Alex R. Horenstein

Eigenvalue Ratio Test for the Number of Factors

This paper proposes two new estimators for determining the number of factors (r) in static approximate factor models. We exploit the well-known fact that the r largest eigenvalues of the variance matrix of N response variables grow unboundedly as N increases, while the other eigenvalues remain bounded. The new estimators are obtained simply by maximizing the ratio of two adjacent eigenvalues. Our simulation results provide promising evidence for the two estimators.

Beta Matrix and Common Factors in Stock Returns

This paper proposes a new method to estimate the rank of the beta matrix in a factor model. We consider the case in which possible factor variables, which we call factor-candidate variables, are observed. The idiosyncratic error terms are allowed to be correlated both over different cross section units and over different time periods. For the factor model, estimating the rank of the beta matrix is equivalent to estimating the number of the relevant factors among the factor-candidate variables. The estimator we propose is easy to use because it is computed with the eigenvalues of the inner product of an estimated beta matrix. Simulation results show that the proposed method works well even in small samples. Our analysis of US individual stock returns is consistent with the notion that the three factors of Fama and French (FF, 1993) capture three different risk sources. The five factors of Chen, Roll, and Ross (CRR, 1985) are correlated with one additional risk factor that is not related to the Fama-French factors. The momentum factors capture a further risk source that is not captured by the FF and CRR factors. In addition, the two factors proposed by Chen, NovyMarx and Zhang (CNZ, 2010) capture a source of risk missed by all the above factors. These results suggest that there are six common risk factors in US individual stock returns among the thirteen factor candidate used. Furthermore, our analysis of portfolio returns reveals that the estimated number of common risk sources changes depending on how the portfolios are constructed. The number of risk sources found from the analysis of portfolio returns is generally smaller than the number of common factors in the individual stock returns.

Understanding Growth Patterns in US Health Care Expenditures

We study the steady upward trend of Health Care Expenditures (HCE) over GDP for a sample of OECD countries between 1970 and 2007. Although the United States is clearly an outlier, almost all of the additional increase in US HCE happened during the 1978–1990 period. We perform two growth accounting exercises to explore sources of variability of HCE over GDP across countries. In the first growth accounting exercise based on value added we find that factor accumulation is unable to replicate the observed growth patterns. We also show that the additional increase in markups in the US corporate medical sector mimics well the ratio of HCE over GDP in the United States. This suggests that differences in the relative price of health care—rather than technology, product quality, and factor accumulation—could explain the divergent growth patterns of HCE over GDP across these countries. In the second growth accounting exercise, we filter out prices from HCE over GDP, and confirm that there is very little variability for the product quality residual to explain the variation in HCE across countries.

Product-Consumer Substitution and Safety Regulation

We develop a theory of safety regulation where product safety and consumer skills are negatively correlated. This correlation, which we call the product-consumer substitution, is driven by demand and supply: consumers with lower skill choose safer products, while sellers offer less safe products only to consumers with high skills. We derive two predictions that are inconsistent with the standard theory: (1) an improvement in safety (regular effect of regulation) can occur in tandem with the standard offsetting behavior, and (2) a deterioration of safety (Peltzman effect) can occur without the standard offsetting behavior. As policy implication, we propose to use simultaneously product and consumer regulation. We validate our theory using a dataset with more than 2 million observations obtained from iRacing, an online racing simulation. Our dataset contains objective measures of product safety and consumer skills which makes it unique among the datasets used in the safety regulation literature.

Betting Against Alpha

I sort stocks on realized alphas and find that they are negatively related with future returns, future alphas, and Sharpe Ratios. These patterns emerged with the development of the CAPM in the 1960s and became more salient as the related literature expanded, especially after 1992 with the increasing popularity of the CAPM anomalies in academia and of factor investing in the private sector. I provide intuition for this counter-intuitive finding by linking it to recent theoretical and empirical work and explore some trading strategies based on it. The results suggest that wide-spread applications of academic research can stem new anomalies.I sort stocks based on realized alphas estimated from the CAPM, Carhart (1997), and Fama-French Five Factor (FF5, 2015) models and find that realized alphas are negatively related with future stock returns, future alpha, and Sharpe Ratios. Thus, I construct a Betting Against Alpha (BAA) factor that buys a portfolio of low-alpha stocks and sells a portfolio of high-alpha stocks. Using rank estimation methods, I show that the BAA factor spans a dimension of stock returns different than Frazzini and Pedersen’s (2014) Betting Against Beta (BAB) factor. Additionally, the BAA factor captures information about the cross-section of stock returns missed by the CAPM, Carhart, and FF5 models. The performance of the BAA factor further improves if the low alpha portfolio is calculated from low beta stocks and the high alpha portfolio from high beta stocks. I call this factor Betting Against Alpha and Beta (BAAB). I discuss several reasons that support the existence of this counter-intuitive low-alpha anomaly.

Asset Pricing and Excess Returns Over the Market Return

This paper provides a new explanation for the failure of the empirical market factor to explain the cross-sectional dispersion of expected stock returns. Specifically, we find the conditions under which all market betas become unitary when the CAPM is augmented with the common factors in the space of excess returns. Based on this finding we propose a simple method that can identify the relevant statistical factors for asset pricing and develop a benchmark model with five statistical factors. Using the model, we examine if the five-factor model of Fama and French (2015) captures all of the factors that are relevant for asset pricing.