Commonly used health-related utility measures fail to account for risk preferences. Generalized Risk-Adjusted Cost-Effectiveness (GRACE) solves this problem, but it currently requires visual analog scale (VAS) measures of health states, which may not be available to the analyst. We develop an empirically based approach for constructing GRACE utilities from existing time trade-off (TTO) utilities. Using nationally representative patient-level data on both VAS and TTO measures of health states, we estimate a mapping from TTO to VAS. Using published estimates of GRACE utility over the VAS domain, we present estimates allowing analysts to map from existing TTO utility to GRACE and to make corresponding willingness-to-pay threshold corrections. Compared with expo-power and hyperbolic absolute risk-aversion models, a linear model provides the best mapping from TTO utilities to VAS measures of health state. We find TTO utilities approximate VAS reasonably well for moderate health state utilities but suffer from more approximation error elsewhere. Notably, TTO underestimates VAS health for the sickest health states, eg, for TTO levels below 0.4. A straightforward linear mapping permits analysts to estimate GRACE using traditional TTO health state utilities. Many existing studies use time-trade-off utilities in place of VAS. Our results suggest that this existing approach underestimates the GRACE value of health improvements for the sickest patients. Our study provides a solution without requiring generation of new data.

Purpose In this study, we aim to show the effect of entrepreneurs’ overconfidence on their required rates of return. Accordingly, we show the implication of two levels of overconfidence: moderate and excessive. Design/methodology/approach We use a hyperbolic absolute risk aversion utility function with a payoff function affected by an ego component to derive different expressions of required rates of return for non-diversified entrepreneurs. Findings Using simulations of these expressions, we show that a confident entrepreneur will require an annual average required return of 76.49%, an entrepreneur with moderate overconfidence will require an average return of 20.80% and an entrepreneur with excessive overconfidence will require an average return of 1.77%. Research limitations/implications Our expressions for the required rate of return depend on the assumption of the hyperbolic utility function. Other expressions will arise from other functions. Practical implications While moderate overconfidence can help overcome the fear of failure, entrepreneurs suffering from excessive overconfidence will underestimate the total risk of a startup. Social implications Excessive overconfidence could lead to bankruptcy. Originality/value This is the first research that addresses overconfidence in relation to required rates of return.

Investors face the challenge of how to incorporate economic and financial forecasts into their investment strategy, especially in times of financial crisis. To model this situation, we consider a financial market consisting of a risk-free asset with a constant interest rate as well as a risky asset whose drift and volatility is influenced by a stochastic process indicating the probability of potential market downturns. We use a dynamic portfolio optimization approach in continuous time to maximize the expected utility of terminal wealth and solve the corresponding HJB equations for the general class of HARA utility functions. The resulting optimal strategy can be obtained in closed form. It corresponds to a CPPI strategy with a stochastic multiplier that depends on the information from the crisis indicator. In addition to the theoretical results, a performance analysis of the derived strategy is implemented. The specified model is fitted using historic market data and the performance is compared to the optimal portfolio strategy obtained in a Black–Scholes framework without crisis information. The new strategy clearly dominates the BS-based CPPI strategy with respect to the Sharpe Ratio and Adjusted Sharpe Ratio.

This article investigates the optimal investment and benefit adjustment problem for the collective defined contribution (CDC) pension plan with long-term care insurance (LTCI). We establish a multistate lifecycle model and consider three health states for members: healthy, severely disabled, and dead. Disabled members are allowed to retire early and may get additional LTCI benefits. The pension fund is allowed to invest in both a risk-free asset and a risky asset. In particular, we consider the stochastic salary rate and the correlation between the salary process and the financial market. We aim to maximize the expected utility of total benefits and terminal wealth under the HARA utility function, which consists of logarithmic utility, CRRA utility, and CARA utility as special cases. By applying dynamic programming and the Legendre transformation, we solve the complex Hamilton–Jacobi–Bellman (HJB) equation and derive optimal investment and benefit adjustment strategies explicitly. Finally, some numerical examples are given to illustrate our result. Numerical analysis indicates that this pension plan enhances protection for disabled members without substantially raising the contribution rate. While the CDC pension plan ensures collective equity, incorporating LTCI promotes individual equity.

In this paper, we study a finite horizon optimal investment stopping problem with an unobservable random variable for the return of a risky asset. Using the Bayesian filter and the dual control approach, we transform the original primal problem into a dual finite horizon optimal stopping problem, which results in the dual value function satisfying a variational inequality with two state variables. For a class of utility functions that includes power utility and non–hyperbolic absolute risk aversion utility, we show that the free boundary satisfies a Volterra-type nonlinear integral equation with expectation over the joint distribution of the dual state process and the filtered probability process, and we simplify and solve the integral equation with the dimension reduction and backward recursive methods. We also construct two simple closed-form approximations for the free boundary using its asymptotic properties and show their accuracy and efficiency with numerical examples. Furthermore, we demonstrate that different model parameters may lead to one, two, or no free boundaries with a simple example. Funding: J. Xing was supported by the National Natural Science Foundation of China [Grant 12101151] and the Youth Foundation of Guizhou University of Finance and Economics [Grant 2022KYQN10]. J. Ma was supported by the National Natural Science Foundation of China [Grant 12071373]. H. Zheng was supported by the Engineering and Physical Sciences Research Council of the United Kingdom [Grant EP/V008331/1].

This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox–Ingersoll–Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton–Jacobi–Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures.

Economic modeling of behavior under uncertainty has almost exclusively used one of two approaches. First comes the mean–variance tradeoff, central to the financial economics literature. The alternative approach uses a specific function to assess expected utility. In this approach, almost all of the economics literature has used either the exponential utility (EU) function, with constant absolute risk aversion (CARA) or the power utility (PU) function with constant relative risk aversion (CRRA). Using a Taylor Series expansion, I show that higher-order terms (skewness and kurtosis) can significantly affect estimates of expected utility. I provide specific formulaic guidance allowing economists to assess when these terms become important. This guidance uses readily observable parameters such as mean, median and variance in a wide array of non-Gaussian statistical distributions. I next review two generalizations, one for CRRA utility, hyperbolic absolute risk aversion (HARA), and one for CARA utility, exponential power (EP). I then introduce the possibility of risk-seeking behavior, both in standard economic theory and in the “value” portion of prospect theory (PT), and provide a new Generalized Logistic Utility (GLU) that automatically incorporates such behavior in its functional form. Next, I introduce issues involved in modeling utility that includes a non-marketable component such as health, the environment, or altruism, and discuss how the choice of utility function alters these models. I then assess the choice between exact utility functions and Taylor Series approximations. I conclude by discussing methods to estimate utility function parameters and methods to choose among alternative estimates.

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.

We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utilities, including many classic and non-standard utilities as examples. A typical application is the composition of a HARA preference and a piecewise linear payoff in asset allocation. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. We conduct a general asymptotic analysis to the optimal portfolio, which directly serves as an analytical tool for financial analysis. We compare this PHARA portfolio with those of other utility families both analytically and numerically. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability of the PHARA utility. Finally, we use financial data to test the performance of the PHARA portfolio in the market.

This paper studies the optimal investment and benefit payment strategies for target benefit (TB) pension plans. The pension fund receives contributions from active members and pays benefits to retirees. Meanwhile, the accumulated wealth is invested in financial market consisting of one risk-free asset and one risky asset, in which the risk-free interest rate is described by the Vasicek model. The general hyperbolic absolute risk aversion (HARA) utility is adopted to describe pension fund managers' risk preferences. Using the dynamic programming approach, we construct the Hamilton–Jacobi–Bellman (HJB) equation and obtain explicit expressions for optimal investment and benefit payment strategies using the Legendre transform-dual technique. Finally, numerical analysis is presented to illustrate the sensitivity of the optimal strategies to model parameters.

This paper studies insurance companies’ optimal reinsurance–investment strategy under the stochastic interest rate and stochastic volatility model, taking the HARA utility function as the optimal criterion. It uses arithmetic Brownian motion as a diffusion approximation of the insurer’s surplus process and the variance premium principle to calculate premiums. In this paper, we assume that insurance companies can invest in risk-free assets, risky assets, and zero-coupon bonds, where the Cox–Ingersoll–Ross model describes the dynamic change in stochastic interest rates and the Heston model describes the price process of risky assets. The analytic solution of the optimal reinsurance–investment strategy is deduced by employing related methods from the stochastic optimal control theory, the stochastic analysis theory, and the dynamic programming principle. Finally, the influence of model parameters on the optimal reinsurance–investment strategy is illustrated using numerical examples.

Optimal reinsurance-investment game for two insurers with SAHARA utilities under correlated markets

<p style='text-indent:20px;'>This paper investigates an optimal asset-liability management problem within the expected utility maximization framework. The general hyperbolic absolute risk aversion (HARA) utility is adopted to describe the risk preference of the asset-liability manager. The financial market comprises a risk-free asset and a risky asset. The market price of risk depends on an affine diffusion factor process, which includes, but is not limited to, the constant elasticity of variance (CEV), Stein-Stein, Schöbel and Zhu, Heston, 3/2, 4/2 models, and some non-Markovian models, as exceptional examples. The accumulative liability process is featured by a generalized drifted Brownian motion with possibly unbounded and non-Markovian drift and diffusion coefficients. Due to the sophisticated structure of HARA utility and the non-Markovian framework of the incomplete financial market, a backward stochastic differential equation (BSDE) approach is adopted. By solving a recursively coupled BSDE system, closed-form expressions for both the optimal investment strategy and optimal value function are derived. Moreover, explicit solutions to some particular cases of our model are provided. Finally, numerical examples are presented to illustrate the effect of model parameters on the optimal investment strategies in several particular cases.</p>

Abstract The 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.

A class of portfolio optimization solvable problems

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.

Inspired by empirical evidence from the oil market, we build a model of an oligopoly facing a fringe as well as competition from renewable resources. We explore different subclasses of HARA utility functions (Cobb–Douglas, power and quadratic utility) to check the robustness of results found in the previous literature. For isoelastic demand, we characterize the equilibrium extraction rates of the fringe and the oligopolists. There always exists a phase of simultaneous supply of the oligopolists and the fringe, implying an inefficient order of use of resources since the oligopolists have smaller unit extraction costs and carbon emissions than the fringe. We calibrate our model to the oil market to quantify this sequence effect. In our benchmark calibration, we find for the three HARA subclasses that the sequence effect is responsible for almost all of the welfare loss compared to the first-best. It becomes smaller as market power decreases. Furthermore, we show that climate damage and Green Paradox effects depend non-monotonically on the degree of market power.

Polynomial affine approach to HARA utility maximization with applications to OrnsteinUhlenbeck [formula omitted] models.

Operationalizing cost-effectiveness analysis (CEA) requires that decisionmakers select maximum willingness to pay thresholds (K). We generalize previous methods used to estimate K using highly flexible hyperbolic absolute risk aversion (HARA) utility functions that encompass a wide range of risk behavior. For HARA utility, we calculate formulas for relative risk aversion (r*) and relative prudence (π∗ ), using literature-based estimates to calibrate our HARA model. We then assess optimal WTP thresholds (K) in absolute value and relative to income (K/M). Across the most-plausible range of risk preference parameters (r* and π∗ ), optimal K/M ratios sit (approximately) in the range of 1 to 3, although we cannot readily rule out larger K/M values. The optimal K always increases with income, while K/M falls with income if utility has increasing relative risk aversion. Results of this more-general model of economic utility are broadly consistent with previous work using more-restrictive Weibull functions. More precision in measuring the key parameters-particularly relative prudence (π∗ ) will narrow down the range of K/M estimates. The highly general HARA structure illuminates why and how optimal CEA thresholds change with income. An appendix illuminates how relative risk aversion and relative prudence relate to each other.

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.