This manuscript introduces deep learning models that simultaneously describe the dynamics of several yield curves. We aim to learn the dependence structure among the different yield curves induced by the globalization of financial markets and exploit it to produce more accurate forecasts. By combining the self-attention mechanism and nonparametric quantile regression, our model generates both point and interval forecasts of future yields. The architecture is designed to avoid quantile crossing issues affecting multiple quantile regression models. Numerical experiments conducted on two different datasets confirm the effectiveness of our approach. Finally, we explore potential extensions and enhancements by incorporating deep ensemble methods and transfer learning mechanisms.

There is mounting pressure on (re)insurers to quantify the impacts of climate change, notably on the frequency and severity of claims due to weather events such as flooding. This is however a very challenging task for (re)insurers as it requires modeling at the scale of a portfolio and at a high enough spatial resolution to incorporate local climate change effects. In this paper, we introduce a data science approach to climate change risk assessment of pluvial flooding for insurance portfolios over Canada and the United States (US). The underlying flood occurrence model quantifies the financial impacts of short-term (12–48 h) precipitation dynamics over the present (2010–2030) and future climate (2040–2060) by leveraging statistical/machine learning and regional climate models. The flood occurrence model is designed for applications that do not require street-level precision as is often the case for scenario and trend analyses. It is applied at the full scale of Canada and the US over 10–25 km grids. Our analyses show that climate change and urbanization will typically increase losses over Canada and the US, while impacts are strongly heterogeneous from one state or province to another, or even within a territory. Portfolio applications highlight the importance for a (re)insurer to differentiate between future changes in hazard and exposure, as the latter may magnify or attenuate the impacts of climate change on losses.

This article introduces a comprehensive framework that effectively combines experience rating and exposure rating approaches in reinsurance for both short-tail and long-tail businesses. The generic framework applies to all nonlife lines of business and products emphasizing nonproportional treaty business. The approach is based on three pillars that enable a coherent usage of all available information. The first pillar comprises an exposure-based generative model that emulates the generative process leading to the observed claims experience. The second pillar encompasses a standardized reduction procedure that maps each high-dimensional claim object to a few weakly coupled reduced random variables. The third pillar comprises calibrating the generative model with retrospective Bayesian inference. The derived calibration parameters are fed back into the generative model, and the reinsurance contracts covering future cover periods are rated by projecting the calibrated generative model to the cover period and applying the future contract terms.

This paper studies optimal defined-contribution (DC) pension management under stochastic interest rates and expected inflation. In addition to financial risk, we consider the risk of pre-retirement death and introduce life insurance to the pension account as an option to manage this risk. We formulate this pension management problem as a random horizon utility maximization problem and derive its explicit solution under the assumption of constant relative risk aversion utility. We calibrate our model to the U.S. data and demonstrate that the pension member’s demand for life insurance has a hump-shaped pattern with age and a U-shaped pattern with the real interest rate and expected inflation. The optimal pension account balance in our model resembles a variable annuity, wherein the death benefits are endogenously determined and depend on various factors including age, mortality, account balance, future contributions, preferences, and market conditions. Our study suggests that offering variable annuities with more flexible death benefits within the DC account could better cater to the bequest demands of its members.

In this paper, we question the traditional independence assumption between mortality risk and financial risk and model the correlation between these two risks, estimating its impact on the price of different life insurance products. The interest rate and the mortality intensity are modelled as two correlated Hull and White models in an affine set-up. We introduce two building blocks, namely the zero-coupon survival bond and the mortality density, calculate them in closed form and perform an investigation about their dependence on the correlation between mortality and financial risk, both with theoretical results and numerical analysis. We study the impact of correlation also for more structured insurance products, such as pure endowment, annuity, term insurance, whole life insurance and mixed endowment. We show that in some cases, the inclusion of correlation can lead to a severe underestimation or overestimation of the best estimate. Finally, we illustrate that the results obtained using a traditional affine diffusive set-up can be generalised to affine jump diffusion by computing the price of the zero-coupon survival bond in the presence of jumps.

This study conducts an optimal surrender analysis of reverse mortgage (RM) loans offered to elderly homeowners as a financing option. Recent market evidence on borrower early surrenders has raised concerns about the marketability of RM products and their impact on the program viability. In this article, we derive the borrower optimal surrender strategy as a function of the underlying value of the home used as collateral for RM contracts with tenure payment option. Using a probabilistic approach to American option pricing, we present a decomposition result for the value of the contract as the sum of its European counterpart without the surrendering provision and an early exercise premium. The methodology allows policymakers to assess the financial incentive of their policy design, from which we explain the existing market evidence about borrower rational lapse by means of the resulting surrender boundary and reference probabilities.

The decision about when and how much to annuitize is an important element of the retirement planning of most individuals. Optimal annuitization strategies depend on the individual’s exposure to annuity risk, meaning the possibility of meeting unfavorable personal and market conditions at the time of the annuitization decision. This article studies optimal annuitization strategies within a life-cycle consumption and portfolio choice model, focusing on stochastic interest rates as an important source of annuity risk. Closing a gap in the existing literature, our numerical results across different model variants reveal several typical structural effects of interest rate risk on the annuitization decision, which may however vary depending on preference specifications and alternative investment opportunities: When allowing for gradual annuitization, annuity risk is temporally diversified by spreading annuity purchases over the whole pre-retirement period, with annuity market participation starting earlier in the life cycle and becoming more extensive with increasing interest rate risk. Ruling out this temporal diversification possibility, as embedded in many institutional settings, incurs significant welfare losses, which are increasing with higher interest rate risk, together with larger overall demand for annuitization.

In the traditional multidimensional credibility models developed by Jewell ((1973) Operations Research Center, pp. 73–77.), the estimation of the hypothetical mean vector involves complex matrix manipulations, which can be challenging to implement in practice. Additionally, the estimation of hyperparameters becomes even more difficult in high-dimensional risk variable scenarios. To address these issues, this paper proposes a new multidimensional credibility model based on the conditional joint distribution function for predicting future premiums. First, we develop an estimator of the joint distribution function of a vector of claims using linear combinations of indicator functions based on past observations. By minimizing the integral of the expected quadratic distance function between the proposed estimator and the true joint distribution function, we obtain the optimal linear Bayesian estimator of the joint distribution function. Using the plug-in method, we obtain an explicit formula for the multidimensional credibility estimator of the hypothetical mean vector. In contrast to the traditional multidimensional credibility approach, our newly proposed estimator does not involve a matrix as the credibility factor, but rather a scalar. This scalar is composed of both population information and sample information, and it still maintains the essential property of increasingness with respect to the sample size. Furthermore, the new estimator based on the joint distribution function can be naturally extended and applied to estimate the process covariance matrix and risk premiums under various premium principles. We further illustrate the performance of the new estimator by comparing it with the traditional multidimensional credibility model using bivariate exponential-gamma and multivariate normal distributions. Finally, we present two real examples to demonstrate the findings of our study.

Modeling multivariate dependence in high dimensions is challenging, with popular solutions constructing multivariate copula as a composition of lower dimensional copulas. Pair-copula constructions do so by using bivariate linking copulas, but their parametrization, in size, being quadratic in the dimension, is not quite parsimonious. Besides, the number of regular vines grows super-exponentially with the dimension. One parsimonious solution is factor copulas, and in particular, the one-factor copula is touted for its simplicity – with the number of parameters linear in the dimension – while being able to cater to asymmetric non-linear dependence in the tails. In this paper, we add nuance to this claim from the point of view of a popular measure of multivariate tail dependence, the tail dependence matrix (TDM). We focus on the one-factor copula model with the linking copula belonging to the BB1 family, pointing out later the applicability of our results to a wider class of linking copulas. For this model, we derive tail dependence coefficients and study their basic properties as functions of the parameters of the linking copulas. Based on this, we study the representativeness of the class of TDMs supported by this model with respect to the class of all possible TDMs. We establish that since the parametrization is linear in the dimension, it is no surprise that the relative volume is zero for dimensions greater than three, and hence, by necessity, we present a novel manner of evaluating the representativeness that has a combinatorial flavor. We formulate the problem of finding the best representative one-factor BB1 model given a target TDM and suggest an implementation along with a simulation study of its performance across dimensions. Finally, we illustrate the results of the paper by modeling rainfall data, which is relevant in the context of weather-related insurance.

Consider a company whose business carries the potential for investment losses and is additionally vulnerable to exogenous shocks. The unpredictability of the shocks makes it challenging for both the company and the regulator to accurately assess their impact, potentially leading to an underestimation of solvency capital when employing traditional approaches. In this paper, we utilize a stylized model to conduct an extreme value analysis of the tail risk of the company under a Fréchet-type and a Gumbel-type shock. Our main results explicitly demonstrate the different roles of investment risk and shock risk in driving large losses. Furthermore, we derive asymptotic estimates for the value at risk and expected shortfall of the total loss. Numerical studies are conducted to examine the accuracy of the obtained estimates.

Reinsurers may default when they have to pay large claims to insurers but are unable to fulfill their obligations due to various reasons such as catastrophic events, underwriting losses, inadequate capitalization, or financial mismanagement. This paper studies the problem of optimal reinsurance design from the perspectives of both the insurer and reinsurer when the insurer faces the potential default risk of the reinsurer. If the insurer aims to minimize the convex distortion risk measure of his retained loss, we prove the optimality of a stop-loss treaty when the promised ceded loss function is charged by the expected value premium principle and the reinsurer offers partial recovery in the event of default. For any fixed premium loading set by the reinsurer, we then derive the explicit expressions of optimal deductible levels for three special distortion functions, including the TVaR, Gini, and PH transform distortion functions. Under these three explicit distortion risk measures adopted by the insurer, we seek the optimal safety loading for the reinsurer by maximizing her net profit where the reserve capital is determined by the TVaR measure and the cost is governed by the expectation. This procedure ultimately leads to the Bowley solution between the insurer and the reinsurer. We provide several numerical examples to illustrate the theoretical findings. Sensitivity analyses demonstrate how different settings of default probability, recovery rate, and safety loading affect the optimal deductible values. Simulation studies are also implemented to analyze the effects induced by the default probability and recovery rate on the Bowley solution.

This paper proposes a theoretical insurance model to explain well-documented loss underreporting and to study how strategic underreporting affects insurance demand. We consider a utility-maximizing insured who purchases a deductible insurance contract and follows a barrier strategy to decide whether she should report a loss. The insurer adopts a bonus-malus system with two rate classes, and the insured will move to or stay in the more expensive class if she reports a loss. First, we fix the insurance contract (deductibles) and obtain the equilibrium reporting strategy in semi-closed form. A key result is that the equilibrium barriers in both rate classes are strictly greater than the corresponding deductibles, provided that the insured economically prefers the less expensive rate class, thereby offering a theoretical explanation to underreporting. Second, we study an optimal deductible insurance problem in which the insured strategically underreports losses to maximize her utility. We find that the equilibrium deductibles are strictly positive, suggesting that full insurance, often assumed in related literature, is not optimal. Moreover, in equilibrium, the insured underreports a positive amount of her loss. Finally, we examine how underreporting affects the insurer’s expected profit.

The theory of utility is a well-known method of constructing insurance premiums (see e.g., Newton et al. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.). Furman and Zitikis ((2008) Insurance: Mathematics and Economics, 42, 459–465.) proposed an alternative method using the mean value of a weighted random variable. According to this approach, for various choices of weighting, popular premiums such as net premium, modified variance premium, Esscher premium, and Kamps premium are obtained. On the other hand, some premiums cannot be obtained with this method, such as the premium of the exponential principle. In this paper, we provide a complementary theory by introducing a family of unimodal weighted distributions for which the mode is a premium principle.