# 4-Variable Mortgage Calculator

4-Variable Mortgage Equation Solver
Fill in 3 fields, leave 1 field blank
Principal \$
Length in Months
Annual Interest Rate     %
Monthly Payment \$

The loan repayment equation is a mathematical expression that relates four variables:

p, principal (amount borrowed)
n, number of months in loan term
i, annual interest rate as a percent
m, monthly repayment amount

If you know 3 of the 4 variables, you can solve for the missing quantity using the mortgage calculator on the left. The calculator can be used for any type of loan, including auto loans, student loans, and business loans. Some examples are given below, along with links to articles that explain each scenario in more mathematical detail (for the curious).

### Example 1: Unknown Monthly Payment

The most common usage of the mortgage equation is figuring a monthly payment based on the amount borrowed, the interest rate, and the length of the lending term. The explicit equation for calculating m in terms of p, n, and i is given here: How Does the Bank Calculate My Monthly Payment?.

Example: Suppose you want to borrow \$20,000 for a new car at an interest rate of 6.9% for 3 years. In the calculator above, enter 20000, 6.9, and 36 for the three known variables and leave the monthly payment blank. When you click the "Calculate" button you will find that the monthly payment for this loan is \$616.63.

If this amount is too high, you must either borrow less, secure a lower interest rate, or borrow for a longer amount of time. Extending the loan period will reduce your monthly payments, but over the course of the loan you will end up paying more in total interest.

### Example 2: Unknown Interest Rate

Suppose you need to borrow \$87,000 for a new home, and you can pay \$550 a month for 25 years. What interest rate will make this possible?

If you plug p = 87000, n = 300, and m = 550 into the calculator above, you get an annual interest rate of 5.8%. This is actually the highest rate you can afford given your constraints. For more details on how to calculate i in terms of p, n, and m, see What is the Maximum Interest Rate I Can Afford?

### Example 3: Unknown Principal

If the best interest rate you can get is 5.5%, and you are able to pay \$750 a month for 15 years, how much money can you borrow? To solve this with the calculator, use m = 750, n = 180, and i = 5.5. The answer is \$91789.89. If you need to borrow more than this, you will need to make larger monthly payments, or be willing to borrow for a longer period of time.

The equation to solve for p in terms of m, n, and i is given in How Much Can I Afford to Borrow?

### Example 4: Unknown Loan Term Length

Suppose you want to take out a loan for \$60,000 and you can get an interest rate of 6.23%. If you can afford to pay \$550 every month, how long will it take to pay off the loan?

For this problem, enter p = 60000, i = 6.23, and m = 550 in the calculator. It turns out it will take you 162 months (13 years and 6 months) to pay off the loan. If this is too long, you must borrow less or pay more each month. To learn how to solve for n in terms of m, p, and i, see How Long Will It Take to Pay Off a Loan?