Loan Balance Calculator

Background.

Five years into a 30-year mortgage, most borrowers are surprised by how little of the debt has gone. On a $250,000 loan at 6%, sixty payments of about $1,499 add up to nearly $90,000, yet the balance has fallen by only about $17,364. The rest went on interest.

That is how an amortizing loan is built. Each month's interest is charged on whatever is still owed, so early payments, made against a large balance, are mostly interest. The first payment on that mortgage includes $1,250 of interest and less than $249 of principal. As the balance shrinks, the split tips the other way, slowly at first and then quickly; half the original $250,000 is still outstanding until the 252nd payment, 21 years in.

Knowing the balance matters whenever the loan might end early. Selling the house, refinancing, comparing a lump-sum payoff with investing the money, or working out how much equity you hold all start from this one figure.

In the United States it also decides when private mortgage insurance on a conventional loan can stop. Under the Homeowners Protection Act, a borrower may ask to cancel it once the balance is scheduled to reach 80% of the home's original value, subject to conditions such as a good payment record, and it must end automatically when the schedule reaches 78%. Both dates come from the original amortization schedule, which is exactly what this formula follows. For a 50,000 loan at 6% on a 75,000 home, the 80% mark of 20,000 is passed with payment 95 and the 78% mark of 14,500 with payment 108.

The estimate follows the schedule as written: a fixed rate, the same payment every month, each one made on time and none extra. Real accounts drift from that. Extra principal payments make the true balance lower; missed payments, fees added to the loan or a rate change make it higher. A lender's payoff quote will also include interest accrued since the last payment date, so the result is the scheduled balance rather than the exact cheque that would close the account.

What is loan balance calculator?

The remaining balance of a loan is the principal still owed after a given number of scheduled payments. For a fully amortizing fixed-rate loan it can be found without building a payment schedule: it equals the original principal grown at the loan's monthly rate, less the accumulated value of the payments made so far. Equivalently, it is the present value of the payments still to come, discounted at the loan rate. The result is in the same currency as the principal you enter. Two more figures follow from it. Principal repaid so far is the original amount minus the balance, and interest paid so far is the total of the payments made minus that repaid principal. The same arithmetic covers any fixed-rate loan repaid in equal monthly instalments, not only mortgages. A $30,000 car loan at 7% over 60 months has a payment of about $594.04, and after 24 payments it still owes $19,238.72, or 64% of the original. Short loans repay principal more evenly than long ones: that borrower is 40% of the way through the term and has cleared 36% of the debt, while the 30-year borrower in the worked example is a sixth of the way through and has cleared about 7%.

How to use this calculator.

1. 01Enter the original principal: the amount borrowed when the loan began, not today's balance or the home's price.
2. 02Enter the nominal annual interest rate from your loan agreement. Use the note rate rather than the APR, which folds in fees and would give a slightly higher balance.
3. 03Enter the full term as a count of monthly payments, such as 360 for 30 years or 180 for 15 years.
4. 04Enter how many monthly payments have been made. Both counts must be whole numbers, and payments made can be 0 (the balance is then the original principal) but must be fewer than the full term. To look ahead, enter the count that will have been made by the date you plan to sell, refinance or pay off.
5. 05Read the remaining balance, then check the model-scope note to see what the estimate leaves out before relying on it for a payoff or refinance. If your statement shows a different figure, the gap is a measure of extra payments, fees or rate changes the schedule does not know about.

The formula.

B_k = P × [(1 + r)ⁿ − (1 + r)ᵏ] ÷ [(1 + r)ⁿ − 1]. P is the original principal, n the total number of monthly payments, k the number already made, and r the monthly rate: the nominal annual percentage divided by 1,200, which is the same as dividing by 100 and then by 12. The bracketed ratio is the fraction of the principal still owed: it equals 1 when no payments have been made (k = 0) and falls to 0 at the final payment (k = n). The monthly payment never appears because it is fixed by P, r and n. Rate changes the answer most: at 6% a 30-year loan still carries 93% of its principal after five years. A rate of 0 is accepted; for an interest-free loan the balance is simply P × (n − k) ÷ n, and the calculator uses exactly that. The equation comes from either of two textbook routes that meet in the middle. Looking back, the balance is the principal grown for k months, P(1 + r)ᵏ, minus the k payments grown to the same date, M[(1 + r)ᵏ − 1] ÷ r. Looking forward, it is the present value of the n − k payments still due, M[1 − (1 + r)^−(n−k)] ÷ r. Substituting the level payment M = P·r(1 + r)ⁿ ÷ [(1 + r)ⁿ − 1] into either gives the single expression above; for the example loan both routes give $232,635.89. The derivation assumes payments at the end of each month and interest compounded monthly at exactly one-twelfth of the annual rate. Loans that accrue interest daily, such as many US car loans, drift slightly with the actual payment dates. Canadian fixed-rate mortgages usually quote a rate compounded semi-annually: 6% quoted that way is equivalent to about 5.9263% compounded monthly, so enter that figure instead of 6.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Consumer Financial Protection Bureau, Your Home Loan Toolkit.
2. [2]Kellison, Theory of Interest, 3rd ed. (PRINT).
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]Federal Reserve, Consumer's Guide to Mortgage Refinancings.