Optimizing Portfolio Returns and Multi-Criteria Decision Analysis
Efficient Solutions via Parametric Constraints
Efficient solutions are obtained by optimizing one objective while considering the other as a parametric constraint. We can obtain an efficient solution by solving the model for each value of the Right-Hand Side (RHS) of the parametric constraint between its ideal and anti-ideal values.
Types of Parametric Constraints
- Total Gross Margin: Represented with the sign ≥.
- Investment Cost: Represented as a parametric constraint ≤; the lower the cost, the better.
Portfolio Optimization and the Efficient Frontier
In a portfolio problem, we change the desired minimum return of the portfolio and solve the resulting model by minimizing the total risk. This allows us to obtain the composition and risk associated with the optimal portfolio for a given total return.
The efficient frontier shows the minimum risk of a portfolio, measured as Standard Deviation (SD), for each value of the portfolio's total return.
The Markowitz Model
The Markowitz model focuses on the objectives of investors: maximizing expected return and minimizing the total risk of the portfolio.
- Expected Return R(X): The weighted sum of the individual average returns from the n assets that compose the portfolio. The weights are the proportions of each asset, which serve as the decision variables of the model.
- Risk of the Portfolio: This is measured by the variance of the portfolio's total return, V(X).
Analytic Hierarchy Process (AHP) Priorities
Local priorities represent the weights or relative contribution of one criterion or alternative with respect to a criterion in the next higher level of the hierarchy.
Global priorities allow for selecting and prioritizing the alternatives within a decision problem.
Pairwise Comparisons: AHP vs. PROMETHEE
There are two types of pairwise comparisons in AHP:
- Between criteria in the same level with respect to a criterion in the next higher level.
- Between alternatives with respect to a criterion in the next upper level of the hierarchy.
In contrast, PROMETHEE is only based on pairwise comparisons between alternatives.
In PROMETHEE, it is necessary to know the behavior of the alternatives for each criterion, either quantitatively or qualitatively, through an evaluation table. AHP only requires the relative contribution of one alternative over another with respect to a criterion, based on expert judgments.