Knowledge‐based scenario tree generation methods and application in multiperiod portfolio selection problem

A scenario tree is an efficient way to represent a stochastic data process in decision problems under uncertainty. This paper addresses how to efficiently generate appropriate scenario trees. A knowledge‐based scenario tree generation method is proposed; the new method is further improved by accounting for subjective judgements or expectations about the random future. Compared with existing approaches, complicated mathematical models and time‐consuming estimation, simulation and optimization problem solution are avoided in our knowledge‐based algorithms, and large‐scale scenario trees can be quickly generated. To show the advantages of the new algorithms, a multiperiod portfolio selection problem is considered, and a dynamic risk measure is adopted to control the intermediate risk, which is superior to the single‐period risk measure used in the existing literature. A series of numerical experiments are carried out by using real trading data from the Shanghai stock market. The results show that the scenarios generated by our algorithms can properly represent the underlying distribution; our algorithms have high performance, say, a scenario tree with up to 10,000 scenarios can be generated in less than a half minute. The applications in the multiperiod portfolio management problem demonstrate that our scenario tree generation methods are stable, and the optimal trading strategies obtained with the generated scenario tree are reasonable, efficient and robust. Copyright © 2013 John Wiley & Sons, Ltd.     

A copula-based heuristic for scenario generation

This paper presents a new heuristic for generating scenarios for two-stage stochastic programs. The method uses copulas to describe the dependence between the marginal distributions, instead of the more common correlations. The heuristic is then tested on a simple portfolio-selection model, and compared to two other scenario-generation methods.     

Cut generation for an employee timetabling problem

Motivated by an industrial application, we study a specific employee timetabling problem. Several investigations are being conducted: a lower bound by Lagrangian relaxation, a heuristic based on a cut generation process and an exact method by Benders decomposition. Experimental results on real and generated instances are reported.     

A hybrid simulation/optimisation scenario model for asset/liability management

In this paper we desciibe a decision support model to sustain management of pension-funds in the strategic planning of the available asset- and liability policy instruments. A main characteristic of the approach is that the relevant risk-drivers are modelled by scenarios, rather than by probability distributions. We will describe the scenario generation methodology, and how the scenarios are used by pension-fund managers to simulate and improve asset/liability strategies until a strategy is identified which is agreed upon by all who carry responsibility for the pension-fund and her sponsors and trustees. Next, we will describe how this process of managerial learning can be improved by a hybrid simulation/optimisation method which applies concepts from the field of non-linear global optimisation to determine the asset-allocations which determine efficient frontiers of contribution rates and downside insolvency risks. We will conclude by showing that the application of the developed models to a particular pension-fund leads to the annulment of the infeasibility of the current asset/liability policy on the one hand, and to a reduction of the expected yearly contributions of US $100 million on the other.     

Monte Carlo scenario generation for retail loan portfolios

Monte Carlo simulation is a common method for studying the volatility of market traded instruments. It is less employed in retail lending, because of the inherent nonlinearities in consumer behaviour. In this paper, we use the approach of Dual-time Dynamics to separate loan performance dynamics into three components: a maturation function of months-on-books, an exogenous function of calendar date, and a quality function of vintage origination date. The exogenous function captures the impacts from the macroeconomic environment. Therefore, we want to generate scenarios for the possible futures of these environmental impacts. To generate such scenarios, we must go beyond the random walk methods most commonly applied in the analysis of market-traded instruments. Retail portfolios exhibit autocorrelation structure and variance growth with time that requires more complex modelling. This paper is aimed at practical application and describes work using ARMA and ARIMA models for scenario generation, rules for selecting the correct model form given the input data, and validation methods on the scenario generation. We find when the goal is capturing the future volatility via Monte Carlo scenario generation, that model selection does not follow the same rules as for forecasting. Consequently, tests more appropriate to reproducing volatility are proposed, which assure that distributions of scenarios have the proper statistical characteristics. These results are supported by studies of the variance growth properties of macroeconomic variables and theoretical calculations of the variance growth properties of various models. We also provide studies on historical data showing the impact of training length on model accuracy and the existence of differences between macroeconomic epochs.     

Shape-based scenario generation using copulas

The purpose of this article is to show how the multivariate structure (the ??shape?? of the distribution) can be separated from the marginal distributions when generating scenarios. To do this we use the copula. As a result, we can define combined approaches that capture shape with one method and handle margins with another. In some cases the combined approach is exact, in other cases, the result is an approximation. This new approach is particularly useful if the shape is somewhat peculiar, and substantially different from the standard normal elliptic shape. But it can also be used to obtain the shape of the normal but with margins from different distribution families, or normal margins with for example tail dependence in the multivariate structure. We provide an example from portfolio management. Only one-period problems are discussed.     

Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure

Mathematical Programming - Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario...     

Multidimensional investment problem

In this paper we demonstrate that the Riesz representation of excessive functions is a useful and enlightening tool to study optimal stopping problems. After a short general discussion of the Riesz representation we concretize to geometric Brownian motions. After this, a classical investment problem, also known as exchange-of-baskets-problem, is studied. It is seen that the boundary of the stopping region in this problem can be characterized as a unique solution of an integral equation arising immediately from the Riesz representation of the value function. The two-dimensional case is studied in more detail and a numerical algorithm is presented.     

Multi-stage scenario generation by the combined moment matching and scenario reduction method

We describe an opportunity to speed up multi-stage scenario generation and reduction using a combination of two well-known methods: the moment matching method (Høyland and Wallace, 2001) and the method for scenario reduction to approximately minimize a metric (Heitsch and Römish, 2009). Our suggestion is to combine them rather than using them in serial by making use of a stage-wise approximation to the moment matching algorithm. Computational results show that combining the methods can bring significant benefits.     

Cut generation for an integrated employee timetabling and production scheduling problem

This paper investigates the integration of the employee timetabling and production scheduling problems. At the first level, we manage a classical employee timetabling problem. At the second level, we aim at supplying a feasible production schedule for a set of interruptible tasks with qualification requirements and time-windows. Instead of hierarchically solving these two problems as in the current practice, we try here to integrate them and propose two exact methods to solve the resulting problem. The former is based on a Benders decomposition while the latter relies on a specific decomposition and a cut generation process. The relevance of these different approaches is discussed here through experimental results.     

Observational data-based quality assessment of scenario generation for stochastic programs

Computational Management Science - In minimization problems with uncertain parameters, cost savings can be achieved by solving stochastic programming (SP) formulations instead of using expected...     

A convex optimisation framework for the unequal-areas facility layout problem

The unequal-areas facility layout problem is concerned with finding the optimal arrangement of a given number of non-overlapping indivisible departments with unequal area requirements within a facility. We present a convex-optimisation-based framework for efficiently finding competitive solutions for this problem. The framework is based on the combination of two mathematical programming models. The first model is a convex relaxation of the layout problem that establishes the relative position of the departments within the facility, and the second model uses semidefinite optimisation to determine the final layout. Aspect ratio constraints, frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are taken into account by both models. We present computational results showing that the proposed framework consistently produces competitive, and often improved, layouts for well-known large instances when compared with other approaches in the literature.     

Parameter estimation in stochastic scenario generation systems

Scenario analysis offers an effective tool for addressing the stochastic elements in multi-period financial planning models. Critical to any scenario generation process is the estimation of the input parameters of the underlying stochastic model for economic factors. In this paper, we propose a new approach for estimation, known as the integrated parameter estimation (IPE). This approach combines the significant features of other well-known estimation techniques within a non-convex multiple objective optimization framework, with the objective weights controlling the relative importance of the features. We solve the non-convex optimization problem using adaptive memory programming – a variation of tabu search. Based on a short interest rate model using UK treasury rates from 1980 to 1995, the integrated approach compares favorably with maximum likelihood and the generalized method of moments. We also evaluate performance with Towers Perrin's CAP:Link scenario generation system.     

The effects of adding objectives to an optimisation problem on the solution set

Suppose that an additional objective function is introduced into a (multicriteria or singlecriterion) optimisation problem. The following questions are considered: How does the set of (properly, weakly, or strictly) efficient points changes when instead of the old problem the new problem is considered? What happens in the value space?     

Numerical solution of an optimal investment problem with proportional transaction costs

This paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite-horizon optimal investment problem with proportional transaction costs. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable. The proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain, a characteristics method for time discretization to deal with the large gradients of the solution, a Newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle (free boundary) feature. Moreover, piecewise linear Lagrange finite elements for spatial discretization are considered. Numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in Dai and Yi (2009) [6].     

Developing concurrent investment plans for power generation and transmission

Decisions on electric power generation and transmission investments may have crucial effects on the development of industrial and residential areas. Decisions made on the infrastructure should have economically beneficial consequences for producers and consumers. The aim of this paper is to propose a model that considers transmission and generation investments simultaneously. The proposed model fills in the gap between models for developing long-term power generation policies and instantaneous power flow models. Unlike other investment models, it explicitly takes the high voltage transmission network into account and the selection of new generation plants located on the interconnected network is made in a more realistic manner considering transmission bottlenecks.The problem subsumes the capacitated network location problem and the network design problem, the former being related to decisions on generation expansion and the latter to decisions on transmission network expansion. The integrated model becomes NP in both feasibility and optimality, because of the sub-problems it contains. Here, a practical procedure is proposed to achieve overall feasibility and also to improve investment decisions when the solution is feasible. The model is tested on the dense interconnected network of an industrialized region in Turkey. The implementation shows how future infeasibilities in the transmission network are highlighted by the model and how generation investment decisions are affected by network expansion alternatives.     

A clustering approach for scenario tree reduction: an application to a stochastic programming portfolio optimization problem

This paper deals with the problem of scenario tree reduction for stochastic programming problems. In particular, a reduction method based on cluster analysis is proposed and tested on a portfolio optimization problem. Extensive computational experiments were carried out to evaluate the performance of the proposed approach, both in terms of computational efficiency and efficacy. The analysis of the results shows that the clustering approach exhibits good performance also when compared with other reduction approaches.     

An optimisation approach for the e-grocery order picking and delivery problem

Central European Journal of Operations Research - Traditional supermarket chains that are adopting an omni-channel approach must now carry out the order picking and delivery processes to serve...