Question

Suppose you plan to save $8,000 per year for the 35 years you are working. In addition to the amount you are saving each year, you expect to sell your house for $600,000 in year 32 and deposit this money into your account. How much can you withdraw in equal amounts each year for the 30 years you are retired. The interest rate you will earn during the 35 years you are saving is 7%. Once you retire, you’ll reduce the amount of stock you have in your portfolio and you will now earn a return of 5% during the 30 years you are retired. Assume that you begin saving in one year and your first withdrawal is in year 36.

Answer 1

Future Value:

Future Value is Value of current asset at future date grown at given int rate or growth rate.

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

Particulars	Amount
Present Value	$       600,000.00
Int Rate	7.0000%
Periods	3

Future Value = Present Value * ( 1 + r )^n
= $ 600000 ( 1 + 0.07) ^ 3
= $ 600000 ( 1.07 ^ 3)
= $ 600000 * 1.225
= $ 735025.8

FV of Annuity :

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here deposits are made at the end of the period. FV of annuity is future value of cash flows deposited at regular intervals grown at specified int rate or Growth rate to future date.

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

Particulars	Amount
Cash Flow	$            8,000.00
Int Rate	7.000%
Periods	35

FV of Annuity = Cash Flow * [ [ ( 1 + r ) ^ n ] - 1 ] /r
= $ 8000 * [ [ ( 1 + 0.07 ) ^ 35 ] - 1 ] / 0.07
= $ 8000 * [ [ ( 1.07 ) ^ 35 ] - 1 ] / 0.07
= $ 8000 * [ [10.6766] - 1 ] / 0.07
= $ 8000 * [9.6766] /0.07
= $ 1105895.03

Amount available after 35 Years:

= Future Value + Future valueof annuity

= $ 735025.8 + $ 1105895.03

= $ 1840920.83

Annual Withdrawl:

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end of the period. PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

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

Particulars	Amount
PV Annuity	$    1,840,920.83
Int Rate	5.0000%
Periods	30

Cash Flow = PV of Annuity / [ 1 - [(1+r)^-n]] /r
= $ 1840920.83 / [ 1 - [(1+0.05)^-6]] /0.05
= $ 1840920.83 / [ 1 - [(1.05)^-6]] /0.05
= $ 1840920.83 / [ 1 - 0.2314 ] /0.05
= $ 1840920.83 / [0.7686 / 0.05 ]
= $ 1840920.83 / 15.3725
= $ 119754.54

Annual withdrawl can be made is $ 119754.54