Question

You just graduated from college and are starting your new job. You realized the importance to save for the future and have figured out that you will save $1,000 per month for the next 14 years; and then increase to $5,000 per month for the following 5 years. The amount accumulated at the end of these investments will be your retirement egg nest. You plan to start retirement and start withdrawing monthly amounts the following month (you will be in retirement for 24 years). If your required rate of return is 12% compounded monthly, how much are your monthly withdrawals?

Answer 1

FV of Annuity :
Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

FV of Annuity of $ 1000 at the end of 14 years:

Particulars	Amount
Cash Flow	$      1,000.00
Int Rate	1.000%
Periods	168

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 1000 * [ [ ( 1 + 0.01 ) ^ 168 ] - 1 ] / 0.01
= $ 1000 * [ [ ( 1.01 ) ^ 168 ] - 1 ] / 0.01
= $ 1000 * [ [5.321] - 1 ] / 0.01
= $ 1000 * [4.321] /0.01
= $ 432096.98

FV of this amount after 5 Years ( 14 + 5= 19):

Future Value:
FV = PV (1+r)^n
Where r is Int rate per period
n - No. of periods

Particulars	Amount
Present Value	$4,32,096.98
Int Rate	1.0000%
Periods	60

Future Value = Present Value * ( 1 + r )^n
= $ 432096.98 ( 1 + 0.01) ^ 60
= $ 432096.98 ( 1.01 ^ 60)
= $ 432096.98 * 1.8167
= $ 784989.16

FV of Annuity of $ 5000 at the end of 19 years:

Particulars	Amount
Cash Flow	$      5,000.00
Int Rate	1.000%
Periods	60

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 5000 * [ [ ( 1 + 0.01 ) ^ 60 ] - 1 ] / 0.01
= $ 5000 * [ [ ( 1.01 ) ^ 60 ] - 1 ] / 0.01
= $ 5000 * [ [1.8167] - 1 ] / 0.01
= $ 5000 * [0.8167] /0.01
= $ 408348.35

Total Value in account after 19 Years = $ 784989.16 + $ 408348.35
= $ 1193337.51

Monthly withdrawl:

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
PV Annuity	$    11,93,337.51
Int Rate	1.000%
Periods	288

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 1193337.51 / [ 1 - [(1+0.01)^-4]] /0.01
= $ 1193337.51 / [ 1 - [(1.01)^-4]] /0.01
= $ 1193337.51 / [ 1 - 0.0569 ] /0.01
= $ 1193337.51 / [0.9431 / 0.01 ]
= $ 1193337.51 / 94.3056
= $ 12653.93

Each month, we can withdraw $ 12653.93