Mr. Wilson is planning to buy himself a new car in 3.5 years’ time. He estimates that the car will cost him R138900 and that the trade-in on his old car 3.5 years from now will be R 15 500. Mr. Wilson decides to open a savings account in which he wants to deposit a fixed amount every month so that he will be able to afford the car. What amount should Mr. Wilson be saving monthly if the bank will pay interest of 13.7% per annum compounded monthly on the savings account?
To calculate the monthly savings amount Mr. Wilson needs to deposit into the savings account, we need to use the future value formula for annuities.
The future value formula for an annuity is:
FV = Pmt * [(1 + r)^n - 1] / r
Where:
FV = Future value of the annuity (the cost of the car in 3.5 years)
Pmt = Monthly savings amount
r = Monthly interest rate
n = Number of months
Given:
FV = R138900
Trade-in value = R15500
r = 13.7% per annum compounded monthly = 13.7 / 12 = 1.1417% monthly
n = 3.5 years * 12 months/year = 42 months
First, calculate the future value of the trade-in value in 3.5 years:
FV_trade-in = R15500 * (1 + 1.1417%)^42 = R15500 * (1 + 0.011417)^42 = R15500 * (1.011417)^42 = R15500 * 1.546222 = R23923.54
Now, calculate the future value needed for the cost of the new car:
FV_needed = R138900 - R23923.54 = R114976.46
Now, substitute the known values into the formula and solve for Pmt:
R114976.46 = Pmt * [(1 + 1.1417%)^42 - 1] / 1.1417%
R114976.46 = Pmt * [1.546222 - 1] / 1.1417%
R114976.46 = Pmt * 0.354222 / 0.011417
Pmt = R114976.46 / (0.354222 / 0.011417)
Pmt = R114976.46 / 31.062469
Pmt ≈ R3702.29
Therefore, Mr. Wilson should be saving approximately R3702.29 per month in the savings account to afford the new car in 3.5 years.