Algorithmic determination of the maximum possible earnings for investment strategies

a r t i c l e i n f o This paper proposes a new method for determining the upper bound of any investment strategy's maximum profit , applied in a given time window [0,T]. This upper bound is defined once all the prices are known at time T and therefore represents the ex-post maximum efficiency of any investment strategy determined during the relevant time interval. This approach allows us to gauge in absolute terms those behaviors defined through atomic " buy " and " sell " actions, and can be extended to more complex strategies. We show that, even in the ex-post framework, establishing this upper bound when transaction costs are implemented is extremely complex. We first describe this problem using a linear programming framework. Thereafter, we propose to embed this question in a graph theory framework and to show that determining the best investment behavior is equivalent to identifying an optimal path in an oriented, weighted, bipartite network or a weighted, directed, acyclic graph. We illustrate this method using real world data and introduce a new theory about absolute optimal behavior in the financial world. Performance gauging in Finance is a complicated issue that generates a series of methodological questions (Jensen [10], Sharpe [19], Elton et al. [6] or Malkiel [14]). In assessing the performance of a sequence of investment/divestment actions relating to a financial asset over time (for example a particular tracker fund), two frameworks can be considered. The first option is to adopt an ex-ante evaluation point of view, answering the following question: " Were the choices of the investor, given his knowledge of the future at that time, optimal or not when they were realized? ". This point of view acknowledges that investment occurs in a stochastic context and that a poor ex-post result does not necessarily indicate that bad decisions were made ex-ante, or during the decision process. Notice that this ex-ante performance assessment requires an awareness of the investor's conception of the future at each stage in the process, and is therefore difficult to achieve in practice. The second option is to adopt an ex-post evaluation approach, which considers only the statistical result of a given investment strategy over time, once price motions are perfectly known. This approach is widely used in professional asset management. For example , the performance of various investment styles is gauged using this technique. Financial journals use …