Unit-linked Life Insurance Contracts Research Articles

Gloss to the Judgment of the Court of Justice of the European Union of 24 February 2020 in Joined Cases C-143/20 and C-213/20 on the Information Disclosure of Life Insurance Undertakings in Unit-linked Insurance Contracts

The gloss refers to the judgment of the Court of Justice of the European Union of 24 February 2020 in joined cases C-143/20 and C-213/20 concerning information disclosure of life insurance undertakings in unit-linked life insurance. This judgment provides guidance on information obligations in this type of insurance with respect to entities whom the information is to be communicated and as regards the scope of the information provided. Said issues seem to be essential for insurance practice, yet they fail to be clearly defined by legislation in force or in case law. The gloss does not address all the questions that have been submitted to the Court, focusing only on the scope of the information obligations in unit-linked life insurance contracts.

Ban on Distributing Certain Unit Linked Life Insurance Contracts as the Supervisory Product Intervention of the Polish Financial Supervision Authority (PFSA)

The article aims at discussing practical aspects of a legal regulation providing the European Supervisory Authorities (ESAs) and national supervisory authorities with the power to apply supervisory product intervention measures, particularly for insurance investment products. In addition to presenting the essence of product decisions, the author outlines the legal basis for product intervention powers of both the EU and national supervisory authorities related to unit linked life insurance contracts. Moreover, the article analyses the reasons and contents of the product decision issued by the Polish Financial Supervision Authority regarding the ban on distributing certain unit linked life insurance contracts, with the ban being effective from January 1, 2022.

Time-consistent mean-variance investment with unit linked life insurance contracts in a jump-diffusion setting

We consider a time-consistent mean–variance portfolio selection problem of an insurer and allow for the incorporation of basis (mortality) risk. The optimal solution is identified with a Nash subgame perfect equilibrium. We characterize an optimal strategy as solution of a system of partial integro-differential equations (PIDEs), a so called extended Hamilton–Jacobi–Bellman (HJB) system. We prove that the equilibrium is necessarily a solution of the extended HJB system. Under certain conditions we obtain an explicit solution to the extended HJB system and provide the optimal trading strategies in closed-form. A simulation shows that the previously found strategies yield payoffs whose expectations and variances are robust regarding the distribution of jump sizes of the stock. The same phenomenon is observed when the variance is correctly estimated, but erroneously ascribed to the diffusion components solely. Further, we show that differences in the insurance horizon and the time to maturity of a longevity asset do not add to the variance of the terminal wealth.

Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

<p style='text-indent:20px;'>This paper studies the hedging problem of unit-linked life insurance contracts in an incomplete market presence of self-exciting (clustering) effect, which is described by a Hawkes process. Applying the local risk-minimization method, we manage to obtain closed-form expressions of the locally risk-minimizing hedging strategies for both pure endowment and term insurance contracts. Besides, we demonstrate the existence of the minimal martingale measure and perform numerical analyses. Our numerical results indicate that jump clustering has a significant impact on the optimal hedging strategies.</p>

A Gloss to the Judgment of the Court of Appeal in Warsaw of 6 November 2020 (I ACa 17/20) Regarding the Invalidity of a Unit-linked Life Insurance Contract

The gloss presents the issue of invalidity of a unit-linked life insurance contract. A characteristic feature of such a contract is the combination of insurance and investment elements in one agreement. Since some contracts have a predominant investment character, their basic function is, in principle, to invest money in a way specified in the agreement – in the form of a clearly distinct unit-linked fund, which is an immanent element of the contract. As far as unit-linked life insurance contracts are concerned, it should be generally assumed that the legislator has left a lot of leeway to form the contents of the agreement, especially with regard to the determination of the amount of benefits, establishment and liquidation of unit-linked funds, terms of premium allocation as well as transfer of the accumulated means between specific unit-linked funds.

Variance and Interest Rate Risk in Unit-Linked Insurance Policies

One of the risks derived from selling long-term policies that any insurance company has arises from interest rates. In this paper, we consider a general class of stochastic volatility models written in forward variance form. We also deal with stochastic interest rates to obtain the risk-free price for unit-linked life insurance contracts, as well as providing a perfect hedging strategy by completing the market. We conclude with a simulation experiment, where we price unit-linked policies using Norwegian mortality rates. In addition, we compare prices for the classical Black-Scholes model against the Heston stochastic volatility model with a Vasicek interest rate model.

A benchmark approach to risk-minimization under partial information

The goal of this paper is to investigate (locally) risk-minimizing hedging strategies under the benchmark approach in a financial semimartingale market model where there are restrictions on the available information. More precisely, we characterize the optimal strategy as the integrand appearing in the Galtchouk–Kunita–Watanabe decomposition of the benchmarked contingent claim under partial information and provide its description in terms of the integrand in the classical Galtchouk–Kunita–Watanabe decomposition under full information via dual predictable projections. Finally we show how these results can be applied to unit-linked life insurance contracts.

Deterministic mean-variance-optimal consumption and investment

In dynamic optimal consumption–investment problems one typically aims to find an optimal control from the set of adapted processes. This is also the natural starting point in case of a mean-variance objective. In contrast, we solve the optimization problem with the special feature that the consumption rate and the investment proportion are constrained to be deterministic processes. As a result we get rid of a series of unwanted features of the stochastic solution including diffusive consumption, satisfaction points and consistency problems. Deterministic strategies typically appear in unit-linked life insurance contracts, where the life-cycle investment strategy is age dependent but wealth independent. We explain how optimal deterministic strategies can be found numerically and present an example from life insurance where we compare the optimal solution with suboptimal deterministic strategies derived from the stochastic solution.

The uncertain mortality intensity framework: Pricing and hedging unit-linked life insurance contracts

We study the valuation and hedging of unit-linked life insurance contracts in a setting where mortality intensity is governed by a stochastic process. We focus on model risk arising from different specifications for the mortality intensity. To do so we assume that the mortality intensity is almost surely bounded under the statistical measure. Further, we restrict the equivalent martingale measures and apply the same bounds to the mortality intensity under these measures. For this setting we derive upper and lower price bounds for unit-linked life insurance contracts using stochastic control techniques. We also show that the induced hedging strategies indeed produce a dynamic superhedge and subhedge under the statistical measure in the limit when the number of contracts increases. This justifies the bounds for the mortality intensity under the pricing measures. We provide numerical examples investigating fixed-term, endowment insurance contracts and their combinations including various guarantee features. The pricing partial differential equation for the upper and lower price bounds is solved by finite difference methods. For our contracts and choice of parameters the pricing and hedging is fairly robust with respect to misspecification of the mortality intensity. The model risk resulting from the uncertain mortality intensity is of minor importance.

Locally risk-minimizing hedging strategies for unit-linked life insurance contracts under a regime switching Lévy model

This paper extends the model and analysis in that of Vandaele and Vanmaele [Insurance: Mathematics and Economics, 2008, 42: 1128–1137]. We assume that parameters of the Levy process which models the dynamic of risky asset in the financial market depend on a finite state Markov chain. The state of the Markov chain can be interpreted as the state of the economy. Under the regime switching Levy model, we obtain the locally risk-minimizing hedging strategies for some unit-linked life insurance products, including both the pure endowment policy and the term insurance contract.

The Effect of Policyholders’ Rationality on Unit-Linked Life Insurance Contracts with Surrender Guarantees

We study the valuation of unit-linked life insurance contracts with surrender guarantees. Instead of solving an optimal stopping problem, we propose a more realistic approach accounting for policyholders’ rationality in exercising their surrender option. The valuation is conducted at the portfolio level by assuming the surrender intensity to be bounded from below and from above. The lower bound corresponds to purely exogenous surrender and the upper bound represents the limited rationality of the policyholders. The valuation problem is formulated by a valuation PDE and solved with the finite difference method. We show that the rationality of the policyholders has a significant effect on average contract value and hence on the fair contract design. We also present the separating boundary between purely exogenous surrender and endogenous surrender. This provides implications on the predicted surrender activity of the policyholders.

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts

We study the valuation and hedging of unit-linked life insurance contracts in a setting where mortality intensity is governed by a stochastic process. We focus on model risk arising from different specifications for the mortality intensity. To do so we assume that the mortality intensity is almost surely bounded under the statistical measure. Further, we restrict the equivalent martingale measures and apply the same bounds to the mortality intensity under these measures. For this setting we derive upper and lower price bounds for unit-linked life insurance contracts using stochastic control techniques. We also show that the induced hedging strategies indeed produce a dynamic superhedge and subhedge under the statistical measure in the limit when the number of contracts increases. This justifies the bounds for the mortality intensity under the pricing measures. We provide numerical examples investigating fixed-term, endowment insurance contracts and their combinations including various guarantee features. The pricing partial differential equation for the upper and lower price bounds is solved by finite difference methods. For our contracts and choice of parameters the pricing and hedging is fairly robust with respect to misspecification of the mortality intensity. The model risk resulting from the uncertain mortality intensity is of minor importance.

Explicit portfolio for unit-linked life insurance contracts with surrender option

Introducing a surrender option in unit-linked life insurance contracts leads to a dependence between the surrender time and the financial market. [J. Barbarin, Risk minimizing strategies for life insurance contracts with surrender option, Tech. rep., University of Louvain-La-Neuve, 2007] used a lot of concepts from credit risk to describe the surrender time in order to hedge such types of contracts. The basic assumption made by Barbarin is that the surrender time is not a stopping time with respect to the financial market. The goal of this article is to make the hedging strategies more explicit by introducing concrete processes for the risky asset and by restricting the hazard process to an absolutely continuous process. First, we assume that the risky asset follows a geometric Brownian motion. This extends the theory of [T. Møller, Risk-minimizing hedging strategies for insurance payment processes, Finance and Stochastics 5 (2001) 419–446], in that the random times of payment are not independent of the financial market. Second, the risky asset follows a Lévy process. For both cases, we assume the payment process contains a continuous payment stream until surrender or maturity and a payment at surrender or at maturity, whichever comes first.

A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market

In [Riesner, M., 2006. Hedging life insurance contracts in a Lévy process financial market. Insurance Math. Econom. 38, 599–608] the (locally) risk-minimizing hedging strategy for unit-linked life insurance contracts is determined in an incomplete financial market driven by a Lévy process. The considered risky asset is not a martingale under the original measure and therefore, a change of measure to the minimal martingale measure is performed. The goal of this paper is to show that the risk-minimizing hedging strategy under the new martingale measure which is found in the paper cited above is not the locally risk-minimizing strategy under the original measure. Finally, the real locally risk-minimizing strategy is derived and a relationship between the number of risky assets held in the proposed portfolio cited in the above-mentioned paper and the one proposed here is given.

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

The present paper investigates the net loss of a life insurance company issuing equity-linked pure endowments in the case of periodic premiums. Due to the untradability of the insurance risk which affects both the in- and outflow side of the company, the issued insurance claims cannot be hedged perfectly. Furthermore, we consider an additional source of incompleteness caused by trading restrictions, because in reality the hedging of the contingent claims is more likely to occur at discrete times. Based on Møller [Møller, T., 1998. Risk-minimizing hedging strategies for unit-linked life insurance contracts. Astin Bull. 28, 17–47], we particularly examine the situation, where the company applies a time-discretized risk-minimizing hedging strategy. Through an illustrative example, we observe numerically that only a relatively small reduction in ruin probabilities is achieved with the use of the discretized originally risk-minimizing strategy because of the accumulated extra duplication errors caused by discretizing. However, the simulated results are highly improved if the hedging model instead of the hedging strategy is discretized. For this purpose, Møller’s [Møller, T., 2001. Hedging equity-linked life insurance contracts. North Amer. Actuarial J. 5 (2), 79–95] discrete-time (binomial) risk-minimizing strategy is adopted.

Fair valuation of insurance contracts under Lévy process specifications

The valuation of options embedded in insurance contracts using concepts from financial mathematics (in particular, from option pricing theory), typically referred to as fair valuation, has recently attracted considerable interest in academia as well as among practitioners. The aim of this article is to investigate the valuation of participating and unit-linked life insurance contracts, which are characterized by embedded rate guarantees and bonus distribution rules. In contrast to the existing literature, our approach models the dynamics of the reference portfolio by means of an exponential Lévy process. Our analysis sheds light on the impact of the dynamics of the reference portfolio on the fair contract value for several popular types of insurance policies. Moreover, it helps to assess the potential risk arising from misspecification of the stochastic process driving the reference portfolio.

Locally Risk-minimizing Hedging of Insurance Payment Streams

For the martingale case Föllmer and Sondermann (1986) introduced a unique admissible risk-minimizing hedging strategy for any square-integrable contingent claim H. Schweizer (1991) developed their theory further to the semimartingale case introducing the notion of local risk-minimization. Møller (2001) extended the theory of Föllmer and Sondermann (1986) to hedge general payment processes occurring mainly in insurance. We expand local risk-minimization to the theory of hedging general payment processes and derive such a hedging strategy for general unit-linked life insurance contracts in a general Lévy process financial market.

Evaluating the performance of Gompertz, Makeham and Lee–Carter mortality models for risk management with unit-linked contracts

The paper compares the performance of three mortality models in the context of optimal pricing and hedging of unit-linked life insurance contracts. Two of the models are the classical parametric results of Gompertz and Makeham, the third is the recently developed method of Lee and Carter [Lee, R.D., Carter, L.R., 1992. Modelling and forecasting U.S. mortality. J. Amer. Statist. Assoc. 87 (14), 659–675] for fitting mortality and forecasting it as a stochastic process. First, quantile hedging techniques of Föllmer and Leukert [Föllmer, H., Leukert, P., 1999. Quantile hedging. Finance Stoch. 3, 251–273] are applied to price a unit-linked contract with payoff conditioned on the client’s survival to the contract’s maturity. Next, the paper analyzes the implications of the three mortality models on risk management possibilities for the insurance firm based on numerical illustrations with the Toronto Stock Exchange/Standard and Poor financial index and mortality data for the USA, Sweden and Japan. The strongest differences between the models are observed in Japan, where the lowest mortality for the next two decades is expected. The general mortality decline patterns, rectangularization of the survival curve and deceleration of mortality at older ages, are well pronounced in the results for all three countries.

Unit-Linked Life Insurance Contracts with Lapse Rates Dependent on Economic Factors

ABSTRACTMany recently introduced unit-linked life insurance policies contain provisions allowing policyholders to lapse the product. The problem of pricing this surrender option is difficult as it involves modelling lapse decisions which may be contingent on different factors. This paper develops a methodology which enables us to model lapse behaviour within a framework provided by developments in financial economics. Using marked point processes with stochastic intensities, we present an approach which accounts for changes in the lapse behaviour of policyholders due to different economic factors. As a result, the model produces more accurate financial values for insurance contracts contingent on financial markets. In the context of unit-linked policies, we illustrate the method by allowing the lapse decision to depend on the stochastic volatility of the underlying asset. Our simulation study indicates that there is a strong relation between the single premiums of these policies and the lapse behaviour.

Hedging life insurance contracts in a Lévy process financial market

Starting from the model of Møller [Risk-minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bulletin 28 (1998) 17–47] we derive analogously, but for an incomplete financial market, a (locally) risk-minimizing hedging strategy for unit-linked life insurance contracts represented by the pure endowment and the term insurance. The incomplete financial market is exemplarily given by a general Lévy-driven model. We investigate the Föllmer–Schweizer decomposition of their intrinsic value. Additionally, we compare our results to the ones obtained by Møller [Risk-minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bulletin 28 (1998) 17–47] and show how they are affected by replacing the complete financial market by an incomplete one.