Hyperbolic Absolute Risk Aversion Utility Research Articles

Optimal investment and benefit payment strategies for TB pension plans with stochastic interest rate under the HARA utility

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.

Optimal investment strategies for asset-liability management with affine diffusion factor processes and HARA preferences

<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>

Estimating optimal willingness to pay thresholds for cost‐effectiveness analysis: A generalized method

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.

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.

Revisiting the Merton Problem: from HARA to CARA Utility

This paper revisits the classical Merton problem on the finite horizon with the constant absolute risk aversion utility function. We apply two different methods to derive the closed-form solution of the corresponding Hamilton–Jacobi–Bellman (HJB) equation. An approximating method consists of two steps: solve the HJB equation with the hyperbolic absolute risk aversion utility function first and then take the limits of the risk aversion parameter to negative infinite. A direct method is also provided to derive another closed-form solution. Finally, we prove that the solutions obtained from different methods are equivalent. In addition, a sufficient condition is proposed to guarantee the optimal consumption is nonnegative and such a condition also leads to the verification theorem. A great advantage of our derived solution is that optimal policies can now be quantitatively scrutinized and discussed from both mathematical and economic viewpoints.

Optimal strategies for a defined contribution pension plan with SAHARA utility under DEV model

This paper investigates the optimal investment problems for a defined contribution pension plan in an incomplete financial market that consists of a risk-free asset anda risky asset whose price process obeys the dynamic elasticity of variance model and a correlated inflation risk model. The pension plan participant aims to maximize the expected utility of the terminal wealth under the symmetric asymptotic hyperbolic absolute risk aversion utility. This paper derives the Hamilton-Jacobi-Bellman equations and the verification results, develops lower and upper bounds for the value function, and computes optimal strategies by applying the dual control Monte Carlo algorithm. Numerical examples are presented to verify the accuracy of the approach.

BEHAVIORAL PORTFOLIO CHOICE UNDER HYPERBOLIC ABSOLUTE RISK AVERSION

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, cumulative prospect theory (CPT), was originally introduced by [A. Tversky & D. Kahneman (1992) Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (4), 297–323]. The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by [H. Jin & X. Y. Zhou (2008) Behavioral portfolio selection in continuous time, Mathematical Finance 18 (3), 385–426]. Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Behavioral Portfolio Choice under Hyperbolic Absolute Risk Aversion

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, Cumulative Prospect Theory (CPT), was originally introduced by Tversky and Kahneman (1992). The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by Jin and Zhou (2008). Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Optimal life-cycle consumption and investment decisions under age-dependent risk preferences

In this article we solve the problem of maximizing the expected utility of future consumption and terminal wealth to determine the optimal pension or life-cycle fund strategy for a cohort of pension fund investors. The setup is strongly related to a DC pension plan where additionally (individual) consumption is taken into account. The consumption rate is subject to a time-varying minimum level and terminal wealth is subject to a terminal floor. Moreover, the preference between consumption and terminal wealth as well as the intertemporal coefficient of risk aversion are time-varying and therefore depend on the age of the considered pension cohort. The optimal consumption and investment policies are calculated in the case of a Black-Scholes financial market framework and hyperbolic absolute risk aversion (HARA) utility functions. We generalize Ye (2008) (2008 American Control Conference, 356-362) by adding an age-dependent coefficient of risk aversion and extend Steffensen (2011) (Journal of Economic Dynamics and Control, 35(5), 659-667), Hentschel (2016) (Doctoral dissertation, Ulm University) and Aase (2017) (Stochastics, 89(1), 115-141) by considering consumption in combination with terminal wealth and allowing for consumption and terminal wealth floors via an application of HARA utility functions. A case study on fitting several models to realistic, time-dependent life-cycle consumption and relative investment profiles shows that only our extended model with time-varying preference parameters provides sufficient flexibility for an adequate fit. This is of particular interest to life-cycle products for (private) pension investments or pension insurance in general.

Utility-Consistent Valuation Schemes for the Own Risk and Solvency Assessment of Life Insurance Companies

Abstract In this paper, we construct new valuation schemes for the liabilities and economic capital of insurance companies. Specifically, we first build a ‘SAHARA’ valuation framework based on Symmetric Asymptotic Hyperbolic Absolute Risk Aversion utility functions. Then, we construct a ‘SAHARA-CPT’ framework that incorporates the previous utility function as a value function and that is based on Cumulative Prospect Theory. The process used for assessing a life insurance company’s own funds consists in replacing the market-consistent parametrization with a utility-consistent parametrization that accounts for the risk aversion of the market and the long-term duration of the company’s commitments. Our illustrations show that this approach leads to a lower value of the Own Risk and Solvency Assessment and to a lower volatility of own funds. The framework that is based on cumulative prospect theory has the advantage over the expected utility theory framework that it considers a precautionary overweighting of extreme events, as a tradeoff for additional model complexity.

Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform

The present paper investigates an optimal reinsurance-investment problem with Hyperbolic Absolute Risk Aversion (HARA) utility. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer. The insurer is allowed to purchase reinsurance from the reinsurer. Both the insurer and the reinsurer are assumed to invest in one risk-free asset and one risky asset whose price follows Heston's SV model. Our aim is to seek optimal investment-reinsurance strategies to maximize the expected HARA utility of the insurer's and the reinsurer's terminal wealth. In the utility theory, HARA utility consists of power utility, exponential utility and logarithmic utility as special cases. In addition, HARA utility is seldom studied in the optimal investment and reinsurance problem due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the insurer. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solutions of optimal investment-reinsurance strategies. Moreover, some special cases are also discussed in detail. Finally, some numerical examples are presented to illustrate the impacts of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategies.

Defined contribution pension planning with a stochastic interest rate and mean-reverting returns under the hyperbolic absolute risk aversion preference

Abstract The interest rate and the market price of risk may be stochastic in a real-world financial market. In this paper, the interest rate is assumed to be driven by a stochastic affine interest rate model and the market price of risk from the stock market is a mean-reverting process. In addition, the dynamics of the stock are simultaneously driven by random sources of interest rate and the stock market itself. In pension fund management, different fund managers may have different risk preferences. We suppose risk preference is described by the hyperbolic absolute risk aversion utility, which is a general utility function describing different risk preferences. Legendre transform-dual theory is presented to successfully obtain explicit expressions for optimal strategies. A numerical example illustrates the sensitivity of optimal strategies to market parameters. Theoretical results imply that the risks from stochastic interest rate and stochastic return may be completely hedged by adopting specific portfolios.

Merton’s portfolio problem including market frictions: A closed-form formula supporting the shadow price approach

This paper formulates a portfolio optimization in a friction market as a suitable stochastic control problem with hyperbolic absolute risk aversion utility functions under a specific functional form of the amount of wealth invested in the risky asset inspired by the Merton optimal dynamics. The state variable is the amount of wealth invested in the risky asset, while the control variable is the share rate. This formulation has two advantages. First, it provides a portfolio where the optimality is obtained by minimizing the probability of large changes in the portfolio composition with consequently low frictions, while allowing for an explicit solution to the optimal control problem. Second, with this formulation, it can be proven that this optimal solution is the optimal solution of the frictionless Merton problem for corrected price dynamics. The corrected dynamics shows that the market friction increases the risk premium by an additional term that is an elementary function of the risk associated with the market frictions, as well the price of the risk, which also depends on the risk aversion of the investor. In addition, the corrected dynamics supports the existence of a shadow price in friction markets as already shown in Kallsen and Muhle-Karbe (2010). Finally, an empirical analysis to assess the model performance is illustrated on five-minute data of four major traded stocks, three market indexes, and four exchange rates including the Bitcoin/USD rate.

Optimal Consumption and Investment Decisions under Time-Varying Preferences

In this article we solve the problem of maximizing the expected utility of future consumption and terminal wealth to determine the optimal pension or life-cycle fund strategy for a cohort of pension fund investors. The setup is strongly related to a DC pension plan where additionally (individual) consumption is taken into account. The consumption rate is subject to a time-varying minimum level and terminal wealth is subject to a terminal floor. Moreover, the preference between consumption and terminal wealth as well as the intertemporal coefficient of risk aversion are time-varying and therefore depend on the age of the considered pension cohort. The optimal consumption and investment policies are calculated in the case of a Black-Scholes financial market framework and hyperbolic absolute risk aversion (HARA) utility functions. We generalize Ye (2008) (2008 American Control Conference, 356-362) by adding an age-dependent coefficient of risk aversion and extend Steffensen (2011) (Journal of Economic Dynamics and Control, 35(5), 659-667), Hentschel (2016) (Doctoral dissertation, Ulm University) and Aase (2017) (Stochastics, 89(1), 115-141) by considering consumption in combination with terminal wealth and allowing for consumption and terminal wealth floors via an application of HARA utility functions. A case study on fitting several models to realistic, time-dependent life-cycle consumption and relative investment profiles shows that only our extended model with time-varying preference parameters provides sufficient flexibility for an adequate fit. This is of particular interest to life-cycle products for (private) pension investments or pension insurance in general.

CAT Bond Spreads via HARA Utility and Nonparametric Tests

After deriving a new utility-based model for pricing catastrophe (CAT) bonds under hyperbolic absolute risk aversion (HARA), we propose two specification tests that use nonparametric estimation techniques to test simultaneously for all possible misspecifications. Existing pricing models, including our new one, are then estimated and tested using CAT bonds primary market data. The utility-based model we propose not only is well-suited for explaining the risk-return relation observed in the CAT bond market but also delivers the best performance among the tested models. We also provide new empirical evidence that the aggregate utility function of CAT bond investors exhibits decreasing absolute risk aversion.

An optimal consumption and investment problem with partial information

AbstractWe consider a finite-time optimal consumption problem where an investor wants to maximize the expected hyperbolic absolute risk aversion utility of consumption and terminal wealth. We treat a stochastic factor model in which the mean returns of risky assets depend linearly on underlying economic factors formulated as the solutions of linear stochastic differential equations. We discuss the partial information case in which the investor cannot observe the factor process and uses only past information of risky assets. Then our problem is formulated as a stochastic control problem with partial information. We derive the Hamilton–Jacobi–Bellman equation. We solve this equation to obtain an explicit form of the value function and the optimal strategy for this problem. Moreover, we also introduce the results obtained by the martingale method.

Optimal Portfolio and Consumption Rule with a CIR Model Under HARA Utility

In the real-world environments, different individuals have different risk preferences. This paper investigates the optimal portfolio and consumption rule with a Cox–Ingersoll–Ross (CIR) model in a more general utility framework. After consumption, an individual invests his wealth into the financial market with one risk-free asset and multiple risky assets, where the short-term rate is driven by the CIR model and stock price dynamics are simultaneously influenced by random sources from both stochastic interest rate and stock market itself. The individual hopes to optimize their portfolios and consumption rules to maximize expected utility of terminal wealth and intermediate consumption. Risk preference of individual is assumed to satisfy hyperbolic absolute risk aversion (HARA) utility, which contains power utility, logarithm utility, and exponential utility as special cases. By using the principle of stochastic optimality and Legendre transform-dual theory, the explicit expressions of the optimal portfolio and consumption rule are obtained. The sensitivity of the optimal strategies to main parameters is analysed by a numerical example. In addition, economic implications are also presented. Our research results show that Legendre transform-dual theory is an effective methodology in dealing with the portfolio selection problems with HARA utility and interest rate risk can be completely hedged by constructing specific portfolios.

HARA utility maximization in a Markov-switching bond–stock market

We present a flexible multidimensional bond–stock model incorporating regime switching, a stochastic short rate and further stochastic factors, such as stochastic asset covariance. In this framework we consider an investor whose risk preferences are characterized by the hyperbolic absolute risk-aversion utility function and solve the problem of optimizing the expected utility from her terminal wealth. For the optimal portfolio we obtain a constant-proportion portfolio insurance-type strategy with a Markov-switching stochastic multiplier and prove that it assures a lower bound on the terminal wealth. Explicit and easy-to-use verification theorems are proven. Furthermore, we apply the results to a specific model. We estimate the model parameters and test the performance of the derived optimal strategy using real data. The influence of the investor’s risk preferences and the model parameters on the portfolio is studied in detail. A comparison to the results with the power utility function is also provided.

Optimal Consumption and Portfolio Decision with Heston’s SV Model Under HARA Utility Criterion

Abstract This paper studies the optimal consumption-investment strategy with Heston’s stochastic volatility (SV) model under hyperbolic absolute risk aversion (HARA) utility criterion. The financial market is composed of a risk-less asset and a risky asset, whose price process is supposed to be driven by Heston’s SV model. The risky preference of the individual is assumed to satisfy HARA utility, which recovers power utility, exponential utility and logarithm utility as special cases. HARA utility is of general framework in the utility theory and is seldom studied in the existing literatures. Legendre transform-dual technique along with stochastic dynamic programming principle is presented to deal with our problem and the closed-form solution to the optimal consumption-investment strategy is successfully obtained. Finally, some special cases are derived in detail.

Generalized Ordered Weighted Reference Dependent Utility Aggregation Operators and Their Applications to Group Decision-Making

To quantify the influence of decision makers’ psychological factors on the group decision process, this paper develops a new class of aggregation operators based on reference-dependent utility functions (RUs) in multi-attribute group decision analysis. RUs include S-shaped RU and non-S-shaped RU. Each RU affords a framework where the psychological factors explicitly enter the decision problem via the basic utility function, reference point and loss aversion coefficient. Under the general framework, we derive a generalized ordered weighted S-shaped RU proportional averaging (GOSP) operator and a generalized ordered weighted non-S-shaped RU proportional averaging (GONSP) operator, respectively. The GOSP operator implies the risk attitude of the DM for relative losses is risk-seeking, while GONSP operator indicates the risk attitude in this case is risk-averse. As a special case, GONSP operator can degenerate into GOWPA operator which means that the attitude of the DM is risk-neutral. Each operator satisfies the desirable properties of general operator, i.e., monotonicity, commutativity, idempotency and boundedness. Furthermore, we consider hyperbolic absolute risk aversion (HARA) function as the basic utility function, and define an S-shaped HARA and a non-S-shaped HARA utility functions. Based on the two new RUs, we propose GOSP–HARA operator and GONSP–HARA operator. Every operator covers many existing aggregation operators. To ascertain weights of such operators, the paper builds an attribute-deviation weight model and a DMs-deviation weight model. Based on these RU operators and weight models, an approach is addressed for solving multiple attribute group decision-making problem. At last, an example is provided to show the feasible of our approach.