Question

Given the following information:

You are 40 years old and you would like to retire at age 70 (assume social security benefits at that age will be $3,000 per month).

You do not have a defined benefit plan; you are just starting to invest in a 401k , current savings are zero (you recently bought a home).

You estimate your life span to 90 yrs of age, and you need annual retirement income of 100,000 per year (use todays dollars) -- from age 70 to age 90.

Use 5% cost of money

a. Calculate the future value of funds required to support the 100,000 annuity for 20 years.

b. Calculate the monthly amount of savings required for the next 30 years to fund the required future value.

again use 5% return

Answer 1

PV of annuity for $ 100,000 for 20 years
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream	P
PMT = the dollar amount of each annuity payment	$         100,000
r = the effective interest rate (also known as the discount rate)	5%
n = the number of periods in which payments will be made	20
PV of annuity=	PMT x (((1-(1 + r) ^- n)) / r)
PV of annuity=	100000* (((1-(1 + 5%) ^-20)) /5%)
PV of annuity=	$ 1,246,221.03
PV of annuity for $ 3,000 per month for 20 years
PV of annuity for making pthly payment
P = PMT x (((1-(1 + r) ^- n)) / i)
Where:
P = the present value of an annuity stream	P
PMT = the dollar amount of each annuity payment	$           36,000	3000*12
r = the effective interest rate (also known as the discount rate)	5.12%	((1+5%/12)^12)-1)
i=nominal Interest rate	5.00%
n = the number of periods in which payments will be made	20
PV of annuity=	PMT x (((1-(1 + r) ^- n)) / i)
PV of annuity=	36000*(((1-(1 + 5.12%) ^-20)) /5%)
PV of annuity=	$    454,575.94
Remaining funds needed by to be accumulated	1246221.03-454575.94
Remaining funds needed by to be accumulated	$    791,645.09
FV of annuity
P = PMT x ((((1 + r) ^ n) - 1) / i)
Where:
P = the future value of an annuity stream	$    791,645.09
PMT = the dollar amount of each annuity payment	P
r = the effective interest rate (also known as the discount rate)	5.12%
i=nominal Interest rate	5.00%
n = the number of periods in which payments will be made	30.00
FV of annuity=	PMT x ((((1 + r) ^ n) - 1) / i)
791645.09=	PMT x ((((1 + 5.12%) ^ 30) - 1) / 5%)
Annual contribution=	791645.09/ ((((1 + 5.12%) ^ 30) - 1) / 5%)
Annual contribution=	$      11,414.41
Monthly contribution=	$           951.20