Beta Is Not Linear: Distributional Effects in Equity Markets
Introduction
The Capital Asset Pricing Model (CAPM) represents an asset's market sensitivity using a single scalar \(\beta\), estimated by ordinary least squares on the full return distribution. Its empirical implementation takes the form:\[ \widehat{E}(R_i) = R_f + \hat{\beta}_i\!\left(\widehat{E}(R_M) - R_f\right), \]where \(\widehat{E}(R_i)\), \(\widehat{E}(R_M)\), and \(R_f\) are the estimated expected return on security \(i\), the estimated expected return on the market portfolio, and the risk-free rate, respectively. Furthermore,\[ \hat{\beta}_i = \frac{\widehat{\mathrm{Cov}}(R_i,\, R_M)}{\widehat{\mathrm{Var}}(R_M)} \qquad \text{(OLS estimated beta)}, \]where \(\widehat{\mathrm{Cov}}(R_i, R_M)\) is the sample covariance between the return on security \(i\) and the market portfolio return, and \(\widehat{\mathrm{Var}}(R_M)\) is the sample variance of the market portfolio return.This is empirically convenient but potentially misleading: if the sensitivity of a stock to the market varies across different market states — rising in crashes, falling in calm periods — then a single OLS beta conflates fundamentally different regimes.
Quantile Regression
Quantile regression, introduced by Koenker and Bassett (1978), estimates the conditional slope \(\beta_\tau\) at each quantile \(\tau\) of the return distribution rather than at the mean alone. Applied to Microsoft (MSFT) against the S&P 500 over 2022–2024 (\(n=751\) daily observations), the resulting quantile beta curve suggests a mild but consistent U-shape: MSFT's estimated market sensitivity is lower in the interior of the distribution and rises toward both tails.
Figure 1. Quantile beta curve for MSFT relative to the S&P 500.
Note: The quantile beta curve is estimated across nineteen conditional return percentiles (\(\tau = 5, 10, \ldots, 95\)), with pointwise 95% bootstrap confidence band (shaded). The dashed red line marks the OLS CAPM beta estimated over the selected window. Controls above the figure adjust the assumed annualised risk-free rate and the trailing estimation window; all estimates update dynamically. The risk-free rate is a constant daily rate \(r_f = r_f^{\text{ann}}/(100\times252)\); default zero. Quantile regression via IRLS (Iteratively Reweighted Least Squares); full-sample CI from pairs bootstrap \(B=1{,}000\); sub-sample CI from \(B=150\). Rolling window is trailing, ending 31 Dec 2024.
Source: Author's own elaboration using daily adjusted closing prices for MSFT and the S&P 500 Index (^GSPC). Sample: 3 Jan 2022 – 31 Dec 2024; \(n = 751\) trading days.
Conclusion
A single number cannot summarise a shape. The quantile beta curve traced in Figure 1 is not flat — it bends, and the bend is precisely where it matters most: in the tails, where markets move sharply and portfolios are most exposed. CAPM assigns MSFT a constant \(\hat{\beta}^{\,\text{OLS}}\) regardless of market conditions; the estimated quantile beta curve suggests that the stock's market sensitivity may increase in the tails of the return distribution relative to that implied by the single OLS beta. An investor who hedges using a single OLS beta is, by construction, hedging average market sensitivity rather than state-dependent market sensitivity.
The analysis complements the discussion in Ma (2025) by dynamically illustrating distributional beta effects using a recent empirical example based on MSFT and the S&P 500.
This topic is also discussed on LinkedIn.
References
Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46(1), 33–50.
Ma, L. (2025). Nonlinear Investing: A Quantamental Approach. Springer Nature Switzerland AG.
