Finance maths formulas

The formulas behind savings, loans and investments, as they appear in maths and business courses: simple and compound interest, continuous compounding, effective rates, loan repayments, present and future value, annuities and the rule of 72. Amounts are plain numbers in any currency. These pages teach the arithmetic; they are not financial advice, and real products add fees and their own rate conventions.

Printable cheat sheet Formula finder

Simple interest

I = Prt

Interest paid only on the original amount, so it grows by the same sum every year.

Compound interest

A = P(1 + r/n)^(nt)

What an amount grows to when interest is added n times a year and then earns interest itself.

Continuous compounding

A = Pe^(rt)

The limit of compound interest as interest is added more and more often: every instant.

Effective annual rate

r_eff = (1 + r/n)ⁿ − 1

Converts a nominal rate compounded n times a year into the single yearly rate that grows money the same amount.

Loan repayment formula

M = Pi/(1 − (1 + i)^( −N)), i = r/k

The fixed payment that pays off a loan, interest included, in N equal instalments.

Present value

PV = FV/((1 + r)^t)

What a future amount is worth today, by discounting it at an interest rate.

Future value of an annuity

FV = PMT·((1 + i)^N − 1)/i

What a series of equal regular deposits grows to, with interest.

Present value of an annuity

PV = PMT·(1 − (1 + i)^( −N))/i

The lump sum today that is worth the same as a series of equal future payments.

Rule of 72

t ≈ 72/(100r)

A mental shortcut for how many years money takes to double at a given annual rate.