Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility,stochastic interest rates and stochastic mortality via the componentwise splitting method

This paper values guaranteed minimum withdrawal benefit (GMWB) riders embedded in variable annuities assuming that the underlying fund dynamics evolve under the influence of stochastic interest rates, stochastic volatility, stochastic mortality and equity risk. The valuation problem is formulated as a partial differential equation (PDE) which is solved numerically by employing the operator splitting method. Sensitivity analysis of the fair guarantee fee is performed with respect to various model parameters. We find that (i) the fair insurance fee charged by the product provider is an increasing function of the withdrawal rate; (ii) the GMWB price is higher when stochastic interest rates and volatility are incorporated in the model, compared to the case of static interest rates and volatility; (iii) the GMWB price behaves non-monotonically with changing volatility of variance parameter; (iv) the fair fee increases with increasing volatility of interest rates parameter, and increasing correlation between the underlying fund and the interest rates; (v) the fair fee increases when the speed of mean-reversion of stochastic volatility or the average long-term volatility increases; (vi) the GMWB fee decreases when the speed of mean-reversion of stochastic interest rates or the average long-term interest rates increase. We investigate both static and dynamic (optimal) policyholder's withdrawal behaviours; we present the optimal withdrawal schedule as a function of the withdrawal account and the investment account for varying volatility and interest rates. When incorporating stochastic mortality, we find that its impact on the fair guarantee fee is rather small. Our results demonstrate the importance of correct quantification of risks embedded in GMWBs and provide guidance to product providers on optimal hedging of various risks associated with the contract.     

Analysis of Participating Life Insurance Contracts: A Unification Approach

Fair pricing of embedded options in life insurance contracts is usually conducted by using risk‐neutral valuation. This pricing framework assumes a perfect hedging strategy, which insurance companies can hardly pursue in practice. In this article, we extend the risk‐neutral valuation concept with a risk measurement approach. We accomplish this by first calibrating contract parameters that lead to the same market value using risk‐neutral valuation. We then measure the resulting risk assuming that insurers do not follow perfect hedging strategies. As the relevant risk measure, we use lower partial moments, comparing shortfall probability, expected shortfall, and downside variance. We show that even when contracts have the same market value, the insurance company's risk can vary widely, a finding that allows us to identify key risk drivers for participating life insurance contracts.     

Fair valuation of cliquet-style return guarantees in (homogeneous and) heterogeneous life insurance portfolios

Participating life insurance contracts allow the policyholder to participate in the annual return of a reference portfolio. Additionally, they are often equipped with an annual (cliquet-style) return guarantee. The current low interest rate environment has again refreshed the discussion on risk management and fair valuation of such embedded options. While this problem is typically discussed from the viewpoint of a single contract or a homogeneous* insurance portfolio, contracts are, in practice, managed within a heterogeneous insurance portfolio. Their valuation must then – unlike the case of asset portfolios – take account of portfolio effects: Their premiums are invested in the same reference portfolio; the contracts interact by a joint reserve, individual surrender options and joint default risk of the policy sponsor. Here, we discuss the impact of portfolio effects on the fair valuation of insurance contracts jointly managed in (homogeneous and) heterogeneous life insurance portfolios. First, in a rather general setting, including stochastic interest rates, we consider the case that otherwise homogeneous contracts interact due to the default risk of the policy sponsor. Second, and more importantly, we then also consider the case when policies are allowed to differ in further aspects like the guaranteed rate or time to maturity. We also provide an extensive numerical example for further analysis.     

Assessing the Risk Potential of Premium Payment Options in Participating Life Insurance Contracts

Most life insurance contracts embed the right to stop premium payments during the term of the contract (paid-up option). Thereby, the contract is not terminated but continues with reduced benefits and often provides the right to resume premium payments later, thus increasing the previously reduced benefits (resumption option). In our analysis, we start with a basic contract with two standard options, namely, an interest rate guarantee and annual surplus participation. Next, in addition to the features of the basic contract, a paid-up and resumption option is included in the framework. The valuation process is not based on assumptions about a particular policyholders' exercise strategy but instead assesses the risk potential from the insurer's viewpoint by providing an upper bound for any possible exercise behavior. This approach provides important information to the insurer about the potential hazard of offering the paid-up and resumption option. Further, the approach allows an analysis of the impact of guaranteed interest rate, annual surplus participation, and investment volatility on the values of the premium payment options.     

Competitive equilibrium with type convergence in an asymmetrically informed market

This article studies an asymmetric information game with 'typeconvergence', in which, under some realizations of a commonuncertainty, inducing informed agents to reveal their typesthrough self-selection by contract choice is either costly orimpossible. Under other realizations, self-selection permitscostless distinctions between informed agents. I obtain sufficientconditions under which contracting with options prior to therealization of the common uncertainty leads to the existenceof a perfectly separating, costless Nash equilibrium. Applicationsto variable rate loan commitments and life insurance contractingare discussed.     

Fair insurance guaranty premia in the presence of risk-based capital regulations,stochastic interest rate and catastrophe risk

A multiperiod model is developed to measure the costs posed to the guaranty fund in a setting that incorporates risk-based capital regulations, interest rate risk and the possibility of catastrophic losses. The guaranty contract is modeled as a put option on the asset of the insurance company with a stochastic strike price and an uncertain maturity. The impacts of the key factors of this model are examined numerically and shown to make material differences in the costs to the guaranty fund.     

Price presentation and consumers’ choice

Currently, regulatory authorities and consumers ask for more cost transparency with respect to financial product components. In life insurance, for instance, the premium for products should be split in its components: A premium for death benefits, the savings premium, the cost of an investment guarantee, and the administration costs. In this regard, it is important for insurance companies and regulators to know to what extent the way of presenting the prices of an offer affects consumer evaluation of the product. Based on a paper by Huber et al. (How do price presentation effects influence consumer choice? The case of life insurance products. Working paper, 2011) as presented at the annual meeting of Deutscher Verein für Versicherungswissenschaft in 2011, this article presents the effects of different forms of presenting the price of life insurance contract components and especially of investment guarantees on consumer evaluation of this product. This is done by means of an experimental study using a representative panel for Switzerland and by focusing on unit-linked life insurance products. The findings reveal that, contrary to consumer products, there is no effect of price bundling and price optic on consumer evaluation and purchase intention for life insurance products. However, there is a significant moderating effect of consumer experience with insurance products on this relationship.     

Uniqueness of the fair premium for equity-linked life insurance contracts

An equity-linked life insurance contract combines an endowment life insurance and an investment strategy with a minimum guarantee. The benefit of this contract is determined by the guaranteed amount plus a bonus equal to a call on the portfolio. This bonus is similar to an Asian option. This article analyzes the relationship between the periodic insurance premium and its proportional share invested into the portfolio. For a general model of the financial risks we show the existence and uniqueness of an insurance premium. Furthermore the premium is strictly increasing and convex as a function of the share invested.     

Implicit Options in Life Insurance: Valuation and Risk Management

Participating life insurance contracts typically contain various types of implicit options. These implicit options can be very valuable and can thus represent a significant risk to insurance companies if they practice insufficient risk management. Options become especially risky through interaction with other options included in the contracts, which makes their evaluation even more complex. This article provides a comprehensive overview and classification of implicit options in participating life insurance contracts and discusses the relevant literature. It points out the potential problems particularly associated with the valuation of rights to early exercise due to policyholder exercise behavior. The risk potential of the interaction of implicit options is illustrated with numerical examples by means of a life insurance contract that includes common implicit options, i.e., a guaranteed interest rate, stochastic annual surplus participation, and paid-up and resumption options. Valuation is conducted using risk-neutral valuation, a concept that implicitly assumes the implementation of risk management measures such as hedging strategies.     

Asset and liability management for insurance products with minimum guarantees: The UK case

Modern insurance products are becoming increasingly complex, offering various guarantees, surrender options and bonus provisions. A case in point are the with-profits insurance policies offered by UK insurers. While these policies have been offered in some form for centuries, in recent years their structure and management have become substantially more involved. The products are particularly complicated due to the wide discretion they afford insurers in determining the bonuses policyholders receive. In this paper, we study the problem of an insurance firm attempting to structure the portfolio underlying its with-profits fund. The resulting optimization problem, a non-linear program with stochastic variables, is presented in detail. Numerical results show how the model can be used to analyze the alternatives available to the insurer, such as different bonus policies and reserving methods.     

Requiem for a market: an analysis of the rise and fall of a financial futures contract

Futures contracts often include a variety of delivery optionsthat allow participants flexibility in satisfying the contract.These options have the potential to broaden the appeal of thecontact. However, if these options are valuable, they may reducethe hedging effectiveness of the contract. This article analyzesthe GNMA CDR futures contract that appears to have failed becauseof flaws in the contract's design. For the first 6 years followingits introduction, the contract attracted significant and increasingvolume, but, subsequently, the volume declined to almost zero.Over the years during which the volume experienced its mostdramatic decline, the Treasury-bond futures contract provideda better hedge for current coupon GNMA securities than did theGNMA CDR futures contract. And, over this same period, the valueof the quality option embedded in the contract often exceeded5 percent of the futures price and reached a level of 19 percentat one point. We interpret the evidence to indicate that thecontract failed because the delivery options reduced the hedgingeffectiveness of the contract for current coupon mortgage securities.     

寿险公司负债价值计量研究

有效的资产负债管理以寿险公司资产和负债价值的正确计量为基础。寿险负债的长期性和多种嵌入期权的特性使寿险公司资产负债管理更具复杂性。寿险负债价值计量的精算方法注重于它的死亡率风险测定的技术秘诀和期限的长期性,但该方法使期权的效应无从体现,也使在利率和被保险人行为的静态假设不成立的前提下,负债价值被低估的情况发生。因此,必须辅之于财务方法来测量期权在寿险合约中的价值。本文对寿险合约中期权价值的测量及其对ALM的作用也做了探讨。     

Hedging and Reserving for Single-Premium Segregated Fund Contracts

Abstract

Three methods for determining suitable provision for maturity guarantees for single-premium segregated fund contracts are compared. Actuarial reserving assumes funds are held in risk-free assets, to give a prescribed probability of meeting the guarantee liability. Dynamic hedging uses the Black-Scholes framework to determine the replicating portfolio. Static hedging assumes a counterparty is willing to sell the options required to meet the guarantee. Using a stochastic cash flow projection, we consider how to assess which approach is most profitable. The example given assumes a typical Canadian segregated fund contract.     

A Finite Difference Approach to the Valuation of Path Dependent Life Insurance Liabilities

This paper sets up a model for the valuation of traditional participating life insurance policies. These claims are characterized by their explicit interest rate guarantees and by various embedded option elements, such as bonus and surrender options. Owing to the structure of these contracts, the theory of contingent claims pricing is a particularly well-suited framework for the analysis of their valuation.The eventual benefits (or pay-offs) from the contracts considered crucially depend on the history of returns on the insurance company's assets during the contract period. This path-dependence prohibits the derivation of closed-form valuation formulas but we demonstrate that the dimensionality of the problem can be reduced to allow for the development and implementation of a finite difference algorithm for fast and accurate numerical evaluation of the contracts. We also demonstrate how the fundamental financial model can be extended to allow for mortality risk and we provide a wide range of numerical pricing results.     

Investment guarantees in unit-linked life insurance from the customer perspective

Interest rate guarantees are a typical contract feature in unit-linked-life insurance products. As the financial crisis of 2007/2008 has shown, these guarantees can be of substantial value for policyholders since they ensure that at least a minimum amount will be paid back even if the mutual fund value falls below a specific guaranteed level. However, from the insurance company’s view, these guarantees can be costly—especially in highly volatile markets—due to the required risk management measures which must be undertaken to secure the guarantees promised to the customers. Thus, the aim of this paper is to investigate whether customers really value these guarantees and if their willingness to pay (WTP) is sufficient to cover the guarantee costs. To elicit customer WTP, we use an online questionnaire and compare these results to the actual guarantee costs calculated with the Black and Scholes option pricing formula. One main finding is that even though most of the participants in the online questionnaire work in the financial industry, subjective prices are difficult to derive and are lower, on average, than the prices obtained using a financial pricing model. However, many participants are still willing to pay a substantially higher price.     

Genetik und Versicherung aus ökonomischer Sicht

For some years direct gene tests have been available for many diseases and nearly every week new gene mutations are found and linked with diseases. By improving the diagnostic possibilities consequences for certain insurance branches will arise. Health and life insurance in particular will perhaps face radical chances, especially concerning the acceptance of contracts. If there is a contract between a policy holder and an insurance company, it is more important whether the cost-effectiveness of genetic screening is given. Exemplary this is calculated for a population-based screening for hemochromatosis. Furthermore it is relevant, whether the population will use available gene tests. The results of a pilot study will be presented. They show that there is a market for such products and that consequences for insurance companies are likely.     

Lapse rate modeling: a rational expectation approach

The surrender option embedded in many life insurance products is a clause that allows policyholders to terminate the contract early. Pricing techniques based on the American Contingent Claim (ACC) theory are often used, though the actual policyholders' behavior is far from optimal. Inspired by many prepayment models for mortgage backed securities, this paper builds a Rational Expectation (RE) model describing the policyholders' behavior in lapsing the contract. A market model with stochastic interest rates is considered, and the pricing is carried out through numerical approximation of the corresponding two-space-dimensional parabolic partial differential equation. Extensive numerical experiments show the differences in terms of pricing and interest rate elasticity between the ACC and RE approaches as well as the sensitivity of the contract price with respect to changes in the policyholders' behavior.     

个体差异、企业特征、制度保护与流动人口社会保险可及性——基于“福利三角”理论模型的实证研究

"福利三角"理论是现代社会政策研究的重要范式,其也可运用于经验研究。本文基于"福利三角"理论视角,从个体差异、企业特征、制度保障三个维度,利用南开大学"2013年流动人口管理和服务对策研究问卷调查"数据,对流动人口社会保险可及性及其影响因素进行实证研究。分析发现,在个体特征方面,受教育程度、工作年限等对流动人口人力资本具有关键影响的因素,企业性质、行业类型等企业特征,劳动合同、合同协商、工会等制度保障因素对流动人口社会保险可及性具有显著影响。在此基础上,应提高流动人口社会保险可及性。     

Interaction effects between dynamic hybrid products and traditional deferred annuities in the German life insurance market

Financial Markets and Portfolio Management - Dynamic hybrid products emerged in 2007 and are now well established in the German life insurance market. In this article, we study interaction effects...