Risk Parity Maximum Diversification and Minimum Variance An Analytic Perspective
STEVEN THORLEY is the H. Taylor Peery Professor of Finance at Brigham Young University in Provo, UT. steven.thorley@byu.edu Portfolio construction techniques based on predicted risk, without expected returns, have become popular in the last decade. In terms of individual asset selection, minimum variance and (more recently) maximum diversification objective functions have been explored, motivated in part by the cross sectional equity risk anomaly first documented in Ang, Hodrick, Xing, and Zhang [2006] Application of these objective functions to large (e.g. 1,000 stock) investable sets requires sophisticated estimation techniques for the risk model. On the other end of the spectrum, the principal of risk parity, traditionally applied to small set (e.g. 2 to 10) asset allocation decisions, has been proposed for large set security selection applications. Unfortunately, most of the published research on these low risk structures is based on standard unconstrained portfolio theory, matched with long only simulations. The empirical results in such studies are specific to the investable set, time period, maximum weight constraints, and other portfolio limitations, as well as the risk model. This article compares and contrasts risk based portfolio construction techniques using long only analytic solutions. We also provide a simulation of risk based portfolios for large cap U.S. stocks, using the CRSP database from 1968 to 2012. We perform this back test using a single index model, standard OLS risk estimates, and no maximum position or other portfolio constraints, which leads to easily replicable results. The concept of risk parity has evolved over time from the original concept that Bridgewater embedded in research in the 1990s. Initially, an asset allocation portfolio was said to be in parity when weights are proportional to asset class inverse volatility. For example, if the equity subportfolio has a forecasted volatility of 15 percent and the fixedincome subportfolio has a volatility of just 5 percent, then a combined portfolio of 75 percent fixed income and 25 percent equity (i.e. three times as much fixed income) is said to be in parity. This early definition of risk parity ignored correlations, even as the concept was applied to more than two asset classes. Qian [2006] formalized a more complete definition that considers correlations, couching the property in terms of a risk budget where weights are adjusted so that each asset has the same contribution to portfolio risk. Maillard, Roncalli, and Teiletche [2010] call this an "equal risk contribution" portfolio, and analyzed properties of an unconstrained analytic solution. Lee's [2011] equivalent "portfolio beta" interpretation says that risk parity is achieved when weights are proportional to the inverse of their beta, with respect to the final portfolio. Anderson, Bianchi, and Goldberg [2012] analyze the historical track record of risk parity in an asset allocation context, while this article focuses on analytic