We graduate the Irish mortality experience from 1950 to 2003 by mathematical formulae from ages 75 years and upwards. The shape of the mortality curve at advanced ages is shown to be different to that recorded in the official tables, with the curve best fitted with Kannisto's version of Perks's Law. Mortality rates show only a modest trend of improvement in the early decades, below improvements in other developed countries. We evaluate the various approaches suggested to date to extend the method of extinct generations so mortality rates for non-extinct generations can be estimated. It is shown that the key advantage of this method is not in correcting for age misstatements but in achieving a close correspondence between death counts and the exposed to risk. This insight allows a rather straightforward approach to estimating the mortality of non-extinct generations. Applying the approach, we show that there has been an acceleration in the rate of improvement in more recent decades, but secular improvements in Irish mortality at advanced ages still lag behind those of England and Wales.

The COVID-19 pandemic requires that actuaries track short-term mortality fluctuations in the portfolios they manage. This demands methods that not only operate over much shorter time periods than a year but that also deal with reporting delays. In this paper, we consider a semi-parametric approach for tracking portfolio mortality levels in continuous time. We identify both seasonal patterns and mortality shocks, thus providing a comparison benchmark for the impact of COVID-19 in terms of a portfolio’s own past experience. A parametric model is presented to allow for the average impact of seasonal variation and also reporting delays. We find that an estimate of mortality reporting delays can be made from a single extract of experience data. This can be used to forecast unreported deaths and improve estimates of recent mortality levels. Results are given for annuity portfolios in France, the UK and the USA.

We examine the value of guaranteed lifetime withdrawal benefit (GLWB) options embedded in variable annuities in two different tax regimes. The New Zealand (NZ) system taxes investment income when it is earned, whereas the system in the United States defers taxes on annuity investment income until it is paid out. We examine the effects of these tax differences on the charges collected by the issuer as well as on the value of the contract to the policyholder. We find that the issuer’s charges are typically lower (higher) in the NZ tax regime when the expected fund earnings are low (high) or the fund volatility is high (low). On the other hand, the value to the policyholder is always lower in the NZ tax regime due to the earlier tax payments.We also find that the value of the GLWB in the NZ tax regime is nearly always below the value of an ordinary payout annuity with the same tax rules.

In this paper, we develop a methodology to automatically classify claims using the information contained in text reports (redacted at their opening). From this automatic analysis, the aim is to predict if a claim is expected to be particularly severe or not. The difficulty is the rarity of such extreme claims in the database, and hence the difficulty, for classical prediction techniques like logistic regression to accurately predict the outcome. Since data is unbalanced (too few observations are associated with a positive label), we propose different rebalance algorithm to deal with this issue. We discuss the use of different embedding methodologies used to process text data, and the role of the architectures of the networks.

This article considers a dynamic version of risk measures for stochastic asset processes and gives a mathematical benchmark for required capital in a solvency regulation framework. Some dynamic risk measures, based on the expected discounted penalty function launched by Gerber and Shiu, are proposed to measure solvency risk from the company’s going-concern point of view. This study proposes a novel mathematical justification of a risk measure for stochastic processes as a map on a functional path space of future loss processes.

Ex post moral hazard arises when the insured has an unobservable influence on the size of a loss after its occurrence. In automobile (property) insurance, ex post moral hazard could increase in the scope of the repairs and/or the value of the repairs. Both vehicle owners and auto repairers could gain from increasing the scope of repairs, while auto repairers would gain from an increase in the value of repairs. An analysis of 994 Australian road traffic crashes found that ex post moral hazard increased the value of repairs by 46.8 per cent of which 9 percentage points was explained by an increase in the scope of the repairs, which was defined as an increased from 2 to 2.4 parts per auto repair.

Executive stock options (ESOs) are widely used to reward employees and represent major items of corporate liability. The International Accounting Standards Board IFRS9 financial reporting standard which came into full effect on 1-Jan 2018, along with its Australian implementation AASB9, requires public corporations to report their fair-value cost in financial statements. Reset ESOs are typically issued to re-incentivise employees by allowing the option to be cancelled and re-issued with a lower exercise price or later maturity. We produce a novel analytical Reset ESO valuation consistent with the IFRS9 financial reporting standard incorporating the simultaneous resetting of vesting period, exercise window, reset level and maturity. We allow for voluntary and involuntary exercise. Our analytical result is expressed solely in terms of standardised European binary power option instruments. Using the multi-state mortality model of Hariyanto (2014, Mortality and disability modelling with an application to pricing a reverse mortgage contract, PhD thesis, University of Melbourne), we estimate longitudinal disability and death transition probabilities from cross-sectional data. We determine survival functions for pre-vesting forfeiture or post-vesting involuntary exercise for use with weighted portfolios of our formulae to illustrate the effect of survival on the fair value. We examine the IFRS9 method of valuation using expected time to option exercise and demonstrate a consistent overestimation of fair value of up to 27% for senior executives.

The global banking system can be shown to be a Complex Adaptive System that exhibits phase transitions from time to time. These phase transitions can result in significant financial losses to the community that we estimate to be much more significant than losses occurring during “business as usual” periods. In this paper, we argue that the significant losses arising from phase transitions in the banking system requires a very different approach to regulation than the current Basel regime, and that there is a need to transition the Basel regime from a Federation of Systems to a System of Systems. We demonstrate that the World Health Organisation’s recent management system for pandemics is ideally suited for management of the global banking system and would have greater potential to control the phase transition losses than the current Basel system.

Traditional techniques for calculating outstanding claim liabilities such as the chain-ladder are notoriously at risk of being distorted by outliers in past claims data. Unfortunately, the literature in robust methods of reserving is scant, with notable exceptions such as Verdonck & Debruyne (2011, Insurance: Mathematics and Economics, 48, 85–98) and Verdonck & Van Wouwe (2011, Insurance: Mathematics and Economics, 49, 188–193). In this paper, we put forward two alternative robust bivariate chain-ladder techniques to extend the approach of Verdonck & Van Wouwe (2011, Insurance: Mathematics and Economics, 49, 188–193). The first technique is based on Adjusted Outlyingness (Hubert & Van der Veeken, 2008. Journal of Chemometrics, 22, 235–246) and explicitly incorporates skewness into the analysis while providing a unique measure of outlyingness for each observation. The second technique is based on bagdistance (Hubert et al., 2016. Statistics: Methodology, 1–23) which is derived from the bagplot; however; it is able to provide a unique measure of outlyingness and a means to adjust outlying observations based on this measure.

Furthermore, we extend our robust bivariate chain-ladder approach to an N-dimensional framework. The implementation of the methods, especially beyond bivariate, is not trivial. This is illustrated on a trivariate data set from Australian general insurers and results under the different outlier detection and treatment mechanisms are compared.

The classical credibility theory is a cornerstone of experience rating, especially in the field of property and casualty insurance. An obstacle to putting the credibility theory into practice is the conversion of available prior information into a precise choice of crucial hyperparameters. In most real-world applications, the information necessary to justify a precise choice is lacking, so we propose an imprecise credibility estimator that honestly acknowledges the imprecision in the hyperparameter specification. This results in an interval estimator that is doubly robust in the sense that it retains the credibility estimator’s freedom from model specification and fast asymptotic concentration, while simultaneously being insensitive to prior hyperparameter specification.

Guaranteed minimum accumulation benefits (GMABs) are retirement savings vehicles that protect the policyholder against downside market risk. This article proposes a valuation method for these contracts based on physics-inspired neural networks (PINNs), in the presence of multiple financial and biometric risk factors. A PINN integrates principles from physics into its learning process to enhance its efficiency in solving complex problems. In this article, the driving principle is the Feynman–Kac (FK) equation, which is a partial differential equation (PDE) governing the GMAB price in an arbitrage-free market. In our context, the FK PDE depends on multiple variables and is difficult to solve using classical finite difference approximations. In comparison, PINNs constitute an efficient alternative that can evaluate GMABs with various specifications without the need for retraining. To illustrate this, we consider a market with four risk factors. We first derive a closed-form expression for the GMAB that serves as a benchmark for the PINN. Next, we propose a scaled version of the FK equation that we solve using a PINN. Pricing errors are analyzed in a numerical illustration.

In this paper, we discuss the estimation of conditional quantiles of aggregate claim amounts for non-life insurance embedding the problem in a quantile regression framework using the neural network approach. As the first step, we consider the quantile regression neural networks (QRNN) procedure to compute quantiles for the insurance ratemaking framework. As the second step, we propose a new quantile regression combined actuarial neural network (Quantile-CANN) combining the traditional quantile regression approach with a QRNN. In both cases, we adopt a two-part model scheme where we fit a logistic regression to estimate the probability of positive claims and the QRNN model or the Quantile-CANN for the positive outcomes. Through a case study based on a health insurance dataset, we highlight the overall better performances of the proposed models with respect to the classical quantile regression one. We then use the estimated quantiles to calculate a loaded premium following the quantile premium principle, showing that the proposed models provide a better risk differentiation.

In this paper, we model the surplus process as a compound Poisson process perturbed by diffusion and allow the insurer to ask its customers for input to minimize the distance from some prescribed target path and the total discounted cost on a fixed interval. The problem is reduced to a version of a linear quadratic regulator under jump-diffusion processes. It is treated using three methods: dynamic programming, completion of square and the stochastic maximum principle. The analytic solutions to the optimal control and the corresponding optimal value function are obtained.

An usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimality criterion of minimising the Conditional Tail Expectation (CTE) risk measure of the insurer’s total risk, this article generalises an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy f(⋅). In the first step, it cuts down the interval [0, ∞) into intervals [0, M1) and [M1, ∞). By shifting the origin of Cartesian coordinate system to (M1, f(M1)), and showing that under the CTE criteria $$f\left( x \right)I_{{[0,M_{{\rm 1}} )}} \left( x \right){\plus}\left( {f\left( {M_{{\rm 1}} } \right){\plus}f\left( {x{\minus}M_{{\rm 1}} } \right)} \right)I_{{[M_{{\rm 1}} ,{\rm }\infty)}} \left( x \right)$$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: (1) the practical applications of our findings and (2) how one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover, it has some other optimal criteria which the original policy does not have. Under optimality criterion of minimising a general translative and monotone risk measure ρ(⋅) of either the insurer’s total risk or both the insurer’s and the reinsurer’s total risks, this article (in its discussion) also extends a given optimal reinsurance contract f(⋅) to a multi-layer and continuous reinsurance policy.

The special tax treatment of United Kingdom pensions means that the decision on how to use pension assets is particularly involved. In particular, the ability to take up to 25% of pension assets as a tax-free cash lump sum at retirement, offers retirees opportunities to enhance their pension above that possible through the purchase of a compulsory purchase annuity (“CPA”). The tax-free cash lump sum can be used to buy a tax-efficient purchased life annuity (“PLA”), or in a phased retirement strategy. Income withdrawal can also be used to defer the purchase of an annuity until age 75 and, potentially, to generate a higher income. In this paper I compare the options available to retirees using stochastic modelling. I compare the expected excess pension and expected shortfall, both relative to the alternative risk-free pension available, to assess the various options. I find that if the maximum amount of tax-free cash is available to be used to enhance retirement income, then phased retirement offers the best risk/reward trade off. The advantage is greatest for higher-rate tax payers. As the level of tax-free cash falls, income withdrawal becomes more attractive to those wishing to take greater risks.

This paper obtains an optimal strategy in a finite horizon time for a portfolio of a defined contribution (DC) pension fund for an investor with the CRRA utility function. It employs the optimal stochastic control method in a financial market with two different asset markets, one risk-free and another one risky asset in which its jump follows either by a finite or infinite activity Lévy process. Sensitivity of jump parameters in an uncertainty financial market has been studied.

This paper provides a review of cyber risk research accomplished in different disciplines, with a primary goal to aid researchers in the field of insurance and actuarial science in identifying potential research gaps as well as leveraging useful models and techniques that have been considered in the literature. We highlight the recent advancements in cyber risk prediction, modeling, management, and insurance achieved in different domains including computer engineering, actuarial science, and business studies. The surveyed works are classified according to their respective modeling approaches, allowing readers to more easily compare the technical aspects of the surveyed works and spot out research gaps based on the research tools of their liking. We conclude this paper with a summary of possible research directions that are identified from the review.