Problem 19 - Applied Calculus

Question

McDonald's Corporation licenses and operates a chain of 31,377 fast-food restaurants throughout the world. Between 2005 and \(2008,\) McDonald's has been generating revenue at continuous rates between 17.9 and 22.8 billion dollars per year. \(^{8}\) Suppose that McDonald's rate of revenue stays within this range. Use an interest rate of \(4.5 \%\) per year compounded continuously. Fill in the blanks: (a) The present value of McDonald's revenue over a five-year time period is between__________________and _____________________ billion dollars. The present value of McDonald's revenue over a twenty-five-year time period is between _____________________and_____________________ billion dollars.

Step-by-Step Solution

Answer

5-year: between 81.36 and 103.56 billion; 25-year: between 311.47 and 396.77 billion.

Step 1: Understand the Problem

We need to find the present value of McDonald's future revenues, given revenue rates and an interest rate compounded continuously. We are provided revenue rates of 17.9 to 22.8 billion dollars per year and an interest rate of 4.5% per year.

Step 2: Formula for Present Value with Continuous Compounding

The present value (PV) of a continuous cash flow is calculated using the formula: \[ PV = \int_{0}^{T}Re^{-rt} \, dt \]where \( R \) is the revenue rate, \( r \) is the interest rate, and \( T \) is the time period. We will compute this for both 5-year and 25-year periods.

Step 3: Calculate Present Value for 5-Year Period

Substitute \( R = 17.9 \) and \( R = 22.8 \), \( r = 0.045 \), and \( T = 5 \) in the formula:\[ PV_{5} = \int_{0}^{5} 17.9e^{-0.045t} \, dt \] and \[ PV_{5} = \int_{0}^{5} 22.8e^{-0.045t} \, dt \]. Calculate these integrals to determine the minimum and maximum present values.

Step 4: Solve the Integral for 5-Year Period

Solving the integral for the lower revenue:\[ PV = \left. -\frac{17.9}{0.045} e^{-0.045t} \right|_{0}^{5} = \frac{17.9}{0.045} (1 - e^{-0.225}) \]. This calculation gives approximately 81.36 billion dollars.For the upper revenue:\[ PV = \left. -\frac{22.8}{0.045} e^{-0.045t} \right|_{0}^{5} = \frac{22.8}{0.045} (1 - e^{-0.225}) \]. This yields approximately 103.56 billion dollars.

Step 5: Calculate Present Value for 25-Year Period

For \( T = 25 \), use the equation:\[ PV_{25} = \int_{0}^{25} 17.9e^{-0.045t} \, dt \] and \[ PV_{25} = \int_{0}^{25} 22.8e^{-0.045t} \, dt \]. Solve these integrals to find the range of present values over 25 years.

Step 6: Solve the Integral for 25-Year Period

Solving the integral for the lower rate:\[ PV = \left. -\frac{17.9}{0.045} e^{-0.045t} \right|_{0}^{25} = \frac{17.9}{0.045} (1 - e^{-1.125}) \]. The calculation results in about 311.47 billion dollars.For the upper rate:\[ PV = \left. -\frac{22.8}{0.045} e^{-0.045t} \right|_{0}^{25} = \frac{22.8}{0.045} (1 - e^{-1.125}) \]. This calculation yields approximately 396.77 billion dollars.

Key Concepts

Continuous Compounding

Continuous compounding is a fascinating concept used in finance to understand the accumulation of interest payments. When we compound interest continuously, it means that the interest is being calculated and added to the principal balance instantly, at every possible moment. This results in a higher amount of accumulated interest compared to regular compounding intervals, such as annually or quarterly.

In essence, with continuous compounding, the formula for calculating the future value of an investment is given by:

• \[ A = Pe^{rt} \]
• Where "\( A \)" is the amount of money accumulated after \( n \) years, including interest.
• "\( P \)" is the initial principal or the original amount invested.
• "\( r \)" is the annual interest rate.
• "\( t \)" is the time in years.

Applying this concept helps businesses and investors to understand how much future cash flows are worth today when the interest rate is compounded continuously. This is especially useful when dealing with inflows or revenues that occur continuously, such as with large corporations like McDonald's that earn massive revenues steadily over time.

Revenue Stream

The term 'revenue stream' refers to the income generated by a business from its various day-to-day operations. For a giant like McDonald's, its revenue streams come from selling food and beverages across thousands of outlets around the world. These revenue streams are crucial because they aid in understanding the financial health of the company and estimating future earnings.

When businesses assess their future revenue streams, they often need to consider how much these revenues will be worth in present terms, especially when making long-term financial decisions. They use calculations of present value to determine this worth. Calculating the present value of a continuous revenue stream involves evaluating these expected revenues using the continuous compounding interest rate, which gives businesses a clear picture of the current worth of these future revenues.

This present value calculation considers not only the projected revenue amount but also the impact of the time period and the interest rate used, which in McDonald's case is a continuous rate of \(4.5\%\). This approach helps them decide on investments, expansions, or other financial strategies effectively.

Integral Calculus

Integral calculus plays a significant role in calculating present values of continuous revenue streams, like those of McDonald's. It helps in finding the total accumulation of these revenues over a specified time. Integral calculus is concerned with the concept of integration, which essentially represents the summation of infinite small data points to find a whole.

When we talk about continuous functions, such as continuous revenue streams, integration allows us to calculate the present value by evaluating the integral of the revenue function multiplied by the exponential decay of the continuous interest rate. This is represented in the formula:

• \[ PV = \int_{0}^{T}Re^{-rt} \, dt \]
• Here, "\( R \)" stands for the revenue rate and "\( r \)" for the interest rate.
• "\( t \)" denotes time, and "\( T \)" is the particular time span over which we are evaluating the revenue.

Integral calculus allows us to solve these complex calculations by breaking down the continuous flow of revenues and applying the compound interest rate to determine their present value. This makes it an indispensable mathematical tool in evaluating long-term financial decisions and accurately forecasting the monetary impact over time.