Question

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 74.00. 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 74.0 when he fully retires, he will begin to make annual withdrawals of $162,481.00 from his retirement account until he turns 93.00. After this final withdrawal, he wants $1.81 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 10.00% interest rate.

Answer format: Currency: Round to: 2 decimal places.

Answer 1

We need to understand the time line for the investment. Derek will pay $ x from his 26th Birthday to 65th year i.e. he will invest for 39 years. Till that time the x contributions would be compounded at 10%.

After that the total amount accumalted will be growing for 10 years at 10%. After that he will make withdrawals of $ 162,481 till age of 93 i.e. for 18 years. After 18 years $ 1.81 million should remain in account.

Now at the age of 93 the amount left in his account would be $ 1.81 million and formula for Compound interest with fixed annual withdrawal 1810000 = P x (1 + i)^n - (W x ((1 + i)^n - 1) / i)

where P is the original amount in the account

i = rate of interest

n = no of periods

w = Annual withdrawals

1810000 = P x (1 + i)^n - (W x ((1 + i)^n - 1) / i)

1810000 = P x (1.1)^18 - (162481 x (1.1)^ 17/ 0.1)

1810000 = P x (1.1)^18 - 82,12,553.87

P x (1.1)^18 = 82,12,553.87 + 1810000

P x (1.1)^18 = 10022553.87

P = 10022553.87/(1.1)^18

P = 1,802,644.41

This is the total amount that would have been collected in his account when he retires. So, to get the yearly payments, we have Future Value of Annuity Formula. Here F is equal to P = 1802644.41

F =

1802644.41 = P (((1.1)^39 -1)/0.1)

1802644.41 = P * 401.4477778925

P = 4,490.358

P = 4490.36