If projects are mutually exclusive, then the NPV and the IRR may disagree on which project adds value to the shareholder's wealth.

Yes, it's possible for the Net Present Value (NPV) and the Internal Rate of Return (IRR) to disagree on which project adds more value to the shareholder's wealth when projects are mutually exclusive. Here's why:

1. Definition: Mutually exclusive projects are those where the acceptance of one project means the rejection of the other.

2. NPV and IRR: NPV is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. IRR is the discount rate that makes the NPV of all cash flows (both positive and negative) from a project equal to zero.

3. Disagreement: The NPV and IRR methods can disagree due to differences in the distribution of cash flows and the scale of the projects. This is known as the "scale problem" or the "timing problem".

   • Scale Problem: If one project is significantly larger than the other, the IRR method may favor the smaller project because it can achieve its return faster, even though the larger project may generate more total profit, which would be favored by the NPV method.

   • Timing Problem: If the projects have different cash flow patterns, the one with earlier cash flows will likely have a higher IRR, while the one with larger later cash flows may have a higher NPV.

4. Shareholder's wealth: The goal of a company is to maximize shareholder's wealth. If the NPV and IRR disagree, the NPV method is generally considered superior because it directly relates to shareholder wealth maximization.

In conclusion, when projects are mutually exclusive, it's possible for the NPV and IRR to disagree on which project adds more value to the shareholder's wealth due to differences in the scale and timing of cash flows.