This paper studies the optimal reinsurance problem when risk is measured by a general risk measure. Necessary and sufficient optimality conditions are given for a wide family of risk measures, including deviation measures, expectation bounded risk measures and coherent measures of risk. The optimality conditions are used to verify whether the classical reinsurance contracts (quota-share, stop-loss) are optimal essentially, regardless of the risk measure used. The paper ends by particularizing the findings, so as to study in detail two deviation measures and the conditional value at risk.Research partially supported by "Welzia Management SGIIC SA", "RD_Sistemas SA", "Comunidad Autónoma de Madrid" (Spain), Grant s-0505/tic/000230, and "MEyC" (Spain), Grant SEJ2006-15401-C04-03Publicad

Two-part models based on generalized linear models are widely used in insurance rate-making for predicting the expected loss. This paper explores an alternative method based on quantile regression which provides more information about the loss distribution and can be also used for insurance underwriting. Quantile regression allows estimating the aggregate claim cost quantiles of a policy given a number of covariates. To do so, a first stage is required, which involves fitting a logistic regression to estimate, for every policy, the probability of submitting at least one claim. The proposed methodology is illustrated using a portfolio of car insurance policies. This application shows that the results of the quantile regression are highly dependent on the claim probability estimates. The paper also examines an application of quantile regression to premium safety loading calculation, the so-called Quantile Premium Principle (QPP). We propose a premium calculation based on quantile regression which inherits the good properties of the quantiles. Using the same insurance portfolio data-set, we find that the QPP captures the riskiness of the policies better than the expected value premium principle.

Climate change refers to persistent alterations to global Earth's climate, such as a rise in global temperatures, which have reached unprecedented peaks in recent years. At the same time, global mean ocean‐and‐sea levels are on an upward trajectory. These climatic shifts significantly influence the frequency, intensity, and duration of extreme weather events, such as heatwaves, heavy precipitations, droughts, floods, and tropical cyclones, which represent substantial risks and challenges for the insurance industry. This paper delves into the profound impact of climate change on the insurance sector, with a particular focus on the agriculture, property, health, and life insurance industries. Our scientific approach consists in measuring climate change through an index composed of a basket of climate and weather‐related extremes, such as the Actuarial Climate Index™ (ACI) defined in and for North America, and its European counterparts, the Iberian ACI (IACI) and French ACI (FACI) climate indices. We discuss how these indices help quantify the impact of climate change on the balance sheets of insurance companies and, therefore, its impact on the sustainability of the insurance business. The paper underscores the pressing need for the insurance industry to adapt and strategically plan for the increasing risks associated with climate change.