Common risk factors in international stock markets

Abstract

A major obstacle for research in international asset pricing and corporate finance has been a lack of reliable and publicly available data on international common risk factors and portfolios. To address this gap, we provide a step-by-step description of how appropriately screened data from Thomson Reuters Datastream and Thomson Reuters Worldscope can be used to construct high-quality systematic risk factors. We provide common risk factors for 23 countries across the globe. To demonstrate the use of this dataset, we present evidence of an “extreme” size premium in a large number of countries. These premia, however, are often not realizable or at least significantly eroded due to transaction costs.

A Appendix

1.1 A.1 Transaction cost estimation

This section explains the trading cost measure employed in Sect. 3.2 in greater detail.

Lesmond et al. (1999) propose a trading costs measure which is based on daily data. They assume that informed trading occurs on nonzero trading days and zero trading days indicate the absence of informed trading (Goyenko et al. 2009).¹⁷

Lesmond et al. (1999) propose a limited dependent variable (LDV) model as follows. First, they model the true (unobserved) returns for each firm j, $R_{jt}^{*}$ as a linear function of the market returns $R_{mt}$:

$$\begin{aligned} R_{jt}^{*} = \beta _j R_{mt} + \varepsilon _{jt}, \end{aligned}$$

(A.1)

where $\beta _j$ denotes the market beta for each firm, and $\varepsilon _{jt}$ an error term with zero expectation and a constant variance $\sigma ^2_j$.

Second, they distinguish three cases to relate the measured returns $R_{jt}$ to the true returns:

$$\begin{aligned} \begin{array}{ll} R_{jt} = R_{jt}^{*} - \alpha _{1j} &{}\quad \text { if } \,\, R_{jt}^{*}< \alpha _{1j} \\ R_{jt} = 0 &{}\quad \text { if } \,\, \alpha _{1j}< R_{jt}^{*} < \alpha _{2j} \\ R_{jt} = R_{jt}^{*} - \alpha _{2j} &{}\quad \text { if } \,\, R_{jt}^{*} > \alpha _{2j}. \end{array} \end{aligned}$$

(A.2)

Therefore, a nonzero return is only observed if either the true return is smaller than the threshold for negative information $\alpha _{1j}$ or bigger than the threshold for positive information $\alpha _{2j}$. To obtain estimates for this model, one has to maximize the following log-likelihood function:¹⁸

$$\begin{aligned} \begin{aligned} \ln {{\mathcal {L}}}&= -\sum _{R_{jt}<0} \ln (2\pi \sigma ^2_j)/2 -\sum _{R_{jt}<0} \frac{1}{2\sigma ^2_j}(R_{jt}+\alpha _{1j}-\beta _j \cdot R_{mt})^2 \\&\quad -\,\sum _{R_{jt}>0} \ln (2\pi \sigma ^2_j)/2 -\sum _{R_{jt}>0} \frac{1}{2\sigma ^2_j}(R_{jt}+\alpha _{2j}-\beta _j \cdot R_{mt})^2 \\&\quad +\,\sum _{R_{jt}=0} \ln \left( \Phi \left( \frac{\alpha _{2j}-\beta _j \cdot R_{mt}}{\sigma _j}\right) - \Phi \left( \frac{\alpha _{1j}-\beta _j \cdot R_{mt}}{\sigma _j}\right) \right) . \end{aligned} \end{aligned}$$

(A.3)

Here, $\Phi$ denotes the cumulative distribution function of the normal distribution. The parameters are estimated by maximizing the log-likelihood function in Eq. (A.3). We impose the following condition: $\alpha _{1j}<0$; $\alpha _{2j}>0$ and $\sigma _j>0$. Note that the three different regions used in this expression are different than in Lesmond et al. (1999). Whereas in the original paper this regions are selected based on the market returns, we select these three regions based on the firm returns, as suggested by Goyenko et al. (2009).

To estimate trading costs, we use half of the estimated round-trip trading costs for each trade: $\frac{\alpha _{2j}-\alpha _{1j}}{2}$.

We estimate the LDV model for all stocks available on the daily TRD files. We use at least four years of daily observations to obtain a trading cost measure for each year. In addition to the daily observations of the year for which the trading costs are estimated, we use all daily observations of the 2 years before and after that year. Like suggested by Lesmond et al. (1999), we use the EW market return as a market return proxy. We assume that trading costs incur whenever a stock enters or leaves a portfolio.