{"title":"Bivariate distribution regression with application to insurance data","authors":"Yunyun Wang , Tatsushi Oka , Dan Zhu","doi":"10.1016/j.insmatheco.2023.08.005","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.08.005","url":null,"abstract":"<div><p>Understanding variable dependence, particularly eliciting their statistical properties given a set of covariates, provides the mathematical foundation in practical operations management such as risk analysis and decision-making given observed circumstances. This article presents an estimation method for modeling the conditional joint distribution of bivariate outcomes based on the distribution regression and factorization methods. This method is considered semiparametric in that it allows for flexible modeling of both the marginal and joint distributions conditional on covariates without imposing global parametric assumptions across the entire distribution. In contrast to existing parametric approaches, our method can accommodate discrete, continuous, or mixed variables, and provides a simple yet effective way to capture distributional dependence structures between bivariate outcomes and covariates. Various simulation results confirm that our method can perform similarly or better in finite samples compared to the alternative methods. In an application to the study of a motor third-party liability insurance portfolio, the proposed method effectively estimates risk measures such as the conditional Value-at-Risk and Expected Shortfall. This result suggests that this semiparametric approach can serve as an alternative in insurance risk management.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"113 ","pages":"Pages 215-232"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49851147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jorge M. Bravo , Mercedes Ayuso , Robert Holzmann , Edward Palmer
{"title":"Intergenerational actuarial fairness when longevity increases: Amending the retirement age","authors":"Jorge M. Bravo , Mercedes Ayuso , Robert Holzmann , Edward Palmer","doi":"10.1016/j.insmatheco.2023.08.007","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.08.007","url":null,"abstract":"<div><p>Continuous longevity improvements and population ageing have led countries to modify national public pension schemes by increasing standard and early retirement ages in a discretionary, scheduled, or automatic way, and making it harder for people to retire prematurely. To this end, countries have adopted alternative retirement age strategies, but our analyses show that the measures taken are often poorly designed and consequently misaligned with the pension scheme's ultimate goals. This paper discusses how to implement automatic indexation of the retirement age to life expectancy developments while respecting the principles of intergenerational actuarial fairness and neutrality among generations of the respective policy scheme design. With stable demographic conditions, we show in policy designs in which extended working lives translate into additional pension entitlements, the pension age must be automatically updated to keep the period in retirement constant. Alternatively, policy designs that pursue a fixed replacement rate are consistent with retirement age policies targeting a constant balance between active years in the workforce and years in retirement. Under conditions of population ageing, the statutory pension age will have to increase at a faster rate to meet the intergenerational equity criteria. The empirical strategy employed a Bayesian Model Ensemble approach to stochastic mortality modelling to address model risk and generate forecasts of intergenerationally and actuarially fair pension ages for 23 countries from 2000 to 2050. The findings show that the pension age increases needed to accommodate the effect of longevity developments on pay-as-you-go equilibrium and to reinstate equity between generations are sizeable and well beyond those employed and/or legislated in most countries. A new wave of pension reforms may be at the doorsteps.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"113 ","pages":"Pages 161-184"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49851127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal retirement savings over the life cycle: A deterministic analysis in closed form","authors":"Marcel Fischer , Bjarne Astrup Jensen , Marlene Koch","doi":"10.1016/j.insmatheco.2023.05.010","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.05.010","url":null,"abstract":"<div><p>In this paper, we explore the life cycle consumption-savings problem in a stylized model with a risk-free investment opportunity, a tax-deferred retirement account, and deterministic labor income. Our closed form solutions show that liquidity constraints can be severely binding; in particular in situations with a high growth rate of labor income, in which retirement saving is optimally postponed. With a tax-deferred account, it is always optimal to save in this (illiquid) account first before saving in the (liquid) taxable account in order to satisfy the needs for consumption smoothing. The optimal retirement savings pattern is far from the widespread practice of contributing a fixed fraction of current labor income over the working life to a tax-deferred environment.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"112 ","pages":"Pages 48-58"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49838102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Carole Bernard , Rodrigue Kazzi , Steven Vanduffel
{"title":"Corrigendum and addendum to “Range Value-at-Risk bounds for unimodal distributions under partial information” [Insurance: Math. Econ. 94 (2020) 9–24]","authors":"Carole Bernard , Rodrigue Kazzi , Steven Vanduffel","doi":"10.1016/j.insmatheco.2023.06.004","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.06.004","url":null,"abstract":"<div><p>In Section 2 of <span>Bernard et al. (2020)</span>, we study bounds on Range Value-at-Risk under the assumption of non-negative risk. However, Proposition 3 is erroneous, and hence Theorems 3, 4, and 5 and Corollary 5 are no longer valid. In this corrigendum, we provide a direct replacement of these theorems and corollary. We note that these results provide generalizations in that there is no longer a constraint on the probability level <em>α</em>.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"112 ","pages":"Pages 110-119"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Diversification quotients based on VaR and ES","authors":"Xia Han , Liyuan Lin , Ruodu Wang","doi":"10.1016/j.insmatheco.2023.08.006","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.08.006","url":null,"abstract":"<div><p>The diversification quotient (DQ) is recently introduced for quantifying the degree of diversification of a stochastic portfolio model. It has an axiomatic foundation and can be defined through a parametric class of risk measures. Since the Value-at-Risk (VaR) and the Expected Shortfall (ES) are the most prominent risk measures widely used in both banking and insurance, we investigate DQ constructed from VaR and ES in this paper. In particular, for the popular models of elliptical and multivariate regular varying (MRV) distributions, explicit formulas are available. The portfolio optimization problems for the elliptical and MRV models are also studied. Our results further reveal favorable features of DQ, both theoretically and practically, compared to traditional diversification indices based on a single risk measure.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"113 ","pages":"Pages 185-197"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49851238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Conditional mean risk sharing of losses at occurrence time in the compound Poisson surplus model","authors":"Michel Denuit , Christian Y. Robert","doi":"10.1016/j.insmatheco.2023.05.008","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.05.008","url":null,"abstract":"<div><p>This paper proposes a new risk-sharing procedure, framed into the classical insurance surplus process. Compared to the standard setting where total losses are shared at the end of the period, losses are allocated among participants at their occurrence time in the proposed model. The conditional mean risk-sharing rule proposed by <span>Denuit and Dhaene (2012)</span> is applied to this end. The analysis adopts two different points of views: a collective one for the pool and an individual one for sharing losses and adjusting the amounts of contributions among participants. These two views are compatible under the compound Poisson risk process. Guarantees can also be added by partnering with an insurer.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"112 ","pages":"Pages 23-32"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49838101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Cramér-Lundberg model with a fluctuating number of clients","authors":"Peter Braunsteins , Michel Mandjes","doi":"10.1016/j.insmatheco.2023.05.007","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.05.007","url":null,"abstract":"<div><p>This paper considers the Cramér-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"112 ","pages":"Pages 1-22"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49838183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotics for a time-dependent by-claim model with dependent subexponential claims","authors":"Meng Yuan , Dawei Lu","doi":"10.1016/j.insmatheco.2023.07.001","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.07.001","url":null,"abstract":"<div><p>Consider a by-claim risk model with a constant force of interest, where each main claim may induce a by-claim after a random time. We propose a time-claim-dependent framework, that incorporates dependence between not only the waiting time and the claim but also the main claim and the corresponding by-claim. Based on this framework, we derive some asymptotic estimates for the finite-time ruin probabilities in the case of subexponential claims. We also provide examples and verify the assumptions on dependence. Numerical studies are conducted to examine the performance of these asymptotic formulas.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"112 ","pages":"Pages 120-141"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49838185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal insurance design under mean-variance preference with narrow framing","authors":"Xiaoqing Liang , Wenjun Jiang , Yiying Zhang","doi":"10.1016/j.insmatheco.2023.06.002","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.06.002","url":null,"abstract":"<div><p>In this paper, we study an optimal insurance design problem under mean-variance criterion by considering the local gain-loss utility of the net payoff of insurance, namely, narrow framing. We extend the existing results in the literature to the case where the decision maker has mean-variance preference with a constraint on the expected utility of the net payoff of insurance, where the premium is determined by the mean-variance premium principle. We first show the existence and uniqueness of the optimal solution to the main problem studied in the paper. We find that the optimal indemnity function involves a deductible provided that the safety loading imposed on the “mean part” of the premium principle is strictly positive. Our main result shows that narrow framing indeed reduces the demand for insurance. The explicit optimal indemnity functions are derived under two special local gain-loss utility functions – the quadratic utility function and the piecewise linear utility function. As a spin-off result, the Bowley solution is also derived for a Stackelberg game between the decision maker and the insurer under the quadratic local gain-loss utility function. Several numerical examples are presented to further analyze the effects of narrow framing on the optimal indemnity function as well as the interests of both parties.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"112 ","pages":"Pages 59-79"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49875785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Valuation of general GMWB annuities in a low interest rate environment","authors":"Claudio Fontana, Francesco Rotondi","doi":"10.1016/j.insmatheco.2023.07.003","DOIUrl":"https://doi.org/10.1016/j.insmatheco.2023.07.003","url":null,"abstract":"<div><p>Variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees.</p></div>","PeriodicalId":54974,"journal":{"name":"Insurance Mathematics & Economics","volume":"112 ","pages":"Pages 142-167"},"PeriodicalIF":1.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49838186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}