How to Use GMM to compare performance of the input model with a capital-only model
According to Hou, Xue and Zhong (2015), the Q Theory is used in the testing of predictions that are attributed to the cross-sectional distribution of stock returns within financial markets. Consequently, the expected return of assets is described primarily by their sensitivity based on the excess of the risk free rate expressed in terms of several factors. These factors are the return on equity which is also referred to as a probability factor, an investment factor as well as a size factor. Consequently, the Q Factor model has been consistently outperformed the Fama and French Model both its terms of its empirical value and its conceptual framework.
As previously mentioned these factors are;
The size factor which is formulated as price multiplied by the outstanding number of shares
Investment to assets ratio which is expressed as the division between annual changes and total assets expressed within a year of lagged assets.
Return on Equity which is expressed as the income attributed to a financial period divided by the lagged book equity within a quarter.
According to Hou, Xue and Zhong (2015), a relatively high investment in stocks earns lower expected returns ceteris paribus based on a low investment strategy. Consequently, high expected return on Equity should be attributed to high expected returns and vice versa. There are several issues that are attributed to this model. These issues include the Internal Rate of Return which is expressed as the expected return over a one year period. Secondly, the expected investment which is most likely positive and finally, the investment performance during past periods which has been described as a poor benchmark in defining future expected returns.
Choice of Testing Assets and Evaluation Criteria
The traditional testing assets that are contained in the Q-factor model are fundamentally constructed on the basis of a 2*3*3 approach. As previously mentioned, these assets are investment-to Assets (I/A), the Return on Equity and the size. However, according to the new Fama and French approach, the traditional Q Factor model is reconstructed by introducing two new testing assets. These assets are the Robust Minus Weak Profitability and the Conservative Minus Aggressive (CMA) factor of investment. The reconstruction of the traditional Q Factor Model to include the two new testing Assets involves a double 2*3 approach in which the size testing factor is interacted directly with the investment factor and the operating profitability factor.
However, it is important to note that the Return on Equity factor under the traditional Q Factor model is expressed on a monthly basis. This basis of construction significantly deviates from the Robust Minus Weak Profitability factor which is essentially expressed on an annualized basis of profitability. This approach in the construction of the Q Factor model is further epitomized by the fact that the Return on Equity under the traditional approach forecasts the stock market returns of assets on the same level on which it forecasts the future performance of these markets, thereby, adopting a historical approach in the process of forecasting and modeling these returns. However, this approach exposes the model to a potential down side. Since the information relating to the future Return on Equity under this approach is pegged on the latest quarterly earnings, the most efficient approach in this regard would entail the use of the most recent earnings information and data based on the ROE testing factor. This approach does not therefore seem efficient in the case of the Robust Minus Weak approach since earnings information and data are entirely derived from the latest financial period year end.
In regards to this analysis, data and information relating to the monthly returns was derived from the Center For Research in Security Prices (CRSP), while information relating to the accounting practices of the underlying firms was derived from The Compustat Annual and Quarterly Fundamental Files. It is important to note that the data set which was used in this analytical process stretched from January 1967 to December 2013, and it did not involve either financial companies or companies with a book value which is negative.
It should be noted that there are four main concerns that are attributed to the traditional Q Factor model and which the FF 5-Factor approach attempts to remedy with its new construction and addition of test factors. The first concern is the deriving of factor information form historical sources which does not give a clear and precise picture of what the future outlook in regards to this returns actually holds.
The second basis of concern raised by Fama and French in relation to the Q Factor model is the redundancy which is associated with the valuation theories that described average returns that are found within this three-factor model. The third basis of concern entails the relationship between the internal rate of return and the expected investment which is primarily negative as is envisaged by this valuation theory. However, according to Fama and French, the reconstruction of the model would most likely yield a positive relationship going forward. The fourth concern entails the use of the CMA in relation to the expected investment as the primary proxy in determining its future performance which may result in coherent inaccuracies and inconsistencies relating to this average returns.
Detailed Description of Construction
As previously mentioned, the reconstruction of the five-factor model entails the addition of two new factors which are the Robust Minus Weak (RMW) profitability factor and the Conservative Minus Aggressive (CMA) investment factor. The RMW refers to the relationship between the weak and robust profitability with diversified portfolio`s returns. The CMA on the other hand refers to the relationship between the low and high stocks of investments and the diversified portfolio’s returns.
Consequently, there are three ways that can be used in the construction of the new factor model which incorporates both the RMW and CMA testing factors. The first approach entails the interaction of the book to market factor with the size factor within a framework of an independent 2 *3 approach. In this regard, the resulting operating profitability is computed as the total revenues less the selling administrative and general expenses, less the cost of goods sold and finally, less the interest expense.
The second approach that can be used in the construction of this model is relatively similar to the first approach in regards to the adoption of the 2*3 approach which replaces a prevailing 2*2 approach with the medians of the data set limits acting as the breakpoints for all the underlying variables. The third approach is slightly different from the first two approaches in the sense that it adopts a quadruple-sorts approach which is independent on the basis of 2*2*2*2. The corresponding factors in this case are book to market, investment, operating profitability and size basing these factors on the breakpoints of the underlying data set (NYSE).
It should be noted that the breakpoint size of the data set (the NYSE) is the median of the book to markets, operating profitability, investment and the market equity based on their 70th and 30th percentiles respectively for share prices listed on the NYSE. The HML on the other hand is the mean of the portfolio returns of the two low book-to-market portfolios of returns.
Power Beyond the HXZ Q Factor Model.
As a result of the evaluations of the data set covering the time period between January 1967 and December 2013, the following observations in relation to the superiority of this model over the traditional Q Factor models can be observed. Firstly, the return on Equity premiums cannot be captured by the by the Fama and French three factor model. However, after the introduction of the two new testing factors, the SMB, RMW, CMA and HML earn an average of 0.285, 0.27%, 0.36% and 0.37% respectively, a fete which could not have been accomplished under the old model.
Secondly, the reconstructed factor model is subsequently capable of capturing mean CMAs and RMWs with a small alpha variation of 0.02 and 0.045 respectively. Consequently, the correlation between the investment factor and the HML is established at 0.69, while the correlation between the UMD and the ROE is also established at 0.5, a fete which was previously not possible to achieve. Therefore, the introduction of the two new testing factors significantly increases the efficiency and accuracy of the tradition Q Factor Model which was formulated and advanced by Hou, Xue and Zhong.
Hou, K., Xue, C. and Zho, L. (2015). A comparison of Factor Models. New York: John Willey