Roger G. Clarke

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.

Option Portfolio Strategies: Measurement and Evaluation

It is well known that options can increase the flexibility of returns available from investment strategies. Papers by Ross (1976), Breeden and Litzenberger (1978), and Arditti and John (1980) point toward spanning opportunities which increase the efficiency of the financial markets in meeting the investment objectives of investors. While these and other studies have demonstrated the potential of option portfolio strategies in molding the return distribution to better fit risk preferences, a number of other studies have used simulation techniques to give an empirical view of just what effect particular option stock and option bond strategies will have in altering risk return patterns. A major study in this area is that of Merton, Scholes, and Gladstein (1978, 1982). Their study, presented in two papers, one on call option portfolio strategies (hereinafter MSGl) and the other on put option portfolio strategies (hereinafter MSG2), presents a broad and insightful discussion of many of the implications of combining options with stock portfolios. The MSG study employs a 12.5-year period for simulations in the first paper on portfolios using call options and a 14-year simulation period in the second paper on portfolios using put options. As they are careful to point out, their results depend on market trends which may not match ex ante expecta-

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.

Performance Attribution and the Fundamental Law

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.

How Much International Exposure is Advantageous in a Domestic Portfolio

The answer to how much international exposure is advantageous in a domestic portfolio depends on what the investor assumes about the long-run risk and expected return of the foreign assets and currency exposure, and on the investors risk/return penalty. The analysis here begins with the investor holding a core position in foreign assets to minimize the risk of the portfolio. Using estimates of volatility and correlation from market history, the authors suggest that a long-run allocation of 29% to 30% in foreign equity to deviate from this core allocation depending on the expected relative returns of domestic and foreign equity and on the expected relative returns of domestic and foreign equity and on the expected currency return.

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.