# Expected Return Model

There’re three topics in the literature about expected return (return predictability).

**Cross-section (CS) vs. Time series (TS).**- Fama & French found that, for factor models, CS factors are better than 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 be better than factor models.

- CBAPMs focus on the
**CAR vs. BHAR**. When evaluation horizon is long (e.g., in years), one need to 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 Cochran’s online courses.
- Challenges in testing a asset pricing model:
- How the LHS assets is formed heavily impacts the results. Usually, To mitigate the multiplicity, one must limit the number of the LHS assets, e.g., 5x5 sort on size and bm.
- The biggest problem for TS is the “multiple comparison” or “multiplicity.” As a result, the GRS test almost always rejects.

- One method to solve 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** of how FBAP (factor-based asset pricing) differs from CBAP (characteristic-based asset pricing), see (Chordia et al., 2017) (recommended!). This paper:

- Show that betas and characteristics are only
*weakly*correlated. - They compare FB and CB to find out which can better explain CS returns.
- They conclude that CBAP explains average returns much more than FBAP

One of the most well-known C-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 of minor importance.

- “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** study and should be used by **matching** to a set of control firms.

- In long-term, the benchmark model estimated before the event may be very biased. Therefore, we need to use
*post-event*benchmark models, 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.