Roger Clarke, Harindra de Silva and Steven Thorley
The strategic perspectives and terminology of the fundamental law is a common framework in the practice of active portfolio management. For tractability, fundamental law theory depends on the simplifying assumption of a diagonal covariance matrix of security returns, though the matrices supplied to numerical optimizers are fully populated. We extend the fundamental law of active management to allow for a full covariance matrix and show that the resulting ex-ante (expected) and ex-post (realized) return equations are exact in contrast to the approximate equality of previous derivations. The exactness of ex-post equations allows for performance attribution of realized returns that completely decomposes the return. Because the various fundamental law parameters we define incorporate all the information in the covariance matrix, they should also provide better ex-ante insights as to the sources and limitations of risk-adjusted active return. In addition to the generalization of the fundamental law, we describe a full covariance matrix alpha generation process and add some comments to the concept of implied breadth. The mathematics and practical application of the full covariance matrix fundamental law parameters are illustrated using an EAFE benchmarked portfolio with the 21 countries as individual securities.