A piece of land may be purchased for $610,000 to be strip-mined for the underlying coal. Annual net income will be $200,000 for 10 years. At the end of 10 years, the surface of the land will be restored as required by a federal law on strip-mining. The reclamation will cost $1.5 million more than

Question

A piece of land may be purchased for $610,000 to be strip-mined for the underlying coal. Annual net income will be $200,000 for 10 years. At the end of 10 years, the surface of the land will be restored as required by a federal law on strip-mining. The reclamation will cost $1.5 million more than the land’s resale value after it is restored. Is this a desirable project, if the minimum attractive rate of return is 10%?

Accepted Answer

In Example 5–11 the NPW was calculated to be $41,000. Since the NPW is positive, the project seems desirable. However, at interest rates below 4.07% or above 18.29% (see Figure 7A–1), the NPW is negative, indicating an undesirable project. This indicates that the project is undesirable at 4% and desirable at 10% and not at 20%. This does not make sense.

The results warn us that NPW may not always be a proper criterion for judging whether or not an investment should be undertaken. In this example the disbursements ($610,000 + $1,500,000) exceed the benefits (10 ×$200,000), which certainly does not portray a desirable investment. Thus Example 5–11 shows that NPW calculations in certain infrequent conditions can lead to unreliable results.

As outlined in this appendix, the problem is in the cash flows. It is not solved by relying on present worth as an economic measure over a confusing double root for the rate of return.

Example 7A–1 demonstrates that cash flows can have multiple solutions to the IRR equation (PW = 0). The example also demonstrates a common feature of such a problem—the cash flows fit neither the pattern of an investment nor the pattern of a loan. Both investments and loans have a single change in the sign of the cash flows over time. In contrast, Example 7A–1 has two sign changes in its series of cash flows.

In Example 7A–1 the open pit mine has a negative initial cash flow, which is followed by a series of positive cash flows—the pattern of an investment. However, the cash flows at the project’s end follow the pattern of a loan—initial cash flows are positive and then the final cash flow for remediation is negative. Thus, Example 7A–1 is neither a loan nor an investment, but rather a combination. Thus it is impossible to know whether it is desirable to have a high interest rate for an investment or a low interest rate for a loan.

Two or more sign changes in the series of cash flows are the distinguishing feature of instances where multiple roots can but may not arise when solving for the rate of return for a set of cash flows.

Examples like the cash flow diagram with a first cost, uniform return, and final remediation expense have been thoroughly analyzed (Eschenbach, Baker, and Whittaker). Only problems with a very large final cost have double positive roots. More important, due to the uncertain remediation expenses being much larger than the initial costs, the sensitivity of the results means that the PW values seem also to be unreliable guides for decision making when there are two positive roots for the PW = 0 equation.

Question 7A.1: A piece of land may be purchased for $610,000 to be strip-mi...

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