Question

You want to withdraw $11,200 per month for 30 years when you retire, then you expect to pass on. You plan to retire in 40 years, and expect to earn a return of 11.2 percent until then. You will make monthly deposits to fund your retirement account. Immediately after you make your last deposit, you plan to withdraw $80,000 to take an around the world trip. You also wish to leave your grandchildren $1,200,000 when you pass on. You will earn an annual return of 5.8 percent after you retire. What is the monthly deposit you must make in order to achieve your financial goals?

Answer 1

Value of inheritance at the time of retirement:

PV = FV/ (1+r) ⁿ

FV = Future value of inheritance = $ 1,200,000

r = Rate of interest = 5.8 % p.a. or 0.058/12 = 0.004833333 p.m.

n = No. of periods = 30 years x 12 months = 360

PV = $ 1,200,000/ (1+0.004833333)³⁶⁰

     = $ 1,200,000/ (1.004833333)³⁶⁰

     = $ 1,200,000/5.673512988

     = $ 211,509.17

PV of required annuity after retirement is:

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

P = Periodic cash flow = $ 11,200

r = Rate per period = 5.7 % p.a. or 0.058/12 = 0.004833333 p.m.

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

Substituting the values in above formula, we get PV as:

PV = $ 11,200 x [1-(1+ 0.004833333)^-360/ 0.004833333]

    = $ 11,200 x [1-(1.004833333)^-360/ 0.004833333]

    = $ 11,200 x [(1-0.176257638)/ 0.004833333]

    = $ 11,200 x (0.823742362/ 0.004833333)

    = $ 11,200 x 170.4294542

    = $ 1,908,809.89

FV of monthly annuity deposits for retirement in 40 years =

Value of world trip + value of monthly annuity + value of inheritance

= $ 80,000 + $ 1,908,809.89 + $ 211,509.17

= $ 2,200,319.06

Periodic payment to accumulate the FV of $ 2,200,319.06 can be computed using formula for FV of annuity as:

FV = P x [(1+r) ⁿ – 1/r]

P = FV/ [(1+r) ⁿ – 1/r]

P = Periodic cash flow

r = Rate per period = 11.2 % p.a. or 0.0112/12 = 0.933333333 semiannually

n = Numbers of periods = 40 years x 12 months = 480 periods

P = $ 2,200,319.06/ [(1+0.933333333)⁴⁸⁰ – 1/0.933333333]

= $ 2,200,319.06/ [(1.933333333)⁴⁸⁰ – 1/0.933333333]

= $ 2,200,319.06/ [(86.4202925236 – 1)/0.933333333]

  = $ 2,200,319.06/ [(85.4202925236/0.933333333]

  = $ 2,200,319.06/ 9,152.1741989588

= $ 240.4149017 or $ 240.41

$ 240.41 need to deposit monthly to achieve the financial goal.