Risk Management.
Measuring — and respecting — the tails.
3 articles · each one with a runnable notebook
Value at Risk Three Ways: Historical, Parametric and Monte Carlo on the DAX
Daily loss limits and regulatory capital hang off one number, and it has three recipes that disagree. We compute 99% and 95% VaR and CVaR on the DAX 2010–2024 by historical simulation, a normal and a fitted Student-t (df 3.22), and 200k seeded Monte-Carlo draws — then let a rolling 250-day Kupiec backtest decide which to believe, and show the same historical quantile falling out of Polars expressions and DuckDB SQL to 1e-15 agreement.
CVaR / Expected Shortfall: Sizing the Losses That Live Beyond VaR
VaR marks where the tail begins; CVaR averages what lives inside it. We estimate both at 97.5% and 99% for HYG, VWO and a 50/50 mix across 2007-2024, show VaR breaking subadditivity in crisis-year samples while CVaR never does, count the GFC blowing through HYG's VaR line at 16x its expected frequency, and spend the same 4% tail budget through rm="MV" vs rm="CVaR" in Riskfolio-Lib - with radically different books as the result.
Copulas & Tail Dependence: Why Markets Crash Together More Often Than Correlation Says
Correlation is one number; dependence is a whole function. On 25 years of weekly S&P 500, FTSE 100 and Nikkei 225 returns we rank-transform to pseudo-observations, fit Gaussian and Student-t copulas by MLE, and show that empirical tail dependence rises exactly where the Gaussian copula sends it to zero — modelling the joint crashes that multi-asset stress scenarios exist for.