Question

You wish to have $250,000 at the end of twenty years. In the last five years, you withdraw $1,000 annually at a rate of 3.8% compounded quarterly. During the middle ten years, you contribute $500 monthly at a rate of 2.8% compounded semi-annually. Given this information, determine the initial deposit that has to be made at the start of the first five years at a rate of 4% compounded monthly?

Answer 1

FVIFA = ((1+r)^n - 1)/r

Payments made during the mid 10 years,

Here, r = 0.028 compounded semi-annually

r on a monthly basis = 0.028/2/12 = 0.001167

n = 12 months x 10 years = 120

FVIFA = 128.73

Future Value = $500 * 128.73 = $64365

Withdrawals during the last 5 years,

Here, r = 0.038 compounded quarterly

r on an annual basis = 0.038/4 = 0.0095

n = 5 years

FVIFA = 21.91

Future Value = -$1000*21.91 = -21910

Total deposits with interest after 5 years = 64365 - 21910 = 42455

Remaining Amount Required with interest = 250000 - 42455 = 207545

Amount required at the beginning of 6th year,

r = 0.04 compounded monthly

r on annual basis = 0.04/12 = 0.0033

P = ??

A = $207545

n = 14

P = A/(1+r)^n = $118662

Amount to be invested at the beginning of 1st 5 years,

A = $118662

FVIFA = 366.299

P = $1789.8

Hence, the monthly amount that needs to be invested = $1789.8.

Amount to be deposited annually for first 5 years = $1789.8 * 5 = $21477.6