Financial Planning Problems Research Articles

Large-scale financial planning via a partially observable stochastic dual dynamic programming framework

The multi-stage stochastic programming (MSP) approach is widely used to solve financial planning problems owing to its flexibility. However, the size of an MSP problem grows exponentially with the number of stages, and such problem can easily become computationally intractable. Financial planning problems often consider planning horizons of several decades, and thus, the curse of dimensionality can become a critical issue. Stochastic dual dynamic programming (SDDP), a sampling-based decomposition algorithm, has emerged to resolve this issue. While SDDP has been successfully implemented in the energy domain, few applications of SDDP are found in the finance domain. In this study, we identify the major obstacle in using SDDP to solve financial planning problems to be the stagewise independence assumption and propose a partially observable SDDP (PO-SDDP) framework to overcome such limitations. We argue that the PO-SDDP framework, which models uncertainties using discrete-valued partially observable Markov states and introduces feasibility cuts, can properly address large-scale financial planning problems.

Optimal chance-constrained pension fund management through dynamic stochastic control

We apply a dynamic stochastic control (DSC) approach based on an open-loop linear feedback policy to a classical asset-liability management problem as the one faced by a defined-benefit pension fund (PF) manager. We assume a PF manager seeking an optimal investment policy under random market returns and liability costs as well as stochastic PF members’ survival rates. The objective function is formulated as a risk-reward trade-off function resulting in a quadratic programming problem. The proposed methodology combines a stochastic control approach, due to Primbs and Sung (IEEE Trans Autom Control 54(2):221–230, 2009), with a chance constraint on the PF funding ratio (FR) and it is applied for the first time to this class of long-term financial planning problems characterized by stochastic asset and liabilities. Thanks to the DSC formulation, we preserve the underlying risk factors continuous distributions and avoid any state space discretization as is typically the case in multistage stochastic programs (MSP). By distinguishing between a long-term PF liability projection horizon and a shorter investment horizon for the FR control, we avoid the curse-of-dimensionality, in-sample instability and approximation errors that typically characterize MSP formulations. Through an extended computational study, we present in- and out-of-sample results which allows us to validate the proposed methodology. The collected evidences confirm the potential of this approach when applied to a stylized but sufficiently realistic long-term PF problem.

분해 및 근사 기법을 통한 대규모 다단 추계적 계획 문제의 해법: 재정 계획문제에 대한 적용

In this dissertation, we study decomposition and approximation techniques to solve a large-scale financial planning problem in multistage stochastic program. First, we propose an extended framework of the state-of-the-art stagewise decomposition algorithm called stochastic dual dynamic programming (SDDP) tailored for large-scale financial planning problems. Our proposed framework addresses the limitations of conventional SDDP in a perspective of finance, making it a viable tool for solving large-scale financial planning problems. Second, we apply the proposed SDDP framework to the asset liability management (ALM) problem of National Pension Service (NPS) of Korea. Furthermore, a sensitivity analysis under various contribution related parameters is conducted to provide insightful information for the sustainability of Korean public pension fund. Last, we introduce a novel stagewise decomposition algorithm called value function gradient learning (VFGL). Throughout three numerical examples, we verify that the VFGL has a great numerical potential compared to the conventional stagewise decomposition algorithms. The findings in this study will provide better understanding and techniques to solve large-scale financial planning problem, and further to the general large-scale multistage stochastic programs.

Optimising data-driven network under limited resource: a partial diversification approach

This paper describes a cardinality constrained network flow structure whose special characteristics are used to analyse different risk aspects under an environment of uncertainty. The network structure developed is a suitable alternative to support financial planning and many other decision-making problems with limited resources. By setting a diversification level, we can manage systematic and non-systematic risks under a stochastic mixed integer linear programming framework. A dual decomposition method, Progressive Hedging (PH), is applied to more efficiently accommodate instances with large numbers of scenarios. We studied the impact of the level of the diversification on transaction costs and considered different factors that influence the performance of the algorithm. In particular, a Lagrangian bound is embedded to enhance the capacity of the method. Numerical results show the effectiveness of the proposed decision support approach.

Financial planning for young households

We analyze the financial planning problems of young households whose main decisions are how to finance the purchase of a house (liabilities) and how to allocate investments in pension savings schemes (assets). The problems are solved using a multi-stage stochastic programming model where the uncertainty is described by a scenario tree generated from a vector auto-regressive process for equity returns and interest rate evolution. We find strong evidence of the importance of taking into account the multi-stage nature of the problem, as well as the need to consider the asset and liability sides jointly.

Solving ALM problems via sequential stochastic programming

In this paper, an approximation of dynamic programming using sequential stochastic programming is introduced to solve long-term dynamic financial planning problems. We prove that by approximating the true asset return dynamics by a set of scenarios and re-solving the problem at every time-step, we can solve in principle the dynamic programming problem with an arbitrarily small error. The dynamic programming algorithm is effected on the approximate sample return dynamics by means of stochastic programming. This method is applied to the problem of a fund that guarantees a minimal return on investments. This minimal return guarantee is the liability of the fund. The dynamic portfolio management problem consists of maximizing the multi-period return while limiting the shortfall with regard to the guaranteed return. The problem is tested in an 8 year out-of-sample backtest from the perspective of a Swiss fund that invests domestically and in the EU markets and faces transaction costs.

Exact methods for large-scale multi-period financial planning problems

A relevant financial planning problem is the periodical rebalance of a portfolio of assets such that the portfolio’s total value exhibits certain characteristics. This problem can be modelled using a transition graph G to represent the future state space evolution of the corresponding economy and mathematically formulated as a linear programming problem. We present two different mathematical formulations of the problem. The first considers explicitly the set of the possible scenarios (scenario-based approach), while the second considers implicitly the whole set of scenarios provided by the graph G (graph-based approach). Unfortunately, for both the formulations the size of the corresponding linear programs can be huge even for simple financial problems. However, the graph-based approach seems to be a more powerful model, since it allows to consider a huge number of scenarios in a very compact formulation. The purpose of this paper is to present both heuristic and exact methods for the solution of large-scale multi-period financial planning problems using the graph-based model. In particular, in this paper we propose lower and upper bounds and three exact methods based on column, row and column/row generation, respectively. Since the methods based on column/row generation exploits simultaneously both the primal and the dual structure of the problem we call it Criss-Cross generation method. Computational results are given to prove the effectiveness of the proposed methods.

Parallel interior-point solver for structured quadratic programs: Application to financial planning problems

Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling.We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra.The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as multistage decision processes and by nature lead to nested block-structured problems. By taking the variance of the random variables into account the problems become non-separable quadratic programs. A reformulation of the problem is proposed which reduces density of matrices involved and by these means significantly simplifies its solution by an interior point method. The object-oriented parallel solver achieves high efficiency by careful exploitation of the block sparsity of these problems. As a result a problem with over 50 million decision variables is solved in just over 2 hours on a parallel computer with 16 processors. The approach is by nature scalable and the parallel implementation achieves nearly perfect speed-ups on a range of problems.

Financial planning via multi-stage stochastic optimization

This paper describes a framework for modeling significant financial planning problems based on multi-stage optimization under uncertainty. Applications include risk management for institutions, banks, government entities, pension plans, and insurance companies. The approach also applies to individual investors who are interested in integrating investment choices with savings and borrowing strategies. A dynamic discrete-time structure addresses realistic financial issues. The resulting stochastic program is enormous by current computer standards, but it possesses a special structure that lends itself to parallel and distributed optimization algorithms. Interior-point methods are particularly attractive. Solving these stochastic programs presents a major challenge for the computational operations research and computer science community. Scope and purpose The globalization of financial markets and the introduction of complex products such as exotic derivatives have increased volatility and risks. Strides in computer and information technology has eliminated any delays between the occurrence of an event and the impact on the markets—within the home country and internationally. Thus there is a great need for an integrative approach to financial analysis and planning that encompasses the decisional environment as well as the stochastic elements in a dynamic fashion. The financial optimization model presented here incorporates several popular approaches to the problem of investment strategies, including stochastic programming and dynamic stochastic control.

Global Asset Liability Management

Dynamic financial analysis (DFA) is a technique which uses Monte Carlo simulation to investigate the evolution over time of financial models of funds, complex liabilities and entire firms. Although of increasing popularity, the drawback of DFA is the dearth of systematic methods for optimising model parameters for strategic financial planning. This paper introduces strategic DFA which employs the only recently mature technology of dynamic stochastic optimisation to fill this gap. The new approach is described in terms of an illustrative case study of a joint university/industry project to create a decision support system for strategic asset liability management involving global asset classes and defined contribution pension plans. Although the application of the system described in the paper is to fund design and risk management, the approach and techniques described here are much more broadly applicable to strategic financial planning problems; for example, to insurance and reinsurance firms, to risk capital allocation in financial institutions and trading firms and to corporate investment and business development involving real options. As well as describing the mathematical and statistical models used in the case study, the paper treats econometric estimation, asset return and liability scenario generation, model specification and optimisation, system evaluation and historical backtesting. Throughout the system visualisation plays an important role.

High-Performance Computing for Asset-Liability Management

Financial institutions require sophisticated tools for risk management. For companywide risk management, both sides of the balance sheet should be considered, resulting in an integrated asset-liability management approach. Stochastic programming models suit these needs well and have already been applied in the field of asset-liability management to improve financial operations and risk management. The dynamic aspect of the financial planning problems inevitably leads to multiple decision stages (trading dates) in the stochastic program and results in an explosion of dimensionality. In this paper we show that dedicated model generation, specialized solution techniques based on decomposition and high-performance computing, are the essential elements to tackle these large-scale financial planning problems. It turns out that memory management is a major bottleneck when solving very large problems, given an efficient solution approach and a parallel computing facility. We report on the solution of an asset-liability management model for an actual Dutch pension fund with 4,826,809 scenarios; 12,469,250 constraints; and 24,938,502 variables; which is the largest stochastic linear program ever solved. A closer look at the optimal decisions reveals that the initial asset mix is more stable for larger models, demonstrating the potential benefits of the high-performance computing approach for ALM.

Introduction to financial optimization: Mathematical Programming Special Issue

Optimization models are effective for solving significant problems in finance, including long-term financial planning and other portfolio problems. Prominent examples include: asset-liability management for pension plans and insurance companies, integrated risk management for intermediaries, and long-term planning for individuals. Several applications will be briefly mentioned. Three distinct approaches are available for solving multi-stage financial optimization models: 1) dynamic stochastic control, 2) stochastic programming, and 3) optimizing a stochastic simulation model. We briefly review the pros and cons of these approaches, discuss further applications of financial optimization, and conclude with topics for future research.

AN EMPIRICAL INVESTIGATION OF THE ORGANISATION LIFE CYCLE MODEL FOR SMALL BUSINESS DEVELOPMENT AND SURVIVAL IN THE SOUTH PACIFIC

Four general stages appear to be common in the organisational life-cycle of small businesses in Papua New Guinea (PNG): Formation, Early Growth, Later Growth and Maturity. Owner/managers confront different set of problems as the life-cycle of their businesses progressed inevitably shifting their operational and management priorities, degree of involvement in running the business, and the scope of management. External environmental problems appear predominant over the life-cycle particularly in the formation and survival stages. Internal problems predominate the late growth and maturity stages. Planning emerged as the biggest problem of small businesses in PNG. At the formation stages, developing a viable business plan for the sourcing of funds and establishing a direction for the business was the major planning problem. Marketing planning problems were more prevalent in the growth stages, and financial planning problems appeared at both the formation and maturity stages. Other problems, such as poor customer contact and weak and fragmented market, spanned the life cycle stages. The organisational life-cycle concept was shown to be adaptable across cultures but problems at different stages of the organisational life-cycle differed across cultures and level of economic development of nations.

Solving long-term financial planning problems via global optimization

Abstract A significant multi-stage financial planning problem is posed as a stochastic program with decision rules. The decision rule — called dynamically balanced — requires the purchase and sale of assets at each time stage so as to keep constant asset proportions in the portfolio composition. It leads to a nonconvex objective function. We show that the rule performs well as compared with other dynamic investment strategies. We specialize a global optimization algorithm for this problem class — guaranteeing finite e-optimal convergence. Computational results demonstrate the procedure's efficiency on a real-world financial planning problem. The tests confirm that local optimizers are prone to erroneously underestimate the efficient frontier. The concepts can be readily extended for other classes of long-term investment strategies.

Financial planning with fractional goals

When solving financial planning problems with multiple goals by means of multiple objective programming, the presence of fractional goals leads to technical difficulties. In this paper we present a straightforward interactive approach for solving such linear fractional programs with multiple goal variables. The approach is illustrated by means of an example in financial planning.

Stochastic Network Programming for Financial Planning Problems

Several financial planning problems are posed as dynamic generalized network models with stochastic parameters. Examples include: asset allocation for portfolio selection, international cash management, and programmed-trading arbitrage. Despite the large size of the resulting stochastic programs, the network structure can be exploited within the solution strategy giving rise to efficient implementations. Empirical results are presented indicating the benefits of the stochastic network approach for the asset allocation case.

Capital budgeting strategies with uncertain cash flows

In financial planning problems the selection of project investment when cash flows are certain has been the subject of considerable work. This paper considers optimal project sets with practical aspects involving the effect of uncertain flows using simulation to determine the confidence limit for required available funds and the probability that a chosen project set is not the best.

Financial Planning in an Inflationary Economy

This paper uses elementary mathematics to obtain a formula that is applicable to a wide variety of financial planning problems for which inflation should be considered. Explicit numerical examples are included to explain the theory.

Rules of Thumb and the Impact of Debt in Capital Budgeting Models

For financial planning problems involving the simultaneous selection of investment and financing decisions, linear programming models are superior from the standpoint of financial theory to simple rules of thumb such as conventional discounting. Nevertheless, from the practical point of view and as the acceptance of programming models by the financial management community is rather sluggish, it is still relevant to ask the question how do the results of the two approaches differ?. The answer to this depends on the complexity of the modelling found necessary. For very simple situations the answer is none at all, but for other situations the difference can be substantial. This paper investigates part of the middle ground of complexity, where the principal extension to the simplest model is the inclusion of debt and debt capacity and proposes that simple rules of thumb can give results sufficiently close to optimal for most purposes.

Compact Formulations of Mathematical Programmes for Financial Planning Problems

A method of converting financial planning problems from mixed-integer to pure-integer programmes is demonstrated on three examples of progressively increasing complexity. The number of constraints and variables is thereby reduced, and the use of reliable integer programming algorithms on small computer systems is facilitated.