Assessment of the capability of the internal risk-measurement model to capture non-linearities
1. When assessing an institution’s compliance with Article 325bh(1), point (b), of Regulation (EU) No 575/2013 in relation to requirements on the effectiveness and capability of the internal-risk measurement model to capture non-linearities of options and other products for an institution using a sensitivity-based approach, competent authorities shall verify whether:
(a)
the internal risk-measurement model captures at least the material first- and second-order terms of Taylor series approximations to reflect the change in the prices due to changes in relevant risk factors, including the cross-gamma risk represented by material joint-moves in risk factors;
(b)
the sensitivity-based approach leads to appropriate results, including where severe shocks are applied to the risk factors.
2. For the purposes of paragraph 1, competent authorities may apply the following steps in the following order:
(a)
identify products for which competent authorities want to test the materiality of the order terms of a Taylor series approximation, and the appropriateness of the sensitivity-based approach under severe shock;
(b)
identify a business day in the stress period where the returns observed for the risk factors in those products were particularly high, where positive, or particularly low, where negative;
(c)
require the institution to calculate the hypothetical and theoretical changes in the values of those products in accordance with Delegated Regulation (EU) 2022/2059, under the scenario identified by the returns on the business day identified in accordance with point (b);
(d)
based on the results of the calculation referred to in point (c), assess whether the sensitivity-based approach leads to appropriate results.
Assessment of the calculation of the expected shortfall risk measure
1. When assessing whether an institution’s internal model is implemented with integrity as required by Article 325bi(1) of Regulation (EU) No 575/2013 in relation to the calculation of the unconstrained expected shortfall measures and of the partial expected shortfall measures for all broad categories of risk factors at a reduced frequency as referred to in Article 325bb(4) of Regulation (EU) No 575/2013, competent authorities shall:
(a)
analyse the process that the institution uses to determine the day of the week when the measures are computed;
(b)
verify that a reduction in the calculation frequency does not lead to an underestimation of risk.
2. For the purposes of paragraph 1, point (b), competent authorities shall:
(a)
verify whether the analysis performed by the institution is adequate to demonstrate that there is no underestimation of risk;
(b)
verify whether the evolution of the daily figures for UES
t
,
,
and
calculated on all portfolio’s positions as required by Article 325bb(1) of Regulation (EU) No 575/2013, does not systematically show a lower risk profile in the day chosen by the institution.
3. For the purposes of paragraph 2, point (b), competent authorities may, where there are hints of a systematically lower risk profile, complement their assessment by:
(a)
requiring the institution to calculate daily and for a given period the unconstrained expected shortfall measures
and the partial expected shortfall measures
,
and
for each broad risk factor categories;
(b)
analysing whether those measures are systematically lower on the day chosen by the institution.
Assessment of the calculation of the partial expected shortfall measures
1. When assessing whether an institution’s internal model is implemented with integrity as required by Article 325bi(1) of Regulation (EU) No 575/2013 in relation to the calculation of the partial expected shortfall measures referred to in Article 325bc of that Regulation, competent authorities shall:
(a)
verify whether the estimator used by the institution to estimate the expected shortfall risk measures is conceptually sound and reasonably accurate;
(b)
verify whether, when calculating partial expected shortfall measures
and
as required by Article 325bc(1) of Regulation (EU) No 575/2013, the institution identifies effective liquidity horizons of the risk factors of a given position, taking into account the maturity of the position in accordance with Article 325bd(4) of that Regulation;
(c)
verify whether, as part of the internal policies referred to in Article 325bi(1), point (e), of Regulation (EU) No 575/2013, the institution has established objective criteria that are appropriate for choosing the risk factors forming the subset of modellable risk factors referred to in Article 325bc(2), point (a), of that Regulation;
(d)
verify whether risk factors that are not part of the subset of modellable risk factors chosen by the institution in accordance with Article 325bc(2), point (a), of Regulation (EU) No 575/2013 are kept constant when computing
and
;
(e)
verify whether the techniques used to calculate
and those used to calculate
are the same, except for those deviations necessary to ensure the fulfilment of requirements set out in Article 325bc(2) to (4) of Regulation (EU) No 575/2013;
(f)
verify whether, when calculating the
, the institution uses equally weighted data in the observation period;
(g)
in relation to the identification of the stress period, verify whether the 12-month rolling windows tested to determine the stress period starts at least from 1 January 2007 as referred to in Article 325bc(2), point (c), of Regulation (EU) No 575/2013 and that the internal policies of the institution specify the frequency of update of the stress period for the calculation of the partial expected shortfall measures, and the other applicable criteria triggering its update.
For the purposes of point (e), competent authorities shall obtain an overview of the differences in the techniques employed by the institution when calculating the partial expected shortfall measures calibrated on the recent 12-month period and on the stress period, and verify whether those differences do not go beyond what is needed to achieve compliance with the requirements set out in Regulation (EU) No 575/2013.
For the purposes of point (g), competent authorities shall verify, also on the basis of past updates, whether the stress period is updated at least with a quarterly frequency and that the institution has followed any possible criteria specified in the internal policies.
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.
3. For the purposes of paragraph 1, point (b), competent authorities shall:
(a)
require the institution to provide the Monte Carlo statistical error at 95 % confidence level, and verify whether the method employed to measure such statistical error is sound;
(b)
require the institution to calculate the expected shortfall risk measures with several different seeds, all other assumptions being equal;
(c)
assess whether the differences in the expected shortfall risk measures with a different seed resulting from the calculation in accordance with point (b) are compatible with the statistical error referred to in point (a).
(d)
where competent authorities deem the results referred to in point (c) incompatible, assess the root cause of such incompatibility, and assess the number of simulations needed to ensure that the statistical error is below 5 %.
4. For the purposes of paragraph 1, point (c), competent authorities shall:
(a)
obtain an overview of the risk factors chosen by the institution and verify:
(i)
whether the criteria ensure a sufficient level of coverage in the modellable risk factors’ types chosen compared to the full set of modellable risk factors to which the institution is exposed;
(ii)
whether the criteria are such that the threshold referred to in 325bc(2), point (a), of Regulation (EU) No 575/2013 is expected to be exceeded over time;
(iii)
whether the institution tests alternative subsets of modellable risk factors to ensure that its choice does not underestimate the own funds requirements;
(iv)
whether the institution, in its choice, favours the selection of risk factors for which data in the stress period exist over risk factors for which proxies are used, and where that is not the case, competent authorities shall assess the reason why the institution did not implement such a criterion, and whether a different choice would improve the quality of the unconstrained and partial expected shortfall measure.
(b)
verify whether, where the position has a maturity of less than 10 days, the effective liquidity horizon of all risk factors is set to 10 days, and that such position does not impact the calculation of
for
;
(c)
where the position has a maturity of Mat days, with
, verify whether:
(i)
all risk factors of that position with a liquidity horizon
have been assigned to an effective liquidity horizon that is the shortest liquidity horizon among the liquidity horizons provided in Table 1 of Article 325bc(1) of Regulation (EU) No 575/2013 that is greater than or equal to Mat ;
(ii)
the position does not impact the calculation of
for all j corresponding to a liquidity horizon that is greater than the shortest liquidity horizon among the liquidity horizons provided in Table 1 of Article 325bc(1) of Regulation (EU) No 575/2013 that is greater than or equal to Mat ;
(d)
verify whether, where the position has a maturity of Mat days, with
, the institution has assigned all risk factors of that position to an effective liquidity horizon corresponding to the liquidity horizon SubCatLH assigned to the risk factors;
(e)
verify whether, when computing
, the institution keeps constant those risk factors with an effective liquidity horizon that is lower than the liquidity horizon corresponding to the index j .
For the purposes of point (a)(ii), competent authority shall assess by which margin the institution exceeded the threshold in the previous quarters.
For the purposes of point (a)(iii), competent authorities may, where they deem insufficient the testing of alternative subsets carried out by the institution, require the institution to test alternative subsets and assess whether alternative choices lead to material differences in terms of own funds requirements.
Assessment of distributional and statistical assumption
1. When assessing whether institution’s internal model is implemented with integrity as required by Article 325bi(1) of Regulation (EU) No 575/2013 in relation to the requirement for the expected shortfall risk measures referred to in Article 325bb of Regulation (EU) No 575/2013 to reflect historically observed data in accordance with Article 325bc(2), point (c), and Article 325bc(4), point (c), of that Regulation, competent authorities shall verify whether:
(a)
the distributional and any other relevant statistical assumptions used in the model, including volatility and correlations, are well justified, including with regard to the tail of the distributions relevant for the expected shortfall calculation;.
(b)
the empirical correlations that the institution uses when applying scenario of future shocks to reflect the joint movement of risk factors in the calculation of the expected shortfall risk measures referred to in Article 325bb of Regulation No 575/2013 are based on historically observed data in accordance with Article 325bc(2), point (c), and Article 325bc(4), point (c), of that Regulation.
For the purposes of point (b) of the first subparagraph, competent authorities may, where appropriate, require the institution to:
(a)
provide a sample of time series;
(b)
calculate the empirical correlations among those time series;
(c)
verify whether the correlations referred to in point (b) do not materially differ from the correlations used by the institution in its internal risk-measurement model.
2. For the purposes paragraph 1, competent authorities shall compare, on the basis of a sample of time series:
(a)
the volatility and other distributional properties of the scenario of future shocks applied to a given risk factor in the calculation of the partial expected shortfall measures;
(b)
the volatility and other distributional properties of the returns observed for the given risk factor.
Competent authorities shall do the assessment referred to in paragraph 1 on the basis of both the period referred to in Article 325bc(4), point (c), of Regulation (EU) No 575/2013 and the period of financial stress referred to in Article 325bc(2), point (c), of that Regulation.
3. For the purposes of paragraph 1, competent authorities may, on a sample of risk factors, perform additional tests, including normality tests, to assess whether the distributions assumed by the institution are adequate. Competent authorities may require the institution to provide the impact that using alternative distributions would have on the expected shortfall risk measures.
Assessment of the stress scenario risk measure
1. When assessing an institution’s compliance with Article 325bi(1), point (e), of Regulation (EU) No 575/2013 in relation to requirements on the determination of the extreme scenario of future shock, competent authorities shall verify whether the internal policies referred to in that point meet all of the following requirements:
(a)
the internal policies comply with Article 21 of Delegated Regulation (EU) 2024/397;
(b)
the internal policies require the production of an up-to-date inventory which, for each non-modellable risk factor:
(i)
describes the risk factor;
(ii)
specifies the liquidity horizon assigned to the risk factor in accordance with Article 325bd of Regulation (EU) No 575/2013;
(iii)
specifies whether the institution calculates the extreme scenario of future shock by the direct method or the stepwise method referred to in Article 2, Article 3, Article 5 and Article 6 of Delegated Regulation (EU) 2024/397 respectively, or determines a regulatory extreme scenario of future shock in accordance with Article 14 of that Delegated Regulation;
(iv)
where the institution uses the stepwise method, specifies whether the historical, asymmetrical sigma, or fallback method is used to calibrate the downward and upward shocks;
(v)
for risk factors for which the institution determines a regulatory extreme scenario of future shock in accordance with Article 14 of Delegated Regulation (EU) 2024/397, justifies that choice;
(vi)
specifies whether the risk factor is part of a bucket and, if so, which one;
(c)
internal policies specify the criteria referred to in Article 1, point (a)(i) and Article 4, point (a)(i), of Delegated Regulation (EU) 2024/397, establishing when either the direct method or the stepwise method referred to in Article 2, Article 3, Article 5 and Article 6 of that Delegated Regulation is used with reference to any non-modellable risk factor or non-modellable standardised bucket;
(d)
internal policies specify the criteria to identify business and non-business days in a way that is consistent across the calculation of the stress scenario risk measure referred to in Article 325bk of Regulation (EU) No 575/2013 and the calculation of the expected shortfall risk measure referred to in Article 325bb of that Regulation;
(e)
internal policies specify the criteria to identify risk factors for which the institution determines the stress scenario risk measure by applying a regulatory extreme scenario of future shock in accordance with Article 14 of Delegated Regulation (EU) 2024/397;
(f)
internal policies require that the institution keeps track of all pricing failures as referred to in Article 13(3) of Delegated Regulation (EU) 2024/397, the cause of the pricing failures, and the remedial actions taken under that Article;
(g)
internal policies specify the frequency of updates, in accordance with Article 12(4) of Delegated Regulation (EU) 2024/397, of the stress period used for the determination of the extreme scenario of future shock, and the other possible criteria triggering an update of such stress period.
2. When assessing whether an institution’s internal model is implemented with integrity as required by Article 325bi(1) of Regulation (EU) No 575/2013 in relation to requirements on the calculation of the stress scenario risk measure referred to in Article 325bk of that Regulation, competent authorities shall:
(a)
where the institution uses the direct method referred to in Article 2 of Delegated Regulation (EU) 2024/397 in relation to non-modellable risk factors:
(i)
verify whether the institution’s processes follow the criteria referred to in Article 1, point (a)(i), of Delegated Regulation (EU) 2024/397 as laid down in the internal policies referred to in Article 325bi(1), point (e), of Regulation (EU) No 575/2013;
(ii)
verify whether, the institution documents and justifies changes in the approach used for the calculation of the stress scenario risk measure, as required by Article 1, point (a)(ii), of Delegated Regulation (EU) 2024/397;
(iii)
verify whether there is any material difference between the stress scenario risk measure resulting, on the one hand, from the direct method and, on the other hand, the stepwise method, for the period of 20 business days referred to in Article 1, point (a)(iii) of Delegated Regulation (EU) 2024/397, and investigate the reasons for any material differences;
(b)
where the institution uses the direct method referred to in Article 5 of Delegated Regulation (EU) 2024/397 in relation to non-modellable standardised buckets:
(i)
verify whether the institution’s processes follow the criteria referred to in Article 4, point (a)(i), of that Delegated Regulation, as laid down in the internal policies referred to in Article 325bi(1), point (e), of Regulation (EU) No 575/2013;
(ii)
verify whether the institution documents and justifies changes in the approach used for the calculation of the stress scenario risk measure, as required by Article 4, point (a)(ii), of Delegated Regulation (EU) 2024/397;
(iii)
verify whether there is any material difference between the stress scenario risk measure resulting, on the one hand, from the direct method and, on the other hand, the stepwise method, for the period of 20 business days referred to in Article 4, point (a)(iii), of Delegated Regulation (EU) 2024/397, and investigate the reasons for any material differences;
(c)
in relation to the determination of the time series of 10 business days returns referred to in Article 7 of Delegated Regulation (EU) 2024/397:
(i)
verify whether the institution does not include more than one observation per business day in the time series used to generate a stress scenario risk measure, and verify whether the time series includes actual market data only, as required by Article 7(1), point (a), of Delegated Regulation (EU) 2024/397;
(ii)
verify whether the criteria referred to in paragraph 1, point (d) to identify business and non-business days are used in the calculation of the 10 business days returns referred to in Article 7 of Delegated Regulation (EU) 2024/397 and in the extension of the stress period by up to 20 business days as referred to in Article 7(1), point (b), of that Regulation, and verify whether the steps to obtain the 10 business days returns, including the determination of D
t ' as referred to in Article 7(1), point (c), of that Delegated Regulation, are performed correctly;
(iii)
verify whether the time series of non-modellable risk factors that the institution previously assessed to be modellable in accordance with Article 325be of Regulation (EU) No 575/2013 include the observations that the institution used for calibrating the scenarios of future shocks referred to in Article 325bc of that Regulation, as required by Article 7(2) of Delegated Regulation (EU) 2024/397;
(d)
in relation to the implementation of the fallback method referred to in Article 10 of Delegated Regulation (EU) 2024/397:
(i)
verify whether institutions can justify the scarce data availability for the non-modellable risk factors or non-modellable standardised buckets for which the institution uses the fallback method;
(ii)
verify whether there is an appropriate identification of risk factors for which the approach referred to in Article 10(2) and (3) of Delegated Regulation (EU) 2024/397 is to be used;
(iii)
verify whether, when applying the method referred to in Article 10(4) of Delegated Regulation (EU) 2024/397, the approach used by the institution to select a risk factor meeting the conditions referred to in paragraph 5 of that Article leads to the determination of upward and downward shocks that are suitable for risk factor for which the fallback approach is applied;
(e)
require the institution to identify non-modellable risk factors or non-modellable standardised buckets for which the value of the non-linearity coefficient referred to in Articles 17 and 18 of Delegated Regulation (EU) 2024/397 is equal either to κ
min
or κ
max
, as referred to in those Articles, and verify whether the extreme scenario of future shock is appropriate or whether, in accordance with Article 325bk(3), point (b), of Regulation (EU) No 575/2013, the institution is to be required to apply a regulatory extreme scenario of future shock in accordance with Article 14 of Delegated Regulation (EU) 2024/397;
(f)
in relation to the determination of the stress period as required by Article 12 of Delegated Regulation (EU) 2024/397:
(i)
where the institution determines the stress period by maximising the value referred to in Article 12(1) of Delegated Regulation (EU) 2024/397 and using sensitivity-based pricing methods in accordance with Article 13(4) of that Delegated Regulation, verify the robustness of the analysis performed by the institution to demonstrate that the price changes that are not captured by the sensitivity-based pricing methods would not modify the stress period;
(ii)
where the institution determines the stress period for the non-modellable risk factors in a broad risk factor category by identifying the 12-month observation period maximising the partial expected shortfall measure PES
RS , i
in accordance with Article 12(2) of Delegated Regulation (EU) 2024/397, verify the robustness of the analysis performed by the institution to demonstrate that the stress period identified is a period of financial stress for its non-modellable risk factors;
(iii)
verify whether the 12-month rolling windows tested to determine the stress period starts at least from 1 January 2007 as referred to in Article 12(3) of Delegated Regulation (EU) 2024/397, and verify whether past updates of the stress period followed the frequency and criteria referred to in paragraph 1, point (g);
(g)
in relation to the computation of losses with sensitivity-based pricing methods under the conditions set out in Article 13(3) of Delegated Regulation (EU) 2024/397:
(i)
assess the robustness of the processes and methods for detecting pricing failures, identifying the financial instruments and commodities for which a pricing failure occurred, identifying the causes of the pricing failures, and determining their material sensitivities;
(ii)
verify whether, following the application of the extreme scenario of future shock to a non-modellable risk factor, the use of sensitivity-based pricing methods is applied only to financial instruments and commodity bearing that risk factor and subject to the pricing failures in accordance with Article 13(3) of Delegated Regulation (EU) 2024/397;
(h)
in relation to the determination of the regulatory extreme scenario of future shock referred to in Article 14 of Delegated Regulation (EU) 2024/397:
(i)
verify the appropriateness of the method that the institution uses to determine whether the maximum loss that may occur due to a change in a non-modellable risk factor or to a change in a non-modellable standardised bucket is finite or not;
(ii)
where the maximum loss corresponding to a non-modellable risk factor or a non-modellable bucket is finite, verify whether the institution identifies such maximum loss accurately;
(iii)
where the maximum loss that may occur due to a change in a non-modellable risk factor or to a change in a non-modellable standardised bucket is not finite, verify whether the distributional and statistical assumptions used in the expert-based approach referred to in Article 14(2), point (a), and Article 14(4) point (a) of Delegated Regulation (EU) 2024/397 are based on objective data and robust tests, and that the extreme scenario of future shock is sufficiently conservative;
(iv)
verify whether:
(1)
the information included in the inventory referred to in paragraph 1, point (b) are consistent with the criteria referred to in point (e) of that paragraph;
(2)
the criteria specified in the inventory referred to in paragraph 1, point (b) to identify those risk factors for which the stress scenario risk measure is obtained by determining a regulatory extreme scenario of future shock are sound;
(v)
verify the robustness of the methodology and the statistical tests that the institution uses to identify risk factors reflecting idiosyncratic risk only in accordance with Article 16(3) and (4) of Delegated Regulation (EU) 2024/397.
For the purposes of point (a)(ii), competent authorities shall verify whether the justification provided fits with the criteria referred to in point (a)(i) and the related changes are not driven by the fact that one method leads to a lower stress scenario risk measure than the other.
For the purposes of point (b)(ii), competent authorities shall verify whether the justification provided fits with the criteria referred to in point (b)(i) and is not driven by the fact that one method leads to a lower stress scenario risk measure than the other.
For the purposes of point (g)(ii), competent authorities shall verify that the institution calculates the losses related to other financial instruments and commodities bearing that risk factor but not subject to a pricing failure with the pricing methods used in the risk measurement model in accordance with Article 13(2) of Delegated Regulation (EU) 2024/397.
3. For the purposes of paragraph 2, points (a)(ii) and (b)(ii), competent authorities may compare the stress scenario risk measure of the risk factors or standardised buckets for which a change in the approach has occurred and assess whether the changes systematically correspond to a lower stress scenario risk measure.
4. For the purposes of paragraph 2, point (c)(i), competent authorities may, on a sample of time series of observations referred to in Article 7(1), point (a), of Delegated Regulation (EU) 2024/397, verify that where observations in the time series are constants over subsequent business days, the actual market data for the risk factor are unchanged. When collecting the sample, competent authorities shall consider time series characterised by a large amount of data without changes over subsequent business days.
5. For the purposes paragraph 2, point (c)(iii), competent authorities may, on a sample of risk factors, compare the risk factors’ observations that the institution uses to calculate the expected shortfall for the risk factor when it was modellable against the risk factors’ observations that the institution uses to calculate the stress scenario risk measure.
6. For the purposes of paragraph 2, point (d)(ii), competent authorities may, on a sample of risk factors for which the institution uses the approaches referred to in Article 10(2) or (3), of Delegated Regulation (EU) 2024/397, verify whether those risk factors meet the conditions for being subject to that methodology.
7. For the purposes of paragraph 2, point (d)(iii), competent authorities may, on a sample of risk factors for which the approach referred to in Article 10(4), of Delegated Regulation (EU) 2024/397 is used, verify whether the corresponding selected risk factors meet the conditions referred to in paragraph 5 of that Article.
When verifying whether the two risk factors are of the same nature in accordance with Article 10(5), point (b), of Delegated Regulation (EU) 2024/397 and whether those risk factors do not differ for features leading to an underestimation of the volatility in accordance with Article 10(5), point (c), of that Delegated Regulation, competent authorities shall verify whether the risk factors share the main characteristics, and whether the selected risk factor attracts name-related specific risk if the non-modellable risk factor attracts it.
8. For the purposes of paragraph 2, point (d)(iii), competent authorities may, on a sample of risk factors for which the methodology referred to in Article 10(4) of Delegated Regulation (EU) 2024/397 is used:
(a)
require the institution to test alternative suitable risk factors meeting the conditions referred to in Article 10(5) of Delegated Regulation (EU) 2024/397 instead of the risk factors selected by the institution;
(b)
compare the extreme scenario of future shock obtained using the risk factors selected by the institution and the extreme scenario of future shock obtained using the alternative risk factors referred to in point (a);
(c)
assess whether the risk factors selected by the institution lead to a systematic underestimation of the extreme scenario of future shock;
9. For the purposes of paragraph 2, point (d)(iii), competent authorities may, on a sample of risk factors for which the methodology referred to in Article 10(4) of Delegated Regulation (EU) 2024/397 is used and for which observations over a 1-year period are more than twelve:
(a)
require the institution to estimate the volatility of those risk factors over that 1-year period;
(b)
require the institution to estimate the volatility over that 1-year period of the risk factors selected in accordance with Article 10(5) of Delegated Regulation (EU) 2024/397 for the risk factors referred to in point (a);
(c)
assess whether the volatility of the risk factors selected by the institution resulting from the estimation referred to in point (b) is systematically lower than the volatility of the risk factors in the institution risk-measurement model resulting from the estimation referred to in point (a).
10. For the purposes of paragraph 2, point (e), competent authorities may, on a sample of risk factors and standardised buckets:
(a)
assess whether the non-linearity coefficient is equal to κ
min
or κ
max
because extremely high or extremely low values characterise the numerator or denominator of the following term
as used in the computation of κ in accordance with Articles 17 and 18 of Delegated Regulation (EU) 2024/397;
(b)
require the institution to plot the loss resulting from risk factor changes in the neighbourhood of the extreme scenario of future shock and assess whether the profile of the loss function is particularly concave or convex in that neighbourhood.
When performing such assessment, competent authorities shall choose the set of non-modellable risk factors and standardised buckets considering their materiality.
11. For the purposes of paragraph 2, point (f)(i), competent authorities may, where appropriate and where losses corresponding to changes in material non-modellable risk factors or non-modellable standardised buckets are highly non-linear:
(a)
require the institution to determine the stress period by maximising the value referred to in Article 12(1) of Delegated Regulation (EU) 2024/397 on a set of non-modellable risk factors or any non-modellable standardised bucket belonging to the same broad category of risk factors, using the pricing methods that the institution uses in the risk-measurement model in accordance with Article 13(2) of that Delegated Regulation;
(b)
require the institution to determine the stress period by maximising the value referred to in Article 12(1) of Delegated Regulation (EU) 2024/397 on the set referred to in point (a) of this paragraph, using sensitivity-based pricing methods in accordance with Article 13(4) of that Delegated Regulation;
(c)
assess whether the stress periods determined in accordance with points (a) and (b) materially differ.
The set of non-modellable risk factors or non-modellable standardised buckets referred to in the first subparagraph, point (a), shall be chosen considering their materiality and the non-linear profile of the loss to changes in their values. To identify non-modellable risk factors or non-modellable standardised buckets with a non-linear loss profile, competent authorities may use as a basis the value of the non-linearity coefficient κ calculated in accordance with Article 17 or Article 18 of Delegated Regulation (EU) 2024/397.
12. For the purposes of paragraph 2, point (f)(ii), competent authorities may:
(a)
require the institution to determine the stress period by maximising the value referred to in Article 12(1) of Delegated Regulation (EU) 2024/397 on a set of non-modellable risk factors or any non-modellable standardised bucket belonging to the same broad category of risk factors, using the pricing methods used in the risk-measurement model in accordance with Article 13(2) of that Delegated Regulation;
(b)
assess whether the stress period determined in accordance with point (a) significantly differs from the stress period identified by the institution when applying the methodology referred to in Article 12(2) of Delegated Regulation (EU) 2024/397.
13. For the purposes of paragraph 2, point (g), competent authorities may, on a sample of pricing failures that the institution may have faced, verify whether the institution followed the processes and methods referred to in paragraph 2, point (g)(i), and assess on that basis the robustness of those processes and methods.
14. For the purposes of paragraph 2, point (h)(iii), competent authorities may, on a sample of non-modellable risk factors or non-modellable standardised buckets:
(a)
require the institution to generate a time series of returns from a fat-tailed statistical distribution prescribed by the competent authority and calculate the extreme scenario of future shock with the stepwise method referred to in Article 3 and Article 6 of Delegated Regulation (EU) 2024/397 combined with the historical method referred to in Article 8 of that Delegated Regulation;
(b)
verify the conservativeness of the institution’s expert-based approach by comparing the regulatory extreme scenario of future shock resulting from that approach with the extreme scenario of future shock calculated in accordance with point (a).
15. By way of derogation from the first subparagraph, point (a), competent authorities may require the institution to use the time series of another similar risk factor instead of generating the time series from a conservative distribution. For the purposes of paragraph 2, point (h)(iv), competent authorities may, on a sample of non-modellable risk factors or non-modellable standardised buckets and at a given reference date, verify whether:
(a)
the risk factors or standardised buckets for which the stress scenario risk measure are determined by applying a regulatory extreme scenario of future shock in accordance with Article 14 of Delegated Regulation (EU) 2024/397 fulfil the criteria identified by the institution to use that method;
(b)
the risk factors or standardised buckets for which the stress scenario risk measure is not determined by applying a regulatory extreme scenario of future shock in accordance with Article 14 of Delegated Regulation (EU) 2024/397 do not fulfil the criteria that the institution identified to use that method.
16. For the purposes of paragraph 2, point (i), competent authorities may, on a sample of non-modellable risk factors:
(a)
verify whether the nature of the risk factor is such that it reflects idiosyncratic risk only by reviewing the description of the risk factor provided in the list referred to in Article 33(1) of this Regulation, and the data inputs used to mark it, as required by Article 16(3), point (a), and Article 16(4), point (a), of Delegated Regulation (EU) 2024/397;
(b)
perform hypothesis testing to assess the significance of correlation coefficients between risk factors in the sample and compare the results of that hypothesis testing with the results that the institution obtained when performing the statistical tests referred to in Article 16(3), point (d), and Article 16(4), point (d), of Delegated Regulation (EU) 2024/397.
