I have series of irregular negative payments followed by one positive payment. I would reorganize the data somewhat if I could solve the more general case where some of the intermediate payments were also positive, but this isn't necessary.

I hadn't suspected, until a few days ago, that this was a difficult problem. All the solutions I have looked at so far recognize that they will not always provide an answer, even when there is one. They generally say little or nothing about when there is or isn't a solution or the circumstances under which they will fail to find an existing answer. It would be helpful to know more about such constraints.

Unfortunately, I am not a mathematician and have difficulty understanding many of the texts I have found. What I need are some simple guidelines and constraints to go along with the algorithms.

To put this in perspective - the Excel XIRR function has been accepted as a solution thus far. But I don't see how to reliably find the maximum and minimum valued solutions where there are more than one and I don't know how likely it is to fail or what characteristics of the data would induce failure (other than leading payments of 0, which it doesn't handle at all).