The weak explanatory power of the Gaussian probability distributions on returns and quadratic investor preferences does not adequately support the Markowitz’s mean-variance criterion under the von Neumann-Morgenstern axioms of choice, ( Markowitz, 1991 ; Loukeris et al., 2009 ). The Power Utility demonstrated a marginal superiority on the Quadratic function, emphasizing skewness ( Loukeris et al., 2009 ). As investors prefer positive skewness, to earn high profits from extreme events, Boyle & Ding (2005) , low kurtosis in lower risk probability because of the extreme outcomes in both sides of the distribution ( Athayde & Flores, 2003 ), Lai, Yu, & Wang (2006) . Thus more accurate detection of preferences requires further higher moments, ( Loukeris et al., 2014a ; Loukeris & Eleftheriadis, 2012 ), that minimize the uncertainty of the information and thus false evaluation of stock prices either endogenous, or exogenous. Actually, the rational utility maximizers lack of descriptive accuracy as they theoretically define the investment behavior, failing to approach the real behavior ( Subrahmanyam, 2007 ). Usual patterns such as overconfidence on private signals cause overreaction, such as the BE/ME effect, the long-run reversals, that cause momentum. The loss aversion ( Barberis & Huang, 2001 ) is a robust cause for price fluctuations.

The prices deviate significantly from their fundamental values according to social mimicking, thus an arbitrage strategy can achieve profits combining contrarian and momentum trading, and as markets are inefficient, prices are very noisy ( Barberis & Sheifer, 2003 ).

Whilst a strong trend of disposition, to sell winners too soon and hold on to losers too long, although past winners do better than losers, is observed ( Shefrin & Statman, 1984a ). Characteristics of gender such as the superiority of women’s conservative tactic, the low frequency of trading, environmental, the good weather, or other non-rational parameters determine the quality of investments. Considering all these parameters we move forth to create an integrated model or portfolio selection, the IPPOS, elaborating models Hybrid neuro-genetic models from the Generalized FeedForward and the Jordan Elma family. The optimal selection problem within a portfolio follows a two-phase process. We investigate the first phase of the optimization problem. The single period model is evaluated, as we introduce six different Generalized FeedForward and Jordan Elman hybrid net models of 11 different topologies each and 4 hybrid forms with Genetic Algorithms, to calculate the efficient frontier surface, in a quintuple scope: i) To analytically investigate the behavior of investors in higher moments; ii) To introduce an advancement of the isoelastic utility; iii) To advance the Markowitz’s portfolio theory, considering apart from in fundamentals evaluation, other available data; iv) To evaluate the performance of the Generalized FeedForward and the Jordan Elman networks in neuro-genetic hybrids or neural network in different topologies in a new learning process; v) To introduce the integrated model IPPOS in optimal portfolio selection problems.

This research in Section 2 provides description on the markets, the higher moments, the utility, the investment behavior, Section 3 describes the methodology. Section 4 describes the data. Section 5 includes the results and Section 6 the conclusions.

The expected returns alter in the cross-section for multiple reasons, one of which is the risk differentials across stocks. We proceed on a further analysis of risk puts emphasis on the connection of loss to risk aversion in our model, considering also non-rational parameters such as, gender, time, firm’s proximity to investor, etc, incorporating the non-linear effects. The loss-aversion and the non-linear constraints are examined into the integrated Intelligent Portfolio Performance Optimization System (IPPOS) we introduce.

The returns distributions are not n.i.i.d., although the Fractal Markets Hypothesis-FMH appears to be quite capable to describe the markets complexity. We model investment preferences including terms of non-linearity, and non-causality. Investors allocate their utility between fears and earnings. They seek a reasonable level of return, under the fear of loss, concluding on doubtful decisions. During bullish periods the fear of losing excess profits, whilst in bearish the fear of maximizing losses, increase non-rational herding reactions on markets. ( Loukeris et al., 2014a , 2014b ; Loukeris & Matsatsinis, 2006 ), ( Loukeris et al., 2016b ; Merton, 2009 ; Odean, 1998 ; Shefrin & Statman, 1984b ) and ( Loukeris et al., 2015a , 2015b ) elaborated further higher moments on the utility function of the HARA family (Hyperbolic Absolute Risk Aversion). Based on the 5^th of hyperskewness and the 6^th of hyperkyrtosis moments ( Loukeris, Bekiros, & Eleftheriadis, 2016a ) as:

$\begin{array}{c}{U}_t\left(R_{t+1}\right)=a{E}_t\left(R_{t+1}\right)-bVa{r}_t\left(R_{t+1}\right)+cSke{w}_t\left(R_{t+1}\right)-dKur{t}_t\left(R_{t+1}\right)\\ +eHypSke{w}_t\left(R_{t+1}\right)–fHypKur{t}_t\left(R_{t+1}\right)\end{array}$ (1)

where

$Kur{t}_t\left(R_{t+1}\right)=Va{r}_t^2\left(R_{t+1}\right)$ (2)

$HypKur{t}_t\left(R_{t+1}\right)=Va{r}_t^4\left(R_{t+1}\right)$ (3)

$Ske{w}_t\left(R_{t+1}\right)=E\left(x_i-\mu \right)Va{r}_t\left(R_{t+1}\right)$ (4)

$HypSke{w}_t\left(R_{t+1}\right)=E\left(x_i-\mu \right)Va{r}_t^2\left(R_{t+1}\right)$ (5)

The general form of the utility function is:

$U_t\left(R_{t+1}\right)={\displaystyle {\sum }_{{\lambda }_{\nu }=1}^{\omega }{\left(-1\right)}^{{\lambda }_{\nu }+1}}\frac{a_{{\lambda }_{\nu }}}{n}{\displaystyle {\sum }_{i=1}^n{\left(r_i-\mu \right)}^n}$ (6)

where ${\lambda }_{\nu }$ is the depth of accuracy on investors utility preferences to risk, $a_{{\lambda }_{\nu }}$ a constant on investors profile: $a_{{\lambda }_{\nu }}=1$ for rational risk averse individuals that follow linear reasoning models with accepted causality levels, $a_{{\lambda }_{\nu }}\ne 1$ for the non-rational, x_i the value of return i in time t.

The Isoelastic Utility, a CRRA function is on the risk averse investors:

$U\left(w\right)=\{\begin{array}{ll}\frac{W^{1-\lambda }-1}{1-\lambda },\hfill & \lambda \in \left(0,1\right)\cup \left(1,+\infty \right]\hfill \\ \log \left(w\right),\hfill & \lambda =1\hfill \end{array}$ (7)

where, W the wealth, λ a measure of risk aversion.

The core problem of Philosophy on the Logic is a source of continuous discussion since the dawn of civilization. Common sense accepts our full responsibility of our choices, but are we? The freedom of selection comes from the freedom of will, in one perspective. But opposing this point many thinkers reject that our will can be managed freely. The Incopatibilists declare the irreconcilability of determinism to freedom, emphasizing in randomness that replaces determinism. Compatibilists support determinism and freedom is consistent. The Libertarians accept freedom, opposing Sceptics who believe the impossibility of freedom. ( Loukeris & Eleftheriadis, 2024 ) noticed the agreement of Epicurus to Aristotle that happiness is the highest good, refusing to identify happiness with pleasure as: i) Subjects are only chasing pleasure, and Epicurean ethical hedonism emphasizes on psychological hedonism; ii) In accordance to the Epicurean empiricism, concludes on the introspective experience: (pleasure is desired and pain is avoided). Thus, Epicurus notices that all our actions maximize the gain of our pleasure. The free will approach, accepts the impossibility of our free will judgments as systematically false ( Latham, 2019 ). ( Colasante & Riccetti, 2021 ) remarked that the research stops to the fourth moment because behaviors at higher orders are often the same as making random choices, ignoring though that the patterns of the unconscious which produces random thinking, thus nature follows random patterns, fed to the subconscious and reflected to ego, altering logical thinking. Hence, higher-order moments reflect the detailed patterns of behavior in gain and loss of non-linearity to the limits of randomness. ( Loukeris & Eleftheriadis, 2024 ) concluded that logic is dynamic in a linear process that adjusts to overriding new challenging ideas that offer higher potentials than the usual series of events. Its non-linearity is consistent to the maximization of utility and investors’ financial welfare. Future work will examine in detail the numerical results of the utilities and wealth impact of the current models incorporating new trends that experience bubble effects.

The convex problem of quadratic utility maximization ( Markowitz, 1952 ), is improved by ( Maringer & Parpas, 2009 ):

${\min }_{x}f\left(x\right)=\lambda Var\left(r_p\right)-\left(1-\lambda \right)E\left(r_p\right)$ (8)

( Loukeris et al., 2014a , 2014b ) and ( Loukeris et al., 2015a , 2015b ) emphasized on further higher moments in the model:

$\begin{array}{l}{\min }_{x}f\left(x\right)=\lambda {\upsilon }_{\gamma }\left[bVa{r}_t\left(r_p\right)+dKur{t}_t\left(r_p\right)+fHypKur{t}_t\left(r_p\right)\right]\\ -\left(1-\lambda \right){\upsilon }_{\gamma }\left[a{E}_t\left(r_p\right)+cSke{w}_t\left(r_p\right)+eHypSke{w}_t\left(r_p\right)\right]+s^{\log }{}^{{\lambda }_{\nu }}\end{array}$ (9)

${\upsilon }_{\gamma }=1-{\epsilon }_{\tau }$ (10)

$r_p={\displaystyle {\sum }_i{x}_i{r}_i^{*}}$ (11)

where υ_γ the company’s financial health, ε_τ the heuristic output (0 healthy, 1 distressed), s the social effect of non-rational features, as gender, local proximity, day of week, weather, frequency of trading, preference of on-line trading etc., $r_i^{*}$ the return of stock i in the efficient. The stocks do not fulfill all the superiority conditions are non-optimal and are exempted from the efficient frontier. As

$E\left(U_P\left(w,\lambda \right)\right)=\max \left\{{\displaystyle {\sum }_i{\left[1+\exp \left(r_i{x}_i\right)\right]}^{1-{\upsilon }_{\gamma }/\lambda }}/\left(1-{\upsilon }_{\gamma }/\lambda \right)\right\}/N$ (19)

let

$Va{r}_t^2\left(R_p\right)=z$ (12)

$Va{r}_t\left(R_{t+1}\right)=y$ (13)

as

$z=y^2={\sigma }^4$ (14)

then

$\begin{array}{l}{\min }_{x}f\left(x\right)=\lambda {\upsilon }_{\gamma }Va{r}_t\left(R_p\right)\left[b+dz+f{z}^2\right]\\ -\left(1-\lambda \right){\upsilon }_{\gamma }\left[a\mu +cE\left(x_i-\mu \right)y+eE\left(x_i-\mu \right)y^2\right]+s^{\log {\lambda }_{\nu }}\end{array}$ (15)

The novel contribution is that we extract hidden weighted social and financial patterns that can make the difference on the stock’s evaluation. The frequency of turbulence in the markets is more compatible to the FMH, because of the extended amount of noise that causes chaotic patterns and the numerous manipulations attempts from other agents. The manipulation of stocks because of internal information is filtered. The evaluation υ_γ, in (10) is more important than the investor’s behavior, because of the reverse influence in υ_γ/λ. The flow chart of processes is described in Figure 1 . On the IPPOS the problem of portfolio optimization is extended to the principal philosophical remark on the Free Will expressed by investors selection.

The Intelligent Portfolio Performance Optimisation System -IPPOS on the first step reads the fundamentals, the accounting data, the market prices, the preferred optimisation period t, and the social sentiments of investors.

In parallel the social sentiments of investors are evaluated between them to define common patterns.

Then if there are common patterns the sentiments are compared to the stock price that are refereed to and are available, in time j during processing.

If they agree then the evaluation of data is preceded by hybrid models, else the sentiments are rejected and new data are examined starting the process from the first step.

Then it proceeds by selecting the initial method to evaluate the companies whose stocks are candidate in the portfolio. On this step the individual investor’s risk profile is given and the λ is selected for the Isoelastic utility.

On the next step the system examines if this is the last firm to be examined, and if the condition for the optimal portfolio as an efficient portfolio is satisfied. Else we proceed on the next the initial evaluation uses a fast Neural Net that gives very accurate evaluations, and creates two subsets: Subset A of the healthy companies, and Subset B of the distressed firms. In the specific model we select the Jordan Elman Neural Net of 1 hidden layer that converges in 4 seconds only. The ${\epsilon }_{\tau ,N}$ value is calculated 0, for the healthy and 1 for the distressed firms. Both firms of subsets A, and B are re-evaluated in a double precision process, by a Hybrid neuro-genetic model of higher performance. Value ${\epsilon }_{\tau ,H}$ is calculated identically through the Hybrid net and it is compared to ${\epsilon }_{\tau ,N}$.

Next step these values are compared and if ${\epsilon }_{\tau ,N}={\epsilon }_{\tau ,H}$ then the decision is final, else the firm is in vague profile and it is re-evaluated in future after more data are available, and cleared.

If ${\epsilon }_{\tau ,N}={\epsilon }_{\tau ,H}=1$ then the firm is a verified distressed firm and it is removed from the overall portfolio, else if ${\epsilon }_{\tau ,N}={\epsilon }_{\tau ,H}=0$ then it is a verified healthy firm and it is included on the Subset C of the healthy firms that are candidate for the optimal efficient portfolio.

On the next step the U_t(R_t(i)) utility function of (22) is calculated per firm.

Next firms are ranked according to their utility score.

Then the Efficient Frontier is calculated.

Next the firms with the higher utility score are selected into the efficient portfolio.

The sub-optimal firms as well as the non-optimal firms are revaluated with potential new data on the step 4 of Neural Nets evaluation, following all the steps.

Next after the efficient portfolio is created, its Utility Function is calculated $U_{Pj}\left(f\right)$.

Then the optimal overall portfolio $U_{Pj}^{*}\left(f\right)$ whose utility is the maximum available, is detected, if possible, by all the available efficient portfolios utilities

$U_{Pj}\left(f\right)$ recorded in $U_{Pj}^{*}\left(f\right)>U_{Pj}\left(f\right)$.

The process stops when the time limit is reached and the IPPOS has the optimal portfolio.

The flow chart of the IPPOS is in above Figure 1 .

The most optimal performance overall wass observed on the Jordan Elman Hybrid models of GA optimization on the input and outputs only of 1 layer where the healthy firms were correctly classified at 99.83% and the distressed at 96.78%, a very low error as MSE was 0.022, the NMSE at 0.052, and the error 3.83%, whilst the fitness of the data to the model was excellent as the correlations coefficient r was the highest 0.973, the model was also impartial as the Akaike was very low at −2481.73, and the processing time quite fast at 55 m. 18 s. The second place was taken by the JE Hybrid models of GA optimization on the input and outputs only no hidden layer with an excellent classification at 99.91% for the healthy companies and 96.78% for the distressed, the error was very low as well in 0.031 for the MSE, 0.075 for the NMSE, in an excellent of the data on the model as r was 0.978, and a great impartiality of AIC in −2416.06, in the fastest time of only 57 m. 29 s., but exposed to over-training phenomena. Similar performance on the third place had the JE hybrid with GA optimization in all layers and Cross Validation in an excellent classification outcome of 99.66% for the healthy, 94.49% for the distressed firms, a very low error as MSE was 0.023, NMSE 0.055, the overall error 12.32% in a very high fitness of the data to the model on r at 0.972, a great impartiality in Akaike at −2439.55, the Cross Validation performance was very similar to the model, whilst it protects from over-fitting hazard thus this model is the most appropriate for complex modelling, and a medium convergence time of 2 h. 35 m. 29 s., to the JE NN of 1 layer that is exposed to overtraining ( Tables 1-3 ).

The GFFs unfortunately had the worst outcomes overall. The best performance overall of the Generalised FeedForward networks was achieved on the GFF Hybrid with GAs on the inputs and outputs only of 1 layer where the healthy firms were correctly classified at 98.90% and the distressed at 88.52%, a very low error as MSE was 0.072, the NMSE at 0.172, and the error 5.67%, very high fitness of the data to the model as the correlations coefficient r was the highest 0.907, the model was also impartial as the Akaike was very low at −1808.33, and the processing time quite fast at 3 h 55 min. 18 s.

The integrated Intelligent Portfolio Performance Optimisation System-IPPOS provides robust approach into the real time portfolio selection problem, as it extracts hidden patterns, avoiding fraud. The Jordan Elman networks have a superior performance that incorporates them in the model. Whilst the Hybrid Jordan Elman neuro-genetic on the inputs and outputs only of 1 layer is a fine model of excellent classification, performance and less processing time, in high risk of overfitting, as the Hybrid Jordan Elman with GAs in all layers and Cross Validation although in a marginal lower rank is the best option in all aspects plus it protects from overtraining. Hence the Jordan Elman models offer an excellent nonlinear regression result.

The core problem of portfolio optimisation as a part of the Logic in humans is answered by this AI imitation of natural neurons within our bodies. The principal philosophical question seeks an answer since the cradle of civilization, is our existence. This paper answers that Logic is dynamic in a linear part that adjusts, overriding new challenging ideas that offer higher potentials than the usual series of events. It may appear non-linear but it is consistent to the maximisation of utility, and investors’ financial welfare. Future work will analytically examine numerical results of utilities and the wealth impact of the models in the new trends of the bubbles.

We are grateful for the funding by the ELKE Fund of the Universities of Macedonia and West Attica.