Problem 28 - Calculus and its applications

Question

A group of entrepreneurs is considering the purchase of a fast-food franchise. Franchise A predicts that it will bring in a constant revenue stream of \(\$ 120,000\) per year for 8 yr. Franchise B predicts that it will bring in a constant revenue stream of \(\$ 112,000\) per year for 10 yr. Based on a comparison of accumulated present values, which franchise is the better buy, assuming the interest rate is \(5.4 \%,\) compounded continuously, and both franchises have the same purchase price?

Step-by-Step Solution

Answer

Franchise B is the better buy with a higher present value of approximately $817,207.

Step 1: Understanding Present Value with Continuous Compounding

The present value (PV) of a constant stream of cash flows can be calculated using the formula for continuous compounding: \[ PV = C \times \left( 1 - e^{-rt} \right) / r \] where \( C \) is the annual revenue, \( r \) is the continuous interest rate (in decimal form), and \( t \) is the number of years.

Step 2: Calculate Present Value for Franchise A

For Franchise A, the annual revenue \( C \) is \( \$120,000 \), \( r = 0.054 \) (5.4% as a decimal), and \( t = 8 \) years. Calculate the present value: \[ PV_A = 120,000 \times \left( 1 - e^{-0.054 \times 8} \right) / 0.054 \].

Step 3: Calculate Present Value for Franchise B

For Franchise B, the annual revenue \( C \) is \( \$112,000 \), \( r = 0.054 \), and \( t = 10 \) years. Calculate the present value: \[ PV_B = 112,000 \times \left( 1 - e^{-0.054 \times 10} \right) / 0.054 \].

Step 4: Solve and Compare Present Values

First, compute the exponential terms: \( e^{-0.054 \times 8} \approx 0.6504 \) and \( e^{-0.054 \times 10} \approx 0.5985 \). Then, calculate for Franchise A: \[ PV_A = 120,000 \times \left( 1 - 0.6504 \right) / 0.054 \approx 772,810.37 \]. For Franchise B: \[ PV_B = 112,000 \times \left( 1 - 0.5985 \right) / 0.054 \approx 817,206.93 \]. Compare \( PV_A \) and \( PV_B \) and choose the higher one.

Key Concepts

Continuous Compounding

When dealing with cash flows that are spread across multiple years, calculating their present value is crucial. Typically, interest can be compounded at intervals such as annually or semi-annually, but continuous compounding treats the interest as if it compounds constantly. This means at every possible instant, offering a more precise value that's slightly higher than traditional compounding techniques.

The formula for present value with continuous compounding is given by \[ PV = C \times \left( 1 - e^{-rt} \right) / r \]where:

• \( C \) is the annual cash flow.
• \( r \) is the interest rate expressed in decimal form.
• \( t \) is the time period in years.

To effectively use this method, it's important to understand the implications of \( e \), the base of natural logarithms, as it plays a key role in the formula, adjusting the cash flow based on the continuous interest application.

Cash Flow Analysis

Cash flow analysis involves evaluating the cash that comes in and out of a business over a certain time frame. For entrepreneurs considering investments such as a franchise, understanding cash flows is vital to assessing profitability.

In the given exercise, Franchise A and Franchise B each provide a steady stream of cash annually. Franchise A gives \( \\(120,000 \) over 8 years, while Franchise B provides \( \\)112,000 \) over 10 years. Such analysis helps determine which franchise yields a better return when these cash flows are discounted to present value terms.

By calculating the present value of these future cash flows using continuous compounding, investors can effectively compare different opportunities, considering the time value of money, and see which investment might offer greater total returns.

Interest Rate Calculations

Interest rate calculations underpin many financial theories and practices. In this context, they help us understand the impact of accrued interest on future cash flows.

When the interest rate is quoted as continuous, it means that interest is theoretically added at every possible moment. This differs from traditional methods where interest is added at discrete intervals.

An interest rate of \( 5.4\% \) becomes \( r = 0.054 \) when expressed as a decimal for formula use. Consistently using this decimal form across the calculation ensures precision, and small deviations can significantly affect results over time. Properly managing and calculating with interest rates, especially when compounded continuously, aids in making informed financial decisions and comparisons, such as selecting the best franchise investment option.