Abstract
We can relate the VaR model’s prediction (tomorrow’s PnL distribution) with actual, later outcomes (the realized and experienced PnL) in a way that is more expressive than the backtesting with its focus on relatively rare VaR violations.
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Notes
- 1.
The probability of a random number falling between two quantiles q d < q u is u − d (see Sect. A.7). The PnL predictions (our interval bounds) correspond to k/1001-quantiles; the probability of a random number falling between two adjacent ones, e.g., d = 17/1001 and u = 18/1001, is u − d = 1/1001.
You will notice that this treatment of quantiles is slightly different from our previous one, where we defined the 10th most negative PnL value to correspond to the 1%-quantile instead of the 10/1001 = 0.99%-quantile. This is largely inconsequential here due to the large number of PnL values involved, and our respective definition choices are solely motivated by convenience and concise, short form notation. In any case, empirical quantiles like these are often handled slightly differently according to a given problem at hand, for example, when it comes to dealing with the very first and last intervals or with questions of quantile interpolation.
- 2.
The confidence interval delimits the outcome of each u between an upper and lower bound (it is thus two-sided); it uses the 5% and the 95% quantiles of the Beta distribution for this, spanning the 90% of outcomes in between (hence it is a 90% confidence interval). You can use Excel’s BETA.INV(0.95;k;n+1-k) and BETA.INV(0.05;k;n+1-k) functions to obtain each day’s upper and lower bounds.
- 3.
While we generally prefer Python and NumPy when doing statistics, the software suite R offers some good support for such tests. The Kolmogorov-Smirnov test is built-in (ks.test(u, "punif")), while Anderson-Darling is available as a separate library (called ADGofTest). Please note that some tests do not allow for identical u-values, which in our case can happen because of the discrete intervals. You can simply add/subtract tiny and different offsets to each u-value to disentangle them, e.g.,
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Auer, M. (2018). Distribution Tests. In: Hands-On Value-at-Risk and Expected Shortfall. Management for Professionals. Springer, Cham. https://doi.org/10.1007/978-3-319-72320-4_16
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