Factor investing has played a significant role in the financial markets over the past few decades, where certain factors have earned a premium through exposure to systematic sources of risk.
An extension of the capital asset pricing model (CAPM), the original factor model developed by Fama and French had three factors: market beta, value and size. As interest in this form of investment analysis has grown, factor models were extended to include momentum through the Carhart model (1997).
Since then, academic studies have explored hundreds of possibilities, including macroeconomic and style factors, contributing to a substantial literature on the topic. However, historical results indicate that only a small subset of factors may possess a persistent ability to generate excess return. Looking across both literature and practice, it is clear that deeper examination of factor phenomena can stimulate new insights and support the development of fresh strategies to capture alpha.
The Quant Research group at Bloomberg is examining ways to combine vast data sets and machine learning techniques to harvest risk premia from factor exposure. Some of the latest research focuses on options-implied factors for equity investing. The team has surveyed recent literature that discusses if and how options trading influences the underlying stock returns. Using a cross-sectional factor model, they test a range of relevant signals for a five-year period, 2013–2018, and they present a data-driven, machine learning approach to constructing composite volatility skews that improves the out-of-sample Sharpe ratio for the resulting factor portfolio.
A look under the hood – how does the factor model work?
Factor models can be macroeconomic or fundamental in nature. Macroeconomic factor models use observable economic time series, such as inflation, oil prices and interest rates, as factors that can help explain security returns. Fundamental factor models are cross-sectional and use the returns of a long/short portfolio associated with observed security characteristics, such as the book-to-market ratio and implied volatility. When constructing a portfolio, factors can be used for selecting and weighting securities in a cross-sectional portfolio and signals will help to determine entry and exit points for the individual securities over time.
In the case of multiple factor testing, steps must be taken to reduce biases and to screen out noise. One solution is out-of-sample testing, which is the cleanest way to rule out spurious factors. This method cannot be used in real time and out-of-sample data will always exist in the future. However, the history of deviation tells a story, and the Bloomberg team has explored this aspect in detail.
In one strand of research, the team has constructed a composite volatility skew factor by combining similar factors to form a group score, given that the weights have been normalized and regularized. On the question of weight optimization, the conventional approach is to select the weightings in a way that maximizes the predictive power of the composite skew in forecasting stock returns. However, an alternative approach is to select the weights to maximize the realized Sharpe ratio over a certain period of time. In factor investing, maximizing for Sharpe is, in fact, the relevant target of optimization. This research combined the factor model and machine learning, with a focus on the Sharpe ratio over the training period.
Studying the skews
Taking a data-driven approach to construct the optimal combination of multiple skew factors, the team could examine the trade-off between training and validation performance. They worked to improve the out-of-sample Sharpe ratio of the pure factor portfolio and curated an interpretable attribution for the composite skew factor exposure. In this process, multiple skews are combined into one composite skew, and only the final composite skew enters the factor model, together with Fama-French-Carhart. Using this composite approach instead of including all skews in a large multi-factor model reduces the dimensions of the factor space and makes the final factor model more parsimonious and robust.
It is important to note that using the realized Sharpe ratio, the target of optimization does not depend on the overall magnitude of the weights. Unlike a conventional regularized linear regression problem, the weights on factor exposures in a linear composite factor are normalized by definition.
In short, the training, validation and testing procedure for the composite skews contains the following steps:
- Pick a security universe and time period for the data.
- Divide the in-sample data into training and validation time periods.
- Parameterize a weighted combination of the skew factor scores.
- Parameterize a regularization penalty.
- Establish the target of optimization: for each set of weights and regularization, this is the Sharpe ratio of the factor portfolio returns for a given time period, adjusted by the Fama-French-Carhart factors.
- Optimize the weights with regard to the training data.
- Optimize the regularization with regard to the validation data.
- Evaluate the performance of the composite factor for out-of-sample data.
The results of the process are intriguing; when studying options volatility skews as an equity factor, the team found that the twisting of volatility skews across maturities is a significant factor in stock returns. Specifically, the factor portfolio is long long-term volatility skews and short short-term volatility skews. Isolating the picture of performance to 2017–2018, it is clear that composite skews have something to offer those interested in factor analysis and performance.
Download the whitepaper to learn more about the evidence that options trading influences the underlying stock returns.