You are the owner of a Jani-King cleaning service franchise. Your accountant has determined that your business will need $27,500 in new equipment in 3 years. If your bank is paying 6% interest compounded monthly, how much must you invest today to meet this financial goal? (Round to the nearest whole dollar.)
P = Po(1+r)^n
P = $27,500
Po = Initial investment.
r = (6%/12)/100% = 0.005 = Monthly % rate expressed as a decimal.
n = 12comp/yr. * 3yrs = 36 Compounding
periods.
P = Po(1.005)^36 = 27500
Po = 27500/(1.005)^36 = $22,980.
To determine how much you must invest today to meet your financial goal of $27,500 in 3 years at an interest rate of 6% compounded monthly, you can use the formula for the future value of a lump sum investment:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = Future value (the desired amount of $27,500)
PV = Present value (the amount you need to invest today)
r = Annual interest rate (6% or 0.06)
n = Number of times interest is compounded per year (monthly, so 12)
t = Number of years (3)
Rearranging the formula to solve for PV, we get:
PV = FV / (1 + r/n)^(n*t)
Substituting the given values into the formula:
PV = $27,500 / (1 + 0.06/12)^(12*3)
Calculating inside the parentheses:
PV = $27,500 / (1 + 0.005)^(36)
Calculating the exponent:
PV = $27,500 / (1.005)^(36)
Calculating (1.005)^(36) ≈ 1.19414:
PV = $27,500 / 1.19414
Dividing $27,500 by 1.19414:
PV ≈ $23,034.87
Therefore, you would need to invest approximately $23,034.87 today to reach your financial goal of $27,500 in 3 years at an interest rate of 6% compounded monthly.