### Problem

The total cost of a car is 15,000. To purchase the car using a loan repaid by monthly installments, you are quoted the following terms.

Initial deposit 2,000

Term 4 years

Interest rate 6% a year

Payments monthly, at the end of each month.

What will your monthly payment be?

### Solution

The value of the loan is the cost of the car less the initial deposit (15,000 – 2,000 = 13,000) (PV). This loan is repaid using monthly installments, so one period is one month, and the term of the loan is 4 years or 48 months (n). The interest rate has been quoted as an annual rate so the periodic (monthly) rate is 6%/12 or 0.5% per month (i)

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

PV = 13,000 n = 48 i = 0.5% PV = Pmt x (1 - 1 / (1 + i)^{n}) / i

Rearranging the formula to solve for Pmt

Pmt = PV x i / (1 - 1 / (1 + i)^{n}) Pmt = 13,000 x 0.5% / (1 - 1 / (1 + 0.5%)^{48}) Pmt = 305.31

### Explanation

For the loan balance to be cleared at the end of 48 payments, the present value of the payments must be equal to the present value of the loan.

The solution to this problem simply uses the present value of an annuity formula to find the 48 payments which, at a discount rate of 0.5%, will give a present value of 13,000.

### AN7 Annuity Examples

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