Question

Ross has decided that he wants to build enough retirement wealth that, if invested at 5 percent per year, will provide him with $5,000 of monthly income for 30 years. To date, he's saved nothing, but he still has 20 years until he retires. How much money does he need to contribute per month to reach his goal?

Answer 1

Formula for PV of annuity can be used to compute required fund size to facilitate the retirement monthly income as:

PV = P x [{1 – (1+r)^-n}/r]

P = Periodic payment = $ 5,000

r = Rate of interest = 5 % p.a. or 0.05/12 = 0.004166667 p.m.

n = Number of periods = 30 years x 12 months = 360

PV = $ 5,000 x [{1 – (1+0.004166667)^-360}/0.004166667]

      = $ 5,000 x [{1 – (1.004166667)^-360}/0.004166667]

     = $ 5,000 x [(1 – 0.223826568893608)/0.004166667]

     = $ 5,000 x (0.776173431106392/0.004166667)

     = $ 5,000 x 186.281608563005

    = $ 931,408.042815027 or $ 931,408.04

$ 931,408.04 is the future value of 20 years monthly savings, which can be computed as:

FV = P x [{(1+r) ⁿ-1}/r]

P = FV/ [{(1+r) ⁿ-1}/r]

P = Periodic payment

r = Rate of interest = 5 % p.a. or 0.05/12 = 0.004166667 p.m.

n = Number of periods = 20 years x 12 months = 240

P = $ 931,408.04/ [{(1+0.004166667) ²⁴⁰-1}/0.004166667]

   = $ 931,408.04/ [{(1.004166667) ²⁴⁰-1}/0.004166667]

= $ 931,408.04/ [(2.71264050159279-1)/0.004166667]

= $ 931,408.04/ (1.71264050159279/0.004166667)

= $ 931,408.04/ 411.033687499575

= $ 2,266.01387751451 or $ 2,266.01

Ross needs to save $ 2,266.01 monthly to reach the goal.