Expected Utility Maximization Research Articles

Expected Utility Maximization with Stochastic Dominance Constraints in Complete Markets

The problem of expected utility maximization with stochastic dominance constraints in complete markets is studied. In the quantile formulation, i.e., changing the decision variable of the problem from a random variable to its quantile function, the problem with a first-order stochastic dominance constraint is easy and can be solved explicitly. For the problem with a second-order stochastic dominance constraint, the optimal solution is characterized explicitly in terms of the solution of a dual problem, which interestingly can be regarded as a problem of two-person efficient risk sharing. Moreover, the optimal solution of the problem of efficient risk sharing is characterized fully and, in some special cases, has closed form.

A Dynamic Semiparametric Characteristics-based Model for Optimal Portfolio Selection

This paper develops a two-step semiparametric methodology for portfolio weight selection for characteristics- based factor-tilt and factor-timing investment strategies. We build upon the expected utility maximization framework of Brandt (1999) and Ait-sahalia and Brandt (2001). We assume that asset returns obey a characteristics- based factor model with time-varying factor risk premia as in Li and Linton (2020). We prove under our return- generating assumptions that in a market with a large number of assets, an approximately optimal portfolio can be established using a two-step procedure. The first step finds optimal factor-mimicking sub-portfolios using a quadratic objective function over linear combinations of characteristics-based factor loadings. The second step dynamically combines these factor-mimicking sub-portfolios based on a time-varying signal, using the investor’s expected utility as the objective function. We develop and implement a two-stage semiparametric estimator. We apply it to CRSP (Center for Research in Security Prices) and FRED (Federal Reserve Economic Data) data and find excellent in-sample and out-sample performance consistent with investors’ risk aversion levels.

Ranking of optimal stock portfolios determined on the basis of expected utility maximization criterion

Aim/purpose – The aim of the paper is to rank the optimal portfolios of shares of com- panies listed on the Warsaw Stock Exchange, taking into account the investor’s propen- sity to risk. Design/methodology/approach – Investment portfolios consisting of varied number of companies selected from WIG 20 index were built. Next, the weights of equity holdings of these companies in the entire portfolio were determined, maximizing portfolio’s expected (square) utility function, and then the obtained structures were compared between investors with various levels of risk propensity. Using Hellwig’s taxonomic development measure, a ranking of optimum stock portfolios depending on the inves- tor’s risk propensity was prepared. The research analyzed quotations from 248 trading sessions. Findings – The findings indicated that whilst there are differences in the weight struc- tures of equity holdings in the entire portfolio between the investor characterized by aversion to risk at the level of γ = 10 and the investor characterized by aversion to risk at the level of γ = 100, the rankings of the constructed optimum portfolios demonstrate strong similarity. The study validated, in conformity with the literature, that with the increase in the number of equity holdings in the portfolio, the portfolio risk initially decreases and then becomes stable at a certain level. Research implications/limitations – The study used data from the past as for which there is no guarantee that they will be adequate for the future. There is sensitivity to the selection of the period from which the historic data come. When changing the period of the analyzed historic data by a small time unit it may prove that the portfolio composi- tion will become totally different. Originality/value/contribution – The paper compares the composition of optimum stock portfolios depending on the investor’s propensity to risk. Their ranking was cre- ated using the taxonomic method for this purpose. Taking advantage of this method also additional variables can be taken into account, which describe and differentiate the port- folio and they can be assigned relevant significance depending on the investor’s prefer- ences. Keywords: optimal portfolio, expected rate of return on the portfolio, portfolio standard deviation, expected utility theory, multidimensional comparative analysis. JEL Classification: G10, G11.

Dynamic Preference “Reversals” and Time Inconsistency

We study identification of time inconsistency when an agent at time 0 makes an advance commitment, and later at time 1 can revise their choice after possibly receiving additional information. Roughly speaking, we prove that the only data that reject time-consistent expected utility maximization is a time-0 choice that is always strictly dominated at time 1. This holds for rich choice sets; if the complete ranking of alternatives is observed in every period and state; when it is natural to assume additional properties like concavity; and with supplementary cardinal information. However, time inconsistency can be point identified from willingness to pay for different alternatives in both periods, if utility from money is plausibly additively-separable and independent of time-1 information.

Pascal's Mugger Strikes Again

AbstractIn a well-known paper, Nick Bostrom presents a confrontation between a fictionalised Blaise Pascal and a mysterious mugger. The mugger persuades Pascal to hand over his wallet by exploiting Pascal's commitment to expected utility maximisation. He does so by offering Pascal an astronomically high reward such that, despite Pascal's low credence in the mugger's truthfulness, the expected utility of accepting the mugging is higher than rejecting it. In this article, I present another sort of high value, low credence mugging. This time, the mugger utilises research on existential risk and the long-term potential of humanity to exploit Pascal's expected-utility-maximising descendant. This mugging is more insidious than Bostrom's original as it relies on plausible facts about the long-term future, as well as realistic credences about how our everyday actions could, albeit with infinitesimally low likelihood, affect the future of humanity.

Robust optimal investment and reinsurance for an insurer with inside information

This paper studies a robust optimal investment–reinsurance problem for an insurer who possesses inside information on the financial market and the insurance business under model uncertainty. The insurer’s surplus process and the risky asset process in the financial market are assumed to be correlated jump diffusion processes with random coefficients. The inside information is modelled by a general random variable related to the future realizations of the surplus process and the risky asset process. Under the criterion of expected utility maximization, the optimal investment–reinsurance problem with inside information is transformed into an adapted stochastic control problem. By incorporating Knightian uncertainty into the model, we establish an adapted stochastic differential game problem with a nonstandard performance functional, which aims to select the robust optimal investment–reinsurance strategy for the insurer with inside information. The stochastic maximum principles are adopted to derive a characterization of the optimal strategy. Some interesting special cases are discussed and the corresponding optimal strategies are derived.

Insurer Optimal Asset Allocation in a Small and Closed Economy: The Case of Iran’s Social Security Organization

We seek to determine the optimal amount of the insurer’s investment in all types of assets for a small and closed economy. The goal is to detect the implications and contributions the risk seeker and risk aversion insurer commonly make and the effectiveness in the investment decision. Also, finding the optimum portfolio for each is the main goal of the present study. To this end, we adopted the optimal asset-liability management (ALM) method to control the firm's risk of financial stability and growth by balancing the assets and liabilities of the firm. In the process, stochastic interest rates and inflation risks were taken into account according to the expected utility maximization framework. All assets were established and calculated by the Kalman Filter with the stochastic interest rate following the Hull-White model; an additional stochastic process models the inflation risk. To consider the stochastic process, we employed the geometric Brownian motion in the liability process to ensure a definite liability value. We chose Iran’s Social Security Organization as our sample insurer company since it has a portfolio of five types of assets and four types of liabilities, and operates in a small and closed economy. By Applying the ALM method with the stochastic control theory approach, we acquire the optimal investment strategies for insurers to minimize their risk. Our findings demonstrate the effects of model parameters, such as the degree of risk-taking on the insurer decision.

Fractional Degree Stochastic Dominance

We develop a continuum of stochastic dominance rules for expected utility maximizers. The new rules encompass the traditional integer-degree stochastic dominance; between adjacent integer degrees, they formulate the consensus of individuals whose absolute risk aversion at the corresponding integer degree has a negative lower bound. By extending the concept of “uniform risk aversion” previously proposed in the literature to high-order risk preferences, we interpret the fractionalized degree parameter as a benchmark individual relative to whom all considered individuals are uniformly no less risk averse in the lottery choices. The equivalent distribution conditions for the new rules are provided, and the fractional degree “increase in risk” is defined. We generalize the previously defined notion of “risk apportionment” and demonstrate its usefulness in characterizing comparative statics of risk changes in fractional degrees. This paper was accepted by David Simchi-Levi, decision analysis.

Joint pricing and stocking decisions for a newsvendor problem with loss aversion and reference point effect

This study investigates the joint pricing and stocking decisions for a loss‐averse retailer with reference point effect under stochastic demand. By solving an expected utility maximization model developed based on the prospect theory, the optimal pricing and stocking decisions are derived and compared with those of the classical newsvendor. The results show that the optimal decisions significantly depend on the loss aversion coefficient, optimism level, and reference effect strength. Smaller loss aversion coefficient and lower optimism level lead to higher expected utilities. In particular, a loss‐averse retailer with a smaller loss aversion coefficient benefits from a higher reference effect strength and vice versa.

Stochastic expected utility for binary choice: a ‘modular’ axiomatic foundation

We present new axiomatisations for various models of binary stochastic choice that may be characterised as “expected utility maximisation with noise”. These include axiomatisations of simple scalability (Tversky and Russo in J Math Psychol 6:1–12, 1969) with respect to a scale having the expected utility (EU) form, and strong utility (Debreu in Econometrica 26(3):440–444, 1958) of the EU form. The latter model features Fechnerian “noise”: choice probabilities depend on EU differences. Our axiomatisations complement the important contributions of Blavatskyy (J Math Econ 44:1049–1056, 2008) and Dagsvik (Math Soc Sci 55:341–370, 2008). Our representation theorems set all models on a common axiomatic foundation, with additional axioms added in modular fashion to characterise successively more restrictive models. The key is a decomposition of Blavatskyy’s (2008) common consequence independence axiom into two parts: one (which we call weak independence) that underwrites the EU form of utility and another (stochastic symmetry) than underwrites the Fechnerian structure of noise. We also show that in many cases of interest (which we call preference-bounded domains) stochastic symmetry can be replaced with weak transparent dominance (WTD). For choice between lotteries, WTD only restricts behaviour when choosing between probability mixtures of a “best” and a “worst” possible outcome.

Minimax regret and failure to converge to efficiency in large markets

This paper studies minimax regret traders in a sealed bid double auction. Unlike the expected utility maximizers that populate typical market models, these traders do not determine their actions using a single prior. The analysis proves that, with no restrictions on beliefs about others' types and strategies, minimax regret traders will not converge to price-taking as the size of the market increases, contrary to standard economic intuition. In fact, minimax regret traders' bids and asks are invariant to the number of traders in the market. Thus, the robustness of the sealed bid double auction is sensitive to traders' preferences.

Incentive compatibility under ambiguity

The paper examines notions of incentive compatibility in an environment with ambiguity-averse agents. In particular, we propose the notion of maxmin transfer coalitional incentive compatibility, which is immune to coalitional manipulations and thus more stable than the individual incentive compatibility condition. The main result characterizes the set of allocations that satisfy the maxmin transfer coalitional incentive compatibility condition. We show that an allocation satisfies maxmin transfer coalitional incentive compatibility if and only if it is maxmin interim efficient. This result extends that of De Castro and Yannelis (J Econ Theory 177:678–707, 2018) in the sense that ambiguity not only resolves the conflict between efficiency and incentive compatibility, it also accommodates stability. Furthermore, this result is false in a finite economy where agents are subjective expected utility maximizers.

Cognitive Imprecision and Small-Stakes Risk Aversion

Abstract Observed choices between risky lotteries are difficult to reconcile with expected utility maximization, both because subjects appear to be too risk averse with regard to small gambles for this to be explained by diminishing marginal utility of wealth, as stressed by Rabin (2000), and because subjects’ responses involve a random element. We propose a unified explanation for both anomalies, similar to the explanation given for related phenomena in the case of perceptual judgments: they result from judgments based on imprecise (and noisy) mental representations of the decision situation. In this model, risk aversion results from a sort of perceptual bias—but one that represents an optimal decision rule, given the limitations of the mental representation of the situation. We propose a quantitative model of the noisy mental representation of simple lotteries, based on other evidence regarding numerical cognition, and test its ability to explain the choice frequencies that we observe in a laboratory experiment.

G-expected utility maximization with ambiguous equicorrelation

This paper studies a class of expected utility maximization problems with respect to a controlled state process with multiple noises, whose pairwise correlations are equal and ambiguous. Using the G-expectation theory, we solve for the robust stochastic controls explicitly from a Hamilton–Jacobi–Bellman–Isaacs equation and deduce a robust choice of the equicorrelation coefficient. We also generalize the results to a block equicorrelation structure, where we consider more than two ambiguous parameters that could be interactive in general. We manage to derive an analytical solution to the robust stochastic controls under an ambiguous two-block equicorrelated structure via the solution to a system of polynomial equations. The results have significant implications for the investment and reinsurance problems among many others.

Indifference pricing of pure endowments via BSDEs under partial information

In this paper, we investigate the pricing problem of a pure endowment contract when the insurance company has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the indifference price of the insurance derivative in terms of the solutions of two backward stochastic differential equations. Finally, we discuss two special cases where we get a more explicit representation of the indifference price process.

Do People Maximize Quantiles?

Payoff quantiles have been used for decision making in banking and investment (in the form of Value-at-Risk) and in the mining, oil and gas industries (in the form of probabilities of exceeding a certain level of production). However, it is unknown how common quantile-based decision making actually is among typical individual decision makers. This paper describes an experiment that aims to (1) compare how common quantile decision making is relative to expected utility maximization, and (2) estimate risk attitude parameters under the assumption of quantile preferences. The experiment has two parts. In the first part, individuals make pairwise choices between risky lotteries, and the competing models are fitted to the choice data. In the second part, we directly elicit a decision rule from a menu of alternatives. The results show that a quantile preference model outperforms expected utility for a considerable minority, 30%--50%, of participants, depending on the metric. The majority of individuals are risk averse, and women are more risk averse than men, under both models.

G-Expected Utility Maximization with Ambiguous Equicorrelation

This paper studies the expected utility maximization problem with respect to a controlled state process with multiple noises, whose pairwise correlations are equal and ambiguous. Using the G-expectation theory, we solve for the robust stochastic controls explicitly from a Hamilton-Jacobi-Bellman-Isaacs equation and deduce a robust choice of the equicorrelation coefficient. We also generalize the results to a block equicorrelation structure, where we strike the balance between efficiency and tractability. Although we face with many ambiguous parameters that may be interactive in the general case, we manage to derive an analytical solution to the robust stochastic controls under the ambiguity of a two-block equicorrelated structure via the solution to a system of polynomial equations. The results have significant implications to the investment and reinsurance problems among many others.

The Survival of the Unfittest

We explore the use of parimutuel markets in a peer- to-peer setting to generate a wide diversity of content offerings while responding adaptively to customer demand. Files are served and paid for through a parimutuel market similar to that used for betting in horse races and in lotteries. Our simulations are based on rational agents, which all act according to a set of simple rules. The results show that a favorite-long-shot bias occurs, where agents tend to bet on long-shots rather than favorites when following simple expected utility maximization. Furthermore, we have confirmed that the long-tail does sustain even when the agents only have a limited view of all files to pick from. If the limited view consists of random subsets of all files, the long tail is enhanced. If the limited view consists of the top most popular items, the long tail slightly decreases. We have also explored the effect of bounded rationality. Our results show that the system is robust in presence of a large fraction of providers that have bounded rationality. If the providers with bounded rationality pick random items, the long tail is enhanced. Conversely, if the providers with bounded rationality only pick their favorite, the long tail slightly decreases.

Optimal and equilibrium execution strategies with generalized price impact

This paper examines the execution problems of large traders with a generalized price impact. Constructing two related models in a discrete-time setting, we solve these problems by applying the backward induction method of dynamic programming. In the first problem, we formulate the expected utility maximization problem of a single large trader as a Markov decision process and derive an optimal execution strategy. Then, in the second model, we formulate the expected utility maximization problem of two large traders as a Markov game and derive an equilibrium execution strategy at a Markov perfect equilibrium. Both of these two models enable us to investigate how the execution strategies and trade performances of a large trader are affected by the existence of other traders. Moreover, we find that these optimal and equilibrium execution strategies become deterministic when the total execution volumes of non-large traders are deterministic. We also show, by some numerical examples, the comparative statics results with respect to several problem parameters.

Expected utility maximization for an insurer with investment and risk control under inside information

This paper studies optimal investment and risk control strategies for an insurer who owns insider information. The insurance risk process is governed by a general jump diffusion process with random parameters and is correlated with the risky asset process in the financial market. We model the inside information by a general random variable related to the insurance risk process and the risky asset process. Under the criterion of expected utility maximization of the terminal wealth, we adopt white noise calculus and BSDE approach to analyze the problem for various utility functions.