Exact Finite-Sample Value-at-Risk Back-Testing.

ExactVaRTest

Fast exact finite-sample back-testing for Value-at-Risk (VaR) models in R.

ExactVaRTest provides fast dynamic-programming algorithms in C++/Rcpp (with pure R fallbacks) for the exact finite-sample distributions and p-values of Christoffersen (1998) independence (IND) and conditional-coverage (CC) VaR backtests. For completeness, it also provides the exact unconditional-coverage (UC) test following Kupiec (1995) via a closed-form binomial enumeration.

In particular, it corrects the severe size distortions from which the usual asymptotic $\chi^2$ approximation suffers in small samples and under extreme coverage rates.

• backtest_lr() returns the LR statistic, its exact p‑value, and a reject / fail‑to‑reject decision for one chosen test type (UC, IND, or CC).

• backtest_all() runs the UC, IND, and CC tests jointly and returns summaries for all three statistics.

Installation

You can install the development version of ExactVaRTest from GitHub with:

# install.packages("pak")
pak::pak("YujianCHEN219/ExactVaRTest")

Example

library(ExactVaRTest)

set.seed(42)
x <- rbinom(300, 1, 0.03)          # synthetic 0/1 exception series

bt <- backtest_lr(x, alpha = 0.05, type = "cc")  # exact LR_cc back-test
print(bt)
#> Exact finite-sample back-test
#> --------------------------------
#> Test           : Conditional coverage (LR_cc)
#> Sample size    : 300
#> Model alpha    : 0.0500
#> Signif. level  : 0.0500
#> LR statistic   : 5.8882
#> Exact p-value  : 0.0442
#> Decision       : REJECT null at 5.00% level

Main features

• Exact finite‑sample distributions and p‑values for LR_ind and LR_cc at any sample size n; for n ≤ 2000 the computation finishes in milliseconds to a few seconds.

• C++ implementation via Rcpp, with automatic fallback to a pure‑R reference engine.

• Minimal dependencies (Rcpp, stats); works on macOS, Linux, and Windows.

Extensions

Freely extends to CoVaR backtesting: pass the institution’s hit sequence on system‑VaR‑breach days into backtest_lr() for exact UC/IND p‑values; for short windows or extreme tails, one option is to treat the systemic‑breach count as random and apply the mixture‑tail test to maintain correct size. (see vignettes and [Francq & Zakoïan 2025]).

Suggested readings

1. Christoffersen, P. F. (1998). Evaluating interval forecasts. International economic review, 841-862.

2. Mehta, C. R., Patel, N. R., & Gray, R. (1985). Computing an exact confidence interval for the common odds ratio in several 2× 2 contingency tables. Journal of the American Statistical Association, 80(392), 969-973.

3. Francq, C., & Zakoïan, J. M. (2025). Inference on dynamic systemic risk measures. Journal of Econometrics, 247, 105936.

Acknowledgements

I greatly appreciate Christian Francq, Christophe Hurlin, and Jean-Michel Zakoïan’s guidance and support.

In particular, Christian Francq generously shared the initial idea; without his help, this package would not exist.

License

This package is free and open source, licensed under GPL.