Risk Management and the Optimal Combination of Equity Market Factors

Abstract

Managing the intertemporal risk of optimally constructed multifactor portfolios adds to performance. The increases in Sharpe ratios are in addition to the utility that investors gain from controlling how much active risk they are exposed to over time. We derive a simple closed-form formula for security weights in optimal multifactor portfolios with an active-risk target. We test the risk control of five well-known factors—value, momentum, small size, low beta, and profitability—and the optimal multifactor portfolio. Our empirical research was carried out on the large-capitalization US equity market for 1966 through 2019. We conclude that for the equity market, more active factors are better than fewer if each subportfolio is “pure” as to factor, anchored to the benchmark, and combined on the basis of forecastable risks. Our portfolio construction methodology allows for transparent performance attribution and replication of the process in other markets and time periods.

Disclosure: The authors report no conflicts of interest.

Editor’s Note

This article was externally reviewed using our double-blind peer-review process. When the article was accepted for publication, the authors thanked the two anonymous reviewers in their acknowledgments.

Submitted 18 November 2019

Accepted 9 April 2020 by Stephen J. Brown.

Acknowledgments

We would like to thank Executive Editor Stephen J. Brown, Managing Editor Heidi Raubenheimer, CFA, and Co-Editor Daniel Giamouridis.

Notes

¹ “Most quantitative mutual funds are failing to beat their benchmarks as US stocks are on track for their best annual performance since 2013, according to research from Bank of America Corp.Large-cap quant funds lagged the Russell 1000 index by an average 3.2 percentage points this year through November” (Idzelis 2019).

² The Novy-Marx (2013) definition of profitability is revenue minus cost of goods sold, all divided by total assets—(REVT – COGS)/AT—using Compustat account codes. For financial sector stocks—for example, commercial banks—we subtracted customer deposits from total assets, so our definition of profitability is (REVT – COGS)/(AT – DPTC). Deposits, which are technically a liability rather than an asset of a bank, represent a source of capital that is neither equity nor long-term debt.

³ Earning positive returns from the volatility premium requires positions in the derivatives market—for example, delta-neutral writing of at-the-money index put and call options—together with a long position in the market portfolio or, recently, derivative contracts known as “volatility swaps.”

⁴ We use the phrase “anchoring to the benchmark” to describe the impact of using scores with cross-sectional cap-weighted means of zero and variances of 1.0, as described in Clarke et al. (2016), in contrast to arithmetic mean-zero unit-variance cross-sectional scores. The benchmark-anchored pure factor returns used in this study are available at https://www.deepcreekm.com.

⁵ Daily factor returns in this study were calculated from beginning-of-month factor exposures and the total daily return to each security in CRSP. In other words, the factor portfolios are not rebalanced during the month, although such rebalancing does not materially change the results.

⁶ The Bollerslev (1986) GARCH(1, 1) model estimates for the daily pure-active-factor returns over the entire 54-year sample are similar to market returns, with highly significant ARCH and GARCH terms. Specifically, the ARCH parameter estimate for the excess market return was 0.088 (t-statistic of 39.0), and the GARCH parameter estimate was 0.898 (t-statistic of 283.8). The ARCH and GARCH estimates for the active factor returns are as follows:

Table

Similar to the market return, the active returns to all five factors have large and significant “theta” coefficients in the EGARCH (“E” for exponential) specification of Nelson and Cao (1992), indicating that negative returns have a larger impact on subsequent volatility than do positive returns.

⁷ Daily returns tend to be positively autocorrelated, so sample variance is less than it would be for more independent observations, such as monthly returns. This issue was first noted by Lo and MacKinlay (1988) in their “volatility ratio” tests.

⁸ The transfer coefficient, introduced in Clarke, de Silva, and Thorley (2002), has various interpretations, including the cross-sectional correlation coefficient between security positions that are constrained to be long only and those positions in an unconstrained optimization. In this context, the TC is defined by the quotient of the long-only portfolio IR and the optimal IR.