A = P(1 + r)[sup n], where P is the principal, r is the interest rate per compounding period (as a decimal) and n is the number of compounding periods. For a nominal rate of R% per year compounded m times a year, r = R100m and n = m × years.

Rearrange for other unknowns: P = A(1 + r)ⁿ, n = ln(A ÷ P)ln(1 + r) and r = (A ÷ P)^1/n − 1.

Worked example. $5000 at 4% per year compounded quarterly for 2 years: r = 0.01, n = 8, A = 5000 × 1.01⁸ = $5414.28, so the interest is $414.28.

Nominal rates with different compounding periods cannot be compared directly. The effective annual rate is the single yearly rate that gives the same growth: i = (1 + [frac R/m])[sup m] − 1, where R is the nominal annual rate (as a decimal) and m is the number of compounding periods per year.

Worked example. 12% per year compounded monthly: i = 1.01¹² − 1 = 0.126825…, so the effective rate is 12.68% per year — the same as 12.68% compounded once a year.

Inflation is the general rise in prices over time. With an average inflation rate f per year, a cost C today becomes C(1 + f)[sup n] after n years, and an amount A received in n years is worth [frac A/(1 + f)[sup n]] in today's dollars.

The real rate of return measures growth in buying power: real rate = [frac 1 + nominal rate/1 + inflation rate] − 1. A quick estimate is nominal rate − inflation rate.

Worked example. A $40 bill today with 2% inflation: in 3 years it costs 40 × 1.02³ = $42.45.