The time value of money (TVM) equation is a key equation in finance. It takes the form of an nth order polynomial having n roots. In finance it is normal to calculate and use only one root (interest rate). The remaining (n-1) roots are mostly complex or negative and they are usually discarded. This fact prompts a research project into whether the discarded, unorthodox roots have financial meaning or utility. The motivation to study the unorthodox roots lies in the econometric dictum that data is valuable and should not be discarded lightly. The research shows that the unorthodox rates matter. This paper contains an example of why they matter. Two of the most important criteria for choosing among capital investment projects are net present value (NPV) and internal rate of return (IRR). Although the two criteria often give the same rank order, in certain circumstances they provide inconsistent rankings and therefore suggest different investment decisions. The inconsistency sparked a debate about which criterion is better. The debate has lasted more than 100 years. The novel approach described in this paper takes into account all n solutions for the IRR. The result is a new equation for NPV. The equation provides a different perspective to the debate and suggests its resolution. The analysis reinforces the traditional view maintained in the literature that NPV is a reliable criterion while the orthodox IRR is not. The reasoning, however, does not rely on the ‘IRR pitfalls’ described in the literature. The analysis shows that two of the pitfalls are not true. Multiple IRRs do not represent a pitfall. This is because they are useful. The excesses of all IRRs over the cost of capital are the components of NPV. It is this fact that resolves the second pitfall of inconsistent ranking. NPV and the orthodox IRR can yield inconsistent rankings but NPV and the multiple-IRR criterion never do because they are identical.