# What loan period length will minimize the total interest paid?

To minimize the total interest paid on a loan you should borrow for the shortest length of time possible. On one end of the spectrum, if you pay the full price of an item up front without taking out a loan, you'll pay zero interest. On the other end, homeowners who take out 30-year mortgages often pay more in interest charges than what they borrow.

If you want to pay no more than $X in total interest, you can use the calculator on the left to determine the ideal length of the borrowing period.

To use the calculator, input the amount borrowed *P*, the annual interest rate *R*, and the desired total interest cap *X*. The calculator will output the number of years and months for which you should take out a loan so that you do not exceed the interest cap, as well as the monthly (or biweekly) payments.

### Borrowing Period Equation

If the amount borrowed is *P*, the annual interest rate *R* (expressed as a decimal), and the number of months in the lending period is *L*, then the monthly payment *M* is given by the formula

M = (PR/12)(1 + R/12)^{L}/[(1 + R/12)^{L} - 1].

The total interest paid over the course of the loan is the sum of the payments minus the amount borrowed, or

LM - P

= L(PR/12)(1 + R/12)^{L}/[(1 + R/12)^{L} - 1] - P.

If you want to set the total interest to a fixed amount *X*, then you must solve the following equation for L:

X = L(PR/12)(1 + R/12)^{L}/[(1 + R/12)^{L} - 1] - P

*L*as both a multiplicative term and exponential term, it cannot be solved for

*L*using straight-forward algebra. However, an iterative numerical algorithm can find the value of

*L*that solves the equation. Such an algorithm is programmed into the calculator above.

Once you solve for the number of months

*L*, you can estimate the monthly payments with the shortcut formula

M = (P+X)/L

### Example Computation

Meg and Joe want to borrow $100000 for a new home. They can secure an annual interest rate of 6.3% and they do not want to pay more than $75,000 in total interest over the course of the loan. How long should their mortgage last?The given parameters are

*P*= 100000,

*R*= 0.063, and

*X*= 75000. Entering these values into the calculator above yields

*L*= 236 months, or equivalently 19 years and 8 months.

From this, you can figure their monthly payment, which is

($100000 + $75000)/236

= $742.

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