You would like to save up for a deposit of $30,000 to buy a home in exactly 8 years. You can invest your savings at an interest rate of 6.5% per year (compounded yearly). Calculate the amount that you must save at the end of each year for the next 8 years to have enough savings for this deposit (to the nearest dollar).
Question
You would like to save up for a deposit of $30,000 to buy a home in exactly 8 years. You can invest your savings at an interest rate of 6.5% per year (compounded yearly). Calculate the amount that you must save at the end of each year for the next 8 years to have enough savings for this deposit (to the nearest dollar).
Solution
To solve this problem, we will use the formula for the future value of a series of payments, or annuities, which is:
FV = P * [(1 + r)^n - 1] / r
Where: FV is the future value of the series of payments, which is $30,000 in this case. P is the amount of each payment, which we are trying to find. r is the interest rate per period, which is 6.5% or 0.065 in this case. n is the number of periods, which is 8 years in this case.
We can rearrange the formula to solve for P:
P = FV * r / [(1 + r)^n - 1]
Substituting the given values:
P = 30000 * 0.065 / [(1 + 0.065)^8 - 1]
Calculating the right side of the equation:
P = 1950 / [(1.065)^8 - 1]
P = 1950 / [1.718186 - 1]
P = 1950 / 0.718186
P = $2715.54
So, you must save approximately $2716 at the end of each year for the next 8 years to have enough savings for the deposit.
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