Question

You plan to retire in exactly 20 years. Your goal is to create a fund that will allow you to receive $20,000 at the end of each year for the next 30 years between retirement and death (a psychic told you that you’d die exactly 30 years after you retire). You know that you will be able to earn 11% per year during the 30-year retirement period. Required:

d. Now assume that you will earn 10% from now through the end of your retirement. You want to make 20 end-of-year deposits into your retirement account that will fund the 30-year stream of $20,000 annual annuity payments. How large do your annual deposits have to be?

Answer 1

First we need to calculate the amount of fund required to facilitate $ 20,000 cash flow using PV of annuity formula as:

PV(at the time of retirement) = P x [1-(1+r)^-n/r]

P = Periodic Payment = $ 20,000

r = Rate per period = 11 % = 0.11

n = Numbers of periods = 30

PV = $ 20,000 x [1-(1+0.11)^-30/0.11]

      = $ 20,000 x [1-(1.11)^-30/0.11]

     = $ 20,000 x [(1- 0.043683)/0.11]

    = $ 20,000 x (0.956317/0.11)

    = $ 20,000 x 8.693793

   = $ 173,875.85

This $ 173,875.85 is the future value of 20 annuity deposits @ 10 %, which can be calculate using formula for FV of ordinary annuity.

FV = P x [(1+r)ⁿ- 1 /r]

FV = Future value of annuity = $ 173,875.85

P = Periodic Payment

r = Rate per period = 10 % = 0.1

n = Numbers of periods = 20

$ 173,875.85 = P x [(1 + 0.1)²⁰ – 1/0.1]

$ 173,875.85 = P x [(1.1)²⁰ – 1/0.1]

$ 173,875.85 = P x [(6.727499949 – 1)/0.1]

$ 173,875.85 = P x (5.727499949/0.1]

$ 173,875.85 = P x 57.27499949

P = $ 173,875.85 / 57.27499949 = $ 3,035.81

Annual deposit should be $ 3,035.81 for 20 years.