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Suppose that “risk” does not concern “what” may happen but “when” something may happen. In “Decision making when things are only a matter of time,” Sebastian Ebert analyzes time risk preferences: that is, preferences toward the risk of something happening sooner or later. The author defines and characterizes “prudence” and “temperance” for discount functions in analogy to the corresponding concepts for utility functions. Prudent discounting and temperate discounting matter for time risk preferences and thus, decisions in which an important event is only a matter of time. For example, the disutility from climate change—which is only a matter of time—is underestimated when ignoring the uncertainty about when its undesired consequences come into effect.

This paper studies a time-inconsistent dividend problem in discrete time with nonex- ponential discounting. Motivated by the decreasing impatience in behaviour economics, a general discount function is used and assumed to be log sub-additive. Using a game-theoretic approach equilibrium barrier strategies are considered. It is shown that in the case of multiple equilibria, there exists an optimal one that pointwisely dominates all the other equilibria. Case studies are conducted where there is no equilibrium, multiple equilibria, and a unique equilibrium.

When solving for optimal strategies, a financial engineer needs to take into consideration of preferences heterogeneities, which involve not only present bias, but also future-focused preferences. We provide a reusable tool (i.e. algorithm) for explicitly solving optimal strategy in the presence of preferences variation over time, decision-makers and goods. In this framework, a new discount function identifies many hitherto unknown preference heterogeneities, a non-standard HJB yields sophisticated solution, a behavior equation produces naive and precommitted solutions. An application of the framework shows that a decision-maker can be neither impatient nor patient, and that sophisticated paradigm engenders immediate gratification in earlier phases of life cycle and self-control in later phases of life cycle.

We develop a novel estimation methodology for an additive nonparametric panel model that is suitable for capturing the pricing of coupon-paying government bonds followed over many time periods. We use our model to estimate the discount function and yield curve of nominally riskless government bonds. The novelty of our approach is the combination of two different techniques: cross-sectional nonparametric methods and kernel estimation for time varying dynamics in the time series context. The resulting estimator is used for predicting individual bond prices given the full schedule of their future payments. In addition, it is able to capture the yield curve shapes and dynamics commonly observed in the fixed income markets. We establish the consistency, the rate of convergence, and the asymptotic normality of the proposed estimator. A Monte Carlo exercise illustrates the good performance of the method under different scenarios. We apply our methodology to the daily CRSP bond market dataset, and compare ours with the popular Diebold and Li (2006) method.

A two-stage non-standard optimal control problem with time inconsistent preferences is studied. In an infinite horizon setting, a time consistent (sophisticated) decision maker chooses the time of switching between two consecutive regimes. The second regime corresponds to the implementation of a new technology, and a cost must be paid at the switching time. Although the problem is formulated for a general discount function, special attention is devoted to models with nonconstant discounting and heterogeneous discounting. The problem is solved by transforming it into a problem in a finite horizon and free terminal time. The corresponding dynamic programming equations are presented, and conditions for the derivation of the switching time by decision makers with different degrees of sophistication are studied. A resource extraction model with technology adoption is solved in detail. Effects of the adoption of different discount functions are illustrated numerically.

This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The investor receives a deterministic income, invests in risky assets, buys insurance and consumes continuously. The objective is to maximize the utility of excessive consumption, heritage and terminal wealth. The non-exponential discounting makes the optimal strategy adopted by a naive person time-inconsistent. The equilibrium for a sophisticated person is Nash subgame perfect equilibrium, and the sophisticated person is time-consistent. We calculate the analytical solution for both the naive strategy and equilibrium strategy in the CRRA case and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less on consumption and insurance and save more than the naive person. The difference in the strategies of the naive and sophisticated person decreases over time. Furthermore, if an individual of either type is more patient in the future or has a greater tendency toward habit formation, he/she will consume less and buy less insurance, and the degree of inconsistency will also be increased. The sophisticated person's consumption and habit level are initially lower than those of a naive person but are higher in later periods.

In this paper, we present a stochastic multi-period multi-echelon $$ \left( {s, S} \right) $$ periodic inventory control model for perishable products in a healthcare environment. We consider age-dependent purchase price, order lead time, and allow for backorder. The model is inspired by a real case where a regional health system operates multiple hospitals in addition to managing a central warehouse. The warehouse centralizes the purchasing of the products and then distributes them among hospitals while demand for products in each hospital is uncertain. We formulate the problem as a stochastic mixed-integer linear programming model and study two purchasing strategies. The first and traditional strategy assumes that the warehouse only orders fresh products, i.e., products with the latest expiry date and most expensive, referred to as purchasing constant remaining life (CRL). The second strategy considers a discount function with a lower purchase price for products with a shorter remaining life, referred to as different remaining life (DRL). The designed models are naturally complex and intractable to obtain optimal solutions by an analytical approach for large instances of the problem. Thus, we developed a genetic algorithm to solve the problem and benchmarked its performance against an exact method. We also verified the results of the stochastic models through simulation. Extensive sensitivity analysis is also conducted to investigate the behaviors of both CRL and DRL strategies about the risk of product shortage, product expiration and the total cost imposed on the system.

PurposeWe investigate causes for the cost overrun and delay of the railway project Stuttgart 21. Besides, we try to forecast the actual costs and completion date at an early stage.Design/methodology/approachThe results of exploratory research show the causes for the cost overrun and delay of Stuttgart 21; we compare our findings with other railway projects. To estimate the costs at an early stage, the reference class forecasting (RCF) model is applied; to estimate the time, we apply an OLS regression.FindingsWe find that the following causes are relevant for the cost overrun and delay of Stuttgart 21: scope changes, geological conditions, high risk-taking propensity, extended implementation, price overshoot, conflict of interests and lack of citizens' participation. The current estimated costs are within our 95% confidence interval based on RCF; our time forecast underestimates or substantially overestimates the duration actually required.Research limitations/implicationsA limitation of our approach is the low number of comparable projects which are available.Practical implicationsThe use of hyperbolic function or stepwise exponential discount function can help to give a clearer picture of the costs and benefits. The straightforward use of the RFC for costs and OLS for time should motivate more decision-makers to estimate the actual costs and time which are necessary in the light of the rising demand for democratic participation amongst citizens.Social implicationsMore realistic estimates can help to reduce the significant distortion at the beginning of infrastructure projects.Originality/valueWe are among the first who use the RCF to estimate the costs in Germany. Furthermore, the hyperbolic discounting function is added as a further theoretical explanation for cost underestimation.

Producción Científica, We consider the time consistent management of a defined benefit stochastic pension plan where the participants have different rates of time preference and the fund manager collects this heterogeneity when discounting the future. The main objective is to select the amortization rate and the investment strategy minimizing both the contribution rate risk and the solvency risk. The problem is formulated as a stochastic control problem with non-constant rate of discount and is solved analytically by means of the dynamic programming approach and the technical interest rate is selected in order to keep stable the fund evolution within prescribed targets. A numerical illustration shows a comparative of the stability of the fund assets and the rate of contribution for a convex combination of exponential functions as discount function and for the constant discount case., Junta de Castilla y León (projects VA056A09 y VA148G18), Ministerio de Economía y Competitividad (projects ECO2014-6056384-P, ECO2017-86261-P, ECO2013-48248P and ECO2017-6382227P)

The preference for incremental gains, or the improving sequence effect, has been presented as one of the most important anomalies in intertemporal choice. Usually, the experiments to verify this effect have been based on sequences of incomes, and most of them have shown a preference for an increasing sequence. The objective of this paper is to analyse this topic in the context of loan repayment plans where a preference for constant and decreasing over rising payments is expected. The methodology applied in this paper is based on the use of questionnaires in the same way as Hoelzl et al. (2011) to evaluate different loan plans with the same interest rates and durations. It clearly shows that people prefer those plans which repay more at the beginning of the loan duration. Thus, the main contribution of this paper is the use of the q -exponential discounting to explain this revealed decreasing sequence effect because the q -exponential discount function can exhibit the inconsistency in intertemporal choices based on a decreasing interest rate. Finally, recommendations are made for banking institutions when considering their strategies of interest rate offers in the context of bank marketing.

In this paper, we present a time-varying and non-stationary but non-hyperbolic discount function that explains dynamic preference reversal. The new discount function emerges from an analysis of intertemporal consumption and savings choices with mortality risk and an altruistic factor. Our analysis shows that the process of updating survival information may also account for dynamic preference reversal.

In this paper, we study general time-inconsistent stochastic control models which are driven by a stochastic differential equation with random jumps. Specifically, the time-inconsistency arises from the presence of a non-exponential discount function in the objective functional. We consider equilibrium, instead of optimal, solution within the class of open-loop controls. We prove an equivalence relationship between our time-inconsistent problem and a time-consistent problem such that the equilibrium controls for the time-consistent problem coincide with the equilibrium controls for the time-inconsistent problem. We establish two general results which characterize the open-loop equilibrium controls. As special cases, a generalized Merton's portfolio problem and a linear-quadratic problem are discussed.

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.

Maximizing a stream of present value stock often requires the use of dynamic economic analysis. To bring all future values into present value terms, we use the discount function. I present a pedagogical supplement to show that this discount function must follow an exponential distribution. This paper serves as a helpful tool to all instructors and graduate students that use discounting functions. It offers to describe the discount function from a different perspective that could make the interpretation of the discount function more easily understandable.

We address optimal rail transit investment issues considering time-inconsistent preferences and population uncertainty. Instead of adopting the typical real options approach which assumes that authorities possess a constant discount rate over time, we propose an extension of real options analysis by modeling authorities’ intertemporal choices with a quasi-hyperbolic discount function. Depending on the assumption about the strategies guiding the behaviors of future authorities, we consider three types of authorities, namely time-consistent authority, naive authority and sophisticated authority, of which the latter two are time-inconsistent. First, an optimal transit investment timing model is proposed. Then, solutions for the above three types of authorities are derived and compared. We demonstrate the performance of the proposed model by conducting numerical tests and applying it to Dalian, China. Main findings include: (1) an authority with time-inconsistent preferences makes decisions earlier than a standard, time-consistent authority; (2) the sophisticated authority invests earlier than the naive authority. Other implications of considering time-inconsistent preferences are also identified.

Purpose The purpose of the paper is to introduce a novel methodology to identify and quantify the difference of financial risks exhibited by listed and unlisted companies in their debt payments from an empirical point of view. Design/methodology/approach The paper attempts to establish the theoretical relationship between the agreed original periods and their corresponding periods of real payments. It is based on Krugman’s curve. This relationship has been implemented using data from listed and unlisted companies of Spain and from Western Europe countries (divided by companies, size and industry). Findings An alternative model has been implemented with the available information about listed and unlisted companies. There is not a significant difference in the financial risk level corresponding to listed and unlisted firms in Spain. Practical/implications The paper could provide a useful guidance in applying the risk in project assessment. Originality/value This paper provides a new methodology to reduce the subjectivity shown in the treatment of risk by traditional approaches. The method allows to including the financial risk in the time parameter of the discount function. Analysis of the delays in debt payments by both listed and unlisted companies; Alternative model able to describe the expected delays from the initial agreed period; Inclusion of the financial risk in the parameter “time” of a discount function.

We study the issue of sustainability using a model with a stock of man-made capital and a stock of exhaustible natural resource that provides a flow of amenity services as well as an input for the production of a consumption good. We ask under what conditions the utility flow will be a constant if infinitesimal households discount their utility using an endogenous utility discount rate that depends some macroeconomic variables. Our main result is that for the utility flow to be constant, the utility discount function must be the marginal product of capital function adjusted for the growth rate of aggregate consumption weighted by the elasticity of the consumer’s marginal rate of substitution between the final good and the amenity services. We demonstrate that Hartwick’s Rule holds but the Hotelling Rule must be modified. We also provide an explicit analytical example to confirm the general result.

People often exhibit prosocial tendencies towards close kin and friends, but generosity decreases as a function of increasing social distance between donor and recipient, a phenomenon called social discounting. Evidence suggests that acute stress affects prosocial behaviour in general and social discounting in particular. We tested the causal role of the important stress neuromodulators cortisol (CORT) and noradrenaline (NA) in this effect by considering two competing hypotheses. On the one hand, it is possible that CORT and NA act in concert to increase generosity towards socially close others by reducing the aversiveness of the cost component in costly altruism and enhancing the emotional salience of vicarious reward. Alternatively, it is equally plausible that CORT and NA exert dissociable, opposing effects on prosocial behaviour based on prior findings implicating CORT in social affiliation, and NA in aggressive and antagonistic tendencies. We pharmacologically manipulated CORT and NA levels in a sample of men (N = 150) and found that isolated hydrocortisone administration promoted prosocial tendencies towards close others, reflected in an altered social discount function, but this effect was offset by concurrent noradrenergic activation brought about by simultaneous yohimbine administration. These results provide inceptive evidence for causal, opposing roles of these two important stress neuromodulators on prosocial behaviour, and give rise to the possibility that, depending on the neuroendocrine response profile, stress neuromodulator action can foster both tend-and-befriend and fight-or-flight tendencies at the same time.

Background Despite evidence of nonproportional trade-offs in time trade-off exercises and the explicit incorporation of exponential discounting in health technology assessment calculations, quality-adjusted life-year (QALY) tariffs are currently still established under the assumption of linear time preferences. Objectives The aim of this study was to introduce a general method of accommodating for nonlinear time preferences in discrete choice experiment (DCE) duration studies and to evaluate its impact on estimated QALY tariffs. Methods A parsimonious utility function is proposed that accommodates any discounting function and preserves linear time preferences as a special case. Based on an efficient DCE design and 1775 respondents from a nationally representative scientific household panel, preferences and QALY tariffs for the Dutch SF-6D were estimated while accommodating for nonlinear time preferences via exponential and hyperbolic discounting functions. Results When the discount rate was estimated directly, we found strong evidence of nonlinear time preferences (with an exponential and hyperbolic discount rate of 5.7% and 16.5%, respectively). When the discount rate was estimated as a function of health state severity, we found that years lived in better health states are discounted minus years lived in impaired health states. Finally, the best statistical fit was obtained when using a hyperbolic discount function, which resulted in smaller QALY decrements and fewer health states classified as worse than immediate death. Conclusions Our results highlight the relevance and even necessity of a paradigm shift in health valuation studies in favor of time-preference corrected QALY tariffs, with potentially important implications for health technology assessment calculations and regulatory decisions.

Disruption risks and quantity discounts have not been explicitly studied in traditional winner determination problems. This paper investigates an innovative winner determination problem with accidental disruptions and quantity discounts for a fourth party logistics provider in a logistics service procurement auction. An effective hybrid strategy that integrates a fortification policy with an outside option policy is developed to mitigate possible disruptions. Integrating the hybrid strategy with a piecewise-linear discount function, a stochastic nonlinear winner determination model is constructed. Utilizing a linearization technique, such a nonlinear winner determination model can be reformulated as a mixed integer linear program. To obtain nearly optimal solutions, a scenario-based approximation method is developed, including a reduced scenario technique and a dual decomposition Lagrangian relaxation technique. Numerical experiments are conducted to illustrate the effectiveness and applicability of the proposed model and method. When the discount parameter decreases or the disruption probability increases, the fortification policy becomes more important for the logistics system. An interesting result indicates that both the auctioneer and bidders may benefit from considering the discount policy. Managerial implications are drawn for fourth party logistics providers to run a reliable logistics system in a cost-effective way.

The phenomenology of delay discounting (e.g. shape of the discount function; relation to mental health) has been reviewed in detail previously, but not its neural substrates. Its neuropsychology is crucial for both theory and clinical practice. So, here, we review the neural underpinnings of delay discounting. We introduce its objective summary measures; provide an atheoretical summary of current findings - linking brain regions to each objectively measurable variable; and then provide a preliminary five-stage summary model of cognitive processing; followed by a mapping of parameters to the flow of information through neural systems. The whole is designed to stimulate future research on the roles of each brain region in delay discounting. Delay and payoff produce activity in many brain areas: thalamus; sensory, parietal, temporal, cingulate, prefrontal, motor, and insular cortex; and basal ganglia. Delay discounting, then, appears to emerge from the interaction of neural systems as they process streams of events in recurrent loops and not to be a simple calculation carried out in a single center in the brain.

For reasons of social influence and social logistics, people in closed networks are expected to experience time compression: The more closed a person's network, the steeper the person's discount function, and the more narrow the expected time horizon within which the person deliberates events and behavior. Consistent with the hypothesis, data on managers at the top of three organizations show network closure associated with a social life compressed into daily contact with colleagues. Further, language in closed networks is predominantly about current activities, ignoring the future. Further still, discount functions employed by executive MBA students show more severe discounting by students in more closed networks. Inattention to the future can be argued to impair achievement, however, I find no evidence across the managers of daily contact diminishing the achievement associated with network advantage. I close with comments on replication and extrapolation to language more generally, within-person variation, and select cognitive patterns (closure bias, end of history, and felt status loss).

We study an optimal stopping problem under non-exponential discounting, where the state process is a multi-dimensional continuous strong Markov process. The discount function is taken to be log sub-additive, capturing decreasing impatience in behavioral economics. On strength of probabilistic potential theory, we establish the existence of an optimal equilibrium among a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfying a boundary condition. This generalizes the existence of optimal equilibria for one-dimensional stopping problems in prior literature.

We derive a statistical mechanical model of the discount function that extends the information theoretic discount function of Brody and Hughston (Proc R Soc Lond A 457:1343–1363, 2001; Quant Finan 2(1):70–80, 2002) with an adaptation of the quantal response statistical equilibrium model of Scharfenaker and Foley (Entropy 19(444):1–15, 2017) and the Weber-Fechner representation of the distinction between perceived time and objective time proposed by Takahashi (Med Hypotheses 65(4):691–693, 2005). We find that our model, among other things, provides an information-theoretic basis for the β-δ quasi-hyperbolic model often employed in behavioral economics. This, together with the ease of incorporating economic and/or psychological constraints, recommends this approach for general use in economics.

People’s health care cost prediction is nowadays a valuable tool to improve accountability in health care. In this work, we study if an interpretable method can reach the performance of black-box methods for the problem of predicting health care costs. We present an interpretable regression method based on the Dempster-Shafer theory, using the Evidence Regression model and a discount function based on the contribution of each dimension. Optimal parameters are learned using gradient descent. The k-nearest neighbors’ algorithm was also used to speed up computations. With the transparency of the evidence regression model, it is possible to create a set of rules based on a patient’s vicinity. When making a prediction, the model gives a set of rules for such a result. We used Japanese health records from Tsuyama Chuo Hospital to test our method, which includes medical checkups, exam results, and billing information from 2016 to 2017. We compared our model to an Artificial Neural Network and Gradient Boosting method. Our results showed that our transparent model outperforms the Artificial Neural Network and Gradient Boosting with an R 2 of 0 . 44 .

Experimental studies suggest that individuals exhibit more risk aversion in choices among prospects when the payment and resolution of uncertainty is immediate relative to when it is delayed. This leads to preference reversals that cannot be attributed to discounting. When data suggests that utility is time independent, probability weighting functions, such as those used to model prospect theory preferences, can accommodate such reversals. We propose a simple descriptive model with a two-parameter probability weighting function where one of these parameters depends on the time at which a prospect is resolved. The time-dependent parameter is responsible for the curvature of the probability weighting function and is regarded as an index of (in)sensitivity towards changes in probabilities. We provide conditions that characterize increased sensitivity towards more distant probabilities; this can account for the observed relatively less risk aversion towards delayed prospects. In our framework, the discount function is unrestricted, such that the model is compatible with empirical findings of non-constant discounting. In a simple application to bargaining we illustrate when it is advantageous for an individual to advance or delay the bargaining resolution time if an opponent displays increased sensitivity towards probability changes with delay.

Spatial discounting is a largely underexplored area of decision-making research, both theoretically and empirically, especially when compared to intertemporal choice, which has received significant attention in psychology and animal behaviour. Spatial decision problems seem to share some of the same features of a temporal decision problem (namely, the risk of reward objects disappearing and the opportunity cost of waiting), but there are several additional factors that affect the appropriate discount function for distant rewards. These include more significant opportunity costs, changes in the distances to all the other available opportunities, the post-reward costs of getting back home, the complex energetics associated with locomotion and all the additional risks faced by travelling itself. This paper organises and explores these factors and suggests some normative models that should predict the adaptive behaviour of animals and humans.

This paper completes Muñoz Torrecillas et al. (1) results and conclusions investigating the relationship between adherence to healthy dietary habits, specifically the Mediterranean Diet (hereinafter, MD), and impulsivity in intertemporal choices. Impulsivity can be defined as the strong preference for small immediate payoffs over larger delayed payoffs, and in the original study this behavior was captured by the parameter k (discount rate of the hyperbolic discount function), calculated using an automated scoring mechanism. Adherence to MD was measured by the KIDMED index and then grouped into three levels: high, medium, and low. While the authors observed that individuals in the high adherence group had the shallowest discounting and individuals in the low adherence group had the steepest discounting, the data were not statistically analyzed in depth. Therefore, the purpose of the present paper is to propose a preliminary quantitative model for this relationship and evaluate its significance. Tests revealed a significant interaction between adherence to MD and magnitude of delayed rewards when predicting discount rates. Specifically, the degree to which impulsivity decreases as adherence to MD increases is strongly influenced by delayed rewards of smaller magnitude. These findings are consistent with the authors' claims that healthy dietary habits may be closely linked with greater self-control when payoffs are small, and thus warrant further examination. The results do not indicate causality though, so future studies could also investigate the directions of this relationship as a means of developing behavioral interventions.

Financial models of PV power plants that are created during the development phase of a project are typically based upon simulations performed at hourly timesteps. Historical irradiance data used in these simulations, such as a Typical Meteorological Year, are created from average irradiances over hour-long periods. We find that when actual weather conditions include significant intra-hour variability in irradiance, the annual energy production estimates based on modeling in hourly timesteps will not account for all inverter clipping which occurs during moments of high irradiance. This effect is particularly significant for systems with high a DC to AC nameplate ratio. At some PV plants already in operation, this phenomenon has been found to result in overestimated production estimates of nearly 5%. In this work, the average amount of this error is quantified for locations within the United States, and a methodology is proposed for compensating for it in post processing. This methodology applies a variable discount to each hour of the year for hourly energy output simulations based on irradiance. The discount function is intended to account for the difference in useful insolation as computed from hourly average irradiance data versus sub-hourly sampling rates.

This paper analyzes performance measurement based on stochastic discount factors, compared to beta models traditionally used in computing funds' (Jensen) alphas. From a theoretical point of view, standard alphas suffer from several limitations. Our paper addresses this issue from an empirical point of view using a sample of Swiss mutual funds from 2000 to 2011. Our results suggest that the key for a "fair” comparison between stochastic discount function (SDF) and beta models is the specification of the set of primitive assets used to calibrate the SDF function. Once this is established, the size of (absolute) performance differences considerably decreases between the two model families. However, there are sizeable performance deviations in the cross-section of funds if conditioning information is incorporated in the tests, up to some 20 basis points per month, or about 2.3% per year. In almost all cases, the SDF-alphas are lower than the standard (Jensen) alphas. In absolute terms, the average SDF-based underperformance of the funds is way larger than the average total expense ratio (TER) of the funds, both in a conditional and unconditional setting.

This paper is focused on searching for the suitable discount rate to be applied to the valuation of a project related to forests in the USA, e.g., a recreational area inside a national park. To do this, we propose a new model based on hazard rate concepts, i.e., based on the risk that waiting time implies. More specifically, we derive the discount function whose instantaneous discount rate is the hazard rate of the system supporting the investment. We determine the rate of failure corresponding to different partition criteria of the whole system, in our case, we can use the information on forest fires caused in different ways, in different states or in different types of forest surfaces. After showing independence between the forest fires by states and causes, we derive a specific discount function for each cause which can be applied to every state or set of states which agree to fight against a concrete cause of forest fire. Additionally, we obtain a unique discount function by weighting the partial discount functions by type of forest surfaces. Our results are in line with the recommendations from several authors about using decreasing discount rates for projects with very long-term impacts.

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.

The experiment assesses the role of Cortisol concentration on bloodstream as correlate of the intertemporal choice and temporal discrimination in Type-2 Diabetes Mellitus (T2DM) patients and smokers. The participants were evaluated in a two independent computerized tasks allowed to obtain the temporal discount function and it's hyperbolic decay parameter (k), which refers to the tendency to discount the subjective value of future goods as a function of the delay to receiving them; and a temporal discrimination index (bisection point), this function relate the response proportion of "Long" stimuli with probe duration. The bisection point is the value at which responses to Short and Long stimuli occur with equal frequency. We analysed both parameters, then a comparisons of the temporal discount parameter [F (2,147) = 79.858, p

A stream of intertemporal payoffs X first-order (second-order) temporarily dominates another stream Y if (the area below) the cumulative payoff function of X up till any moment of time is always at least as high as that of Y; or, equivalently, any rank-dependent discounted utility maximizer with a strictly increasing (concave) utility function and a strictly decreasing (convex) discount function prefers X over Y. An individual A is relatively more patient than an individual B if, whenever B is willing to undertake some investment project, A is moreover so; or, equivalently, for any stream of payoffs, A’s present-day equivalent is at least as high as that of B; or, equivalently, A’s discount factor is at least as high as that of B and B’s utility function is “more concave” than that of A. Related concepts of absolute patience and comparative probabilistic patience are briefly discussed.

I characterize the entire class of consumption rules for finite-horizon models in which consumption is proportional to lifetime wealth. Any such rule can be obtained from a preference model with CRRA period utility. In a steady state with constant interest rates, a proportional consumption rule can be derived from a model with time-consistent preferences or from a model with possibly time-inconsistent preferences in which a household continually reoptimizes future utility discounted relative to the present instant. These two preference models will only coincide for the special case when the discount function is exponential. More generally, there will be two distinct yet observationally equivalent preference models. Hyperbolic-like discounting may arise because that is a simpler way for the brain to process a standard exponential discount function after accounting for mortality risk.

The aim of this paper is to present a methodology to generate a partially subadditive (respectively superadditive) discount function starting from an entirely subadditive (respectively superadditive) discount model. To do this, we are going to use the concept of deformation of time in a discount function, focusing on Stevens' power law. A deformation of time is a function that mathematically represents the perception from an individual or a group of individuals about how the calendar time evolves. This approach will be important when describing the treatment of addictions and other diseases in patients who show a certain degree of impulsivity in their intertemporal choice.

This paper studies the optimal consumption–investment–reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.

We establish a direct connection between time preference and risk about an attribute (health) of the instantaneous utility function. In doing so, we derive a risk-induced discount function that corresponds to a normalized expectation of that attribute. We provide several results characterizing this risk-induced discount function depending on the stochastic properties of the risk, which we model as a discrete Markov process. When it is well-defined, which we refer to as full approximation, the risk-induced discount function coincides with exponential discounting if the Markov process is stationary. However, a slight perturbation of the beliefs can trigger time-inconsistent discounting. When considering non-stationary Markov processes, time-inconsistency also emerges in situations where individuals’ beliefs change in a non-anticipated fashion over time, as exemplified by quasi-hyperbolic discounting. Results are illustrated via several applications.

In the dual risk model, we study the periodic dividend problem with a non-exponential discount function which results in a time-inconsistent control problem. Viewing it within the game theoretic framework, we extend the Hamilton-Jacobi-Bellman (HJB) system of equations from the fixed terminal to the time of ruin and derive the verification theorem, and we generalize the theory of classical optimal periodic dividend. Under two special non-exponential discount functions, we obtain the closed-form expressions of equilibrium strategy and the corresponding equilibrium value function in a compound Poisson dual model. Finally, some numerical examples are presented to illustrate the impact of some parameters.

Traditionally, the interval and delay effects have been identified and considered as the same anomaly in the context of intertemporal choice, when individuals or groups of individuals make their decisions about reward preferences. This has supposed that most studies on this topic have been focused on the delay effect and, consequently, that the discount functions provided by the existing literature have considered only this effect. This is the case of hyperbolic discounting, which has been used to describe the delay, but not the interval effect. Therefore, the main objective of this paper is to carry out a detailed analysis of both anomalies, which will allow us to mathematically relate them, thus finding their analogies and differences. To do this, we will first analyze the concept of delay effect and later the different definitions of the interval effect. The main conclusion of this paper is twofold. On the one hand, if the benchmark for valuation is fixed, the delay effect coincides with the so-called decreasing interval effect. On the other hand, if the assessment reference point is the beginning of each interval, both anomalies are different. These findings make necessary to redefine the concept of interval effect. Finally, we will analyze the relationship between the interval effect, the delay effect and the subadditivity

The effectiveness of medical therapies depends crucially on patients’ adherence. To gain deeper insight into the behavioral mechanisms underlying adherence, we present a microeconomic model of the decision-making process of an individual who is initially in an asymptomatic clinical state and to whom a prophylactic therapy is offered with the aim of preventing damage to health in the future. The focus of modeling is the optimization of an intertemporal utility function, where time-inconsistent preferences are incorporated by a quasi-hyperbolic discount function. The predictions of the model concur with experience in clinical practice. Moreover, the introduction of time-inconsistency reveals a self-control problem of the individuals where resolutions made before may be given up at a later time. A more pronounced present bias leads to a decrease in adherence and, consequently, the gain in societal welfare resulting from the prophylactic therapy declines. Developing effective strategies to improve adherence is a major challenge in health care. As an example, the impact of financial incentives offered to the patients on adherence and welfare are investigated on the basis of the model. The results are consistent with empirical findings. The approach presented contributes to a better understanding of the complex interaction of the relevant determinants for adherence, particularly regarding the individuals’ self-control problem.