Equity-linked Life Insurance Contracts Research Articles

CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs

<p style='text-indent:20px;'>This paper analyzes Conditional Value-at-Risk (CVaR) based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs. A nonlinear partial differential equation (PDE) that an option value process inclusive of transaction costs should satisfy is provided. In particular, the closed-form expression of a European call option price is given. Meanwhile, the CVaR-based partial hedging strategy for a call option is derived explicitly. Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility. We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method. Furthermore,our results are implemented to derive target clients’ survival probabilities and age of equity-linked life insurance contracts.</p>

Quantile hedging in a defaultable market with life insurance applications

The paper is devoted to quantile hedging in a market with defaultable securities. Both perfect and quantile hedging strategies are given for a European call option on a vulnerable equity. Application of quantile methodology to pricing the equity-linked life insurance contracts is demonstrated. A numerical example is provided to illustrate the effect of a default on the option price, on the probability of successful hedging, and on the insurance-related variables.

Quantile hedging in models with dividends and application to equity-linked life insurance contracts

The paper demonstrates the effect of the dividends on pricing and hedging the European contingent claims under a budget constraint and presents insurance applications. Explicit formulae for the quantile pricing and hedging of the European call option are derived assuming the jump-diffusion model of the financial market. These results are used to determine the premium of the pure endowment with fixed guarantee equity-linked life insurance contract as well as the survival probability of the insured. A numerical example is given to illustrate the role of dividends in valuation and risk management of such insurance contracts.

Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.

Pricing and Hedging Equity-Linked Life Insurance Contracts Beyond the Classical Paradigm: The Principle of Equivalent Forward Preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.

Constrained non-concave utility maximization: An application to life insurance contracts with guarantees

We study a problem of non-concave utility maximization under a fair pricing constraint. The framework finds many applications in, for example, the optimal design of managerial compensation or equity-linked life insurance contracts. Deriving closed-form solutions, we observe that the fair pricing constraint will reduce the riskiness of the optimal strategies substantially. In an extensive numerical section, we analyze innovative retirement products that adapt the investment strategy of the premium pool according to the policyholder’s preferences, modeled as constant relative risk aversion (CRRA). Such products are a response to the loss of attractiveness of traditional life insurance contracts with guarantees that are negatively affected by increasing solvency requirements for return guarantees and a general decrease in interest rate levels. Taking into account that retirement products are usually tax-privileged, we find that fairly priced guarantee contracts that follow this optimal investment strategy lead to a higher expected utility level than asset investments.

Funding Life Insurance Contracts with Guarantees: How Can We Optimally Respond to the Policyholder's Needs?

We study a problem of non-concave utility maximization under a fair pricing constraint. The framework finds many applications in, for example, the optimal design of managerial compensation or equity-linked life insurance contracts. Deriving closed-form solutions, we observe that the fair pricing constraint will reduce the riskiness of the optimal strategies substantially. In an extensive numerical section, we analyze innovative retirement products that adapt the investment strategy of the premium pool according to the policyholder's preferences, modeled as constant relative risk aversion (CRRA). Such products are a response to the loss of attractiveness of traditional life insurance contracts with guarantees that are negatively affected by increasing solvency requirements for return guarantees and a general decrease in interest rate levels. Taking into account that retirement products are usually tax-privileged, we find that fairly priced guarantee contracts that follow this optimal investment strategy lead to a higher expected utility level than asset investments.

Quadratic hedging of equity-linked life insurance contracts under the real-world measure in discrete time

Quadratic hedging of equity-linked life insurance contracts under the real-world measure in discrete time

Valuation of the surrender option in equity-linked life insurance contracts

Valuation of the surrender option in equity-linked life insurance contracts

Valuation of the surrender option in equity-linked life insurance contracts

In this paper, we consider the surrender option, a specific embedded option in unit-linked life insurance contracts that gives the insured the right to terminate early the contract before maturity and receive a cash amount called surrender value. We use the no arbitrage approach to value the surrender option, seen as an US option. The valuation is based on the Cox et al. (1979) binomial model. The aim of this contribution is to expose the methodological approach in the particular case of the single premium contract, with focus on the surrender option valuation and to allow the exploitation of this intuitive and flexible method in order to deal with more realistic and complex valuation frameworks.

OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION

AbstractIn a typical equity-linked life insurance contract, the insurance company is entitled to a share of return surpluses as compensation for the return guarantee granted to the policyholders. The set of possible contract terms might, however, be restricted by a regulatory default constraint — a fact that can force the two parties to initiate sub-optimal insurance contracts. We show that this effect can be mitigated if regulatory policy is more flexible. We suggest that the regulator implement a traffic light system where companies are forced to reduce the riskiness of their asset allocation in distress. In a utility-based framework, we show that the introduction of such a system can increase the benefits of the policyholder without deteriorating the benefits of the insurance company. At the same time, default probabilities (and thus solvency capital requirements) can be reduced.

Pricing of long dated equity-linked life insurance contracts

This article adopts an approach to pricing of equity-linked life insurance contracts, which only requires the existence of the numéraire portfolio. An equity-linked life insurance contract is equivalent to a sum of the guaranteed amount and the value of an option on the equity index with some mortality risk attached. The numéraire portfolio equals the growth optimal portfolio and is used as numéraire or benchmark, where the real-world probability measure is taken as pricing measure. To obtain tractable solutions the short rate is modelled as a quadratic form of some Gaussian factor processes. Furthermore, the dynamics of the mortality rate is modelled as a threshold life table. The dynamics of the discounted equity market index or benchmark is modelled by a time transformed squared Bessel process. The equity-linked life insurance contracts are evaluated analytically.

EFFICIENT HEDGING FOR DEFAULTABLE SECURITIES AND ITS APPLICATION TO EQUITY-LINKED LIFE INSURANCE CONTRACTS

The paper deals with efficient hedging problem for defaultable securities with multiple default times and nonzero recovery rates. First, we convert the efficient hedging problem into a Neyman–Pearson problem with composite hypothesis against a simple alternative. Then we apply nonsmooth convex duality to provide a solution in the framework of a “defaultable” Black–Scholes model. Moreover, in the case of zero recovery rates, we find a closed form solution for the problem. As an application, it is shown how to use such type of results in pricing equity-linked life insurance contracts. The results are also demonstrated by some numerical examples.

Risk-Shifting and Optimal Asset Allocation in Life Insurance: The Impact of Regulation

In a typical equity-linked life insurance contract, the insurance company is entitled to a share of return surpluses as compensation for the return guarantee granted to the policyholders. The set of possible contract terms might, however, be restricted by a regulatory default constraint - a fact that can force the two parties to initiate sub-optimal insurance contracts. We show that this effect can be mitigated if regulatory policy is more flexible. We suggest that the regulator implement a traffic light system where companies are forced to reduce the riskiness of their asset allocation in distress. In a utility-based framework, we show that the introduction of such a system can increase the benefits of the policyholder without deteriorating the benefits of the insurance company. At the same time, default probabilities (and thus solvency capital requirements) can be reduced.

Quantile hedging on equity-linked life insurance contracts with transaction costs

This paper analyzes the application of quantile hedging on equity-linked life insurance contracts in the presence of transaction costs. Following the time-based replication strategy, we present the explicit expressions for the present values of expected hedging errors and transaction costs. The results are derived by using the adjusted hedging volatility σ̄ proposed by Leland. Furthermore, the estimated values of expected hedging errors, transaction costs and total costs are obtained from a simulation approach for comparison. Finally, the costs of maturity guarantee for equity-linked life insurance contracts inclusive of transaction costs are discussed.

Pricing Equity Index Annuities with Surrender Options in Four Models

Equity-indexed annuities (EIAs) have generated a great deal of interest since Keyport Life first launched “Key Index” in February 1995. They are considered to be the most innovative products to appear on the market in years. EIAs are, essentially equity-linked deferred annuities which provide the policyholder with a guaranteed accumulation rate on their premium, and also at maturity, benefits from an additional return based on the performance of an equity mutual fund or a family of mutual funds or a stock index, typically of the Standard and Poor’s (S&P500) index, so the customer can profit from the growth of the stock market. Product designs of EIAs can vary, depending on the companies that sell them. In this article, we focus on the pricing of one of the product designs in the market which gives the possibility to surrender their policy before maturity. Such options can be valued using Monte Carlo simulation method proposed for pricing American options but, here, we use the least squares Monte Carlo suggested by Longstaff and Schwartz added to the control variate tool in order to construct efficient estimators. We analyze these equity-linked life insurance contracts under four different models. The frameworks differ from the way we model the price of the fund associated with the contract. The first setting is the usual Black and Scholes model, the second is the environment of jump diffusions, especially a Kou process, the third is the regime switching log normal model (RSLN) developed by Hamilton and the fourth is a mixed of a regime switching and a jump diffusion. The surrender option is priced using the Longstaff and Schwartz methodology.

Dynamic hedging of conditional value-at-risk

In this paper, the problem of partial hedging is studied by constructing hedging strategies that minimize conditional value-at-risk (CVaR) of the portfolio. Two dual versions of the problem are considered: minimization of CVaR with the initial wealth bounded from above, and minimization of hedging costs subject to a CVaR constraint. The Neyman–Pearson lemma approach is used to deduce semi-explicit solutions. Our results are illustrated by constructing CVaR-efficient hedging strategies for a call option in the Black–Scholes model and also for an embedded call option in an equity-linked life insurance contract.

Valuation of Equity-Linked Life Insurance Contracts with Surrender Guarantees in a Regime-Switching Rational Expectation Model

We develop a two-state regime-switching rational expectation model with the economic states recession and expansion for the valuation of equity-linked life insurance contracts with surrender guarantees. Our valuation model jointly captures the empirically observed macroeconomical regime-dependency of both the lapsing and the fi nancial market dynamics. More precisely, our surrender model incorporates both the emergency fund and the interest rate hypothesis in an intensity-based approach via a countercyclical exogenous and a procyclical endogenous surrender intensity. On the equity side the reference fund evolution allows for a countercyclical volatility and a procyclical risk-free rate. The contract value of a representative policyholder is characterized as solution of a system of two coupled PDEs, which we solve numerically by the Crank-Nicolson scheme combined with a penalty method. In extensive numerical experiments we study the effect of the model parameters on the contract values and particularly surrender option values and also compare our regime-switching rational expectation model with the regime-switching American style surrender model.

Pricing Equity Index Annuities with Surrender Options in Four Models

Since Keyport Life first launched ”Key Index” in February 1995, Equity-indexed annuities are considered to be the most innovative products to appear on the market in years. EIAs are, essentially equity-linked deferred annuities which provide the policyholder with a guaranteed accumulation rate on their premium, and also at maturity, benefits from an additional return based on the performance of an equity mutual fund or a family of mutual funds or a stock index, typically of the Standard and Poor’s (S&P 500) index, so the customer can profit from the growth of the stock market. Products designs of EIAs can vary , depending on the companies that sell them.In this paper, we focus on the pricing of one of the product design in the market which gives the possibility to surrender their policy before maturity. Such options can be valued using Monte Carlo simulation method proposed for pricing American options but, here, we use the least squares Monte-Carlo suggested by Longstaff and Schwartz added to the control variate tool in order to construct efficient estimators. We analyze these equity-linked life insurance contracts under four different models. The frameworks differ from the way we model the price of the fund associated with the contract. The first setting is the usual Black and Scholes model, the second is the environment of jump diffusions, especially a Kou process, the third is the regime switching log normal model developed by Hamilton and the fourth is a mixed of a regime switching and a jump diffusion. The surrender option is priced using the Longstaff and Schwartz methodology.

Quantile hedging for equity-linked life insurance contracts with stochastic interest rate

This paper studies the problem of pricing equity-linked life insurance contracts, and also focuses on the valuation of insurance contracts with stochastic guarantee. The contracts under consideration are based on two risky assets which satisfy a two-factor jump-diffusion model: one asset is responsible for future gains, and the other one is a stochastic guarantee. As most life insurance products are long-term contracts, it is more practical to consider the problem in a stochastic interest rate environment. In our setting, the stochastic interest rate behaviour is also described by a jump-diffusion model. In addition, quantile hedging technique is developed and exploited to price such finance/insurance contracts with initial capital constraints. Explicit formulas for both the price of the contracts and the survival probability are obtained. Our results are illustrated by numerical example based on financial indexes Russell 2000 and S&P 500.