Analyzing the future values of anticipated claims is essential in order for insurance companies to avoid major losses caused by prospective future claims. This study proposes a novel three-parameter compound Lomax extension. The new density can be "monotonically declining", "symmetric", "bimodal-asymmetric", "asymmetric with right tail", "asymmetric with wide peak" or "asymmetric with left tail". The new hazard rate can take the following shapes: "J-shape", "bathtub (U-shape)", "upside down-increasing", "decreasing-constant", and "upside down-increasing". We use some common copulas, including the Farlie-Gumbel-Morgenstern copula, the Clayton copula, the modified Farlie-Gumbel-Morgenstern copula, Renyi's copula and Ali-Mikhail-Haq copula to present some new bivariate quasi-Poisson generalized Weibull Lomax distributions for the bivariate mathematical modelling. Relevant mathematical properties are determined, including mean waiting time, mean deviation, raw and incomplete moments, residual life moments, and moments of the reversed residual life. Two actual data sets are examined to demonstrate the unique Lomax extension's usefulness. The new model provides the lowest statistic testing based on two real data sets. The risk exposure under insurance claims data is characterized using five important risk indicators: value-at-risk, tail variance, tail-value-at-risk, tail mean-variance, and mean excess loss function. For the new model, these risk indicators are calculated. In accordance with five separate risk indicators, the insurance claims data are employed in risk analysis. We choose to focus on examining these data under five primary risk indicators since they have a straightforward tail to the left and only one peak. All risk indicators under the insurance claims data are addressed for numerical and graphical risk assessment and analysis.