Question

A) 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 $120,165.00 from his retirement account until he turns 94.00. 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 7.00% interest rate.

B) Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 75.00. 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 75.0 when he fully retires, he will begin to make annual withdrawals of $102,458.00 from his retirement account until he turns 95.00. After this final withdrawal, he wants $1.51 million remaining in his account. 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

Amount required on 72^nd birthday is equal to present value of future withdrawals

= 120,165*PVAF(7%, 22 years)

= 120,165*11.0612

= $1,329,169.10

Amount required on 65^th birthday = 1,329,169.10/(1.07)⁷

= $827,739.71

Future value of Annuity = Periodic Amount*[{(1+r)ⁿ – 1}/r]

Let annual deposits be x

827,739.71= x*[{(1.07)⁴⁰ -1}/0.07]

827,739.71= 199.63511197x

X = $4,146.26

Hence, annual contribution required = $4,146.26

Amount required on 75^th birthday is equal to present value of future withdrawals

= 102,458*PVAF(7%, 20 years) + 1,510,000*PVF(7%, 20 years)

= 102,458*10.5940 + 1,510,000*0.2584

= $1,475,624.05

Amount required on 65^th birthday = 1,475,624.05/(1.07)¹⁰

= $750,132.44

Future value of Annuity = Periodic Amount*[{(1+r)ⁿ – 1}/r]

Let annual deposits be x

750,132.44 = x*[{(1.07)⁴⁰ -1}/0.07]

750,132.44= 199.63511197x

X = $3,757.52

Hence, annual contribution required = $3,757.52