Question

Meadow brook manor would like to buy some additional land and rebuild a new assisted living center. The anticipated total cost is 24 million. The ceo of the firm is quite conservative and will only do this when the company has sufficent funds to pay cash for the entire construction project. Managment has decided to save 1.4 million a quarter for this purpose. The firm earns 5 percent compounded quartlery on the funds it saves. how long does the company have to wait before expanding its operations.

Answer 1

Number of quarters needed for this fund can be computed using formula for FV of annuity as:

FV = C x [(1+r) ⁿ – 1/r]

FV = Future value of annuity = $ 24,000,000

C = Periodic cash flow = $ 1,400,000

r = Rate per period = 5 % p.a. or 0.05/4 = 0.0125 p. q.

n = Numbers of periods

$ 24,000,000 = $ 1,400,000 x [(1+0.0125) ⁿ – 1/0.0125]

    = $ 1,400,000 x [(1.0125) ⁿ – 1/0.0125]

  = $ 1,400,000 x [(1.0125) ⁿ – 1/0.0125]

[(1.0125) ⁿ – 1/0.0125] = $ 24,000,000 / $ 1,400,000

[(1.0125) ⁿ – 1/0.0125] = 17.1428571

(1.0125) ⁿ – 1 = 0.0125 x 17.1428571

(1.0125) ⁿ – 1 = 2.142857143

(1.0125) ⁿ = 2.142857143 + 1

(1.0125) ⁿ = 3.142857143

Taking log of both sides, we get:

n x log 1.0125 = log 3.142857143

n x 0.00539503188 = 0.49732464082

n = 0.49732464082/0.00539503188

n = 92.18196516

n = 92.18 periods or 23.05 years

The company has to wait 23.05 years to start the expansion.