BANKING AND FINANCE

Banking and Finance



Banking and Finance

Introduction

The practical relevance of portfolio selection models has constantly increased, since their introduction in the financial literature, due to the structural transfer of big private capitals toward investments generally not required by non institutional operators (Acerbi, 2001).

As a consequence, the interest of private and institutional investors for techniques and tools aimed at a more efficient forecast of the dynamics of securities prices and to a rational management of investment capital, is hugely increased (Acerbi, 2001).

The comparisons among these three approaches consist in an ex-post analysis on the results obtained by each of them, which proves how the strategies obtained by means of the robust approach have a definitely better performance.

Robust optimization

Deterministic optimization problems are formulated assuming that the values to assign to the parameters are known; in practice, however, it is very difficult to find examples of systems that do not include some level of uncertainty about the values to assign to some of the parameters or about the actual design of some of the components of the system.

In a rather large number of cases not much is lost by assuming that these uncertain quantities are actually known, either because the level of uncertainty is low, or because they play a less significant role in the process that must be analyzed or controlled.

But most frequently the uncertain parameters play a central role in the analysis of the decision making process (an example can be the value to assign to the return of a financial asset at a future time T) and so the peculiarity of these parameters cannot be ignored without the risk of invalidating the possible implications of the analysis (Wets, 1991) (Alotto, 2001).

When the parameters are only known to belong to certain intervals or it is possible to suppose that, with a certain confidence level, their value can cover certain variation ranges, an approach that can overcome this drawback of stochastic programming is given by a recent methodology of mathematical programming, called robust optimization. In extreme synthesis, the goal is to find a solution which is feasible for all possible data realizations and optimal in some respect (Alotto, 2001).

In spite of this uncertainty, the decision x must satisfy the actual constraints whether it is possible to know them or not; the only way to meet the requirements is to restrict ourselves to robust feasible candidate solutions, i.e. those which satisfy all possible realizations of the uncertain constraints. So the admissible region is given byAx-b-[A,b]:-c:(c,A,b)-I.With the aim to choose the best among these robust feasible solutions it is necessary to decide how to “aggregate” the different realizations of the objective (cTx) into a single quality characteristic; in order to be methodologically consistent, it seems to be better to use the worst-case-oriented approach3 and take as an objective function the maximum over all possible realizations of the objective cTx, that isf(x)=sup{cTx|-[A,b]:(c,A,b)-I}.So it is possible to associate to the original uncertain linear programming problem (or more precisely to the family of all certain linear ...