Using the four–equity factor model with market, size, value, and momentum as factors, the authors find that linear correlations overstate the benefits of diversification. They model nonlinear factor dependence and demonstrate that significant economic value can be earned when recognizing nonlinear dependence in an alternative model.
Factors based on size, value, and momentum are used in modeling because of their ability to explain observed stock returns and also because they appear to have a low correlation with each other. The authors believe that focusing only on linear factor dependence is dangerous. They show that nonlinear factor dependence is significant and, if ignored, will lead to underestimation of extreme risks. Nonlinear dependence between the four factors (including market) can add value for risk-averse investors who allocate capital using the four-factor model. The authors present a skewed t copula model, which, in one case, improves returns by more than 1% per year versus the benchmark multivariate normal model.
How Is This Research Useful to Practitioners?
Portfolio managers who use factor models for stock selection or allocate between style portfolios will find this research useful for refining their modeling approach or in managing portfolio risk.
The authors examine time-series returns for each factor and observe the behavior of each factor. They find that the long-term returns (1965–2010) on momentum are large and quite remarkable. They note that the size factor accumulated losses during 1995–2000 as the market significantly increased and that momentum slumped during 2009 while the overall market was recovering. They also show that all four-factor distributions have fat tails.
The estimation of extreme risk concerns both portfolio managers and risk managers. During the quant meltdown in August 2007, the joint nonnormality of the four factors was apparent, with increased correlations between value, size, and momentum returns. The authors believe standard risk management techniques based on constant correlations are misleading and that ignoring multivariate nonnormality in equity factors leads to a large underestimation of risk.
The authors develop a model using a constant skewed t copula instead of the multivariate normal model. Their model performs the best in all but one case. More importantly, their model performance is not the result of more trading, which would lower realized returns because of higher transaction costs.
How Did the Authors Conduct This Research?
The authors examine almost 50 years of weekly equity factor returns. Market, size, and value factors are constructed. Size factors are constructed by using median market capitalization to form size portfolios. Lower and upper terciles of book-to-market ratios are used to construct the value factors. Market factor is the return on all stocks (on the NYSE, Amex, and NASDAQ) minus the one-month T-bill rate.
Bivariate analysis is conducted on the four-factor model and descriptive statistics are produced. Threshold correlations are calculated and the authors find clear evidence of asymmetry in the threshold correlations. Although linear correlations are almost zero, their threshold correlations are almost always positive and often increase as the thresholds become more extreme. In the case of a portfolio that is long small stocks and long momentum, the simple correlation is low, which suggests high diversification, when, in fact, correlation is very high when momentum performs poorly. An asymmetrical copula model is able to capture this factor dependence.
The authors assess the cost of ignoring time-varying and nonlinear dependence between factors using a real-time portfolio selection experiment. They find that when leverage is large (margin requirement of 20%), the nonnormal factor models offer significant economic gains.
Additionally, the authors investigate the effect of different models on portfolio risk measurement. The expected shortfall of an equally weighted portfolio of the four factors was 20%–50% higher during 2006–2010 when using an asymmetrical copula model for risk management instead of a multivariate normal distribution.
This research is valuable, particularly for any professional who uses size, value, and momentum factors in quantitative portfolio management for optimization purposes. For fund-of-funds managers who actively choose managers, an understanding of nonnormal models that use dynamic correlations may result in better diversification and risk estimation. Future research could look at additional factors, such as country or sector momentum or equity listed on non-US exchanges.