In the previous posts, we have seen the way to compute Net Present Value (NPV) and Internal rate of return (IRR) for cash flow streams and how they affect investment decisions, continuing that discussion, in this post we will see an example of how incremental internal rate of return can be used to analyze alternate investment options, for this we will look into an example from Investment Science by Luenberger (Problem 2.8) which goes as follows.

Problem 2.8

Two copy machines are available, both have useful lives of 5 years. One machine can be either leased or purchased outright; the other must be purchased. Hence there are a total of three options: A, B and C. The details are shown in the below table (The first year's maintenance is included in the initial cost. Then there are four additional maintenance payments, occurring at the beginning of each year, followed by revenues from resale). The present values of the expenses of these three options using a 10% interest rate are also indicated in the table. According to a present value analysis, the machine of least cost, as measured by the present value, should be selected, that is option B from the below table.

It is not possible to compute the IRR for any of these alternatives, because all cash flows are negative (except for the resale values). However it is possible to calculate the IRR on an incremental basis, Find the IRR corresponding to a change from A to B. Is a change from A to B justified on the basis of the IRR?

Problem 2.8

Two copy machines are available, both have useful lives of 5 years. One machine can be either leased or purchased outright; the other must be purchased. Hence there are a total of three options: A, B and C. The details are shown in the below table (The first year's maintenance is included in the initial cost. Then there are four additional maintenance payments, occurring at the beginning of each year, followed by revenues from resale). The present values of the expenses of these three options using a 10% interest rate are also indicated in the table. According to a present value analysis, the machine of least cost, as measured by the present value, should be selected, that is option B from the below table.

It is not possible to compute the IRR for any of these alternatives, because all cash flows are negative (except for the resale values). However it is possible to calculate the IRR on an incremental basis, Find the IRR corresponding to a change from A to B. Is a change from A to B justified on the basis of the IRR?

Analysis:

Since IRR cannot be calculated for any of the above cash flow streams (remember IRR for a cash flow stream has unique positive solution only if the sign changes from negative to positive or vice versa, in the cash flow stream, see here). But we can analyze the IRR corresponding to a change from A to B and a change from A to B is justified if the IRR(Change from A to B) > 10% (interest rate).

To calculate the IRR from A to B, we subtract the cash flow streams A and B (or B and A) and then compute the IRR of the resulting cash flow stream as shown in the below figure.

**Spreadsheet computation:**Incremental IRR on moving from option A to B

Note: In the below figure, each cash flow stream has 6 cash flows, the first year's maintenance + initial cost (negative cash flow), followed by 4 additional maintenance cost payments (all negative cash flows) followed by a resale value (>=0), for example, for option A, the cash flow stream would be like (-6000, -8000, -8000, -8000, -8000, 0) based on the above data in the table.

From the spreadsheet computation, we find that the internal rate of return for a change from A to B is 11.84%, since this value is greater than the 10% interest rate, therefore a change from A to B is justified based on IRR, in general, incremental IRR calculation an be used to analyze different investment alternatives like the above example, this will be handy in cases where IRR cannot be calculated for individual cash flow streams.

From the spreadsheet computation, we find that the internal rate of return for a change from A to B is 11.84%, since this value is greater than the 10% interest rate, therefore a change from A to B is justified based on IRR, in general, incremental IRR calculation an be used to analyze different investment alternatives like the above example, this will be handy in cases where IRR cannot be calculated for individual cash flow streams.

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