Risk-neutral valuation of participating life insurance contracts in a stochastic interest rate environment

Over the last years, the valuation of life insurance contracts using concepts from financial mathematics has become a popular research area for actuaries as well as financial economists. In particular, several methods have been proposed of how to model and price participating policies, which are characterized by an annual interest rate guarantee and some bonus distribution rules. However, despite the long terms of life insurance products, most valuation models allowing for sophisticated bonus distribution rules and the inclusion of frequently offered options assume a simple Black–Scholes setup and, more specifically, deterministic or even constant interest rates.We present a framework in which participating life insurance contracts including predominant kinds of guarantees and options can be valuated and analyzed in a stochastic interest rate environment. In particular, the different option elements can be priced and analyzed separately. We use Monte Carlo and discretization methods to derive the respective values.The sensitivity of the contract and guarantee values with respect to multiple parameters is studied using the bonus distribution schemes as introduced in [Bauer, D., Kiesel, R., Kling, A., Ruß, J., 2006. Risk-neutral valuation of participating life insurance contracts. Insurance: Math. Econom. 39, 171–183]. Surprisingly, even though the value of the contract as a whole is only moderately affected by the stochasticity of the short rate of interest, the value of the different embedded options is altered considerably in comparison to the value under constant interest rates. Furthermore, using a simplified asset portfolio and empirical parameter estimations, we show that the proportion of stock within the insurer’s asset portfolio substantially affects the value of the contract.     

Asset management and surplus distribution strategies in life insurance: An examination with respect to risk pricing and risk measurement

In this paper, we investigate the impact of different asset management and surplus distribution strategies in life insurance on risk-neutral pricing and shortfall risk. In general, these feedback mechanisms affect the contract’s payoff and hence directly influence pricing and risk measurement. To isolate the effect of such strategies on shortfall risk, we calibrate contract parameters so that the compared contracts have the same market value and same default-value-to-liability ratio. This way, the fair valuation method is extended since, in addition to the contract’s market value, the default put option value is fixed. We then compare shortfall probability and expected shortfall and show the substantial impact of different management mechanisms acting on the asset and liability side.     

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.     

A joint valuation of premium payment and surrender options in participating life insurance contracts

In addition to an interest rate guarantee and annual surplus participation, life insurance contracts typically embed the right to stop premium payments during the term of the contract (paid-up option), to resume payments later (resumption option), or to terminate the contract early (surrender option). Terminal guarantees are on benefits payable upon death, survival and surrender. The latter are adapted after exercising the options. A model framework including these features and an algorithm to jointly value the premium payment and surrender options is presented. In a first step, the standard principles of risk-neutral evaluation are applied and the policyholder is assumed to use an economically rational exercise strategy. In a second step, option value sensitivity on different contract parameters, benefit adaptation mechanisms, and exercise behavior is analyzed numerically. The two latter are the main drivers for the option value.     

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.     

Long-tail longitudinal modeling of insurance company expenses

The insurance industry is known to have high operating expenses in the financial services sector. Insurers, investors and regulators are interested in models to understand the behavior of expenses. However, the current practice ignores skewness, occasional negative values as well as their temporal dependence.Addressing these three features, this paper develops a longitudinal model of insurance company expenses that can be used for prediction, to identify unusual behavior, and to measure firm efficiency. Specifically, we use a three-parameter asymmetric Laplace density for the marginal distribution of insurers’ expenses in each year. Copula functions are employed to accommodate their temporal dependence. As a function of explanatory variables, the location parameter allows us to analyze an insurer’s expenses in light of the firm’s characteristics. Our model can be interpreted as a longitudinal quantile regression.The analysis is performed using property-casualty insurance company data from the National Association of Insurance Commissioners of years 2001-2006. Due to the long-tailed nature of insurers’ expenses, two alternative approaches are proposed to improve the performance of the longitudinal quantile regression model: rescaling and transformation. Predictive densities are derived that allow one to compare the predictions for individual insurers in a hold-out-sample. Both predictive models are shown to be reasonable with the rescaling method outperforming the transformation method. Compared with standard longitudinal models, our model is shown to be superior in identifying insurers’ unusual behavior.     

A sensitivity analysis concept for life insurance with respect to a valuation basis of infinite dimension

A sensitivity analysis concept is introduced for prospective reserves of individual life insurance contracts as deterministic mappings of the actuarial assumptions interest rate, mortality probability, disability probability, etc. Upon modeling these assumptions as functions on a real time line, the prospective reserve is here a mapping with infinite dimensional domain. Inspired by the common idea of interpreting partial derivatives of first order as local sensitivities, a generalized gradient vector approach is introduced in order to allow for a sensitivity analysis of the prospective reserves as functionals on a function space. The capability of the concept is demonstrated with an example.     

Valuation and hedging of participating life-insurance policies under management discretion

The valuation and hedging of participating life insurance policies, also known as with-profits policies, is considered. Such policies can be seen as European path-dependent contingent claims whose underlying security is the investment portfolio of the insurance company that sold the policy. The fair valuation of these policies is studied under the assumption that the insurance company has the right to modify the investment strategy of the underlying portfolio at any time. Furthermore, it is assumed that the issuer of the policy does not setup a separate portfolio to hedge the risk associated with the policy. Instead, the issuer will use its discretion about the investment strategy of the underlying portfolio to hedge shortfall risks. In that sense, the insurer’s investment portfolio serves simultaneously as the underlying security and as the hedge portfolio. This means that the hedging problem can not be separated from the valuation problem. We investigate the relationship between risk-neutral valuation and hedging of these policies in complete and incomplete financial markets.     

A sensitivity analysis of typical life insurance contracts with respect to the technical basis

In [Christiansen, M.C., 2007. A sensitivity analysis concept for life insurance with respect to a valuation basis of infinite dimension. Insurance: Math. Econom. doi:10.1016/j.insmatheco.2007.07.005] a sensitivity analysis concept was introduced for the prospective reserve of individual life insurance contracts as functional of the technical basis parameters such as interest rate, mortality probability, disability probability, et cetera. On the basis of that concept, the present paper gives in addition the sensitivities of the premium level.Applying these approaches, an extensive sensitivity analysis is carried out: A study of the basic life insurance contract types ‘pure endowment insurance’, ‘temporary life insurance’, ‘annuity insurance’ and ‘disability insurance’ identifies their diverse characteristics, in particular their weakest points concerning fluctuations of the technical basis. An investigation of combinations of these insurance contract types shows what synergy effects can be expected by creating insurance packages.     

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.     

Analytical approximations for prices of swap rate dependent embedded options in insurance products

Life insurance products have profit sharing features in combination with guarantees. These so-called embedded options are often dependent on or approximated by forward swap rates. In practice, these kinds of options are mostly valued by Monte Carlo simulations. However, for risk management calculations and reporting processes, lots of valuations are needed. Therefore, a more efficient method to value these options would be helpful. In this paper analytical approximations are derived for these kinds of options, based on an underlying multi-factor Gaussian interest rate model. The analytical approximation for options with direct payment is almost exact while the approximation for compounding options is also satisfactory. In addition, the proposed analytical approximation can be used as a control variate in Monte Carlo valuation of options for which no analytical approximation is available, such as similar options with management actions. Furthermore, it’s also possible to construct analytical approximations when returns on additional assets (such as equities) are part of the profit sharing rate.     

Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints

We consider that the surplus of an insurance company follows a Cramér-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks.[Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215-228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890-907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.     

Risk comparison of different bonus distribution approaches in participating life insurance

The fair pricing of explicit and implicit options in life insurance products has received broad attention in the academic literature over the past years. Participating life insurance (PLI) contracts have been the focus especially. These policies are typically characterized by a term life insurance, a minimum interest rate guarantee, and bonus participation rules with regard to the insurer’s asset returns or reserve situation. Researchers replicate these bonus policies quite differently. We categorize and formally present the most common PLI bonus distribution mechanisms. These bonus models closely mirror the Danish, German, British, and Italian regulatory framework. Subsequently, we perform a comparative analysis of the different bonus models with regard to risk valuation. We calibrate contract parameters so that the compared contracts have a net present value of zero and the same safety level as the initial position, using risk-neutral valuation. Subsequently, we analyze the effect of changes in the asset volatility and in the initial reserve amount (per contract) on the value of the default put option (DPO), while keeping all other parameters constant. Our results show that DPO values obtained with the PLI bonus distribution model of Bacinello (2001), which replicates the Italian regulatory framework, are most sensitive to changes in volatility and initial reserves.     

The influence of corporate taxes on pricing and capital structure in property-liability insurance

A change in the corporate tax level can have a significant impact on rate making and capital structure for insurance companies. The purpose of this paper is to study this effect on competitive equity-premium combinations for different asset and liability models while retaining a fixed safety level. This is a crucial consideration as a change in the tax rate leads, in general, to a different risk of insolvency. Hence, fixing the safety level serves to isolate the effect of taxes without shifting the insurer’s risk situation whenever taxes are varied. The model framework includes stochastic assets as well as stochastic claims costs. We further compare the results for liability models with and without a jump component. Insurance rate making is conducted using option pricing theory.     

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.     

Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance

New regulations, stronger competitions and more volatile capital markets have increased the demand for stochastic asset-liability management (ALM) models for insurance companies in recent years. The numerical simulation of such models is usually performed by Monte Carlo methods which suffer from a slow and erratic convergence, though. As alternatives to Monte Carlo simulation, we propose and investigate in this article the use of deterministic integration schemes, such as quasi-Monte Carlo and sparse grid quadrature methods. Numerical experiments with different ALM models for portfolios of participating life insurance products demonstrate that these deterministic methods often converge faster, are less erratic and produce more accurate results than Monte Carlo simulation even for small sample sizes and complex models if the methods are combined with adaptivity and dimension reduction techniques. In addition, we show by an analysis of variance (ANOVA) that ALM problems are often of very low effective dimension which provides a theoretical explanation for the success of the deterministic quadrature methods.     

Longevity risk management for life and variable annuities: The effectiveness of static hedging using longevity bonds and derivatives

For many years, the longevity risk of individuals has been underestimated, as survival probabilities have improved across the developed world. The uncertainty and volatility of future longevity has posed significant risk issues for both individuals and product providers of annuities and pensions. This paper investigates the effectiveness of static hedging strategies for longevity risk management using longevity bonds and derivatives (q-forwards) for the retail products: life annuity, deferred life annuity, indexed life annuity, and variable annuity with guaranteed lifetime benefits. Improved market and mortality models are developed for the underlying risks in annuities. The market model is a regime-switching vector error correction model for GDP, inflation, interest rates, and share prices. The mortality model is a discrete-time logit model for mortality rates with age dependence. Models were estimated using Australian data. The basis risk between annuitant portfolios and population mortality was based on UK experience. Results show that static hedging using q-forwards or longevity bonds reduces the longevity risk substantially for life annuities, but significantly less for deferred annuities. For inflation-indexed annuities, static hedging of longevity is less effective because of the inflation risk. Variable annuities provide limited longevity protection compared to life annuities and indexed annuities, and as a result longevity risk hedging adds little value for these products.     

A general asset-liability management model for the efficient simulation of portfolios of life insurance policies

New regulations and a stronger competition have increased the importance of stochastic asset-liability management (ALM) models for insurance companies in recent years. In this paper, we propose a discrete time ALM model for the simulation of simplified balance sheets of life insurance products. The model incorporates the most important life insurance product characteristics, the surrender of contracts, a reserve-dependent bonus declaration, a dynamic asset allocation and a two-factor stochastic capital market. All terms arising in the model can be calculated recursively which allows an easy implementation and efficient simulation. Furthermore, the model is designed to have a modular organization which permits straightforward modifications and extensions to handle specific requirements. In a sensitivity analysis for sample portfolios and parameters, we investigate the impact of the most important product and management parameters on the risk exposure of the insurance company and show that the model captures the main behaviour patterns of the balance sheet development of life insurance products.     

Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates

We study indifference pricing of mortality contingent claims in a fully stochastic model. We assume both stochastic interest rates and stochastic hazard rates governing the population mortality. In this setting we compute the indifference price charged by an insurer that uses exponential utility and sells k contingent claims to k independent but homogeneous individuals. Throughout we focus on the examples of pure endowments and temporary life annuities. We begin with a continuous-time model where we derive the linear pdes satisfied by the indifference prices and carry out extensive comparative statics. In particular, we show that the price-per-risk grows as more contracts are sold. We then also provide a more flexible discrete-time analog that permits general hazard rate dynamics. In the latter case we construct a simulation-based algorithm for pricing general mortality-contingent claims and illustrate with a numerical example.     

Heath–Jarrow–Morton modelling of longevity bonds and the risk minimization of life insurance portfolios

This paper has two parts. In the first, we apply the Heath–Jarrow–Morton (HJM) methodology to the modelling of longevity bond prices. The idea of using the HJM methodology is not new. We can cite Cairns et al. [Cairns A.J., Blake D., Dowd K, 2006. Pricing death: framework for the valuation and the securitization of mortality risk. Astin Bull., 36 (1), 79–120], Miltersen and Persson [Miltersen K.R., Persson S.A., 2005. Is mortality dead? Stochastic force of mortality determined by arbitrage? Working Paper, University of Bergen] and Bauer [Bauer D., 2006. An arbitrage-free family of longevity bonds. Working Paper, Ulm University]. Unfortunately, none of these papers properly defines the prices of the longevity bonds they are supposed to be studying. Accordingly, the main contribution of this section is to describe a coherent theoretical setting in which we can properly define these longevity bond prices. A second objective of this section is to describe a more realistic longevity bonds market model than in previous papers. In particular, we introduce an additional effect of the actual mortality on the longevity bond prices, that does not appear in the literature. We also study multiple term structures of longevity bonds instead of the usual single term structure. In this framework, we derive a no-arbitrage condition for the longevity bond financial market. We also discuss the links between such HJM based models and the intensity models for longevity bonds such as those of Dahl [Dahl M., 2004. Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts, Insurance: Math. Econom. 35 (1) 113–136], Biffis [Biffis E., 2005. Affine processes for dynamic mortality and actuarial valuations. Insurance: Math. Econom. 37, 443–468], Luciano and Vigna [Luciano E. and Vigna E., 2005. Non mean reverting affine processes for stochastic mortality. ICER working paper], Schrager [Schrager D.F., 2006. Affine stochastic mortality. Insurance: Math. Econom. 38, 81–97] and Hainaut and Devolder [Hainaut D., Devolder P., 2007. Mortality modelling with Lévy processes. Insurance: Math. Econom. (in press)], and suggest the standard pricing formula of these intensity models could be extended to more general settings.In the second part of this paper, we study the asset allocation problem of pure endowment and annuity portfolios. In order to solve this problem, we study the “risk-minimizing” strategies of such portfolios, when some but not all longevity bonds are available for trading. In this way, we introduce different basis risks.