The Fundamental Law of Active Management : Time Series Dynamics and Cross-Sectional Properties

I derive a generalized version of the fundamental law of active management under some weak conditions. I show that the original fundamental law of Grinold and various extensions are special cases of the result presented in this paper. I also show that cross-sectional ICs are usually different from time series ICs even if the time series ICs are all the same across securities. The fundamental law derived in this paper is robust to forecast model specification. My results show that average signal IC and IC standard deviation are the two most important variables in determining the potential IR a strategy can achieve. I extend the fundamental law to models with multiple factors and study the impact of missing one or more return or risk factors to portfolio IR