5.1.2. Expected Shortfall

Artzner et al. [47,48] recommended the use of conditional VaR instead of VaR, famously called Expected Shortfall (ES). The ES is a metric that quantifies the average loss in situations where the VaR level is exceeded. It is defined by the following expression

$$ES\_p = \frac{1}{p} \int\_0^p VaR\_x dx\_\prime \qquad \text{where} \quad 0 < p < 1.$$

The ES of OGE2Fr is given by

$$\, \, ES\_p = \frac{1}{p} \int\_0^p \left[ a \left\{ -\log \left( 1 - \left[ 1 - \log \left( 1 - p^{1/\beta} \right) \right]^{-1/a} \right) \right\} \right]^{-1/b} d\mathbf{x}.\tag{28}$$

Figure 12 illustrates VaR and ES for some random parameter combinations of OGE2Fr.

### 5.1.3. Numerical Calculation of VaR and ES

The results of OGE2Fr presented in Section 4 allowed us to further explore its application to these risk measures. From Table 5, we take the values of MLEs of PD1 and PD2, respectively, to measure the volatility associated with these measures. Higher values

of these risk measures signify heavier tails while lower values indicate a much lighter tail behavior of the model. It is worth mentioning that the OGE2Fr model produced substantially more significant results than its counterparts, indicating that the model has a heavier tail. In Table 7, we show the numerical results of VaRs and ESs of PD1 and PD2, respectively, of the proposed model. For the convenience of the reader, Figure 13 show the results graphically.

**Figure 12.** Plots of (**a**) VaR (**b**) ES for some parameter values.

**Figure 13.** Estimated VaR (**<sup>a</sup>**,**<sup>c</sup>**) with ES (**b**,**d**) for PD1 & PD2.


**Table 7.** Numerical measures of VaRs and ESs of PD1 & PD2.
