GPU parallel implementation for asset-liability management in insurance companies
Introduction
Asset liability management (ALM) is a broad denomination for the models that are used to forecast the evolution of a company along time, jointly projecting its assets and liabilities portfolios and computing the predicted cash inflows and outflows in the future. Such a company can be either a bank, an insurance company, or more generally any financial institution, a state pension fund or even a non financial corporation. Depending on the business model of the company, the specific definition of the underlying models for the assets and liabilities may vary. ALM tools aim to cover the liquidity and interest rate risks to ensure the solvency of the company, i.e. its capability to meet all its financial obligations. Another additional relevant goal is to increase the company profit. Thus, ALM can be generally understood as a management tool to maximize the investment returns, while minimizing the reinvestment risks. These models have a particular relevance in the insurance industry, because one central problem in the insurance business is precisely to guarantee the solvency of the company. Some ALM models for life insurance have been presented, for example, in [1] and the references therein. Additional references are [2], [3], [4], [5], [6], [7], [8], for example. Their numerical solution can be carried out by using Monte Carlo techniques [1]. Also high dimensional integration with sparse grids can be used to reduce the computational cost [9]. For certain simple cases, following Ito theory, one can obtain Kolmogorov partial differential equations. These equations can be solved using finite differences, see [5], [10] or [8], for example.
Traditionally, the analysis of the cash flows has been computed for some previously designed (adverse) scenarios (feasible economic situations) to stress the ALM model of the company. However, nowadays the importance of stochastic ALM models for insurance companies has increased, mainly due to new regulations and a stronger competition. With Solvency II, see [11], [12], insurance companies are allowed, and even encouraged, to develop their own in house ALM models and simulators to asses their risks. In the case of banks, the ALM is also required to manage liquidity risk in the Basel III regulation. The increase in computational power, thanks to the modern hardware architectures, has also made feasible the computation of more accurate approximations of the portfolio evolution using models of increasing complexity.
Our ALM model mainly consists in two coupled models: one for the asset portfolio and one for the liabilities.
Liability model. On the side of the liabilities, our portfolio comprises only with-profit life insurance policies, which are contracts between the insurer company and the insured (policyholder). The policyholder pays a monthly premium, while the insurer pays monthly a variable rate benefit in a policyholder saving account, and also pays a benefit, either at maturity (or policy expiration) or if the insured dies before the expiry date of the policy. For the liability model, several issues are taken into account, three of the most important being: the saving account, the surrender and the mortality of the policyholder. In order to compute the projected liabilities cash outflows, some of them are known on beforehand, as they do not depend on stochastic economic variables: for example, the final payments due to the maturity of the policies. Others are uncertain and depend on the market evolution or on the stochastic behavior of policyholders biometry.
As we are considering with-profit policies, also known as participating life insurance contracts, that allow the policyholder to participate in the earnings of the company, the premiums paid by the policyholder are incorporated into a policyholder saving account, equipped with an interest rate guarantee and a surplus associated to the participation in the company benefits. So, we need to incorporate a model for the policyholder account evolution. In this respect, we follow the usual procedure in European countries, where the interest rate guarantee takes the form of a cliquet option (see [13], for example). Recently, some attention has been addressed to the optimization of the interest rate guarantee by the insurance company (see [14] or [13], for example). In particular, we mainly follow [13] concerning the model for the policyholder saving account evolution. Note that as we merge the policies of the same characteristics in a model point, actually we consider a model for the model point saving account.
We also include a model for the possibility of surrender the policy before maturity. We assume that surrender can happen when the rates in the market are higher than the profit offered by the company at some moment. So, the surrender model is parameterized by considering at any time the difference between the profit of the company and the profit provided by risk free bonds (see [15], [16], [17], for example).
We also consider a biometric model for the benefit that the company must pay in case of death of the policyholder before the policy's maturity.
Asset model. On the side of the asset, the portfolio involves fixed interest rate instruments (bonds), equities and cash. Insurance companies have a conservative investing strategy, and thus the biggest part of the monthly incomes are indexed to fixed rate assets. We assume that the remaining part is invested in equity. The bond and equity returns are related to the corresponding stochastic models of the interest rates and the asset value evolution, respectively.
Asset allocation model. We note that the assets portfolio is not constant in time, its composition evolves according to the investment strategy of the company. Thus, the asset portfolio is restructured (rebalanced) monthly to maintain the proportion of each asset (cash, equity and bonds) equal to a given target proportion in the asset mix. For this purpose, the investment strategy decides whether to buy or sell assets, and which kind of bonds or equities must be sold or bought. This target asset mix can be parameterized and represents an input for the ALM model. So, at each rebalance date we ensure that there is enough money to meet the obligations with the policyholders and the company's shareholders, and also to fit the pre-specified asset mix. In the sense of [18], we pose a simplified fixed asset mix modeling approach.
We note that in the stochastic programming setting, this is not understood as a dynamic asset allocation model, as the paths are not interconnected. The stochastic programming approach is mainly based on an event tree for the involved random variables, thus looking for the optimal decision for each node of the event tree, given the information available at that point. The optimal policy fits the conditions at each state and optimizes the long term. In the present work, we just rebalance the portfolio to maintain a target asset mix, however we do not solve the optimal allocation problem. Our objective is just to illustrate the advantages of GPU computations to improve the computational performance of the simple model, which does not include the dynamic optimal allocation through stochastic programming. For this approach, we address the readers to the books [19], [20] or the references [21], [18], for example. We are are exploring to incorporate the stochastic programming approach to the here proposed GPU implementation. In this respect, a useful starting reference could be the work in [22], that uses parallel computing techniques based on a PVM protocol for ALM with stochastic programming in a real Dutch pension fund.
Balance sheet projection. The balance sheet of the company must be computed for any given economic scenario of the previously defined models. Each scenario is given by the evolution of several economic variables. If we know all the values of the variables at any time in the scenario, we can supply them to the assets and liabilities models, compute the surrendered policies, etc. Next, we compute the cash flows and the value of the assets and liabilities portfolios.
Numerical method and parallel implementation. The numerical method to compute the projected balance sheet of the company is based on a Monte Carlo framework. We generate thousands of economic scenarios provided by the stochastic models for the evolution of interest rates and assets.
Computing the balance sheet of the company for all times at each scenario can be a quite demanding computational task, that can take even a day or more, mainly depending on the number of policies in the portfolio, the number of scenarios, the temporal length of the forecast and the number of time steps per scenario. Taking all this into account, the model has been parallelized in multi-CPU clusters using OpenMP and also in heterogeneous systems, more precisely using GPUs. The current implementation has been developed for Nvidia GPUs using CUDA. We show performance results for the Multi-CPU and GPU implementations.
The plan of the article is as follows. In Section 2 we describe a with-profit life insurance company and the composition of the assets and liabilities portfolios. In Section 3 we describe the computations related to the balance sheet of the book of the company. In Section 4 we introduce the models for the asset, the liabilities and the asset allocation model. In Section 5 we describe the numerical scheme, while in Section 6 we present its implementation and its parallelization using GPUs. Finally, in the last section we show some results with the GPU implementation of a test example.
Section snippets
Insurance company: with-profit life insurance product
For any company we have two portfolios, an asset portfolio (a set of resources with the expectation that they will provide future benefits) and a liability one (debts or obligations that arise during the course of business operations). In the case of our with-profit life insurance company, the asset portfolio comprises bonds, equities and cash (see Table 1), while the liability portfolio contains with-profit life insurance policies.
Although there are many different types of insurance contracts,
Balance sheet
In financial accounting, a balance sheet is a summary of the financial balances of a corporation or any business organization. A balance sheet can be understood as a “snapshot of a company's financial condition”, taking into account the values of the assets and the liabilities at a particular closing date t. There is a large amount of standard calculus that a company computes to understand its financial position at a certain date (see Table 2). In this section we explain the overall structure
ALM model
As indicated in the introduction, our ALM model can be understood as a fixed mix model, based on two models (see Table 3): one for the assets portfolio and another one for the liabilities portfolio. Assets and liabilities evolve together in time, interacting each other.
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Asset model: it mainly consists of the models for the evolution of the interest rates and the assets.
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Liability model: this is the model for the evolution of the liabilities. It includes a model for the policy holder account, for
Numerical method: MC balance sheet projection
So far, we have shown how to compute the balance sheet of an insurance company, at only one given time step. Thus, only the usual actuarial calculus to compute the balance for the books of the enterprise has been described. As everything was assumed to be known at that time, no simulations were needed.
Next, we are going to start simulating (predicting the future) the balance sheet evolution for the economic scenarios in time [T0, T]; this is the so called projection of the balance sheet. These
Implementation details
Computing the balance sheet of the company at all times for each scenario can be a quite demanding computational task, that can take even a day or more, mainly depending on the number of policies, the number of scenarios, the temporal duration of the forecast and the number of time steps per each scenario Therefore, the model has been parallelized in multi CPU clusters using OpenMP and also in heterogeneous systems, more precisely using GPUs. Thus, the code has been parallelized both in
Numerical results
In this section we show the numerical results and the performance of the parallel implementations by using OpenMP for Multi-CPU and CUDA for GPUs.
Conclusions
In this work we have developed an efficient GPU and multiCPU parallel implementation of a stochastic asset liability management model for a (with-profit) Life Insurance Company.
The library has been developed in an object oriented way using C++ and CUDA, for the Nvidia GPUs version, and using OpenMP for the Multi-CPU version. The most usual characteristics of a with profit life insurance policy have been considered, like interest rate guarantee with cliquet option style, mortality benefits,
Acknowledgements
This work is related to a project between Instituto Tecnológico de Matemática Industrial (ITMATI) and the consultancy company Analistas Financieros Internacionales (AFI). This work has been partially funded by Spanish MINECO (Research Project MTM2013-47800-C2-1-P). We also would like to thank Paul MacManus, Samuel Soláns, José Antonio Puertas, Iratxe Galdeano and Janire Alonso for their most valuable comments and help.
Finally, the authors would like to thank three anonymous referees whose
José L. Fernández. Full professor in Mathematical Analysis at the Faculty of Mathematics in the Autonomous University of Madrid. He got a Ph.D. in Mathematics from Washington University at St. Louis. During his stay in USA, he has been main researcher of 2 grants from the National Science Foundation and a Special Grant in 1986. He has also been main researcher of a EU project involving research groups from universities in 6 European countries. He has more than 50 scientific publications. Main
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2020, Journal of Computational ScienceCitation Excerpt :The ALM problem can be viewed: (i) dynamically because of its current portfolio position results from previous actions [11], (ii) stochastically due to the uncertain nature of asset prices and liabilities [63], and (iii) as a multistage problem as regular discrete time periods are defined to review the asset allocation and guarantee the liability payment [49]. ALM models may be applied in many different financial environments, for instance, banks [51,23], insurance [12,38,20], pension funds [61,48] and debt management [13,55]. Furthermore, the stochastic multistage ALM problem has been addressed before; some recent examples are in Yao et al. [60], Duarte et al. [18], and Li et al. [44].
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José L. Fernández. Full professor in Mathematical Analysis at the Faculty of Mathematics in the Autonomous University of Madrid. He got a Ph.D. in Mathematics from Washington University at St. Louis. During his stay in USA, he has been main researcher of 2 grants from the National Science Foundation and a Special Grant in 1986. He has also been main researcher of a EU project involving research groups from universities in 6 European countries. He has more than 50 scientific publications. Main research areas are related financial mathematics and complex analysis. He has been editor in chief of Revista Matemática Iberoamericana for a long time. He has advised 8 Ph.D. defended thesis. Also he has been Associate Member of the company Analistas Financieros Internacionales (AFI), a Spanish financial consultancy firm with worldwide activity, in which he got a lot of experience in the collaboration with the financial and insurance sectors, and also was the Director of the AFI Financial School and the AFI Master on Quantitative Finance. In 2009 h3 was nominated Doctor Honoris Causa by Universidad de La Laguna (Spain) and in 2015 he was one of the three first awarded with the medal of the Spanish Royal Society of Mathematics.
Ana M. Ferreiro-Ferreiro. Associate Professor of Applied Mathematics at the Department of Mathematics of the University of A Coruña since 2006, member of the research group “Models and numerical methods in engineering and applied sciences”, M2NICA and affiliated Researcher of Technological Institute for Industrial Mathematics, ITMATI. She obtained her M.Sc. degree in Mathematics from University of Santiago de Compostela, Spain, and in 2006 received the Ph.D. degree in mathematics from University of Sevilla, Spain. Her research interests include mathematical modeling, numerical methods, high-performance computing and parallel computing. She has worked in numerical schemes for solving hyperbolic equations arising in fluid mechanics and her current research interests are financial mathematics, numerical methods for global optimization and numerical simulation using HPC hybrid architectures with GPUs accelerators. Her scientific activity has produced more than 15 publications in relevant journals.
José A. García-Rodríguez. Associate Professor of Applied Mathematics at the Department of Mathematics of the University of A Coruña since 2006, member of the research group “Models and numerical methods in engineering and applied sciences”, M2NICA and affiliated researcher of Technological Institute for Industrial Mathematics, ITMATI. He obtained his M.Sc. degree in Mathematics from University of Málaga, Spain, and received the Ph.D. degree in mathematics from the same university in 2005. He has worked in numerical schemes for solving hyperbolic equations arising in fluid mechanics and its parallel implementation. His current research interests are financial mathematics, numerical methods for global optimization and numerical simulation using HPC hybrid architectures with GPUs accelerators. He has participated in projects and technology transfer contracts with enterprises and public institutions. His scientific activity has produced more than twenty publications in relevant journals.
Carlos Vázquez. Full professor in Applied Mathematics at the Faculty of Informatics in the University of A Coruña (Spain) since 2000.Formerly he was associated professor at universities of A Coruña, Vigo and Santiago, and invited professor at universities of Lyon and Bologna. He has a B.Sc. in Mathematics (5 year degree), a B.Sc. in Economics (5 year degree) and a Ph.D. in Applied Mathematics from University of Santiago. He got a Doctorat Troisieme Cicle at University Claude Bernard de Lyon (France). He is coauthor of more than 90 articles in scientific journals (80 in JCR) related to modeling, numerical methods and scientific computing for problems arising in finance, insurance, fluid and solid mechanics, tribology, glaciology and biology. Main researcher of national and international grants, and involved in more than 15 contracts with industry. He has been invited speaker in more than 30 international congresses and author of more than 120 national and international congress communications. He has advised 11 Ph.D. defended thesis. He currently belongs to editorial boards of 7 international scientific journals. He is also main researcher of the group “Models and numerical methods in engineering and applied sciences”, M2NICA, and affiliated researcher of Technological Institute for Industrial Mathematics, ITMATI.