Interest Rate Finder

This works backwards from what actually happened. You know what a sum was worth at the start, what it is worth now, and how long that took. What compound rate would have produced that, growing steadily the whole way?

The formula:

r = (Future Value / Present Value)^(1/n) − 1

Variable definitions:

• Present Value = what it was worth at the start
• Future Value = what it is worth at the end
• n = the number of periods between the two, usually years
• r = the constant per-period rate that connects them

This is the compound annual growth rate (CAGR), and the “annual” part is only true if your periods are years. Enter quarters and you get a quarterly rate.

Worked example: An investment of $10,000 is worth $15,000 after 5 years.

• 15,000 / 10,000 = 1.5
• 1.5^(1/5) = 1.08447
• r = 0.08447, so 8.45% per year

Check it: 10,000 grown at 8.447% five times over gives 10,000 × 1.5 = 15,000 exactly.

What “steadily” hides. The rate you get back is the single constant rate that would have arrived at the same endpoint. It says nothing about the path. A fund that returned +60%, −30%, +40%, −10% and +20% might land on exactly the same 8.45%, and so might a savings account that paid 8.45% every year without moving. Both descriptions are true and only one of them tells you anything about risk. This is why fund factsheets quote CAGR and standard deviation side by side, and why quoting the first without the second is a well-worn way to make a volatile product look calm.

Why the arithmetic average is worse. Take $100 that falls 50% and then rises 100%. Average those two yearly returns and you get +25%. The money is back at $100, so the real rate is 0%. Averaging percentage changes is simply the wrong operation on multiplicative data; the root in the formula above is the right one.

Reading the doubling time. Once you have r, the time to double is ln(2) / ln(1 + r). The Rule of 72 estimates the same thing as 72 divided by the percentage rate, which is close for single-digit rates and drifts once you get past about 20%. The calculator shows both so you can see the gap.

Where this is used:

• Comparing two investments held for different lengths of time
• Turning a multi-year revenue figure into the annual growth number a board expects
• Checking a quoted return against the beginning and ending balances on a statement
• Working out the implied rate on a zero-coupon bond or a discounted note