Question

(Try to work this question WITHOUT using Excel, get answer by using the formula and show the calculations in detail)

Question (Retirement planning)

You have just graduated Hofstra University at age 22. You hard work has paid off as you already have a job as an investment banker at Goldman Sachs waiting for you. You plan to work continuously until age 65 and retire exactly on that day. You expect to live until exactly 90 and enjoy your golden years and leave you heirs NOTHING. Assume your investments earn 8% per year.

You plan to contribute $10,000 to your retirement fund every year on your birthday starting at age 23. Your last deposit will be at exactly age 65 and your first withdrawal will be at age 66. Your last withdrawal will be at the moment you die at age 90.

Ignore all tax considerations for this problem. Consider the effect of inflation. Assume inflation averages 4% per year.
(i) How much you will be able to spend each year in retirement?

FV (deposits) = PV (withdrawals)
NOTE: This could be at any time period but t=65 is particularly convenient

(ii) How much will you be able to spend each year in retirement if you begin deposits at age 30?

(iii) How much larger do your deposits have to be if deposits start at age 30 to equal your answer in part (i)?

Answer 1

Total number of payment=(65-22)=43

Annuity (Payment)=$10000

Interest Rate=8%

Future value at the age of 66=P*((1+r)^n-1)/r (Here r= interest rate, n=number of payment), P= payment amount

=10000*((1+8%)^43-1)/8%=$3295830

Now, Formula for calculating annuity for a present value (PMT)=P*r/((1 - (1 / (1 + r) ^ n)) (Here P=present value, r=rate, n=number of withdrawal)

Here, After retirement P=3295830, n=(90-66)+1=25, r=8%

Hence, PMT= 3295830*8%/((1 - (1 / (1 + 8%) ^ 25))=$308749.3

i) Hence, you will be able to spend $308749.3 at the age of 66 to age 90 which is equivalent to $308749/1.04^(66-22)=$54971.3 as per todays's age.

ii) Similarly, here n=(65-30)=25

Hence, retirement corpus=  10000*((1+8%)^25-1)/8%=$731059.4

and your PMT= 731059.4*8%/((1 - (1 / (1 + 8%) ^ 25))=$68484.75

Hence, you will be able to spend $68484.75 at the age of 66 to age 90 which is equivalent to $68484.75/1.04^(66-22)=$12193.46 as per todays's age

iii) To get the same amount as of part i) your retirement corpus should be $3295830

Hence, you needed PMT= F*r/((1+r)^n-1) (Here F=Future value, r=interest rate, n=number of payment)

=3295830*8%/((1+8%)^25-1)=$45082.93