Problem 12 - Calculus Concepts

Question

Cola Sales In \(1993,\) PepsiCo installed a new soccer scoreboard for Alma College in Alma, Michigan. The terms of the installation were that Pepsi would have sole vending rights at Alma College for the next 7 years. It is estimated that in the 3 years after the scoreboard was installed, Pepsi sold 36.4 thousand liters of Pepsi products to Alma College students, faculty, staff, and visitors. Suppose that the average yearly sales and associated revenue remained constant and that the revenue from Alma College sales was reinvested at \(4.5 \%\) APR. Also assume that during that time PepsiCo received revenue of \(\$ 0.80\) per liter of Pepsi. a. The vending of Pepsi products on campus can be considered a continuous process. Assuming that the revenue was invested in a continuous stream and that interest on that investment was compounded continuously, how much did Pepsi make from its 7 years of sales at Alma College? b. Assuming a continuous stream, how much would PepsiCo have had to invest in 1993 to create the same 7 -year future value?

Step-by-Step Solution

Answer

Pepsi made approximately $10,897 from the 7 years of sales and would have had to invest about $8,650 in 1993.

Step 1: Calculate Total Revenue for Three Years

Over three years, Pepsi sold 36.4 thousand liters at $0.80 per liter, thus: \( \text{Total Revenue} = 36,400 \times 0.80 \). Calculate this to find total revenue.

Step 2: Determine Annual Revenue

Divide the total revenue calculated in Step 1 by 3 to determine the constant yearly revenue: \( \text{Annual Revenue} = \frac{36,400 \times 0.80}{3} \).

Step 3: Use Continuous Compounding Formula

To find the future value with continuous compounding, use \( FV = R \times \frac{e^{rt} - 1}{r} \), where \( R \) is the annual revenue, \( r = 0.045 \) (4.5% annual rate), and \( t = 7 \).

Step 4: Solve for Future Value

Substitute the values from previous steps into the equation from Step 3 to compute the future value: \( FV = \left( \frac{(36,400 \times 0.80)}{3} \right) \times \frac{e^{0.045 \times 7} - 1}{0.045} \).

Step 5: Calculate Present Value for Continuous Stream

Use the present value formula for continuous compounding: \( PV = \frac{FV}{e^{rt}} \). Substitute \( FV \) computed in Step 4, \( r = 0.045 \), and \( t = 7 \) into the equation.

Step 6: Solve for Present Value

Calculate the present value using \( PV = \frac{\left( \frac{(36,400 \times 0.80)}{3} \right) \times \frac{e^{0.045 \times 7} - 1}{0.045}}{e^{0.045 \times 7}} \) to find out how much PepsiCo would need to invest in 1993.

Key Concepts

Future Value

The concept of future value revolves around the growth of money over time due to interest or other gains. It's the amount to which a current sum of money will grow after a certain period when it's invested at a given interest rate. In continuous compounding, where the interest is calculated and added back to the principal continuously, the future value can be computed using this formula:

• \( FV = R \times \frac{e^{rt} - 1}{r} \)

where:

• \( FV \) is the future value.
• \( R \) is the constant revenue or cash flow per year.
• \( r \) is the annual interest rate.
• \( t \) is the time in years.

Continuous compounding applies here because Pepsi's revenue from vending is reinvested and grows continuously over the years.
The formula helps to compute the total amount Pepsi would earn after 7 years of continuous sales on campus.

Present Value

Present value is essentially the opposite of future value. It answers how much you'd need to invest now to achieve a future financial goal, considering a specific interest rate. The idea is that money today is worth more than the same amount in the future due to its potential earning capacity.
The formula used for finding the present value when compounding continuously is:

• \( PV = \frac{FV}{e^{rt}} \)

where:

• \( PV \) is the present value.
• \( FV \) is the future value computed earlier.
• \( r \) is the annual rate of return (or interest rate).
• \( t \) is the duration in years.

For PepsiCo, this calculation helps in understanding how much they would have needed to invest initially in 1993 to match the future earnings from the sales today.

Continuous Stream

A continuous stream refers to a regular, ongoing flow of revenue or cash without any interruptions. It's a strategy often used to describe constant income or investments that are expected to accrue throughout a period. In this context, Pepsi's agreement to sell their products continuously to Alma College can be viewed as such a stream.
By considering this continuous flow, you use integration in calculus to factor in all possible values over time. In financial terms, it's important as it provides a realistic representation of earnings, as if money is flowing in perpetually, rather than in discrete intervals.
This approach allows us to use specific finance formulas like those for present and future value with continuous compounding, ensuring that all fractions of time within the period are accounted for, which effectively maximizes growth potentials.

Compound Interest

Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan.
This powerful concept leads to exponential growth of invested money, especially when using continuous compounding. The formula for compound interest when compounded continuously is expressed as:

• \( A = P \times e^{rt} \)

where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (the initial amount).
• \( r \) is the annual interest rate (as a decimal).
• \( t \) is the time the money is invested for in years.

The key advantage of compound interest is that you earn interest on both the money you invest and the interest that accumulates. In Pepsi's sales arrangement, using continuous compounding ensures the most efficient growth over a longer period.