This paper studies optimal investment and benefit payment strategies for a target benefit pension plan under loss aversion. The plan manager maximizes discounted S-shaped utility from benefit payments, subject to a downside constraint. The target benefit is determined by an actuarial equivalence condition linking contributions and benefits, while deviations from the target reflect intergenerational risk sharing. The pension fund invests in a risk-free asset and multiple risky assets. Using the martingale approach, we derive explicit solutions for the optimal investment and benefit payment strategies. Numerical illustrations and sensitivity analyzes highlight the effects of key market parameters on the optimal policies.
Quantifying uncertainty in survival outcomes is fundamental to demographic and actuarial research. In this context, entropy offers a rigorous framework for capturing the underlying dynamics of mortality. This study investigates how entropy can be used to characterize uncertainty and predictability in survival outcomes, with particular relevance to longevity risk assessment. We propose entropy measures derived from small perturbations to the survival function and interpret them within the additive mortality framework, a flexible class encompassing many widely used mortality models. This approach allows us to examine how entropy reflects age-specific mortality dynamics and long-term trends. Through numerical illustrations based on empirical data, we demonstrate that entropy-based measures provide meaningful insights into the structure and evolution of mortality, offering a complementary perspective to traditional actuarial tools and deepening the understanding of survival uncertainty.
Variable annuities (VAs) are long-term insurance contracts that offer guaranteed minimum benefits for the purpose of retirement planning. The valuation of VA attracts increasing attention over the past decade. In this paper, we propose a dynamic drawdown-dependent fee structure for VA, under which fees are charged only when the underlying fund experiences a drawdown from its running maximum, that is, when the embedded guarantees are most likely to be exercised. By reducing the fee when the guarantee has little value, our fee design aligns fee collection with periods of policyholder's needs for guarantee and insurer's liability, as well as helps to reduce the policyholder's surrender incentive. Under this framework, we derive analytical expressions for the contract value of GMDB, which further allows the valuation for GMMB via a numerical approach. To illustrate the merit of the proposed fee structure, we analyze its impact on policyholder surrender incentives through a comparison with the constant fee structure. Numerical examples and sensitivity analysis are provided to illustrate our results.
We consider a continuous-time optimization problem for a life-insurance buyer under a general mean-second moment criterion, which generalizes the mean-variance and the mean-modified variance criteria. The buyer invests in a Black-Scholes financial market and purchases (instantaneous) term life insurance. We assume that the buyer aims to optimize a general mean-second moment criterion evaluated at her discounted wealth at death. Because this general criterion does not satisfy the property of iterated expectations, the individual's problem is time-inconsistent. To overcome this difficulty, the buyer chooses her life insurance death benefit and investment strategy from a time-consistent perspective, as in Bj & ouml;rk and Murgoci [(2010). A general theory of Markovian time inconsistent stochastic control problems. Working paper, available at SSRN 1694759]. We first provide a verification lemma for solving the general mean-second moment problem. Then, under three specific mean-second moment criteria, we find the buyer's equilibrium control strategies explicitly, along with her value function, by solving the corresponding extended Hamilton-Jacobi-Bellman system of equations. We also present two numerical examples to illustrate our results.
This paper investigates the robust optimal investment for an ambiguity averse member of a defined contribution (DC) pension plan in a fully-fledged, time consistent mean-variance modeling framework. In particular, the paper extends the literature on defined contribution pension plans in three directions: (1) We relax its assumption of purely continuous stock and/or contribution processes, which allows to introduce the effects of news, job loss, macroeconomic conditions, etc., into the model; (2) Unlike most studies in DC pension plans, we allow for ambiguity about both the mean arrival rate and jump size distribution of the stock returns and contribution rate processes of the member; (3) Ambiguity in our setting is time-varying. The model thus features stochastic stock volatility, stochastic interest rate, stochastic contribution rate, jumps in both stock and contribution rate processes, and time-varying ambiguity about diffusion parameters. Welfare analysis indicates that ignoring ambiguity can be very costly to the member. The framework proposed in this paper is general and adds significant realism to existing models in the literature.
This paper introduces a support condition approach for mixed insurance risk control using linear programming inspired by occupation measures. The model includes reinsurance - continuous control, capital injection, and dividends - singular controls. Building upon [Goreac et al. (2022). Linearization techniques and the dual algorithm for a class of mixed singular/continuous control problems in reinsurance. Part I: Theoretical aspects. Applied Mathematics and Computation, 431, Article 127321], we propose: (i) a dual formulation akin to Fenchel-Legendre transforms that characterizes optimal measure support and enables efficient policy extraction once the value function is known; (ii) validation against benchmarks from [Goreac et al. (2024). Linearization techniques and the dual algorithm for a class of mixed singular/continuous control problems in reinsurance. Part II: Numerical aspects. Applied Mathematics and Computation, 473, Article 128655], including exponential claims without reinsurance and general distributions with reinsurance but no injections. Numerical results recover threshold-based dividend and capital injection policies for various claim distributions, demonstrating the method's efficiency in solving Hamilton-Jacobi-Bellman variational inequalities via measure relaxation.
Extreme cold temperature events have long been associated with excess mortality via many different causes of death. Climate change is expected to intensify the frequency and severity of these extreme temperature events. To quantify and model cold-related excess deaths and, in turn, to better understand the potential impact of climate change on future mortality levels, we propose a new approach based on the state-of-the-art stationary vine copulas. We adopt the S-vine model for the first time in the context of climate-driven mortality risk, and introduce a special case of the model to aid model comparison and enhance interpretability of the results. This model is referred to as a (stationary) centrally connected C-vine (CCC-vine). Three types of dependence are captured by the proposed models, which are temporal dependence, contemporaneous cross-sectional dependence, and non-contemporaneous cross-sectional dependence. We fit the CCC-vine model to the US regional cause-specific death data over the period 1999-2018 and conclude that the model outperforms various benchmark models including the Gaussian copula model and the VAR model. Based on the fitted models, we generate several temperature scenarios and assess cause-specific excess deaths and overall excess deaths due to extreme cold temperatures. We also analyze and compare the geographical differences in cold-related excess deaths across six continental US regions. The results from our study can help public health interventions during extreme cold events to reduce temperature-driven excess deaths.
Actuarial literature contains countless formulations and analytical results of optimal reinsurance for a single risk, but there is limited research on optimal solutions when the cedent runs many lines of business and is asked to manage the total risk in a certain integrated sense. In this paper, we extend the problem of optimal reinsurance to a multivariate framework where the cedent has multiple risks that cannot be bundled together into one. More specifically, we optimize layer contracts using a more industrially based criterion, where risk and profit are balanced through a ratio between the value-at-risk and the expected surplus. The effects of the marginal risk distributions and the dependence structure, as well as of the premium principle, are investigated. Using convexity theory, we show that the optimal solution is strongly related to the monotonicity property of the hazard rate functions of the risks. For decreasing hazard rates the solution can typically be found by Lagrangian optimization, while for increasing hazard rates, one often ends up with an extreme point solution, where at least one of the contracts is a stop-loss one. Further, the joint optimization of the contracts results in a better balance between risk and expected gain than when the contracts are optimized separately, even when the risks are independent. This advantage increases with the dependence between the risks, as well as when the marginal risk distributions become more heavy-tailed.
Building upon a hierarchical Bayesian random walk with drift (HBS-RW) model proposed by T. Lin, and Tsai ((2022). Hierarchical Bayesian modeling of multi-country mortality rates. Scandinavian Actuarial Journal, 2022(5), 375-398), this paper further incorporates jump components to account for extreme events in the data. Using the mortality data for both genders of seventeen countries, we demonstrate that the proposed models with jump components generally perform better than the HBS-RW model without a jump component in most conducted tests. Finally, applying three grouping methods to mortality data of these seventeen countries, we find that mortality rate improvements in developing countries tend to be faster than those in developed countries over a specific period; additionally, except for the U.S., the expected improvement rates for all of the other countries eventually converge to an average of about 2%, which confirms the finding in T. Lin, and Tsai ((2022). Hierarchical Bayesian modeling of multi-country mortality rates. Scandinavian Actuarial Journal, 2022(5), 375-398).
We derive exact tail asymptotics of the Parisian ruin probability for Gaussian risk models driven by locally self-similar Gaussian processes with a power-type deterministic trend. The considered setting includes non-stationary Gaussian processes whose local correlation structure is governed by a self-similar limiting process, extending classical fractional Brownian motion models. The asymptotic behavior is shown to depend on the interplay between the local variance decay, the self-similarity index, and the trend exponent, leading to several distinct regimes. In each regime, the ruin probability admits an explicit asymptotic representation involving Parisian Pickands-type constants. The analysis relies on a uniform Pickands lemma allowing for families of limiting Gaussian fields, extending existing double-sum techniques and enabling the treatment of locally self-similar Gaussian risk models.
Laplace approximation provides a Gaussian approximation of a posterior distribution via a second-order Taylor expansion. Although the Bernstein-von Mises theorem guarantees asymptotic normality as the sample size approaches infinity, the Gaussian approximation may be unreliable when the sample size is finite. This is particularly true when the posterior distribution is skewed, which is a common occurrence in the insurance ratemaking process, where the use of a Gaussian distribution may not yield an accurate approximation. In this study, by utilizing the generalized version of Taylor expansion [Widder (1928). A generalization of Taylor's series. Transactions of the American Mathematical Society, 30(1), 126-154], we introduce a generalized version of Laplace approximation where the posterior distribution is approximated by various parametric distributions in the exponential family. We apply this method to random effects models, connecting it to credibility premium, in the insurance context. While credibility premium provides an affine posterior mean approximation, it lacks further distributional information. Our method introduces the ability to approximate the posterior distribution, while still providing the same point approximation as credibility premium. Numerical analysis confirms the effectiveness of the proposed approach.
In this paper, we address an optimal stochastic asset allocation and reinsurance problem in continuous-time contagious financial and insurance markets. The insurer is subject to contagious claims, which are modeled using an enhanced dynamic contagion process. This process generalizes the externally-exciting Cox process with shot noise and the self-exciting Hawkes process while also capturing the dependence structure between the financial and insurance markets. Furthermore, the insurer is assumed to be ambiguity-averse, with distinct modeling risk aversion preferences for the risky asset, extreme external events, and contagious insurance claims. The insurer aims to maximize the expected utility of the terminal surplus and a specified penalty function at a fixed terminal date under the worst-case scenario. Using the dynamic programing principle, we derive the extended Hamilton-Jacobi-Bellman (HJB) equation and develop an iterative numerical scheme to compute the value function and optimal controls. The convergence of the numerical method is rigorously proven. To support our quantitative analysis, we provide several numerical examples illustrating the impact of the dependence structure and ambiguity aversion on optimal controls.
This article studies a mean-variance Stackelberg game between an insured and an insurer under a self-protection model, where the insured's effort is observable by the insurer and reflected in the insurance premium. We adopt a proportional insurance model under a quadratic premium principle and incorporate a cost-sharing mechanism for self-protection. The existence of equilibrium solutions for insurance demand, self-protection effort, and cost-sharing proportions is established, and their interactions are analyzed theoretically. Numerical analysis reveals conditions under which self-protection and market insurance act as complementary or substitutable risk management tools.
This paper investigates the optimal control problem of ratcheting dividend with capital injection under a Brownian risk model. We show that the value function is the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation. Explicit analytical expressions of the value function and the optimal strategy are obtained when the general ratcheting dividend strategies are restricted as finite ratcheting dividend strategies (i.e. the dividend rate takes only a finite number of values), where the optimal dividend strategy is the threshold-type finite ratcheting dividend strategy, and the optimal capital injection strategy is the bailout strategy. Some numerical illustrations are provided at the end.
This paper examines portfolio insurance (PI) problems for investors who require a minimum level of consumption. Its key contribution is introducing the Option-Based Portfolio and Consumption Insurance (OBPCI) strategy, which extends the popular OBPI framework. As the optimal solution to a constrained utility maximization problem under a general local covariance model, OBPCI is derived using the martingale approach and can be interpreted as a portfolio of options. Given the lack of valid benchmarks involving consumption in the literature, we also introduce and formalize two competitive strategies, optimal on their own, albeit suboptimal to our main problem: the Synthetic Constant Proportion Portfolio and Consumption Insurance (SCPPCI) and the Synthetic Option-Based Portfolio and Consumption Insurance (SOBPCI). Our numerical analysis shows that, under realistic parameters, SCPPCI and SOBPCI can lead to equivalent welfare losses of up to $ 9\% $ 9% for a short investment horizon and $ 5\% $ 5% for low risk aversion, respectively, relative to OBPCI.
We investigate optimal proportional portfolio insurance (PPI) strategies aimed at reducing exposure to carbon intensive stocks. PPI strategies enable investors to mitigate downside risk while retaining the potential for upside gains. In this paper we determine the PPI strategies to maximise the expected utility of the terminal cushion, where the terminal cushion is penalised proportionally to the realised volatility of stocks issued by firms operating in carbon-intensive sectors. We model the risky assets' dynamics using geometric Brownian motions whose drift rates are modulated by an unobservable common stochastic factor to capture market-specific or economy-wide state variables that are typically not directly observable. Using the classical stochastic filtering theory, we formulate a suitable optimisation problem and solve it for the CRRA utility function. We characterise optimal carbon-penalised PPI strategies and optimal value functions under full and partial information. We also carry a numerical analysis showing that the proposed strategy reduces carbon-emissions intensity without compromising financial performance.
To address the critical challenges faced by the elderly population in the field of long-term care, this paper proposes a novel insurance product that flexibly combines a variable annuity with a guaranteed lifelong withdrawal benefit (GLWB) rider and long-term care (LTC) insurance. Specifically, the proposed insurance product incorporates a conversion period during which the policyholder can flexibly decide whether to convert the standard annuity contract into a combo product that integrates LTC benefits. Using the Fourier cosine (COS) method combined with Markov chain approximation techniques, we develop a unified framework to price the GLWB annuity under general stochastic volatility models. By improving the algorithm through cubic spline interpolation, we achieve efficient valuation of this contract. The stochastic volatility models considered in this study include the Heston model, 3/2 model, 4/2 model, Hull-White model, and Scott model. The impact of annuity model parameters, conversion timing, and the policyholder's gender on the GLWB value is analyzed.
In this paper, we provide generalizations of the functional equations that characterize the lack-of-memory properties: more specifically, we extend the univariate functional equation introduced by Kaminsky (1983, An aging property of the Gompertz survival function and a discrete analog (Tech. Rep.). Department of Mathematical Statistics, University of Ume & aring;, Sweden) and the corresponding bivariate strong and weak versions studied in Marshall and Olkin (2015, A bivariate Gompertz-Makeham life distribution. Journal of Multivariate Analysis, 139, 219-226. https://doi.org/10.1016/j.jmva.2015.02.011) by allowing the conditional survival distribution to be a fully general time dependent distortion of the unconditional one. Since the univariate functional equation leads only to a trivial case and the solutions of the strong bivariate functional equation have been already studied in the literature, the analysis focuses on the weak bivariate case, where joint residual lifetimes are conditioned on survival beyond a common threshold t. In view of potential applications to insurance risk analysis, we study the impact of the time dependent distortion on the aging properties and on the dependence structure of the residual lifetimes via time-varying Kendall's function and tail dependence coefficients: moreover, we provide some illustrative examples showing that these distributions can model both broken hearth effect as well as its reverse version.
