# Internal Rate of Return

The internal rate of return (IRR) is the discount rate for which the net present value of a project is zero. In other words, the sum of discounted costs is equal to the sum of discounted benefits when discounted by the IRR. This method is appropriate when there is only one alternative to the *status quo*. If the IRR is higher than the rate of return on alternative investments, then the project is a good investment. In some cases a minimum rate of return is used to determine which projects should be implemented. As with the benefit-cost ratio, the IRR can be calculated directly or incrementally.

### Calculating the Simple Internal Rate of Return

This can easily be done with a spreadsheet, which allows the discount rate to be varied until it results in a net present value of 0.

n = the number of years over which benefits and costs are analyzed

B_{i} = the benefits of the project in year i, i=0 to n

C_{i} = the costs of the project in year i

d = the discount rate

First, discount the costs and benefits in future years.

The discounted benefits of the project in year i are equal to B_{i}/(1+d)^{i}

The discounted costs of the project in year i are equal to C_{i}/(1+d)^{i}

Then calculate the net present value; sum both the discounted benefits and the discounted costs over all years (0 through n) and subtract the sum of the discounted costs from the sum of the discounted benefits: Σ (B_{i}/(1+d)^{i} ) - Σ (C_{i}/(1+d)^{i}) summed over i = 0 to n.

Then vary the discount rate, d, until the net present value equals 0. The resulting d is the IRR of the rate of return that the project will deliver.

### Calculating the Incremental Internal Rate of Return

This method is applicable if there are two or more alternative projects to compare to the base case.

B_{k} = the total discounted benefits of an alternative k, calculated as shown above

C_{k} = the total discounted costs of an alternative k, calculated as shown above

This method is analogous to the Incremental Benefit-Cost Ratio Method. Alternatives are considered in increasing order of total discounted costs until all are considered. The procedure is to follow the challenger-defender logic, at each step calculating the discount rate for which the increment in total discounted benefits (B_{c}-B_{f}) are equal to the increment in discounted costs (C_{c}-C_{f}). If the incremental IRR is greater than 0, the challenger becomes the defender. Ultimately, the surviving defender, f, is the economically preferred alternative.

Like the Incremental Benefit-Cost Ratio Method, this procedure is mathematically equivalent to the Net Present Value Method, and it always gives the same result. Howeverm since this method is mathematically tedious, it is rarely used.