S-shaped Utility Function Research Articles

Portfolio choice with narrow framing and loss aversion: a simplified approach

This paper considers portfolio construction issues for a ‘mental accountant’, who exhibits an S-shaped utility function with loss aversion and narrowly-frames their asset allocation decision. We argue that the presence of narrow framing does not circumvent the existence of a budget constraint, and explicitly incorporate this into the investor's portfolio selection problem. The assumption of narrow framing allows us to derive relatively simple expressions for the mental accountant's optimal asset allocation decision. Our findings indicate that the mental accountant operates similarly to a high-conviction investor, leading to a time-series momentum approach to investment.

POLYNOMIAL UTILITY

We approximate the utility function by polynomial series and solve the related dynamic portfolio optimization problems. We study the quality of the Taylor and Bernstein series approximation in response to the points and degrees of the expansions and generalize from earlier expansions applied to portfolio optimization. The issue of time inconsistency, arising from a dynamically adapted center of the expansion, is approached by equilibrium theory. We present new ways of constructing polynomial utility functions and study their pitfalls and potentials. In the numerical study, we focus on two specific utility functions: For power utility, access to the optimal portfolio allows for a complete illustration of the approximations; for the S-shaped utility function of prospect theory, the use of equilibrium theory allows for approximating the solution to the (obviously interesting but yet unsolved) case of current wealth as a dynamic reference point.

An integrated sustainable bio-fuel and bio-energy supply chain: A novel approach based on DEMATEL and fuzzy-random robust flexible programming with [formula omitted] measure

The viability of the Indian government’s green initiative to blend 20% fuel grade ethanol with gasoline is investigated by modelling a sustainable integrated bio-fuel and bio-energy supply chain with multiple echelons in order to address the dual crisis of fossil fuels, i.e., environmental as well as gradually diminishing reserves. In order to find the best strategic and planning decisions, a multi-objective mixed integer programming model is developed that analyses the supply chain from a technological, economic, environmental, and social standpoints while handling various feedstocks, supply zones, products, and multiple periods. The drawbacks in addressing the uncertainty in the parameter as well as risk in decision making are overcome by introducing a novel fuzzy-random robust flexible programming approach based on Me measure and probabilistic programming. The presented model incorporates hybrid uncertainty, flexible constraints and a convex mixture of optimistic-pessimistic perspective of decision maker. A hybrid strategy combining the fuzzy Decision Making Trial and Evaluation Laboratory (DEMATEL) method and the multi-choice reference point goal programming with S-shaped utility function is presented to solve the generated multi-objective model. A case study in West Bengal, India is then performed that envisages the economic, environmental and social aspects of a bio-fuel and bio-energy supply chain.

The importance of dynamic risk constraints for limited liability operators

Previous literature shows that prevalent risk measures such as value at risk or expected shortfall are ineffective to curb excessive risk-taking by a tail-risk-seeking trader with S-shaped utility function in the context of portfolio optimisation. However, these conclusions hold only when the constraints are static in the sense that the risk measure is just applied to the terminal portfolio value. In this paper, we consider a portfolio optimisation problem featuring S-shaped utility and a dynamic risk constraint which is imposed throughout the entire trading horizon. Provided that the risk control policy is sufficiently strict relative to the Sharpe ratio of the asset, the trader’s portfolio strategies and the resulting maximal expected utility can be effectively constrained by a dynamic risk measure. Finally, we argue that dynamic risk constraints might still be ineffective if the trader has access to a derivatives market.

Fuzzy ELECTRE II Location Preselection Model of Temporary Marine Dumping Area Based on S-Shaped Utility Function and Combination Weight

Fuzzy ELECTRE II Location Preselection Model of Temporary Marine Dumping Area Based on S-Shaped Utility Function and Combination Weight

How Do Investors Value Sustainability? A Utility-Based Preference Optimization

We investigate how an investor’s preference for sustainable assets in the portfolio varies for differing levels of risk aversion. Using a sample of 411 publicly listed firms in the S&P 500, we calculate financial and sustainability returns, on which the investor’s utility depends. We approximate the investor’s preference by the exponential and s-shaped utility function and optimize with regard to the sustainability preference. We find that with increasing levels of risk aversion, both minimum-variance and maximum Sharpe ratio type investors seek to incorporate sustainable assets in the portfolio.

A three-way decision approach with S-shaped utility function under Pythagorean fuzzy information

Three-way decision, as opposed to conventional two-way decision-making, can more effectively reduce decision-making risks by adding a postponed decision-making. With respect to the three-way decision problem consisting of Pythagorean fuzzy information, considering the risk attitudes of decision makers(DMs), we exploit Pythagorean fuzzy sets (PFNs), S-shaped utility and three-way decision to establish a new Pythagorean three-way decision model. The model realizes the effective integration of Pythagorean fuzzy theory and three-way decision, and enriches three-way decision based on risk preference. Firstly, we define a new PFN score function and construct a new condition probability by using Pythagorean fuzzy similarity, which not only reflects the difference in distance but conforms to human cognitive habits. Subsequently, considering the impact of the changes in risk preferences on the measurement of decision risks, we define an action utility function and construct the risk measurement matrix function based on S-shaped utility function, calculate the thresholds by employing optimization formulation. According to key steps and algorithms, the Pythagorean fuzzy-three-way decision rules based on S-shaped utility function can be obtained. Finally, we utilize an example of selecting film projects to validate the interpretability and applicability of our proposed model.

Community decision-makers’ choice of multi-objective scheduling strategy for integrated energy considering multiple uncertainties and demand response

• Construct the RE output uncertainty model by the probability box theory. • Use complex networks to classify the community residents. • A CDR model is built to mobilize various users to join in IESs scheduling. • Propose a utility model for decision-makers based on the S-shaped utility function. • Select the second optimal strategy by Muirhead mean operator. To address the impact of uncertainty on scheduling strategy in community integrated energy systems, a multi-objective optimal scheduling model under the constraint of various uncertainties and demand response is suggested. Firstly, a source-load uncertainty model and a comprehensive demand response model are built. Secondly, construct a multi-objective satisfaction model and utility model based on the energy supplier profit, resident cost, carbon treatment amount, and renewable energy utilization rate. Then, the best strategy is determined using the entropy weight method and the Muirhead mean operator. Finally, a multi-scene case of a residential area is used to validate the model's effectiveness. The results show that: 1) The source-load uncertainty model improves overall robustness. 2) The resident cost is reduced by 7.59%-9.84%, the carbon treatment amount is reduced by 17.71%-95.635%, and the energy supplier profit and the renewable energy utilization rate are increased. 3) The strategy with the best satisfaction or the best utility may not be the best choice among the objective functions of different relationships, so decision-makers need to choose according to the actual situation. The model is beneficial to the economy and environment, and it serves as a guide for community decision-makers in selecting a comprehensive energy multi-objective scheduling strategy.

Safety evaluation of urban rail transit operation considering uncertainty and risk preference: A case study in China

Safety evaluation of urban rail transit (URT) operation can not only have a great impact on passenger travel options, but is also related to planning of urban development. The evaluation results of URT operational safety are important for transit operators and passengers. In this study, we develop an integrated multi-stage evaluation framework to assess URT operation from the perspective of safety, which considers information uncertainty of key influence factors (KIFs), and risk preference of decision-makers (DMs). Firstly, to identify and select KIFs, this study makes systematic analysis of impacts on URT operational safety from five aspects: passengers, management, equipment, environment, and disaster respectively. Vensim software is used to build a stock flow model based on system dynamics method, and a two-level index system for assessing URT operational safety is established. Secondly, DMs’ weights are specified based on interval valued triangular fuzzy (IVTF)-TOPSIS method, and IVTF-AHP-entropy is introduced to determine the combined weight of each indicator. Thirdly, an S-shaped utility function under IVTF environment is developed to get the ranking order of research objects, and operational safety levels are determined based on cloud model. Finally, Changsha subway network is selected as a case study to test our proposed evaluation framework, and some related suggestions for operators and managers have been put forward in the future.

Violations at the Reference Point of Discontinuity: Limitations of Prospect Theory and an Alternative Model of Risk Choices

The tilted S-shaped utility function proposed in Prospect Theory (PT) relied fundamentally on the geometrical notion that there is a discontinuity between gains and losses, and that individual preferences change relative to a reference point. This results in PT having three distinct parameters; concavity, convexity and the reference point represented as a disjoint between the concavity and convexity sections of the curve. The objective of this paper is to examine the geometrical violations of PT at the zero point of reference. This qualitative study adopted a theoretical review of PT and Markowitz’s triply inflected value function concept to unravel methodological assumptions which were not fully addressed by either PT or cumulative PT. Our findings suggest a need to account for continuity and to resolve this violation of PT at the reference point. In so doing, an alternative preference transition theory, was proposed as a solution that includes a phase change space to cojoin these three separate parameters into one continuous nonlinear model. This novel conceptual model adds new knowledge of risk and uncertainty in decision making. Through a better understanding of an individual’s reference point in decision making behaviour, we add to contemporary debate by complementing empirical studies and harmonizing research in this field. Doi: 10.28991/ESJ-2022-06-01-03 Full Text: PDF

Optimal investment problem under behavioral setting: A Lagrange duality perspective

In this paper, we consider the optimal investment problem with both probability distor- tion/weighting and general non-concave utility functions with possibly finite number of inflection points. Our model contains the model under cumulative prospect theory (CPT) as a special case, which has inverse S-shaped probability weighting and S-shaped utility function (i.e. one inflection point). Existing literature have shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, we combine both features and propose a step-wise relaxation method to handle general non-concave utility functions and probability distortion functions. The necessary and sufficient conditions on eliminating the duality gap for the Lagrange method based on the step-wise relaxation have been provided under this circumstance. We have applied this solution method to solve in closed-form several representative examples in mathematical behavioral finance including the CPT model, Value-at-Risk based risk management (VAR-RM) model with probability distortions, Yarri’s dual model and the goal reaching model. We obtain a closed-form optimal trading strategy for a special example of the CPT model, where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the non-concavity of the utility function.

Sustainability evaluation of public–private partnership microgrid considering uncertainty and risk preference

Abstract Making sustainability evaluation of public–private partnership (PPP) microgrid can not only produce a great impact on distributed generation but also be related to the planning and operation of distribution network. In consideration of the identification of key influence factors (KIFs) about the sustainability of PPP microgrid, information loss, and risk attitude of decision-makers (DMs), an integrated decision framework is proposed to evaluate PPP microgrid from the perspective of sustainability. Firstly, a three-stage method based on interval valued triangular fuzzy (IVTF)-Delph is developed to identify and select KIFs, and a sustainable evaluation criteria system is established from the economic aspect, social aspect, environmental aspect, technical aspect, and managerial aspect, respectively. Secondly, DMs’ weights are determined based on the IVTF-TOPSIS method, and combined weights of criteria are determined based on IVTF-AHP and IVTF-entropy. Thirdly, an S-shaped utility function under IVTF environment is utilized to get the ranking of PPP microgrids. Moreover, a case study demonstrates that the proposed sustainability evaluation framework is very suitable for PPP microgrid. Finally, sensitivity analysis shows that the utility values are influenced by risk preference and criterion weight fluctuation.

BEHAVIORAL PORTFOLIO CHOICE UNDER HYPERBOLIC ABSOLUTE RISK AVERSION

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, cumulative prospect theory (CPT), was originally introduced by [A. Tversky & D. Kahneman (1992) Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (4), 297–323]. The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by [H. Jin & X. Y. Zhou (2008) Behavioral portfolio selection in continuous time, Mathematical Finance 18 (3), 385–426]. Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Behavioral Portfolio Choice under Hyperbolic Absolute Risk Aversion

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, Cumulative Prospect Theory (CPT), was originally introduced by Tversky and Kahneman (1992). The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by Jin and Zhou (2008). Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Optimal risk asset allocation of a loss-averse bank with partial information under inflation risk

The paper aims to solve the asset allocation problem of a loss-averse commercial bank in an environment with macroeconomic risks where inflation rates obey a mean-reverting process. The bank has only partial information about the appreciation rate of inflation. Its risk preference is described using an S-shaped utility function and then a semi-analytical investment strategy is derived through the filtering theory and inverse Fourier Transformation method. The paper finds that the investment proportion which decreases with a higher degree of information incompleteness is positively correlated with inflation risks and has a close but not monotonic relationship with the reference point.

Stochastic Maximum Principle Under Probability Distortion

Within the framework of the cumulative prospective theory of Kahneman and Tversky, this paper considers a continuous-time behavioral portfolio selection problem whose model includes both running and terminal terms in the objective functional. Despite the existence of S-shaped utility functions and probability distortions, a necessary condition for optimality is derived. The results are applied to various examples.

Stochastic Maximum Principle Under Probability Distortion

Within the framework of the cumulative prospective theory of Kahneman and Tversky, this paper considers a continuous-time behavioral portfolio selection problem whose model includes both running and terminal terms in the objective functional. Despite the existence of S-shaped utility functions and probability distortions, a necessary condition for the optimality is derived. The results are applied to a few examples.

Testing for prospect and Markowitz stochastic dominance efficiency

We develop non-parametric tests for prospect stochastic dominance Efficiency (PSDE) and Markowitz stochastic dominance efficiency (MSDE) using block bootstrap resampling. Under the appropriate conditions we show that they are asymptotically conservative and consistent. We employ Monte Carlo experiments to assess the finite sample size and power of the tests. We use the tests to empirically establish whether the value-weighted market portfolio is the best choice of every individual with preferences exhibiting certain patterns of local attitudes towards risk. Our results indicate that we cannot reject the hypothesis of prospect stochastic dominance efficiency for the market portfolio. This is supportive of the claim that the particular portfolio can be rationalized as the optimal choice for any S-shaped utility function. Instead, we reject the hypothesis for Markowitz stochastic dominance, which could imply that there exist reverse S-shaped utility functions that do not rationalize the market portfolio.

Optimal investment strategies for participating contracts

Participating contracts are popular insurance policies, in which the payoff to a policyholder is linked to the performance of a portfolio managed by the insurer. We consider the portfolio selection problem of an insurer that offers participating contracts and has an S-shaped utility function. Applying the martingale approach, closed-form solutions are obtained. The resulting optimal strategies are compared with portfolio insurance hedging strategies (CPPI and OBPI). We also study numerical solutions of the portfolio selection problem with constraints on the portfolio weights.

Theories of Risk: Testing Investor Behaviour on the Taiwan Stock and Stock Index Futures Markets

Investor behavior towards risk lies at the heart of economic decision making in general and modern investment theory and practice in particular. This paper uses both the mean-variance (MV) criterion and stochastic dominance (SD) procedures to analyze the preferences for four of the most widely studied investor types in the Taiwan stock and stock index futures market. We find that risk averters (concave utility function) prefer spot to futures, whereas risk seekers (convex utility function) prefer futures to spot. Our findings also show that investors with S-shaped utility functions prefer spot (futures) to futures (spot) when markets move upward (downward). Finally, our results imply that investors with reverse S-shaped utility functions prefer futures (spot) to spot (futures) when markets move upward (downward). These results are robust with respect to sub-periods, spot returns including dividends and diversification. Although we do not check whether risk averters, risk seekers, and investors with S-shaped and reverse S-shaped utility functions actually exist in the market, we do show that their existence is plausible. The implications of our findings on market efficiency and the existence of arbitrage opportunities are also discussed in this study.