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I have this simple problem about Net Present Value (NPV) and I was wondering why should we maximise it. Suppose you have a firm and you can either sell it now for $ \\\$ $ 200.000 or invest $ \\\$ $30.000 in a project that will bring $ \\\$ $50.000 at the end of the year then sell the firm for $ \\\$ $200.000. The interest rate is 10%.

The NPV of the first choice is simply $ \\\$ $200.000, while the NPV of the second choice is $250.000/1.10 - 30.000=197.272$. So we must pick the first choice.

However, for me both projects can lead us to have $ \\\$ $220.000 at the end (since we can invest the 200.000 at 10% interest rate in the first choice), so why is the second project worse ?

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    $\begingroup$ You said... both projects can lead us to have $220.000 at the end. If you have to choose between 220k and 220k, you should pick 220k. There is no problem so I'm voting to close until you proof-read your question. $\endgroup$
    – H2ONaCl
    Oct 7, 2023 at 7:17
  • $\begingroup$ @H2ONaCl I agree that the wording of the question could be improved, but I think the problem is clear enough, ie NPV analysis and comparison of the respective end positions seem to lead to different conclusions. $\endgroup$ Oct 7, 2023 at 8:27

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The second project involves investing \$30,000, but you haven't specified where this money comes from. Let's consider two cases: a) the owner of the firm has no cash initially and must borrow the \$30,000; b) they have enough cash to fund the investment.

In case (a), the owner will need to pay interest at 10% to borrow the \$30,000, an outflow of \$3,000. They will also need to repay the \$30,000 at the end of the year. So the second project will leave them after one year with cash of: $$200,000 - 3,000 - 30,000 + 50,000 = \\\$217,000$$ The first project, as you say, will yield cash after one year of \$220,000. In this case, therefore, it is not true that both projects result in cash of \$220,000 after one year. The first project comes out better, just as it does on the NPV criterion.

In case (b), there is no interest to be paid as the owner can fund the \$30,000 investment. Under the second project there is however an opportunity cost, the \$3,000 interest that could have been earned on the \$30,000 if it were not invested in the project that will bring \$50,000. The first project, because it does not make that investment, can use the \$30,000 to earn \$3,000 interest, which is additional to the \$200,000 from sale of the firm and \$20,000 interest on the sale proceeds. So again the first project comes out better, again consistently with the conclusion from the NPV criterion.

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