A Dynamic Programming Algorithm for the Valuation of Guaranteed Minimum Withdrawal Benefits in Variable Annuities

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 levels of rationality in the policyholder’s withdrawal behaviour 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

Special Issue "Quantitative Risk Assessment in Life, Health and Pension Insurance"

1noThis is an Editorial to the Special Issue "Quantitative Risk Assessment in Life, Health and Pension Insurance".openopenAnna Rita BacinelloBacinello, ANNA RIT

Design and pricing of equity-linked life insurance under stochastic interest rates

A valuation model for equity-linked life insurance contracts incorporating stochastic interest rates is presented. Our model generalizes some previous pricing results based on deterministic interest rates. Moreover, a design of a new equity-linked product with some appealing features is proposed and compared with the periodical premium contract of Brennan and Schwartz (1976). Our new product is very simple to price and may easily be hedged either by long positions in the mutual fund of linkage or by European call options on the same fund

Fair Valuation of a Guaranteed Life Insurance Participating Contract Embedding a Surrender Option

In this article we deal with the problem of pricing a guaranteed life insurance participating policy, sold in the Italian market, which embeds a surrender option. This feature is an American-style put option that enables the policyholder to sell back the contract to the insurer at the cash surrender value. Employing a recursive binomial formula patterned after the Cox, Ross, and Rubinstein (1979) discrete option pricing model we compute, first of all, the total price of the contract, which also includes a compensation for the participation feature ("participation option," henceforth). Then this price is split into the value of three components: the "basic contract", the "participation option", and the "surrender option". The numerical implementation of the model allows us to catch some comparative statics properties and to tackle the problem of suitably fixing the contractual parameters in order to obtain the premium computed by insurance companies according to standard actuarial practice. Copyright The Journal of Risk and Insurance.

Fair pricing of life insurance participating policies with a minimum interest rate guaranteed

In this paper we analyse, in a contingent-claims framework, one of the most common life insurance policies sold in Italy during the last two decades. The policy, of the endowment type, is initially priced as a standard one, given a mortality table and a technical interest rate. Subsequently, at the end of each policy year, the insurance company grants a bonus, which is credited to the mathematical reserve and depends on the performance of a special investment portfolio. More precisely, this bonus is determined in such a way that the total interest rate credited to the insured equals a given percen-tage (participation level) of the annual return on the reference portfolio and anyway does not fall below the technical rate (minimum interest rate guaran-teed, henceforth). Moreover, if the contract is paid by periodical premiums, it is usually stated that the annual premium is adjusted at the same rate of the bonus, and thus the benefit is also adjusted in the same measure. In such policy the variables controlled by the insurance company (control-variables, henceforth) are the technical rate, the participation level and, in some sense, the riskiness of the reference portfolio measured by its volatility. However, as it is intuitive, not all sets of values for these variables give rise to a fair contract, i.e. to a contract priced consistently with the usual assump-tions on financial markets and, in particular, with no-arbitrage. We derive then necessary and sufficient conditions under which each control-variable is determined by a fair pricing of the contract, given the remaining two ones

On the Market-Consistent Valuation of Participating Life Insurance Heterogeneous Contracts under Longevity Risk

The purpose of this paper is to conduct a market-consistent valuation of life insurance participating liabilities sold to a population of partially heterogeneous customers under the joint impact of biometric and financial risk. In particular, the heterogeneity between groups of policyholders stems from their offered minimum interest rate guarantees and contract maturities. We analyse the effects of these features on the company’s insolvency while embracing the insurer’s goal to achieve the same expected return for different cohorts of policyholders. Within our extensive numerical analyses, we determine the fair participation rates and other key figures, and discuss the implications for the stakeholders, taking account of various degrees of conservativeness of the insurer when pricing the contracts

Modelling the Surrender Conditions in Equity-Linked Life Insurance

We propose a model for pricing a unit-linked life insurance policy embedding a surrender option. We consider both single and annual premium contracts. First we analyse a quite general contract, for which we obtain a backward recursive valuation formula based on the Cox, Ross and Rubinstein (1979) binomial model. Then we concentrate upon a particular case, that is the famous model with exogenous minimum guarantees. In this case we extend our previous analysis in order to take into account the possibility that the guarantees at death or maturity and the surrender values are endogenously determined, and provide necessary and sucient conditions for the premiums to be well defined.surrender option, equity-linked life insurance, exogenous and endogenous guarantees, single and annual premium contracts, binomial trees

An Efficient Monte Carlo Based Approach for the Simulation of Future Annuity Values

In this paper we propose a methodology for valuing future annuity contracts based on the Least-Squares Monte Carlo approach. We adopt, as first step, a simplified computational framework where just one risk factor is taken into account, and then we extend it introducing other sources of risk. We give a brief description of the valuation procedure and provide some numerical illustrations. Furthermore, to test the efficiency of the proposed methodology, we compare our results with those obtained by applying a straightforward and time-consuming approach based on nested simulations. Finally, we present some possible applications in the context of de-risking strategies for pension plans and in the valuation of guaranteed annuity options