Rao’s Method: A novel approach to Internal Rate of Return (IRR)

Those familiar with my work know that I have been researching IRR calculation for over a decade. In 2020, I explored a boundary-based approach but found it insufficiently robust. More recently, I developed an NPV-NFV-based approach and collaborated with Dr. A. C. S. Rao, a professor at the Indian Institute of Technology, to formalize it for presentation and publication as a conference paper at the RAIT 2025 conference. Here, I provide a brief overview and will share the link to the paper once it is published in IEEE Xplore.

Introduction

The Internal Rate of Return (IRR) is a widely used financial metric for assessing the profitability of investments. Traditional methods, such as Newton-Raphson and Brent’s method, struggle with irregular cash flows, multiple IRR scenarios, and convergence issues. Many common tools, including Microsoft Excel, Google Sheets, and numpy_financial, fail in certain cases, leading to incorrect or undefined IRR values.

In our research, we introduce Rao’s Method, a novel approach that integrates Net Present Value (NPV) and Net Future Value (NFV) to improve the accuracy and stability of IRR calculations. By leveraging residual minimization, our method ensures a more reliable IRR estimation across diverse cash flow scenarios.

Challenge the Norm

Traditionally the IRR is defined as the discount rate that equates the Net Present Value (NPV) of future cash flows to zero. Mathematically, IRR is the rate r that satisfies the equation:

IRR in terms of NPV

Theoretically at IRR even Net Future Value (NFV) should be zero

IRR in terms of NFV

In theory, at IRR, both NPV and NFV should be zero. However, in practical scenarios, due to the effects of compounding, particularly in cases with extreme cash flows, even a minor rate change can cause significant variations between NPV and NFV. Therefore, IRR can be better defined as the rate that sufficiently minimizes NPV or NPV, bringing either close to zero.

Foundation for Rao’s Method

Rao’s Method

Rao’s Method revolutionizes IRR computation by addressing the shortcomings of traditional approaches. Unlike conventional methods that rely solely on Net Present Value (NPV), Rao’s Method integrates both NPV and Net Future Value (NFV) to enhance accuracy and stability.

Key Factors:

Evaluates IRR from both NPV and NFV perspectives. Uses residual minimization to select the most stable IRR. Employs a centroid-based initial guess to improve Newton-Raphson convergence. Handles irregular and oscillatory cash flows more effectively.

Rao’s Method

By utilizing a dual-validation approach, Rao’s Method ensures that the IRR remains consistent, even in cases where standard algorithms struggle with multiple IRRs, extreme values, or slow convergence. Through a residual minimization technique, Rao’s Method selects the most reliable IRR by effectively reducing both NPV and NFV errors. Additionally, it employs a centroid-based initial guess to enhance the efficiency of Newton-Raphson iterations, making it more robust against unpredictable cash flow patterns. While no method, including Rao’s, can entirely eliminate the issue of multiple IRRs, this innovative approach significantly enhances the reliability of IRR computation, particularly in complex financial scenarios.

Test It Yourself: Challenge Your IRR Calculation Skills!

This is not a debate on whether IRR is the best metric, as it can yield multiple values satisfying the equation. Instead, the goal of Rao’s Method is to improve the probability of finding a rate that satisfies the IRR equation. Try these simple Chit Fund cash flows Cash flow 1: -5000,-5000,-5000,-5000,-5000,-5000,-5000,-5000,98000,-6000,-6000,-6000,-6000,-6000,-6000,-6000,-6000,-6000,-6000,-6000 Cash Flow 2: -5000,-5000,-5000,-5000,-5000,-5000,-5000,-5000,-5000,99000,-6000,-6000,-6000,-6000,-6000,-6000,-6000,-6000,-6000,-6000

Test them in Google Sheets or Excel: Copy from here Evaluate IRR using Rao’s Method: Try it here

Future Work

There is ongoing debate about IRR’s limitations and whether NPV or IRR is the better approach for evaluating cash flows. I believe this debate arises from an incomplete picture of cash flows — when all relevant cash flows are considered, IRR, NPV, and NFV should ideally converge on the same preferred choice. My research will focus on validating this principle and addressing the gaps that cause discrepancies between these metrics.

Final Thoughts

Rao’s Method provides a structured, stable, and improved approach to IRR computation by considering both NPV and NFV, ensuring a higher probability of accurate results, especially in complex financial scenarios. I look forward to further discussions and feedback on this approach.