Document Type : English paper for special issue on "climate change & insurance industry"
Authors
1 Shahid Beheshti University Faculty of Mathematical Sciences Department of Statistics
2 Insurance Research Center, Tehran, Iran.
Abstract
Catastrophic risk is fundamentally defined by the simultaneous exposure of a large population to substantial financial or physical losses caused by a single high‑severity peril. Such events—whether natural disasters like earthquakes, floods, and droughts, or human‑induced hazards such as large‑scale terrorism—result in severe casualties and extensive infrastructural damage. Iran is recognized as one of the most disaster-prone countries globally, having experienced numerous catastrophic events, particularly seismic disasters, with over 80,000 fatalities reported in the past three decades. The pronounced vulnerability of Iran’s arid and semi‑arid regions highlights the need for rigorous catastrophe risk management and the application of actuarially sound reinsurance pricing methodologies.
In the insurance context, extreme events are characterized by their low frequency and high severity. These events can produce losses that exceed the financial capacity of primary insurers, potentially driving them toward insolvency. A catastrophic risk is fundamentally characterized by the simultaneous exposure of a large population to the potential for significant financial or physical loss stemming from a single, high-severity peril. Such events often manifest as natural calamities, including earthquakes, floods, or droughts, or as human-induced hazards such as large-scale terrorism, all leading to extensive loss of life and widespread infrastructure destruction.
Therefore, transferring catastrophe (CAT) risk through reinsurance serves as a crucial strategic tool for maintaining solvency and capital stability. The central question addressed in this study is how to accurately determine the CAT reinsurance premium.
To this end, the paper proposes a hybrid reinsurance framework that combines Quota‑Share (QS) and Excess‑of‑Loss (XL) contracts for catastrophe coverage. Catastrophe loss data—specifically earthquake fatalities in Iran from 1950 to 2023—are modeled using the Peaks‑Over‑Threshold (POT) method within the framework of Extreme Value Theory (EVT). The parameters of the Generalized Pareto Distribution (GPD) are estimated via Maximum Likelihood Estimation (MLE), and the price is computed based on the Standard Deviation Premium Principle.
Sensitivity analysis across multiple threshold percentiles reveals that premium estimation is highly dependent on the selected threshold: higher thresholds consistently yield higher reinsurance premiums, confirming the threshold’s critical role in negotiations between the ceding company and the reinsurer. The proposed QS–XL framework therefore provides an integrated actuarial approach for pricing catastrophic risk in high‑vulnerability regions, offering enhanced financial resilience and underwriting efficiency.
The pronounced heavy-tailed characteristic inherent in catastrophe loss distributions significantly dampens the reinsurance companies’ willingness to underwrite direct Catastrophe contracts. To effectively incentivize underwriting participation, the pricing for such contracts typically relies on the Standard Deviation Premium Principle rather than the simpler net premium principle, as this approach better accounts for extreme outcomes. Crucially, to accurately model and capture the aforementioned heavy-tail behavior of catastrophic risk, this paper employs the POT model for fitting the loss data. In this paper, we used the POT model to fit the catastrophic loss data. Using the standard deviation premium principle and considering excess-of-loss reinsurance after quota-share reinsurance contract, we calculated the contract price under different thresholds. To empirically assess the influence of the threshold value on the resultant reinsurance premium, we conducted sensitivity analyses across several predetermined percentile levels for threshold selection. The analysis definitively demonstrated that the chosen threshold value is a critical determinant in establishing the final premium (P). Once an appropriate threshold is established, the Generalized Pareto Distribution (GPD) parameters are subsequently estimated using the Maximum Likelihood Estimation (MLE) method. The findings consistently indicated that selecting a threshold corresponding to a higher data percentile leads to a higher reinsurance premium.
Therefore, the selection of the optimal threshold becomes a crucial negotiation point, enabling both the ceding company and the reinsurer to agree on a premium level that is mutually acceptable and statistically justified.
Future studies may consider the occurrence of events generating sequences of large claims within a group.
The pronounced heavy-tailed characteristic inherent in catastrophe loss distributions significantly dampens the reinsurance companies’ willingness to underwrite direct Catastrophe contracts. To effectively incentivize underwriting participation, the pricing for such contracts typically relies on the Standard Deviation Premium Principle rather than the simpler net premium principle, as this approach better accounts for extreme outcomes. Crucially, to accurately model and capture the aforementioned heavy-tail behavior of catastrophic risk, this paper employs the POT model for fitting the loss data. In this paper, we used the POT model to fit the catastrophic loss data. Using the standard deviation premium principle and considering excess-of-loss reinsurance after quota-share reinsurance contract, we calculated the contract price under different thresholds. To empirically assess the influence of the threshold value on the resultant reinsurance premium, we conducted sensitivity analyses across several predetermined percentile levels for threshold selection. The analysis definitively demonstrated that the chosen threshold value is a critical determinant in establishing the final premium (P). Once an appropriate threshold is established, the Generalized Pareto Distribution (GPD) parameters are subsequently estimated using the Maximum Likelihood Estimation (MLE) method.
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