In this paper we discuss the development of a valuation system of asset-liability management of portfolios of life insurance policies on advanced architectures. According to the new rules of the Solvency II project, numerical simulations must provide reliable estimates of the relevant quantities involved in the contracts; therefore, valuation processes have to rely on accurate algorithms able to provide solutions in a suitable turnaround time. Our target is to develop an effective valuation software. At this aim we first introduce a change of numéraire in the stochastic processes for risks sources, thus providing estimates under the forward risk-neutral measure that result in a gain in accuracy. We then parallelize the Monte Carlo method to speed-up the simulation process
This work focuses on the development of parallel algorithms for the evaluation of profit-sharing life insurance policies (PS policies). This research activity is mainly motivated by the Solvency II project [12], the European project involving the outstanding Control Authorities, which aims to establish a revised set of capital requirements and risk management standards for insurance companies. The new rules of the Solvency II Directive Proposal are increasing more and more the request of stochastic Asset-Liability Management (ALM) models. The ALM, in the Professional Actuarial Specialty Guide [29], is “the practice of managing a business so that decisions on assets and liabilities are coordinated; it can be defined as the ongoing process of formulating, implementing, monitoring and revising strategies related to assets and liabilities in an attempt to achieve financial objectives for a given set of risk tolerances and constraints”. This can be obtained by stochastic modelling and simulation.
In this context, we investigate the computational issues in the ALM of PS policies. In these contracts, the benefits which are credited to the policyholder are indexed to the annual return of an investment portfolio: the company invests the reserve in a fund, called the segregated fund, and shares the yearly return with the policyholder. A profit-sharing policy is then a derivative contract, with underlying the segregated fund. In Italian insurance market, the crediting mechanism typically guarantees a minimum to the policyholder. It is worth emphasizing that PS policies have been widely analysed in the Solvency II report, since the minimum guarantee feature results in a risk mitigation which allows insurers to reduce the sum needed at the beginning of each year in order to meet the future liabilities. Profit-sharing policies require mark-to-market valuations in order to properly compute all the quantities related to risk management, thus obtaining reliable estimates. The literature on this topic is very rich, we recall [1], [2], [3], [16], [17], [20] and [22] among the others. In particular, we refer to [16] and [17]. The numerical simulation of these financial instruments leads to large-scale computational problems. Our target is to develop a valuation system capable of being properly scaled in order to balance accuracy and efficiency.
The typical computational kernels in the application we consider are Stochastic Differential Equations (SDE) and multidimensional integrals. In [13] we investigated the use of different methods for the numerical solution of the mentioned kernels. Starting from the analysis we carried out in [14] we focused on the development of a parallel algorithm for the evaluation of participating life insurance policies in distributed environments.
In the present paper, we deal with the numerical simulation of a real ALM portfolio; in this framework, we analyse a change of numéraire in the stochastic processes for risk sources, since the flexibility of this approach can be particularly valuable in a model with stochastic interest rates. In particular, we analyse the use of the numéraire which defines the forward risk-neutral measure [6], [18], [19] and [21]. Pricing under the forward measure can provide considerable gains in accuracy, since it allows to discount at a deterministic price deflator, even though the short rate is stochastic [18]. Moreover, we use parallel computing environments to obtain efficient simulation processes.
We propose parallel algorithms for asset-liability management of PS policies portfolios, under both risk-neutral and forward risk-neutral measure, based on the parallelization of Monte Carlo method.
This paper is organized as follows. In Section 2 we outline the asset-liability framework for the evaluation of PS policies; in Section 3 we introduce the stochastic processes for the risk sources; in particular, in Section 3.1 we present the risk-neutral setting, while in Section 3.2 we discuss the change of numéraire, describing the mathematical framework under the forward risk-neutral measure. In Section 4 we discuss the parallel Monte Carlo algorithm. In Section 5 we report the numerical results of a valuation of a real portfolio, in terms of accuracy and efficiency. We test both sequential algorithms based on risk-neutral and forward measure respectively, and the parallel ones implemented on a blade server with twelve processors. Finally, in Section 6 we give some conclusions.
In this work we discussed the development of parallel algorithms for the simulation of portfolios of profit-sharing policies. Asset-liability management of these financial instruments requires insurance companies to be equipped with effective computational tools in order to be able to compute reliable estimates of all the relevant quantities in the contracts in suitable turnaround times. We presented parallel algorithms for the valuation of portfolios of PS policies, based on the parallelization of Monte Carlo method. We considered the standard risk-neutral setting for the simulation of stochastic processes for risk sources as well as the forward risk-neutral measure, that allows to significantly reduce the standard error in the estimate of the stochastic reserve, thus resulting in a considerable gain in terms of both accuracy and efficiency. We showed that the combined use of forward risk-neutral measure and parallel computing methodologies enables the production of effective simulation software which meets both the needs of insurance companies and the requirements of the Control Authorities. The developed software is thus a tool from which insurance companies can actually benefit.
