You are planning to retire at the age of 65. You think you live for 5 years afte
ID: 2614593 • Letter: Y
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
Explanation / Answer
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
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
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