Tracking Error Models for Multiple Benchmarks

We propose a new multiple-benchmark tracking-error model for portfolio selection problem. The tracking error of a portfolio from a set of benchmark portfolios is defined as the difference between its return and the highest return from the set of benchmarks. We derive closedform solution of our portfolio strategy, whose main component is the sum of the benchmark portfolios weighted by their respective probabilities of attaining the highest return among the portfolios in the benchmark. These probabilities, also known as the persistency values, are less sensitive to estimation errors in the means and covariances. These features help to stabilize the computational performance of our portfolio strategy against estimation errors. We use the proposed model to address several pertinent issues in active portfolio management: (1) What are the benefits in tracking performance of multiple benchmarks? We demonstrate that under suitable conditions, multiple benchmarks tracking error model can actually produce portfolio strategy that has less variability in portfolio returns, compared to the portfolio strategy constructed using single benchmark model, given a fixed target rate of returns. This addresses the agency issue in this problem, as portfolio managers are more concerned with variability of the excess returns above the benchmark, whereas the investors are more concerned with the variability of the total returns. (2) How and when to rebalance the portfolio allocation when prices and asset returns change over time, taking into account transaction cost? We show that our model can control for transaction cost by adding the buy-and-hold strategy into the set of benchmark portfolios. This approach reduces drastically the transaction volume of several popular static portfolio rules executed dynamically over time. Last but not least, we perform comprehensive numerical experiments with various empirical data sets to demonstrate tha our approach can consistently provide higher net Sharpe ratio (after accounting for transaction cost), higher net aggregate return, and lower turnover rate, compared to ten different benchmark portfolios proposed in the literature, including the equally weighted portfolio (the 1/N strategy). ∗Department of Decision Sciences, NUS Business School, National University of Singapore, Singapore. Email: [email protected]. Part of this work was done when the author was at the Department of Mathematics, Faculty of Science, National University of Singapore, Singapore. †Department of Decision Sciences, NUS Business School, National University of Singapore, Singapore. Email: [email protected] ‡Department of Engineering Systems and Design, Singapore University of Technology and Design, Singapore. Email: natarajan [email protected] §Department of Decision Sciences, NUS Business School, National University of Singapore, Singapore. Email: [email protected]