless of how often it compounds within that year.
Important Assumptions and Interpretation
- The calculation assumes no deposits, withdrawals, or payments occur between the Present Value and the Future Value — only pure compounding.
- The compounding rate is assumed constant across every period; the calculator does not solve for a rate that changes over time.
- Make sure Duration divides evenly by Interest Interval so the number of compounding periods is a whole number.
- The result describes the rate implied by the two values and the time between them; it is not a forecast of future returns.