Question

(Future value of a complex annuity) Springfield mogul Montgomery Burns, age 90, wants to retire at age 100 so he can steal candy from babies full time. Once Mr. Burns retires, he wants to withdraw $0.8 billion at the beginning of each year for 8 years from a special offshore account that will pay 19 percent annually. In order to fund his retirement, Mr. Burns will make 10 equal end-of-the-year deposits in this same special account that will pay 19 percent annually. How much money will Mr. Burns need at age 100, and how large of an annual deposit must he make to fund this retirement account?

Answer 1

To calculate how much money Mr. Burns will need at age 100 and the size of the annual deposit, we can use the formula for the future value of an ordinary annuity:

Future Value = A * ((1 + r)^n - 1) / r

Where:

A = Annual deposit amount

r = Interest rate per period

n = Number of periods

Given information:

Mr. Burns wants to withdraw $0.8 billion annually for 8 years.

The interest rate is 19% per year.

Mr. Burns will make 10 equal end-of-the-year deposits.

Let's solve for the future value to find out how much money Mr. Burns will need at age 100:

Future Value = $0.8 billion * ((1 + 0.19)^8 - 1) / 0.19

Future Value = $0.8 billion * (1.19^8 - 1) / 0.19

Future Value ≈ $0.8 billion * (6.8481 - 1) / 0.19

Future Value ≈ $0.8 billion * 5.8481 / 0.19

Future Value ≈ $24.716 billion

Therefore, Mr. Burns will need approximately $24.716 billion at age 100.

To find the size of the annual deposit, we rearrange the formula to solve for A:

A = Future Value * (r / ((1 + r)^n - 1))

A = $24.716 billion * (0.19 / ((1 + 0.19)^10 - 1))

A ≈ $24.716 billion * 0.19 / (1.19^10 - 1)

A ≈ $24.716 billion * 0.19 / (6.8481 - 1)

A ≈ $24.716 billion * 0.19 / 5.8481

A ≈ $80 million

Therefore, Mr. Burns must make an annual deposit of approximately $80 million to fund this retirement account.