Final answer:
To save $16,000 for a down payment over four years at a 5.7% APR, you should deposit approximately $297.58 each month. This calculation is based on the future value of an annuity formula used in financial mathematics.
Step-by-step explanation:
To save $16,000 for a down payment on a home by making regular monthly deposits over four years with an APR of 5.7%, you will need to calculate the monthly deposit required using the future value of an annuity formula. This is a typical financial mathematics problem that deals with the concept of time value of money.
The future value of an annuity formula is given by:
FV = P x ((1 + r)^n - 1) / r
Where,
FV is the future value of the annuity (total amount saved),
P is the monthly payment,
r is the monthly interest rate,
n is the total number of payments.
Rearranging the formula to solve for P, we get:
P = FV / ((1 + r)^n - 1) / r
To apply this formula, we first need to convert the annual interest rate to a monthly rate by dividing by 12. Then, we identify that the number of payments n is equal to 48 (4 years x 12 months per year).
Let's compute the monthly deposit:
r = 5.7% / 12 = 0.00475
n = 4 x 12 = 48
FV = $16,000
P = $16,000 / (((1 + 0.00475)^48 - 1) / 0.00475)
After calculating this, we find that the monthly deposit should be approximately $297.58.
Thus, in order to save $16,000 over four years at an APR of 5.7%, you would need to deposit about $297.58 each month.