There has been a growing interest among pension plan sponsors in envisioning how the mortality experience of their active and deferred members may turn out to be if a pandemic similar to the COVID-19 occurs in the future. To address their needs, we propose in this paper a stochastic model for simulating future mortality scenarios with COVID-alike effects. The proposed model encompasses three parameter levels. The first level includes parameters that capture the long-term pattern of mortality, whereas the second level contains parameters that gauge the excess age-specific mortality due to COVID-19. Parameters in the first and second levels are estimated using penalised quasi-likelihood maximisation method which was proposed for generalised linear mixed models. Finally, the third level includes parameters that draw on expert opinions concerning, for example, how likely a COVID-alike pandemic will occur in the future. We illustrate our proposed model with data from the United States and a range of expert opinions.

The COVID-19 pandemic interrupts the relatively steady trend of improving longevity observed in many countries over the last decades. We claim that this needs to be addressed explicitly in many mortality modelling applications, for example, in the life insurance industry. To support this position, we provide a descriptive analysis of the mortality development of several countries up to and including the year 2020. Furthermore, we perform an empirical and theoretical investigation of the impact a mortality jump has on the parameters, forecasts and implied present values of the popular Lee–Carter mortality model. We find that COVID-19 has resulted in substantial mortality shocks in many countries. We show that such shocks have a large impact on point and interval forecasts of death rates and, consequently, on the valuation of mortality-related insurance products. We obtain similar findings under the Cairns–Blake–Dowd mortality model, which demonstrates that the effects caused by COVID-19 show up in a variety of models. Finally, we provide an overview of approaches to handle extreme mortality events such as the COVID-19 pandemic in mortality modelling.

Estimating tail risk measures for portfolios of complex variable annuities is an important enterprise risk management task which usually requires nested simulation. In the nested simulation, the outer simulation stage involves projecting scenarios of key risk factors under the real-world measure, while the inner simulations are used to value pay-offs under guarantees of varying complexity, under a risk-neutral measure. In this paper, we propose and analyse an efficient simulation approach that dynamically allocates the inner simulations to the specific outer scenarios that are most likely to generate larger losses. These scenarios are identified using a proxy calculation that is used only to rank the outer scenarios, not to estimate the tail risk measure directly. As the proxy ranking will not generally provide a perfect match to the true ranking of outer scenarios, we calculate a measure based on the concomitant of order statistics to test whether further tail scenarios are required to ensure, with given confidence, that the true tail scenarios are captured. This procedure, which we call the dynamic importance allocated nested simulation approach, automatically adjusts for the relationship between the proxy calculations and the true valuations and also signals when the proxy is not sufficiently accurate.

We examine the impact of asset allocation and contribution rates on the risk of defined benefit (DB) pension schemes, using both a run-off and a shorter 3-year time horizon. Using the 3-year horizon, which is typically preferred by regulators, a high bond allocation reduces the spread of the distribution of surplus. However, this result is reversed when examined on a run-off basis. Furthermore, under both the 3-year horizon and the run-off, the higher bond allocation reduces the median level of surplus. Pressure on the affordability of DB schemes has led to widespread implementation of the so-called de-risking strategies, such as moving away from predominantly equity investments to greater bond investments. If the incentives produced by shorter term risk assessments are contributing to this shift, they might be harming the long-term financial health of the schemes. Contribution rates have relatively lower impact on the risk.

Understanding core statistical properties and data features in mortality data are fundamental to the development of machine learning methods for demographic and actuarial applications of mortality projection. The study of statistical features in such data forms the basis for classification, regression and forecasting tasks. In particular, the understanding of key statistical structure in such data can aid in improving accuracy in undertaking mortality projection and forecasting when constructing life tables. The ability to accurately forecast mortality is a critical aspect for the study of demography, life insurance product design and pricing, pension planning and insurance-based decision risk management. Though many stylised facts of mortality data have been discussed in the literature, we provide evidence for a novel statistical feature that is pervasive in mortality data at a national level that is as yet unexplored. In this regard, we demonstrate in this work a strong evidence for the existence of long memory features in mortality data, and second that such long memory structures display multifractality as a statistical feature that can act as a discriminator of mortality dynamics by age, gender and country. To achieve this, we first outline the way in which we choose to represent the persistence of long memory from an estimator perspective. We make a natural link between a class of long memory features and an attribute of stochastic processes based on fractional Brownian motion. This allows us to use well established estimators for the Hurst exponent to then robustly and accurately study the long memory features of mortality data. We then introduce to mortality analysis the notion from data science known as multifractality. This allows us to study the long memory persistence features of mortality data on different timescales. We demonstrate its accuracy for sample sizes commensurate with national-level age term structure historical mortality records. A series of synthetic studies as well a comprehensive analysis of real mortality death count data are studied in order to demonstrate the pervasiveness of long memory structures in mortality data, both mono-fractal and multifractal functional features are verified to be present as stylised facts of national-level mortality data for most countries and most age groups by gender. We conclude by demonstrating how such features can be used in kernel clustering and mortality model forecasting to improve these actuarial applications.

Raw mortality data often exhibit irregular patterns due to randomness. Graduation refers to the act of smoothing crude mortality rates. In this paper, we propose a flexible and robust methodology for graduating mortality rates using adaptive P-splines. Since the observed data at high ages are often sparse and unreliable, we use an exponentially increasing penalty. We use mortality data of England and Wales and model male and female mortality rates jointly by means of penalties, achieving borrowing of information between the two sexes.

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.

In most industrialised countries, one of the major societal challenges is the demographic change coming along with the ageing of the population. The increasing life expectancy observed over the last decades underlines the importance to find ways to appropriately cover the financial needs of the elderly. A particular issue arises in the area of health, where sufficient care must be provided to a growing number of dependent elderly in need of long-term care (LTC) services. In many markets, the offering of life insurance products incorporating care options and LTC insurance products is generally scarce. In our research, we therefore examine a life annuity product with an embedded care option potentially providing additional financial support to dependent persons. To evaluate the care option, we determine the minimum price that the annuity provider requires and the policyholder’s willingness to pay for the care option. For the latter, we employ individual utility functions taking account of the policyholder’s condition. We base our numerical study on recently developed transition probability data from Switzerland. Our findings give new and realistic insights into the nature and the utility of life annuity products proposing an embedded care option for tackling the financing of LTC needs.

Initial insurance losses are often reported with a textual description of the claim. The claims manager must determine the adequate case reserve for each known claim. In this paper, we present a framework for predicting the amount of loss given a textual description of the claim using a large number of words found in the descriptions. Prior work has focused on classifying insurance claims based on keywords selected by a human expert, whereas in this paper the focus is on loss amount prediction with automatic word selection. In order to transform words into numeric vectors, we use word cosine similarities and word embedding matrices. When we consider all unique words found in the training dataset and impose a generalised additive model to the resulting explanatory variables, the resulting design matrix is high dimensional. For this reason, we use a group lasso penalty to reduce the number of coefficients in the model. The scalable, analytical framework proposed provides for a parsimonious and interpretable model. Finally, we discuss the implications of the analysis, including how the framework may be used by an insurance company and how the interpretation of the covariates can lead to significant policy change. The code can be found in the TAGAM R package (github.com/scottmanski/TAGAM).

A new method of forecasting mortality is introduced. The method is based on the continuous-time dynamics of the Lexis diagram, which given weak assumptions implies that the death count data are Poisson distributed. The underlying mortality rates are modelled with a hidden Markov model (HMM) which enables a fully likelihood-based inference. Likelihood inference is done by particle filter methods, which avoids approximating assumptions and also suggests natural model validation measures. The proposed model class contains as special cases many previous models with the important difference that the HMM methods make it possible to estimate the model efficiently. Another difference is that the population and latent variable variability can be explicitly modelled and estimated. Numerical examples show that the model performs well and that inefficient estimation methods can severely affect forecasts.

If the general level of house prices falls a long way, policymakers may introduce new policies which seek to support prices. This paper considers the effect of such interventions on the valuation of no-negative-equity guarantees (NNEG) in equity release mortgages. I model interventions by a reflecting barrier expressed as a fraction of the current level of house prices. Reflection at the barrier is instantaneous, so the no-arbitrage property is preserved, and hence risk-neutral valuation of NNEG is possible. The reflecting barrier can alternatively be justified as a representation of the different economic nature of the underlying housing (and particularly freehold land) assets in NNEG valuations, compared with the underlying equity assets in many other option valuations.

We generalise Quasi-Linear Means by restricting to the tail of the risk distribution and show that this can be a useful quantity in risk management since it comprises in its general form the Value at Risk, the Conditional Tail Expectation and the Entropic Risk Measure in a unified way. We then investigate the fundamental properties of the proposed measure and show its unique features and implications in the risk measurement process. Furthermore, we derive formulas for truncated elliptical models of losses and provide formulas for selected members of such models.

This paper studies estimation of the conditional mean squared error of prediction, conditional on what is known at the time of prediction. The particular problem considered is the assessment of actuarial reserving methods given data in the form of run-off triangles (trapezoids), where the use of prediction assessment based on out-of-sample performance is not an option. The prediction assessment principle advocated here can be viewed as a generalisation of Akaike’s final prediction error. A direct application of this simple principle in the setting of a data-generating process given in terms of a sequence of general linear models yields an estimator of the conditional mean squared error of prediction that can be computed explicitly for a wide range of models within this model class. Mack’s distribution-free chain ladder model and the corresponding estimator of the prediction error for the ultimate claim amount are shown to be a special case. It is demonstrated that the prediction assessment principle easily applies to quite different data-generating processes and results in estimators that have been studied in the literature.

Traditionally, actuaries make their predictions based on simple, robust methods. Stochastic models become increasingly popular because they can enrich the point estimates with error estimates or even provide the whole probability distribution. Here, we construct such a model for German inpatient health expenses per age using the functional data approach. This allows us to see in which age groups the expenses change the most and where predictions are most uncertain. Jumps in the derived model parameters indicate that 3 years might be outliers. In fact, they can be explained by changes in the reimbursement system and must be dealt with. As an application, we compute the probability distribution of the total health expenses in the upcoming years.