A person aged 30 wishes to accumulate a fund for retirement by making deposits o

Question

A person aged 30 wishes to accumulate a fund for retirement by making deposits of $100 at the beginning of each month for 15 years followed by deposits of $200 at the beginning of each month for the next 15 years. Starting at age 65 withdrawals will be made at the beginning of each month for 25 years. Assuming all payments are certain to be made, find the amount of each withdrawal if the monthly effective rate of interest is 0.75% during the first 35 years and 0.50% thereafter. (Answer should be: $2234.07)

Explanation / Answer

FV of immediate annuity = [PMT(((1+r)n)-1)/r]*(1+r).

FV of immediate annuity of $100 per month for 15 years, where r = 0.0075, n = 12*15 i.e. 180 & PMT = 100 is:

=[100(((1+0.0075)180)-1)/0.0075]*(1+0.0075)

=$38,124.38

FV of above amount invested @0.75% p.m. for next 20 yrs = 38124.38 * (1+0.0075)240 ...........[ FV = PV (1+r)n ]

= $2,29,095.20

FV of immediate annuity of $200 per month for 15 years, where r = 0.0075, n = 12*15 i.e. 180 & PMT = 200 is:

=[200(((1+0.0075)180)-1)/0.0075]*(1+0.0075)

=$76,248.76

FV of above amount invested @0.75% p.m. for next 5 yrs = 38,124.38 * (1+0.0075)60 ...........[ FV = PV (1+r)n ]

= $1,19,381.20

Total FV after investment for 35 years is $ 3,48,476.40 ($ 2,29,095.2 + $ 1,19,381.2) which is also PV for immediate annuity of $ C for next 25 years when r = 0.005. Here n = 25*12 = 300

PV = C*[1-(1+r)-n]/r*(1+r)

So, C = PV*r / {[1-(1+r)-n]*(1+r)}

C = 3,48,476.40*0.005 / {[1-(1+0.005)-300]*(1+0.005)}

C = $ 2,234.07

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