Problem 17 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Mike L. and Mike C. have decided to establish the Mike and Mike Math Millenium Miracle Prize. The M\&M \(M^{3}\) prize is worth \(\$ 2000\) to the lucky winner. Due to limited funds, Mike and Mike have decided to award the prize once every 4 years, starting 10 years from now and going on inde nitely. (It s like the Fields Medal in Math, only more accessible.) They have begun to go door-to- door to take collections in order to establish the fund. How much money should the M\&M \(M^{3}\) Prize Fund contain right now in order to start payments 10 years from today? Assume a guaranteed interest rate of \(5 \%\) per year compounded annually.

Step-by-Step Solution

Answer

To establish the Mike and Mike Math Millenium Miracle Prize Fund to start payments 10 years from today with the given conditions, Mike and Mike should gather approximately $16000.

Step 1: Understanding the Situation

This is a problem of present value of a deferred annuity. An annuity is a series of equal payments at regular intervals. In this case, the annuity is deferred, which means that payments will not start immediately but after a certain period of time. The payments for the annuity are the $2000 prizes, the regular interval is every 4 years, and the deferment period is 10 years.

Step 2: Applying the Formula for Present Value of a Deferred Annuity

The formula for the present value of a deferred annuity is P = PMT * [(1 - (1 + r)^ -n) / r] * (1 + r)^ -d, where P is the present value (the amount needed in the fund now), PMT is the amount of each payment ($2000), n is the number of payments, r is the annual interest rate (5%), and d is the deferment period. Considering that the account will be indefinite, the perpetual annuity formula will be used, which simplifies to P = PMT / r, and then the deferment period is applied.

Step 3: Calculating the Amount in the Fund

PMT is $2000, r is 5 % or 0.05, and d is 10 years. The annuity's period is 4 years, which means n needs adjustment. Since payments are made every 4 years, the effective rate should be calculated as 0.05/4 = 0.0125. As n approaches infinity and only one payment occurs every 4 years, the formula becomes: P = $2000 / 0.0125 * (1 + 0.0125)^ -10.

Key Concepts

Annuity Payments

Annuity payments are the consistent, periodic amounts paid or received over time, central to the concept of an annuity. These payments can be monthly, quarterly, annually, or at any regular interval agreed upon in the contract.

For instance, when a person retires and opts to receive income from a retirement fund, they may choose to receive it in the form of an annuity, guaranteeing them regular payments rather than a lump sum. In the case of the Mike and Mike Math Millennium Miracle Prize, annuity payments are made to the prize winner every four years.

The most common types of annuities are:

• Ordinary annuities: Payments occur at the end of each period.
• Annuities due: Payments occur at the beginning of each period.
• Deferred annuities: Payments start after a certain deferment period, as seen in our example with the prize starting after 10 years.

Understanding the timing and frequency of these payments is crucial for calculating the present value or future value of an annuity.

Perpetual Annuity Formula

A perpetual annuity, as the name implies, is an annuity that continues forever. The perpetual annuity formula is a financial tool used to calculate the present value of an endless series of cash flows.

In the context of the Mike and Mike Math Millennium Miracle Prize, the perpetual annuity formula helps to determine how much money should be contained in the prize fund today to ensure the \(2000 prize can be paid out every four years indefinitely. The formula itself is quite simple: \[ \frac{PMT}{r} \. \]

Here, \(PMT\) represents the annuity payment (in our problem, \)2000), and \(r\) is the periodic (annual in this case) interest rate expressed as a decimal (for instance, 5% would be expressed as 0.05). The resulting value gives us the present value of the annuity that would satisfy all future payments.

It's worth noting that the formula for a perpetual annuity differs from the one for a regular annuity due to the fact that, theoretically, a perpetual annuity has an infinite number of payments, which simplifies the regular annuity formula.

Compounded Interest Rate

The compounded interest rate is a crucial concept in finance, as it refers to the process by which interest is earned on both the initial principal and the accumulated interest from previous periods.

For example, if you invest \(100 at an annual compounded interest rate of 5%, you'll have \)105 after one year. If the interest is compounded annually, in the second year, you earn interest not just on your original \(100, but also on the \)5 interest from the first year, resulting in a total of $110.25 by the end of the second year.

In our exercise involving the M&M \(M^{3}\) Prize Fund, the interest rate is compounded annually at 5%. To factor this into the calculation of the required present value, we need to adjust the perpetual annuity formula to account for the deferment period using this compounded rate.

The adjusted formula becomes \[ P = \frac{PMT}{r} \times (1 + r)^{-d}\], where \(d\) represents the deferment period. Compounding impacts how the prize fund grows over time until the start of the annuity payments and affects the overall amount required to establish the fund.