Question

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 72.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 72.0 when he fully retires, he will begin to make annual withdrawals of $117,576.00 from his retirement account until he turns 93.00. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 9.00% interest rate.

Answer 1

First of all lets find Present value of annuity at end of year 72
PV(annuity) = Annuity x PVIFA(r%,n)
r = rate of interest = 9%
n = number of years = 93-72 = 21 years
Annuity = 117576$
PVIFA(r%,n) = [1-(1/(1+r)^n / r ]
PVIFA(9%,21) = [1-(1/(1+9%)^21 / 9%]
=[1-(1/(1+0.09)^21 / 0.09]
=[1-(1/(1.09)^21 / 0.09]
=[1-0.163698 / 0.09]
=0.8363/0.09
=9.2922
Thus PV(annuity) = 117576 x 9.2922
=1092545 $
Now lets calculate PV of above amount at age of 65
Thus n = 72-65 = 7 years
PV = FV/(1+r)^n
=1092545/(1+9%)^7
=1092545/(1+0.09)^7
=1092545/1.09^7
=1092545/1.8280
=597660 $

Now lets find annuity amount he must make between 26th birth day to 65th birthday

FV(annuity) = Annuity[(1+r)^n - 1 /r]

r = 9%

n = 40 years

597660 = Annuity[(1+9%)^40 - 1 /9%]

597660 = Annuity [(1+0.09)^40 - 1 /0.09]

597660 = Annuity [(1.09)^40 - 1 /0.09]

597660 = Annuity [31.4094 - 1 /0.09]

597660 = Annuity [30.4094/0.09]

597660 = Annuity[337.8824]

Annuity = 1768.84 $

Thus he need to make contributions of $1768.84 to his retirement account from his 26th birthday to his 65th birthday