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.