Portfolio optimization via stochastic programming: Methods of output analysis

Solutions of portfolio optimization problems are often in ̄uenced by errors or misspeci®cations due to approximation, estimation and incomplete information. Selected methods for analysis of results obtained by solving stochastic programs are presented and their scope illustrated on generic examples ± the Markowitz model, a multiperiod bond portfolio management problem and a general strategic investment problem. The approaches are based on asymptotic and robust statistics, on the moment problem and on results of parametric optimization. Key words: Portfolio optimization, stochastic programming, stability, postoptimality, worst-case analysis 1 Some early contributions The main feature of the investment and ®nancial problems is the necessity to make decisions under uncertainty and over more than one time period. The uncertainties concern the future level of interest rates, yields of stock, exchange rates, prepayments, external cash ̄ows, in ̄ation, future demand, liabilities, etc. There exist various stochastic models describing or explaining these parameters and they represent an important part of various procedures used to generate the input for decision models. To build a decision model, one has to decide ®rst about the purpose or goal; this includes identi®cation of the uncertainties or risks one wants to hedge, of the hard and soft constraints, of the time horizon and its discretization, etc. The next step is the formulation of the model and generation of the data input. An algorithmic solution concludes the ®rst part of the procedure. The subsequent interpretation and evaluation of the results may lead to model changes and, consequently, to a new solution or it may require a ``what-if '' analysis to get information about robustness of the results. Example 1 ± The Markowitz model. In conclusions of his famous paper [49] on portfolio selection, Markowitz stated that ``what is needed is essentially a `probabilistic' reformulation of security analysis''. He developed a model for portfolio optimization in an uncertain environment under various simpli®cations. It is a static, single period model which assumes a frictionless market. It applies to small rational investors whose investments cannot in ̄uence the market prices and who prefer higher yields to lower ones and smaller risks to larger ones. Let us recall the basic formulation: The composition of portfolio of I assets is given by weights of the considered assets, xi, i ˆ 1; . . . ; I , P i xi ˆ 1. The unit investment in the i-th asset provides the random return ri over the considered ®xed period. The assumed probability distribution of the vector r of returns of all assets is characterized by a known vector of expected returns Er ˆ r and by a covariance matrix V ˆ ‰cov…ri; rj†; i; j ˆ 1; . . . ; I Š whose main diagonal consists of variances of individual returns. This allows to quantify the ``yield from the investment'' as the expectation r…x† ˆPi xiri ˆ r>x of its total return and the ``risk of the investment'' as the variance of its total return, s2…x† ˆPi; j cov…ri; rj†xixj ˆ x>Vx. According to the assumptions, the investors aim at maximal possible yields and, at the same time, at minimal possible risks ± hence, a typical decision problem with two criteria, ``max'' fr…x†;ÿs2…x†g. The mean-variance e1⁄2ciency introduced by Markowitz is fully in line with general concepts of multicriteria optimization. Hence, mean-variance e1⁄2cient portfolios can be obtained by solving various optimization problems, e.g., max x AX flr>xÿ 1=2x>Vxg …1† where the value of parameter lV 0 re ̄ects investor's risk aversion. In classical theory, the set X is de®ned by P i xi ˆ 1 without nonnegativity constraints, which means that short sales are permitted. It was the introduction of risk into the investment decisions which was the exceptional feature of this model and a real breakthrough. The Markowitz model became a standard tool for portfolio optimization. It has been applied not only to portfolios of shares, but also to bonds [57], to international loans [68], etc., even to asset and liability management with portfolio returns replaced by the surplus, cf. [51]. However, there are many questions to be answered: Modeling the random returns to get their expectations, variances and covariances and analysis of sensitivity of the investment strategy on these estimated input values, the choice of the value of l, etc. From the point of view of optimization, an inclusion of linear regulatory constraints does not cause any serious problems. This, however does not apply to minimal transaction unit constraints which introduce 0±1 variables; e.g., [71]. In the interpretation and application of the results one has to be aware of the model assumptions (not necessarily ful®lled in real-life), namely, that it is a oneperiod model based on the buy-and-hold strategy applied between the initial investment and the horizon of the problem so that decisions based on its repeated use over more that one period can be far from a good, suboptimal dynamic decision, cf. [9]. At the same time, Roy [67] proposed to use the Safety-First criterion max x AX P…r>xV rp† …2† 246 J. DupacÏovaÂ