Constant Relative Risk Aversion Research Articles

Altruistic giving and risk taking in human affairs

AbstractThe purpose of this paper is to provide a general proposition of the relationship between altruism and risk taking. As explained in the body of the paper, we diverge from a result reported in Stark et al. (2022) and provide an expansion and a generalization of a preliminary result reported in Stark (2024). In a broad utility framework, we study the risk aversion of an altruistic person who is an active donor (benefactor) and the risk aversion of a beneficiary of an altruistic transfer. In both cases, we find that altruism lowers risk aversion. The specific case in which the utility functions of the benefactor and of the beneficiary are constant relative risk aversion (CRRA) functions constitutes a vivid example of lesser risk aversion characterization. We conclude that in terms of risk-taking behavior, a “population” endowed with altruism is uniformly more willing to take risks than a comparable “population” devoid of altruism.

Scalarized utility-based multi-asset risk measures

We introduce a risk measure that simultaneously allows to minimize hedging costs and maximize expected utility in the presence of a risk constraint. We call it the scalarized utility-based multi-asset (SUBMA) risk measure. For the SUBMA risk measure we state the following results: If the utility function has constant relative risk aversion and the risk constraint is coherent, then the SUBMA risk measure is coherent. In a one-period financial market setup, we present a sufficient condition for the SUBMA risk measure to be finite-valued. We derive results about the existence of optimal payoffs. Finally, we present a dual representation for a map that generalizes classical risk measures and has not been analyzed so far. In particular, it gives us the dual representation for the SUBMA risk measure.

Portfolio problem for the α−hypergeometric stochastic volatility model with consumption

This paper solves the finite horizon optimal consumption-investment problem for an investor that trades in the α−hypergeometric stochastic volatility model, with preferences based on a power utility function. To find the optimal strategy, we follow the dynamic programming approach. First, we perform a suitable change of variables in order to fully transform the non-linear Hamilton-Jacobi-Bellman (HJB) equation into a semilinear one. Next, we apply the Girsanov theorem to deduce an implicit Feynman-Kac formula depending upon the volatility process. The operator defined by the Feynman-Kac representation is shown to be a contraction operator on a designed function space. Finally, the Banach fixed point theorem yields the existence of a solution to the HJB equation in the designed space. Moreover, we apply a verification theorem to guarantee that the solution of the HJB equation coincides with the value function.

The role of health in consumption and portfolio decision-making: Insights from state-dependent models

In this paper, we study an optimal consumption and asset allocation problem accounting for the fact that individuals’ utility differs across various health states. Our study is based on the assumption that an individual’s health status evolves through a semi-Markov process, where the transition probabilities are contingent on both the present state and the duration spent in that state. The optimal form of stock investment and of consumption is determined analytically. The optimal consumption level is significantly shaped by the integration of health-related factors and can be represented as the inverse of the marginal utility function with respect to time and state price density. Additionally, we introduce a Lagrangian multiplier that can be derived by solving a fixed point problem. While our optimal solutions are applicable to a wide range of utility functions, we provide numerical illustrations specifically using the power utility function in a model encompassing three distinct health states. Transition probabilities are estimated from real data collected by the China Insurance Regulatory Commission (CIRC), using a high-order polynomial Perks formula. We find that the investor’s consumption is higher when health is good, but lower when care is needed.

Pairs trading with costly short-selling

We study an optimal pairs trading model with costly short-selling. When the investor has logarithm utility function, we derive the solution in closed form, which shows that: the optimal allocation functions are piece-wise linear in the pair's relative price; stock borrowing fees asymmetrically reduce the optimal size of the long/short position; and for risk-hedging purpose, it can be optimal to short sell even when the stock borrowing fees outweigh the expected return earned from short selling. When the investor has power utility function, we propose analytical allocation functions adopting which only causes a small utility loss. When the investor is constrained from borrowing funds, the margin requirement for short selling can significantly affect the trading strategy. Empirically, we demonstrate the importance of incorporating short selling costs when trading pairs in China's stock market and verify the model-implied relation between short selling costs and profitability of pairs trading.

General equilibrium and dynamic inconsistency

We study the role of different sources of naivete in a general equilibrium version of the Ramsey model with quasi-hyperbolic discounting. When agents are aware of others’ naivete, as strongly suggested by empirical evidence, they revise consumption paths, correctly anticipating prices in a resulting sliding equilibrium (perfect foresight). When agents are unaware of others’ naivete, as is typically assumed in the theoretical literature, they revise both consumption paths and price expectations (quasi-perfect foresight). We prove the existence of sliding equilibrium under perfect foresight for the class of isoelastic utility functions. We show that generically quasi-hyperbolic discounting matters for saving behavior: sliding equilibrium under perfect foresight is observationally equivalent to some optimal path in the standard Ramsey model only if utility is logarithmic. By comparing sliding equilibria under different types of foresight we show that perfect foresight implies a higher saving rate, long-run capital stock and consumption level than quasi-perfect foresight.

Wealth shocks and portfolio choice

We use new euro area representative data from the Consumer Expectations Survey (CES) to elicit household-specific propensities to invest and consume out of positive wealth shocks. Using a randomized assignment of hypothetical lottery gains ranging from €5,000 to €50,000 and a realistic menu of consumption, saving and asset choices, we estimate the causal effect of wealth shocks on risky asset ownership and conditional asset shares. Wealth shocks have a positive effect on stockholding (between 8.4 and 12.8 percentage points increase in participation for the largest wealth shock). The majority of households do not participate in the stock market, even after a large increase in wealth. The conditional asset share invested in risky assets is constant for wealth shocks up to €20,000, and edges up slightly (by at most 2 %) for larger prizes. Our evidence is consistent with constant relative risk aversion for the majority of risky asset investors, while we also find important heterogeneity in the level of risk aversion across individuals.

Duality in optimal consumption–investment problems with alternative data

This study investigates an optimal consumption–investment problem in which the unobserved stock trend is modulated by a hidden Markov chain that represents different economic regimes. In the classic approach, the hidden state is estimated using historical asset prices, but recent technological advances now enable investors to consider alternative data in their decision-making. These data, such as social media commentary, expert opinions, COVID-19 pandemic data and GPS data, come from sources other than standard market data sources but are useful for predicting stock trends. We develop a novel duality theory for this problem and consider a jump-diffusion process for alternative data series. This theory helps investors identify “useful” alternative data for dynamic decision-making by providing conditions for the filter equation that enable the use of a control approach based on the dynamic programming principle. We apply our theory to provide a unique smooth solution for an agent with constant relative risk aversion once the distributions of the signals generated from alternative data satisfy a bounded likelihood ratio condition. In doing so, we obtain an explicit consumption–investment strategy that takes advantage of different types of alternative data that have not been addressed in the literature.

Bayesian Learning in an Affine GARCH Model with Application to Portfolio Optimization

This paper develops a methodology to accommodate uncertainty in a GARCH model with the goal of improving portfolio decisions via Bayesian learning. Given the abundant evidence of uncertainty in estimating expected returns, we focus our analyses on the single parameter driving expected returns. After deriving an Uncertainty-Implied GARCH (UI-GARCH) model, we investigate how learning about uncertainty affects investments in a dynamic portfolio optimization problem. We consider an investor with constant relative risk aversion (CRRA) utility who wants to maximize her expected utility from terminal wealth under an Affine GARCH(1,1) model. The corresponding stock evolution, and therefore, the wealth process, is treated as a Bayesian information model that learns about the expected return with each period. We explore the one- and two-period cases, demonstrating a significant impact of uncertainty on optimal allocation and wealth-equivalent losses, particularly in the case of a small sample size or large standard errors in the parameter estimation. These analyses are conducted under well-documented parametric choices. The methodology can be adapted to other GARCH models and applications beyond portfolio optimization.

Navigating Risk Aversion and Regret

This study investigates the distinctive modeling of regret utility when compared with common utility. I also introduce the interplay between common utility and regret utility. Using this model, I examine the differences in decision making, which encompasses issues such as risk sharing and principal–agent dilemmas. Regret utility is set so that its risk aversion shows common utility’s prudence (i.e., downside risk aversion). This paper reveals, both qualitatively and quantitively and with a concrete model, that regret utility leads to a more balanced and optimal ratio of agent payouts to outputs compared with common utility, meaning when major outputs are kept by principal, there are relatively larger agent payouts, and when major outputs are kept by the agent, there are relatively smaller agent payouts. This means that regret makes a more balanced distribution, and regret utility is more conservative (not biased). In addition, preliminary empirical research was performed in which people were asked risk preference or averseness questions, and their risk averseness was calculated by using the CRRA (Constant Relative Risk Aversion) utility function. The regret condition leads to a more conservative attitude. Furthermore, the regret model can be used in other areas, like in conservative investment portfolio optimization.

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.