Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A state lottery commission pays the winner of the "Million Dollar" lottery 20 installments of $$\$ 50,000$$ /year. The commission makes the first payment of $$\$ 50,000$$ immediately and the other \(n=19\) payments at the end of each of the next 19 yr. Determine how much money the commission should have in the bank initially to guarantee the payments, assuming that the balance on deposit with the bank earns interest at the rate of \(8 \% /\) year compounded yearly. Hint: Find the present value of an annuity.

Step-by-Step Solution

Answer

The commission should have an initial amount of approximately $629,067.60 in the bank to guarantee the payments for the "Million Dollar" lottery. This is calculated using the present value of an annuity formula and considering the first $50,000 payment made immediately.

Step 1: Convert the interest rate to decimal form

To convert the given annual interest rate of 8% to its decimal equivalent form, we simply divide 8 by 100: \(r = \frac{8}{100} = 0.08\)

Step 2: Find the present value of the annuity

We can now plug the given values into the present value of an annuity formula: \(PVA = 50,000 \cdot \frac{1 - (1 + 0.08)^{-19}}{0.08}\)

Step 3: Calculate the expression within the formula

First, we need to calculate the expression within the parentheses: \((1 + 0.08)^{-19}\) Now, we can compute the rest of the formula, which is: \(PVA = 50,000 \cdot \frac{1 - (1 + 0.08)^{-19}}{0.08} \approx 579067.60\)

Step 4: Consider the first $50,000 payment

Since the first payment of $50,000 is made immediately, we need to add this amount to the present value of the annuity. We get: \(Total = PVA + 50,000 = 579067.60 + 50,000 = 629067.60\)

Step 5: Determine the initial amount the commission should have in the bank

The commission should have an initial amount of approximately $629,067.60 in the bank to guarantee the payments for the "Million Dollar" lottery.

Key Concepts

Time Value of Money

The concept of the time value of money is foundational in financial mathematics. It is based on the idea that money available today is worth more than the same amount in the future due to its potential earning capacity. This core principle acknowledges that if you have money in your hand today, you can invest it and earn interest, making it worth more in the future.

For example, receiving \(1,000 today is more valuable than receiving \)1,000 five years from now because you can invest the $1,000 now and accumulate additional wealth over those five years. Understanding this concept is crucial because it affects decisions regarding investments, loans, savings, and annuity payments.

Financial Mathematics

Financial mathematics is a field of applied mathematics concerned with financial markets. It focuses on tools and techniques that are used to solve problems related to finance. One of the primary areas within this discipline is the calculation of present and future values of cash flows.

Financial mathematics covers topics like simple and compound interest, annuities, amortization schedules, bonds, and options pricing. Professionals in finance use these mathematical concepts to analyze and predict market behavior and to optimize financial portfolios for risk and return.

Interest Rate Calculations

Interest rate calculations are at the heart of many financial transactions. They determine the amount of interest that accumulates over time on savings, loans, bonds, and other financial products. One key aspect is understanding how to convert interest rates into various forms, such as from an annual percentage rate (APR) to a monthly or daily rate, or from nominal to effective interest rates.

For instance, in the problem at hand, an annual interest rate of 8% is first converted to a decimal form (0.08). This is a necessary step because subsequent calculations, like finding the present value of an annuity, require the interest rate to be expressed as a decimal.

Annuity Payments

An annuity is a financial product that provides regular payments over a certain period of time. Annuity payments can be seen in products like pensions, structured settlements, and lottery winnings, such as the case in the exercise. There are two main types of annuities: ordinary annuities and annuities due. With ordinary annuities, payments are made at the end of each period, whereas with annuities due, payments are made at the beginning.

When calculating the worth or present value of these payments, a proper understanding of the annuity formula and its application is crucial. The present value of an annuity formula considers both the periodic payment amount and the time value of money to determine how much a series of future payments is worth today. This allows for a correct valuation of the annuity given the interest rate and the number of payments.