Exact methods for large-scale multi-period financial planning problems |
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Authors: | R Baldacci M A Boschetti N Christofides S Christofides |
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Institution: | (1) DEIS, University of Bologna, Bologna, Italy;(2) Department of Mathematics, University of Bologna, Via Sacchi 3, 47023 Cesena, Italy;(3) Centre for Quantitative Finance, Imperial College, London, UK |
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Abstract: | 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.
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Keywords: | Portfolio management Large scale optimization Linear programming Column/row generation |
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