Question

You are planning to retire at the age of 65. You think you live for 5 years after you retire, and you will need $50,000 per year (today’s value) at the beginning of each year for your retirement. While you are working, you will put your money in funds which give you average return of 8%. After you retire, you will put your money to safer funds which give you 3.5% retune. The inflation rate is 2%. If you just turned to 25 years old today and you will contribute to the funds at the end of each month until you retire with initial investment of $5,000 today, how much do you have to put in your retirement account every month?

a. $109

b. $112

c. $119

d. $123

Answer 1

We have following information –

Your age today = 25 years

Retirement age = 65 years

Working age (remaining) = 65 – 25 = 40 years

Average return for working years (40 years) = 8% per year

Average return after retirement (5 years) = 3.5% per year

The inflation rate = 2% per year

Year (beginning) (n)	Remaining years for retirement (t= n – 25 -1)	Amount (today's value) (A)	Value for that year at 2% inflation [B =A* (1+2%)^t]	Value at the end of retirement (65th year) {=B/(1+3.5%)^(t -40)}
66	40	$ 50,000.00	$    110,401.98	$ 110,401.98
67	41	$ 50,000.00	$    112,610.02	$ 108,801.95
68	42	$ 50,000.00	$    114,862.22	$ 107,225.11
69	43	$ 50,000.00	$    117,159.47	$ 105,671.13
70	44	$ 50,000.00	$    119,502.66	$ 104,139.66
Total				$ 536,239.84

Now we know that total fund requirement at the time of retirement is $ 536,239.84

But you have initial investment of $5,000 today, its value at retirement

FV = PV * (1+ r) ^n

Where,

Future value FV =?

Present value PV =$5,000

Interest rate = 8% per annum or 8%/12 = 0.67% per month

Time period n = 40 years or 40 * 12 = 480 months

Therefore,

FV = $5,000 * (1+ 0.67%) ^ 480

= $121,366.93

Now additional fund requirement = total fund requirement at the time of retirement – future value of the initial investment of $5,000

=$ 536,239.84 - $121,366.93

= $414,872.91

Now $414,872.91 is the future value of the amount that you have to put in your retirement account every month and monthly deposit we can calculate with the help of following formula

FV of deposits = PMT [(1+r) ^n – 1] /r

Where,

Future value of deposits (FV) = $414,872.91

Monthly deposits PMT =?

Number of deposits n = 40 year *12 = 480 monthly deposits

Annual interest rate I =8%, therefore monthly rate = 8%/12 =0.67%

Therefore          

$414,872.91 = PMT * [(1+0.67%) ^480 -1]/0.67%

OR PMT = $118.84 or $119

Therefore correct answer is option c. $119