I assume that, by "beta" / $\beta$, you mean:
a statistic that measures the expected increase or decrease of an individual stock price in proportion to movements of the stock market as a whole. $\mathbf\beta$ can be used to indicate the contribution of an individual asset to the market risk of a portfolio when it is added in small quantity. It refers to an asset's non-diversifiable risk, systematic risk, or market risk. $\mathbf\beta$ is not a measure of idiosyncratic risk.
(from wikipedia)
Beta ($\mathrm\beta$) is often used in asset pricing, portfolio management, and risk assessment (for references consult any mainstream Finance textbook, such as Reilly et al, 2025 or CFA Institute, 2021). It's general formula is:
$$
\beta_i = \frac{\mathrm{Cov}(R_i, R_m)}{\mathrm{Var}(R_m)} = \rho_{i,m} \cdot \frac{\sigma_i}{\sigma_m}
$$
where:
- $R_i$: Return on asset $i$
- $R_m$: Return on the market (eg., QQQ)
- $\rho_{i,m}$: Pearson's correlation between $R_i$ and $R_m$
- $\sigma_i$: Standard deviation of $R_i$
- $\sigma_m$: Standard deviation of $R_m$
Now let's consider both approaches in the OP.
Method 1: Asset-Specific Calculation
This approach calculates individual rolling correlationas and variance/SD for each asset and market pair. Each $\mathbf\beta$ is computed separately. It is straightforward and easy to debug, but computationally inefficient for large datasets due to repetitive calculations.
Method 2: Vectorised Calculation
The second method uses pandas vectorised operations to simultaneously compute rolling correlations and standard deviations across all assets. $\mathbf\beta$ for each asset ($\mathbf\beta_i$ ) and is derived in a single pass which significantly improves computational performance by reducing redundancy and exploiting pandas' internal numerical optimisations (McKinney, 2017).
Performance Enhancement Using Polars
For further performance optimisation, particularly beneficial with large-scale datasets or intensive rolling computations, the use of Polars, an efficient DataFrame library implemented in Rust, with Python bindings, is especially recommended, as an alternative to Pandas. Polars utilises parallel computation, lazy evaluation, and superior memory efficiency, usually outperforming pandas in these tasks involving large rolling window calculations or frequent numerical operations. Transitioning from pandas to Polars can significantly improve $\mathbf\beta$ calculation workflows, reducing runtimes and improving scalability. Polars is not a drop-in, plug-and-play, replacements for pandas, and therefore a learning curve for the polars API, but here is an example using polars:
import polars as pl
# Example data preparation
df = pl.DataFrame({
"date": pl.date_range(start="2023-01-01", end="2024-01-01", interval="1d"),
"AAPL": pl.Series([0.01, -0.02, 0.015, ...]), # daily returns
"QQQ": pl.Series([0.008, -0.018, 0.012, ...]) # daily returns
})
# Rolling window size (eg., 252 trading days ~ 1 year)
window_size = 252
# Calculate rolling covariance and variance
df = df.with_columns(
rolling_cov=pl.cov("AAPL", "QQQ").rolling_mean(window_size),
rolling_var_market=pl.col("QQQ").rolling_var(window_size)
)
# Calculate Beta as rolling covariance divided by rolling market variance
df = df.with_columns(
beta=(pl.col("rolling_cov") / pl.col("rolling_var_market"))
)
# Display result
print(df.select(["date", "beta"]))
Comparative Analysis and Recommendation
Both methods provide mathematically equivalent results, provided return definitions (log vs. simple) remain consistent. Method 2 (vectorised) is thus recommended due to its efficiency, scalability, and suitability for analytical workflows and production environments.
Also, Method 2 maintains the explicit correlation matrices, making it easier to perform subsequent analyses, such as filtering based on $\mathbf\beta$ thresholds or aggregating $\mathbf\beta$ values by asset groups.
A useful consistency check is verifying that the $\mathbf\beta$ of the benchmark against itself equals exactly 1:
$$
\beta_{\text{QQQ}} = \frac{\mathrm{Cov}(R_{\text{QQQ}}, R_{\text{QQQ}})}{\mathrm{Var}(R_{\text{QQQ}})} = 1
$$
(Note: slight numerical deviations due to floating-point precision may occur.)
References
CFA Institute. (2021). Alternative Investments. Wiley.
Reilly, F. K., Brown, K. C., & Leeds, S. J. (2025). Investment Analysis and Portfolio Management (12th ed.). Cengage Learning.
McKinney, W. (2017). Python for data analysis: Data wrangling with pandas, NumPy, and IPython (2nd ed.). O'Reilly Media.
Wikipedia contributors. (Accessed June 2025.). Beta (finance). Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Beta_(finance)
