Aggregation of the stress scenario risk measures
1. For the purposes of aggregating the stress scenario risk measures as referred to in Article 325bk(3), point (d), of Regulation (EU) No 575/2013, institutions shall, for each stress scenario risk measure they have computed, determine the corresponding rescaled stress scenario risk measure as follows:
(a)
where institutions determined the extreme scenario of future shock for a single risk factor in accordance with the stepwise method set out to in Article 3, the corresponding rescaled stress scenario risk measure shall be calculated in accordance with the following formula:
where:
—
RSS is the rescaled stress scenario risk measure for the non-modellable risk factor;
—
SS is the stress scenario risk measure for the non-modellable risk factor;
—
, and where LH is the liquidity horizon referred to in Article 325bd(1) of Regulation (EU) No 575/2013 for the non-modellable risk factor;
—
κ is the non-linearity coefficient for the non-modellable risk factor calculated in accordance with Article 17;
(b)
where institutions determined a stress scenario risk measure for more than one risk factor by determining an extreme scenario of future shock in accordance with the stepwise method set out in Article 6 for a non-modellable standardised bucket comprising those risk factors, the corresponding rescaled stress scenario risk measure shall be calculated in accordance with the following formula:
where:
—
RSS is the rescaled stress scenario risk measure for the non-modellable standardised bucket;
—
SS is the stress scenario risk measure for the non-modellable standardised bucket;
—
, and where LH is the liquidity horizon referred to in Article 325bd(1) of Regulation (EU) No 575/2013 for the risk factors within the non-modellable standardised bucket;
—
κ is the non-linearity coefficient for the non-modellable standardised bucket calculated in accordance with Article 18;
(c)
where institutions determined the extreme scenario of future shock for a single risk factor in accordance with the direct method set out in Article 2, the corresponding rescaled stress scenario risk measure shall be calculated with the following formula:
where:
—
RSS is the rescaled stress scenario risk measure for the non-modellable risk factor;
—
SS is the stress scenario risk measure for the non-modellable risk factor;
—
, and where LH is the liquidity horizon referred to in Article 325bd(1) of Regulation (EU) No 575/2013 for the non-modellable risk factor;
—
UCF is the uncertainty compensation factor to be calculated in accordance with Article 20;
(d)
where institutions determined a stress scenario risk measure for more than one risk factor by determining an extreme scenario of future shock in accordance with the direct method set out in Article 5 for the non-modellable bucket comprising those risk factors, the corresponding rescaled stress scenario risk measure shall be calculated in accordance with the following formula:
where:
—
RSS is the rescaled stress scenario risk measure for the non-modellable standardised bucket;
—
SS is the stress scenario risk measure for the non-modellable standardised bucket;
—
, and where LH is the liquidity horizon referred to in Article 325bd(1) of Regulation (EU) No 575/2013 for the risk factors within the non-modellable bucket;
—
UCF is the uncertainty compensation factor to be calculated in accordance with Article 20;
(e)
where institutions determined a stress scenario risk measure by determining a regulatory extreme scenario of future shock in accordance with Article 14, the corresponding rescaled stress scenario risk measure shall be calculated in accordance with the following formula:
where:
—
RSS is the rescaled stress scenario risk measure;
—
SS is the stress scenario risk measure.
2. Institutions shall aggregate the stress scenario risk measures in accordance with the following formula:
where:
—
ICSR denotes the set of non-modellable risk factors or non-modellable standardised buckets for which institutions determined a stress scenario risk measure that was classified as reflecting idiosyncratic credit spread risk only, in accordance with paragraph 3;
—
k is an index denoting the non-modellable risk factors or non-modellable standardised buckets belonging to ICSR ;
—
EIR denotes the set of non-modellable risk factors or non-modellable standardised buckets for which institutions determine a stress scenario risk measure that was classified as reflecting idiosyncratic equity risk only, in accordance with paragraph 4;
—
l is an index denoting the non-modellable risk factors or non-modellable standardised buckets belonging to EIR ;
—
OR denotes a non-modellable risk factor or non-modellable standardised bucket for which institutions determine a stress scenario risk measure that was neither classified as reflecting idiosyncratic credit spread risk only, in accordance with paragraph 3, nor idiosyncratic equity risk only, in accordance with paragraph 4;
—
j is an index denoting the non-modellable risk factors or non-modellable standardised buckets belonging to OR ;
—
are respectively the rescaled stress scenario risk measures for the non-modellable risk factors or the non-modellable standardised buckets k , l , j calculated in accordance with paragraph 1;
—
.
3. The non-modellable risk factors that institutions classify as reflecting idiosyncratic credit spread risk only shall meet all of the following conditions:
(a)
the nature of the risk factor is such that it reflects idiosyncratic credit spread risk only;
(b)
the value taken by the risk factor is not driven by systematic risk components;
(c)
the correlation among risk factors is negligible;
(d)
institutions perform and document the statistical tests used to verify the condition set out in point (c).
4. The non-modellable risk factors that institutions classify as reflecting idiosyncratic equity risk only shall meet all of the following conditions:
(a)
the nature of the risk factor is such that it reflects idiosyncratic equity risk only;
(b)
the value taken by the risk factor is not driven by systematic risk components;
(c)
the correlation among risk factors is negligible;
(d)
institutions perform and document the statistical tests used to verify the condition set out in point (c).
Non-linearity coefficient for a single risk factor
Where the stress scenario risk measure for which institutions are determining the non-linearity coefficient has been determined for a single risk factor, such non-linearity coefficient shall be determined as follows:
(a)
where the extreme scenario of future shock for the non-modellable risk factor does not coincide with either the downward calibrated shock or the upward calibrated shock determined in accordance with Article 3(1), point (b), institutions shall set
for that non-modellable risk factor;
(b)
where the extreme scenario of future shock for the non-modellable risk factor coincides with the downward calibrated shock determined in accordance with Article 3(1), point (b), institutions shall calculate the non-linearity coefficient in accordance with the following formula:
where:
—
;
—
;
—
ɸ is the estimate of the tail parameter for the non-modellable risk factor calculated in accordance with Article 19;
—
loss 0 is the loss that occurs when the downward shock CS
down determined in accordance with Article 3(1), point (b) is applied to the non-modellable risk factor;
—
is the loss that occurs when a downward shock equal to
is applied to the non-modellable risk factor, where CS
down is the downward shock determined in accordance with Article 3(1), point (b);
—
is the loss that occurs when a downward shock equal to
is applied to the non-modellable risk factor, where CS
down is the downward shock determined in accordance with Article 3(1), point (b);
(c)
where the extreme scenario of future shock for the non-modellable risk factor coincides with the upward calibrated shock determined in accordance with Article 3(1), point (b), institutions shall calculate the non-linearity coefficient in accordance with the following formula:
where:
—
;
—
—
ɸ is the estimate of the tail parameter for the non-modellable risk factor calculated in accordance with Article 19;
—
loss 0 is the loss that occurs when the the upward shock CS
up determined in accordance with Article 3(1), point (b) is applied to the non-modellable risk factor;
—
is the loss that occurs when an upward shock equal to
is applied to the non-modellable risk factor, where CS
up is the upward shock determined in accordance with Article 3(1), point (b);
—
is the loss that occurs when an upward shock equal to
is applied to the non-modellable risk factor, where CS
up is the upward shock determined in accordance with Article 3(1), point (b).
Non-linearity coefficient for a bucket
Where the stress scenario risk measure for which institutions are determining the non-linearity coefficient has been determined for a non-modellable standardised bucket, the non-linearity coefficient shall be determined as follows:
(a)
where the extreme scenario of future shock does not correspond to a scenario identified in accordance with Article 6(1), point (b), where the value of the parameter β referred to in Article 6(1), point (c), it set equal to 1, institutions shall set the non-linearity coefficient
for that non-modellable bucket;
(b)
where the extreme scenario of future shock is a scenario where the corresponding downward shock determined in accordance with Article 6(1), point (b) is applied to each risk factor in the non-modellable bucket, institutions shall calculate the non-linearity coefficient in accordance with the following formula:
where:
—
;
—
—
is the median of the estimates of the tail parameters calculated in accordance with Article 19 for each of the risk factors within the bucket;
—
loss 0 is the loss occurring when the corresponding downward shock determined in accordance with Article 6(1), point (b), is applied to each risk factor in the non-modellable bucket;
—
is the loss occurring when the corresponding downward shock determined in accordance with Article 6(1), point (b), multiplied by
is applied to each risk factor in the non-modellable bucket;
—
is the loss occurring when the corresponding downward shock determined in accordance with Article 6(1), point (b), multiplied by
is applied to each risk factor in the non-modellable bucket;
(c)
where the extreme scenario of future shock is a scenario where the corresponding upward shock determined in accordance with Article 6(1), point (b), is applied to each risk factor in the non-modellable bucket, institutions shall calculate the non-linearity coefficient in accordance with the following formula:
where:
—
;
—
—
is the median of the estimates of the tail parameters calculated in accordance with Article 19 for each of the risk factors within the bucket;
—
loss 0 is the loss occurring when the corresponding upward shock determined in accordance with Article 6(1), point (b), is applied to each risk factor in the non-modellable bucket;
—
is the loss occurring when the corresponding upward shock determined in accordance with Article 6(1), point (b), multiplied by
is applied to each risk factor in the non-modellable bucket;
—
is the loss occurring when the corresponding upward shock determined in accordance with Article 6(1), point (b), multiplied by
is applied to each risk factor in the non-modellable bucket.
Calculation of the estimate of the tail parameter
Institutions shall calculate the estimate of the tail parameter for a given non-modellable risk factor as follows:
(a)
where institutions used the historical method set out in Article 8 for determining the downward and upward calibrated shock of that non-modellable risk factor and the extreme scenario of future shock is the downward calibrated shock, they shall calculate the estimate of the tail parameter in accordance with the following formula:
where:
—
;
—
Ret is the time series of 10 business days returns for the non-modellable risk factor used in the historical method set out in Article 8;
—
represents the i-th smallest return in the time series Ret ;
—
denotes the integer part of
;
—
is the estimate of the left-tail expected shortfall for the time series Ret calculated in accordance with Article 11(1);
(b)
where institutions used the historical method set out in Article 8 for determining the downward and upward calibrated shock of that non-modellable risk factor and the extreme scenario of future shock is the upward calibrated shock, they shall calculate the estimate of the tail parameter in accordance with the following formula:
where:
—
;
—
Ret is the time series of 10 business days returns for the non-modellable risk factor used in the historical method set out in Article 8;
—
represents the i-th smallest return in the time series
;
—
denotes the integer part of
;
—
is the estimate of the right-tail expected shortfall for the time series Ret calculated in accordance with Article 11(2);
(c)
in all other cases institutions shall set the estimate of the tail parameter
.
Calculation of the uncertainty compensation factor
1. Where the stress scenario risk measure for which the institutions are determining the uncertainty compensation factor (UCF) has been determined for a single risk factor, the uncertainty compensation factor shall be equal to:
where:
—
N is the number of losses in the time series referred to in Article 2(1), point (a)(iii), from which the extreme scenario of future shock has been determined for the non-modellable risk factor in accordance with that Article.
2. Where the stress scenario risk measure for which the institutions are determining the uncertainty compensation factor has been determined for a non-modellable standardised bucket, the uncertainty compensation factor shall be equal to:
where:
—
N is the number of losses in the time series referred to in Article 5(1), point (a)(iv), from which the extreme scenario of future shock has been determined for the non-modellable bucket in accordance with that Article.
Source: EUR-Lex (Publications Office of the EU), © European Union, reuse permitted under Commission Decision 2011/833/EU.
