Skip to content
All library documents

GARCH, Extreme Value Theory, and Copulas for VaR Estimation

Article Quant Q&A · Author: Kondo
Read the original

Summary

The document outlines a proposed workflow for estimating portfolio value-at-risk with ARMA-GARCH models, extreme value theory, and a copula. It fits conditional volatility to return series, models the tails of standardized residuals with generalized Pareto distributions and the interior with a kernel estimate, transforms the margins to uniforms, and fits a copula to represent dependence. The proposed time-varying dependence estimate is forecast, simulated, and transformed back through the marginal models to generate portfolio returns and VaR estimates.

The author reports that the procedure produced far more exceedances than expected across several confidence levels, but the document contains no resolution or diagnostic results. It is therefore a description of an attempted method and an open calibration problem, not evidence that the workflow is reliable. Choices such as tail thresholds, dependence specification, forecasting, return sign conventions, and validation design may all need scrutiny.

Key ideas

  • The proposed workflow combines conditional volatility models, tail distributions, and a copula for joint risk estimation.
  • Generalized Pareto distributions model residual tails while a kernel estimate represents the interior.
  • A time-varying copula is estimated in a rolling window and forecast before simulating joint returns.
  • The author reports excessive VaR exceedances but provides no confirmed cause or corrected procedure.
  • Model choices and backtesting design need careful review when observed violations exceed expectations.

Tags

Risk managementVolatilityStatisticsBacktesting
Full text
# 'GARCH - extreme value theory - copula' approach to estimate risk measures in R


# 'GARCH - extreme value theory - copula' approach to estimate risk measures in R












I'm reading about this approach of using GARCH-EVT-copula methodology to separate univariate and joint estimation and then estimate for example VaR and ES. I wanted to try something similar, but my attempt failed miserably. Anyone of you sees a mistake in the methodology outlined below? Any hint is appreciated!

I took a few series, computed log returns, and multiplied them by -1 (so that I have losses on the right and returns on the left). Then I used this approach:

> 1) ARMA-GARCH fitting according to AIC (using the `rugarch` package); 2) obtain standardized residuals $z_t = \frac{\epsilon_t}{\sigma_t}$; 3) semi-parametric modeling of $z_t$ using a generalized Pareto distribution for upper and lower tails (thresholds at 10% and 90%) and Gaussian kernel for the interior part (using the `spd` package); 4) transformation in uniform margins using the `pspd` command from the previous package; 5) copula fitting using an Archimedean copula model via maximum likelihood (since ML could not be evaluated in all points, I transformed the uniform margins in pseudo-observations using the `pobs` command before fitting); 6) since I want a time-varying copula approach, I use a moving window of 1000 observations to estimate a time-varying Archimedean copula parameter, and fit an ARIMA in order to do forecasting; 7) given predicted copula parameters (thanks to previous ARIMA specification), N copula realizations are simulated from day $t+1$ to $T$; 8) using the `qspd` command, I transform copula realizations back in standardized residuals $z_{t}$; 9) I insert estimated $z_t$ back in ARMA-GARCH specification, compute log returns, transform in simple returns, and calculate return of equally weighted portfolio; 10) now I have N returns for each day, and can compute required VaR as the $\alpha$-quantile (90, 95, 97.5, 99, ...).

Now the staggering surprise: this methodology failed miserably for me for relatively moderate quantiles (90, 95 and 97.5) and more extreme quantiles (99, 99.5, 99.9). I obtain a number of violations which is WAY more than expected. Any reason why? Is there a mistake in the steps mentioned?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.