What is my interest rate?

How the interest rate is worked out

Inputs. The number of payments N, the loan amount L, the payment P, the balance left at the end B (a balloon; 0 for a loan paid off), the payments a year P/Y and the compounding times a year C/Y (whole numbers from 1 to 365; both 12 by default), and whether payments are made at the end (usual for loans) or the start of each period. Amounts are positive; an empty payment or balance counts as 0. Find the picks the value to work out: the rate by default, or the payment, the term (the number of payments), the loan amount or the balloon (the balance left at the end).

The loan equation. With i the rate per payment period (a decimal), g = (1 + i)^N, t = 0 for payments at the end and 1 at the start, and a = (1 + i × t) × (g − 1) ÷ i (a = N when i = 0):

L × g = P × a + B

The loan grows with interest to L × g; the payments, each grown to the end, plus the balance left must match it. This is the time value of money equation of a financial calculator, with the loan as money received and the payments and balance as money paid.

Rate per period and yearly rate. I/Y is a nominal yearly rate compounded C/Y times a year: i = (1 + I/Y ÷ (100 × C/Y))^(C/Y ÷ P/Y) − 1. The other way round, I/Y = 100 × C/Y × ((1 + i)^(P/Y ÷ C/Y) − 1); with C/Y = P/Y, I/Y = 100 × P/Y × i.

Finding the rate. The rate has no formula. The calculator checks every yearly rate above −100% and at most 1,000% where L × g − P × a − B changes sign: every 0.01% from −20% to 50%, every 0.1% from −99.9% to −20% and from 50% to 1,000%, and at −99.95%, −99.99%, −99.999%, −99.9999%, −99.99999% and −99.999999%. It refines each sign change by bisection. For a loan (money received first, then paid) there is exactly one such rate.

The other values. P = (L × g − B) ÷ a. L = (P × a + B) ÷ g. B = L × g − P × a. N = ln(g) ÷ ln(1 + i) with g = (P × (1 + i × t) − B × i) ÷ (P × (1 + i × t) − L × i); at 0%, N = (L − B) ÷ P.

Other results. Total of payments = N × P. Total interest = N × P + B − L. Rate per period = 100 × i. Effective yearly rate = 100 × ((1 + i)^(P/Y) − 1).

Precision and overflow. The rate search reads the sign of the equation from the form above when (1 + i)^N is at most 1, and from the same equation divided by (1 + i)^N when it is above 1, so no step overflows; a rate at which (1 + i)^N is not a finite number above 0 in 64-bit floating point is dropped. With the loan, the payment and the balance all 0 (or empty) the equation holds at every rate, so there is no rate. The other computed values use the same divided equation when (1 + i)^N is above 1, so an answer inside the limits never fails on a larger number on the way. The total of payments and the total interest are exact decimal arithmetic on the typed amounts and the computed value, each read as the decimal it prints as, rounded once at the end. The computed value, the rates and the powers are 64-bit floating point, so a value that is exactly half a cent in decimals can land just below it and show rounded down.

Limits and no answer. N is above 0 and at most 12,000; the rate is above −100% and at most 1,000%; amounts are from $0 to $1 trillion. A computed amount below $0 or over $1 trillion, or a computed N of 0 or less or over 12,000, is no answer. So is a (1 + i)^N that is not a finite number above 0 in 64-bit floating point, N at 0% with no payment, N when g ≤ 0 or g = 1 (the payment does not cover the interest), and a rate when none fits.

Display. Money shows to the cent, rounded half up from the value worked out; rates at most 4 decimals (the rate per period 6); the number of payments at most 2 decimals.

Assumptions

• The rate stays the same for the whole loan and every payment is the same size.
• No fees are included: with fees, this is not the APR.
• This is an estimate for planning, not financial advice.

Worked examples by hand

$20,000 at $400 a month for 60 months. The rate that fits is i = 0.618341% a month: g = 1.00618341^60 = 1.447536, a = (g − 1) ÷ i = 72.376775, and 400 × 72.376775 ÷ 1.447536 = 20,000.00. So the rate is 12 × 0.618341% = 7.4201% a year. The payments total 60 × 400 = $24,000, so the interest is $4,000.

Excel's RATE example. $8,000 repaid at $200 a month for 48 months: i = 0.770147% a month (g = 1.445210, a = 57.808389, 200 × 57.808389 ÷ 1.445210 = 8,000.00), so the rate is 9.2418% a year.

The CFPB auto loan (find the payment). $20,000 at 4.75% for 36 months. i = 0.0475 ÷ 12 = 0.00395833; g = 1.00395833^36 = 1.152829; a = 38.609370. P = 20,000 × 1.152829 ÷ 38.609370 = $597.18 a month, and the interest is 36 × 597.18 − 20,000 = $1,498.32.

A balloon loan. $30,000, 60 payments of $450, $10,000 still owed at the end. i = 0.562880% a month fits: g = 1.400432, a = 71.139915, and (450 × 71.139915 + 10,000) ÷ 1.400432 = 30,000.00. The rate is 6.7546% a year.