*5.1. Actuarial Measures*

One of the most important duties of actuarial sciences organizations is to assess market risk in a portfolio of instruments, which originates from changes in underlying factors such as equities prices, interest rates, or currency rates.One of the most important duties of actuarial sciences organizations is to assess market risk in a portfolio of instruments, which originates from changes in underlying factors such as equities prices, interest rates, or currency rates. We compute several important risk measures for the suggested distribution in this section, such as Value at Risk (VaR) and Expected Shortfall (ES), which are important in portfolio optimization under uncertainty.

**Figure 11.** Profiles of the log-likelihood function for the parameters *α*, *β*, *a* and *b*, respectively, of the OGE2Fr for the PD2.

### 5.1.1. Value at Risk

The quantile premium principal of the distribution of aggregate losses, commonly known as Value at risk (VaR), is the most widely used measure to evaluate exposure to risk in finance. VaR of a rv is the *p*th quantile of its cdf. If *X* ∼ OGE2Fr denotes a random variable with cdf (17), then its VaR is

$$VaR\_p = \left[ a \left\{ -\log \left( 1 - \left[ 1 - \log \left( 1 - p^{1/\beta} \right) \right]^{-1/a} \right) \right\} \right]^{-1/b}.\tag{27}$$
