Question

What is the present value of a 1000 USD annuity for 10 years, with the first payment occurring at the end of year 10 (that is, ten 1000 USD payments occurring at the end of year 10 through year 19), given a discount rate of 10 percent?

Given a 10 percent discount rate, what is the present value of a perpetuity of 1000 USD per year if the first payment does not begin until the end of year 10?

Please explain clearly.

Answer 1

Present value of annuity is $ 2,605.90

Present value of perpetuity is $ 4,240.98

Explanation:

1.

Discounted value of annuity at the end of year 9 can be computed using formula for PV of annuity as:

PV = P x [1- (1+r)^-n/r]

P = Periodic cash flow = $ 1,000

r = Rate of return = 0.1

n = Number of periods = 10

PV (in years 9) = $ 1,000 x [(1-(1+0.1)^-10/0.1]

   = $ 1,000 x [(1-(1.1)^-10/0.1]

= $ 1,000 x [(1- 0.385543289429531)/0.1]

   = $ 1,000 x (0.614456710570469/0.1)

   = $ 1,000 x 6.14456710570469

   = $ 6,144.56710570469 or $ 6,144.57

Present value of $ 6,144.57 can be computed as:

PV = FV/(1+r) ⁿ

    = $ 6,144.57/ (1+0.1)⁹

    = $ 6,144.57/ (1.1)⁹

   = $ 6,144.57/ 2.357947691

    = $ 2,605.89750292302 or $ 2,605.90

2.

Discounted value of perpetuity at the end of year 9 can be computed as:

PV = Periodic cash flow/Rate of return

PV (in years 9) = $ 1,000/0.1

= $ 10,000

PV = $ 10,000/ (1+0.1)⁹

       = $ 10,000/ (1.1)⁹

= $ 10,000/2.357947691

= $ 4,240.97618372485 or $ 4,240.98