# Giacomo di Tollo, PhD Student

## Science Department - D'Annunzio University

### Chieti- Pescara (I)

## Research Project

My research project is concerned with the application of hybrid metaheuristics to problems arising from the economic and financial areas. The specimen of problem we are taking into account is the Portfolio Selection Problem, consisting in selecting a set of assets, and the share invested in each asset, that provides the investor a minimum required return and minimizes the risk. One of the main contributions for this problem is the seminal work by Markowitz (1952), who introduced the so-called mean-variance model, which takes thevariance of the portfolio returns as the measure of investorâ€™s risk. According to Markowitz, the portfolio selection problem can be formulated as an optimization problem over real-valued variables with a quadratic objective function and linear constraints (Mean-Variance Portfolio Selection Problem).

This formulation can be enriched, aiming at making it more realistic, by introducing other sort of constraints (linear and non-linear) in order to reflect the actual behavior of investor and real-worls trade. Furthermore, objective functions other than variance can be introduced (di Tollo, Roli (2008)). When taking into account those features, the portfolio selection problem becomes NP-Hard, hence the need of approximated and hybrid algorithms for tackling this problem. Approximated algorithms encompass a wide range of strategies providing us with a solution which cannot be guaranteed to be optimal; nevertheless they are effective in providing a solution in a shorter time then exact algorithm, so their use is upheld in problems whose search space is big and cannot be explored by exact methods in a reasonable amount of time. In the last years, there has been a wide amount of research aimed to combine principles stemming from different stategies, as well as to combine approximated algorithms (Meta-Heuristics) with OR and IA grounded techniques. We proposed an hybrid algorithm combining meta-heuristics (Tabu Search, Best Improvement and First Improvement) with Quadratic Programming, where the local search explores the search space composed of assets, whilst QP determine the best configuration for the asset selected by the mataheuristic. With this approach we were able to solve known instances of the Portfolio Selection Problem, and the approaches compares favourably with the literature about this topic in both terms of time and quality of the obtained solutions. Then we further step got into the following aspects:

- Allowing Short Sales;
- Tackling Index Tracking Problem;
- Combining the Portfolio Selection Problem with Index Tracking in a multiobjective framework.
- Estimation Local Search for Portfolio Selection Problem
- Statistical dominance through Pareto Efficient Portfolios

### Allowing Short Sales;

With the term short—selling we refer to the situation where the investor borrow the stock to be sold, and actually sell it.

The investor usually borrow the stock from another investor, typically a large institutional investor. The lender retains the right to call back the stock; if he does, and another lender is not promptly available, the investor must cover the short position (i.e., buy back the stock) and deliver it to the lender. The proceeds of a short sale are used as collateral for the lender of the stock. Furthermore the proceeds of the stock sale are invested in cash instruments such as short-term Treasury bills. The broker and the stock lender retain a portion of the interest earned on the proceeds. A large institutional investor that shorts stock typically receives a portion of the interest (referred to as a short rebate), while a small retail customer who sells short typically receives no part of the interest.

Short sales are excluded by assumption in most of the writings on portfolio optimization, though this restrictive assumption is arbitrary for some purposes at least. Indeed, these aspects of financial trading can be modeled with constraints, introducing several models can be introduced to handle the different policies of escrows, short proceeds and short rebate. These policies are to be investigated by using local search techniques, as the complexity of the model increases adding these features. Furthermore, the size of world-trade markets, make the

conclusion gained in this field apt to be applied to large scale dynamic solving problem.

Short selling PSP by metaheuristics has been tackled together with Prof Manfred Gilli, Evis Kellezi, Enrico Schumann, Gerda Cabej. Outcome of our joint research will be available soon.

### Tackling Index Tracking Problem;

The main shortcoming of the previous approach (Portfolio Selection Problem) consists on the fact that tackling it requires to rely on the estimates of mean and variance of assets return. This estimates have been proven to be anything else than perfect forecasts of the future values ot these aggregate, and are furthermore sujbect to evaluation errors. One way to overcome this shortcoming would be to tackle a formulation that does not rely on these estimates. For this reasons, a new problem can be tackled: the Index Tracking Problem. In its canonical formulation we want to find a portfolio that reproduces the performance of a stock market index, yet without full replication of the index having no knowledge about the index composition. Indeed, such kind of knowledge could lead us, in case we are neglecting transaction costs, to fully replicate the index. Nevertheless, the index structure is often re-adjusted (for example, because one of the stocks has been suspended from stock exchange regulation board or because its price has fallen and, in market capitalisation terms, another stock merits inclusion in the index; because a company has grown enough to be included in the index; because of mergers amongs companies), so a full replication could make us bearing undesirable transaction costs, apart from being cumbersome to manage and monitor. As a tracking portfolio needs to be rebalanced over time, there is plenty of scope for applying metaheuristics approaches to tackle dynamic versions of the problem.

A survey of the literature about Index Tracking by Metaheuristics has been carried out together with Prof. Dietmar Maringer

### Combining the Portfolio Selection Problem with Index Tracking in a multiobjective framework.

The two problems we are talking about can be combined in a multiobjective framework in order to replicate an index

(frequently requested by investors and fund managers) and to introduce another criterion to be maximized (in order to

express preferences. So far, a framework for dealing with Index Tracking and Mean Variance Portfolio has been introduced in a joint work with Mauro Birattari and Thomas Stuetzle (IRIDIA, BRUXELLES, B).

### Estimation Local Search for Portfolio Selection Problem

A new local search approach for dealing with optimized portfolio choice, based on estimation local search through Monte Carlo simulation and with a sound parameter tuning through F-Race. Joint work with Prasanna Balaprakash (IRIDIA,BRUXELLES, B).

### Statistical dominance through Pareto Efficient Portfolios

In this work we outline a procedure in order to really determine if portfolios belonging to the same efficient frontier are non-dominated. Togheter with Mauro Birattari, Thomas Stuetzle, Eliseo Ferrante (IRIDIA, BRUXELLES, B), Antonietta Di Salvatore (Sapienza, Roma, I).