This paper studies a hierarchical risk model where an accident can cause a combination of different types of claims, whose sizes could be dependent. In addition, the frequencies of accidents that cause the different combinations of claims are dependent. We first derive formulas for computing risk measures, such as the Tail Conditional Expectation and Tail Variance of the aggregate losses for a portfolio of businesses. Then, we present formulas for performing the associated capital allocation to different types of claims in the portfolio. The main tool we used is the moment (or size-biased) transform of the multivariate distributions.

This paper presents the AffineMortality R package which performs parameter estimation, goodness-of-fit analysis, simulation, and projection of future mortality rates for a set of affine mortality models for use in pricing and reserving. The computational routines build on the univariate Kalman Filtering approach of Koopman and Durbin ((2000). Journal of Time Series Analysis, 21(3), 281–296.) along other numerical methods to enhance the robustness of the results. This paper provides a discussion of how the package works in order to effectively estimate and project survival curves, and describes the available functions. Illustration of the package for mortality analysis of the US male data set is provided.

This study investigates the benefits and drawbacks of pension plan consolidation by quantifying the impact of mergers of heterogeneous plans on different stakeholders in a unique Canadian implementation of defined benefit plans. Using a comprehensive framework that combines a realistic economic scenario generator, a stochastic mortality model that captures differences among subpopulations, a cost model with economies of scale, and a dynamic asset allocation methodology, we evaluate the combined effect of asset- and liability-side changes on three groups of measures: plan-related risk measures assessing profits from an economic capital perspective, consumption-based metrics to understand the impact on members, and contribution risk measures capturing the risk from the employer’s viewpoint. We apply the framework to a hypothetical and empirically relevant merger and find that consolidation is favorable under most circumstances: the positive impacts of better diversification and economies of scale continue to outweigh the negative effects of heterogeneity even when the merging plans have different mortality expectations, different maturity levels, or modest differences in initial funded ratios.

Risk measurement and econometrics are the two pillars of actuarial science. Unlike econometrics, risk measurement allows taking into account decision-makers’ risk aversion when analyzing the risks. We propose a hybrid model that captures decision-makers’ regression-based approach to study risks, focusing on explanatory variables while paying attention to risk severity. Our model considers different loss functions that quantify the severity of the losses that are provided by the risk manager or the actuary. We present an explicit formula for the regression estimators for the proposed risk-based regression problem and study the proposed results. Finally, we provide a numerical study of the results using data from the insurance industry.

The extended $3/2$ short rate model is a mean reverting model of the short rate which, for suitably chosen parameters, permits a sensible term structure of bond yields and closed-form valuation formulae of zero-coupon bonds and options. This article supplies proofs of the formulae for the expected present values of future cash flows under the real-world probability measure, known as actuarial valuation. Finally, we give formulae for asymptotic levels of bond yields and formulae for bond option prices for the extended $3/2$ model, under particular conditions on its parameters.

We design the insurance contract when the insurer faces arson-type risks. We show that the optimal contract must be manipulation-proof. Therefore, it is continuous, has a bounded slope, and satisfies the no-sabotage condition when arson-type actions are free. Any contract that mixes a deductible, coinsurance, and an upper limit is manipulation-proof. A key feature of our models is that we provide a simple, general, and entirely elementary proof of manipulation-proofness that is easily adapted to different settings. We also show that the ability to perform arson-type actions reduces the insured’s welfare as less coverage is offered in equilibrium.

Generalized Poisson (GP) distribution was introduced in Consul & Jain ((1973). Technometrics, 15(4), 791–799.). Since then it has found various applications in actuarial science and other areas. In this paper, we focus on the distributional properties of GP and its related distributions. In particular, we study the distributional properties of distributions in the $\mathcal{H}$ family, which includes GP and generalized negative binomial distributions as special cases. We demonstrate that the moment and size-biased transformations of distributions within the $\mathcal{H}$ family remain in the same family, which significantly extends the results presented in Ambagaspitiya & Balakrishnan ((1994). ASTINBulletin: the Journal of the IAA, 24(2), 255–263.) and Ambagaspitiya ((1995). Insurance Mathematics and Economics, 2(16), 107–127.). Such findings enable us to provide recursive formulas for evaluating risk measures, such as Value-at-Risk and conditional tail expectation of the compound GP distributions. In addition, we show that the risk measures can be calculated by making use of transform methods, such as fast Fourier transform. In fact, the transformation method showed a remarkable time advantage over the recursive method. We numerically compare the risk measures of the compound sums when the primary distributions are Poisson and GP. The results illustrate the model risk for the loss frequency distribution.

We investigate bottom-up risk aggregation applied by insurance companies facing reserve risk from multiple lines of business. Since risk capitals should be calculated in different time horizons and calendar years, depending on the regulatory or reporting regime (Solvency II vs IFRS 17), we study correlations of ultimate losses and correlations of one-year losses in future calendar years in lines of business. We consider a multivariate version of a Hertig’s lognormal model and we derive analytical formulas for the ultimate correlation and the one-year correlations in future calendar years. Our main conclusion is that the correlation coefficients that should be used in a bottom-up aggregation formula depend on the time horizon and the future calendar year where the risk emerges. We investigate analytically and numerically properties of the ultimate and the one-year correlations, their possible values observed in practice, and the impact of misspecified correlations on the diversified risk capital.