BAYESIAN LEARNING FOR THE MARKOWITZ PORTFOLIO SELECTION PROBLEM

We study the Markowitz portfolio selection problem with unknown drift vector in the multi-dimensional framework. The prior belief on the uncertain expected rate of return is modeled by an arbitrary probability law, and a Bayesian approach from filtering theory is used to learn the posterior distribution about the drift given the observed market data of the assets. The Bayesian Markowitz problem is then embedded into an auxiliary standard control problem that we characterize by a dynamic programming method and prove the existence and uniqueness of a smooth solution to the related semi-linear partial differential equation (PDE). The optimal Markowitz portfolio strategy is explicitly computed in the case of a Gaussian prior distribution. Finally, we measure the quantitative impact of learning, updating the strategy from observed data, compared to nonlearning, using a constant drift in an uncertain context, and analyze the sensitivity of the value of information with respect to various relevant parameters of our model.

Dealing with Drift Uncertainty: A Bayesian Learning Approach

One of the main challenges investors have to face is model uncertainty. Typically, the dynamic of the assets is modeled using two parameters: the drift vector and the covariance matrix, which are both uncertain. Since the variance/covariance parameter is assumed to be estimated with a certain level of confidence, we focus on drift uncertainty in this paper. Building on filtering techniques and learning methods, we use a Bayesian learning approach to solve the Markowitz problem and provide a simple and practical procedure to implement optimal strategy. To illustrate the value added of using the optimal Bayesian learning strategy, we compare it with an optimal nonlearning strategy that keeps the drift constant at all times. In order to emphasize the prevalence of the Bayesian learning strategy above the nonlearning one in different situations, we experiment three different investment universes: indices of various asset classes, currencies and smart beta strategies.

Kovariantiškumas ir kodiferencija sudarant optimalų vertybinių popierių portfelį

Formuojant vertybinių popierių portfelį svarbu nustatyti ryšius tarp atskirų akcijų grąžų. Tačiau laikantis stabilumo prielaidos (modeliuojant akcijų grąžų sekas stabiliaisiais dėsniais) klasikiniai ryšio matai (kovariacija, koreliacija) negali būti taikomi. Todėl apibendrintasis Markovitzo uždavinys yra sprendžiamas su apibendrintais ryšio matais (kovariantiškumas, kodiferencija). Parodyta, kad kodiferencijos tarp atskirų finansinių instrumentų koeficientas gerokai supaprastina portfelio formavimą.Buvo sudaryti Baltijos šalių dešimties vertybinių popierių optimalūs portfeliai.On covariation and Codifference in optimal portfolio constructionIgoris Belovas, Audrius Kabašinskas, Leonidas Sakalauskas SummaryConstructing an optimal portfolio it is essential to determine possible relationships between different stock returns. However, under the assumption of stability (stock returns are modelled with stable laws) accustomed relationship measures covariance, correlation) can not be applied. Thus generalized Markowitz problem is solved with generalized relationship measures (covariation, codifference). Portfolio construction strategies with and without codifference coefficients matrix are given. We show thatthe codifference application strongly simplifies the construction of the optimal portfolio. Optimal stock portfolios (with 10 most realizable Baltic States stocks) with and without codifference coefficients matrix are constructed.