Constant Absolute Risk Aversion Research Articles

Risky choice and diminishing sensitivity in MaaS context: A nonlinear logit analysis of traveler behavior

In the rapidly transforming landscape of urban transportation, Mobility-as-a-Service (MaaS) has emerged as a potentially transformative solution to adapt to fluctuating travel demand. However, despite arguments in the literature suggesting that uncertainty in travel plans may influence travelers’ preferences for MaaS, the extent and mechanisms of this impact remain relatively underexplored. This study aims to bridge this gap by evaluating travelers’ preferences toward MaaS subscriptions, mode choices, and their reactions to uncertainty within a risky-choice modeling framework. We incorporate a linear-specified benchmark model and two nonlinear models (one based on Expected Utility Theory with Constant Absolute Risk Aversion adjustments, and the other on Prospect Theory), to estimate travelers’ risk perceptions and their valuation of gains and losses associated with travel time and costs caused by occasional activities during commute trips. The data utilized in our study originates from a specifically designed stated preference survey carried out in Hong Kong. Our findings present insightful observations into MaaS bundle configurations, socio-demographic patterns, and notably, behaviors under uncertain circumstances. In the face of uncertainty, we observe a noticeable increase in reluctance to adopt MaaS, and a heightened preference for taxis. The models also underscore significant risk aversion and diminishing sensitivity concerning travel time and cost, with these tendencies proving more pronounced for travel time compared to cost. These insights are valuable for policymakers and MaaS operators, offering an enhanced understanding of users’ risk attitudes and the intricate interplay between travelers’ preferences for MaaS and the uncertainties inherent in daily travel.

Robust CARA Optimization

Decision making under uncertainty involves both ambiguity and risk. In “Robust CARA Optimization,” Chen and Sim have developed innovative optimization models designed for ambiguity-averse decision makers whose risk preference is consistent with constant absolute risk aversion (CARA). The research delves into maximizing the worst-case expected exponential utility amid uncertainties from independent factors with ambiguous marginals. To enhance computational feasibility, the authors developed a series of approximations: starting with tractable concave functions in affinely perturbed cases, advancing to concave piecewise affinely perturbed scenarios, and culminating in novel multi-deflected linear decision rules for adaptive optimization. This comprehensive framework extends to a multi-period consumption model, ultimately forming an exponential conic optimization problem efficiently solvable with existing solvers. Practical applications demonstrated in project and multiperiod inventory management highlight the models’ potential to surpass existing stochastic optimization methods, especially under high risk aversion.

Optimal reinsurance-investment problem with default risk for an insurer under the constant elasticity of variance model

Abstract Accepted by: Giorgio Consigli In this paper we study the optimal reinsurance-investment problem for an insurer with constant absolute risk aversion (CARA) utility. Because the insurer needs to avoid large claims and make profits in the actual operation process, he/she is allowed to buy proportional reinsurance and invest in a risk-free bond, a stock and a default bond. Moreover, the price process of the stock follows the constant elasticity of variance (CEV) model and the correlation between the risk process and the stock’s price is considered. For the objective of expected utility maximization, we establish the Hamilton–Jacobi–Bellman equation and derive the optimal reinsurance-investment strategy explicitly for the pre-default case and the post-default case via the dynamic programming. Finally, numerical examples and sensitivity analyses are provided to illustrate the effects of model parameters on the optimal strategy. We find that (i) the default risk has no impact on the optimal strategy of the stock and the parameters of the stock’s price have no impact on the optimal money amount invested in the defaultable bond. (ii) The optimal reinsurance strategy depends on the volatility of the stock’s price under the CEV model. (iii) The correlation between risk model and the stock’s price does have effects on the insurer’s optimal reinsurance-investment strategy. These findings may provide guidance to the management of insurers in practice.

Safety versus Compensation for Professional Athletes Who Face the Prospect of Career-Ending Injuries: An Economic Risk Analysis

The National Football League and National Hockey League have instituted several rule changes, equipment improvements and medical protocols in response to the frequency of serious career-ending injuries. These two professional leagues and others have litigated lawsuits by former players who have sought financial compensation. This paper constructs a simple economic model of a risk averse athlete who faces the uncertain prospect of a career-ending injury. Improving the athlete’s welfare can be accomplished by reducing the probability of a serious injury or providing increased compensation in the event of such an injury. The net marginal preference for safety is high in sports with moderate to high probabilities of serious injury. Compensation plans are favored in sports with moderate to low probabilities. Significant increases in player salaries have little effect on a players net marginal preference for safety. These results are robust to constant or decreasing absolute risk aversion.

Optimal portfolio with relative performance and partial information: A mean‐field game approach

AbstractWe use a mean‐field game (MFG) formulation to investigate a family of portfolio management problems with relative performance in a partially observable financial market. The stock price process is characterized by a factor model and investors can only observe past stock price data. All investors share a common investment period and have constant absolute risk aversion (CARA) utility functions. We construct explicit linear feedback equilibria for both a finite population game and a corresponding MFG. Filtering theory and dynamic programming method are involved to solve two kinds of optimization problems with partial information. Our results show that the limiting strategy for ‐agent game can be derived as equilibrium of suitable MFG in a partially observable financial market.

Equilibrium investment with random risk aversion.

We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by itsexpectation.

Calibration of quantum decision theory: aversion to large losses and predictability of probabilistic choices

We present the first calibration of quantum decision theory (QDT) to a dataset of binary risky choice. We quantitatively account for the fraction of choice reversals between two repetitions of the experiment, using a probabilistic choice formulation in the simplest form without model assumption or adjustable parameters. The prediction of choice reversal is then refined by introducing heterogeneity between decision makers through their differentiation into two groups: ‘majoritarian’ and ‘contrarian’ (in proportion 3:1). This supports the first fundamental tenet of QDT, which models choice as an inherent probabilistic process, where the probability of a prospect can be expressed as the sum of its utility and attraction factors. We propose to parameterize the utility factor with a stochastic version of cumulative prospect theory (logit-CPT), and the attraction factor with a constant absolute risk aversion function. For this dataset, and penalising the larger number of QDT parameters via the Wilks test of nested hypotheses, the QDT model is found to perform significantly better than logit-CPT at both the aggregate and individual levels, and for all considered fit criteria for the first experiment iteration and for predictions (second ‘out-of-sample’ iteration). The distinctive QDT effect captured by the attraction factor is mostly appreciable (i.e. most relevant and strongest in amplitude) for prospects with big losses. Our quantitative analysis of the experimental results supports the existence of an intrinsic limit of predictability, which is associated with the inherent probabilistic nature of choice. The results of the paper can find applications both in the prediction of choice of human decision makers as well as for organizing the operation of artificial intelligence.

The Generalized Risk-Adjusted Cost-Effectiveness (GRACE) Model for Measuring the Value of Gains in Health: An Exact Formulation

AbstractThe generalized risk-adjusted cost-effectiveness (GRACE) analysis method modifies standard cost-effectiveness analysis (CEA), the primary method currently used worldwide to value health improvements arising from healthcare interventions. Generalizing standard CEA, GRACE allows for decreasing or even increasing returns to health. Previous presentations of GRACE have relied extensively on Taylor Series expansion methods to specify key model parameters, including those that properly adjust for illness severity and preexisting disability, consequences of uncertain treatment outcomes, and the marginal rate of substitution between life expectancy and health-related quality of life. Standard CEA cannot account for these sources of value or cost in its valuation of medical treatments. However, calculations of GRACE measures based on Taylor Series are approximations, which may be poorly behaved in some contexts. This paper provides a new approach for implementing GRACE, using exact utility functions instead of Taylor Series approximations. While any proper utility function will suffice, we illustrate with three well-known functions: constant relative risk aversion (CRRA) utility; hyperbolic absolute risk aversion (HARA) utility, of which CRRA is a special case; and expo-power (EP) utility, of which constant absolute risk aversion (CARA) is a special case. The analysis then extends from two-period to multiperiod models. We discuss methods to estimate parameters of HARA and EP functions using two different types of data, one from discrete choice experiments and the other from “happiness economics” methods. We conclude with some reflections on how this analysis might affect benefit-cost analysis studies of healthcare interventions.

Risk aversion in share auctions: Estimating import rents from TRQs in Switzerland

This paper analyzes risk aversion in discriminatory share auctions. I generalize the k‐step share auction model of Kastl (2011, 2012) and establish that marginal profits are set‐identified for any given coefficient of constant absolute risk aversion. I also derive necessary conditions for best‐response behavior, which allows determining risk preferences from bidding data. Further, I show how the bidders' optimality conditions allow computing bounds on the marginal profits that are tighter than those currently available. I use my results to estimate import rents from Swiss tariff‐rate quotas on high‐quality beef. Rents are overestimated when ignoring risk aversion, and rent extraction is underestimated. Small bidders (small, privately owned butcheries) are more risk averse than large bidders (general retailers). Best response violations are few and uniform across bidder sizes.

A dynamic Heston local–stochastic volatility model and Legendre transform dual-asymptotic solution for optimal investment strategy problems with CARA utility

In this paper, we consider a Heston local–stochastic volatility (HLSV) model to study an optimal investment strategy problem, and analyze the optimal strategy when the volatility component of the model obeys a slow varying process and a fast varying process, respectively. For the optimal investment objective with a constant absolute risk aversion (CARA) utility function, the analytical solution under the HLSV model cannot be obtained due to the complicated nonlinearity of the partial differential equation. In this paper we employ a dual method, Legendre transformation, and an asymptotic expansion technique to derive an asymptotic solution. We also apply a Monte Carlo method to compute the optimal strategy, which can be compared with the asymptotic solution. Finally, numerical examples are provided to support our theoretical results.

Optimizing a single-product production-inventory system under constant absolute risk aversion

Optimizing a single-product production-inventory system under constant absolute risk aversion

Effects of Supply Reliability, Risk Aversion, and Wealth on Retailer’s Optimal Order Strategy

This study examines the effects of supply reliability, risk aversion, and wealth on the optimal order strategy of retailers in the case of uncertain demand by measuring the degree of risk aversion. A more practical model of optimal ordering strategy is proposed, considering supply reliability, demand uncertainty, risk aversion, and retailer wealth, in which two random variables, supply reliability factors and demand, are introduced into the retailer’s function of expected utility. To avoid nonconvergence at both ends, the demand follows a triangular rather than a normal distribution. It is found that the optimal order quantity will increase with the improvement of supply reliability when the risk-averse degree is fixed. The results also show that the optimal order quantity of risk-averse retailers is smaller than that of risk-neutral retailers. Additionally, the optimal order quantity for the risk-averse retailer decreases as the degree of risk aversion increases, when supply reliability is fixed. Further research shows that the retailer is a constant absolute risk aversion (CARA). That means retailer’s wealth has nothing to do with the risk aversion and changes in the retailer’s wealth will not affect the retailer’s optimal order quantity. This study provides valuable insights for sustainable supply chain management and marketing.

Solvability of Differential Riccati Equations and Applications to Algorithmic Trading with Signals

ABSTRACT We study a differential Riccati equation (DRE) with indefinite matrix coefficients, which arises in a wide class of practical problems. We show that the DRE solves an associated control problem, which is key to provide existence and uniqueness of a solution. As an application, we solve two algorithmic trading problems in which the agent adopts a constant absolute risk-aversion (CARA) utility function, and where the optimal strategies use signals and past observations of prices to improve their performance. First, we derive a multi-asset market making strategy in over-the-counter markets, where the market maker uses an external trading venue to hedge risk. Second, we derive an optimal trading strategy that uses prices and signals to learn the drift in the asset prices.

Nonparametric identification and estimation of all-pay auction and contest models

AbstractIn this paper, I study the nonparametric identification and estimation of multi-unit all-pay auctions of incomplete information. First, I consider the setting where multiple goods are allocated among several risk-neutral participants with independent private values (IPV). I prove the nonparametric identification of the model and derive two different consistent estimators of the distribution of bidder valuations. The first estimator is based on the classical structural approach similar to that of Guerre et al. (Econometrica 68(3):525–574, 2000). The second estimator, instead, allows estimation of the quantile function of the bidders’ valuations directly using the quantile density of the bids. Monte Carlo simulations show good small sample property under various assumptions of the number of players and goods. Next, I consider a variety of model extensions: the case of affiliated private values (APV), asymmetric players, the addition of random noise, as well as the case of risk-averse bidders. In contrast to all other scenarios, I prove that the general model with risk-averse bidders is not identified even in the semi-parametric case in which utility function is restricted to belong to the class of functions with constant absolute risk aversion (CARA). On the other hand, I show that the model with risk aversion can be identified if the distribution of valuations is restricted to having fixed support.

Equilibrium policy portfolios when some investors are restricted from holding certain assets

PurposeThe authors analyze the equilibrium effects of non-tradable assets on optimal policy portfolios. They study how the existence of non-tradable assets impacts optimal asset allocation decisions of investors who own such assets and of investors who do not have access to non-tradable assets.Design/methodology/approachIn this theoretical analysis, the authors analyze a model with tradable and non-tradable asset classes whose cash flows are jointly normally distributed. There are two types of investors, with and without access to non-tradable assets. All investors have constant absolute risk aversion preferences. The authors derive closed form solutions for optimal investor demand and equilibrium asset prices. They calibrated the model using US data for listed equity, bonds and private equity. Further, the authors illustrate the sensitivities of quantities and prices with respect to the main parameters.FindingsThe study finds that the existence of non-tradable assets has a large impact on optimal asset allocation. Investors with (without) access to non-tradable assets tilt their portfolios of tradable assets away from (toward) assets to which non-tradable assets exhibit positive betas.Practical implicationsThe model provides important insights not only for investors holding non-tradable assets such as private equity but also for investors who do not have access to non-tradable assets. Investors who ignore the effect of non-tradable assets when reverse-engineering risk premia from asset covariances and market capitalizations might severely underestimate the equity risk premium.Originality/valueThe authors provide the first comprehensive analysis of the equilibrium effects of non-tradability of some assets on optimal policy portfolios. Thus, this paper goes beyond analyzing the effects of market imperfections on individual portfolio choices.

Moment conditions for fractional degree stochastic dominance

The fractional degree stochastic dominance provides a continuum of stochastic dominance rules for ranking uncertain prospects, which encompasses usual integer-degree cases. This work presents its moment conditions. Specifically, the necessary conditions are provided based on the ordering of the moment-generating functions of distributions, which can be related to expected utility with constant absolute risk aversion preferences. We consider numerically many well-known distributions including normal distribution, gamma distribution and so on to show that the moment conditions are simple and computationally feasible in practical applications. We also apply our conditions to detect portfolio efficiency.

Informational efficiency and welfare

In a continuous-time market with a safe rate and a risky asset that pays a dividend stream depending on a latent state of the economy, several agents make consumption and investment decisions based on public information–prices and dividends–and private signals. If each investor has constant absolute risk aversion, equilibrium prices do not reveal all the private signals, but lead to the same estimate of the state of the economy that one would hypothetically obtain from the knowledge of all private signals. Accurate information leads to low volatility, ostensibly improving market efficiency, but also reduces each agent’s consumption through a decrease in the price of risk. Thus, informational efficiency is reached at the expense of agents’ welfare.

Growth-at-risk and macroprudential policy design

This paper explores the foundations for the application of the empirical growth-at-risk (GaR) approach to the assessment and design of macroprudential policies. It starts considering a stylized benchmark linear specification of the empirical GaR approach in combination with a linear–quadratic social welfare criterion that rewards expected GDP growth and penalizes the gap between expected GDP growth and GaR. If the growth rate follows a normal distribution, this welfare criterion can be microfounded as consistent with expected utility maximization under preferences for GDP levels exhibiting constant absolute risk aversion. The benchmark formulation implies an optimal policy rule linear in the risk indicator and an optimal gap between expected growth and GaR that does not depend on the time-varying risk indicator and is inversely related to the cost-effectiveness of macroprudential policy and the risk preference parameter. Extensions of the benchmark formulation show the potential to adapt the analysis and its insights to the richer specifications typically considered in empirical work.

Optimal Asset Allocation for CRRA and CARA Insurers under the Vasicek Interest Rate Model

This paper considers the reinsurance-investment problem with interest rate risks under constant relative risk aversion and constant absolute risk aversion preferences, respectively. Stochastic control theory and dynamic programming principle are applied to investigate the optimal proportional reinsurance-investment strategy for an insurer under the Vasicek stochastic interest rate model. Solving the corresponding Hamilton-Jacobi-Bellman equation via the Legendre transform approach, the optimal premium allocation strategies maximizing the expected utilities of terminal wealth are derived. In addition, several sensitivity analyses and numerical illustrations are given to analyze the impacts of different risk preferences and interest rate fluctuation on the optimal strategies. We find that the asset allocation and reinsurance ratio of the insurer are correlated with risk preference coefficient and interest rate fluctuation, and the insurance company may adjust the reinsurance-investment strategy to deal with interest rate risk.

Expanding the Gordon-Loeb model to cyber-insurance

We present an economic model for decisions on competing cyber-security and cyber-insurance investment based on the Gordon-Loeb model for investment in information security. We consider a one-period scenario in which a firm may invest in information security measures to reduce the probability of a breach, in cyber-insurance or in a combination of both. The optimal combination of investment and insurance under the assumptions of the Gordon-Loeb model is investigated via consideration of the costs and benefits of investment in security alongside purchasing insurance at an independent premium rate. Under both exponential (constant absolute risk aversion) and logarithmic (constant relative risk aversion) utility functions it is found that when the insurance premium is below a certain value, utility is maximised with insurance and security investment. These results suggest that cyber-insurance is a worthwhile undertaking provided it is not overly costly. We believe this model to be the first attempt to integrate the Gordon-Loeb model into a classical microeconomic analysis of insurance, particularly using the Gordon-Loeb security breach functions to determine the probability of an insurance claim. The model follows the tradition of the Gordon-Loeb model in being accessible to practitioners and decision makers in information security.