Question

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 70.00. 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 70.0 when he fully retires, he will begin to make annual withdrawals of $138,803.00 from his retirement account until he turns 90.00. 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.

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

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 70.00. 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 70.0 when he fully retires, he will begin to make annual withdrawals of $159,433.00 from his retirement account until he turns 85.00. After this final withdrawal, he wants $1.43 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 5.00% interest rate.

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

Answer 1

First Question:

Exactly one year after the day he turns 70.0 when he fully retires, he will begin to make annual withdrawals of $138,803.00 from his retirement account until he turns 90.00.

PV of all the withdrawls when he turns 70, P = PV of anuity = A / r x [1 - (1 + r)^-n] = 138,803 / 7% x [1 - (1 + 7%)^{-(90 - 70)}] =  1,470,481

So, kitty size required when he turns 65 = Q = P x (1 + r)^-n =  1,470,481 x (1 + 7%)^{-(70 - 65)} =  1,048,433

If B be the anual contribution from his 26th birthday to his 65th birthday, then FV of B as annuity = Q

Hence, B / r x [(1 + r)ⁿ - 1] = Q

Or, B/7% x [(1 + 7%)⁴⁰ - 1] = 1,048,433

Or, 199.6351B = 1,048,433

Hence, the annual contribution = B = 1,048,433 / 199.635 = $  5,251.74

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Second question:

Exactly one year after the day he turns 70.0 when he fully retires, he will begin to make annual withdrawals of $159,433.00 from his retirement account until he turns 85.00. After this final withdrawal, he wants $1.43 million remaining in his account.

PV of all the withdrawls when he turns 70, P = PV of anuity + PV of value remaining in the account = A / r x [1 - (1 + r)^-n] + V x (1 + r)^-n = 159,433 / 5% x [1 - (1 + 5%)^{-(85 - 70)}] + 1,430,000 x (1 + 5%)^{-(85 - 70)} =   2,342,714

So, kitty size required when he turns 65 = Q = P x (1 + r)^-n =  2,342,714 x (1 + 5%)^{-(70 - 65)} =   1,835,578

If B be the anual contribution from his 26th birthday to his 65th birthday, then FV of B as annuity = Q

Hence, B / r x [(1 + r)ⁿ - 1] = Q

Or, B/5% x [(1 + 5%)⁴⁰ - 1] =  1,835,578

Or, 120.7998B = 1,835,578

Hence, the annual contribution = B =  1,835,578 / 120.7998 = $  15,195.21