The NPV (Net Present Value) rule and the IRR (Internal Rate of Return) rule are two popular methods used in capital budgeting to evaluate investment projects. Both methods are used to determine whether an investment is profitable or not, but they differ in their approach and calculation.

The NPV Rule:

The NPV rule is based on the concept of time value of money, which states that a dollar today is worth more than a dollar in the future due to inflation and the opportunity cost of investing. The NPV rule calculates the present value of all cash inflows and outflows of a project and compares it to the initial investment. If the present value of the cash inflows is greater than the initial investment, the project is considered profitable and should be accepted.

The NPV formula is as follows:

NPV = Σ [CFt / (1 + r)t] - C0

Where CFt is the cash flow in year t, r is the discount rate, and C0 is the initial investment.

The IRR Rule:

The IRR rule is based on the concept of the internal rate of return, which is the discount rate that makes the present value of the cash inflows equal to the initial investment. The IRR rule calculates the rate of return that a project will generate and compares it to the required rate of return. If the IRR is greater than the required rate of return, the project is considered profitable and should be accepted.

The IRR formula is as follows:

0 = Σ [CFt / (1 + IRR)t] - C0

Where CFt is the cash flow in year t, IRR is the internal rate of return, and C0 is the initial investment.

Key Differences:

The NPV rule and the IRR rule differ in the following ways:

1. Approach: The NPV rule is an absolute approach, while the IRR rule is a relative approach.

2. Discount Rate: The NPV rule uses a discount rate to calculate the present value of cash flows, while the IRR rule calculates the rate of return that makes the present value of cash flows equal to the initial investment.

3. Reinvestment Assumption: The NPV rule assumes that cash inflows are reinvested at the required rate of return, while the IRR rule assumes that cash inflows are reinvested at the IRR.

4. Multiple IRRs: The IRR rule can have multiple rates of return, which can lead to confusion and incorrect decision making.

In conclusion, both the NPV rule and the IRR rule are useful methods to evaluate investment projects. However, the NPV rule is more widely used in practice due to its simplicity and accuracy in determining the profitability of a project.