“GMDB+GMAB”型变额年金投资组合保险策略绩效比较

参照我国目前保险市场上的三款变额年金产品,本文设计了具有最低身故利益保证 (GMDB)和最低累计利益保证(GMAB)的变额年金保单,并从内部收益率、退保率及保证成本三方面,综合评估固定乘数CPPI及TIPP、动态乘数CPPI及TIPP四种策略的绩效。研究结果显示,对照固定乘数投资组合保险策略,动态风险乘数策略能有效提高内部收益率;在考虑动态退保情形下,无锁利机制的CPPI策略会导致更多的退保率;交易成本在很大程度上影响保证成本,采用市场波动调整法可以在对内部收益率影响较小的情况下有效降低交易成本。此外,动态乘数调整中参数的设定对投资策略绩效的影响大小不一。     

对我国开展变额年金产品试点的几点思考

变额年金保险，是指包含保险保障功能，保单利益与连续的投资账户投资单位价格相关联，同时按照保单约定具有最低保单利益保证需要的人身保险。变额年金保险应当约定年金给付保险责任，或提供满期保险金转换为年金的选择权。     

变额年金动态对冲策略的研究与实施

自2010年中国保监会允许国内试点变额年金市场之后,国内还没有相关文章对内部组合对冲模式下变额年金产品在精算领域面临的问题做出系统的阐述,本文在这方面给出了一套方法并结合实际进行了验证。通过经济情景发生器产生合理的经济情景,并基于假设的保单数据分析变额年金保证利益的合理定价区间;然后深入地讨论了动态对冲过程中各环节的设计并对结果进行分析,结果表明动态对冲可以显著降低各期损益以及累积损益的波动性;最后对变额年金的准备金和资本计算进行了国际比较,并给出了反映对冲与不反映对冲情况下的结果比较。     

我国寿险保单退保影响因素的实证研究——基于公司层面的微观数据

从微观的角度,利用中国寿险市场的统计数据考察了寿险公司经营信息对保单持有人退保行为的影响,考虑到不同类型公司受影响程度的差异性,将样本数据分成大型寿险公司和中小寿险公司两类,并建立非线性平滑转换面板数据模型来研究不同类型的寿险公司退保率受各因素影响的差异。实证分析的结果表明：是否销售投连险显著地影响中小寿险公司的退保率,对大型寿险公司则无影响;与寿险公司规模高增长相伴随的是保单的高退保,这在中国寿险行业普遍存在;保单分红水平对于退保的影响程度要低于预期,“利率替代”效应并不是导致退保的最主要原因;保单持有人会比较信赖历史悠久的公司,而保费水平的高低并不是影响保单持有人是否退保的重要因素。     

固定乘数平衡管理模式下变额年金的风险管理

本文在固定乘数平衡管理模式下讨论了最低生存保证给付保险(GMLB)和最低死亡保证给付保险(GMDB)两类变额年金的风险管理问题。结果表明,如果采用固定乘数平衡管理模式管理变额年金,适当的最低收益率保证不会带来风险,但当最低收益率超过一定水平时,变额年金业务必然破产。另外当变额年金业务有外部现金流时,保险公司可以综合衡量市场、收益和风险三个因素,决定资本回报率、外部现金流规模和保证收益率。     

我国分红保险监管模式的探讨

分红保险是世界各国寿险公司规避利率风险，保证自身稳健经营的有效手段。相对于传统保证型的寿险保单，分红保单向保单持有人提供的是非保证的保险利益，红利的分配还会影响保险公司的负债水平、投资策略以及偿付能力。为了保障保单持有人的利益和保证保险公司的持续经营，各国保险监管机构都非常重视对分红保险的监管，除了将保险监管的重点集中在分红产品的红利演示、分红基金的红利分配、分红基金的信息披露、保单持有人的合理预期和分红基金的负债确认等方面外，对不同的红利分配方式形成了不同的监管模式。     

马尔科夫体制转换模型下变额年金的风险管理

本文在马尔科夫体制转换模型下探究了变额年金的风险管理问题.保险产品的长期性使其易受到经济周期的影响,马尔科夫体制转换模型因在描述经济周期变化的卓越表现而受到学界和业界的广泛关注.分析显示,在我国马尔科夫体制转换模型相对于Black-Scholes模型在捕获资产变化特征上具有优势.本文利用体制转换模型建模投资账户价值,推导最低身故利益保证和最低到期利益保证债务的在险价值和条件尾部期望的解析表达式.将两个估计模型应用到最低利益保证年金上,发现Black-Scholes模型往往低估最低利益保证年金需要的风险资本,这种结果与Black-Scholes模型下资产收益的薄尾特征是一致的.另外,通过将模型参数分为与体制转换相关参数和与体制转换无关参数,本文也分析了风险测度对两类参数的敏感度,极大地丰富了现存关于最低利益保证年金风险管理的研究.     

浅议变额年金的发展

变额年金的产生及发展变额年金最早出现于1952年,由美国教师保险和年金协会下设的大学退休股权基金推出,是大学教师退休金计划的一部分。得益于最低利益保证(GMxB)的引进,20世纪末其销量急剧膨胀,并且亚太、欧洲多国保险市     

变额年金在我国的应用及风险管理探讨

采用对比法和归纳法,从变额年金的风险因素入手,通过对比国外比较成熟的风险管理评估模式和风险管理需要的外部条件等因素的探讨,为变额年金在风险管理模式、最低保证利益设计等方面提出一定的建议。     

变额年金保险:一场革命性的变革

变额年金保险兼顾养老、投资、最低保证三方面功能,符合我国保险消费者的偏好。2011年5月,中国保监会发布《关于开展变额年金保险试点的通知》(简称《通知》)和《变额年金保险管理暂行办法》(简称《暂行办法》),这标志着国内寿险产品创新在经过多年的沉寂之后,一种新型的寿险理财产品终于要面世了。近十年来,养老年金类产品发展     

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

We present regression-based Monte Carlo simulation algorithm for solving the stochastic control models associated with pricing and hedging of the guaranteed lifelong withdrawal benefit (GLWB) in variable annuities, where the dynamics of the underlying fund value is assumed to evolve according to the stochastic volatility model. The GLWB offers a lifelong withdrawal benefit, even when the policy account value becomes zero, while the policyholder remains alive. Upon death, the remaining account value will be paid to the beneficiary as a death benefit. The bang-bang control strategy analysed under the assumption of maximization of the policyholder’s expected cash flow reduces the strategy space of optimal withdrawal policies to three choices: zero withdrawal, withdrawal at the contractual amount or complete surrender. The impact on the GLWB value under various withdrawal behaviours of the policyholder is examined. We also analyse the pricing properties of GLWB subject to different model parameter values and structural features.     

The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours

In this paper, we present a dynamic programming algorithm for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general Lévy processes framework. The GMWB gives the policyholder the right to make periodical withdrawals from her policy account even when the value of this account is exhausted. Typically, the total amount guaranteed for withdrawals coincides with her initial investment, providing then a protection against downside market risk. At each withdrawal date, the policyholder has to decide whether, and how much, to withdraw, or to surrender the contract. We show how different policyholder’s withdrawal behaviours can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for different contractual and market parameters, policyholder behaviours and different types of Lévy processes.     

Mitigating Interest Rate Risk in Variable Annuities: An Analysis of Hedging Effectiveness under Model Risk

Variable annuities are investment vehicles offered by insurance companies that combine a life insurance policy with long-term financial guarantees. These guarantees expose the insurer to market risks, such as volatility and interest rate risks, which can be managed only with a hedging strategy. The objective of this article is to study the effectiveness of dynamic delta-rho hedging strategies for mitigating interest rate risk in variable annuities with either a guaranteed minimum death benefit or guaranteed minimum withdrawal benefit rider. Our analysis centers on three important practical issues: (1) the robustness of delta-rho hedging strategies to model uncertainty, (2) the impact of guarantee features (maturity versus withdrawal benefits) on the performance of the hedging strategy, and (3) the importance of hedging interest rate risk in either a low and stable or rising interest rate environment. Overall, we find that the impact of interest rate risk is equally felt for the two types of products considered, and that interest rate hedges do lead to a significant risk reduction for the insurer, even when the ongoing low interest rate environment is factored in.     

Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even to surrender the contract. For numerical valuation of the GMWB rider, we use willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve a similar level of numerical accuracy. The design of our pricing algorithm also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine the effectiveness of delta hedging when the fund dynamics exhibit various jump levels.     

Discrete hedging of American-type options using local risk minimization

Local risk minimization and total risk minimization discrete hedging have been extensively studied for European options [e.g., Schweizer, M., 1995. Variance-optimal hedging in discrete time. Mathematics of Operation Research 20, 1–32; Schweizer, M., 2001. A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanic, J., Musiela, M., Option pricing, interest rates and risk management, Cambridge University Press, pp. 538–574]. In practice, hedging of options with American features is more relevant. For example, equity linked variable annuities provide surrender benefits which are essentially embedded American options. In this paper we generalize both quadratic and piecewise linear local risk minimization hedging frameworks to American options. We illustrate that local risk minimization methods outperform delta hedging when the market is highly incomplete. In addition, compared to European options, distributions of the hedging costs are typically more skewed and heavy-tailed. Moreover, in contrast to quadratic local risk minimization, piecewise linear risk minimization hedging strategies can be significantly different, resulting in larger probabilities of small costs but also larger extreme cost.     

Variable annuities in a Lévy-based hybrid model with surrender risk

This paper proposes a market consistent valuation framework for variable annuities (VAs) with guaranteed minimum accumulation benefit, death benefit and surrender benefit features. The setup is based on a hybrid model for the financial market and uses time-inhomogeneous Lévy processes as risk drivers. Further, we allow for dependence between financial and surrender risks. Our model leads to explicit analytical formulas for the quantities of interest, and practical and efficient numerical procedures for the evaluation of these formulas. We illustrate the tractability of this approach by means of a detailed sensitivity analysis of the fair value of the VA and its components with respect to the model parameters. The results highlight the role played by the surrender behaviour and the importance of its appropriate modelling.     

Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks

Recent variable annuities offer participation in the equity market and attractive protection against downside movements. Accurately quantifying this additional equity market risk and robustly hedging options embedded in the guarantees of variable annuities are new challenges for insurance companies. Due to sensitivities of the benefits to tails of the account value distribution, a simple Black–Scholes model is inadequate in preventing excessive liabilities. A model which realistically describes the real world price dynamics over a long time horizon is essential for the risk management of the variable annuities. In this article, both jump risk and volatility risk are considered for risk management of lookback options embedded in guarantees with a ratchet feature. We evaluate relative performances of delta hedging and dynamic discrete risk minimization hedging strategies. Using the underlying as the hedging instrument, we show that, under a Black–Scholes model, local risk minimization hedging can be significantly better than delta hedging. In addition, we compare risk minimization hedging using the underlying with that of using standard options. We demonstrate that, under a Merton's jump diffusion model, hedging using standard options is superior to hedging using the underlying in terms of the risk reduction. Finally, we consider a market model for volatility risks in which the at‐the‐money implied volatility is a state variable. We compute risk minimization hedging by modeling at‐the‐money Black–Scholes implied volatility explicitly; the hedging effectiveness is evaluated, however, under a joint model for the underlying price and implied volatility. Our computational results suggest that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options, compared to hedging using the underlying, can potentially be more effective in risk reduction under both jump and volatility risks.     

Efficient Greek Calculation of Variable Annuity Portfolios for Dynamic Hedging: A Two-Level Metamodeling Approach

The financial risk associated with the guarantees embedded in variable annuities cannot be addressed adequately by traditional actuarial techniques. Dynamical hedging is used in practice to mitigate the financial risk arising from variable annuities. However, a major challenge of dynamical hedging is to calculate the dollar Deltas of a portfolio of variable annuities within a short time interval so that rebalancing can be done on a timely basis. In this article, we propose a two-level metamodeling approach to efficiently estimating the partial dollar Deltas of a portfolio of variable annuities under a multiasset framework. The first-level metamodel is used to estimate the partial dollar Deltas at some well-chosen market levels, and the second-level metamodel is used to estimate the partial dollar Deltas at the current market level based on the precalculated partial dollar Deltas. Our numerical results show that the proposed approach performs well in terms of accuracy and speed.     

Indifference Pricing of a GLWB Option in Variable Annuities

We investigate the valuation problem of variable annuities with guaranteed lifelong/lifetime withdrawal benefit (GLWB) options, which give the policyholder the right to withdraw a specified amount as long as he or she lives, regardless of the performance of the investment. We assume the static approach that the policyholder’s withdrawal rate is a constant throughout the life of the contract. We apply the principle of equivalent utility to find the indifference price for a variable annuity with a GLWB contract with an equity-indexed death benefit. Using an exponential utility function, Hamilton-Jacobi-Bellman (HJB) type partial differential equations (PDEs) are derived for the pricing functions. We first assume the mortality is deterministic, and the pricing PDE is solved numerically using a finite difference method. The effects of various parameters are investigated, including the age at inception of the policyholder, withdrawal rate, risk-free rate, and volatility of the underlying asset. We also consider a roll-up option and analyze the effect of delaying the start of the withdrawals. Another pricing PDE is derived with a stochastic mortality, when the force of mortality is modeled with a stochastic differential equation. A finite difference method is used again to solve the pricing PDE numerically, and the sensitivities of the GLWB contracts with respect to the withdrawal rate and the risk-free rate are explored.     

Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method

This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Lévy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. [Scand. Actuar. J., 2014, 1–20], and Luo and Shevchenko [Int. J. Financ. Eng., 2014, 2, 1–24], to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. [op. cit.]. We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging intra-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Lévy framework. While the second moment and variance have been considered in existing literature, we show that the Value-at-Risk (VaR) may also be of interest as a risk measure to minimise risk in variable annuities portfolios.