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arxiv: 1906.11831 · v1 · pith:FIW4XAQ3new · submitted 2019-06-27 · 💱 q-fin.PM

A portfolio choice problem in the framework of expected utility operators

Irina Georgescu , Louis Aim\'e Fono This is my paper

Pith reviewed 2026-05-25 14:17 UTC · model grok-4.3

classification 💱 q-fin.PM
keywords portfolio choicepossibilistic riskexpected utility operatorsfuzzy numbersrisk aversionprudencetemperancepossibilistic moments
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The pith

Two formulas approximate the optimal allocation in a possibilistic portfolio choice problem using risk aversion, prudence, temperance and possibilistic moments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets up a portfolio choice problem in which asset returns are fuzzy numbers and employs expected utility operators to define possibilistic expected utility and associated risk attitudes. It derives two approximate formulas that express the optimal fraction invested in the risky asset in terms of the investor's risk aversion and prudence together with the first three possibilistic moments, with the second formula also bringing in temperance and the fourth moment. A sympathetic reader would care because the formulas supply explicit, computable expressions for near-optimal decisions when uncertainty is described by possibility distributions rather than probabilities.

Core claim

Using expected utility operators, the authors formulate a general possibilistic portfolio choice problem and derive two approximate formulas for the optimal allocation. The first depends on the investor's risk aversion and prudence along with the first three possibilistic moments of the returns. The second includes the temperance index and the fourth moment as well.

What carries the argument

Expected utility operators that associate a notion of possibilistic expected utility to model choice under fuzzy risk.

If this is right

  • The optimal share in the risky asset is approximated directly from risk aversion, prudence and the first three possibilistic moments without solving the full optimization.
  • Adding the temperance index and the fourth possibilistic moment yields a refined approximation.
  • The formulas apply uniformly across any expected utility operator in the abstract framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approximations could be checked for accuracy by comparing them against numerical solutions for standard triangular or trapezoidal fuzzy numbers.
  • The approach suggests that portfolio rules in fuzzy settings may systematically involve higher-order risk attitudes in the same way probabilistic models use skewness and kurtosis.
  • If reliable, the formulas could be embedded in practical decision tools that accept possibility distributions as input.

Load-bearing premise

Risk is modelled by fuzzy numbers and expected utility operators can be used to define a suitable notion of possibilistic expected utility for the choice problem.

What would settle it

For a concrete utility function and a specific fuzzy-number return, compute the exact optimal allocation by direct maximization and compare it to the value produced by each approximate formula.

read the original abstract

Possibilistic risk theory starts from the hypothesis that risk is modelled by fuzzy numbers. In particular, in a possibilistic portfolio choice problem, the return of a risky asset will be a fuzzy number. The expected utility operators have been introduced in a previous paper to build an abstract theory of possibilistic risk aversion. To each expected utility operator one can associate a notion of possibilistic expected utility. Using this notion, we will formulate in this very general context a possibilistic choice problem. The main results of the paper are two approximate calculation formulas for corresponding optimization problem. The first formula approximates the optimal allocation with respect to risk aversion and investor's prudence, as well as the first three possibilistic moments. Besides these parameters, in the second formula the temperance index of the utility function and the fourth possibilistic moment appear.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper formulates a portfolio choice problem in possibilistic risk theory, where asset returns are modeled as fuzzy numbers. Building on prior work introducing expected utility operators, it associates these with a possibilistic expected utility notion and derives two approximate formulas for the optimal allocation. The first approximation depends on the investor's risk aversion and prudence together with the first three possibilistic moments; the second additionally incorporates the temperance index and the fourth possibilistic moment. These approximations are obtained via expansion of the possibilistic expected utility operator.

Significance. If the expansions are correctly derived and the operators are well-defined, the explicit formulas supply practical, closed-form approximations for optimization under possibilistic uncertainty. This extends classical Taylor-based portfolio approximations (Arrow-Pratt, prudence, temperance) to the fuzzy setting and could support numerical work in finance once validated against exact solutions or simulations. The contribution is primarily translational rather than foundational.

minor comments (3)
  1. The abstract refers to 'the corresponding optimization problem' without stating the precise objective (e.g., whether it is maximization of the possibilistic expected utility or a constrained version). A short statement of the optimization program in §2 or §3 would clarify the scope of the approximations.
  2. Notation for the possibilistic moments (e.g., how the first four moments are defined from the fuzzy-number membership function) should be introduced explicitly before the expansion formulas, preferably with a reference to the prior paper on expected utility operators.
  3. The manuscript would benefit from a brief numerical illustration or comparison of the two approximations against an exact solution for a simple utility function and triangular fuzzy return, to indicate the order of the error.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the paper and for recommending minor revision. No specific major comments or requested changes appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines a possibilistic portfolio choice problem via expected utility operators introduced in prior work, then derives two approximate formulas for the optimal allocation by Taylor expansion of the operator (involving risk aversion, prudence, temperance, and the first four possibilistic moments). These expansions are standard decision-theoretic calculations once the operator is given; they do not reduce by construction to fitted parameters or self-citations. The central results are independent derivations in the new setting rather than renamings or self-definitional steps. No load-bearing uniqueness theorem or ansatz is smuggled via self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the domain assumption that risk is modeled by fuzzy numbers and on expected utility operators introduced in a prior paper by the same authors; no free parameters or new invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Risk is modelled by fuzzy numbers
    Opening hypothesis of possibilistic risk theory stated in the abstract.

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Reference graph

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