Value at Risk backtesting
Period-by-period comparison of actual losses with the stated VaR, to judge whether the model breaches its threshold as often as its confidence level promises, and no more often.
VaR backtesting compares each day's actual loss with the figure the model stated the day before. Each breach is called an exception. At 99% over 250 business days, 2.5 are expected; the question is how many it takes before the model is wrong. Paul Kupiec answered in 1995 in The Journal of Derivatives (“Techniques for verifying the accuracy of risk measurement models”): a likelihood ratio test on the share of exceptions, the POF (proportion of failures) test, which follows a chi-square with one degree of freedom, critical value 3.84 at 5%. It only counts exceptions: seven bunched in one crash week count the same as seven spread across the year. Peter Christoffersen closed that gap in 1998 (“Evaluating interval forecasts,” International Economic Review, vol. 39, no. 4) with an independence test based on a Markov chain of exceptions, then a conditional coverage test that joins the two, chi-square with two degrees. In January 1996, the Basel Committee's backtesting framework places the model in three zones by the number of exceptions over 250 days. Green from 0 to 4, yellow from 5 to 9, red from 10. In the yellow zone the capital multiplier, set at 3, gets a plus factor of 0.40 to 0.85; in the red zone it goes to 4. In insurance the horizon gets in the way. An internal model's SCR is a 99.5% VaR over one year: one exception expected every two centuries, with a single observation a year. Article 124 of the Solvency II Directive still requires the model's results to be tested against experience. So insurers backtest submodules on short steps, market risk daily or monthly, and relies on the profit and loss attribution of Article 123. On the shape of the tail beyond the threshold, the test says nothing.
Textbook case: an investment portfolio tracked by a one-day 99% VaR records 7 exceptions over 250 days. The model promised 250 × 0.01 = 2.5. Kupiec test: under a 1% rate, the log-likelihood is 243 × ln(0.99) + 7 × ln(0.01) = −2.442 − 32.236 = −34.678. At the observed rate, 7 / 250 = 2.8%, it is 243 × ln(0.972) + 7 × ln(0.028) = −6.901 − 25.029 = −31.930. The statistic is −2 × (−34.678 + 31.930) = 5.50, above 3.84: the model is rejected at the 5% level. The binomial distribution agrees: the probability of at least 7 exceptions with a true 1% rate is 1 − 0.9863 = 1.4%. On the Basel grid, 7 exceptions put the model in the yellow zone, plus factor 0.65, and the multiplier rises from 3 to 3.65.
VaR backtesting, value at risk backtesting, Kupiec test, Kupiec POF test, Christoffersen test, conditional coverage test, Basel traffic light test, traffic light approach, VaR exceptions, how to backtest VaR
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