# Selected Problems

Some specific problems we studied

## Scenario Generation Techniques

The main challenging issue in portfolio selection problems is to provide evidence on how effective the mathematical models are as decision tools in determining reliable solutions to face uncertain future events. Nevertheless, the effectiveness of the models strongly depends on the data input required by the models and on the methods used to generate it. Many mathematical models for portfolio optimization require the availability of a set of scenarios to compute an estimate of both the expected portfolio risk and the expected portfolio return. Different methods can be used to generate scenarios. They range from the simple historical approach, based on the assumption that past realizations are representative of future outcomes, to more complex parametric and non-parametric methods. Each method used to generate scenarios is called a scenario generation technique. When dealing with scenario generation techniques practitioners are mainly concerned on how reliable and effective such methods are when embedded into portfolio selection models.

References:

## Index Tracking Problem and Enhanced Indexation

The index tracking problem is the problem of creating a portfolio of assets whose performance replicates, as close as possible, that of a financial market index provided by statistical bureaus like Standard & Poor's. Enhanced index tracking, sometimes also referred to as enhanced indexation, aims at slightly outperforming the financial market index with minimal additional risk. Therefore, on the one hand, in enhanced index tracking the portfolios cannot significantly deviate from the financial market index. On the other hand, in enhanced indexation the investor can engage limited additional risk (i.e., larger deviations from the financial market index) in order to yield a larger return.

References:

## Asset-Backed Securitization

Asset-Backed Securitization (ABS) is a well-stated financial mechanism which allows an institution (either a commercial bank or a firm) to get funds through the conversion of unmarketable assets (e.g. lease assets, mortgage assets or commercial papers) into capital market products called notes or asset-backed securities. More precisely, the proceeds of the notes market issuance become a long term loan (outstanding principal) for the assets owner (the originator). We analyze the combinatorial problem faced by the financial institution which has to optimally select the set of assets to be converted into notes. Issued notes yield an interest payable on periodic bases and are divided into tranches characterized by different maturity dates. The reimbursement to the holders of a tranche of notes corresponds to a reimbursement installment of the main outstanding principal. Hence, the outline of the outstanding principal has as many installments (steps) as the number of tranches of notes with different maturity issued on the market. If the sum of the outstanding principals of the selected assets has a global reimbursement profile which decreases more rapidly than that of the main outstanding principal, then the originator gets funds from its customers in advance with respect to the deadline at which it should pay the capital installment of his loan. The problem objective function is thus to minimize the gap between the two profiles and stresses the importance of studying alternative shapes for the outstanding principals which allow the sum of the outstanding principals of the assets to better fit the main outstanding principal. We have investigated heuristic and approximation algorithms to solve the problem under different amortization rules for both assets and principal investment. The particular shape of the assets outstanding principal has been exploited both in the mathematical formulations of the problem and in its solution.

References: