A Continuous-Time Framework for Time-Weighted Returns, Money-Weighted Returns, Internal Rates of Return, and Dietz Return Estimators
William C. Lindsey （ Professor Emeritus, University of Southern California, Los Angeles, California, USA ）
Gita Govahi （ Assistant Dean Marshall School of Business, University of Southern California, Los Angeles, California ）
https://doi.org/10.37155/2972-4813-gep0403-1Abstract
This paper develops a unified continuous-time framework for portfolio return measurement in the presence of external cash flows. Beginning with a stochastic differential equation describing portfolio value dynamics, we demonstrate how several widely used return measures—including the Time-Weighted Return (TWR), Simple Dietz Return, Modified Dietz Return, and Internal Rate of Return (IRR), also known as the Money-Weighted Return (MWR) or Dollar-Weighted Return (DWR)—may be interpreted as alternative reductions or approximations of a common portfolio evolution process.
The framework clarifies the relationships among these methodologies and identifies the role of invested-capital approximation in performance measurement. In particular, the Modified Dietz Return denominator is interpreted as a first-order approximation to average invested capital within a continuous-time setting, providing a theoretical explanation for its practical effectiveness. The analysis further examines the sources of divergence among return measures and demonstrates how differences arise from cash-flow timing, cash-flow magnitude, valuation frequency, and portfolio volatility.
The framework is subsequently extended to AUM fee arrangements, establishing a direct relationship between fee revenue and average invested capital through time. Numerical examples illustrate both the convergence and divergence of the major return measures under alternative cash-flow scenarios.
The principal contribution of this paper is the development of a unified continuous-time interpretation of portfolio return measurement. By identifying average invested capital as a common underlying quantity, the framework clarifies the mathematical and economic relationships among Time-Weighted Returns, Money-Weighted Returns, Internal Rates of Return, Dietz Return estimators, and AUM fee economics.
Keywords
Time-Weighted Return; Money-Weighted Return; Internal Rate of Return; Dollar-Weighted Return; Modified Dietz Return; Portfolio Performance Measurement; Continuous-Time Finance; Assets Under Management (AUM)Full Text
PDFReferences
[2] CFA Institute. (2020). GIPS® Standards Handbook for Firms: Explanation of the Provisions in Sections 1–8. CFA Institute, Charlottesville, VA. November 2020.
[3] Dietz, P. O. (1966). “Pension Fund Investment Performance—What Method to Use When.” Financial Analysts Journal, Vol. 22, No. 1, pp. 83–86.
[4] Feibel, B. J. (2003). Investment Performance Measurement: Theory and Practice. John Wiley & Sons, Hoboken, NJ.
[5] Spaulding, D. (2011). Investment Performance Measurement, 2nd Edition. John Wiley & Sons, Hoboken, NJ.
[6] Karatzas, I., & Shreve, S. E. (1998). Methods of Mathematical Finance. Springer-Verlag, New York.
[7] Merton, R. C. (1969). “Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Case.” Review of Economics and Statistics, Vol. 51, No. 3, pp. 247–257.
[8] Merton, R. C. (1971). “Optimum Consumption and Portfolio Rules in a Continuous-Time Model.” Journal of Economic Theory, Vol. 3, No. 4, pp. 373–413.
[9] Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications, 6th Edition. Springer-Verlag, Berlin.
[10] Sharpe, W. F. (1966). “Mutual Fund Performance.” Journal of Business, Vol. 39, No. 1, Part 2, pp. 119–138.
[11] Treynor, J. L. (1965). “How to Rate Management of Investment Funds.” Harvard Business Review, Vol. 43, No. 1, pp. 63–75.
[12] William C. Lindsey, Robert Grados, Gita Govahi. The Modified Dietz Return Estimator: A Unified Framework for Portfolio Return Measurement. Global Economic Perspectives Vol 4 Issue 2 2026.
Copyright © 2026 William C. Lindsey, Gita Govahi
Publishing time:2026-07-22
This work is licensed under a Creative Commons Attribution 4.0 International License