Anna Rita Bacinello

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 percentage (participation level) of the annual return on the reference portfolio and anyway does not fall below the technical rate (minimum interest rate guaranteed, 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 assumptions 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.

Pricing equity-linked life insurance with endogenous minimum guarantees

Abstract This paper analyses the problem of pricing insurance contracts in which the benefits are linked to the realization of a portfolio of equities and a minimum amount guaranteed is provided. Building on the models of Brennan and Schwartz (1976, 1979) and Delbaen (1990) for endowment policies, we extend them to the case in which the minimum guarantees are endogenous , i.e. they are functions of the premium(s) paid. In this framework we give sufficient conditions for both the single and the periodic premium to be well defined and present some significant examples of endogenous minimum guarantees satisfying these conditions. We also consider the problem of pricing insurance contracts different from the endowment one, and conclude with some numerical results.

Pricing Guaranteed Life Insurance Participating Policies with Annual Premiums and Surrender Option

Abstract This paper analyzes a life insurance endowment policy, paid by annual premiums, in which the benefit is annually adjusted according to the performance of a special investment portfolio and a minimum return is guaranteed to the policyholder. In particular, the author considers both the case in which the annual premium is constant and the case in which the premium also is adjusted according to the performance of the reference portfolio. Moreover, the policy under scrutiny is characterized by the presence of a surrender option, that is, of an American-style put option that enables the policyholder to give up the contract and receive the surrender value. The aim of the paper is to give sufficient conditions under which there exists a (unique) fair premium. This premium is implicitly defined by an equation (or, alternatively, can be viewed as a fixed point of a suitable function) based on a recursive binomial tree àla Cox, Ross, and Rubinstein (1979). An iterative algorithm is then implemented in order to compute it.

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.

Pricing life insurance contracts with early exercise features

In this paper we describe an algorithm based on the Least Squares Monte Carlo method to price life insurance contracts embedding American options. We focus on equity-linked contracts with surrender options and terminal guarantees on benefits payable upon death, survival and surrender. The framework allows for randomness in mortality as well as stochastic volatility and jumps in financial risk factors. We provide numerical experiments demonstrating the performance of the algorithm in the context of multiple risk factors and exercise dates.

Regression-Based Algorithms for Life Insurance Contracts with Surrender Guarantees

We present a general framework for pricing life insurance contracts embedding a surrender option. The model allows for several sources of risk, such as uncertainty in mortality, interest rates and other financial factors. We describe and compare two numerical schemes based on the Least Squares Monte Carlo method, emphasizing underlying modeling assumptions and computational issues.

Single and Periodic Premiums for Guaranteed Equity-Linked Life Insurance under Interest-Rate Risk: The “Lognormal + Vasicek” Case

Interest-rate risk, while significantly affecting the pricing of almost all life-insurance products, has been up to now disregarded in the analysis of equity-linked policies for which a minimum-amount-guaranteed provision operates. The purpose of the present paper is to build on the work of Brennan and Schwartz(1976,1979a,b) and Delbaen(1990) to show how uncertainty in interest rates influences both single and periodic premiums for equity-linked life insurance. To this end, we consider a model in which the unit price of the fund to which benefits are referred follows a lognormal process, while the spot rate of interest is described as in Vasicek(1977), and we employ the martingale approach to contingent-claims pricing introduced by Harrison and Kreps(1979) to obtain pricing formulae for guaranteed equity-linked policies that account for interest-rate risk. The paper includes a detailed comparative static analysis of our extended formulae, as well as some numerical examples.

Fixed income linked life insurance policies with minimum guarantees: Pricing models and numerical results

Abstract We propose a pricing model for life insurance policies in which the benefits are linked to the performance of a portfolio of interest rate sensitive assets (reference fund), and a minimum guarantee provision is present. The model is cast in the celebrated term structure framework developed by Cox, Ingersoll and Ross (1985). As for the behaviour of the investment component, we analyse two polar cases. In the first one the payments due on the reference fund when the contract is still “alive” are not reinvested, while in the second case we propose a reinvestment policy. We show how to obtain a closed form solution for the single premium in the no-reinvestment case, and how to implement a simulation approach to calculate numerically the single premium in the reinvestment case. We illustrate our analysis with numerical results that help in understanding the comparative static properties of the models proposed.

Valuation of contingent-claims characterising particular pension schemes

Abstract The benefits promised by pension schemes are often rather complex contingent-claims whose valuation requires the specification of the stochastic behaviour of several state-variables such as salaries, inflation rates, rates of return on investments, and so on. The object of this paper is, first of all, to present a valuation model suitable for their pricing, and then to apply this model to the valuation of very peculiar options embedded in the benefits offered by some “hybrid” pension plans.

The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours

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 policyholder’s withdrawal behaviours 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.