Question

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 73.00. 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 73.0 when he fully retires, he will begin to make annual withdrawals of $171,973.00 from his retirement account until he turns 91.00. 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 8.00% interest rate.

Answer 1

Step 1 - Find out the Present value of yearly withdrawals (from 74 year to 91 year ) at the end of his 73rd birthday
We can use the present value of annuity formula to calculate this value.
Present value of annuity = P x {[1 - (1+r)^-n]/r}
Present value of annuity = value of yearly wothdrawals at the of 73rd year = ?
P = yearly withdrawals = $171973
r = interest rate = 8%
n = number of yearly withdrawals = 18
Present value of annuity = 171973 x {[1 - (1+0.08)^-18]/0.08}
Present value of annuity = 171973 x 9.371887
Present value of annuity = 16,11,711.55
Present Value of yearly withdrawals at the end of his 73rd birthday = $16,11,711.55
Step 2 - Find of the present value of value derived in step 1 at the end of his 65th birthday i.e.exactly 8 year before
Present value = F / (1+r)^n
F = Value derived in step 1 = $16,11,711.55
r = interest rate = 8%
n = number of years from 65 years to 73rd Year = 8
Present value = 1611711.55 / (1+0.08)^8
Present value = 870757.60
Present value of value derived in step 1 at the end of his 65th birthday = $8,70,757.60
Step 3 - Find out the yearly conribution to retirement account from his 26th birthday to his 65th birthday
We can use the present value of annuity formula to calculate the yearly contribution.
Present value of annuity = P x {[1 - (1+r)^-n]/r}
Present value of annuity = value derived in step 2 = $870757.60
P = yearly contribution = ?
r = interest rate = 8%
n = number of yearly contributions = 40
870757.60 = P x {[1 - (1+0.08)^-40]/0.08}
870757.60 = P x 11.92461
P = 73021.87
Yearly contribution to retirement account = $73,021.87