Question

1. Twenty years from now a local municipality will have to replace their wastewater treatment plant. The best estimates from engineers are that this project will cost $12,500,000 to complete at that time. A forward-thinking member of the city council has proposed that city council budget an annual annuity for each of the next twenty years to ensure the municipality will have sufficient cash in the future to complete the project without incurring debt. If the city can lock in an interest rate of 3.5% for the next twenty years how much money will the town need to annually invest in an annuity to ensure they have $12,500,000 in twenty years?

Answer 1

Formula for FV of annuity can be used to compute annual deposit as:

FV = P x [(1+r) ⁿ – 1/r]

P = FV/ [(1+r) ⁿ – 1/r]

P = Periodic cash investment

FV = Future value of investment = $ 12,500,000

r = Rate per period = 3.5 % p.a.

n = Number of periods = 20

P = $ 12,500,000 / [(1+0.035)²⁰ – 1/0.035]

     = $ 12,500,000 / [(1.035)²⁰ – 1/0.035]

     = $ 12,500,000 / [(1.98978886346584 – 1/0.035]

     = $ 12,500,000 / (0.98978886346584/0.035)

     = $ 12,500,000 / 28.2796818133098

     = $ 442,013.459787829 or $ 442,013.46

City council needs to invest annually $ 442,013.46 to accumulate the fund in 20 years.