Insurance Mathematics & Economics最新文献

This paper develops an incomplete equilibrium model with multi-agents' different risk attitudes and heterogeneous income/payout profiles. Particularly, we apply its concrete and computationally tractable model to reinsurance derivatives pricing and life-cycle investment, which are important for insurance and asset management companies in practice. In numerical experiments, we explicitly obtain endogenously determined expected returns of the risky asset in equilibrium, agents' specific reinsurance prices with their stochastic discount factors (SDF) and optimal life-cycle trading strategies. Moreover, we investigate how each agent's degree of risk aversion and income/payout profile, and correlations between an insurance or economic factor and the risky asset price affect reinsurance claims pricing and optimal portfolios in life-cycle investment.

We propose several methods for obtaining endogenous and positive ultimate forward rates (UFRs) for risk-free interest rate curves based on the Smith-Wilson method. The Smith-Wilson method, which is adopted by Solvency II, can both interpolate the market price data and extrapolate to the UFR. However, the method requires an exogenously-chosen UFR. To obtain an endogenous UFR, de Kort and Vellekoop (2016) proposed an optimization framework based on the Smith-Wilson method. In this paper, we prove the existence of an optimal endogenous UFR to their optimization problem under the condition that the cash flow matrix is square and invertible. In addition, to ensure the positivity of the optimal endogenous UFR during extreme time periods such as the COVID-19 pandemic, we extend their optimization framework by including non-negative constraints. Furthermore, we also propose a new optimization framework that can not only generate endogenous and positive UFRs but also incorporate practitioners' prior knowledge. We prove the feasibility of our frameworks, and conduct empirical studies for both the Chinese government bonds and the EURIBOR swaps to illustrate the capabilities of our methods.

The accuracy and interpretability of a (non-life) insurance pricing model are essential qualities to ensure fair and transparent premiums for policy-holders, that reflect their risk. In recent years, classification and regression trees (CARTs) and their ensembles have gained popularity in the actuarial literature, since they offer good prediction performance and are relatively easy to interpret. In this paper, we introduce Bayesian CART models for insurance pricing, with a particular focus on claims frequency modelling. In addition to the common Poisson and negative binomial (NB) distributions used for claims frequency, we implement Bayesian CART for the zero-inflated Poisson (ZIP) distribution to address the difficulty arising from the imbalanced insurance claims data. To this end, we introduce a general MCMC algorithm using data augmentation methods for posterior tree exploration. We also introduce the deviance information criterion (DIC) for tree model selection. The proposed models are able to identify trees which can better classify the policy-holders into risk groups. Simulations and real insurance data will be used to illustrate the applicability of these models.

This paper studies a non-zero-sum stochastic differential game for multiple mean-variance insurers. Insurers can purchase proportional reinsurance and invest in a risk-free asset, a market index, a defaultable bond and multiple pairs of mispriced stocks. The dynamics of the mispriced stocks satisfy a “cointegrated system” where the expected returns follow the mean reverting processes, and the bond is defaultable with a recovering proportional value at default. In particular, we assume that the investment opportunities in mispriced stocks are only available for a few insurers, which is more realistic and in line with the superiority of information in the competitive market. Each insurer's objective is maximizing a function of her terminal wealth and competitors' relative wealth under the mean-variance criterion. Using techniques in stochastic control theory, we establish the extended Hamilton-Jacobi-Bellman equations and obtain the equilibrium strategies. Note that the derived solutions are analytical and time-consistent, and we verify the competitive advantages gained from investment opportunities in mispriced stocks. We represent our results in terms of the M-matrices, which help us prove the existence and uniqueness of the solutions and further explicitly analyze how the crucial arguments in the model affect the equilibrium strategies. Numerical examples with detailed sensitivity analyses are presented to support our conclusions.

Stress testing, and in particular, reverse stress testing, is a prominent exercise in risk management practice. Reverse stress testing, in contrast to (forward) stress testing, aims to find an alternative but plausible model such that under that alternative model, specific adverse stresses (i.e. constraints) are satisfied. Here, we propose a reverse stress testing framework for dynamic models. Specifically, we consider a compound Poisson process over a finite time horizon and stresses composed of expected values of functions applied to the process at the terminal time. We then define the stressed model as the probability measure under which the process satisfies the constraints and which minimizes the Kullback-Leibler divergence to the reference compound Poisson model.

We solve this optimization problem, prove existence and uniqueness of the stressed probability measure, and provide a characterization of the Radon-Nikodym derivative from the reference model to the stressed model. We find that under the stressed measure, the intensity and the severity distribution of the process depend on time and state, and hence the stressed model is not a compound Poisson process. We illustrate the dynamic stress testing by considering stresses on VaR and both VaR and CVaR jointly and provide illustrations of how the stochastic process is altered under these stresses. We generalize the framework to multivariate compound Poisson processes and stresses at times other than the terminal time. We illustrate the applicability of our framework by considering “what if” scenarios, where we answer the question: What is the severity of a stress on a portfolio component at an earlier time such that the aggregate portfolio exceeds a risk threshold at the terminal time? Furthermore, for general constraints, we propose an algorithm to simulate sample paths under the stressed measure, thus allowing to compare the effects of stresses on the dynamics of the process.

In this paper, we focus on the asymptotic behavior of a recently popular risk measure called the tail moment (TM), which has been extensively applied in the field of risk theory. We conduct the study under the framework in which the individual risks of a financial or insurance system follow convolution equivalent or Gamma-like distributions. Precise asymptotic results are obtained for the TM when the individual risks are mutually independent or have a dependence structure of the Farlie-Gumbel-Morgenstern type. Moreover, based on some specific scenarios, we give an asymptotic analysis on the relative errors between our asymptotic results and the corresponding exact values. Since the model settings in this paper are not covered by traditional ones, our work fills in some gaps of the asymptotic study of the TM for light-tailed risks.

We consider the situation when the number of claims is unavailable, and a Tweedie's compound Poisson model is fitted to the observed pure premium. Currently, there are two different models based on the Tweedie distribution: a single generalized linear model (GLM) for mean and a double generalized linear model (DGLM) for both mean and dispersion. Although the DGLM approach facilitates the heterogeneous dispersion, its soundness relies on the accuracy of the saddlepoint approximation, which is poor when the proportion of zero claims is large. For both models, the power variance parameter is estimated by considering the profile likelihood, which is computationally expensive. We propose a new approach to fit the Tweedie model with the EM algorithm, which is equivalent to an iteratively re-weighted Poisson-gamma model on an augmented data set. The proposed approach addresses the heterogeneous dispersion without needing the saddlepoint approximation, and the power variance parameter is estimated during the model fitting. Numerical examples show that our proposed approach is superior to the two competing models.

In this paper we propose a model for pricing GLWB variable annuities under a stochastic mortality framework. Our set-up is very general and only requires the Markovian property for the mortality intensity and the asset price processes. The contract value is defined through an optimization problem which is solved by using dynamic programming. We prove, by backward induction, the validity of the bang-bang condition for the set of discrete withdrawal strategies of the model. This result is particularly remarkable as in the insurance literature either the existence of optimal bang-bang controls is assumed or it requires suitable conditions. We assume constant interest rates, although our results still hold in the case of a Markovian interest rate process. We present extensive numerical examples, modelling the mortality intensity as a non mean reverting square root process and the asset price as an exponential Lévy process, and compare the results obtained for different parameters and policyholder behaviours.

The financial return of equity-indexed annuities depends on an underlying fund or investment portfolio complemented by an investment guarantee. We discuss a so-called cliquet-style or ratchet-type guarantee granting a minimum annual return. Its path-dependent payoff complicates valuation and risk management, especially if interest rates are modelled stochastically. We develop a novel scenario-matrix (SM) method. In the example of a Vasicek-Black-Scholes model, we derive closed-form expressions for the value and moment-generating function of the final payoff in terms of the scenario matrix. This allows efficient evaluation of values and various risk measures, avoiding Monte-Carlo simulation or numerical Fourier inversion. In numerical tests, this procedure proves to converge quickly and outperforms the existing approaches in the literature in terms of computation time and accuracy.

In non-life insurance, it is essential to understand the serial dynamics and dependence structure of the longitudinal insurance data before using them. Existing actuarial literature primarily focuses on modeling, which typically assumes a lack of serial dynamics and a pre-specified dependence structure of claims across multiple years. To fill in the research gap, we develop two diagnostic tests, namely the serial dynamic test and correlation test, to assess the appropriateness of these assumptions and provide justifiable modeling directions. The tests involve the following ingredients: i) computing the change of the cross-sectional estimated parameters under a logistic regression model and the empirical residual correlations of the claim occurrence indicators across time, which serve as the indications to detect serial dynamics; ii) quantifying estimation uncertainty using the randomly weighted bootstrap approach; iii) developing asymptotic theories to construct proper test statistics. The proposed tests are examined by simulated data and applied to two non-life insurance datasets, revealing that the two datasets behave differently.