Deep neural networks have become an important tool for use in actuarial tasks, due to the significant gains in accuracy provided by these techniques compared to traditional methods, but also due to the close connection of these models to the generalized linear models (GLMs) currently used in industry. Although constraining GLM parameters relating to insurance risk factors to be smooth or exhibit monotonicity is trivial, methods to incorporate such constraints into deep neural networks have not yet been developed. This is a barrier for the adoption of neural networks in insurance practice since actuaries often impose these constraints for commercial or statistical reasons. In this work, we present a novel method for enforcing constraints within deep neural network models, and we show how these models can be trained. Moreover, we provide example applications using real-world datasets. We call our proposed method ICEnet to emphasize the close link of our proposal to the individual conditional expectation model interpretability technique.

This work presents a $\textsf{Python}$ EMD package named AdvEMDpy that is both more flexible and generalises existing empirical mode decomposition (EMD) packages in $\textsf{Python}$, $\textsf{R}$, and $\textsf{MATLAB}$. It is aimed specifically for use by the insurance and financial risk communities, for applications such as return modelling, claims modelling, and life insurance applications with a particular focus on mortality modelling. AdvEMDpy both expands upon the EMD options and methods available, and improves their statistical robustness and efficiency, providing a robust, usable, and reliable toolbox. Unlike many EMD packages, AdvEMDpy allows customisation by the user, to ensure that a broader class of linear, non-linear, and non-stationary time series analyses can be performed. The intrinsic mode functions (IMFs) extracted using EMD contain complex multi-frequency structures which warrant maximum algorithmic customisation for effective analysis. A major contribution of this package is the intensive treatment of the EMD edge effect which is the most ubiquitous problem in EMD and time series analysis. Various EMD techniques, of varying intricacy from numerous works, have been developed, refined, and, for the first time, compiled in AdvEMDpy. In addition to the EMD edge effect, numerous pre-processing, post-processing, detrended fluctuation analysis (localised trend estimation) techniques, stopping criteria, spline methods, discrete-time Hilbert transforms (DTHT), knot point optimisations, and other algorithmic variations have been incorporated and presented to the users of AdvEMDpy. This paper and the supplementary materials provide several real-world actuarial applications of this package for the user’s benefit.

Recently, there has been an increasing interest from life insurers to assess their portfolios' mortality risks. The new European prudential regulation, namely Solvency II, emphasized the need to use mortality and life tables that best capture and reflect the experienced mortality, and thus policyholders' actual risk profiles, in order to adequately quantify the underlying risk. Therefore, building a mortality table based on the experience of the portfolio is highly recommended and, for this purpose, various approaches have been introduced into actuarial literature. Although such approaches succeed in capturing the main features, it remains difficult to assess the mortality when the underlying portfolio lacks sufficient exposure. In this paper, we propose graduating the mortality curve using an adaptive procedure based on the local likelihood. The latter has the ability to model the mortality patterns even in presence of complex structures and avoids relying on expert opinions. However, such a technique fails to offer a consistent yet regular structure for portfolios with limited deaths. Although the technique borrows the information from the adjacent ages, it is sometimes not sufficient to produce a robust life table. In the presence of such a bias, we propose adjusting the corresponding curve, at the age level, based on a credibility approach. This consists in reviewing the assumption of the mortality curve as new observations arrive. We derive the updating procedure and investigate its benefits of using the latter instead of a sole graduation based on real datasets. Moreover, we look at the divergences in the mortality forecasts generated by the classic credibility approaches including Hardy–Panjer, the Poisson–Gamma model and the Makeham framework on portfolios originating from various French insurance companies.

Farm Service Agency (FSA) direct loans are intended to provide transitory credit to creditworthy borrowers unable to obtain conventional credit at reasonable terms. Farm loan program (FLP) effectiveness is measured in part by how readily direct loan borrowers graduate to conventional credit. A survey of FSA borrowers originating direct loans during fiscal years 1994-1996 is used to estimate graduation rates. A majority of 1994-1996 loan originators did exit the direct FLP by November 2004. A multinomial logit model indicates financial strength at origination resulted in greater likelihood of farming without direct loans approximately 9 years after loan origination.

This is the first of two papers in which we estimate transition probabilities amongst levels of disability as defined in the Australian Survey of Disability, Ageing and Carers. In this paper we describe both the main tools of our estimation and the estimation of the numbers of individuals in different disability categories at annual intervals using survey data that are available at five year intervals. In Paper II we describe our estimation procedure, followed by its implementation, discussion of results and graduation of the estimated transition probabilities.

This is the second of two papers in which we estimate transition probabilities amongst levels of disability as defined in the Australian Survey of Disability, Ageing and Carers. In this paper we describe our estimation procedure, followed by its implementation, discussion of results and graduation of the estimated transition probabilities.

This paper presents a new method of graduation which uses parametric formulae together with Bayesian reversible jump Markov chain Monte Carlo methods. The aim is to provide a method which can be applied to a wide range of data, and which does not require a lot of adjustment or modification. The method also does not require one particular parametric formula to be selected: instead, the graduated values are a weighted average of the values from a range of formulae. In this way, the new method can be seen as an automatic graduation method which we believe can be applied in many cases without any adjustments and provide satisfactory graduated values. An advantage of a Bayesian approach is that it allows for model uncertainty unlike standard methods of graduation.

The provision of graduated mortality rates, for the United Kingdom pensioners' experience, based on the so-called ‘amounts’ data sets is addressed. Specifically a methodology is investigated, building on the existing methods practiced by the CMI Bureau, which takes a more detailed account of the underlying structure of the data involved. The method is applied to the U.K. pensioners' experience and recent mortality trends in this experience revealed.

Our research deals with the way that calendar time affects mortality patterns in the Spanish population, and how this information can be used to elaborate predictions. A description of the observed mortality evolution has been worked out using data from 1975 to 1993. We have used Heligman-Pollard Law number two to model the evolution of Spanish mortality over the period and using univariate time series analysis, we have obtained a prognosis for years 1994 to 2010.

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.

We investigate the implications of a dual approach to the graduation of the force of mortality based on the modelling of the exposures as gamma random variables, as opposed to the modelling of the numbers of deaths as Poisson random variables.

This paper considers cross-validation as an objective and risk-based method for selecting the smoothing parameter in a non-parametric graduation. In addition, the relative merits of two kernel estimators are compared in the context of mortality graduation. Finally, it is well known in the statistical literature that the use of theoretically superior kernels is not as important as the choice of bandwidth. Our results support this conclusion, suggesting that the focus on such weights is misguided in the actuarial textbooks on moving weighted averages.

This paper extends the theory of graduation by parametric formulae to include dynamic estimation methods. This is an application of the Kalman filter and allows the parameters of the curve fitted to vary with age. The amount of variation is determined by the amount of smoothing required, and the method can be regarded as a combination of curve fitting and sequential smoothing, each of which has been used separately for performing graduations. In practice, a dynamic straight line can always be used for the graduation and the method has a sensible logical interpretation.

The consequences of graduating truncated English Life Table data by either the Gompertz or truncated Heligman and Pollard ‘laws of mortality’, recently advocated by Thatcher (1990), are examined in greater detail.

In this paper it is demonstrated how recently developed statistical techniques designed to facilitate the joint modelling of the mean and dispersion are well suited to model the presence of duplicate policies in graduation.

It is demonstrated how existing actuarial graduation practice, used in the construction of life tables, can be extended to considerable effect by formulating the techniques within the generalised linear and non-linear modelling framework.

In this paper we present an approach to the graduation of select mortality data and we illustrate this approach by graduating the CMIB's data for Female Permanent Assurances 1979–1982. The difference between our approach and that of the CMIB is that we graduate simultaneously by attained age and duration since selection.