Multiscale risk analysis with information entropy applied to portfolio optimization

Brought to my Attention.   Regards portfolio investment, Risk and entropy.    Authors are colleagues of mine.  Technical.

Multiscale risk analysis with information entropy applied to portfolio optimization By George G. Polak and David F. Rogers b,

Department of Information Systems and Supply Chain Management, Raj Soin College of Business, Wright State University, Dayton, OH, 45435-0001, United States. ORCID: 0000-0002-6222-7468 b Racers Consulting & Management, 30 Silver Avenue, Ft. Mitchell, KY, 41017-2909, United States.

ORCID: 0000-0002-3676-4075

* Corresponding author. E-mail: George.Polak@Wright.edu (G. G. Polak), RogersDavidF10@gmail.com (D. F. Rogers).

Abstract The overall risk assumed in making a decision or constructing a portfolio to maximize returns in a probabilistic setting includes both a return value-based component and a probabilistic information-

based component. Each is independent of the other, and each plays an important role in our approach to decision analysis and portfolio optimization. We introduce the concept of an information entropy profile for any discrete and finite probability distribution for the purpose of fully quantifying the information based component of risk. The profile is based on the partitioning of the state space into planning cells and is employed within the framework of Multiscale Risk Analysis to optimize decision-making across the full array of possibilities between the maximin and expected value approaches. We formulate mixed- integer nonlinear optimization models to find decisions without a priori enumeration of the partitions where the information-based risk as measured by entropy is expressed as both 1) an objective to be minimized subject to a constraint on expected returns and 2) an upper-bounded constraint coupled with an objective of maximizing expected return. We also present bounding models that are formulated without logarithmic functions. Although all the models are nonconvex, we demonstrate that realistically-sized instances can be solved to optimality by off-the-shelf global optimization software.

Keywords Risk analysis; Investment analysis; Integer programming; Nonlinear programming;

Global optimization.  Submitted: March 31, 2021

Declarations

Funding: The authors did not receive funding from any public, commercial, or not-for-profit agency.

Conflicts of interest/Competing interests: There are no conflicts of interest/competing interests involving funding; employment; financial or non-financial interests, directly or indirectly related to this work. Availability of data and material: Data used for this work is available in Polak et al. (2010). Code availability (software application or custom code): Custom GAMS code available from the authors.

1 Introduction

“Only those who will risk going too far can possibly find out how far one can go.” T. S. Eliot (1931)

 A fundamental goal of decision analysis is to determine a best strategy from among several alternatives for implementation in a risk-filled future. We consider an individual Decision Maker (DM) who is charged with constructing a financial portfolio from a discrete and finite set of possible investment opportunities actuated within a discrete and finite probabilistic state space for which alternative potential outcomes for the investments are defined. These outcomes may be posited according to particular economic circumstances during the planning period, e.g., 1) reflecting historical returns during previous time periods or 2) projected returns put forth by consultants. Winston (2008) refers to this as a scenario-based setting and how to best make decisions in this type of straightforward setting remains a pervasive issue for an individual DM as well as for corporations, banks, and governments.

The most basic criteria for making sound decisions in this environment are:

 MaxiMin (MM) – The most conservative strategy where decisions are made to maximize the minimum returns across the state space, often employed for completely mitigating risk. This is a nonparametric criterion that is more appropriate for periods of higher uncertainty and greater volatility such as the downturn in the world economy in 2008 for which the DM may be highly risk-averse.

 Expected Value (EV) – A much more aggressive strategy where decisions are made to maximize

expected returns regardless of the level of potential market volatility and of the characteristics of the probability distribution.

Employing either the MM or EV criteria may be eminently viable approaches for specific applications as posited by Manski (2009, 2011, 2014) and Brock & Manski (2011) in the disparate areas of, respectively,

1) social welfare planning, 2) medical treatment selection for noninfectious diseases, 3) selecting the sizeof government, and 4) endowment allocation. However, the MM and EV criteria may both be regarded as extremes, resulting in too much risk being assumed with an EV approach or too little return being gained with a MM approach. For example, in a healthcare setting, planning with respect to the EV criteria may result in an unacceptable scarcity of services and planning with respect to a MM strategy can lead to excess capacity.   ....