Problem 42 - 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: $$\begin{array}{c|c|c|c|c}\hline \text { Ycar } & 1 & 2 & 3 & 4 \\\\\hline \text { Cash flow } & 550.000 & \$ 40.000 & \$ 25,000 & 520,000 \\\\\hline\end{array}$$ (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

Yes, purchase the machine as the present value of cash flows ($116,177.94) exceeds the machine's cost ($97,000).

Step 1: Understand the Cash Flows

The cash flows for the machine over 4 years are as follows: - Year 1: $50,000 - Year 2: $40,000 - Year 3: $25,000 - Year 4: $20,000 These are the amounts that will be received at the end of each respective year.

Step 2: Formula for Present Value Calculation

The formula to calculate the present value (PV) of each cash flow is given by \[PV = \frac{CF}{(1 + r)^t}\]where: - \(CF\) is the cash flow in each year.- \(r\) is the interest rate (7.5% or 0.075).- \(t\) is the time period (year).

Step 3: Calculate Present Value for Each Year

Calculate the present value for the cash flow of each year:- Year 1: \[PV_1 = \frac{50,000}{(1 + 0.075)^1} = \frac{50,000}{1.075} \approx 46,511.63\]- Year 2: \[PV_2 = \frac{40,000}{(1 + 0.075)^2} = \frac{40,000}{1.155625} \approx 34,603.67\]- Year 3: \[PV_3 = \frac{25,000}{(1 + 0.075)^3} = \frac{25,000}{1.24270859375} \approx 20,111.73\]- Year 4: \[PV_4 = \frac{20,000}{(1 + 0.075)^4} = \frac{20,000}{1.33773164} \approx 14,950.91\]

Step 4: Calculate Total Present Value

Add all the present values calculated in the previous step to get the total present value of the cash flows:\[\text{Total PV} = PV_1 + PV_2 + PV_3 + PV_4 \approx 46,511.63 + 34,603.67 + 20,111.73 + 14,950.91 = 116,177.94\]

Step 5: Compare Present Value with Machine Cost

The total present value of the cash flows is approximately $116,177.94. The cost of the machine is $97,000. Since the present value of the cash flows is greater than the cost of the machine, the investment is financially viable.

Key Concepts

Cash Flow Analysis

Cash flow analysis is a crucial financial tool that helps us determine if an investment is worth pursuing. It involves evaluating the anticipated cash inflows and outflows over a set period. In our example, the machine's purchase generates cash flows over four years. Understanding each cash flow is the first step:

• Year 1: Cash inflow of \(50,000\)
• Year 2: Cash inflow of \(40,000\)
• Year 3: Cash inflow of \(25,000\)
• Year 4: Cash inflow of \(20,000\)

These figures represent the amounts expected at each year's end. The analysis allows us to assess the pattern of inflows, identify shortfalls, and anticipate potential profitability. Regular evaluations help investors and businesses make informed financial decisions.

Net Present Value

Net Present Value (NPV) is a critical concept in financial management. It helps determine whether an investment will add value to a company. When calculating NPV, we discount future cash flows to the present using an interest rate. Here, we apply our given interest rate of 7.5% annually.Using the formula:\[PV = \frac{CF}{(1 + r)^t}\]where:

• \(CF\) is cash flow for each year,
• \(r\) is the interest rate (7.5% or 0.075),
• \(t\) is the time period in years.

By calculating each year's present value and summing them up, we arrive at the total present value. For our scenario, that turns out to be approximately \(116,177.94\). This figure is crucial as it gives us a singular value that allows comparison with the cost of the machine, thus determining the investment's potential profitability.

Investment Decision Making

Investment decision making involves analyzing and comparing the present value of cash flows to the investment cost. This practice ensures that resources are used effectively and investments are profitable.In our scenario, the total present value of cash flows, \(116,177.94\), exceeds the machine's cost of \(97,000\). This positive difference indicates that the investment will generate more income than its purchase price, making it a financially sound decision.Making such decisions requires attention to:

• The reliability of projected cash flows,
• The chosen discount rate reflecting the risk and opportunity cost,
• The comparison of NPV to the initial investment cost.

By carefully considering these factors, investors and businesses can make choices that support sustainable growth and profitability. When the NPV is positive, as in our example, it signals a green light for investment.