Presentation of EAJ Issue 11/2 - December 6th
Authors: Gabrielli Andrea , Rödel Karen Tanja , Rödel Karen Tanja , Christiansen Marcus , Gerber Hans , Cadena Meitner 
Published: November 2023

An Individual Claims Reserving Model for Reported Claims

We present a claims reserving technique that uses claim-specific feature and past payment information in order to estimate claims reserves for individual reported claims. We design one single neural network allowing us to estimate expected future cash flows for every individual reported claim. We introduce a consistent way of using dropout layers in order to fit the neural network to the incomplete time series of past individual claims payments. A proof of concept is provided by applying this model to synthetic as well as real insurance data sets for which the true outstanding payments for reported claims are known.

Measuring Profitability of Life Insurance Products under Solvency II

In this paper, we propose an enhanced method for the measurement of profitability of life insurance products. In contrast to most of the existing literature, we consider the development of the insurance contracts over their entire lifetime under the real-world probability measure and distinguish between different sources of capital. We study the pathwise realization of random variables describing shareholder profitability to obtain and analyze their distribution. These distributions are more versatile than single statistics such as expected values since they additionally allow for the analysis of extreme outcomes. Moreover, we specifically consider the strain on shareholders arising from the solvency capital requirement under Solvency II. We use a cost of capital approach based on the explicit computation of the solvency capital requirement and the interrelated capital required from shareholders for each year of the projection period. To demonstrate the feasibility of our profit measures, we provide a concrete application to products with interest rate guarantees including an internal model approach for market risks under Solvency II. Our numerical application shows that our proposed profit measures are particularly suitable for revealing the profitability of different life insurance products in today’s regulatory environment.

Multi-Year Analysis of Solvency Capital in Life Insurance

With the commencement of the Solvency II directive, insurers in the European Union need to provide a projection of their solvency figures into the future (as part of the Own Risk and Solvency Assessment, ORSA). This is a highly complex task since future solvency figures depend on the development of numerous (stochastic) risk factors. The required evaluations are numerically challenging, which in practice forces companies to limit their analyses to only a few selected deterministic scenarios. These deterministic scenarios clearly cannot describe the full probability distribution of a company’s future solvency situation. The focus of this paper is on financial guarantees in participating life insurance products. In particular, we study two major types of interest rate guarantees in life insurance, a maturity guarantee and a (path-dependent) cliquet-style guarantee. In order to derive entire probability distributions of future solvency ratios, we limit the model framework to two sources of risk (a Hull–White model for interest rates and a geometric Brownian motion for stocks). This partly leads to closed-form solutions of the market-consistent valuation of the liabilities, ensures higher accuracy in computations and less numerical effort. Furthermore, the model allows for a detailed analysis of the impact of the different types of interest rate guarantees on the future solvency situation. Our results suggest that the type of guarantee has a significant impact on the long-term solvency of the company.

On the calculation of prospective and retrospective reserves in non-Markov models

Almost all life and health insurance models in the actuarial literature use either a Markov assumption or a semi-Markov assumption. This paper shows that non-Markov modelling is also feasible and presents suitable numerical and statistical tools for the calculation of prospective and retrospective reserves. A central idea is to base the calculation of reserves on forward and backward transition rates. Feasible estimators for the forward transition rates have been recently suggested in the medical statistics literature. This paper slightly extends them according to insurance needs and newly introduces symmetric estimators for backward transition rates. Only few adjustments are actually needed in the classical insurance formulas when switching from Markov modelling to as-if-Markov evaluations in order to avoid model risk.

Equivalence principle and Jewell’s inequality

In his address to the 21st International Congress of Actuaries, the late Professor William S. Jewell pointed out that, for a whole life insurance policy with level premiums payable continuously and a death benefit of constant amount payable at the moment of death, even though the equivalence principle stipulates that the insurer’s expected gain at issue is zero, the insurer’s expected gain at the moment of death of the insured is positive. This seemingly surprising result turns out to be true for more general life insurance policies. We present a simple derivation of and some elaboration on this fact.

A new measure of mortality differentials based on precedence probability

This short note aims to propose a new comparison measure for life tables: the age shift needed to restore equality of survival chance as measured by the precedence probability. Mortality by household income in France and by generation in Belgium is used as an illustration.