Risk Aversion Utility Function Research Articles

An optimal investment consumption model for retirees with no health insurance

Retirees meet a number of problems as they are growing older which needs persistent attention. Hence, without a doubt, the outcomes of the financial markets influence the choices that people make when nearing retirement. In our model, the stock price dynamics follow Geometric Brownian motion (GBM) and our goal was to optimize the expected discounted utility of consumption and terminal wealth whilst considering health expenses. The investment return process comprises risk free asset and risky assets, and the health expenses. We choose power utility functions where comprehensive solutions for Hyperbolic Absolute Risk Aversion (HARA) utility functions are obtained and optimal investment, consumption and health expenditure strategies are derived by applying dynamic programming and variable change technique on the Hamilton-Jacobi-Bellman (HJB) equations. In our numerical results it showed various effects of some economic and market parameters on the optimal investment, consumption and health expense strategies. The inflation price market risk governs the amount invested in stock, bond and also how much to be put in health to sustain a given period of the retiree's lifetime. As the health welfare rate R increases, the proportion of wealth invested in the stock increases. We also investigated the effects of the high correlation coefficients and low correlation coefficients on consumption and income rate respectively. As the constant variance discounting coefficient increases, seasoned enterprise annuity retirees decrease their allocation to the risky assets. Finally, a numerical example is presented to depict the effects of financial parameters on the optimal investment strategy with health expenditure.

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.

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.

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 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.

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.

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.

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.

Downside risk aversion vs decreasing absolute risk aversion: an intuitive exposition

Downside risk aversion (downside RA) and decreasing absolute risk aversion (DARA) are different concepts that describe preferences for which the harm from bearing risk is lessened by an increase in wealth. This note presents some intuitive explanations of the difference between the two concepts using simple lotteries and graphical analysis. All risk-averse utility functions exhibit downside risk aversion, except those that exhibit sufficiently strong increasing absolute risk aversion. In a sense, downside RA is to be expected: adding downside risk to a baseline lottery is analogous to increasing risk while adding upside risk is analogous to decreasing risk. The difference between the two concepts can be attributed to the use of different measures of the harm from risk bearing: downside RA measures harm using the utility premium and DARA measures harm using the risk premium. The two premia can change at different rates and even in different directions as wealth increases.

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.

Low-carbon strategies in dual-channel supply chain under risk aversion.

In a low-carbon supply chain (LCSC) constructed by a single manufacturer and a single retailer, three decision-making models are established by introducing channel preference attributes. That is, a single sales channel model, an online and offline dual channel model, and a dual channel model in which the manufacturer share revenue with her retailer. Using the mean variance (MV) method to characterize the risk aversion utility function of the manufacturer and the retailer, the following r are found. i) Consumers' preference for low-carbon products is conducive to raising the price of low-carbon products and the firms' profits. ii) The deepening of the retailer's risk aversion promotes the increase of the manufacturer's price, while the impact of the manufacturer's risk aversion has an opposite effects. Although the arbitrage behavior of the retailer can not be completely avoided in the risk aversion environment. However in a limited risk aversion environment, dual-channel development is conducive to increasing the profits of the manufacturer, but may harm the interests of the retailer. iii) Consumers' preference for online channel helps the manufacturer increase prices and profit. iv) The sharing of revenue by the manufacturer to the retailer is conducive to increasing the revenue of the retailer. Based on this, in order to improve corporate profits and promote long-term transactions, the manufacturer can carry out low-carbon promotion and implement a dual-channel strategy. The retailer can actively seek channel cooperation with the manufacturer.

Portfolio optimization and intergenerational risk sharing for a collective defined contribution pension plan

Abstract In this paper, we use a continuous time stochastic model to study a collective defined contribution pension plan when the interest rate is stochastic, and where the benefit levels are adjusted depending on the performance of the plan, and with risk sharing between different generations. The nominal interest rate is characterized by the Vasicek model, and the pension fund is invested in a financial market consisting of three assets: one risk-free asset, one bond and one risky asset. The participants of the pension plan are the risk bearers, and the plan seeks optimal investment and risk-sharing arrangements for plan sponsors and participants that maximize the expected accumulated discount utility of intermediate benefit adjustments and terminal wealth. Closed-form solutions are derived via the stochastic optimal control approach under constant relative risk aversion utility function. Numerical results show the effects of financial market parameters on the optimal investment strategy and how the optimal benefit changes with respect to different risk aversions and wage increase rates.

Expected Utility Theory on General Affine GARCH Models

ABSTRACT Expected utility theory has produced abundant analytical results in continuous-time finance, but with very little success for discrete-time models. Assuming the underlying asset price follows a general affine GARCH model which allows for non-Gaussian innovations, our work produces an approximate closed-form recursive representation for the optimal strategy under a constant relative risk aversion (CRRA) utility function. We provide conditions for optimality and demonstrate that the optimal wealth is also an affine GARCH. In particular, we fully develop the application to the IG-GARCH model hence accommodating negatively skewed and leptokurtic asset returns. Relying on two popular daily parametric estimations, our numerical analyses give a first window into the impact of the interaction of heteroscedasticity, skewness and kurtosis on optimal portfolio solutions. We find that losses arising from following Gaussian (suboptimal) strategies, or Merton's static solution, can be up to and 5%, respectively, assuming low-risk aversion of the investor and using a five-years time horizon.

Mean-variance versus utility maximization revisited: The case of constant relative risk aversion

We examine if mean-variance (M-V) is a good proxy for portfolios based on the Constant Relative Risk Aversion (CRRA) utility function. M-V portfolios are considered good proxies for portfolios from several utility functions which is why they are routinely used in the portfolio theory literature as the benchmark. Our results clearly show this is not the case. Low risk CRRA portfolios are in many cases very different to M-V portfolios, especially with respect to downside risk. If a risk-free asset is available, in many cases, no M-V portfolio is an adequate proxy for CRRA portfolios. The implications of our findings are that: i) M-V portfolios are a poor proxy for investors with CRRA preferences, ii) CRRA portfolios are more suited to investors who care about downside risk than M-V portfolios, and iii) the efficacy of M-V to proxy for utility maximization should be examined more thoroughly.

A First-Price Sealed-Bid Asymmetric Auction When Two Bidders Have Respective CRRA and General Utility Functions

We study a first-price auction with two bidders where one bidder is characterized by a constant relative risk aversion utility function (i.e., a concave power function) while the other has a general concave utility function. We establish the existence and uniqueness of the optimal strategic markups and analyze the effects of one bidder’s risk aversion level on the optimal strategic markups of him and his opponent’s, the allocative efficiency of the auction, and the seller’s expected revenue, respectively.

Optimal investment strategy with constant absolute risk aversion utility under an extended CEV model

In this paper, we develop an extended constant elasticity of variance (CEV) model with stochastic volatility to study an optimal investment strategy problem. This extended CEV model remedies the shortcoming of classical CEV model. In CEV model, the volatility term is an power function of stock price, which only covers firm-specific risks. Consequently, we consider the coefficient of volatility with the mean reverting process to make the volatility involve the market risks to improve the classical CEV model. For the optimal investment objective with a constant absolute risk aversion (CARA) utility function, the analytical solution under the extended CEV model cannot be obtained due to the complicated nonlinearity of the partial differential equation. In this paper we successfully employ a dual method, Legendre transformation, and an asymptotic expansion technique to approach an asymptotic solution. The numerical examples indicate the optimal strategy is an increasing function of the expectation of stock returns and correlations between the two market risks. Besides, it is a decreasing function of interest rate and risk aversion coefficient. In addition, by statistical analysis, we find that the power parameter, the expectation of stock returns and interest rate are all significant factors affecting the investment strategy.

Optimal consumption and portfolios with the hyperbolic absolute risk aversion preference under the CEV model

This paper studies the optimal investment-consumption decision under the constant elasticity of variance (CEV) model for an individual seeking to maximize the expected utility from cumulative consumption plus the expected utility from terminal wealth. Due to the fact that different individuals may have different risk preferences, we assume that the risk preference of an individual satisfies a hyperbolic absolute risk aversion (HARA) utility function. Generally speaking, power utility function, logarithmic utility function and exponential utility function widely used in investment theory are usually special cases of HARA utility function. By using the principle of dynamic programming and Legendre transform-dual technique, we obtain the explicit expression of the optimal investment-consumption decision. In addition, we derive the results under other utility functions as well and analyze some characteristics of the optimal portfolios and the optimal consumption decisions. A numerical simulation is presented to illustrate our results. Research results suggest that the optimal investment decisions between with consumption behavior and without it have considerable differences.

An Optimal Compensation Agency Model for Sustainability under the Risk Aversion Utility Perspective

This paper explores how to construct a fair and optimal compensation system between the principal and the agent in the face of financial compensation agency problems during a limited period in relation to the concept of sustainability. In the construction of the principal’s compensation system, the agent’s degree of operational financial effort will affect the overall revenue function for reaching sustainability. Both the principal and the agent have a maximum expected utility in the negative exponential pattern of the general hyperbolic absolute risk aversion (HARA) utility function that satisfies their respective objective functions. The proposed model and numerical example analysis results prove that the compensation system for sustainability can provide a fair and optimal financial system, from a sustainability perspective. The main contribution of this study is the construction and development of an optimal compensation agency model for risk management, which is derived by considering the effect of risk aversion utility on revenue. The proposed model can provide a fair and feasible approach within the issue of compensation, from the viewpoint of sustainability, for an optimal compensation agency problem.

Optimal annuitisation in a deterministic financial environment

The global reforms to public pension schemes over the last thirty years have progressively reduced individuals’ post-retirement social security income. In order to compensate for this, individuals join pension funds and individual plans to increase their wealth at retirement. These types of fully funded plans generally give individuals the opportunity to withdraw the capital accumulated into their scheme or to convert it into an annuity. In this paper, we analyse individuals’ post-retirement choices to allocate the wealth at retirement between consumption, risk-free investments and a life annuity. We develop a discrete time optimisation model, in a deterministic framework, with a constant relative risk aversion (CRRA) utility function. We study the effect of a bequest motive and the annuity rate used by the insurer on the optimal choice. Several numerical applications are presented to illustrate the optimal annuitisation decision results and the optimal consumption paths.