A Two-Period Portfolio Selection Model for 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 assets into capital market products called notes or asset-backed securities. In this paper, we analyze the combinatorial problem faced by the financial institution which has to optimally select the set of assets to be converted into notes. We assume that assets follow an amortization rule characterized by constant periodic principal installments (Italian amortization). The particular shape of the assets outstanding principal is exploited both in the mathematical formulation of the problem and in its solution. In particular, we study a model formulation for the special case where assets selection occurs at two dates during the securitization process. We introduce two heuristic approaches based on Lagrangian relaxation and analyze their worst-case behavior compared to the optimal solution value. The performance of the algorithms is tested on a large set of problem instances generated according to two real-world scenarios provided by a leasing company. The proposed approximation algorithms turn out to yield solutions of high quality within very short computation time. The comparison to the solution approach applied by practitioners yields an average improvement of roughly 10% of the objective function value

On a Stackelberg Subset Sum Game

This contribution deals with a two-level discrete decision problem, a so-called Stackelberg strategic game: A Subset Sum setting is addressed with a set $N$ of items with given integer weights. One distinguished player, the leader, may alter the weights of the items in a given subset $L\subset N$, and a second player, the follower, selects a solution $A\subseteq N$ in order to utilize a bounded resource in the best possible way. Finally, the leader receives a payoff from those items of its subset $L$ that were included in the overall solution $A$, chosen by the follower. We assume that the follower applies a publicly known, simple, heuristic algorithm to determine its solution set, which avoids having to solve NP-hard problems. Two variants of the problem are considered, depending on whether the leader is able to control (i.e., change) the weights of its items (i) in the objective function or (ii) in the bounded resource constraint. The leader's objective is the maximization of the overall weight reduction, for the first variant, or the maximization of the weight increase for the latter one. In both variants there is a trade-off for each item between the contribution value to the leader's objective and the chance of being included in the follower's solution set. We analyze the leader's pricing problem for a natural greedy strategy of the follower and discuss the complexity of the corresponding problems. We show that setting the optimal weight values for the leader is, in general, NP-hard. It is even NP-hard to provide a solution within a constant factor of the best possible solution. Exact algorithms, based on dynamic programming and running in pseudopolynomial time, are provided. The additional cases, in which the follower faces a continuous (linear relaxation) version of the above problems, are shown to be straightforward to solve.Comment: 13 pages, 1 figur