You want to purchase a new car in 8 years and expect the car to cost $21,000. Your bank offers a plan with a guaranteed APR of 5.5 % if you make regular monthly deposits. How much should you deposit each month to end up with $21,000 in 8 years?
You should invest $_____ each month.
Assuming your interest rate of 5.5% is compounded monthly to match the payment period,
i = .055/12 = .0045833..
n = 8(12) = 96
monthly payment --- p
p(1.00458333..^96 - 1)/.00458333... = 21000
solve for p
To find out how much you should deposit each month, we can use the formula for future value of an annuity.
Future Value = Payment × [(1 + interest rate)^(number of periods) -1] / interest rate
In this case, the future value is $21,000, the interest rate is 5.5% (or 0.055 as a decimal), and the number of periods is 8 years. We need to solve for the payment.
$21,000 = Payment × [(1 + 0.055)^(8) - 1] / 0.055
First, let's simplify the formula a bit:
$21,000 × 0.055 = Payment × [(1 + 0.055)^(8) - 1]
$1,155 = Payment × [(1.055)^(8) - 1]
Next, let's calculate [(1.055)^(8) - 1]:
[(1.055)^(8) - 1] ≈ 1.503
Now, let's substitute this value back into the equation:
$1,155 = Payment × 1.503
To solve for the payment, divide both sides of the equation by 1.503:
Payment ≈ $1,155 / 1.503
Payment ≈ $768.69
Therefore, you should deposit approximately $768.69 each month to end up with $21,000 in 8 years.