Question

Assume your goal is to retire at age 65 and you estimate you will live until age 90. Your income at age 30 is $50,000 and you expect this to increase at a rate of 8% per year. The nominal rate of return on your investment portfolio is 6% and you plan to save 15% of your income per year. The expected rate of inflation is 3%. How much will your fund pay per year during your retirement? I would like to see it worked out so I can understand it.

Answer 1

First let’s compute future value of savings using formula for FV of growing annuity.

FV_GA = P x [(1+r) ⁿ – (1+g) ⁿ/r-g]

P = First payment = $ 50,000 x 0.15 = $ 7,500

r = Rate per period = nominal rate – inflation = 6 % - 3 % = 3 %

g = Growth rate = 8 % or 0.08

n = Number of periods = 65 – 30 = 35 years

FV_GA = $ 7,500 x [(1+0.03) ³⁵ – (1+0.08) ³⁵/0.03-0.08]

         = $ 7,500 x [(1.03) ³⁵ – (1.08) ³⁵/-0.05]

        = $ 7,500 x [(2.81386245437152 – 14.7853442943206)/-0.05]

        = $ 7,500 x [(-11.97148183994908)/-0.05]

        = $ 7,500 x 239.429636798981

       = $ 1,795,722.27599236 or $ 1,795,722.28

The fund $ 1,795,722.28 will facilitate annual cash outflow for retirement life which can be computed using formula for PV of annuity as:

PV = P x [1-(1+r)^-n/r]      

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

P = Periodic cash flow

r = Rate per period = 3 % p.a.

n = Numbers of periods = 90 - 65 = 25

P = $ 1,795,722.28/[1-(1+0.03)^-25/0.03]

   = $ 1,795,722.28/[1-(1.03)^-25/0.03]

   = $ 1,795,722.28/ [(1-0.47760556926166)/0.03]

   = $ 1,795,722.28/ (0.52239443073834/0.03)

   = $ 1,795,722.28/17.413147691278

= $ 103,124.507517929

= $ 103,124.51

The fund can pay $ 103,124.51 annually during retirement life.