内部收益率几个基本问题的证明

Mathematical Proof of Several Basic Problems of Internal Rate of Return

为了使项目内部收益率的计算和应用具有一个完整和严格的理论基础，利用数学手段对它的唯一性和概率分布以及相关问题进行了分析证明，得到了内部收益率在一定范围内是唯一的以及其概率在某些条件下近似服从正态分布的理论结果，亦修正了终值方程存在唯一正根则为内部收益率的错误说法．最后，讨论了项目净现值的解析特性，得出了存在内部收益率时净现值在其大于零的范围内为严格单调下降函数的结论，从理论上证明了其使用的合理性．

In order to lay a complete and strict foundation for computation and application of the internal rate of return (IRR) of a project, the uniqueness and probability distribution of IRR were proved by means of mathematical theories. The result shows that the IRR is unique in a certain range, and the probability is approximately in accordance with the normal distribution theory in some conditions. That the IRR of project exists only if the final value equation has a unique positive root is proved incorrect. Additionally, the analytic properties of the net present value (NPV) are discussed. It can be concluded that the NPV, in the range greater than zero is a strictly monotone decreasing function when IRR exists. So the application of NPV is rational.