Expected Return Model
There are three topics in the literature on expected returns (return predictability).
- Cross-section (CS) vs. Time series (TS).
- Fama & French find that, for factor models, CS factors outperform TS factors (Fama & Kenneth, 2020).
- Factor vs. Characteristic.
- CBAPMs focus on the unique characteristics of individual assets, while factor-based models aggregate the effects of many underlying characteristics into a smaller set of common factors (generated by ChatGPT).
- Characteristic models are found to outperform factor models.
- CAR vs. BHAR. When the evaluation horizon is long (e.g., in years), one should use BHAR.
1 Cross-section (CS) vs. Time series (TS)
- For an introductory explanation of how CS and TS models are used and tested in asset pricing, watch Prof. John Cochrane’s online courses.
- Challenges in testing an asset pricing model:
- How the LHS assets are formed heavily impacts results. To mitigate multiplicity, one must limit the number of LHS assets, e.g., a 5x5 sort on size and BM.
- The biggest problem for TS is the multiple comparisons, or multiplicity. As a result, the GRS test almost always rejects.
- One method to address the above issues is to use a panel regression framework (Campbell R. Harvey, Yan Liu, 2021).
2 Factor (FBAP) vs. Characteristic (CBAP)
For an introduction to how FBAP (factor-based asset pricing) differs from CBAP (characteristic-based asset pricing), see (Chordia et al., 2017) (recommended!). This paper:
- Shows that betas and characteristics are only weakly correlated.
- Compares FB and CB to see which better explains cross-sectional returns.
- Concludes that CBAP explains average returns much more than FBAP.
One well-known characteristic-based method is the DGTW method (Daniel & Titman, 1997). It’s matching-based.
A more recent paper is the “C-14” (fourteen) characteristics (Bessembinder et al., 2019). It’s regression-based.
3 3 CAR vs. BHAR
Definitions of CAR and BHAR: $$ \begin{align} AR_{i,t} &= R_{i,t}-R_{base,t} \\ CAR(t_{1},t_{2}) &= \sum_{t=t_{1}}^{t_{2}} AR_{i,t} \\ BHAR(t_{1},t_{2}) &= \prod_{t=t_{1}}^{t_{2}} (1+R_{i,t}) - \prod_{t=t_{1}}^{t_{2}} (1+R_{base,t}) \end{align} $$
In the short run, the difference between CAR and BHAR is minor.
- “Imperfect assessment of normal benchmark returns is of minor importance over short horizons such as a few days. In contrast, as Kothari and Warner (2007) emphasize, the issue can be of first-order importance over the horizons considered in long-run return studies.”
BHAR is preferred over CAR in long-run studies and should be implemented by matching to a set of control firms.
- Over longer horizons, the benchmark model estimated before the event may be biased. Therefore, we need to use a post-event benchmark model, i.e., matching-based.
4 References
- Fama, E. F., & French, K. R. (2020). Comparing Cross-Section and Time-Series Factor Models. The Review of Financial Studies, 33(5), 1891–1926. https://doi.org/10.1093/rfs/hhz089
- Chordia, T., Goyal, A., & Shanken, J. A. (2017). Cross-Sectional Asset Pricing with Individual Stocks: Betas versus Characteristics (SSRN Scholarly Paper No. 2549578). https://doi.org/10.2139/ssrn.2549578
- Daniel, K., & Titman, S. (1997). Evidence on the characteristics of cross sectional variation in stock returns. The Journal of Finance, 52(1), 1–33.
- Bessembinder, H., Cooper, M. J., & Zhang, F. (2019). Characteristic-Based Benchmark Returns and Corporate Events. The Review of Financial Studies, 32(1), 75–125.