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FRM Part II · FRM Exam Part II · Beyond Exceedance-Based Backtesting of Value-at-Risk Models

A risk manager has PIT values u_t from a VaR model and wants to test the full distribution rather than just tail exceedances. She transforms each value by z_t = Φ^-1(u_t), where Φ^-1 is the inverse standard normal CDF. Under a correctly specified model, what should be true of z_t, and what is the main practical benefit of this transformation?

Under a correct model the inverse-normal transformed PIT values are i.i.d. standard normal. This allows standard parametric tests, such as likelihood ratio tests of zero mean, unit variance and no autocorrelation, to evaluate the whole distribution. Independence must still be tested.

  1. Az_t should be i.i.d. standard normal, which allows standard normal-based tests such as likelihood ratio tests for mean, variance and autocorrelationCorrect
  2. Bz_t should be uniform on (0,1), which allows a simple count of exceedances
  3. Cz_t should be i.i.d. standard normal, which removes the need to test independence
  4. Dz_t should have mean 0.5 and variance 1/12, which allows a binomial test

Explanation

Applying the inverse normal CDF to uniform PIT values yields i.i.d. N(0,1) variables under a correct model. This lets the analyst use well-developed normal-based tests on mean, variance and serial correlation. Independence still has to be tested. Mean 0.5 and variance 1/12 describe the uniform before transformation.

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