Question

A rich donor gives a hospital $1,040,000 one year from today. Each year after that, the hospital will receive a payment 6% larger than the previous payment, with the last payment occurring in ten yearsʹ time. What is the present value (PV) of this donation, given that the interest rate is 11% g?

Answer 1

Answer:

$7,681,257.74

Explanation:

Since the hospital will receive a payment 6% larger than the previous payment each year after the first payment, the formula for the Present Value of a Growing Annuity is used to obtain the present value.

The present value of a growing annuity formula is meant for the estimation of the present day value different payments hat grow at a proportionate rate which will be received for a period of time. This formula is stated as follows:

PV = {P ÷ (r - g)} × {1 - [(1+g)÷(1+r)]^n] ...................................... (1)

Where

PV = Present value

P = First payment = $1,040,000

r = interest rate = 11% = 0.11

g = growth rate = 6% = 0.06

n = number of years = 10 years

Substuiting all the values into equation (1), we have:

PV = {$1,040,000 ÷ (0.11 - 0.06)} × {1 - [(1+0.6)÷(1+0.11)]^10]

     = {$1,040,000 ÷ (0.05)} × {1 - [(1.06)÷(1.11)]^10]

     = {$1,040,000 ÷ (0.05)} × {1 - [(1.06)÷(1.11)]^10]

     = $20,800,000 × (1 - 0.630708763)

     = $20,800,000 × 0.369291237  

     = $7,681,257.74  

 I wish you the best.