Constant Relative Risk Aversion Research Articles

Navigating Risk Aversion and Regret

Navigating Risk Aversion and Regret

ALM for insurers with multiple underwriting lines and portfolio constraints: a Lagrangian duality approach

We investigate how an insurance firm can optimally allocate its assets to back up liabilities from multiple non-life business lines. The insurance risks are modeled by a multidimensional jump-diffusion process that accounts for simultaneous claims in different insurance lines with policy limits. We use Lagrangian convex duality techniques to derive optimal investment-underwriting strategies that maximize the expected utility from dividends and final wealth over a finite horizon. We examine how risk aversion, prudence, portfolio constraints, and multivariate insurance risk affect the firm’s earnings retention. We obtain explicit solutions for optimal strategies under constant relative risk aversion preferences. Finally, we illustrate our results with numerical examples and show the impact of co-integration for asset-liability management with multiple sources of insurance risk.

Prediction and Allocation of Stocks, Bonds, and REITs in the US Market

AbstractThis study employs dynamic model averaging and selection of Vector Autoregressive and Time-Varying Parameters Vector Autoregressive models to forecast out-of-sample monthly returns of US stocks, bonds, and Real Estate Investment Trusts (REITs) indexes from October 2006 to December 2021. The models were recursively estimated using 17 additional predictors chosen by a genetic algorithm applied to an initial list of 155 predictors. These forecasts were then used to dynamically choose portfolios formed by these assets and the riskless asset proxied by the 3-month US treasury bills. Although we did not find any predictability in the stock market, positive results were obtained for REITs and especially for bonds. The Bayesian-based approaches applied to just the returns of the three risky assets resulted in portfolios that remarkably outperform the portfolios based on the historical means and covariances and the equally weighted portfolio in terms of certainty equivalent return, Sharpe ratio, Sortino ratio and even Conditional Value-at-Risk at 5%. This study points out that Constant Relative Risk Averse investors should use Bayesian-based approaches to forecast and choose the investment portfolios, focusing their attention on different types of assets.

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.

Benchmark-driven investment for DC pension plans

Abstract We investigate whether a benchmark and non-constant risk aversion affect the probability density distribution of optimal wealth at retirement. We maximize the expected utility of the ratio of pension wealth at retirement to an inflation-indexed benchmark. Together with a threshold and a lower bound, we are able to generate closed-form solutions. We find that this non-constant risk aversion type of utility could shift the probability density distribution of optimal wealth more towards the benchmark, and that the probability of achieving a certain percentage of the desired benchmark could be increased. The probability density distribution generated under constant relative risk aversion (CRRA) risk preference is more widely spread along the benchmark.

Optimal investment strategy for asset-liability management with a defaultable bond under stochastic default intensity

This paper studies a continuous-time asset-liability management problem with a defaultable bond under stochastic interest rates and stochastic default intensity. Specifically, an asset-liability manager is allowed to invest in a cash, a treasury bond, a defaultable bond and a stock. Using the stochastic control scheme and partial differential equation approach, we derive closed-form expressions for optimal investment strategies and corresponding value functions under the power and exponential utility functions. In addition, we also provide numerical experiments to illustrate the impact of relevant parameters in the default process and recovery rate on the optimal asset allocation strategy.

Optimal investment in ambiguous financial markets with learning

We consider the classical multi-asset Merton investment problem under drift uncertainty, i.e. the asset price dynamics are given by geometric Brownian motions with constant but unknown drift coefficients. The investor assumes a prior drift distribution and is able to learn by observing the asset prize realizations during the investment horizon. While the solution of an expected utility maximizing investor with constant relative risk aversion (CRRA) is well known, we consider the optimization problem under risk and ambiguity preferences by means of the KMM (Klibanoff et al., 2005) approach. Here, the investor maximizes a double certainty equivalent. The inner certainty equivalent is for given drift coefficient, the outer is based on a drift distribution. Assuming also a CRRA type ambiguity function, it turns out that the optimal strategy can be stated in terms of the solution without ambiguity preferences but an adjusted drift distribution. To the best of our knowledge an explicit solution method in this setting is new. We rely on some duality theorems to prove our statements.Based on our theoretical results, we are able to shed light on the impact of the prior drift distribution as well as the consequences of ambiguity preferences via the transfer to an adjusted drift distribution, i.e. we are able to explain the interaction of risk and ambiguity preferences. We compare our results with the ones in a pre-commitment setup where the investor is restricted to deterministic strategies. It turns out that (under risk and ambiguity aversion) an infinite investment horizon implies in both cases a maximin decision rule, i.e. the investor follows the worst (best) Merton fraction (over all realizations of it) if she is more (less) risk averse than a log-investor. We illustrate our findings with an extensive numerical study.

Bankruptcy and retirement: a comparison in an optimal stopping times ordered framework

We deal with an agent who optimally decides the consumption-portfolio-leisure policy to maximize a power utility function, considering two optimal stopping times: bankruptcy and retirement from labor. Predetermining the order of these two stopping times, we find two independent multiple stopping times models: the optimization problem of retirement before bankruptcy, and the one of bankruptcy before retirement. In the numerical results part, we perform the sensitivity analysis with respect to key parameters, and conduct a comparative analysis.

Constant relative risk aversion utility and consumption CAPM: discount factors and risk aversions for Norway, Sweden, and the UK

Constant relative risk aversion utility and consumption CAPM: discount factors and risk aversions for Norway, Sweden, and the UK

Investigating the Impact of Consumption Distribution on CRRA Estimation: Quantile-CCAPM-Based Approach

Abstract Using quantile maximization decision theory, this paper considers a quantile-based Euler equation that states that the asset price is a function of the quantiles of the payoff, consumption growth, the stochastic discount factor, risk aversion, and the distribution of the consumption growth rate. We use a more general distribution assumption (log-elliptical distributions) than the log-normality of the consumption growth rate assumed in the literature. The simulation results show that: (1) the higher the downside risk aversion, the lower the constant relative risk aversion; (2) the heavier the tails of the Student-t distribution, the higher the risk aversion for each level of downside risk aversion; and (3) the curve of the relationship between risk aversion and downside risk aversion shifts upward when the normality assumption is dropped, and the magnitude of this shift is high even for high degrees of freedom of the Student-t distribution. Our results suggest that using normally distributed errors to model stock returns and consumption growth rates could lead to an underestimation of the risk aversion coefficient.

Stable paretian distribution, return generating processes and habit formation—The implication for equity premium puzzle

We study an asset pricing model with stable Paretian shocks to solve the equity premium puzzle. We extend the model with different return generating processes for consumption and dividends, relaxing the strong assumption made in previous studies that aggregate dividends are equal to consumption. The model derives solutions for asset prices and returns. Solutions are provided for the CRRA (constant relative risk aversion) utility function and habit formation utility function. We also prove the convergence of the solution under the maximally negative skewness condition and show that this new model can generate higher risk premia. With more realistic assumptions about the exogenous endowments and utility functions, our new model improves the explanatory power of fat-tailed models with stable shocks for the equity premium, thus providing an alternative approach to resolving the equity premium puzzle.

Optimal investment in defined contribution pension schemes with forward utility preferences

Optimal investment strategies of an individual worker during the accumulation phase in the defined contribution pension scheme have been well studied in the literature. Most of them adopted the classical backward model and approach, but any pre-specifications of retirement time, preferences, and market environment models do not often hold in such a prolonged horizon of the pension scheme. Pre-commitment to ensure the time-consistency of an optimal investment strategy derived from the backward model and approach leads the supposedly optimal strategy to be sub-optimal in the actual realizations. This paper revisits the optimal investment problem for the worker during the accumulation phase in the defined contribution pension scheme, via the forward preferences, in which an environment-adapting strategy is able to hold optimality and time-consistency together. Stochastic partial differential equation representation for the worker's forward preferences is illustrated. This paper constructs two of the forward utility preferences and solves the corresponding optimal investment strategies, in the cases of initial power and exponential utility functions.

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.

Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer

AbstractWe study the optimal investment-reinsurance problem in the context of equity-linked insurance products. Such products often have a capital guarantee, which can motivate insurers to purchase reinsurance. Since a reinsurance contract implies an interaction between the insurer and the reinsurer, we model the optimization problem as a Stackelberg game. The reinsurer is the leader in the game and maximizes its expected utility by selecting its optimal investment strategy and a safety loading in the reinsurance contract it offers to the insurer. The reinsurer can assess how the insurer will rationally react on each action of the reinsurer. The insurance company is the follower and maximizes its expected utility by choosing its investment strategy and the amount of reinsurance the company purchases at the price offered by the reinsurer. In this game, we derive the Stackelberg equilibrium for general utility functions. For power utility functions, we calculate the equilibrium explicitly and find that the reinsurer selects the largest reinsurance premium such that the insurer may still buy the maximal amount of reinsurance. Since in the equilibrium the insurer is indifferent in the amount of reinsurance, in practice, the reinsurer should consider charging a smaller reinsurance premium than the equilibrium one. Therefore, we propose several criteria for choosing such a discount rate and investigate its wealth-equivalent impact on the expected utility of each party.

Better bunching, nicer notching

This paper studies the bunching identification strategy for an elasticity parameter that summarizes agents’ responses to changes in slope (kink) or intercept (notch) of a schedule of incentives. We show that current bunching methods may be very sensitive to implicit assumptions in the literature about unobserved individual heterogeneity. We overcome this sensitivity concern with new non- and semi-parametric estimators. Our estimators allow researchers to show how bunching elasticities depend on different identifying assumptions and when elasticities are robust to them. We follow the literature and derive our methods in the context of the iso-elastic utility model and an income tax schedule that creates a piece-wise linear budget constraint. We demonstrate bunching behavior provides robust estimates for self-employed and not-married taxpayers in the context of the U.S. Earned Income Tax Credit. In contrast, estimates for self-employed and married taxpayers depend on specific identifying assumptions, which highlight the value of our approach. We provide the Stata package bunching to implement our procedures.

Optimal Consumption and Robust Portfolio Choice for the 3/2 and 4/2 Stochastic Volatility Models

This manuscript derives optimal consumption and investment strategies for risk-averse investors under the 4/2 stochastic volatility class of models. We work under an expected utility (EUT) framework and consider a Constant Relative Risk Aversion (CRRA) investor, who may also be ambiguity-averse. The corresponding Hamilton–Jacobi–Bellman (HJB) and HJB–Isaacs (HJBI) equations are solved in closed-form for a subset of the parametric space and under some restrictions on the portfolio setting, for complete markets. Conditions for proper changes of measure and well-defined solutions are provided. These are the first analytical solutions for the 4/2 stochastic volatility model and the embedded 3/2 model for the type of excess returns established in the literature. We numerically illustrate the differences between the 4/2 model and the embedded cases of the 1/2 model (Heston) as well as the 3/2 model under the same data, and for two main cases: risk-averse investor in a complete market with consumption, and ambiguity-averse investor in a complete market with no consumption. In general, the 4/2 and 1/2 models recommend similar levels of consumption and exposure, while the 3/2 leads to significantly different recommendations.

Optimal Consumption and Investment Problem under 4/2-CIR Stochastic Hybrid Model

In this paper, we investigate the optimal consumption and investment problem under the expected utility maximization criterion. It is supposed that the financial market consists of a risky asset and a risk-free asset, and the risky asset prices follow the 4/2 Cox–Ingersoll–Ross (CIR) stochastic hybrid model. The investment objective is to obtain an optimal consumption–investment strategy by maximizing the objective function. The closed-form expression of the optimal consumption–investment strategy is obtained by using optimal control theory and the corresponding Hamilton–Jacobi–Bellman (HJB) equation under the power utility function. In addition, we present a numerical example to illustrate the influence of model parameters on the optimal consumption–investment strategy.

Stochastic control methods for optimization problems in Ornstein-Uhlenbeck spread models

We develop stochastic optimal control methods for spread financial models defined by the Ornstein–Uhlenbeck (OU) processes. To this end, we study the Hamilton - Jacobi - Bellman (HJB) equation using the Feynman - Kac (FK) probability representation. We show an existence and uniqueness theorem for the classical solution of the HJB equation, a quasi-linear partial derivative equation of parabolic type. Then we show a special verification theorem and, as a consequence, construct optimal consumption/investment strategies for power utility functions. Moreover, using fixed point tools we study the numeric approximation for the HJB solution and we establish the convergence rate which, as it turns out in this case, is super geometric, i.e., more rapid than any geometric one. Finally, we illustrate numerically the behavior of the obtained strategies.

Power utility maximization problem under rough stochastic local volatility models with continuous-time Markov chain approximation

This paper studies the portfolio optimization problem based on rough stochastic local volatility models. Due to the challenge that this model is neither a semimartingale nor a Markov process, we first introduce the semimartingale approximate optimization problem and present the convergence analysis. We use the classic stochastic control method and continuous-time Markov chain approximation approach to solve the approximate optimization problem respectively. Then the optimal investment strategy and the HJB equation satisfied by the value function are obtained respectively. Furthermore, for the classic stochastic control method, we demonstrate the existence of solutions for the associated partial differential equation via the nonlinear Feynman–Kac formula. For continuous-time Markov chain approximation method, we prove the convergence of the value function. Finally, we provide a numerical analysis to illustrate the above results by using the power utility function. The numerical analysis shows that continuous-time Markov chain approximation method is effective in solving the portfolio optimization problem under rough stochastic local volatility models.

Optimal Investment Strategies for DC-Pension Fund under Combined Stochastic Volatility Models

We investigate the optimal investment strategies of DC pension under stochastic volatility model using combined Heston-Hull-White (HHW) model with a constant income drawdown. The pension fund manager (PFM) aims to maximize the expected terminal utility of wealth in a complete market setting under constant relative risk aversion (CRRA). The goal of the PFM is to maintain the standard of living of the participants after retirement. We derive the HJB equation associated with the control problem and finally established the close form solution using stochastic dynamic programming principle (SDPP). The results show that the optimal investment and benefit payment strategies converge uniquely with time.