Affine mortality models, developed in continuous time, are well suited to longevity risk applications including pricing and capital management. A major advantage of this mortality modeling approach is the availability of closed-form cohort survival curves, consistent with the assumed time dynamics of mortality rates. This article makes new contributions to the estimation of multifactor continuous-time affine models including the canonical Blackburn-Sherris, the arbitrage-free Nelson-Siegel (AFNS), and the Cox-Ingersoll-Ross (CIR) mortality models. We discuss and address numerical issues with model estimation. We apply the estimation methods to age–cohort mortality data from five different countries, providing insights into the dynamics of mortality rates and the fitting performance of the models. We show how the use of maximum likelihood with the univariate Kalman filter turns out to be faster and more robust compared to traditional estimation methods that heavily use large matrix multiplication and inversion. We present graphical and numerical goodness-of-fit results and assess model robustness. We project cohort survival curves and assess the out-of-sample performance of the models for the five countries. We confirm previous results by showing that, across these countries, although the CIR mortality model fits the historical mortality data well, particularly at older ages, the canonical and AFNS affine mortality models provide better out-of-sample performance. We also show how these affine mortality models are robust with respect to the set of age–cohort data used for parameter estimation. R code is provided.

Economic growth has been shown to be an important factor that explains changes in mortality probabilities. Economic growth is commonly measured via the gross national product per capita (GDP), but this article argues that the Consumer Price Index (CPI) is a more natural factor to explain mortality dynamics. It is the CPI that approximates the affordability of health care, food and housing. We augment the well-known Lee-Carter model with the observable CPI factor and test this model using data from the United States, Canada, Australia, and France. We show that the in-sample model fit of our proposed model improves compared with the Lee-Carter model (either augmented with the gross national product factor or not). We also show that the out-of-sample forecasting performance of our proposed model, as measured by the mean squared forecast error, is a considerable improvement. Also, the Lee-Carter model augmented with both the gross national product and CPI factors performs even better in sample and out of sample.

This article investigates a dividend optimization problem with a positive creeping-associated terminal value at ruin for spectrally negative Lévy processes. We consider an insurance company whose surplus process evolves according to a spectrally negative Lévy process with a Gaussian part and a finite Lévy measure. Its objective function relates to dividend payments until ruin and a creeping-associated terminal value at ruin. The positive creeping-associated terminal value represents the salvage value or the creeping reward when creeping happens. Owing to formulas from fluctuation theory, the objective considered is represented explicitly. Under certain restrictions on the terminal value and the surplus process, we show that the threshold strategy should be the optimal one over an admissible class with bounded dividend rates.

In recent years, improvements in all-cause mortality rates and life expectancies for males and females in England and Wales have slowed down. In this article, cause-specific mortality data for England and Wales from 2001 to 2018 are used to investigate the cause-specific contributions to the slowdown in improvements. Cause-specific death counts in England and Wales are modeled using negative binomial regression, and a breakpoint in the linear temporal trend in log mortality rates is investigated. Cause-specific scenarios are generated, where the post-breakpoint temporal trends for certain causes are reverted to pre-breakpoint rates and the effects of these changes on age-standardized mortality rates and period life expectancies are explored. These scenarios are used to quantify cause-specific contributions to the mortality improvement slowdown. Reductions in improvements at older ages in circulatory system diseases, as well as the worsening of mortality rates due to mental and behavioral illnesses and nervous system diseases, provide the greatest contributions to the reduction of improvements in age-standardized mortality rates and period life expectancies. Future period life expectancies scenarios are also generated, where cause-specific mortality rate trends are assumed to either persist or be reverted. In the majority of scenarios, the reversion of cause-specific mortality trends in a single cause of death results in the worsening of period life expectancies at birth and age 65 for both males and females. This work enhances the understanding of cause-specific contributions to the slowdown in all-cause mortality rate improvements from 2001 to 2018, while also providing insights into causes of death that are drivers of life expectancy improvements. The findings are of benefit to researchers, policymakers, and insurance professionals.

This study proposes a new approach to modeling subnational mortality that relies on individual features (e.g., sex, geographical region, socioeconomic status) instead of dealing directly with subpopulations. Our strategy leads to more parsimonious models because fewer parameters are needed to explain mortality. Also, data providers might aggregate data over privacy concerns, and our framework allows for the use of such data, unlike the common subnational mortality modeling approach. A general one-step Bayesian estimation methodology that works well with most age–period–cohort mortality models proposed thus far in the literature is presented; it uses Markov chain Monte Carlo techniques by combining deterministic filtering with adaptive Metropolis steps and is well-suited for high-dimensional cases like the one investigated in this article. In a case study using real data, the framework is applied to Canadian mortality data from three datasets that encompass three features: sex, geographic region, and socioeconomic status. We show that the proposed approach combined with a reasonable mortality model provides realistic, coherent, and plausible mortality projections and that it fits the data reasonably.

The frequency and intensity of weather-related catastrophes, such as storms and floods, have been increasing due to climate change. This leads to rising storm catastrophe risks to the property–casualty insurance and reinsurance sector. In this article, we propose an index-based storm catastrophe (CAT) bond for reinsurers to hedge catastrophe risks related to storm losses. Storm loss data have a large portion of zero values and a continuous positive right-skewed distribution, together with high-dimensional spatial dependence. We address these unique properties by proposing a general two-part autoregressive (2PAR) distribution as the marginal model and a semicontinuous spatiotemporal vine copula as the dependence model. We investigate the CAT bond market equilibrium and endogenously solve for the optimal market price of risk and coupon rates. Our empirical results using historical loss data at the county level in Florida show that the proposed CAT bond can stabilize the reinsurer’s cash flows and create attractive returns to investors by offering high coupon rates. Our framework can be generalized to design and price other catastrophe financing facilities.

By modeling reserves with microlevel models, individual claims information is better preserved and can be more easily handled in the fitting process. Some of the claim information is available immediately at the report date and remains known until the closure of the claim. However, other helpful information changes as claims develop; for example, the previously observed number of payments. In this article, we seek to model payment counts discretely based on past information, in terms of both claim characteristics and previous payment counts. We use a dynamic score that weighs the risk of the claim based on previous claim behavior and that gets updated at the end of each discrete interval. In the model used in this article, we will also distinguish between the different types of payments. We evaluate our model by fitting it into a dataset from a major Canadian insurance company. We will also discuss estimation procedures, make predictions, and compare the results with other models.

Consider an insurer who both sells catastrophe insurance policies and makes risky investments. Suppose that insurance claims arrive according to a Poisson process and the price of the investment portfolio evolves according to a general stochastic process independent of the insurance claims. In the focus of catastrophe risk management are catastrophe insurance losses. For the case of heavy-tailed claims, we derive a simple asymptotic formula for the tail probability of the present value of future claims. The transparent expression of our formula explicitly reflects the different roles of the various underlying risks in driving catastrophic losses. Our work is distinguished from most other works in this strand of research in that we carry out the asymptotic study over the whole class of subexponential distributions. Thus, our work allows both very heavy-tailed distributions such as Pareto-type distributions and moderately heavy-tailed distributions such as Lognormal and Weibull distributions.

This article studies the Bowley solution for a sequential game within the expected utility framework. We assume that the policyholders are expected utility maximizers and there exists a representative policyholder who faces a fixed loss with given probability and no loss otherwise. This policyholder selects the optimal indemnity function in response to the pricing kernel set by the insurer. Knowing the policyholder’s choice of indemnity function, the insurer adjusts the pricing kernel to maximize its expected net profit. This pricing kernel is of our central interest in this article, and in our setting the pricing kernel can be evaluated via the safety loading factor in an expected value premium principle. For a wide class of utility functions, we show that the optimal safety loading factor increases with respect to both the policyholder’s risk aversion level and the probability of zero loss. We also show that the insurance contract corresponding to the Bowley solution is Pareto dominated in the sense that both parties’ interests can be further improved, which shows the inefficiency of the Bowley solution. Some numerical examples are presented to illustrate the main results, and it is shown that both the policyholder and insurer can strictly benefit from the Bowley solution.

The Value at Risk (VaR) and the expected shortfall (ES) are the two most popular risk measures in banking and insurance regulation. To bridge between the two regulatory risk measures, the probability equivalent level of VaR-ES (PELVE) was recently proposed to convert a level of VaR to that of ES. It is straightforward to compute the value of PELVE for a given distribution model. In this article, we study the converse problem of PELVE calibration; that is, to find a distribution model that yields a given PELVE, which may be obtained either from data or from expert opinion. We discuss separately the cases when one-point, two-point, n-point, and curve constraints are given. In the most complicated case of a curve constraint, we convert the calibration problem to that of an advanced differential equation. We apply the model calibration techniques to estimation and simulation for datasets used in insurance. We further study some technical properties of PELVE by offering a few new results on monotonicity and convergence.