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 $188,527.00 from his retirement account until he turns 92.00. 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 10.00% interest rate.

Annual deposit = $3,077.99

Explanation:

Giving the following information:

Derek plans to retire on his 65th birthday.

10 years after he retires, he will neither make deposits to nor take withdrawals from his retirement account.

At his 75 birthday, he will begin to make annual withdrawals of $188,527.00 from his retirement account until he turns 92.00. He will make contributions to his retirement account from his 26th birthday to his 65th birthday.

First, we need to calculate the final value required at his 75 birthday:

Number of years = 92 - 75 = 17 years

Annual withdrawl = 188,527

FV = 188,527*17 = 3,204,959

Now, we need to calculate the amount needed 10 years before his retirement where he won't deposit or withdraw:

PV = FV / (1+i) ^n

PV = 3,204,959/1.10^10 = 1,235,650.44

Now, we can calculate the annual deposit from his 26th birthday to his 65:

FV = {A*[ (1+i) ^n-1]}/i

A = annual deposit

Isolating A:

A = (FV*i) / {[ (1+i) ^n]-1}

A = (1,235,650.44*0.10) / [ (1.10^39) - 1]

A = 3,077.99