Market Capitalization and Equity Tail Risk
Empirical Calibration of Stress-Test Buckets for Portfolio Margin
Summary
This paper presents the first empirical calibration of market-capitalization-differentiated stress-test buckets for US equity portfolio margin. Using 5.37 million return observations across 2,509 validated US equities over 21 years (2005–2026), we document a strictly monotonic relationship between market capitalization and 3-day tail risk at the 99th and 99.9th percentiles.
The relationship is robust to alternative bucket specifications (4 through 12 buckets), sub-period analysis (including COVID and the Global Financial Crisis), filter choices, and lookback windows.
Main Results (Table 1)
8 equal-observation quantile buckets (~671K observations each):
| Bucket | Cap Range | VaR 99% | CVaR 99% | CVaR 99.9% | Recommended Stress |
|---|---|---|---|---|---|
| 1 | < $280M | -17.86% | -25.67% | -47.52% | -50% |
| 2 | $280M – $630M | -16.76% | -23.87% | -43.54% | -44% |
| 3 | $630M – $1B | -15.79% | -22.26% | -40.15% | -40% |
| 4 | $1B – $2B | -14.52% | -20.42% | -36.72% | -37% |
| 5 | $2B – $4B | -13.24% | -18.66% | -33.94% | -34% |
| 6 | $4B – $8B | -12.14% | -17.30% | -32.39% | -32% |
| 7 | $8B – $23B | -11.29% | -16.17% | -29.90% | -30% |
| 8 | ≥ $23B | -9.86% | -14.28% | -27.35% | -27% |
The "Recommended Stress" column rounds the CVaR 99.9% values to whole percentages, with Bucket 1 set to -50%.
Key Findings
- Strict monotonicity — tail risk decreases at every adjacent-bucket transition from micro-cap to large-cap, with no inversions
- Uniform fat tails — CVaR/VaR ratio is stable at ~1.4–1.5× across all buckets, indicating self-similar tail shape across the capitalization spectrum
- Robust across regimes — COVID crash approximately doubled all bucket VaRs with uniform multiplicative scaling
- Robust across specifications — monotonicity preserved from 4 through 12 buckets
Implementation
The portfolio builder's Cap Stress Grid uses these empirical results as the default preset. The grid supports:
- Editable market cap boundaries between buckets
- Per-bucket long stress override
- Custom stress column with a top-level multiplier (e.g., enter 150 to apply 1.5× the base stress)
- Separate short stress column
See the Risk Methodology page for how the Cap % scenario integrates with the overall stress framework.
Full Paper
The complete paper with methodology details, robustness analysis, sub-period results, and quantile regression is available at docs/cap_stress_paper.md in the project repository.
Supporting data files:
Cap_Stress_Study_20260316_0103.xlsx— Primary results with 8-bucket quantile specificationCap_Stress_Robustness_20260316_0130.xlsx— Full robustness analysis across bucket counts and sub-periods