Showing papers in "Insurance Mathematics & Economics in 2014"
TL;DR: In this article, the authors investigated the case of M -quantiles as the minimizers of an asymmetric convex loss function, in contrast to Orlicz quantiles that have been considered in Bellini and Rosazza Gianin (2012).
Abstract: In the statistical and actuarial literature several generalizations of quantiles have been considered, by means of the minimization of a suitable asymmetric loss function. All these generalized quantiles share the important property of elicitability , which has received a lot of attention recently since it corresponds to the existence of a natural backtesting methodology. In this paper we investigate the case of M -quantiles as the minimizers of an asymmetric convex loss function, in contrast to Orlicz quantiles that have been considered in Bellini and Rosazza Gianin (2012). We discuss their properties as risk measures and point out the connection with the zero utility premium principle and with shortfall risk measures introduced by Follmer and Schied (2002). In particular, we show that the only M -quantiles that are coherent risk measures are the expectiles , introduced by Newey and Powell (1987) as the minimizers of an asymmetric quadratic loss function. We provide their dual and Kusuoka representations and discuss their relationship with CVaR. We analyze their asymptotic properties for α → 1 and show that for very heavy tailed distributions expectiles are more conservative than the usual quantiles. Finally, we show their robustness in the sense of lipschitzianity with respect to the Wasserstein metric.
165 citations
TL;DR: In this paper, the authors introduce the admissible risk class to study risk aggregation with dependence uncertainty and derive a new convex ordering lower bound over this class and give a sufficient condition for this lower bound to be sharp in the case of identical marginal distributions.
Abstract: Risk aggregation with dependence uncertainty refers to the sum of individual risks with known marginal distributions and unspecified dependence structure. We introduce the admissible risk class to study risk aggregation with dependence uncertainty. The admissible risk class has some nice properties such as robustness, convexity, permutation invariance and affine invariance. We then derive a new convex ordering lower bound over this class and give a sufficient condition for this lower bound to be sharp in the case of identical marginal distributions. The results are used to identify extreme scenarios and calculate bounds on Value-at-Risk as well as on convex and coherent risk measures and other quantities of interest in finance and insurance. Numerical illustrations are provided for different settings and commonly-used distributions of risks.
139 citations
TL;DR: In this article, a new probability density function with bounded domain is presented, which is based on the generalized Lindley distribution proposed by Zakerzadeh and Dolati (2010).
Abstract: In this paper a new probability density function with bounded domain is presented. The new distribution arises from the generalized Lindley distribution proposed by Zakerzadeh and Dolati (2010). This new distribution that depends on two parameters can be considered as an alternative to the classical beta distribution. It presents the advantage of not including any special function in its formulation. After studying its most important properties, some useful results regarding insurance and inventory management applications are obtained. In particular, in insurance, we suggest a special class of distorted premium principles based on this distribution and we compare it with the well-known power dual premium principle. Since the mean of the new distribution can be normalized to give a simple parameter, this new model is appropriate to be used as a regression model when the response is bounded, being therefore an alternative to the beta regression model recently proposed in the statistical literature.
95 citations
TL;DR: In this paper, the authors considered the optimal reinsurance and investment problem in an unobservable Markov-modulated compound Poisson risk model, where the intensity and jump size distribution are not known but have to be inferred from the observations of claim arrivals.
Abstract: We consider the optimal reinsurance and investment problem in an unobservable Markov-modulated compound Poisson risk model, where the intensity and jump size distribution are not known but have to be inferred from the observations of claim arrivals. Using a recently developed result from filtering theory, we reduce the partially observable control problem to an equivalent problem with complete observations. Then using stochastic control theory, we get the closed form expressions of the optimal strategies which maximize the expected exponential utility of terminal wealth. In particular, we investigate the effect of the safety loading and the unobservable factors on the optimal reinsurance strategies. With the help of a generalized Hamilton–Jacobi–Bellman equation where the derivative is replaced by Clarke’s generalized gradient as in Bauerle and Rieder (2007), we characterize the value function, which helps us verify that the strategies we constructed are optimal.
85 citations
TL;DR: In this paper, the authors investigated an optimal investment strategy of DC pension plan in a stochastic interest rate framework, where the goal of the fund manager is to maximize the expectation of the constant relative risk aversion utility of the terminal value of the pension fund over a guarantee which serves as an annuity after retirement.
Abstract: This paper investigates an optimal investment strategy of DC pension plan in a stochastic interest rate and stochastic volatility framework. We apply an affine model including the Cox–Ingersoll–Ross (CIR) model and the Vasicek mode to characterize the interest rate while the stock price is given by the Heston’s stochastic volatility (SV) model. The pension manager can invest in cash, bond and stock in the financial market. Thus, the wealth of the pension fund is influenced by the financial risks in the market and the stochastic contribution from the fund participant. The goal of the fund manager is, coping with the contribution rate, to maximize the expectation of the constant relative risk aversion (CRRA) utility of the terminal value of the pension fund over a guarantee which serves as an annuity after retirement. We first transform the problem into a single investment problem, then derive an explicit solution via the stochastic programming method. Finally, the numerical analysis is given to show the impact of financial parameters on the optimal strategies.
82 citations
TL;DR: Two different methods of modeling multivariate claim counts using copulas based on the negative binomial distribution are considered and the results demonstrate the superiority of the copula-based approaches over the common shock model.
Abstract: It is no longer uncommon these days to find the need in actuarial practice to model claim counts from multiple types of coverage, such as the ratemaking process for bundled insurance contracts. Since different types of claims are conceivably correlated with each other, the multivariate count regression models that emphasize the dependency among claim types are more helpful for inference and prediction purposes. Motivated by the characteristics of an insurance dataset, we investigate alternative approaches to constructing multivariate count models based on the negative binomial distribution. A classical approach to induce correlation is to employ common shock variables. However, this formulation relies on the NB-I distribution which is restrictive for dispersion modeling. To address these issues, we consider two different methods of modeling multivariate claim counts using copulas. The first one works with the discrete count data directly using a mixture of max-id copulas that allows for flexible pair-wise association as well as tail and global dependence. The second one employs elliptical copulas to join continuitized data while preserving the dependence structure of the original counts. The empirical analysis examines a portfolio of auto insurance policies from a Singapore insurer where claim frequency of three types of claims (third party property damage, own damage, and third party bodily injury) are considered. The results demonstrate the superiority of the copula-based approaches over the common shock model. Finally, we implemented the various models in loss predictive applications.
76 citations
TL;DR: In this article, the authors investigated an optimal reinsurance and investment problem for an insurer whose surplus process is approximated by a drifted Brownian motion, and employed the stochastic dynamic programming to derive the closed-forms of the optimal reins insurance and investment strategies under the constant relative risk aversion (CRRA) utility maximization.
Abstract: In this paper, we investigate an optimal reinsurance and investment problem for an insurer whose surplus process is approximated by a drifted Brownian motion. Proportional reinsurance is to hedge the risk of insurance. Interest rate risk and inflation risk are considered. We suppose that the instantaneous nominal interest rate follows an Ornstein–Uhlenbeck process, and the inflation index is given by a generalized Fisher equation. To make the market complete, zero-coupon bonds and Treasury Inflation Protected Securities (TIPS) are included in the market. The financial market consists of cash, zero-coupon bond, TIPS and stock. We employ the stochastic dynamic programming to derive the closed-forms of the optimal reinsurance and investment strategies as well as the optimal utility function under the constant relative risk aversion (CRRA) utility maximization. Sensitivity analysis is given to show the economic behavior of the optimal strategies and optimal utility.
70 citations
TL;DR: In this article, generalized additive models for location, scale and shape define a flexible, semi-parametric class of regression models for analyzing insurance data in which the exponential family assumption for the response is relaxed.
Abstract: Generalized additive models for location, scale and, shape define a flexible, semi-parametric class of regression models for analyzing insurance data in which the exponential family assumption for the response is relaxed. This approach allows the actuary to include risk factors not only in the mean but also in other key parameters governing the claiming behavior, like the degree of residual heterogeneity or the no-claim probability. In this broader setting, the Negative Binomial regression with cell-specific heterogeneity and the zero-inflated Poisson regression with cell-specific additional probability mass at zero are applied to model claim frequencies. New models for claim severities that can be applied either per claim or aggregated per year are also presented. Bayesian inference is based on efficient Markov chain Monte Carlo simulation techniques and allows for the simultaneous estimation of linear effects as well as of possible nonlinear effects, spatial variations and interactions between risk factors within the data set. To illustrate the relevance of this approach, a detailed case study is proposed based on the Belgian motor insurance portfolio studied in Denuit and Lang (2004).
68 citations
TL;DR: In this paper, a simple decomposed annuity structure that enables cost transparency and could be linked to any investment fund is proposed, and participants can leave before death without financial penalty.
Abstract: The financial industry has recently seen a push away from structured products and towards transparency The trend is to decompose products, such that customers understand each component as well as its price Yet the enormous annuity market combining investment and longevity has been almost untouched by this development We suggest a simple decomposed annuity structure that enables cost transparency and could be linked to any investment fund It has several attractive features: (i) it works for any heterogeneous group; (ii) participants can leave before death without financial penalty; and (iii) participants have complete freedom over their own investment strategy
63 citations
TL;DR: In this paper, the authors show that the default loss of each debtor and the total default of all debtors are both approximately equal to a discrete compound Poisson distribution, and they give Le Cam's error bound between total default and a DCP distribution.
Abstract: Probability generating function (p.g.f.) is a powerful tool to study discrete compound Poisson (DCP) distribution. By applying inverse Fourier transform of p.g.f., it is convenient to numerically calculate probability density and do parameter estimation. As an application to finance and insurance, we firstly show that in the generalized CreditRisk+ model, the default loss of each debtor and the total default of all debtors are both approximately equal to a DCP distribution, and we give Le Cam’s error bound between the total default and a DCP distribution. Next, we consider geometric Brownian motion with DCP jumps and derive its r th moment. We establish the surplus process of the difference of two DCP distributions, and numerically compute the tail probability. Furthermore, we define the discrete pseudo compound Poisson (DPCP) distribution and give the characterizations and examples of DPCP distribution, including the strictly decreasing discrete distribution and the zero-inflated discrete distribution with P ( X = 0 ) > 0.5 .
61 citations
TL;DR: In this article, the authors analyzed the optimal dividend payment problem in the dual model under constant transaction costs and showed that for a general spectrally positive Levy process, an optimal strategy is given by a ( c 1, c 2 ) -policy that brings the surplus process down to c 1 whenever it reaches or exceeds c 2 for some 0 ≤ c 1 c 2.
Abstract: We analyze the optimal dividend payment problem in the dual model under constant transaction costs. We show, for a general spectrally positive Levy process, an optimal strategy is given by a ( c 1 , c 2 ) -policy that brings the surplus process down to c 1 whenever it reaches or exceeds c 2 for some 0 ≤ c 1 c 2 . The value function is succinctly expressed in terms of the scale function. A series of numerical examples are provided to confirm the analytical results and to demonstrate the convergence to the no-transaction cost case, which was recently solved by Bayraktar et al. (2013).
TL;DR: In this article, an optimal investment and reinsurance problem with delay for an insurer under the mean-variance criterion is concerned. But the problem is not addressed in this paper, since it is difficult to obtain the optimal solution in practice.
Abstract: This paper is concerned with an optimal investment and reinsurance problem with delay for an insurer under the mean–variance criterion. A three-stage procedure is employed to solve the insurer’s mean–variance problem. We first use the maximum principle approach to solve a benchmark problem. Then applying the Lagrangian duality method, we derive the optimal solutions for a variance-minimization problem. Based on these solutions, we finally obtain the efficient strategy and the efficient frontier of the insurer’s mean–variance problem. Some numerical examples are also provided to illustrate our results.
TL;DR: In this article, an insurer who wants to maximize its expected utility of terminal wealth by selecting optimal investment and risk control strategies is modeled by a jump-diffusion process and is negatively correlated with the capital gains in the financial market.
Abstract: Motivated by the AIG bailout case in the financial crisis of 2007–2008, we consider an insurer who wants to maximize his/her expected utility of terminal wealth by selecting optimal investment and risk control strategies. The insurer’s risk process is modeled by a jump-diffusion process and is negatively correlated with the capital gains in the financial market. We obtain explicit solutions of optimal strategies for various utility functions.
TL;DR: In this paper, the authors consider a periodic barrier strategy with exponential inter-dividend-decision times and continuous monitoring of solvency and show that it is the periodic strategy that maximizes the expected present value of dividends paid until ruin.
Abstract: The dual model with diffusion is appropriate for companies with continuous expenses that are offset by stochastic and irregular gains Examples include research-based or commission-based companies In this context, Bayraktar et al (2013a) show that a dividend barrier strategy is optimal when dividend decisions are made continuously In practice, however, companies that are capable of issuing dividends make dividend decisions on a periodic (rather than continuous) basis In this paper, we consider a periodic dividend strategy with exponential inter-dividend-decision times and continuous monitoring of solvency Assuming hyperexponential gains, we show that a periodic barrier dividend strategy is the periodic strategy that maximizes the expected present value of dividends paid until ruin Interestingly, a ‘liquidation-at-first-opportunity’ strategy is optimal in some cases where the surplus process has a positive drift Results are illustrated
TL;DR: This paper proposes several new notions of dependence to model dependent risks and give their characterizations through the probability measures or distributions of the risks or through the expectations of the transformed risks.
Abstract: Dependence structures of multiple risks play an important role in optimal allocation problems for insurance, quantitative risk management, and finance. However, in many existing studies on these problems, risks or losses are often assumed to be independent or comonotonic or exchangeable. In this paper, we propose several new notions of dependence to model dependent risks and give their characterizations through the probability measures or distributions of the risks or through the expectations of the transformed risks. These characterizations are related to the properties of arrangement increasing functions and the proposed notions of dependence incorporate many typical dependence structures studied in the literature for optimal allocation problems. We also develop the properties of these dependence structures. We illustrate the applications of these notions in the optimal allocation problems of deductibles and policy limits and in capital reserves problems. These applications extend many existing researches to more general dependent risks.
TL;DR: In this paper, the authors developed a stochastic model for individual claims reserving using observed data on claim payments as well as incurred losses, called the individual paid and incurred chain (iPIC) reserving method.
Abstract: This paper develops a stochastic model for individual claims reserving using observed data on claim payments as well as incurred losses. We extend the approach of Pigeon et al. (2013), designed for payments only, towards the inclusion of incurred losses. We call the new technique the individual Paid and Incurred Chain (iPIC) reserving method. Analytic expressions are derived for the expected ultimate losses, given observed development patterns. The usefulness of this new model is illustrated with a portfolio of general liability insurance policies. For the case study developed in this paper, detailed comparisons with existing approaches reveal that iPIC method performs well and produces more accurate predictions.
TL;DR: In this paper, an asset allocation problem for defined contribution pension funds with stochastic income and mortality risk under a multi-period mean-variance framework was investigated, where synthetically both to enhance the return and to control the risk by the mean variance criterion.
Abstract: This paper investigates an asset allocation problem for defined contribution pension funds with stochastic income and mortality risk under a multi-period mean–variance framework. Different from most studies in the literature where the expected utility is maximized or the risk measured by the quadratic mean deviation is minimized, we consider synthetically both to enhance the return and to control the risk by the mean–variance criterion. First, we obtain the analytical expressions for the efficient investment strategy and the efficient frontier by adopting the Lagrange dual theory, the state variable transformation technique and the stochastic optimal control method. Then, we discuss some special cases under our model. Finally, a numerical example is presented to illustrate the results obtained in this paper.
TL;DR: In this paper, the authors investigated the optimal risk-tolerant and counterparty default risk in the presence of the regulatory initial capital and the counterpart default risk, and derived optimal strategies that maximize the expected utility of an insurer's terminal wealth or minimize the value-at-risk (VaR) of its promised indemnity.
Abstract: In a reinsurance contract, a reinsurer promises to pay the part of the loss faced by an insurer in exchange for receiving a reinsurance premium from the insurer. However, the reinsurer may fail to pay the promised amount when the promised amount exceeds the reinsurer’s solvency. As a seller of a reinsurance contract, the initial capital or reserve of a reinsurer should meet some regulatory requirements. We assume that the initial capital or reserve of a reinsurer is regulated by the value-at-risk (VaR) of its promised indemnity. When the promised indemnity exceeds the total of the reinsurer’s initial capital and the reinsurance premium, the reinsurer may fail to pay the promised amount or default may occur. In the presence of the regulatory initial capital and the counterparty default risk, we investigate optimal reinsurance designs from an insurer’s point of view and derive optimal reinsurance strategies that maximize the expected utility of an insurer’s terminal wealth or minimize the VaR of an insurer’s total retained risk. It turns out that optimal reinsurance strategies in the presence of the regulatory initial capital and the counterparty default risk are different both from optimal reinsurance strategies in the absence of the counterparty default risk and from optimal reinsurance strategies in the presence of the counterparty default risk but without the regulatory initial capital.
TL;DR: In this paper, the authors proposed a technique to derive the optimal surrender strategy for a variable annuity (VA) as a function of the underlying fund value, based on splitting the value of the VA into a European part and an early exercise premium.
Abstract: This paper proposes a technique to derive the optimal surrender strategy for a variable annuity (VA) as a function of the underlying fund value. This approach is based on splitting the value of the VA into a European part and an early exercise premium following the work of Kim and Yu (1996) and Carr et al. (1992) . The technique is first applied to the simplest VA with GMAB (path-independent benefits) and is then shown to be possibly generalized to the case when benefits are path-dependent. Fees are paid continuously as a fixed percentage of the fund value. Our approach is useful to investigate the impact of path-dependent benefits on surrender incentives.
TL;DR: In this article, the authors investigated the time-consistent dynamic mean-variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality.
Abstract: This paper investigates the time-consistent dynamic mean–variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality. Using an HJB framework, we solve the hedging problem in which insurance liabilities follow a doubly stochastic Poisson process with an intensity rate that is correlated and cointegrated to the index mortality rate. The derived closed-form optimal hedging policy articulates the important role of cointegration in longevity hedging. We show numerically that a time-consistent hedging policy is a smoother function in time when compared with its time-inconsistent counterpart.
TL;DR: In this article, a portfolio selection problem in which security returns are given by experts' evaluations instead of historical data is discussed, and a factor method for evaluating security returns based on experts' judgment is proposed and a mean-chance model for optimal portfolio selection is developed taking transaction costs and investors preference on diversification and investment limitations on certain securities into account.
Abstract: This paper discusses a portfolio selection problem in which security returns are given by experts’ evaluations instead of historical data. A factor method for evaluating security returns based on experts’ judgment is proposed and a mean-chance model for optimal portfolio selection is developed taking transaction costs and investors’ preference on diversification and investment limitations on certain securities into account. The factor method of evaluation can make good use of experts’ knowledge on the effects of economic environment and the companies’ unique characteristics on security returns and incorporate the contemporary relationship of security returns in the portfolio. The use of chance of portfolio return failing to reach the threshold can help investors easily tell their tolerance toward risk and thus facilitate a decision making. To solve the proposed nonlinear programming problem, a genetic algorithm is provided. To illustrate the application of the proposed method, a numerical example is also presented.
TL;DR: In this paper, a bidimensional renewal risk model with constant interest force and dependent subexponential claims was considered and an explicit asymptotic formula for the finite-time ruin probability was derived.
Abstract: This paper considers a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution, we derive for the finite-time ruin probability an explicit asymptotic formula.
TL;DR: A goodness-of-fit test for bivariate copula models is constructed, using a canonical local Gaussian correlation and Gaussian pseudo-observations which make the test independent of the margins, so that it is a genuine test of the copula structure.
Abstract: In this paper we examine the relationship between a newly developed local dependence measure, the local Gaussian correlation, and standard copula theory. We are able to describe characteristics of the dependence structure in different copula models in terms of the local Gaussian correlation. Further, we construct a goodness-of-fit test for bivariate copula models. An essential ingredient of this test is the use of a canonical local Gaussian correlation and Gaussian pseudo-observations which make the test independent of the margins, so that it is a genuine test of the copula structure. A Monte Carlo study reveals that the test performs very well compared to a commonly used alternative test. We also propose two types of diagnostic plots which can be used to investigate the cause of a rejected null. Finally, our methods are applied to a “classical” insurance data set.
TL;DR: In this paper, two alternative extensions of the classical univariate conditional-tail-expectation (CTE) in a multivariate setting are introduced. And the proposed multivariate CTE-s satisfy natural extensions of positive homogeneity property, the translation invariance property and the comonotonic additivity property.
Abstract: In this paper, we introduce two alternative extensions of the classical univariate Conditional-Tail-Expectation (CTE) in a multivariate setting. The two proposed multivariate CTEs are vector-valued measures with the same dimension as the underlying risk portfolio. As for the multivariate Value-at-Risk measures introduced by Cousin and Di Bernardino (2013), the lower-orthant CTE ( resp. the upper-orthant CTE ) is constructed from level sets of multivariate distribution functions ( resp. of multivariate survival distribution functions). Contrary to allocation measures or systemic risk measures, these measures are also suitable for multivariate risk problems where risks are heterogeneous in nature and cannot be aggregated together. Several properties have been derived. In particular, we show that the proposed multivariate CTE-s satisfy natural extensions of the positive homogeneity property, the translation invariance property and the comonotonic additivity property. Comparison between univariate risk measures and components of multivariate CTE is provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Sub-additivity of the proposed multivariate CTE-s is provided under the assumption that all components of the random vectors are independent. Illustrations are given in the class of Archimedean copulas.
TL;DR: In this paper, the authors provide an extensive study of how different sets of financial and demographic parameters affect the fair guaranteed fee charged for a guaranteed lifetime withdrawal benefits (GLWB) as well as the profit and loss distribution, using tractable equity and stochastic mortality models in a continuous time framework.
Abstract: Guaranteed lifetime withdrawal benefits (GLWB) embedded in variable annuities have become an increasingly popular type of life annuity designed to cover systematic mortality risk while providing protection to policyholders from downside investment risk. This paper provides an extensive study of how different sets of financial and demographic parameters affect the fair guaranteed fee charged for a GLWB as well as the profit and loss distribution, using tractable equity and stochastic mortality models in a continuous time framework. We demonstrate the significance of parameter risk, model risk, as well as the systematic mortality risk component underlying the guarantee. We quantify how different levels of equity exposure chosen by the policyholder affect the exposure of the guarantee providers to systematic mortality risk. Finally, the effectiveness of a static hedge of systematic mortality risk is examined allowing for different levels of equity exposure.
TL;DR: In this article, a model for price calculations based on three components: a fair premium, price loadings reflecting general expenses and solvency requirements, and profit is proposed, where the first two components are typically evaluated on a yearly basis, while the third is viewed from a longer perspective.
Abstract: We consider a model for price calculations based on three components: a fair premium; price loadings reflecting general expenses and solvency requirements; and profit The first two components are typically evaluated on a yearly basis, while the third is viewed from a longer perspective When considering the value of customers over a period of several years, and examining policy renewals and cross-selling in relation to price adjustments, many insurers may prefer to reduce their short-term benefits so as to focus on their most profitable customers and the long-term value We show how models of personalized treatment learning can be used to select the policy holders that should be targeted in a company’s marketing strategies An empirical application of the causal conditional inference tree method illustrates how best to implement a personalized cross-sell marketing campaign in this framework
TL;DR: In this article, the authors further studied the capital allocation concerning mutually interdependent random risks and established that risk-averse insurers incline to evenly distribute the total capital among multiple risks.
Abstract: This paper further studies the capital allocation concerning mutually interdependent random risks. In the context of exchangeable random risks, we establish that risk-averse insurers incline to evenly distribute the total capital among multiple risks. For risk-averse insurers with decreasing convex loss functions, we prove that more capital should be allocated to the risk with the larger reversed hazard rate when risks are coupled by an Archimedean copula. Also, sufficient conditions are developed to exclude the worst capital allocations for random risks with some specific Archimedean copulas.
TL;DR: In this article, the authors investigate the impact of correlation risk on the optimal asset-liability management of an insurer, and derive the explicit closed-form solution to the optimal portfolio policy.
Abstract: Consider an insurer who invests in the financial market where correlations among risky asset returns are randomly changing over time. The insurer who faces the risk of paying stochastic insurance claims needs to manage her asset and liability by taking into account of the correlation risk. This paper investigates the impact of correlation risk to the optimal asset–liability management (ALM) of an insurer. We employ the Wishart process to model the stochastic covariance matrix of risky asset returns. The insurer aims to minimize the variance of the terminal wealth given an expected terminal wealth subject to the risk of paying out random liabilities of compound Poisson process. This ALM problem then becomes a linear–quadratic stochastic optimal control problem with stochastic volatilities, stochastic correlations and jumps. The recognition of an affine form in the solution process enables us to derive the explicit closed-form solution to the optimal ALM portfolio policy, obtain the efficient frontier, and identify the condition that the solution is well behaved.
TL;DR: In this paper, the problem of pricing and hedging variable annuity contracts for which the fee deducted from the policyholder's account depends on the account value is investigated, and an equation from which the guaranteed benefit can be calculated and a strategy which allows the insurer to hedge the benefit is characterized.
Abstract: We investigate the problem of pricing and hedging variable annuity contracts for which the fee deducted from the policyholder’s account depends on the account value. It is believed that state-dependent fees are beneficial to policyholders and insurers since they reduce policyholders’ incentives to lapse the policies and match the costs incurred by policyholders with the pay-offs received from embedded guarantees. We consider an incomplete financial market which consists of two risky assets modelled with a two-dimensional Levy process. One of the assets is a security which can be traded by the insurer, and the second asset is a security which is the underlying fund for the variable annuity contract. In our model we derive an equation from which the fee for the guaranteed benefit can be calculated and we characterize a strategy which allows the insurer to hedge the benefit. To solve the pricing and hedging problem in an incomplete financial market we apply a quadratic objective.
TL;DR: In this paper, the authors derived the optimal solutions with premium constraint under two special distortion risk measures (VaR and TVaR) under a general feasible region which requires the retained loss function to be increasing and left-continuous.
Abstract: Recently distortion risk measure has been an interesting tool for the insurer to reflect its attitude toward risk when forming the optimal reinsurance strategy. Under the distortion risk measure, this paper discusses the reinsurance design with unbinding premium constraint and the ceded loss function in a general feasible region which requiring the retained loss function to be increasing and left-continuous. Explicit solution of the optimal reinsurance strategy is obtained by introducing a premium-adjustment function. Our result has the form of layer reinsurance with the mixture of normal reinsurance strategies in each layer. Finally, to illustrate the applicability of our results, we derive the optimal reinsurance solutions with premium constraint under two special distortion risk measures—VaR and TVaR.