Pierre Devolder

Stochastic mortality under measure changes

We provide a self-contained analysis of a class of continuous-time stochastic mortality models that have gained popularity in the last few years. We describe some of their advantages and limitations, examining whether their features survive equivalent changes of measures. This is important when using the same model for both market-consistent valuation and risk management of life insurance liabilities. We provide a numerical example based on the calibration to the French annuity market of a risk-neutral version of the model proposed by Lee & Carter (1992).

Stochastic optimal control of annuity contracts

The purpose of this paper is to show how stochastic optimal control theory can be applied to find an optimal investment policy before and after retirement in a defined contribution pension plan where benefits are paid under the form of annuities; annuities are supposed to be guaranteed during a certain fixed period of time. Using different kinds of utility functions we try to look at different strategies on the one hand in the investment part (i.e. before retirement) and on the other hand in the payment part (i.e. after retirement). The needed change of strategy after retirement can be interpreted in this model as a typical ALM constraint taking into account a guaranteed technical interest rate used by the insurer.

Opérations stochastiques de capitalisation

This paper presents a stochastic model of capitalization which takes mto account the financial risk in the actuarial processes. We first introduce a stochastic differential equation which allows us to define the capitalization and actuahzation processes. We use these concepts to present a new principle of premium calculation for the capitalization operations, based on the equality between backward reserve and conditional expectation of the forward reserve. A generalization of the classical Thiele equation in life insurance is also given. Numerical examples dlustrate the model.

Life annuitization : why and how much

This paper addresses some of the problems a majority of retired individuals face: Why and in what proportion should they invest in a life annuity to maximize the utility of their future consumption or a bequest? The market considered in this work is made up of three assets: a life annuity, a risky asset and a cash account. As this problem doesn’t accept any suitable explicit solution, it is numerically solved by the Markov Chain approximation developed by Kushner and Dupuis. Without a bequest motive, we observe that the optimal planning of consumption is divided into two periods and that optimal asset allocation should include the risky asset. Next, the influence of a bequest on consumption and investment pattern is developed. We demonstrate that even with a bequest motive, pensioners should allocate a part of their wealth to the purchase of life annuities.

Automatic balancing mechanisms for Notional Defined Contribution Accounts in the presence of uncertainty

The notional defined contribution model combines pay-as-you-go financing and a defined contribution pension formula. This paper aims to demonstrate the extent to which liquidity and solvency indicators are affected by fluctuations in economic and demographic conditions and to explore the introduction of an automatic balancing mechanism (ABM) into the pension scheme. We demonstrate that the introduction of an ABM reduces the volatility of the buffer fund and that, in most cases, the automatic mechanism that re-establishes solvency produces the highest value of the risk-adjusted notional factor.

Iterated VaR or CTE Measures: a False Good Idea?

The purpose of this paper is twofold. Firstly, we consider different risk measures in order to determine the solvency capital requirement of a pension fund. Secondly, we illustrate the impact of the time horizon of long-term guarantee products on these capital. We consider a financial market modelled by a common Black–Scholes–Merton model. We neglect the mortality and underwriting risks by assuming that the pension fund is fully hedged against these risks, which allows us to keep understandable and tractable formulæ (the longevity risk will be a part of future researches). A portfolio is built in this market according to different strategies and the pension fund offers a fixed guaranteed rate on a certain time horizon. We begin with well-known static risk measures (value at risk and conditional tail expectation measures) and then we consider their natural dynamic generalization. In order to be time consistent, we consider their iterated versions by a backward iterations scheme. Within the dynamic setting, we show that solvency capital can be expensive and that attention must be paid to the safety level considered.

Risk measures versus ruin theory for the calculation of solvency capital for long-term life insurances

Abstract The purpose of this paper is twofold. First we consider a ruin theory approach along with risk measures in order to determine the solvency capital of long-term guarantees such as life insurances or pension products. Secondly, for such products,we challenge the definition of the Solvency Capital Requirement (SCR) under the Solvency II (SII) regulatory framework based on a yearly viewpoint. Several methods for the calculation of the solvency capital are presented. We start our study with risk measures as considered in the SII framework and then proceed with the ruin theory approach. Instead of considering the continuous time setting of the ruin theory,we consider the discrete time—the yearly basis—of the accounting viewpoint.We finally give an illustration with a fixed guaranteed rate product along with the equity, interest rate and longevity risks. The latter risk brings us to consider zero-coupon longevity bonds in which we invest the capital. We show that long-term guarantees might be overloaded under the SII regulation.

Guarantee Valuation in Notional Defined Contribution Pension Systems

The notional defined contribution pension scheme combines pay-as-you-go financing and a defined contribution pension formula. The return on contributions is based on an index set by law, such as the growth rate of GDP, average wages or contribution payments. The volatility of this return compromises the systems pension adequacy and therefore guarantees may be needed. Here, we provide a minimum return guarantee to the pension contributions. The price is calculated in a utility indifference framework. We obtain a closed-form solution for a general dependence structure with exponential preferences and in presence of stochastic short interest rates.

Fair valuation of various participation schemes in life insurance

Fair valuation is becoming a major concern for actuaries, especially in the perspective of IAS norms. One of the key aspects in this context is the simultaneous analysis of assets and liabilities in any sound actuarial valuation. The aim of this paper is to illustrate these concepts, by comparing three common ways of giving bonus in life insurance with profit: reversionary, cash or terminal. For each participation scheme, we compute the fair value of the contract taking into account liability parameters (guaranteed interest rate and participation level) as well as asset parameters (market conditions and investment strategy). We find some equilibrium conditions between all those coefficients and compare, from an analytical and numerical point of view, the systems of bonus. Developments are made first in the classical binomial model and then extended in a Black and Scholes economy.

Adequacy, Fairness and Sustainability of Pay-As-You-Go-Pension-Systems: Defined Benefit Versus Defined Contribution

There are three main challenges facing public pension systems. First, pension systems need to provide an adequate income for pensioners in the retirement phase. Second, participants wish a fair level of benefits in relation to the contributions paid. Last but no least, the pension system would need to be financially sustainable in the long run. In this paper, we analyse defined benefit versus defined contribution schemes in terms of adequacy, fairness and sustainability jointly. Also, risk sharing mechanisms, that involve changes in the key variables of the system, are designed to restore the financial sustainability at the same time that we study their consequences on the adequacy and fairness of the system.