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The framework of this paper was intertemporal choice, that is to say, the process whereby people were required to choose between a smaller-sooner reward and a larger-later income. In this study, the selection of rewards was supported by a discount function instead of direct preferences between the involved rewards. The objective of this paper was to measure the discounting concentration of a discount function through a variant of the Gini index and the Lorenz curve usually used in statistics. Both measures allowed for the comparison of the discounting concentration corresponding to two discount functions. The methodology employed in this paper was based on the parallelism between a discount function and the distribution function of an absolutely continuous random variable. This similarity allowed us to export the measures of concentration from the field of statistics to finance. The main result of this work was the analysis of the discounting concentration depending on other characteristics of the shape of a discount function (regularity and super-additivity) and the total area under the discount function curve.

A general methodology for loan amortization under arbitrary discount functions is discussed. It is shown that it is always possible to uniquely define a scheme for constructing the loan amortization schedule with an arbitrary assigned discount function. It is also shown that, even if the loan amortization is carried out from the sequence of principal payments and the sequence of accrued interest, the underlying discount function can be uniquely determined at the maturities corresponding to the installment payment dates. As a special case of the proposed approach, we derive the amortization method according to the law of simple interest. [ABSTRACT FROM AUTHOR]

The framework of this paper was intertemporal choice, that is to say, the process whereby people were required to choose between a smaller-sooner reward and a larger-later income. In this study, the selection of rewards was supported by a discount function instead of direct preferences between the involved rewards. The objective of this paper was to measure the discounting concentration of a discount function through a variant of the Gini index and the Lorenz curve usually used in statistics. Both measures allowed for the comparison of the discounting concentration corresponding to two discount functions. The methodology employed in this paper was based on the parallelism between a discount function and the distribution function of an absolutely continuous random variable. This similarity allowed us to export the measures of concentration from the field of statistics to finance. The main result of this work was the analysis of the discounting concentration depending on other characteristics of the shape of a discount function (regularity and super-additivity) and the total area under the discount function curve. [ABSTRACT FROM AUTHOR]

Background: Intertemporal decision-making, which involves making choices between outcomes at different time points, is a fundamental aspect of human behavior. Understanding the underlying mental processes is vital for comprehending the complexities of human decision-making and choice behavior. Objective: The main objective of this study is to investigate the interplay of mental processes, specifically cognitive evaluation, subjective valuation, and comparison, in the context of intertemporal decision-making, with a specific focus on understanding the discounting process. Methodology: Development of a mathematical representation of the discounting process that incorporates the mental processes associated with intertemporal decision-making. Result: Our findings indicate that hyperbolic discounting aligns well with the cognitive processes underlying intertemporal decision-making. Subsequent research will employ qualitative questionnaires to establish the discount function relevant to specific groups, thereby enhancing our comprehension of the discounting process within intertemporal decision-making.

The framework of this paper is behavioral finance and, more specifically, intertemporal choice when individuals exhibit decreasing impatience in their decision-making processes. After characterizing the two main types of decreasing impatience (moderately and strongly decreasing impatience), the main objective of this paper is to generalize these concepts when the criterion of time increase is given by an arbitrary function which describes such increments. In general, the methodology is mathematical calculus but particularly the concept of derivative according to the function which rules the increase of time. The main contribution of this paper is the characterization of this extension of the concept of decreasing impatience by using the aforementioned novel derivative and the well-known Prelec's index.

Background: Intertemporal decision-making, which involves making choices between outcomes at different time points, is a fundamental aspect of human behavior. Understanding the underlying mental processes is vital for comprehending the complexities of human decision-making and choice behavior. Objective: The main objective of this study is to investigate the interplay of mental processes, specifically cognitive evaluation, subjective valuation, and comparison, in the context of intertemporal decision-making, with a specific focus on understanding the discounting process. Methodology: Development of a mathematical representation of the discounting process that incorporates the mental processes associated with intertemporal decision-making. Result: Our findings indicate that hyperbolic discounting aligns well with the cognitive processes underlying intertemporal decision-making. Subsequent research will employ qualitative questionnaires to establish the discount function relevant to specific groups, thereby enhancing our comprehension of the discounting process within intertemporal decision-making. [ABSTRACT FROM AUTHOR]

The framework of this paper is behavioral finance and, more specifically, the analysis of the main anomalies (delay, magnitude and sign effects) present in the processes of intertemporal choice. To the extent of our knowledge, only the delay effect (also known as decreasing impatience) has been discriminated between moderately and strongly decreasing impatience. However, taking into account that anomalies must be explained from a psychological point of view, the main objective of this paper is to relate the aforementioned paradoxes with the four categories of temperaments (artisan, guardian, idealist and rational) by using the sixteen personality types derived from the Myers–Briggs Type Indicator and the Behavioral Investor Types. To do this, we will use the Analytic Hierarchy Process methodology in order to detect the different levels of impatience through the so-called hyperbolic factor. Indeed, the main contribution of this paper refers to an empirical application which complements the theoretical analysis. [ABSTRACT FROM AUTHOR]

The framework of this paper is behavioral finance and, more specifically, intertemporal choice when individuals exhibit decreasing impatience in their decision-making processes. After characterizing the two main types of decreasing impatience (moderately and strongly decreasing impatience), the main objective of this paper is to generalize these concepts when the criterion of time increase is given by an arbitrary function which describes such increments. In general, the methodology is mathematical calculus but particularly the concept of derivative according to the function which rules the increase of time. The main contribution of this paper is the characterization of this extension of the concept of decreasing impatience by using the aforementioned novel derivative and the well-known Prelec's index. [ABSTRACT FROM AUTHOR]

Framework and justification: The content of this paper is located on the intersection of two fields: Finance and Algebra. In effect, the current dynamism shown by most financial instruments makes it necessary to endow the foundations of finance with, as general as possible, algebraic structures. Therefore, the objective of this paper is to provide a novel view of the fundamentals of finance by using purely algebraic concepts and structures, more specifically the properties of separability and additivity of the involved discount functions and their corresponding operators. This approach provides more flexibility to the axioms of financial mathematics, so anticipating potential changes in the behavior of the so-called "rational" decision makers. Methodologically, this paper uses a variety of algebraic tools which fit the intuition behind the financial logic. Indeed, the main contribution of the paper is the wide variety of algebraic concepts belonging to the abstract algebra which can be applied to describe the behavior of intertemporal choices.

We investigate the effects of resetting mechanisms when valuing the future in economic terms through the discount function. Discounting is specially significant in addressing environmental problems and in evaluating the sense of urgency to act today to prevent or mitigate future losses due to climate change effects and other disasters. Poissonian resetting events can be seen in this context as a way to intervene the market, it modifies the discount function and it can facilitate a specific climate policy. We here obtain the exact expression of the discount function in Laplace space and attain the expression of the long-run interest rate, a crucial value in environmental economics and climate policy. Both quantities are obtained without assuming any model for the evolution of the market. Model specific results are achieved for diffusion processes and in particular for the Ornsteinâ€"Uhlenbeck and Feller processes. The effect of Poissonian resetting events is non-trivial in these cases. The overall lesson we can learn from the obtained results is that effective policies to favor climate action should be resolute and frequent enough in time: the frequency of the interventions is critical for actually observing the desired consequences in the long-run interest rate. [ABSTRACT FROM AUTHOR]

Purpose – The main purpose of this paper is to determine the discount function which better fits the individuals' preferences through the empirical analysis of the different functions used in the field of intertemporal choice. Design/methodology/approach – After an in-depth revision of the existing literature and unlike most studies which only focus on exponential and hyperbolic discounting, this manuscript compares the adjustment of data to six different discount functions. To do this, the analysis is based on the usual statistical methods, and the non-linear least squares regression, through the algorithm of Gauss-Newton, in order to estimate the models' parameters; finally, the AICc method is used to compare the significance of the six proposed models. Findings – This paper shows that the so-called q-exponential function deformed by the amount is the model which better explains the individuals' preferences on both delayed gains and losses. To the extent of the authors' knowledge, this is the first time that a function different from the general hyperbola fits better to the individuals' preferences. Originality/value – This paper contributes to the search of an alternative model able to explain the individual behavior in a more realistic way.

This study proposes a mixed-integer nonconvex programming (MINP) model for the winner determination problem (WDP) considering two discount functions in a combinatorial auction to save shipper's transportation cost. For the WDP, the shipper allows carriers to submit bids for a bundle of lanes. Then the winning carries are selected by solving the WDP. Specifically, this study considers the shipment distance-based and volume-based discounts for transportation cost, simultaneously. The state-of-the-art linearization technique is available to convert the MINP model into a mixed-integer linear program (MILP) to obtain a global optimum, but the solution time becomes inefficient when the problem size becomes large. To find efficient and effective linearization techniques for large-scale WDP, this study (1) proposes a novel WDP model with discount policies, (2) utilizes superior encoding formulation to avoid the unbalanced branch-and-bound trees in solving MILP, and (3) reduces big-M constraints to speed up the solving time. The proposed method leads to significant savings in computational efforts. Numerical experiments with real-world-sized truckload service procurement problems are solved by the proposed method and further confirmed the drastic reduction in computational time for solving the large-size WDP. [ABSTRACT FROM AUTHOR]

Online Shopping is a phenomenon that is growing rapidly at this time and consumers are an important element in the buying and selling competition in the market and consumers who make a difffference in determining the profifits of the sellers. This research discusses the problem of online shopping using the Robust Optimization method. Robust Optimization Method is a process to get optimal results with an uncertainty. Based on the demand model to optimize the buying price, an Integer Linear Programming model with discount functions is built which will be converted into Robust Optimization. In this study also used a tool that is the Maple application in the numerical calculation process.

Discounted utility theory and its generalizations (e.g., quasihyperbolic discounting, generalized hyperbolic discounting) use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity–concavity of discount function. This test can be used with any utility function (which can be linear or not) and any preferences over risky lotteries (expected utility theory or not). The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function (for delays up to two month). [ABSTRACT FROM AUTHOR]

We develop the process of discounting when underlying rates follow a jump-diffusion process, that is, when, in addition to diffusive behavior, rates suffer a series of finite discontinuities located at random Poissonian times. Jump amplitudes are also random and governed by an arbitrary density. Such a model may describe the economic evolution, specially when extreme situations occur (pandemics, global wars, etc.). When, between jumps, the dynamical evolution is governed by an Ornstein–Uhlenbeck diffusion process, we obtain exact and explicit expressions for the discount function and the long-run discount rate and show that the presence of discontinuities may drastically reduce the discount rate, a fact that has significant consequences for environmental planning. We also discuss as a specific example the case when rates are described by the continuous time random walk.

Greenhouse peppers are one of the most important crops globally. However, as in any production activity, especially agricultural, they are subject to important risk factors such as price fluctuations, pests, or the use of bad quality water. This article aims to evaluate the viability of these types of crops by using discounted cash flows. Risk evaluation has been carried out through the analysis of pepper plantations for 2016 and 2017. The traditional application of this tool has significant limitations, such as the discount rate to be used or the estimation of future cash flows. However, by using discount functions that decrease over time in combination with decoupled net present value, these limitations are expected to improve. The use of decoupled net present value has permitted an increase in the accuracy and quantification of risks, isolating the main risks such as price drops (EUR 3720 ha−1 year−1) and structural risks (EUR 1622 € ha−1 year−1). The use of decreasing discount functions has permitted a more realistic investment estimation. Finally, the sensitivity analysis shows that decoupled net present value (DNPV) is little affected by changes in interest rates in contrast to traditional net present value (NPV).

The framework of this paper is intertemporal choice and, more specifically, the so-called delay effect. Traditionally, this anomaly, also known as decreasing impatience, has been revealed when individuals reverse their preferences over monetary or non-monetary rewards. In this manuscript, we will analyze the delay effect by using preference relations and discount functions. The treatment of the delay effect with discount functions exhibits several scenarios for this paradox. Thus, the objective of this paper is to deduce the different expressions of the delay effect and their mathematical characterizations by using discount functions in stationary and dynamic settings. In this context, subadditivity will be derived as a particular case of decreasing impatience. Finally, we will introduce a new discount function, the so-called asymmetric exponential discount function, able to describe decreasing impatience.

In this paper, we mathematically formalize the concept of improving sequence effect, which is one of the main anomalies of the discounted utility model [1]. The improving sequence effect implies a preference for a given sequence of outcomes, which increase over time, and has been empirically demonstrated for both monetary and nonmonetary results (hedonic experiences and health-related outputs). Nevertheless, to date, there is no mathematical treatment of this anomaly in the context of intertemporal choice, which allows us to relate this paradox to other anomalies, such as the delay and magnitude effects. In this way, the present manuscript has filled this gap. More specifically, we have proved that the improving sequence effect for monetary rewards cannot be rationalized by using a separable discount function but only by considering a non-separable discount function. Moreover, under certain conditions, we have proved that the delay and magnitude effects are necessary conditions for the existence of the improving sequence effect.

The magnitude effect is one of the main anomalies shown by people in the context of intertemporal choice. Thus, the main objective of this paper is to provide a general procedure to obtain a general discount function able to explain this paradox. By this methodology, we will to mathematically deform time in a discount function, more specifically to multiply the variable “time” by a certain power of the reward amount. However, in most cases, the existing literature presents some ad hoc discounting models to describe a unique anomaly. In order to solve this limitation, we will apply the former methodology to the q -exponential discount function introduced by Cajueiro (2006). In this way, we will obtain a mathematical setting able to jointly explain the delay and the magnitude effects, depending on the values of parameters q and α involved in the model. Moreover, another advantage of this amount-based q -exponential function is its convergence to the time-deformed exponential, hyperbolic and linear discounting, depending on the value of q in the joint interval ( − ∞ , 1 ) ∪ ( 1 , + ∞ ) . [ABSTRACT FROM AUTHOR]

Background: This paper aims to characterize the absence of arbitrage in the context of the Arbitrage Theory proposed by Kreps (1981) and Clark (2000) which involves a certain number of well-known financial markets. More specifically, the framework of this model is a linear (topological) space X in which a (convex) cone C defines a vector ordering. There exist markets for only some of the contingent claims of X which assign a price p i to the marketed claim m i . The main purpose of this paper is to provide some novel algebraic characterizations of the no arbitrage condition and specifically to derive the decomposability of discount functions with this approach. Methods: Traditionally, this topic has been focused from a topological or probabilistic point of view. However, in this manuscript the treatment of this topic has been by using purely algebraic tools. Results: We have characterized the absence of arbitrage by only using algebraic concepts, properties and structures. Thus, we have divided these characterizations into those concerning the preference relation and those involving the cone. Conclusion: This paper has provided some novel algebraic properties of the absence of arbitrage by assuming the most general setting. The additivity of discount functions has been derived as a particular case of the general theory.