This paper investigates a well-known downside protection strategy called the constant proportion portfolio insurance (CPPI) in defined contribution (DC) pension fund modeling. Under discrete time trading CPPI, an investor faces the risk of portfolio value hitting the floor which denotes the process of guaranteed portfolio values. In this paper, we question how to deal with so-called ‘gap risk’ which may appear due to uncontrollable events resulting in a sudden drop in the market. In the market model considered, the risky asset price and the labor income are assumed to be continuous-time stochastic processes, whereas trading is restricted to discrete-time. In this setting, an exotic option (namely, the ‘cushion option’) is proposed with the aim of reducing the risk that the portfolio value falls below the defined floor. We analyze the effectiveness of the proposed exotic option for a DC plan CPPI strategy through Monte Carlo simulations and sensitivity analyses with respect to the parameters reflecting different setups.

The conditional expectation $m_{X}(s)=\mathrm{E}[X|S=s]$, where X and Y are two independent random variables with $S=X+Y$, plays a key role in various actuarial applications. For instance, considering the conditional mean risk-sharing rule, $m_X(s)$ determines the contribution of the agent holding the risk X to a risk-sharing pool. It is also a relevant function in the context of risk management, for example, when considering natural capital allocation principles. The monotonicity of $m_X(\!\cdot\!)$ is particularly significant under these frameworks, and it has been linked to log-concave densities since Efron (1965). However, the log-concavity assumption may not be realistic in some applications because it excludes heavy-tailed distributions. We consider random variables with regularly varying densities to illustrate how heavy tails can lead to a nonmonotonic behavior for $m_X(\!\cdot\!)$. This paper first aims to identify situations where $m_X(\!\cdot\!)$ could fail to be increasing according to the tail heaviness of X and Y. Second, the paper aims to study the asymptotic behavior of $m_X(s)$ as the value s of the sum gets large. The analysis is then extended to zero-augmented probability distributions, commonly encountered in applications to insurance, and to sums of more than two random variables and to two random variables with a Farlie–Gumbel–Morgenstern copula. Consequences for risk sharing and capital allocation are discussed. Many numerical examples illustrate the results.

Conditional risk measures and their associated risk contribution measures are commonly employed in finance and actuarial science for evaluating systemic risk and quantifying the effects of risk interactions. This paper introduces various types of contribution ratio measures based on the multivariate conditional value-at-risk (MCoVaR), multivariate conditional expected shortfall (MCoES), and multivariate marginal mean excess (MMME) studied in [34] (Ortega-Jiménez, P., Sordo, M., & Suárez-Llorens, A. (2021). Stochastic orders and multivariate measures of risk contagion. Insurance: Mathematics and Economics, vol. 96, 199–207) and [11] (Das, B., & Fasen-Hartmann, V. (2018). Risk contagion under regular variation and asymptotic tail independence. Journal of Multivariate Analysis 165(1), 194–215) to assess the relative effects of a single risk when other risks in a group are in distress. The properties of these contribution risk measures are examined, and sufficient conditions for comparing these measures between two sets of random vectors are established using univariate and multivariate stochastic orders and statistically dependent notions. Numerical examples are presented to validate these conditions. Finally, a real dataset from the cryptocurrency market is used to analyze the spillover effects through our proposed contribution measures.

We use recent advances in polynomial diffusion processes to develop a continuous-time joint mortality model for the actuarial valuation and risk analysis of life insurance liabilities. The model considers the stochastic nature of future mortality improvements and introduces a common subordinator for the marginal survival processes, resulting in a nontrivial dependence structure between the survival of pairs of individuals. Polynomial diffusion processes can be used to derive closed-form formulae for standard actuarial quantities. The model fits well with a classic dataset provided by a Canadian insurer and can be used to evaluate products issued to multiple lives, as shown through numerical applications.

We analyse the effect of natural catastrophes on insurance demand in a developing economy and the role of insurance regulation in this relationship. The analysis is based on a theoretical model and a panel regression using data for Vietnam. What makes Vietnam especially interesting is the fact that it is strongly affected by natural catastrophes and experienced a change in insurance regulation in recent years. The theoretical results indicate that a loss experience likely has a less positive effect on demand in developing economies than in developed economies. A higher insurance penetration and a tighter insurance regulation, however, can make the impact of a loss event more positive. These findings are mirrored by our empirical analysis: overall natural catastrophes decrease insurance demand of affected households in Vietnam. The enhancement of regulation was not only accompanied by increased insurance demand but it also reverses the effect of natural catastrophes on demand.

In this paper, we present experimental evidence on the effect adverse selection has on coverage choices and pricing in corporate insurance markets. Two sets of experimental data, each generated by experiments utilizing a specific parameterization of a corporate insurance decision, are presented to gauge these effects. In the first, subject behavior conforms to a unique equilibrium in which high risk firms choose higher coverage and contracts are priced accordingly. Insurers act competitively and convergence to equilibrium behavior is marked. In the second set, there is little evidence that subject behavior is consistent with either of the two equilibrium outcomes supported by the experimental setting—pooling by fully insuring losses and pooling by self insuring.

The tonuity, proposed by Chen et al. ((2019) ASTIN Bulletin: The Journal of the IAA, 49(1), 530.), is a combination of an immediate tontine and a deferred annuity. However, its switching time from tontine to annuity is fixed at the moment the contract is closed, possibly becoming sub-optimal if mortality changes over time. This article introduces an alternative tonuity product, wherein a dynamic switching condition is pivotal, relying on the observable mortality trends within a reference population. The switching from tontine to annuity then occurs automatically once the condition is satisfied. Using data from the Human Mortality Database and UK Continuous Mortality Investigation, we demonstrate that, in a changing environment, where an unforeseen mortality or longevity shock leads to an unexpected increase or decrease in mortality rates, the proposed dynamic tonuity contract can be preferable to the regular tonuity contract.

The association between economic variables and the frequency and duration of disability income insurance (DII) claims is well established. Across many jurisdictions, heightened levels of unemployment have been associated with both a higher incidence and a longer duration of DII claims. This motivated us to derive an asset portfolio for which the total asset value moves in line with the level of unemployment, thus, providing a natural match for the DII portfolio liabilities. To achieve this, we develop an economic tracking portfolio where the asset weights in the portfolio are chosen so that the portfolio value changes in a way that reflects, as closely as possible, the level of unemployment. To the best of our knowledge, this is the first paper applying economic tracking portfolios to hedge economic risk in DII. The methodology put forward to establish this asset-liability matching portfolio is illustrated using DII data from the UK between 2004 and 2016. The benefits of our approach for claims reserving in DII portfolios are illustrated using a simulation study.

The emergence of COVID-19 has resulted in a notable rise in mortality rates, consequently affecting various sectors, including the insurance industry. This paper analyzes the reflections of a sudden increase in mortality rates on the financial performance of a survival benefit scenario under the International Financial Reporting Standard 17. For this purpose, we thoroughly examined a single insurance scenario under four different states by modifying the interest and jump elements. We use Poisson-log bilinear Lee–Carter and Vasicek models for mortality and stochastic interest rate, respectively. Integrating the mortality model with a jump model that incorporates COVID-19 deaths we constructed a temporary mortality jump model. As a result, the temporary mortality jump model reflects the effects of the pandemic more realistically. We observe that even in this case mortality has a minor impact, whereas interest rates significantly still affect the financial position and performance of insurance companies.

During the late eighteenth and early nineteenth centuries, mutual associations predominated in insuring the large fleet of ships that carried coal from Britain's northeast to London and other ports. The number of associations grew rapidly from the late 1770s, initially on the Tyne, then spreading to other ports on the east coast. They largely saw off the challenge from joint-stock companies created after the liberalisation of the marine insurance market in 1824. Low administrative and legal costs and the ability to mobilise local knowledge to minimise risks allowed the associations to offset the disadvantage of insuring vessels in the same trade facing similar adversities. This article discusses how mutual associations were organised and operated, traces their development on the Tyne and the competition they encountered there from Lloyd's of London and joint-stock insurance companies, and examines the incidence of mutual associations elsewhere in Britain.

We use Benford's law to examine the non-random elements of health care costs. We find that as health care expenditures increase, the conformity to the expected distribution of naturally occurring numbers worsens, indicating a tendency towards inefficient treatment. Government insurers follow Benford's law better than private insurers indicating more efficient treatment. Surprisingly, self-insured patients suffer the most from non-clinical cost factors. We suggest that cost saving efforts to reduce non-clinical expenses should be focused on more severe, costly encounters. Doing so focuses cost reduction efforts on less than 10% of encounters that constitute over 70% of dollars spent on health care treatment.

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.

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.

High-cardinality categorical features are pervasive in actuarial data (e.g., occupation in commercial property insurance). Standard categorical encoding methods like one-hot encoding are inadequate in these settings.

In this work, we present a novel Generalised Linear Mixed Model Neural Network (“GLMMNet”) approach to the modelling of high-cardinality categorical features. The GLMMNet integrates a generalised linear mixed model in a deep learning framework, offering the predictive power of neural networks and the transparency of random effects estimates, the latter of which cannot be obtained from the entity embedding models. Further, its flexibility to deal with any distribution in the exponential dispersion (ED) family makes it widely applicable to many actuarial contexts and beyond. In order to facilitate the application of GLMMNet to large datasets, we use variational inference to estimate its parameters—both traditional mean field and versions utilising textual information underlying the high-cardinality categorical features.

We illustrate and compare the GLMMNet against existing approaches in a range of simulation experiments as well as in a real-life insurance case study. A notable feature for both our simulation experiment and the real-life case study is a comparatively low signal-to-noise ratio, which is a feature common in actuarial applications. We find that the GLMMNet often outperforms or at least performs comparably with an entity-embedded neural network in these settings, while providing the additional benefit of transparency, which is particularly valuable in practical applications.

Importantly, while our model was motivated by actuarial applications, it can have wider applicability. The GLMMNet would suit any applications that involve high-cardinality categorical variables and where the response cannot be sufficiently modelled by a Gaussian distribution, especially where the inherent noisiness of the data is relatively high.

The distribution-free chain ladder of Mack justified the use of the chain ladder predictor and enabled Mack to derive an estimator of conditional mean squared error of prediction for the chain ladder predictor. Classical insurance loss models, that is of compound Poisson type, are not consistent with Mack’s distribution-free chain ladder. However, for a sequence of compound Poisson loss models indexed by exposure (e.g., number of contracts), we show that the chain ladder predictor and Mack’s estimator of conditional mean squared error of prediction can be derived by considering large exposure asymptotics. Hence, quantifying chain ladder prediction uncertainty can be done with Mack’s estimator without relying on the validity of the model assumptions of the distribution-free chain ladder.

Sickness insurance companies were developed in Spain by doctors and healthcare professionals, remaining outside the interests of general insurance companies. Their management was hardly professional, with limited actuarial techniques and they only accounted for a small percentage of total insurance business premiums. From the 1970s onwards, various factors changed this situation, driving processes of concentration, with numerous takeovers and mergers, first reducing the number of local and regional companies to the benefit of companies of national scope. Subsequently, the growth in demand for this type of coverage sparked the interest of national general insurance companies and multinationals, leading to a restructuring of the sector which has progressively acquired greater weight within the insurance business and become increasingly internationalised. This last stage immersed the health sector in Spain in the great processes of globalisation of the sector, characterised by a financialisation of capital promoted by the bank investment funds. These processes are little known and are the focus of analysis of this paper, with the aim of enabling comparison at international level.

In this paper, we explore how to design the optimal insurance contracts when the insured faces insurable, counterparty, and additive background risk simultaneously. The target is to minimize the mean-variance of the insured’s loss. By utilizing the calculus of variations, an implicit characterization of the optimal ceded loss function is given. An explicit structure of the optimal ceded loss function is also provided by making full use of its implicit characterization. We further derive a much simpler solution when these three kinds of risk have some special dependence structures. Finally, we give a numerical example to illustrate our results.

Community Rating System (CRS) incentivizes investments in risk reduction above NFIP standards using discounts on insurance premiums. These discounts are cross-subsidized by increasing premiums in non-CRS communities. We examine the distribution of these subsidies and find that redistribution does occur, but the gains and losses are not economically large with 95% of households gaining or losing no more than 0.3% of household income. We also examine their relationship with other community characteristics and find that the strongest predictor of premium reductions is the underlying flood risk level within the community. Thus, CRS appears to reduce the cost of living in the riskier communities.

This paper considers variable annuity (VA) contracts embedded with guaranteed minimum accumulation benefit (GMAB) riders when policyholder’s proceeds are taxed upon early surrender or maturity. These contracts promise the return of the premium paid by the policyholder, or a higher rolled-up value, at the end of the investment period. A partial differential equation valuation framework which exploits the numerical method of lines is used to determine fair fees that render the policyholder and insurer breakeven. Two taxation regimes are considered: one where capital gains are allowed to offset losses and a second where gains do not offset losses. Most insurance providers highlight the tax-deferred features of VA contracts. We show that the regime under which the insured is taxed significantly impacts prices. If losses are allowed to offset gains then this enhances the market, increasing the policyholder’s willingness to participate in the market compared to the case when losses are not allowed to offset gains. With fair fees from the policyholder’s perspective, we show that the net profit is generally positive for insurance companies offering the contract as a naked option without any hedge. We also show how investment policy, as reflected in the Sharpe ratio, impacts and interacts with policyholder persistency.