Constant Relative Risk Aversion Utility Research Articles

Navigating Risk Aversion and Regret

Navigating Risk Aversion and Regret

Constant relative risk aversion utility and consumption CAPM: discount factors and risk aversions for Norway, Sweden, and the UK

Constant relative risk aversion utility and consumption CAPM: discount factors and risk aversions for Norway, Sweden, and the UK

Stable paretian distribution, return generating processes and habit formation—The implication for equity premium puzzle

We study an asset pricing model with stable Paretian shocks to solve the equity premium puzzle. We extend the model with different return generating processes for consumption and dividends, relaxing the strong assumption made in previous studies that aggregate dividends are equal to consumption. The model derives solutions for asset prices and returns. Solutions are provided for the CRRA (constant relative risk aversion) utility function and habit formation utility function. We also prove the convergence of the solution under the maximally negative skewness condition and show that this new model can generate higher risk premia. With more realistic assumptions about the exogenous endowments and utility functions, our new model improves the explanatory power of fat-tailed models with stable shocks for the equity premium, thus providing an alternative approach to resolving the equity premium puzzle.

The impact of ambiguity on dynamic portfolio selection in the epsilon-contaminated binomial market model

We consider dynamic portfolio selection under ambiguity in the classical multi-period binomial market model. Ambiguity is incorporated in the real-world probability measure through an epsilon-contamination, that gives rise to a completely monotone capacity conveying a pessimistic investor’s ambiguous beliefs. The dynamic portfolio selection problem is formulated as a Choquet expected utility maximization problem on the final wealth. Then, the optimal final wealth is proved to be a function of the final stock price: this allows a dimension reduction of the problem, switching from an exponential to a linear size with respect to the number of periods. Finally, an explicit characterization of the optimal final wealth is given in the case of a constant relative risk aversion utility function and the interaction between the ambiguity and the relative risk aversion parameters is investigated.

Optimal consumption, investment and life insurance selection under robust utilities

We study the problem faced by a wage earner with an uncertain lifetime who has access to a Black–Scholes-type financial market consisting of one risk-free security and one risky asset. His preferences relative to consumption, investment and life insurance purchase are described by a robust expected utility. We rewrite this problem in terms of a two-player zero-sum stochastic differential game and we derive the wage earner optimal strategies for a general class of utility functions, studying the case of discounted constant relative risk aversion utility functions with more detail.

A comprehensive functional form of the optimal harvest control rule for multiple fishery management objectives

For many of the world's fisheries, harvest control rules (HCRs) are the main tools for supporting decision-making. We previously clarified the optimal shape of the HCR to achieve multiple fisheries management objectives (maximising average catch, reducing variation in yields, and avoiding stock collapse) and ensure robustness to estimation errors in biomass by numerically estimating the optimal values of the 21 biological reference points (BRPs) comprised in the HCR. However, for actual management, a simple but comprehensive functional form to emulate the optimal HCR is desirable, as numerical HCR optimisation with many BRPs is time-consuming. Here, we introduced three objective utility functions ( U1– U3) representing HCR performance for composite management objectives: mean–variance utility functions, where the performance indicator for variation in yields is the standard deviation ( U1) or the annual average variance ( U2) of yields, and the constant relative risk aversion utility function ( U3). We derived two equations to emulate the optimal HCRs with three adjusting parameters corresponding to the management objectives and different magnitudes of estimation errors. These equations will help stakeholders discuss desired management strategies by showing expected catch and risk by adjusting the parameter values.

Analysis of Farmers' Risk Preference Factors on the Determination of Rice Micro Insurance Premiums Using the Utility Theory

Business activities in the agricultural sector, especially rice farming, will always be faced with a high risk of uncertainty. The risks experienced by farmers come from the natural environment, natural disasters, climate, and plant-disturbing organisms. To avoid this situation, the government is currently providing the best solution in the form of a Rice Farming Insurance program (AUTP), which is expected to provide protection against the risk of crop failure that farmers may experience. The purpose of this research is to analyze rice farmers' risk preferences in determining rice farming insurance premiums. The research method for rice farmers' risk preferences was analyzed using constant relative risk averse (CRRA) utility theory. Based on the research results, rice farmers in Majalaya District, Bandung Regency has a very risk averse risk preference. The risk preferences of farmers participating in AUTP and non-AUTP are very risk averse. The policy implications that can be explained based on the results of this study are increasing farmers' understanding regarding the description and benefits of agricultural insurance through counseling and assistance by the Agriculture Service and PT. Jasindo, so that rice farmers in Majalaya District, Bandung Regency have awareness of the benefits of insurance. Encouraging the participation of rice farmers in Majalaya District in the AUTP program can also be carried out by prioritizing rice farmers with very risk averse risk preferences. The policy implications that can be explained based on the results of this study are increasing farmers' understanding regarding the description and benefits of agricultural insurance through counseling and assistance by the Agriculture Service and PT. Jasindo, so that rice farmers in Majalaya District, Bandung Regency have awareness of the benefits of insurance. Encouraging the participation of rice farmers in Majalaya District in the AUTP program can also be carried out by prioritizing rice farmers with very risk averse risk preferences. The policy implications that can be explained based on the results of this study are increasing farmers' understanding regarding the description and benefits of agricultural insurance through counseling and assistance by the Agriculture Service and PT. Jasindo, so that rice farmers in Majalaya District, Bandung Regency have awareness of the benefits of insurance. Encouraging the participation of rice farmers in Majalaya District in the AUTP program can also be carried out by prioritizing rice farmers with very risk averse risk preferences.

Understanding the impact of occasional activities on travelers’ preferences for mobility-as-a-service: A stated preference study

Mobility-as-a-Service (MaaS) is a promising solution for sustainable mobility and has therefore received considerable attention from researchers. People have complicated travel diaries. Larger households, especially with children, with a limited number of cars, are essentially prone to delays or travel cancellations. MaaS options fill this gap for such families by offering mobility options accessible at low ownership costs. This paper investigates the impact of non-mandatory trips (mandatory trips include work and school) on MaaS subscription preferences. Different likelihoods of occurrence and the corresponding travel pattern shifts are examined using a stated preference survey of 997 individuals in Hong Kong. A two-stage model is developed: we first calibrate individual risk attitudes using the Constant Relative Risk Aversion (CRRA) utility function. Then we develop several mixed logit models that incorporate risks associated with activities and the response of travelers to the scale of such uncertainty on mode shift decisions. The results reveal that respondents tend to be more multimodal after subscribing to a MaaS bundle, and the use of a taxi is greatly encouraged after subscription. In terms of users’ risk preference, we find that more risk-averse individuals tend to be reluctant to subscribe to MaaS bundles when the level of uncertainty increases. Moreover, our findings demonstrate that travelers are more willing to pay for flexible travel options and have a higher value of time when facing uncertainties (i.e., in the presence of occasional activities). These findings can provide valuable insights for both policymakers and MaaS operators, reflecting users’ perceptions regarding the features of MaaS.

Optimal Consumption Under a Habit-Formation Constraint: The Deterministic Case

We formulate and solve a deterministic optimal consumption problem to maximize the discounted constant relative risk aversion utility of an individual’s consumption-to-habit process assuming they only invest in a riskless market and that they are unwilling to consume at a rate below a certain proportion of their consumption habit. Increasing increases the degree of addictiveness of habit formation, with (respectively, ) corresponding to nonaddictive (respectively, completely addictive) model. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) consume at the minimum rate if their wealth-to-habit ratio is below a threshold and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either (1) larger than a high threshold or (2) below a low threshold and the agent is more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.

Weak Instruments, Degree of Risk Aversion and Equity Premium: Evidence from Singapore, South Korea and Taiwan

AbstractUsing data from Singapore, South Korea, and Taiwan, we estimate the coefficient of relative risk aversion (RRA) in the constant relative risk aversion utility specification of the consumption‐based capital asset pricing model. Conventional instrumental variables methods find that the coefficient of RRA is low but the inverse of it—the elasticity of intertemporal substitution in consumption—is also low. Such contradictory findings could be attributed to instruments being weak. Using weak‐instrument robust tests, we find from the equity market data that the coefficient of RRA is rather high, which could potentially explain the high equity premiums in these three East Asian economies.

Optimal investment strategy for DC pension plan with inflation risk under the hybrid stochastic volatility model

In this paper, we investigate an optimal investment strategy for defined-contribution (DC) pension plan under hybrid stochastic volatility (Heston–Hull–White) model, taking account of the inflation risk and the stochastic salary. The fund wealth is invested in financial market consisting of a risk-free asset, an inflation-indexed bond and a stock with hybrid Heston–Hull–White model. The goal of the pension fund manager is to maximize the expected utility of the terminal real wealth. We derive the Hamilton–Jacobi–Bellman (HJB) equation through the dynamic programming principle, under the constant relative risk aversion (CRRA) utility function, the optimal investment strategy is obtained. Finally, a numerical example is presented to characterize the impacts of financial parameters on the optimal investment strategy.

Facing a Risk: To Insure or Not to Insure—An Analysis with the Constant Relative Risk Aversion Utility Function

The decision to transfer or share an insurable risk is critical for the decision maker’s economy. This paper deals with this decision, starting with the definition of a function that represents the difference between the expected utility of insuring, with or without deductibles, and the expected utility of not insuring. Considering a constant relative risk aversion (CRRA) utility function, we provide a decision pattern for the potential policyholders as a function of their wealth level. The obtained rule applies to any premium principle, any per-claim deductible and any risk distribution. Furthermore, numerical results are presented based on the mean principle, a per-claim absolute deductible and a Poisson-exponential model, and a sensitivity analysis regarding the deductible parameter and the insurer security loading was performed. One of the main conclusions of the paper is that the initial level of wealth is the main variable that determines the decision to insure or not to insure; thus, for high levels of wealth, the decision is always not to insure regardless of the risk aversion of the decision maker. Moreover, the parameters defining the deductible and the premium only have an influence at low levels of wealth.

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.

CRRA utility and the sustainability of cooperation in infinitely-repeated games

In a symmetric infinitely-repeated game, where players have constant relative risk aversion (CRRA), or constant elasticity of intertemporal substitution, utility functions, it is shown that the critical discount factor required to sustain full cooperation is decreasing in the coefficient of relative risk aversion (increasing in the elasticity of intertemporal substitution). An application to cooperation in international environmental agreements (IEA) is presented and it is shown that the limit of the critical discount factor as the number of countries goes to infinity is equal to one (zero) if the coefficient of intertemporal inequality aversion is less (greater) than one. • Analyses infinitely-repeated game when players have CRRA utility functions. • Cooperation in infinitely-repeated games is easier the more risk-averse are players. • Application to International Environmental Agreement with many countries.

Optimal consumption, investment and life-insurance purchase under a stochastically fluctuating economy

We study the optimal consumption, investment and life-insurance purchase and selection strategies for a wage-earner with an uncertain lifetime with access to a financial market comprised of one risk-free security and one risky-asset whose prices evolve according to linear diffusions modulated by a continuous-time stochastic process determined by an additional diffusive nonlinear stochastic differential equation. The process modulating the linear diffusions may be regarded as an indicator describing the state of the economy in a given instant of time. Additionally, we allow the Brownian motions driving each of these equations to be correlated. The life-insurance market under consideration herein consists of a fixed number of providers offering pairwise distinct contracts. We use dynamic programming techniques to characterize the solutions to the problem described above for a general family of utility functions, studying the case of discounted constant relative risk aversion utilities with more detail.

Quasi-hyperbolic discounting under recursive utility and consumption–investment decisions

This paper examines an Epstein–Zin recursive utility with quasi-hyperbolic discounting in continuous time. I directly define the utility process supporting the Hamilton–Jacobi–Bellman (HJB) equation in the literature and consider Merton's optimal consumption–investment problem for application. I show that a solution to the HJB equation is the value function. The numerical and mathematical analyses show that unlike in the constant relative risk aversion utility, present bias in the Epstein–Zin utility causes economically significant overconsumption, maintaining a plausible attitude toward risks. Additionally, the sophisticated agent's preproperation occurs if and only if the elasticity of intertemporal substitution is larger than one.

Optimal HARA Investments with Terminal VaR Constraints

This paper studies the impact of Value at Risk (VaR) constraints on investors with hyperbolic absolute risk aversion (HARA) risk preferences. We derive closed-form representations for the “triplet”: optimal investment, terminal wealth, and value function, via extending the Bellman-based methodology from constant relative risk aversion (CRRA) utilities to HARA utilities. In the numerical part, we compare our solution (HARA-VaR) to three critical embedded cases, namely, CRRA, CRRA-VaR, and HARA, assessing the influence of key parameters like the VaR probability and floor on the optimal wealth distribution and allocations. The comparison highlights a stronger impact of VaR on a CRRA-VaR investor compared to a HARA-VaR (HV). This is in terms of not only lower Sharpe ratios but also higher tail risk and lower returns on wealth. The HV analysis demonstrates that combining both, capital guarantee and VaR, may lead to a correction of the partially adverse effects of the VaR constraint on the risk appetite. Moreover, the HV portfolio strategy also does not show the high kurtosis observed for the PV strategy. A wealth-equivalent loss (WEL) analysis is also implemented demonstrating that, for a HV investor, losses would be more serious if adopting a CRRA-VaR strategy as compared to a HARA strategy.

Optimal social welfare policy within financial and life insurance markets

We consider a continuous lifetime model for investor whose lifetime is a random variable. We assume the investor has an access to the social welfare system, the financial market and the life insurance market. The investor aims to find the optimal strategies that maximize the expected utility obtained from consumption, investing in the financial market, buying life insurance, registering in the social welfare system, the size of his estate in the event of premature death and the size of his fortune at time of retirement if he lives that long. We use dynamic programming techniques to derive a second-order nonlinear partial differential equation whose solution is the maximum objective function. We use special case of discounted constant relative risk aversion utilities to find an explicit solutions for the optimal strategies. Finally, we have shown a numerical solution for the problem under consideration and study some properties for the optimal strategies.

OPTIMAL INVESTMENT POLICY AND CAPITAL MANAGEMENT IN A FINANCIAL INSTITUTION

This research work considered an asset optimization problem where we examine how a financial institution can optimally allocate its total wealth among three assets namely; treasury, security and loan in stochastic interest rate setting and also determined how a financial institution can manage its capital. The optimal investment policy was derived through the application of stochastic optimization theory for the case of constant relative risk aversion (CRRA) utility function. Also, the Stochastic Differential Equation (SDE) for the capital adequacy ratio under Basel Accord, the SDE for the Total Risk – Weighted Assets (TRWA), the SDEs for the capital required to maintain the capital adequacy ratio under Basel II and Central Bank of Nigeria (CBN) standards were derived and solved numerically to study the capital management problem of the financial institution. Numerical examples using published data obtained from Central Bank of Nigeria (CBN) statistical bulletin and Nigeria Stock Exchange were presented to illustrate the dynamics of the optimal investment policy and how a financial institution can manage its capital. From the results, the optimal investment strategy can be achieved by shifting the financial institution investment away from the risky assets (security and loan) towards the riskless asset (treasury). It was also observed that if a financial institution observes the Basel II standard or Nigeria CBN standard of capital requirement, the financial institution would be considered to be strongly capitalized and guarantees the ability to absorb unexpected losses.

A Certainty Equivalent Merton Problem

The Merton problem is the well-known stochastic control problem of choosing consumption over time, as well as an investment mix, to maximize expected constant relative risk aversion (CRRA) utility of consumption. Merton formulated the problem and provided an analytical solution in 1970; since then a number of extensions of the original formulation have been solved. In this note we identify a certainty equivalent problem, i.e., a deterministic optimal control problem with the same optimal value function and optimal policy, for the base Merton problem, as well as a number of extensions. When time is discretized, the certainty equivalent problem becomes a second-order cone program (SOCP), readily formulated and solved using domain specific languages for convex optimization. This makes it a good starting point for model predictive control, a policy that can handle extensions that are either too cumbersome or impossible to handle exactly using standard dynamic programming methods.