2. For the purposes of paragraph 1, point (a), competent authorities:
(a)
shall verify how the institution choses the estimator it uses and the analysis made to support such choice;
(b)
verify whether the expected shortfall estimator corresponds either to the integral of the estimator for the Value-at-Risk numbers referred to in Article 325bf of Regulation (EU) No 575/2013 understood as a function of the tail probability from zero to one minus the relevant confidence level, and dividing by one minus the relevant confidence level, or to a more conservative choice;
(c)
may compare the Value-at-Risk and expected-shortfall estimators used by the institution against the estimators included in Table 1.
Table 1
Value-at-risk estimators
Expected shortfall estimators
;
Where:
denotes the i-th lowest value in the sample used for the estimation, i.e. is the most severe loss in a profit-and-loss sample and typically a large negative number;
N denotes the number of values in the sample
used for the estimation;
α denotes the tail probability, i.e. one minus the confidence level;
denotes the integer part of an argument;
and for the expected shortfall assuming , i.e. computing for a left tail of a sample;
and for the expected shortfall assuming that , i.e. more than one loss is in the α-tail of a profit-and-loss sample;
.
(d)
where the calculation of the expected shortfall risk measures is based on Monte Carlo simulations, verify whether the number of simulations ensures convergence towards stable results.
For the purposes of point (d), competent authorities shall review the tests performed by the institution to set the number of simulations, and the statistical tests ensuring that the randomness properties of the sequences used to generate the simulation are appropriate. Where the competent authority deems those tests insufficient, it may use the assessment method referred to in paragraph 4.
