A company will need $50,000 in five years for a new addition.To meet this goal the company deposits the money in an account today that pays 4% annual interest compounded quarterly. Find the amount that should be invested to total $50,000 in 5 years.
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To find the amount that should be invested to total $50,000 in 5 years, we can use the future value formula for compound interest:
FV = P(1 + r/n)^(nt)
Where:
FV = future value of the investment
P = principal amount (initial investment)
r = annual interest rate (expressed as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case:
FV = $50,000
r = 4% = 0.04 (annual interest rate)
n = 4 (quarterly compounding, so 4 times per year)
t = 5 (5 years)
Now, we rearrange the formula to solve for the principal amount (P):
P = FV / (1 + r/n)^(nt)
Plugging in the values:
P = $50,000 / (1 + 0.04/4)^(4*5)
Simplifying the equation inside the brackets:
P = $50,000 / (1 + 0.01)^20
Calculating the value inside the brackets:
P = $50,000 / (1.01)^20
Using a calculator, we find that (1.01)^20 is approximately 1.2202.
P = $50,000 / 1.2202
P ≈ $41,006.57
Therefore, the company should invest approximately $41,006.57 to have $50,000 in 5 years by depositing the money in an account that pays 4% annual interest compounded quarterly.