Performance Attribution and the Fundamental Law

Abstract

The reported study operationalized the “fundamental law of active management” in the context of a factor-based performance attribution system. The system incorporates factor payoffs in the linear regression framework that many portfolio managers and external reviewers use to judge what is being rewarded in the market. The study indicates that parameters of the fundamental law can be used to approximate and interpret the results of the regression-based performance attribution system. The procedure is illustrated by the use of security holdings, returns, and factor exposure data for two portfolios benchmarked to the S&P 500 Index for April 1995 to March 2004.

The study reported here operationalized the “fundamental law of active management” by using a factor-based performance attribution system that identifies the sources of benchmark-relative returns in actively managed portfolios. Some of the relative return can be ascribed to marketwide factor exposures that differ from the benchmark, such as beta, company size, and company sector membership, and the realized payoffs to those factors. Relative performance not captured by these marketwide factors is generally attributed to security selection. In practice, the information content of the security-ranking system is often measured by an information coefficient or the performance of stocks grouped within quantile rankings, with little attempt to relate the success of the security-ranking system to its actual basis point contribution to performance. In this article, we show how a regression-based attribution system can be extended to decompose the active return associated with stock selection into the information content of the rankings and constraint-induced noise.

The fundamental law of active management shows that, in addition to the forecasting power of the ranking system, performance is also influenced by how well the manager is able to structure the portfolio to capture the most attractive securities. The relationship between the security rankings and actual over- and underweight positions in the portfolio is measured by the transfer coefficient. A previous extension of the fundamental law demonstrated that the lower the transfer coefficient, the more noise in the active return. The procedures we discuss here allow the contribution from the security rankings to be separated from the noise component and give the manager insight into the determinants of portfolio performance.

To illustrate the attribution procedure and test the accuracy of the fundamental law, we collected data on two portfolios benchmarked to the S&P 500 Index for the 108 months of April 1995 to March 2004. We examined performance attribution results for both a long-only portfolio and a long-short portfolio constructed on the basis of the same signal. The results illustrate the advantages in implementation efficiency of long-short strategies. Despite the simplifying assumptions used in the fundamental law mathematics, our estimates of signal and noise contributions were within a basis point per month of the contributions from regression analysis. We next used the 108 monthly time-series observations to test two key predictions of the fundamental law: an ex ante or expectational relationship for the information ratio and an ex post relationship describing the sources of realized variance in active returns.

The fundamental law yields predictions about the expected value and variance of active returns under the assumption of fixed parameter values. Thus, the perfect empirical test of the fundamental law predictions requires repeated observations of the same month (or a time series without any structural changes in the market). In practice, covariance matrices and the underlying effectiveness of security-ranking procedures change over time, so our nine years of monthly observations provided only a rough check on the fundamental law predictions. Nevertheless, using the time-series averages as proxies for fixed parameter values, we found that the average information ratio in our sample is reasonably close to the value predicted by using the ex ante fundamental law equation with a transfer coefficient. In addition, the proportions of realized performance variance attributable to signal success and to constraint-induced noise are related to the squared transfer coefficient but with a bias toward more signal contribution than the ex post fundamental law equation predicts. Our subperiod analysis suggests that this bias results from nonstationarities inherent in real markets over time.

We thank Steven Sapra for technical assistance in conducting the optimizations.