Home > Current

Vol 4 No 3 (2026)

  • A Continuous-Time Framework for Time-Weighted Returns, Money-Weighted Returns, Internal Rates of Return, and Dietz Return Estimators

    William C. Lindsey, Gita Govahi

    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.

    PDF