Alonzo plans to retire as soon as he has accumulated $250,000 through quarterly payments of $2,500. If Alonzo invests his money at 5.4% interest, compounded quarterly, when (to the nearest year) can he retire?
To solve this problem, we need to use the formula for the future value of a series of equal payments, also known as an annuity. The formula is:
FV = P * [(1 + r)^n - 1] / r
where:
FV = future value
P = periodic payment
r = interest rate per period
n = number of periods
In this case, Alonzo's periodic payment is $2,500, the interest rate is 5.4% (or 0.054) per quarter, and he wants to accumulate a total of $250,000.
We can rearrange the formula to solve for n:
n = (log(1 + (FV * r) / P)) / log(1 + r)
Let's plug in the values:
n = (log(1 + ($250,000 * 0.054) / $2,500)) / log(1 + 0.054)
Using a calculator:
n ≈ 33.52
So Alonzo needs approximately 33.52 quarters to accumulate $250,000. Since there are 4 quarters in a year, we can convert the number of quarters to years:
n_years = n / 4
n_years ≈ 8.38
Rounding to the nearest year, Alonzo can retire in about 8 years.
8943854
56
84.787
=85 yrs