Harindra de Silva

Minimum-Variance Portfolios in the U.S. Equity Market

In the minimum-variance portfolio, far to the left on the efficient frontier, security weights are independent of expected security returns. Portfolios can be constructed using only the estimated security covariance matrix, without reference to equilibrium expected or actively forecasted returns. Empirical results illustrate the practical value of large-scale numerical optimizations using return-based covariance matrix estimation methodologies, providing new perspective on the factor characteristics of low-volatility portfolios. Optimizations that go back to 1968 reveal that the long-only minimum-variance portfolio has about three-fourths the realized risk of the capitalization-weighted market portfolio, with higher average returns.

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

Know Your VMS Exposure

One of the ongoing debates in equity market research is the set of common factors that explains the cross section of individual stock returns. With the influential backing of Fama and French [1993], a three-factor model that includes the market, size, and value factors is frequently cited in academic research and widely used in portfolio management. More recently, momentum has joined the list of accepted factors, resulting in references to a four-factor model. Lately, security volatility has begun to be used, along with the factors just mentioned, in describing portfolio risk. The authors introduce a specific measure of the idiosyncratic volatility factor that mirrors the Fama–French methodology, calling it VMS for volatile-minus-stable stocks. VMS is calculated for the entire span of the CRSP database and found to have strong credentials. VMS seems to be more important than SMB (small-minus-big market capitalization) and HML (high-minus-low book-to-market ratio), and similar to UMD (up-minus-down past return) in explaining the covariance structure of stock returns. The relative importance of VMS holds over the entire history for which it can be measured in the U.S. market (1931–2008) and continues to be an important factor in the covariance structure of stock returns in recent decades (1983–2008). Volatility, however, is not very orthogonal to the more well-known factors, a desirable property for new factors. Specifically, VMS is highly correlated to the general market (e.g., volatile stocks outperform stable stocks when the general equity market goes up) despite the fact that the authors measure security volatility in a market-idiosyncratic setting. VMS is also positively correlated with SMB (e.g., volatile stocks tend to outperform when small-cap stocks outperform) despite the Fama–French process of double sorting on market capitalization. Finally, VMS is negatively correlated with HML (e.g., volatile stocks tend to outperform when growth stocks outperform) although this correlation was not pronounced until the last few decades. In contrast to the other Fama–French factors, the average return of the VMS factor has been close to zero over time and negative in recent decades.

Toward More Information-Efficient Portfolios

The long-only constraint imposed in traditional portfolios is one of the more severe constraints in terms of its impact on potential value-added, particularly for portfolios benchmarked to a capitalization-weighted benchmark such as the S&P 500; it can reduce the effectiveness of the managers information by 50% or more. This loss can be avoided to a great degree by eliminating the long-only constraint or by creating a market-neutral portfolio with a derivatives overlay to restore market exposure. The information ratio can also be increased considerably using only underlying securities by allowing modest short positions and using the cash generated to purchase an equivalent amount of long positions, thus maintaining full market exposure.

Risk Allocation versus Asset Allocation

Most investors are exposed to both systematic and active risks in their portfolios. Systematic risks stem from consistent exposure to marketwide factors, and are usually associated with marketwide risk premiums. Active risk comes from actively managing underlying security and/or systematic risk exposures. Traditional long-only investment strategies are usually dominated by systematic risk, while alternative investment strategies typically have more active risk than systematic risk. A risk allocation framework that explicitly differentiates between these two sources of risk enables investors to improve the risk/return profile of their portfolios. Such a framework also enables investors to better incorporate non-traditional or alternative investment strategies into portfolios by characterizing them in terms of their systematic and active risk rather than thinking of them as a separate asset class.

A Factor Approach to Asset Allocation

The typical asset allocation decision focuses on gaining exposure to systematic market risks such as equity, interest rate, and credit risk. Investors also often explicitly manage their exposure to firm-specific characteristics like size, book-to-market, or momentum. For a global portfolio, we can add another category of exposures not correlated with systematic market risks and firm-specific characteristics: global market factors that explain the cross-section of returns across individual equity, fixed-income, and currency markets. Portfolios constructed to include exposures to each of these three categories of risk and return seem to be more efficient at producing diversified returns than those limited to just systematic market risks. Using this factor-based approach to asset allocation results in optimal portfolios with significantly less exposure to equity market risk than the typical institutional portfolio generated using the traditional asset allocation approach.

Pure Factor Portfolios and Multivariate Regression Analysis

Linking factor portfolio construction to cross-sectional regressions of security returns on standardized factor exposures leads to a transparent and investable perspective on factor performance. Under capitalization weighting, multivariate regression coefficients translate to portfolio returns that are benchmark relative and cleared of secondary factor exposures. The methodological contributions in this article are illustrated using a 50-year data set of 1,000 large U.S. stocks and five factor exposures: value, momentum, small size, low beta, and profitability. Using two case studies in factor portfolio analysis, the authors focus on cheapness, as measured by earnings yield, and interest rate risk, as measured by sensitivity to the 10-year Treasury bond return.

Fundamentals of Efficient Factor Investing

Even in the absence of security-specific alphas, constructing a total portfolio from factor sub-portfolios is not generally mean-variance efficient. For example, an optimal combination of four fully-invested factor sub-portfolios, Low Beta, Small Size, Value, and Momentum, captures only about 45 percent of the potential improvement over the market Sharpe ratio. In contrast, a long-only portfolio of individual securities, using the same risk model and return forecasts, captures about 90 percent of the potential improvement. In this paper, we adapt general portfolio theory to factor-based investing, and investigate optimal combinations of factor portfolios using the largest one thousand common stocks in the U.S. equity market from 1968 to 2014.

The Not-So-Well-Known Three-and-One-Half Factor Model

Equity analysts conceptualize the Fama-French framework as a tool for studying the size and value characteristics of equity portfolios along with the market return. But the market return is not the return to market beta. In fact, commercial providers of equity risk models typically include both a market factor and a beta factor, along with variations of the size and value factors. In other words, in equity risk modeling practice, the basic Fama-French framework includes four factors not just three. Unlike the other three factors, the intercept term (i.e., market factor) does not have a coefficient that varies across securities so can be described as just half a factor. We clarify the nature and role of the “first�? factor in equity return models and explain that the distinction between the market portfolio return and the return to the cross-sectional variation in security beta also applies to portfolio performance measurement. Specifically, the realized alphas of low (high) beta portfolios are reduced (increased) when a beta factor is included. The problem of ignoring the beta factor in performance measurement pertains to fully invested portfolios that have a low or high beta based on security selection, not to changes in portfolio beta induced by cash or leverage.

Toward More Information‐Efficient Portfolios

When certain constraints are imposed on portfolio managers, their ability to efficiently capture alpha is diminished. Managers may dramatically improve expected risk-adjusted performance by relaxing the short-sale constraint on long-only portfolios. Although a portfolio’s target tracking error may dictate different levels of short-selling activity, even a modest introduction of short selling can result in dramatic improvement.