HARA Utility Research Articles

Endowments, patience types, and uniqueness in two-good HARA utility economies

AbstractThis paper establishes a link between endowments, patience types, and the parameters of the HARA Bernoulli utility function that ensure equilibrium uniqueness in an economy with two goods and two impatience types with additive separable preferences. We provide sufficient conditions that guarantee uniqueness of equilibrium for any possible value of $$\gamma $$ γ in the HARA utility function $$\frac{\gamma }{1-\gamma }\left( b+\frac{a}{\gamma }x\right) ^{1-\gamma }$$ γ 1 - γ b + a γ x 1 - γ . The analysis contributes to the literature on uniqueness in pure exchange economies with two-goods and two agent types and extends the result in Loi and Matta (2022).

Optimal Reinsurance–Investment Strategy Based on Stochastic Volatility and the Stochastic Interest Rate Model

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.

The Optimal Reinsurance-Investment Problem Considering the Joint Interests of an Insurer and a Reinsurer under Hara Utility

The Optimal Reinsurance-Investment Problem Considering the Joint Interests of an Insurer and a Reinsurer under Hara Utility

Benefits of Fluctuating Exchange Rates on the Investor’s Wealth

We consider a problem of maximizing the utility of an agent who invests in a stock and a money market account incorporating proportional transaction costs λ > 0 and foreign exchange rate fluctuations. Assuming a HARA utility function U c = c p / p for all c ≥ 0 , p < 1 , p ≠ 0 , we suggest an approach of determining the value function. Contrary to fears associated with exchange rate fluctuations, our results show that these fluctuations can bring about tangible benefits in one’s wealth. We quantify the level of these benefits. We also present an example which illustrates an investment strategy of our agent.

An Oligopoly-Fringe Model with HARA Preferences

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.

This paper designs a numerical methodology, named PAMH, to approximate an investor’s optimal portfolio strategy in the contexts of expected utility theory (EUT) and mean-variance theory (MVT). Thanks to the use of hyperbolic absolute risk aversion utilities (HARA), the approach produces optimal solutions for decreasing relative risk aversion (DRRA) investors, as well as for increasing relative risk aversion (IRRA) agents. The accuracy and efficiency of the approximation is examined in a comparison to known closed-form solutions for a one dimensional (n=1) geometric Brownian motion with a CIR stochastic volatility model (i.e. GBM 1/2 or Heston model), and a high dimensional (up to n=35) stochastic covariance model. The former confirms the method works even when the theoretical candidate is not well-defined, while the latter illustrates low errors (up to 8% in certainty equivalent rate (CER)) and feasible computational time (less than one hour in a PC).Given the potential of this method, we investigate a relevant practical setting with no closed-form solution, namely when assets follow an OrnsteinUhlenbeck 4/2 stochastic volatility (SV, i.e. OU 4/2) model. We conduct sensitivity analyses of the optimal strategies for DRRA and IRRA investors with respect to key parameters; (e.g. risk aversion, minimum capital guarantee and 4/2’s parameters). In particular, the efficient frontier for the IRRA case is presented. A comparison to important sub-optimal strategies in terms of CER is performed, indicating low CER performances due to ignorance of stochastic volatility for CRRA investors, i.e. a myopic strategy would be even better than ignoring SV. The analyses highlight the importance of efficient and precise numerical methods to obtain substantially higher CERs.

Portfolio optimization with wealth-dependent risk constraints

Regulatory risk constraints as in the European Solvency II standard formula for insurance companies may lead to wealth-dependent constraints on the investment strategy. We develop two solution approaches for portfolio optimization problems in continuous time with wealth-dependent constraint sets. In the first approach, we reduce the optimization problem to an associate problem with constraints independent of wealth and a different utility function. The associate problem is then solved using known convex duality results. In the second approach, we use a change of control. We apply these results to Solvency II constraint sets and find that even for an investor with HARA utility who inherently reduces risk in times of distress, the constraints help to prevent the investor from taking too much risk in an optimistic market. Furthermore, we measure significant loss in utility and reduction in risk caused by the constraints, and we also evaluate the trade-off between these two effects.

OPTIMAL DIVIDEND POLICY AND STOCK PRICES

We model a corporation dividend as an exchange option on stochastic cash flow and capital budge. Then we solve optimal dividend policy problem completely based on the dividend model under the assumption that the cash reservoir of a corporation follows a mean reverting process from empirical evidence and economic arguments. Our optimal dividend controls depend on explicitly with the cash flow and the capital budget of the corporation, and maximizes the HARA utility performance. We specify the unique optimal dividend control for the cash flow and the capital budge. Multiplicity or absence of optimal dividend policies are given. The stock price of the corporation is studied in terms of our stochastic dividend model. We find an explicit relation among the volatility of the stock price, the volatility of the cash flow and the volatility of the capital budget. The ex-dividend stock price is positively proportional to the stochastic cash flow and the probability of the dividend delta with respect to the cash flow, and negatively proportional to the capital budget and the probability of the dividend delta with respect to the capital budget. Hence, our approach provides another passage through which countercyclical volatility of the stock price can arise from the countercyclical cash flow and capital budget directly.

Unequal returns: Using the Atkinson index to measure financial risk

We apply the Atkinson (1970) inequality index to time series of asset returns to offer a novel measure of financial risk consistent with expected-utility theory. This measure is converted to a certainty-equivalent return serving as a performance measure. We extend the Atkinson index to HARA utility and derive closed-form solutions to our measures for a number of preference-return combinations. Further, we establish relationships between risk aversion and the weights assigned to the cumulants of the return distribution for our performance measure. Using data from hedge funds and asset-pricing anomalies, we find that our performance measure contains additional, economically meaningful information.

Wealth Effect on Portfolio Allocation in Incomplete Markets

This paper provides a novel five-component decomposition of optimal dynamic portfolio choice. It reveals the simultaneous impacts from market incompleteness and wealth-dependent utilities. The decomposition leads to implementation via either closed-form solutions or Monte Carlo simulations. With a nonrandom but time-varying interest rate, we can explicitly solve for the optimal policy under the wealth-dependent HARA utility. It is a combination of a bond holding scheme and the corresponding simpler CRRA strategy. Under incomplete markets with a stochastic volatility model estimated on US equity data, wealth-dependence introduces sophisticated cycle-dependence for optimal policy and hysteresis effect in Sharpe ratio.

Optimal reinsurance and investment policies under HARA utility

Optimal reinsurance and investment policies under HARA utility

Risk‐Taking‐Neutral Background Risks

AbstractThis article examines how decision making under uncertainty is affected by the presence of a linearly dependent background risk, for individuals with HARA utility. A linearly dependent background risk is a background risk that increases linearly in the chosen tradable outcome. In order to do this, we construct a parametric class of background risks that we label as risk‐taking‐neutral (RTN). These background risks have the property that they will not alter the decision made with respect to the market risk. As such, these RTN background risks provide a benchmark. In many situations, a background risk that is faced by an investor can be compared to one from the RTN class in order to predict qualitative changes in the investor's choice decision. As this benchmark is easily available, it is convenient to use to predict these changes.

A note on the worst case approach for a market with a stochastic interest rate

We solve robust optimization problem and show the example of the market model for which the worst case measure is not a martingale measure. In our model the instantaneous interest rate is determined by the Hull-White model and the investor employs the HARA utility to measure his satisfaction.To protect against the model uncertainty he uses the worst case measure approach. The problem is formulated as a stochastic game between the investor and the market from the other side. PDE methods are used to find the saddle point and the precise verification argument is provided.

Unequal Returns: Using the Atkinson Index to Measure Financial Risk

We apply the Atkinson (1970) inequality index to time series of asset returns to offer a novel measure of financial risk consistent with expected-utility theory. This measure is converted to a certainty-equivalent return serving as a performance measure. We extend the Atkinson index to HARA utility and derive closed-form solutions to our measures for a number of preference-return combinations. Further, we establish relationships between risk aversion and the weights assigned to the cumulants of the return distribution for our performance measure. Using data from hedge funds and asset-pricing anomalies, we find that our performance measure contains additional, economically meaningful information.

Semi-analytical solutions for dynamic portfolio choice in jump-diffusion models and the optimal bond-stock mix

This paper studies the optimal portfolio selection problem in jump-diffusion models where an investor has a HARA utility function, and there are potentially a large number of assets and state variables. More specifically, we incorporate jumps into both stock returns and state variables, and then derive semi-analytical solutions for the optimal portfolio policy up to solving a set of ordinary differential equations to greatly facilitate economic insights and empirical applications of jump-diffusion models. To examine the effect of jump risk on investors’ behavior, we apply our results to the bond-stock mix problem and particularly revisit the bond/stock ratio puzzle in jump-diffusion models. Our results cast new light on this puzzle that unlike pure-diffusion models, it cannot be rationalized by the hedging demand assumption due to the presence of jumps in stock returns.

Optimization problem for a portfolio with an illiquid asset: Lie group analysis

Management of a portfolio that includes an illiquid asset is an important problem of modern mathematical finance. One of the ways to model illiquidity among others is to build an optimization problem and assume that one of the assets in a portfolio cannot be sold until a certain finite, infinite or random moment of time. This approach arises a certain amount of models that are actively studied at the moment.Working in the Merton's optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. Our goal in this paper is to carry out a complete Lie group analysis of PDEs describing value function and investment and consumption strategies for a portfolio with an illiquid asset that is sold in an exogenous random moment of time with a prescribed liquidation time distribution. The problem of such type leads to three dimensional nonlinear Hamilton–Jacobi–Bellman (HJB) equations. Such equations are not only tedious for analytical methods but are also quite challenging form a numeric point of view. To reduce the three-dimensional problem to a two-dimensional one or even to an ODE one usually uses some substitutions, yet the methods used to find such substitutions are rarely discussed by the authors.We find the admitted Lie algebra for a broad class of liquidation time distributions in cases of HARA and log utility functions and formulate corresponding theorems for all these cases. We use found Lie algebras to obtain reductions of the studied equations. Several of similar substitutions were used in other papers before whereas others are new to our knowledge. This method gives us the possibility to provide a complete set of non-equivalent substitutions and reduced equations.

Malliavin Calculus for Stochastic Strings with Applications To Barrier Options and Optimal Portfolios

We apply the Malliavin calculus to the stochastic string framework and obtain a Clark-Ocone-like formula. This result allows us to rewrite the hedging portfolio explicitly in terms of the Malliavin derivative of the discounted payoff. We illustrate this new result with two applications. Firstly, we obtain a closed-form expression for the hedging portfolio of a barrier option on a bond. Secondly, we solve explicitly the optimal portfolio problem in our framework. Related to this issue, we also find a mutual fund theorem and provide an example with HARA utility functions.

CAPM with various utility functions: Theoretical developments and application to international data

This paper presents an extension of the Capital Assets Pricing Model (hereafter CAPM) where various utility functions are applied. Specifically, we propose an overall CAPM beta that accounts for the higher order moments and reflects the investor preferences and attitudes toward risk. We particularly develop CAPM betas for different classes of utility function: the negative exponential utility function, power utility function or “Constant Relative Risk Aversion (CRRA) Utilities” and hyperbolic utility function or “HARA Utilities” (hyperbolic absolute risk aversion). In order to validate our theoretical results, we analyze the impact of investors’ preferences on the valuation equation. Applying the International CAPM, the results indicate that our utilities-based betas differ largely from the traditional CAPM betas. Moreover, the results confirm the importance of higher order moments on the pricing equation. Finally, the results both empirically and theoretically post to the consistent effect of the risk av...

Expected Utility Maximization for Exponential Lévy Models with Option and Information Processes

We consider the expected utility maximization problem for exponential Levy models and HARA utilities in the presence of illiquid assets in a portfolio. This illiquid asset is modeled by an option of European type on another risky asset which is correlated with the first one. Under some hypotheses on Levy processes, we give the expressions of information processes figured in the maximum utility formula. As applications, we consider Black--Scholes models with correlated Brownian motions and also Black--Scholes models with jump part represented by a Poisson process.

Higher-order risk vulnerability

We add an independent unfair background risk to higher-order risk-taking models in the current literature and examine its interaction with the main risk under consideration. Parallel to the well-known concept of risk vulnerability, which is defined by Gollier and Pratt (Econometrica 64:1109–1123, 1996), an agent is said to have a type of higher-order risk vulnerability if adding an independent unfair background risk to wealth raises his level of this type of higher-order risk aversion. We derive necessary and sufficient conditions for all types of higher-order risk vulnerabilities and explain their behavioral implications. We find that as in the case of risk vulnerability, all familiar HARA utility functions have all types of higher-order risk vulnerabilities except for a type of third-order risk vulnerability corresponding to a downside risk aversion measure called the Schwarzian derivative.