Problem 40 - Applied Calculus

Question

A company is considering whether to buy a new machine, which costs $$\$ 97,000$$. The cash flows (adjusted for taxes and depreciation) that would be generated by the new machine are given in the following table: $$ \begin{array}{c|c|c|c|c} \hline \text { Year } & 1 & 2 & 3 & 4 \\ \hline \text { Cash flow } & \$ 50,000 & \$ 40,000 & \$ 25,000 & \$ 20,000 \\ \hline \end{array} $$ (a) Find the total present value of the cash flows. Treat each year's cash flow as a lump sum at the end of the year and use an interest rate of \(7.5 \%\) per year, compounded annually. (b) Based on a comparison of the cost of the machine and the present value of the cash flows, would you recommend purchasing the machine?

Step-by-Step Solution

Answer

The present value of the cash flows is approximately \$116,198.87, which exceeds the machine's cost of \$97,000, so purchasing the machine is recommended.

Step 1: Identify Cash Flows and Interest Rate

First, note the cash flows from the machine for each year, which are \\(50,000 for Year 1, \\)40,000 for Year 2, \\(25,000 for Year 3, and \\)20,000 for Year 4. Also note the interest rate is given as 7.5\%, compounded annually.

Step 2: Calculate Present Value for Year 1 Cash Flow

Use the present value formula \( PV = \frac{FV}{(1 + r)^n} \), where \( FV \) is the future value, \( r \) is the interest rate, and \( n \) is the number of years. Substitute \( FV = 50,000 \), \( r = 0.075 \), and \( n = 1 \):\[ PV_1 = \frac{50,000}{(1+0.075)^1} = \frac{50,000}{1.075} \approx 46511.63 \]

Step 3: Calculate Present Value for Year 2 Cash Flow

Using the same formula, substitute \( FV = 40,000 \), \( r = 0.075 \), and \( n = 2 \):\[ PV_2 = \frac{40,000}{(1+0.075)^2} = \frac{40,000}{1.155625} \approx 34598.37 \]

Step 4: Calculate Present Value for Year 3 Cash Flow

Again use the formula, with \( FV = 25,000 \), \( r = 0.075 \), and \( n = 3 \):\[ PV_3 = \frac{25,000}{(1+0.075)^3} = \frac{25,000}{1.2421728} \approx 20123.02 \]

Step 5: Calculate Present Value for Year 4 Cash Flow

Substitute into the formula for the fourth year with \( FV = 20,000 \), \( r = 0.075 \), and \( n = 4 \):\[ PV_4 = \frac{20,000}{(1+0.075)^4} = \frac{20,000}{1.3365519} \approx 14965.85 \]

Step 6: Calculate Total Present Value of Cash Flows

Add the present values of each year's cash flow together:\[ PV_{total} = PV_1 + PV_2 + PV_3 + PV_4 \approx 46511.63 + 34598.37 + 20123.02 + 14965.85 = 116198.87 \]

Step 7: Compare Cost and Present Value of Cash Flows

The total present value of the cash flows is \\(116,198.87, which is greater than the cost of the machine, \\)97,000. Since the present value of cash flows exceeds the cost, the investment in the machine is financially beneficial.

Key Concepts

Cash Flow Analysis

Cash flow analysis is the process of examining the cash inflows and outflows from a business over a certain period of time. In the context of making investment decisions, like purchasing a new machine, cash flow analysis helps determine if the expected financial benefits would justify the initial expenditure.
For the exercise at hand, the cash flows generated by the machine are analyzed over a period of four years. These cash flows are initially high in Year 1 and diminish over the subsequent years. This is a common pattern in investment cash flows, reflecting the initial strong performance of an asset that gradually declines.
Using the information from the exercise, each year's cash flow is treated as a lump sum at the end of the year. This allows for a straightforward calculation of the present value for each year’s cash flow, thus providing a clear picture of the investment's financial viability.

Investment Decision

An investment decision involves evaluating the potential returns of an asset against its cost. In this problem, the company must decide whether buying the machine is worthwhile, based on the comparison of its cost to the present value of the expected cash flows.
The decision-making process is guided by the principle that an investment is considered beneficial if the present value of its future cash flows is greater than its cost. In this problem, the machine costs $97,000, while the calculated total present value of the cash flows is approximately $116,198.87.
Given that the present value of the cash flows exceeds the machine's cost, it indicates a profitable investment, justifying the purchase.

Interest Rate

The interest rate plays a crucial role in present value calculations. It reflects the time value of money — the principle that money today is worth more than the same amount in the future due to its earning potential. In this exercise, a 7.5% interest rate is used, compounded annually.
Interest rates are used to discount future cash flows to their present value. The higher the interest rate, the lower the present value of those future cash flows. This requires careful consideration, as small changes in the interest rate can significantly affect the present value and, consequently, the investment decision.
By applying the interest rate to each year's cash flow, we determine how much today's money, when invested at this rate, would equal the future cash flows from the machine.

Depreciation

Depreciation is the process of allocating the cost of an asset over its useful life. For this exercise, while the impact of depreciation is not directly calculated, understanding it is important, as it impacts cash flow analysis indirectly.
As a component of operating expenses, depreciation reduces taxable income, thus influencing the actual cash flows recorded for tax purposes. In the context of the exercise, the cash flows considered already account for depreciation and taxes, providing a more accurate picture of the net benefit of the investment.
Depreciation can affect the net cash flow by altering the timing and amount of tax benefits received, which can ultimately change the present value of the cash flows from a potential investment. Recognizing its impact can lead to more informed investment decisions.