Document Type : Original Research Paper

Authors

• ali Sadeghkhani ¹
• Mohsen gharahkhani ²

¹ Actuarial &Mathematics, University of Windsor: Windsor, Ontario, CA

² Assistant Professor, Faculty of Finance & Accounting, Iranian eUniversity, Tehran, Iran

Abstract

BACKGROUND AND OBJECTIVES: Assessing the rate adequacy of excess-of-loss (XoL) reinsurance treaty layers, encompassing both non-catastrophe risk and catastrophe exposures, has become a critical challenge in contemporary reinsurance supervision. Increasing loss volatility, climate-driven catastrophe risk, growing concentration of insured values, and heightened regulatory emphasis on internal risk-based supervision have exposed structural weaknesses in traditional XoL pricing practices. In many reinsurance markets, pricing decisions continue to rely on single-family severity distributions, historical average loss ratios, or prevailing market-leading terms. While such approaches offer operational simplicity, they frequently underestimate tail risk, ignore parameter uncertainty, and fail to reflect company-specific characteristics such as portfolio composition, exposure growth, capital constraints, and retrocession structure. These shortcomings are particularly pronounced in mid and upper XoL layers, where data scarcity and heavy-tailed loss behavior materially affect solvency outcomes.
The primary objective of this study is to develop an internal supervisory framework for evaluating the rate adequacy of XoL treaty layers that moves beyond conventional pricing benchmarks and explicitly links technical rates to portfolio risk and capital resilience. Rather than treating rate adequacy as a standalone actuarial exercise, the proposed framework embeds pricing analysis within the broader context of internal risk and solvency assessment (IRSA), providing a forward-looking tool for identifying rate inadequacy and its capital implications.
METHODS: The proposed framework adopts a coordinated, multi-perspective methodology that distinguishes between non-catastrophe risk layers and catastrophe layers while integrating exposure dynamics across both. For non-catastrophe risk XoL treaties, loss severity is modeled using a spliced distribution, combining a lognormal distribution for the body of losses with a generalized Pareto distribution for the tail. Model parameters are estimated using maximum likelihood techniques, and Monte Carlo simulation is applied to generate synthetic loss samples. From these simulations, key technical measures are derived, including limited expected values, layer-specific technical premiums, and technical rate-on-line benchmarks.
For catastrophe XoL layers, where historical loss data are typically sparse, a Bayesian frequency–severity modeling approach is employed. Industry-informed prior distributions are combined with limited company-specific experience to obtain posterior distributions that explicitly capture parameter uncertainty. This structure enables the estimation of predictive loss distributions, exceedance probabilities, and credibility intervals relevant for solvency-oriented decision-making.
A key innovation of the framework is the integration of an exposure-based supervisory module. Recognizing that premium growth is an unreliable proxy for tail risk, interval-valued data are used to represent the evolution of high-value insured exposures in the tail of the portfolio. Exposure growth is translated into scaling factors that adjust loss frequency and tail behavior, ensuring consistency between pricing outcomes and the underlying risk profile. In addition, a marginal participation mechanism is introduced to quantify the incremental contribution of accepting specific layers or participation shares to aggregate portfolio risk and capital consumption.
FINDINGS: The framework is illustrated through a case study of a multi-layer property risk XoL treaty implemented in the domestic reinsurance market. The results demonstrate that conventional pricing approaches based on single-family severity distributions materially understate tail losses, particularly in mid and upper layers. While the first layer is broadly aligned with its internally derived technical rate, several higher layers exhibit substantial rate inadequacy relative to technical benchmarks. The spliced severity model produces significantly higher and more realistic technical rates for layers exposed to heavy-tailed loss behavior, correcting the optimistic bias observed under lognormal-only specifications.
The Bayesian catastrophe module yields stable parameter estimates and explicit uncertainty measures, highlighting the range of plausible loss outcomes relevant for internal solvency assessment. Exposure-based analysis further reveals that observed premium growth fails to capture the true expansion of tail exposure. The marginal participation analysis shows that the existing layer structure constrains the reinsurer’s ability to accept homogeneous participation shares without disproportionately increasing capital strain, causing rate inadequacy in upper layers to translate directly into elevated capital consumption.
CONCLUSION: This study demonstrates that rate adequacy and capital resilience are inseparable dimensions of reinsurance risk management. The proposed internal supervisory framework integrates stochastic pricing, heavy-tailed loss modeling, Bayesian uncertainty quantification, and exposure dynamics within a single analytical structure. By embedding rate adequacy assessment within the IRSA process, the framework enables reinsurers and supervisors to move beyond static market benchmarks toward risk-sensitive, forward-looking decision-making. Deviations between technical rates and market terms are translated into actionable supervisory signals, supporting informed decisions on participation shares, coverage limits, treaty acceptance, and portfolio rebalancing.

Keywords

• Reinsurance supervision
• Excess-of-loss (XoL)
• Rate adequacy
• Exposure-based pricing
• Solvency assessment

Main Subjects

• Insurance pricing