S-shaped Utility Function Research Articles

A Periodic Review Inventory System with S-Shaped Utility Function

Prospect theory shows that human behavior in decision making can often be characterized by an S-shaped utility function. In this paper, we consider a periodic review inventory system with an exponential-type S-shaped utility function. The objective is to maximize the total expected utility over the planning horizon. We show that, for the single-period case without setup cost, a state-dependent order-up-to policy is optimal, to which it is not difficult to obtain the optimal order-up-to level. For the multi-period case without setup costs, we partially characterize the optimal policy when the demand distributions are log-concave, and present an iterative algorithm for computing the optimal policy. Numerical studies are conducted to illustrate the impact of parameters on the system performance and optimal policies. The results show that the ordering behavior of the inventory manager can be significantly different from those derived from the classical models. It is also shown that the approaches in this paper can be applied to both single-period and multi-period systems with setup costs for obtaining optimal policies.

S-Shaped Utility Explains the Subprime Crash

Author propose an S-shaped utility function of consumption which, combined with a heterogeneous agents and external habit setting, it’s well the first order moments of the American financial and macroeconomic time series relevant for the equity premium puzzle in the second half of XX century. The average relative risk aversion of the agents remains in the 0-3 range. The shape of the utility and its relative risk aversion function of consumption suggest an explanation for the 2008 suprime crash.

Measuring the Behavioral Component of Financial Fluctuations: An Analysis Based on the S&P 500

We study the evolution of the behavioral component of the financial market by estimating a Bayesian mixture model in which two types of investors coexist: one rational, with standard subjective expected utility theory (SEUT) preferences, and one behavioral, endowed with an S-shaped utility function. We perform our analysis by using monthly data on the constituents of the S&P 500 index from January 1962 to April 2012. We assume that agents take investment decisions by ranking the alternative assets according to their performance measures. A tuning parameter blending the rational and the behavioral choices can be estimated by using a criterion function. The estimated parameter can be interpreted as an endogenous market sentiment index. This is confirmed by a number of checks controlling for the correlation of our endogenous index with measures of (implied) financial volatility, market sentiments and financial stress. Our results confirm the existence of a significant behavioral component that reaches its peaks during periods of recession. Moreover, after controlling for a number of covariates, we observe a significant correlation between the estimated behavioral component and the S&P 500 return index.

House selection via the internet by considering homebuyers’ risk attitudes with S-shaped utility functions

The widespread use of the Internet has significantly changed the behavior of homebuyers. Using online real estate agents, homebuyers can rapidly find some modern houses that meet their needs; however, most current online housing systems provide limit features. In particular, existing systems fail to consider homebuyers’ housing goals and risk attitudes. To increase effectiveness, online real estate agents should provide an efficient matching mechanism, personalized service and house ranking with the aim of increasing both buyers’ satisfaction and deal rate. An efficient online real estate agent should provide an easy way for homebuyers to find (rank) a suitable house (alternatives) with consideration of their different housing philosophies and risk attitudes. In order to comprehend these ambiguous housing goals and risk attitudes, it is also indispensable to determine a satisfaction level for each fuzzy goal and constraint.In this study, we propose fuzzy goal programming with an S-shaped utility function as a decision aid to help homebuyers in choosing their preferred house via the Internet in an easy way. With the use of a decision aid, homebuyers can specify their housing goals and constraints with different priority levels and thresholds as a matching mechanism for a fuzzy search, while the matching mechanism can be translated into a standard query language for a regular relational database. Moreover, a laboratory experiment is conducted on a real case to demonstrate the effectiveness of the proposed approach. The results indicate that the proposed method provides better customer satisfaction than manual systems in housing selection service.

Discrete-Time Behavioral Portfolio Selection Under Prospect Theory

We formulate and study a general multi-period behavioral portfolio selection model under Kahneman and Tversky's prospect theory, featuring an incomplete market and an S-shaped utility function. We first discuss the ill-posedness issue under a multi-period framework and identify the conditions for the well-posedness under which infinitely leveraging an asset is not optimal for the investor. Moreover, we show that the well-posedness of the multi-period portfolio selection problem can be characterized in terms of an induced loss-aversion measure, which is an increasing function of time. Under the conditions for well-posedness, we solve the multi-period behavioral portfolio selection problem completely by deriving its semi-analytical optimal policy. In particular, we identify two cases: the case with one risky asset and the case with multiple risky assets that are jointly elliptically distributed, under which the optimal behavioral portfolio policy takes a piecewise linear feedback form. For the multiple risky assets case, we further demonstrate that the two-fund separation is still valid under the S-shaped utility. We also discuss the implications of our findings to the well documented phenomena of non-participation effect and horizon effect.

Aggregate investor preferences and beliefs: A comment

A recent study in this journal presents encouraging results of a daunting simulation analysis of the statistical properties of a centered bootstrap approach to stochastic dominance efficiency analysis. However, by relying on the first-order optimality condition in a situation where multiple optima may occur, the empirical analysis draws the questionable conclusion that some of the toughest data sets in empirical asset pricing can be rationalized by the representative investor maximizing an S-shaped utility function, consistent with the so-called Prospect Stochastic Dominance criterion. Further research could be directed to developing global optimization algorithms and consistent re-sampling methods for statistical inference for general risky choice problems. • Standard portfolio and asset pricing theory assume risk aversion. • Risk seeking introduces the need for global optimization method. • First-order conditions are not sufficient for the global maximum. • A proper global optimality test finds market portfolio inefficiency. • The aggregated investor does not seem to obey the 'PSD' rule.

Book-to-Market Ratio and Skewness of Stock Returns

ABSTRACT:This study demonstrates that stocks with low book-to-market ratios, also known as glamour stocks, have significantly more positive skewness in their return distributions compared to the return distributions of value stocks with high book-to-market ratios. The premium (discount) investors apply to these glamour (value) stocks also correlates significantly with the difference in return skewness. These findings suggest that the value/glamour-stock puzzle is partially explained by investor preference for positive skewness in stock returns. Such preference for skewness, which is consistent with investors having inverse S-shaped utility functions, is observed in such consumer behaviors as lottery purchases and gambling. This paper further documents significant predictive power of accounting-based measures, such as the book rate of return, with respect to the skewness of stock returns.Data Availability: Data are available from sources identified in the paper.

Optimal portfolio choice for a behavioural investor in continuous-time markets

The aim of this work consists in the study of the optimal investment strategy for a behavioural investor, whose preference towards risk is described by both a probability distortion and an S-shaped utility function. Within a continuous-time financial market framework and assuming that asset prices are modelled by semimartingales, we derive sufficient and necessary conditions for the well-posedness of the optimisation problem in the case of piecewise-power probability distortion and utility functions. Finally, under straightforwardly verifiable conditions, we further demonstrate the existence of an optimal strategy.

Asset-pricing implications of biologically based non-expected utility

Results in population ecology suggest that evolutionary successful species should have an adaptive (reference-based) S-shaped utility function that is intrinsically more sensitive to aggregate than uninsured idiosyncratic shocks—the former cannot be diversified demographically. To test the asset-pricing relevance of these ideas, I embed the non-expected utility specification implied by evolutionary theory into an economy with partial risk sharing due to limited commitment. For the benchmark specification (CRRA=6 over gains), Monte Carlo simulations of a Markov growth economy produce the following results: (i) matching the degree of consumption-smoothing in the cross section, the Sharpe ratio for a Lucas tree is 0.33, an increase of 44 percent relative to expected utility; (ii) the risk-free rate is low, stable and counter-cyclical, hence equity returns, unlike in the expected utility case, have the correct pattern of predictability; (iii) in the cross section, excess returns across equity classes exhibit both a value premium and a size discount with risk adjusted returns that are at least two times higher than their expected utility counterparts.

Global Shapes of Preference Scaling Functions

In the expected utility framework, concavity (or convexity) of the preference scaling function corresponds to risk-aversion (or risk-seeking) preferences. Friedman and Savage ( 1948 ) and Markowitz ( 1952 ) indicate that individuals may be risk-averse at some wealth levels but risk-seeking at others. Prospect theory ( Kahneman & Tversky, 1979 ) proposes an S-shaped utility function exhibiting risk-aversion in the domain of gain and risk-seeking in the loss domain. In this study, we use lottery experiments to infer global shapes of the preference scaling functions that determine risk attitudes in interaction with uncertainty. The global shapes that emerge are—fully concave, fully convex, linear, Reverse-S and S-shaped preference scaling functions. Using annual income as a proxy for wealth, we examine the relationship between global shapes and wealth. The data largely supports the Friedman–Savage hypothesis; we find that at low levels of wealth people are concave, followed by Reverse-S and then a second upper concave segment at higher levels of wealth. JEL: C91, D81

STOCHASTIC DOMINANCE AND BEHAVIOR TOWARDS RISK: THE MARKET FOR ISHARES

Prospect theory suggests that risk seeking can occur when investors face losses and thus an S-shaped utility function can be useful in explaining investor behavior. Using stochastic dominance procedures, Post and Levy (2015) find evidence of reverse S-shaped utility functions. This is consistent with investors exhibiting risk-seeking tendencies in bull markets and risk aversion in bear markets. We use both ascending and descending stochastic dominance procedures to test for risk-averse and risk-seeking behavior. By partitioning iShares' return distributions into negative and positive return regions, we find evidence of all four utility functions: concave, convex, S-shaped and reverse S-shaped.

Test statistics for prospect and Markowitz stochastic dominances with applications

Summary Levy and Levy (2002, 2004) extend the stochastic dominance (SD) theory for risk averters and risk seekers by developing the prospect SD (PSD) and Markowitz SD (MSD) theory for investors with S-shaped and reverse S-shaped (RS-shaped) utility functions, respectively. Davidson and Duclos (2000) develop SD tests for risk averters whereas Sriboonchitra et al. (2009) modify their statistics to obtain SD tests for risk seekers. In this paper, we extend their work by developing new statistics for both PSD and MSD of the first three orders. These statistics provide a tool to examine the preferences of investors with S-shaped utility functions proposed by Kahneman and Tversky (1979) in their prospect theory and investors with RS-shaped investors proposed by Markowitz (1952a). We also derive the limiting distributions of the test statistics to be stochastic processes. In addition, we propose a bootstrap method to decide the critical points of the tests and prove the consistency of the bootstrap tests. To illustrate the applicability of our proposed statistics, we apply them to study the preferences of investors with the corresponding S-shaped and RS-shaped utility functions vis-a-vis returns on iShares and vis-a-vis returns of traditional stocks and Internet stocks before and after the Internet bubble.

Book-to-Market Ratio and Skewness of Stock Returns

This study demonstrates that stocks with low book-to-market ratios, also known as glamour stocks, have significantly more positive skewness in their return distributions compared to the return distributions of value stocks with high book-tomarket ratios. The premium (discount) investors apply to these glamour (value) stocks also correlates significantly with the difference in return skewness. These findings suggest that the value/glamour-stock puzzle is partially explained by investor preference for positive skewness in stock returns. Such preference for skewness, which is consistent with investors having inverse S-shaped utility functions, is observed in such consumer behaviors as lottery purchases and gambling. This paper further documents significant predictive power of accounting-based measures, such as the book rate of return, with respect to the skewness of stock returns.

Constant Proportion Portfolio Insurance: Discrete-time Trading and Gap Risk Coverage

A practical implementation of constant proportion portfolio insurance (CPPI) strategies must inevitably take market frictions into account. I study a CPPI in a setting with trading costs, fees and borrowing restrictions, and relax the assumption of continuous portfolio rebalancing. The main goals are to cover issuer's gap risk and to maximize CPPI performance according to investor's preferences over possible multipliers: the proportionality factor that determines the risky exposure of a CPPI. Investment objectives are described by the Sortino ratio and alternatively by an S-shaped utility function known from behavioral finance. Investors with either objective will choose a lower multiplier than if CPPI performance is measured by the expected return. Discrete-time trading requires a portfolio rebalancing rule, which affects both performance and gap risk. Two commonly applied strategies, rebalancing at equidistant time steps and rebalancing based on fixed market moves, are compared to a new rule, which takes trading costs into account. While the new and the market-based rules deliver similar CPPI performance, the new rebalancing rule achieves this by fewer trading interventions. Issuer's gap risk can be covered by a fee charge, by hedging or by an artificial floor. A new approach to determine the artificial floor is introduced. All three methods reduce losses from gap events effectively at only a small cost to the investor.

Prospect Theory, Indifference Curves, and Hedging Risks

The prospect theory is one of the most popular decision-making theories. It is based on S-shaped utility functions, unlike the von Neumann and Morgenstern (NM) theory, which is based on concave utility functions. The S-shaped functions bring challenges, and extensions and generalizations of the NM theory into the prospect theory are not always possible. For example, in the prospect theory, the monotonicity of indifference curves depends on the underlying mean, unlike in the NM theory. Risk-hedging decisions also become more complex within the prospect theory. In this paper, we discuss these topics and establish general results concerning certain covariances from which we can in turn infer properties of indifference curves and hedging decisions within the prospect theory.

Prospect Theory and Hedging Risks

The prospect theory is one of the most popular decision-making theories. It is based on the S-shaped utility function, unlike the von Neumann and Morgenstern (NM) theory, which is based on the concave utility function. The S-shape brings in mathematical challenges: simple extensions and generalizations of NM theory into the prospect theory cannot be frequently achieved. For example, the nature of monotonicity of the indifference curve depends on the underlying mean. Price hedging decisions also become more complex within the prospect theory. We discuss these topics in detail and offer a general result concerning the sign of a covariance from which we then infer desired properties of the indifference curve and also justify hedging decisions within the prospect theory. We illustrate our general considerations with a thoroughly worked out example.

Prospect Theory, Indifference Curves, and Hedging Risks

The prospect theory is one of the most popular decision-making theories. It is based on the S-shaped utility function, unlike the von Neumann and Morgenstern (NM) theory, which is based on the concave utility function. The S-shape brings in mathematical challenges: simple extensions and generalizations of NM theory into the prospect theory cannot be frequently achieved. For example, the nature of monotonicity of the indifference curve depends on the underlying mean. Price hedging decisions also become more complex within the prospect theory. We discuss these topics in detail and offer a general result concerning the sign of a covariance from which we then infer desired properties of the indifference curve and also justify hedging decisions within the prospect theory. We illustrate our general considerations with a thoroughly worked out example.

A single-period inventory system with a general S-shaped utility and exponential demand

We discuss a single-period inventory system with a general S-shaped utility function and exponential demand. It is shown that there exists a state-dependent order-up-to policy to be optimal. Behaviors of optimal policy are illustrated by a numerical example.

An Approximation Approach for Representing S-Shaped Membership Functions

In general, to formulate a fuzzy-linear-programming problem with n S-shaped utility (membership) functions, traditional methods require n or more extra binary variables because S-shaped curves are neither convex nor concave in all places. Adding binary variables does not improve the bound of the linear-programming relaxation. On the contrary, added binary variables increase the computational burden in the solution process if problems get large. Therefore, a formulation without binary variables should be more efficient. Accordingly, this study proposes a piecewise-linear approach to formulate an S-shaped membership function (MF) without adding any extra binary variables, which improves the efficiency of fuzzy-linear programming in solving decision/management problems with S-shaped MFs. Finally, a computational experiment is provided to demonstrate the superiority of the proposed models. An illustrative example is also provided to show the usefulness of the proposed method.

Stochastic Dominance and Behavior towards Risk: The Market for iShares

Prospect theory suggests that risk seeking can occur when investors face losses and thus an S-shaped utility function can be useful in explaining investor behavior. Using stochastic dominance procedures, Post and Levy (2005) find evidence of reverse S-shaped utility functions. This is consistent with investors exhibiting risk-seeking tendencies in bull markets and risk aversion in bear markets. We use both ascending and descending stochastic dominance procedures to test for risk averse and risk seeking behavior. By partitioning iShares’ return distributions into negative and positive return regions, we find evidence of all four utility functions: concave, convex, S-shaped and reverse S-shaped.