The loan and interest formulas

mortgage mathsborrowing formulasamortization formulas

Payment, total interest, effective rate and doubling time — everything that decides what borrowing actually costs.

Loan Payment (Amortized Loan or Mortgage)

M = \frac{P\,i}{1 - (1+i)^{-n}}

Level payment that retires a loan of principal P in exactly n payments at periodic interest rate i — the mortgage, truck loan and equipment finance formula.

Total Interest Paid Over a Loan

I = M\,n - P

Everything a loan costs beyond the amount borrowed: total of all payments minus the principal.

Effective Annual Rate from a Nominal Rate

\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1

What a quoted nominal annual rate really costs or earns once it compounds m times a year — 12% compounded monthly is 12.68% effective.

Real Interest Rate (Fisher Equation)

r_{\text{real}} = \frac{1 + i}{1 + f} - 1

What a return is worth after inflation is taken out, done exactly rather than by the rough subtraction that overstates it.

Rule of 72 (Doubling Time)

n \approx \frac{0.72}{i}

Mental-arithmetic estimate of how many periods it takes money to double at rate i — with the rate written as a percentage R, it is the familiar 72 divided by R.

How they fit together

Every one of these turns on the same idea: money has a rent, and the rent compounds. The payment formula is the anchor, because it answers the only question a lender is really asked — what is the cheque each month. The rest exist to stop the payment number from fooling you.

Reach for total interest paid before signing anything: a longer amortization always shrinks the payment and always raises the total, and the two facts are easy to hold separately until you see them side by side. Reach for the effective annual rate whenever two offers are quoted on different compounding periods, because a nominal rate is not comparable across frequencies. Reach for the real rate when the money is invested rather than borrowed, since a 5% return during 3% inflation is a 1.94% gain in purchasing power, not 2%. The Rule of 72 is the mental-arithmetic cousin of all of them, accurate enough for a conversation and never intended for a contract.

One caution on the payment formula: a rate quoted per year must be divided into the payment period before it enters the equation, and Canadian fixed mortgages compound semi-annually rather than monthly, which makes the periodic rate slightly lower than a naive division suggests.