What is the difference between nominal and effective rate?

A nominal annual rate is just the periodic rate multiplied by the number of periods, ignoring compounding within the year. The effective rate accounts for it. At 1% a month, the nominal annual rate is 12% but the effective rate is 12.68%. Loans are quoted nominally; investment returns are usually quoted effectively.

Can I use this for irregular cash flows?

No — this equation requires a level payment. For uneven cash flows use the NPV and IRR calculator, which handles a different amount in every period, or the XIRR tool for flows on irregular dates.

Why can solving for N give a fractional answer?

Because the equation is continuous. A result of 115.87 periods means the balance is cleared partway through the 116th payment, so the final payment is smaller than the rest. Lenders handle this with an adjusted final payment.

Time Value of Money Solver

Corporate Finance7 min readSanguine Straphanger

One Equation Behind Every Loan, Annuity and Savings Plan

Mortgages, bonds, leases and pensions are all the same calculation with different labels.

The master equation

Every fixed cash flow problem reduces to one identity: the present value, the payment stream and the future value must net to zero once everything is moved to the same point in time.

Time value of money

PV(1+r)ⁿ + PMT(1 + r·S)·[((1+r)ⁿ − 1)/r] + FV = 0

S = 0 for payments at period end (ordinary annuity)
S = 1 for payments at period start (annuity due)
r is the rate PER PERIOD, not per year

Four of the five variables determine the fifth. PV, FV and PMT have closed-form solutions. N comes out of a logarithm. RATE has no closed form at all, which is why this tool solves it numerically — Newton-Raphson first, with a bracketed bisection fallback for the awkward cases where Newton diverges.

Signs are not cosmetic

The equation nets cash flows to zero, so they cannot all point the same way. If you enter the loan and the repayments both as positive numbers there is no rate that balances them, and the honest answer is an error rather than a plausible-looking figure.

Think of it from your own account's perspective. You receive the mortgage: positive. You send the payments: negative. Get that right and the rest follows.

Ordinary annuity versus annuity due

An ordinary annuity pays at the end of each period; an annuity due pays at the start. Rent and insurance premiums are typically due; loan payments and coupons are typically ordinary.

The relationship is exact and worth remembering: an annuity due is worth precisely (1 + r) times the equivalent ordinary annuity, because every payment has one extra period to compound. Switch the timing selector above and watch the result move by exactly that factor.

Rate per period, not per year

The most frequent input error after signs. For a 6.5% mortgage paid monthly, the periodic rate is 6.5 ÷ 12 = 0.5417%, and N is 360, not 30. Enter 6.5 with N = 360 and you have modelled a 78% loan.

The periods-per-year selector only affects how the effective annual rate is displayed. The rate you type is always the rate per period.

Frequently Asked Questions & Quantitative Reference

Lenders round the payment to the cent and often adjust the final payment to clear the balance exactly. Day-count conventions also vary — some use actual days rather than equal months. Differences of a few units over a full term are normal; differences of hundreds mean an input is wrong.