Why CAPM Cross-Sectional Regressions Have Low R-Squared

# CAPM yields very poor fit (low R-squared). Is that normal?


# CAPM yields very poor fit (low R-squared). Is that normal?












I am playing around with the CAPM for a small European stock market (about 100 stocks). First, I use five years of monthly data (January 2017 to December 2021) to estimate betas for each firm using time series regressions $$ (r_{i,t}-r_{f,t})=\alpha_i+\beta_i (r_{m,t}-r_{f,t})+\varepsilon_{i,t} $$ or more briefly, $$ r^*_{i,t}=\alpha_i+\beta_i r^*_{m,t}+\varepsilon_{i,t} $$ where $r^*_i:=(r_{i,t}-r_{f,t})$ is firm's $i$ excess return and $r^*_m:=(r_{m,t}-r_{f,t})$ is the market's excess return. Second, I take the vector of estimated betas and use it as a regressor in a cross-sectional regression for January 2022, $$ r^*_{i,\tau}=\lambda_\tau \hat\beta_i+\varepsilon_{i,\tau} $$ where $\tau$ denotes January 2022. I get an extremely poor fit: $0<R^2<0.1$. Being unsure if that is a "normal" result, I repeat the cross-sectional regression for February and other months in 2022 and keep getting equally poor fit; in all cases, $0<R^2<0.1$.

Question 1: Is that "normal"? E.g. if I were to run the model on data from a major stock market (perhaps NYSE), would I get a similarly poor fit? References would be welcome.

My original aim was to illustrate how the CAPM works (first for myself and then hopefully for a class I am teaching). I was hoping to observe a cloud of data approximating a hypothetical security market line (SML), but the data does not seem to cooperate.

Question 2: Are there any tricks (in an honest sense) to make the SML more "visible"? I have tried using multi-month compound returns (e.g. the entire year 2022) in the cross-sectional regression to see if a linear pattern emerges, but this did not work.

## Answer by Richard Hardy (score 6, accepted)

https://quant.stackexchange.com/a/74455

#### Question 1

There may be at least two reasons for this (aside from possible programming errors or poor data):

- The model is actually a very poor approximation of reality, as Matthew Gunn indicates in his comments. (He mentions post 1980 period for the U.S.)

- Noise completely clouds the signal, and using $\hat\beta$s in place of $\beta$s in the cross-sectional regression makes matters worse. This is explained in Cochrane "Asset Pricing" (2005) Section 20.2 "The Cross Section: CAPM and Multifactor Models":

> The first tests of the CAPM such as Lintner (1965b) were not a great success. If you plot or regress the average returns versus betas of individual stocks, you find a lot of dispersion, and the slope of the line is much too flat — it does not go through any plausible risk-free rate. Miller and Scholes (1972) diagnosed the problem. Betas are measured with error, and measurement error in right-hand variables biases down regression coefficients. Fama and MacBeth (1973) and Black, Jensen, and Scholes (1972) addressed the problem by grouping stocks into portfolios. Portfolio betas are better measured because the portfolio has lower residual variance. Also, individual stock betas vary over time as the size, leverage, and risks of the business change. Portfolio betas may be more stable over time, and hence easier to measure accurately.

> There is a second reason for portfolios. Individual stock returns are so volatile that you cannot reject the hypothesis that all average returns are the same. $\frac{\sigma}{\sqrt{T}}$ is big when $\sigma = 40–80%$. By grouping stocks into portfolios based on some characteristic (other than firm name) related to average returns, you reduce the portfolio variance and thus make it possible to see average return differences. Finally, I think much of the attachment to portfolios comes from a desire to more closely mimic what actual investors would do rather than simply form a statistical test. <...>

> The CAPM proved stunningly successful in empirical work. <...>

#### Question 2

Using sensibly constructed portfolios instead of individual assets will make the estimates of $\beta$ more precise, reducing the problem mentioned under point 2 above. Within each portfolio, the asset betas should be similar. Across portfolios, they should be dissimilar. Thus you could rank the assets by their (estimated) betas and group the nearby assets into portfolios, such that large-beta assets go together in a portfolio and small-beta assets also go together in a different portfolio. This is one of the main insights behind the Fama-MacBeth two-stage procedure, for example.

## Answer by Marcus Opp (score 3)

https://quant.stackexchange.com/a/80704

THE CAPM makes no PREDICTION whatsoever about the R2. The CAPM could hold perfectly and the R2 is zero. This happens if all volatility is idiosyncratic. It is a common misconception that people think they ought to get a high R2.