We present a methodology to forecast mortality rates and estimate longevity and mortality risks. The methodology uses generalized dynamic factor models fitted to the differences in the log-mortality rates. We compare their prediction performance with that of models previously described in the literature, including the traditional static factor model fitted to log-mortality rates. We also construct risk measures using vine-copula simulations, which take into account the dependence between the idiosyncratic components of the mortality rates. The methodology is applied to forecast mortality rates for a population portfolio for the UK and to estimate longevity and mortality risks.

The Erlang mixture model has been widely used in modeling insurance losses due to its desirable distributional properties. In this paper, we consider the problem of efficient estimation of the Erlang mixture model. We present a new thresholding penalty function and a corresponding EM algorithm to estimate model parameters and to determine the order of the mixture. Using simulation studies and a real data application, we demonstrate the efficiency of the EM algorithm.

Solvency regulations require financial institutions to hold initial capital so that ruin is a rare event. An important practical problem is to estimate the regulatory capital so the ruin probability is at the regulatory level, typically with less than 0.1% over a finite-time horizon. Estimating probabilities of rare events is challenging, since naive estimations via direct simulations of the surplus process is not feasible. In this paper, we present a stratified sampling algorithm for estimating finite-time ruin probabilities. We further introduce a sequence of measure changes to remove the pathwise discontinuities of the estimator, and compute unbiased first and second-order derivative estimates of the finite-time ruin probabilities with respect to both distributional and structural parameters. We then estimate the regulatory capital and its sensitivities. These estimates provide information to insurance companies for meeting prudential regulations as well as designing risk management strategies. Numerical examples are presented for the classical model, the Sparre Andersen model with interest and the periodic risk model with interest to demonstrate the speed and efficacy of our methodology.

The quantification of operational risk has to deal with various concerns regarding data, much more than other types of risk which banks and insurers are obliged to manage. One of the main questions that worries both researchers and practitioners is the bias in the data on the operational losses amounts recorded. We support the assertions made by several authors and defend that this concern is serious when modeling operational losses data and, typically, is presented in all the databases. We show that it's possible, based on mild assumptions on the internal procedures put in place to manage operational losses, to make parametric inference using loss data statistics, that is, to estimate the parameters for the losses amounts, taking in consideration the bias that, not being considered, generates a two fold error in the estimators for the mean loss amount and the total loss amount, the former being overvalued and the last undervalued. In this paper, we do not consider the existence of a threshold for which, all losses above, are reported and available for analysis and estimation procedures. In this sense, we follow a different approach to the parametric inference. Here, we consider that the probability that a loss is reported and ends up recorded for analysis, increases with the size of the loss, what causes the bias in the database but, at the same time, we do not consider the existence of a threshold, above which, all losses are recorded. Hence, no loss has probability one of being recorded, in what we defend is a realist framework. We deduce the general formulae, present simulations for common theoretical distributions used to model (operational reported) losses amounts, estimate the impact for not considering the bias factor when estimating the value at risk and estimate the true total operational losses the bank incurred.

A simple technique for continuity estimation for ruin probability in the compound Poisson risk model is proposed. The approach is based on the contractive properties of operators involved in the integral equations for the ruin probabilities. The corresponding continuity inequalities are expressed in terms of the Kantorovich and weighted Kantorovich distances between distribution functions of claims. Both general and light-tailed distributions are considered.

Gender and age are the top two risk factors considered in pricing life insurance products. Although it is believed that mortality rates are also related to other factors (e.g. smoking, overweight, and especially marriage), data availability and marketing often limit the possibility of including them. Many studies have shown that married people (particularly men) benefit from the marriage, and generally have lower mortality rates than unmarried people. However, most of these studies used data from a population sample; their results might not apply to the whole population. In this study, we explore if mortality rates differ by marital status using mortality data (1975–2011) from the Taiwan Ministry of the Interior. In order to deal with the problem of small sample sizes in some marital status groups, we use graduation methods to reduce fluctuations in mortality rates. We also use a relational approach to model mortality rates by marital status, and then compare the proposed model with some popular stochastic mortality models. Based on computer simulation, we find that the proposed smoothing methods can reduce fluctuations in mortality estimates between ages, and the relational mortality model has smaller errors in predicting mortality rates by marital status. Analyses of the mortality data from Taiwan show that mortality rates differ significantly by marital status. In some age groups, the differences in mortality rates are larger between marital status groups than between smokers and non-smokers. For the issue of practical consideration, we suggest modifications to include marital status in pricing of life insurance products.

Optimal reinsurance from an insurer's point of view or from a reinsurer's point of view has been studied extensively in the literature. However, as two parties of a reinsurance contract, an insurer and a reinsurer have conflicting interests. An optimal form of reinsurance from one party's point of view may be not acceptable to the other party. In this paper, we study optimal reinsurance designs from the perspectives of both an insurer and a reinsurer and take into account both an insurer's aims and a reinsurer's goals in reinsurance contract designs. We develop optimal reinsurance contracts that minimize the convex combination of the Value-at-Risk (VaR) risk measures of the insurer's loss and the reinsurer's loss under two types of constraints, respectively. The constraints describe the interests of both the insurer and the reinsurer. With the first type of constraints, the insurer and the reinsurer each have their limit on the VaR of their own loss. With the second type of constraints, the insurer has a limit on the VaR of his loss while the reinsurer has a target on his profit from selling a reinsurance contract. For both types of constraints, we derive the optimal reinsurance forms in a wide class of reinsurance policies and under the expected value reinsurance premium principle. These optimal reinsurance forms are more complicated than the optimal reinsurance contracts from the perspective of one party only. The proposed models can also be reduced to the problems of minimizing the VaR of one party's loss under the constraints on the interests of both the insurer and the reinsurer.

Optimal reinsurance indemnities have widely been studied in the literature, yet the bargaining for optimal prices has remained relatively unexplored. Therefore, the key objective of this paper is to analyze the price of reinsurance contracts. We use a novel way to model the bargaining powers of the insurer and reinsurer, which allows us to generalize the contracts according to the Nash bargaining solution, indifference pricing and the equilibrium contracts. We illustrate these pricing functions by means of inverse-S shaped distortion functions for the insurer and the Value-at-Risk for the reinsurer.