Institutions shall calculate the estimate of the tail parameter for a given non-modellable risk factor as follows:
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:
= \frac{\frac{1}{\alpha \times \mathrm{N}} \times \left\{\sum_{\mathrm{i}=1}^{\lfloor \alpha \times \mathrm{N} \rfloor} \mathrm{Ret}_{(\mathrm{i})}^2 + (\alpha \times \mathrm{N} - \lfloor \alpha \times \mathrm{N} \rfloor) \times \mathrm{Ret}_{(\lfloor \alpha \times \mathrm{N} \rfloor + 1)}^2\right\}}{\{\mathrm{ES}_{\mathrm{Left}} (\mathrm{Ret})\}^2}
where:
\alpha = 2.5\%
;
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;
\mathrm{Ret}_{(i)}
represents the i-th smallest return in the time series Ret;
[\alpha \times N]
denotes the integer part of;
\alpha \times N
\mathrm{ES}_{\mathrm{Left}} (\mathrm{Ret})
is the estimate of the left-tail expected shortfall for the time series Ret calculated in accordance with Article 11(1);
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:
= \frac{\frac{1}{\alpha \times N} \times \left\{ \sum_{i=1}^{[\alpha \times N]} (-\mathrm{Ret})_{(i)}^{2} + (\alpha \times N - [\alpha \times N]) (-\mathrm{Ret})_{([\alpha \times N]+1)}^{2} \right\}}{\left\{ \mathrm{ES}_{\mathrm{Right}}(\mathrm{Ret}) \right\}^{2}}
where:
\alpha = 2.5\%
;
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;
-\mathrm{Ret}_{(i)}
represents the i-th smallest return in the time series;
-\mathrm{Ret}
[\alpha \times N]
denotes the integer part of;
\alpha \times N
\mathrm{ES}_{\mathrm{Right}} (\mathrm{Ret})
is the estimate of the right-tail expected shortfall for the time series Ret calculated in accordance with Article 11(2);
in all other cases institutions shall set the estimate of the tail parameter.
= 1.04