## Problem

A car is financed using a three-year loan. The loan has a 8% nominal annual interest rate, compounded monthly. The price of the car is 7,000, and a deposit of 2,000 is paid in cash. Calculate the monthly auto loan payments, assuming that the payments start one month after the purchase.

## Solution

The value of the loan is the price of the car less the initial deposit (7,000 – 2,000 = 5,000) (PV). This loan is repaid using monthly installments, so one period is one month, and the term of the loan is 3 years or 36 months (n). The interest rate has been quoted as an annual rate so the periodic (monthly) rate is 8%/12 per month (i)

As the auto loan payments (Pmt) are made at the end of each period, and we know the present value, the problem is solved using the annuity payment formula PV as follows:

Auto loan payment = PV x i / (1 - 1 / (1 + i)^{n}) PV = Value of the auto loan 5,000 n = number of months = 3 x 12 = 36 i = nominal rate = 8%/12 per month Auto loan payment = 5000 x (8%/12) / (1 - 1 / (1 + 8%/12)^{36}) Auto loan payment = 156.68

## Auto Loan Payment Explanation

For the auto loan balance to be cleared at the end of 36 payments, the present value of the payments must be equal to the present value of the auto loan.

The solution to this problem simply uses the present value of an annuity formula to find the 36 payments which, at a discount rate of 8%/12, will give a present value of the auto loan of 5,000.

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a degree from Loughborough University.