# Economic Intuitions Behind The Q-Factors

## Summary

- Intuitively, investment predicts returns because given expected cash flows, high (low) costs of capital mean low (high) net present values of new projects and low (high) investment.
- Profitability predicts stock returns because high expected cash flows relative to low investment must mean high discount rates.
- Investment and profitability (ROE) factors are almost totally uncorrelated, meaning that they are independent, or unique.
- There is a significant correlation between q-factors and shocks in state variables.

In their groundbreaking paper "Digesting Anomalies: An Investment Approach," published in the March 2015 issue of The Review of Financial Studies, Kewei Hou, Chen Xue and Lu Zhang proposed a new four-factor asset pricing model that went a long way toward explaining many of the anomalies that neither the Fama-French three-factor model nor subsequent four-factor models could explain. The authors called their model the " *q*-factor" model. Specifically, their four factors are:

The market excess return (beta). The difference between the return on a portfolio of small-cap stocks and the return on a portfolio of large-cap stocks (size). The difference between the return on a portfolio of low-investment stocks and the return on a portfolio of high-investment stocks. The difference between the return on a portfolio of high return-on-equity (ROE) stocks and the return on a portfolio of low ROE stocks.

In our book Your Complete Guide to Factor-Based Investing, Andy Berkin and I established five criteria that should be required before you consider allocating to a factor. The criteria are: persistence across long periods of time; pervasiveness across industries, countries, regions and even asset classes; robustness to various definitions; implementability (survives transactions costs); and intuitive risk- or behavioral-based explanations that provide reasons for believing the premium should persist in the future. We prefer risk-based explanations because risk cannot be arbitraged away, although popularity and the resulting cash flows can reduce premiums. However, we are willing to accept behavioral explanations because of limits to arbitrage which, along with the tendency for human behavior to remain unchanged, allow anomalies, such as the poor performance of small growth stocks with high investment and low profitability, to persist.

**Intuitive Risk-Based Explanation**

Given that one of the five required criteria is having an intuitive explanation for the persistence of the premium (with a preference for a risk-based explanation), it is important to note that Hou, Xue and Zhang provided theoretical underpinnings for the investment and profitability factors. They explained: "Intuitively, investment predicts stock returns because given expected cash flows, high costs of capital mean low net present values of new projects and low investment, and low costs of capital mean high net present values of new projects and high investment. Profitability predicts stock returns because high expected cash flows relative to low investment must mean high discount rates. The high discount rates are necessary to offset the high expected cash flows to induce low net present values of new projects and low investment."

Among their important findings was that the investment and profitability (ROE) factors are almost totally uncorrelated, meaning that they are independent, or unique. In addition, the authors found that the alphas of the value and momentum factors in the *q*-factor model are small and insignificant. These two factors, and the role they play, have been replaced by the investment and ROE factors. They also found that the *q*-factor model outperforms the Fama-French three-factor and four-factor models in its ability to explain most anomalies. In fact, most anomalies become insignificant at the 5 percent level of statistical significance. In other words, "Many anomalies are basically different manifestations of the investment and ROE effects."

The authors did acknowledge, however, that "the *q*-factor model is by no means perfect in capturing all the anomalies." Like all models, even the *q*-factor model is flawed, or wrong. If a model were perfect, it would be called a law (as we have in physics).

Because of its empirical success, the use of the *q*-factor model by practitioners for performance evaluation and portfolio management has been increasing. In addition, Eugene Fama and Kenneth French built on the concepts from the *q*-factor model, incorporating a profitability factor and an investment factor into their five-factor model (market beta, size and value being the other factors) and a six-factor model, which added momentum.

Given the risks of data mining, it's important that any factor model has intuitive explanations for why the factors explain the variation in returns across diversified portfolios. Otherwise, there will be questions about the theoretical soundness of the model.

### **Q****-Factors and Economic Risks**

**Q**

Suresh Rajput and Muhammad Ilyas contribute to the literature on factor models with their January 2020 study "Do the Q-Factors Proxy for Surprises in Economic State Variables? They investigated whether the q-factors of ME (size), I/A (investment/assets) and ROE correlate with surprises in economic variables within the framework of the intertemporal capital asset pricing model (ICAPM). They ask if the *q*-factors are in line with the intertemporal asset pricing theory, which theorizes that average returns are explained by the responsiveness of returns to the changes in expected investment opportunities. They chose a set of important economic state variables that have been shown in the literature to help explain the mean and variance of stock returns, such as term spread, short-term T-bill yield, default spread and dividend yield, to describe the changes in expected investment opportunities. The term spread is the spread of 10-year and one-year government bonds. The dividend yield is the sum of the last 12 months' dividends divided by the level of the index. The default spread is the spread of long-term Baa and long-term government bonds. And the risk-free rate is the yield of a one-month Treasury bill. Their data sample covered the U.S. market over the period January 1967 to December 2018. Following is a summary of their findings, each of which was significant at least at the 5 percent confidence level:

There is a significant correlation between *q*-factors and shocks in state variables. ME correlates with the surprises in default spread, term spread and Treasury bill yield. I/A is related to the surprises in aggregate dividend yield. ROE correlates with surprises in term spread and default spread.

These findings led the authors to conclude: "These findings suggest that the *q*-factors may act as a proxy for the surprises in economic state variables that describe the changes in the investment opportunity set."

Summarizing, Rajput and Ilyas provide evidence that the *q*-factors are related to surprises to economic variables that have been shown in the literature to help explain the mean and variance of stock returns, providing the important theoretical risk-based explanation for why investors should expect the premiums to persist, as risk cannot be arbitraged away.

**Conclusion**

Through their research, financial economists continue to advance our understanding of how financial markets work and how prices are set. The Fama-French three-factor model was a significant improvement on the single-factor capital asset pricing model (CAPM). Mark Carhart moved the needle further by adding momentum as a fourth factor. The authors of the *q*-theory made further significant advancements, which in turn motivated the development of the competing Fama-French five- and six-factor models.

The competition to find superior models is what helps advance our understanding not only of the markets but also our understanding about which factors to focus on when selecting the most appropriate investment vehicles and developing portfolios.

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