Constant Risk Aversion Research Articles

Age-dependent risk aversion: Re-evaluating fiscal policy impacts of population aging

The existing literature on fiscal unsustainability in the United States often evaluates three alternative policies: increasing the payroll tax rate, reducing pension benefits, and extending the retirement age. Studies suggest that reducing the replacement rate and increasing the retirement age provide higher welfare for future generations. However, these studies often assume constant risk aversion (CRA), which contradicts the empirical evidence that suggests risk aversion tends to increase with age. To provide a more comprehensive understanding of risk aversion, life-cycle behavior, and welfare under uncertainties, this study integrates age-dependent increasing risk aversion (IRA) into an overlapping generations model with risk-sensitive preferences. The quantitative analysis shows that individuals who exhibit IRA tend to adjust hours worked to reduce income risk and accumulate more precautionary savings to ensure future consumption. However, reducing uncertainty consumes resources that could otherwise have been used to increase overall consumption and leisure. Individuals who expect to become more risk averse in old age may prefer an increase in the payroll tax rate over the other two options, as the latter would result in relatively higher uncertainty.

Power Mixture Forward Performance Processes

We consider the forward investment problem in market models where the stock prices are continuous semimartingales adapted to a Brownian filtration. We construct a broad class of forward performance processes with initial conditions of power mixture type, $u(x) = \int_{\mathbb{I}} \frac{x^{1-\gamma}}{1-\gamma }\nu(\mathrm{d} \gamma)$. We proceed to define and fully characterize two-power mixture forward performance processes with constant risk aversion coefficients in the interval $(0,1)$, and derive properties of two-power mixture forward performance processes when the risk aversion coefficients are continuous stochastic processes. Finally, we discuss the problem of managing an investment pool of two investors, whose respective preferences evolve as power forward performance processes.

Time-consistent mean-variance reinsurance-investment problem with long-range dependent mortality rate

This paper investigates the time-consistent mean-variance reinsurance-investment (RI) problem faced by life insurers. Inspired by recent findings that mortality rates exhibit long-range dependence (LRD), we examine the effect of LRD on RI strategies. We adopt the Volterra mortality model proposed in Wang et al. [(2021). Volterra mortality model: actuarial valuation and risk management with long-range dependence. Insurance: Mathematics and Economics 96, 1–14] to incorporate LRD into the mortality rate process and describe insurance claims using a compound Poisson process with intensity represented by the stochastic mortality rate. Under the open-loop equilibrium mean-variance criterion, we derive explicit equilibrium RI controls and study the uniqueness of these controls in cases of constant and state-dependent risk aversion. We simultaneously resolve difficulties arising from unbounded non-Markovian parameters and sudden increases in the insurer's wealth process. While the exiting literature suggests that LRD has a significant effect on longevity hedging, we find that reinsurance is a risk management strategy that is robust to LRD.

A new behavioral finance mean variance framework

PurposeThe author proposes an update to the mean variance (MV) framework that replaces a constant risk aversion parameter using a dynamic risk aversion indicator. The contribution to the literature is made through making the static risk aversion parameter operational using an indicator of market sentiment. Results suggest that Sharpe ratios improve when the author replaces the traditional risk aversion parameter with a dynamic sentiment indicator from the behavioral finance literature when allocating between a risky portfolio and a risk-free asset. However, results are mixed when using the behavioral framework to allocate between two risky assets.Design/methodology/approachThe author includes a dynamic risk aversion parameter in the mean variance framework and back test using the traditional and updated behavioral mean variance (BMV) framework to see which framework leads to better performance.FindingsThe author finds that the behavioral framework provides superior performance when allocating between a risky and risk-free asset; however, it under performs when allocating between risky assets.Research limitations/implicationsThe research is based on back testing; therefore, it cannot be concluded that this strategy will perform well in real-time circumstances.Practical implicationsPortfolio managers may use this strategy to optimize the allocation between a risky portfolio and a risk-free asset.Social implicationsAn improved allocation between risk-free and risky assets that could lead to less leverage in the market.Originality/valueThe study is the first to use such a sentiment indicator in the traditional MV framework and show the math.

Optimal Auctions: Non-expected Utility and Constant Risk Aversion

Abstract We study auction design for bidders equipped with non-expected utility preferences that exhibit constant risk aversion (CRA). The CRA class is large and includes loss-averse, disappointment-averse, mean-dispersion, and Yaari’s dual preferences as well as coherent and convex risk measures. Any preference in this class displays first-order risk aversion, contrasting the standard expected utility case which displays second-order risk aversion. The optimal mechanism offers “ full-insurance” in the sense that each agent’s utility is independent of other agents’ reports. The seller excludes less types than under risk neutrality and awards the object randomly to intermediate types. Subjecting intermediate types to a risky allocation while compensating them when losing allows the seller to collect larger payments from higher types. Relatively high types are willing to pay more, and their allocation is efficient.

Asset Pricing with Fading Memory

Abstract Building on evidence that lifetime experiences shape individuals’ macroeconomic expectations, we study asset prices in an economy in which a representative agent learns with fading memory about unconditional mean endowment growth. With IID fundamentals, constant risk aversion, and memory decay calibrated to microdata, the model generates a high and strongly countercyclical objective equity premium, while the subjective equity premium is virtually constant. Consistent with this theory, experienced payout growth (a weighted average of past growth rates) is negatively related to future stock market excess returns and subjective expectations errors in surveys, and positively to analysts’ forecasts of long-run earnings growth.

Time-Varying Risk Aversion and Dynamic Portfolio Allocation

Despite the overwhelming evidence of time-varying risk aversion documented in recent literature, standard dynamic portfolio theories often adopt an assumption of constant (relative) risk aversion due to analytical tractability. In “Time Varying Risk Aversion and Dynamic Portfolio Allocation,” Li et al. explicitly consider the implications of time-varying risk aversion for dynamic portfolio allocation under the framework of regime-switching models. An investor with regime-dependent utility exhibits a decreasing relative risk aversion (DRRA) and has higher risk aversion when a bear market regime is more likely in the future. They develop an efficient dynamic programming algorithm that overcomes the challenges imposed by regime-dependent preference in obtaining time-consistent portfolio policies. The empirical results show that VIX is an important predictor of regime shifts and that investors with regime-dependent risk aversion achieve better investment performance than those with constant risk aversion.

Dynamic optimal portfolio choice under time-varying risk aversion

In this paper, we empirically analyze the possible advantages of modelling a time-varying risk aversion that best fits investors’ behavior in the context of the optimal portfolio choice. We build optimal dynamic portfolios by focusing on the estimation of a time-varying relative risk aversion parameter (RRA). Conditional univariate and multivariate models, such as GARCH, GARCH-M and DCC-GARCH, for modelling the optimal portfolio choice and the RRA parameter are implemented. As a model validation tool, the realized performance and downside risk exposure of these portfolios one month ahead is compared to that resulting from implementing a constant risk aversion parameter. The Ledoit and Wolf (2008) test provides robustness to our results and reveals the average outperformance of the dynamic risk aversion strategy over others as the constant risk aversion or the passive management strategies.

Uniqueness of equilibrium strategies in dynamic mean-variance problems with random coefficients

This paper is concerned with the uniqueness issue of open-loop equilibrium investment strategies of dynamic mean-variance portfolio selection problems with random coefficients. A unified method is developed to treat both the problems with deterministic risk-free return rate, state-dependent risk aversion, and those with full random coefficients, constant risk aversion. To do so, some new necessity conditions for the existence of equilibrium investment strategies are established, which considerably extends the analogue in [9] with distinctive ideas. Some new interesting facts are revealed. For example, if risk-free return rate is random, it is shown that there exists a unique open-loop equilibrium investment strategy relying on initial wealth, even when risk aversion is merely a constant but not state-dependent.

Optimal Auctions for Dual Risk Averse Bidders: Myerson meets Yaari

We study auction design for bidders equipped with non-expected utility preferences that exhibit constant risk aversion (CRA). The CRA class is large and includes loss-averse, disappointment-averse, mean-dispersion and Yaari's dual preferences as well as coherent and convex risk measures. The optimal mechanism offers full-insurance in the sense that each agent's utility is independent of other agents' reports. The seller excludes less types than under risk neutrality, and awards the object randomly to intermediate types. Subjecting intermediate types to a risky allocation while compensating them when losing allows the seller to collect larger payments from higher types. Relatively high types are anyway willing to pay more, and their allocation is efficient. (A previous version has been circulated under the title Optimal Auctions for Dual Risk Averse Bidders: Myerson meets Yaari.)

Asymptotic Optimality of a First-Order Approximate Strategy for an Exponential Utility Maximization Problem with a Small Coefficient of Wealth-Dependent Risk Aversion

In Delong [8] we investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. We assume that the insurer’s risk aversion coefficient consists of a constant risk aversion and a small amount of wealth-dependent risk aversion. We apply perturbation theory and expand the equilibrium value function of the optimization problem on the parameter epsilon controlling the degree of the insurer’s risk aversion depending on wealth. We derive a candidate for the first-order approximation to the equilibrium investment strategy. In this paper we formally show that the zeroth-order investment strategy pi _0^* postulated by Delong (Math Methods Oper Res 89:73–113, 2019) performs better than any strategy pi _0 when we compare the asymptotic expansions of the objective functions up to order {mathcal {O}}(1) as epsilon rightarrow 0, and the first-order investment strategy pi _0^*+pi _1^*epsilon postulated by Delong (Math Methods Oper Res 89:73–113, 2019) is the equilibrium strategy in the class of strategies pi ^*_0+pi _1epsilon when we compare the asymptotic expansions of the objective functions up to order {mathcal {O}}(epsilon ^2) as epsilon rightarrow 0, where epsilon denotes the parameter controlling the degree of the insurer’s risk aversion depending on wealth.

Wealth Fluctuations and Risk Preferences: Evidence from U.S. Investor Portfolios

Using data on the portfolio holdings and income of millions of U.S. retirement investors, I find that positive and persistent shocks to income lead to a significant increase in the equity share of investor portfolios, while increases in financial wealth due to realized returns lead to a small decline in the equity share. In a standard homothetic life-cycle model with human capital and constant risk aversion, the portfolio responses to these two wealth shocks should be of equal magnitude and opposite sign. The positive net effect in the data is evidence for risk aversion that decreases in total wealth. I estimate a structural life-cycle consumption and portfolio choice model that accounts for inertia in portfolio rebalancing and matches the reduced-form estimates with a significant degree of non-homotheticity in risk preferences, such that a 10% permanent income growth leads to an average decrease in risk aversion by 1.7%. Decreasing relative risk aversion preferences concentrate equity in the hands of the wealthy and double the share of wealth at the top of the wealth distribution.

Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility

This paper investigates the open-loop equilibrium reinsurance-investment (RI) strategy under general stochastic volatility (SV) models. We resolve difficulties arising from the unbounded volatility process and the non-negativity constraint on the reinsurance strategy. The resolution enables us to derive the existence and uniqueness result for the time-consistent mean variance RI policy under both situations of constant and state-dependent risk aversions. We apply the general framework to popular SV models including the Heston, the 3/2 and the Hull–White models. Closed-form solutions are obtained for the aforementioned models under constant risk aversion, and the non-leveraged models under state-dependent risk aversion.

Optimal dynamic reinsurance with common shock dependence and state-dependent risk aversion

This paper studies an optimal dynamic proportional reinsurance in a risk model with two dependent classes of insurance business. Under the criterion of maximizing the mean–variance utility of the terminal wealth with state-dependent risk aversion, we formulate the time-inconsistent problem within a game theoretic framework. By the technique of stochastic control theory, explicit expressions of the optimal results are derived not only for diffusion risk model but also for compound Poisson risk model. Furthermore, the similar problem with constant risk aversion is studied as well. Finally, some numerical examples are presented to show the impact of model parameters on the optimal strategies for both compound Poisson and diffusion cases.

Optimal investment for insurance company with exponential utility and wealth-dependent risk aversion coefficient

We investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. The insurer’s risk aversion coefficient changes in time and depends on the current insurer’s net asset value (the excess of assets over liabilities). We use the notion of an equilibrium strategy and derive the HJB equation for our time-inconsistent optimization problem. We assume that the insurer’s risk aversion coefficient consists of a constant risk aversion and a small amount of a wealth-dependent risk aversion. Using perturbation theory, the equilibrium value function, which solves the HJB equation, is expanded on the parameter controlling the degree of risk aversion depending on wealth. We find the first-order approximations to the equilibrium value function and the equilibrium investment strategy. Some new results for exponential utility maximization problem with constant risk aversion are derived in order to approximate the solution to our exponential utility maximization problem with wealth-dependent risk aversion.

Dynamic portfolio choice without cash

We consider the dynamic mean–variance portfolio choice without cash under a game theoretic framework. The mean–variance criterion is investigated in the situation where an investor allocates the wealth among risky assets while keeping no cash in a bank account. The problem is solved explicitly up to solutions of ordinary differential equations by applying the extended Hamilton–Jacobi–Bellman equation system. Given a constant risk aversion coefficient, the optimal allocation without a risk-free asset depends linearly on the current wealth, while that with a risk-free asset turns out to be independent of the current wealth. We also study the minimum-variance criterion, which can be viewed as an extension of the mean–variance model when the risk aversion coefficient tends to infinity. Calibration exercises demonstrate that for large investments, the mean–variance model without cash yields the highest certainty equivalent return for both short-term and long-term investments. Furthermore, the mean–variance portfolio choices with and without cash yield almost the same Sharpe ratio for an investment with large initial wealth.

Robust time-inconsistent stochastic control problems

This paper establishes a general analytical framework for continuous-time stochastic control problems for an ambiguity-averse agent (AAA) with time-inconsistent preference, where the control problems do not satisfy Bellman’s principle of optimality. The AAA is concerned about model uncertainty in the sense that she is not completely confident in the reference model of the controlled Markov state process and rather considers some similar alternative models. The problems of interest are studied within a set of dominated models and the AAA seeks for an optimal decision that is robust with respect to model risks. We adopt a game-theoretic framework and the concept of subgame perfect Nash equilibrium to derive an extended dynamic programming equation and extended Hamilton–Jacobi–Bellman–Isaacs equations for characterizing the robust dynamically optimal control of the problem. We also prove a verification theorem to theoretically support our construction of robust control. To illustrate the tractability of the proposed framework, we study an example of robust dynamic mean–variance portfolio selection under two cases: 1. constant risk aversion; and 2. state-dependent risk aversion.

Time-Consistent Mean-Variance Reinsurance-Investment Problems Under Unbounded Random Parameters: BSDE and Uniqueness

To strike the best balance between insurance risk and profit, insurers transfer insurable risk through reinsurance and enhance yield by participating into the financial market. The long-term commitment of insurance contracts makes insurers necessary to consider time-consistent (TC) reinsurance-investment policies. Using the open-loop TC mean-variance (MV) reinsurance-investment framework, we investigate the equilibrium reinsurance-investment problems for the financial market with unbounded random coefficients or, specifically, an unbounded risk premium. We characterize the problem via a backward stochastic differential equation (BSDE) framework. An explicit solution to the equilibrium strategies is derived for a constant risk aversion under a general class of stochastic models, embracing the constant elasticity of variance (CEV) and Ornstein-Uhlenbeck (OU) processes as special cases. For state-dependent risk aversions, the problem is related to the existence of a solution to a quadratic BSDE with unbounded parameters. A semi-closed form solution is derived, up to the solution to a nonlinear partial differential equation. By examining properties of the equilibrium strategies numerically, we find that the reinsurance decision is greatly affected by the market situation under the state-dependent risk aversion case. We prove the uniqueness of equilibrium strategies for both cases.

Robust Time-Inconsistent Stochastic Control Problems

This paper establishes a general analytical framework for continuous-time stochastic control problems for an ambiguity-averse agent (AAA) with time-inconsistent preference, where the control problems do not satisfy Bellman's principle of optimality. The AAA is concerned about model uncertainty in the sense that she is not completely confident in the reference model of the controlled Markov state process and rather considers some similar alternative models. The problems of interest are studied within a set of dominated models and the AAA seeks for an optimal decision that is robust with respect to model risks. We adopt a game-theoretic framework and the concept of subgame perfect Nash equilibrium to derive an extended dynamic programming equation and extended Hamilton -- Jacobi -- Bellman -- Isaacs equations for characterizing the robust dynamically optimal control of the problem. We also prove a verification theorem to theoretically support our construction of robust control. To illustrate the tractability of the proposed framework, we study an example of robust dynamic mean-variance portfolio selection under two cases: 1. constant risk aversion; and 2. state-dependent risk aversion.

Constant risk aversion in stochastic contests with exponential completion times

AbstractThis article analyzes a class of stochastic contests among multiple players under risk‐averse exponential utility. In these contests, players compete over the completion of a task by simultaneously deciding on their investment, which determines how fast they complete the task. The completion time of the task for each player is assumed to be an exponentially distributed random variable with rate linear in the player's investment and the completion times of different players are assumed to be stochastically independent. The player that completes the task first earns a prize whereas the remaining players earn nothing. The article establishes a one‐to‐one correspondence between the Nash equilibrium of this contest with respect to risk‐averse exponential utilities and the nonnegative solution of a nonlinear equation. Using the properties of the latter, it proves the existence and the uniqueness of the Nash equilibrium, and provides an efficient method to compute it. It exploits the resulting representation of the equilibrium investments to determine the effects of risk aversion and the differences between the outcome of the Nash equilibrium and that of a centralized version.© 2016 Wiley Periodicals, Inc. Naval Research Logistics 66:4–14, 2019