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Normalized Fractional Order Entropy-Based Decision-Making Models under Risk

Poulami Paul, Chanchal Kundu

TL;DR

This work extends fractional entropy-based decision-making by introducing a normalized Ubriaco entropy $NS_q$ to capture risk attitudes in portfolio choices. The authors formulate two risk measures, NEU-FE and NEU-FEV, that blend $NS_q$ with normalized expected utility and, for the variance-augmented version, with return variance under a risk-tradeoff $\lambda$, enabling a continuum from risk-averse to risk-seeking behavior. They analyze theoretical properties, derive sensitivity to variance, and validate the framework on NIFTY50 data using multiple ML predictors (Lasso, Ridge, Random Forest, ANN), finding normalization improves numerical stability and that NEU-FE often matches or outperforms its non-normalized counterpart, while NEU-FEV highlights heavytail effects. The results support using normalized fractional entropy as a robust, risk-aware component in portfolio selection, with practical implications for investors seeking controlled exposure to uncertainty.

Abstract

Constructing efficient portfolios requires balancing expected returns with risk through optimal stock selection, while accounting for investor preferences. In a recent work by Paul and Kundu (2026), the fractional-order entropy due to Ubriaco was introduced as an uncertainty measure to capture varying investor attitudes toward risk. Building on this foundation, we introduce a novel normalized fractional order entropy aligned with investors' risk preferences that combines normalized fractional entropy with expected utility and variance. Risk sensitivity is modeled through the fractional parameter, interpolating between conservative or risk aversion and adventurous or high risk tolerance attitudes. Furthermore, the robustness and statistical significance of the fractional order entropy-based risk measure, termed normalized expected utility-fractional entropy (NEU-FE) and normalized expected utility-fractional entropy-variance (NEU-FEV) risk measures are explained with the help of machine learning tools, including Random forest, Ridge regression, Lasso Regression and artificial neural networks by using Indian stock market (NIFTY50). The results confirm that the proposed decision models support investors in making high-quality portfolio investments.

Normalized Fractional Order Entropy-Based Decision-Making Models under Risk

TL;DR

This work extends fractional entropy-based decision-making by introducing a normalized Ubriaco entropy to capture risk attitudes in portfolio choices. The authors formulate two risk measures, NEU-FE and NEU-FEV, that blend with normalized expected utility and, for the variance-augmented version, with return variance under a risk-tradeoff , enabling a continuum from risk-averse to risk-seeking behavior. They analyze theoretical properties, derive sensitivity to variance, and validate the framework on NIFTY50 data using multiple ML predictors (Lasso, Ridge, Random Forest, ANN), finding normalization improves numerical stability and that NEU-FE often matches or outperforms its non-normalized counterpart, while NEU-FEV highlights heavytail effects. The results support using normalized fractional entropy as a robust, risk-aware component in portfolio selection, with practical implications for investors seeking controlled exposure to uncertainty.

Abstract

Constructing efficient portfolios requires balancing expected returns with risk through optimal stock selection, while accounting for investor preferences. In a recent work by Paul and Kundu (2026), the fractional-order entropy due to Ubriaco was introduced as an uncertainty measure to capture varying investor attitudes toward risk. Building on this foundation, we introduce a novel normalized fractional order entropy aligned with investors' risk preferences that combines normalized fractional entropy with expected utility and variance. Risk sensitivity is modeled through the fractional parameter, interpolating between conservative or risk aversion and adventurous or high risk tolerance attitudes. Furthermore, the robustness and statistical significance of the fractional order entropy-based risk measure, termed normalized expected utility-fractional entropy (NEU-FE) and normalized expected utility-fractional entropy-variance (NEU-FEV) risk measures are explained with the help of machine learning tools, including Random forest, Ridge regression, Lasso Regression and artificial neural networks by using Indian stock market (NIFTY50). The results confirm that the proposed decision models support investors in making high-quality portfolio investments.
Paper Structure (8 sections, 3 theorems, 25 equations, 3 figures, 2 tables)

This paper contains 8 sections, 3 theorems, 25 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Normalized fractional Ubriaco entropy
  3. Normalized fractional entropy-based decision analysis models
  4. Properties of the NEU-FE and NEU-FEV measures of risk
  5. Mathematical sensitivity of NEU-FEV to variance.
  6. Statistical Validation of the proposed models using real data
  7. Performance metrics comparison of the proposed Risk Measures
  8. Conclusion

Key Result

Theorem 3.1

Let $A_{11}$ and $A_{12}$ be two different actions in the action space associated with a general decision model $G = (\Theta,\mathcal{A}_1,\vartheta)$ having equal expected utility, $NS_q(A_{11},\theta)$ and $NS_q(A_{12},\theta)$ denote normalized fractional entropies of the corresponding states of

Figures (3)

  • Figure 1: Heatmap of variations of NEU-FEV measure with changes in $q$ for NIFTY50 stocks.
  • Figure 2: Heatmap variations of NEU-FEV measure with changes in $q$ for a single NIFTY50 stock.
  • Figure 3: Comparison of $R^2$ for all the fractional order entropy-based models.

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3