XLE rolling 20-day Brent beta across Brent realized-volatility quintiles
The convexity thesis fails on arrival. The question was whether XLE's rolling 20-day beta to Brent crude rises when crude's own realized volatility spikes—the idea being that energy equities re-couple to oil in violent regimes as supply-shock pass-through dominates, and drift on company-specific flows when oil is calm. Over roughly three years of daily data, that pattern does not show up. The top Brent-volatility quintile produced a mean XLE beta of 0.220, versus 0.241 in the calmest quintile—a gap with the wrong sign. The median gap flipped mildly positive, which is less a regime effect than a sign of just how noisy rolling betas get in overlapping windows.
The full analysis below lays out the quintile distributions, the tests, and why this reads as a null result rather than a missed signal.
Over the past ~3 years, is XLE's rolling 20-day beta to daily Brent crude returns significantly higher when Brent's 20-day realized volatility is in its top quintile than when it is in its bottom quintile? I expect energy equities to have convex crude beta: in high-vol oil regimes, XLE re-couples to Brent faster and with greater magnitude as supply-shock pass-through dominates, while in calm regimes it trades more on idiosyncratic company flows.
How this was measured
Daily XLE close-to-close returns were computed from minute bars and joined with daily Brent crude returns from brent_daily_df on overlapping trading dates over 2023-09-29 to 2026-08-28. For each trading day, a trailing 20-session beta of XLE on Brent was estimated as the rolling covariance divided by rolling variance of Brent returns. Brent's realized volatility was measured as the annualized rolling 20-day standard deviation of daily Brent returns. Days were then grouped into quintiles of that realized-volatility series; top and bottom quintile beta distributions were compared with a Welch t-test, Mann-Whitney U, and a Newey-West HAC regression to account for overlapping 20-day windows. The analysis window respects the available data range.
The key numbers
Reading the numbers
High-vol Brent days gave XLE an average beta of 0.220 versus 0.241 on calm days, a gap of -0.021. None of the tests reached significance (Welch p=0.35, HAC p=0.78, Mann-Whitney p=0.95), and the continuous correlation was near zero (rho=0.049).
The charts
This box plot splits days into five Brent volatility groups, from calmest (Q1) to most volatile (Q5). The mean beta climbs from 0.241 in Q1 to a peak of 0.460 in Q4, then drops to 0.220 in Q5, so the most volatile crude regime does not produce the strongest oil-equity coupling. The eye should land on that Q5 drop, which is the opposite of the expected convex pattern: the highest-volatility quintile is not materially different from the lowest-volatility one.
Each dot is one day's rolling 20-day XLE-Brent beta plotted against annualized Brent realized volatility. The 720 points span beta values from about -0.30 to 1.32 and volatility from about 15% to 109%, but the cloud is essentially flat rather than rising toward the right. The Spearman correlation of 0.049 confirms there is no meaningful tendency for beta to increase as crude volatility increases, so the scatter does not support the convexity idea.
Quintile summary
| vol_quintile | N | mean_beta | median_beta | std_beta | mean_brent_rv_ann |
|---|---|---|---|---|---|
| Q1 | 144 | 0.241 | 0.21 | 0.229 | 0.2035 |
| Q2 | 144 | 0.319 | 0.255 | 0.191 | 0.2605 |
| Q3 | 144 | 0.324 | 0.282 | 0.167 | 0.3109 |
| Q4 | 144 | 0.46 | 0.348 | 0.322 | 0.3943 |
| Q5 | 144 | 0.22 | 0.247 | 0.135 | 0.7029 |
The takeaway
No — over the past ~3 years, XLE's rolling 20-day Brent beta is not significantly higher when Brent's 20-day realized volatility is in its top quintile than when it is in its bottom quintile. The top-quintile mean beta was 0.220 versus 0.241 in the bottom quintile—a gap of about -0.021, the opposite sign of the convexity thesis. The median gap was positive (+0.038), but that flip is a sign of how noisy and skewed these rolling betas are, not a real regime effect. All formal tests agree there is no meaningful difference: the naive Welch p-value was 0.35, the overlap-corrected HAC p-value was 0.78, and the non-parametric test was essentially 1.0. Across the full volatility range, the Spearman correlation between beta and Brent realized volatility was just 0.05, and the quintile means bounced around erratically—Q4 loaded highest at 0.46, while Q5 fell back to 0.22. This is best read as a null result: with 144 days per bucket and heavily overlapping 20-day windows, the data do not support the claim that energy equities re-couple to crude with greater magnitude in high-vol regimes. If such convexity exists, it is too weak or intermittent to show up here.
The fine print
- Rolling beta and rolling Brent volatility share the same 20-day window, so observations are serially dependent; the HAC p-value (0.78) matters more than the naive one.
- Only ~144 trading days per quintile — and overlapping windows cut effective sample size much further.
- Top vs bottom buckets are defined by this specific window's RV distribution, not an external high/low-vol threshold.
- Betas are based on daily close-to-close returns, not intraday timing, so they measure daily association, not fast re-coupling.