Problem 10 - Applied Calculus

Question

A company is expected to earn \(\$ 50,000\) a year, at a continuous rate, for 8 years. You can invest the earnings at an interest rate of \(7 \%\), compounded continuously. You have the chance to buy the rights to the earnings of the company now for \(\$ 350,000 .\) Should you buy? Explain.

Step-by-Step Solution

Answer

No, you should not buy because the present value of the earnings is less than the cost.

Step 1: Understand the Cash Flows

The company will earn $50,000 each year, continuously, for 8 years. This means that the earnings are evenly spread throughout each of these years, and we will calculate their present value as a single lump sum due to continuous compounding.

Step 2: Formula for Present Value with Continuous Compounding

The present value for continuous compounding can be calculated using the formula: \[ PV = C \times \frac{1 - e^{-rt}}{r} \]Where:- \( PV \) is the present value,- \( C \) is the annual cash flow ($50,000),- \( r \) is the continuous compounding interest rate (0.07),- \( t \) is the total period (8 years).

Step 3: Calculate the Present Value of Earnings

Substitute the values into the formula:\[ PV = 50000 \times \frac{1 - e^{-0.07 \times 8}}{0.07} \]This uses the natural exponent function to adjust the value of continuous earnings over the investment horizon.

Step 4: Perform the Calculation

Calculate \( e^{-0.56} \), which is approximately 0.5712. Thus,\[ 1 - e^{-0.56} \approx 1 - 0.5712 = 0.4288 \]Now, compute:\[ PV \approx 50000 \times \frac{0.4288}{0.07} \approx 50000 \times 6.1257 \approx 306,285 \]

Step 5: Compare Present Value with Cost

The calculated present value of the earnings, \( \\(306,285 \), is less than the asking price of \( \\)350,000 \).

Step 6: Make the Decision

Since the present value \( (\\(306,285) \) is less than the cost \( (\\)350,000) \), buying the company's earnings would result in a loss if purchased at this price because the earnings value does not cover or exceed the initial investment.

Key Concepts

Present Value

Present value (PV) is a crucial concept when it comes to understanding the value of money received in the future today. It allows you to determine how much a series of future cash flows is currently worth, given a specific interest rate. When money is continuously compounded, it means that interest is added to the principal at every possible instant, optimizing growth.
For calculating the present value with continuous compounding, the formula used is: \[PV = C \times \frac{1 - e^{-rt}}{r}\]Here, \(C\) represents the constant annual cash flow you expect, \(r\) is the continuous compounding interest rate, and \(t\) is the time period in years. This formula accounts for the time value of money, adjusting future earnings by diminishing them back to present terms.
Understanding the PV concept is essential for making effective investment decisions as it reveals whether the future earnings justify the current investment or not.

Investment Decision

Investment decisions are choices made by an investor, determining where and how much to invest based on future gains and current costs. It often involves comparing potential earnings with the investment's present value. In the scenario, the decision revolves around whether the \(\\(350,000\) asking price for the company's earnings rights is worth it, considering the calculated present value of those future earnings.
By calculating the present value of the company's annual earnings, we determine whether the expected future cash flows are worth more or less than the upfront investment. This is a common method used in investment appraisal, helping to assess the profitability and feasibility of a potential investment. In this case, since the present value of the cash flows (\(\\)306,285\)) is less than the \(\$350,000\) cost, it suggests that buying the earnings rights would not be a profitable decision.

Cash Flow Analysis

Cash flow analysis involves evaluating the expected inflows and outflows of cash over a given period, helping investors or business owners understand their liquidity, profitability, and long-term financial health. This is done by assessing all the expected future cash earnings a company brings in.
In our example, the company has a steady cash flow of \(\$50,000\) annually over eight years. By analyzing these flows, investors can use formulas like the present value with continuous compounding to adjust these flows back to their current value. This adjustment helps them measure if these earnings surpass the initial costs.
Engaging in a rigorous cash flow analysis allows for more strategic decision-making, ensuring that investors are putting their resources into worthwhile ventures. It balances expected gains and current investment, underlining the importance of using present value calculations to achieve a clear financial picture.