Long–Short Extensions: How Much Is Enough?

Abstract

Long–short extension strategies, such as 130–30, allow portfolio managers to reduce the implementation inefficiencies associated with the long-only constraint. Ample research using benchmark-specific and time period–specific numerical analyses indicates that long–short extensions increase expected information ratios. What is lacking is a general theory or mathematical model of long–short extensions based on underlying assumptions about benchmark composition, the security covariance matrix, and the portfolio optimization process. The analytical model developed here identifies the roles various parameters play in determining the size of the long–short extension. The impact of changes in the model parameters over time and across markets is illustrated with the use of historical and current equity benchmark data.

Long–short extension ratios, such as 130–30, are an increasingly common way for the investment management industry to describe portfolios that have been released from the long-only constraint. The ratio of a portfolio’s long and short positions to net notional value is often the primary description of the strategy. Unfortunately, managers and their clients may not understand the underlying parameters associated with the value of the long–short ratio beyond generally recognizing that the size of the extension (e.g., 30 percent) and active risk are positively related. This study develops a mathematical model to identify the underlying parameters that determine the size of the long–short extension. The relationships are illustrated with historical data on 500 large-capitalization U.S. stocks and current data on a variety of U.S. and international equity benchmarks.

The analytical model enhances perspectives from previous studies that depended on benchmark- and period-specific numerical examples or on insights from simulations. The model assumes a generic security-ranking process, a simple covariance matrix, a single measure of benchmark concentration, and unconstrained portfolio optimization. The model confirms the basic intuition that the size of the long–short extension increases with the active-risk target chosen by the manager and decreases with the estimated costs of shorting. In addition, the model shows that the unconstrained short extension decreases with individual security risk and increases with the following parameters: security correlation, the weight concentration of the benchmark, the number of securities in the benchmark, and the assumed accuracy of security return forecasts.

The model provides important perspectives on long–short extension strategies. For example, three of the model parameters—individual security risk, security correlation, and benchmark weight concentration—change over time, which suggests that to maintain a constant level of active risk, the exact size of the long–short extension should be allowed to vary. Application of the analytical model to a variety of U.S. and international benchmarks indicates that the number of securities and the weight concentration of the chosen benchmark have a substantial impact on the size of the unconstrained long–short extension.

Note: Analytic Investors applies a disciplined quantitative process to manage a variety of equity strategies for institutional and individual investors.