Risk Aversion Parameter Research Articles

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.

Expected Surplus Growth Compared with Mean–Variance Optimization

In this article, the authors revisit a neglected paper written by Rubinstein in which he advocated a generalized logarithmic utility function. The profession has rather ignored his model in favor of the mean–variance approach, for reasons that are well known. The authors note that generalized logarithmic utility allows us to identify absolute risk aversion more directly and intuitively than the approaches adopted in mean–variance portfolio construction. They demonstrate that, for top-level asset allocation in which the conditions required by mean–variance are satisfied and mean–variance gives a good approximation to expected utility, generalized logarithmic utility gives essentially the same answer. In situations not suited to mean–variance, such as the inclusion of derivatives in portfolios or, more generally, when higher moments in return and utility probability distributions are unavoidable, Rubinstein’s model appears superior and seems particularly well suited to multi-asset investment. The authors’ simplified Bayesian model shows out-of-sample advantages for broad US asset allocation for the 2000 through mid-2020 period. <b>TOPICS:</b>Statistical methods, portfolio theory, portfolio construction <b>Key Findings</b> ▪ Maximizing the expected Rubinstein utility increases expected surplus growth for conservative investors and for those who do not rebalance frequently. Markowitz’s risk aversion parameter is replaced without requiring subjective judgement of aversion to return variance. ▪ Bayesian prediction can incorporate both scenario priors and observed return data in a simple grid form. This increases model flexibility and decreases sensitivity to errors in estimates of return probability distribution parameters. ▪ We demonstrate out-of-sample advantages for Rubinstein utility with simplified Bayesian prediction in broad US stock, bond, and cash allocation for the most recent decades. This is based on a prior inferred from earlier history, including the 1929 crash.

Reserve price of risk-averse search engine in keyword auctions with advertisers’ endogenous investment

Motivated by vigorous development of keyword auctions, this paper analyzes the reserve price policies in keyword auction with advertisers’ endogenous investment and risk-averse search engine. We explore advertisers’ optimal investment and equilibrium bidding strategies and derive the determination functions where utility-maximizing reserve price and efficient reserve price which maximizes the social welfare satisfy respectively. The results show that advertisers’ equilibrium bidding is monotonously increasing in bidders’ valuations, the number of advertisers, as well as the reserve price. Meanwhile, advertisers’ optimal investment is negatively correlated with reserve price and the number of advertisers. By numerical examples, the utility-maximizing reserve price decreases with the risk aversion parameter and the number of advertisers. Search engine’s expected utility increases with risk aversion parameter and decreases with the number of advertisers. Moreover, we declare that search engine can use reserve price as a regulatory tool to increase the utility. But there is an upper bound on search engine’s utility. It is interesting to find the efficient reserve price equals to zero. Hence there is a trade-off between total efficiency and search engine’s utility by search engine that has incentive to withhold reserve price that would benefit social welfare.

Proper solutions for Epstein–Zin Stochastic Differential Utility

In this article, we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein--Zin stochastic differential utility (EZ-SDU) who invests in a constant-parameter Black-Scholes-Merton market over the infinite horizon. The parameter combinations that we consider in this paper are such that the risk aversion parameter $R$ and the elasticity of intertemporal complementarity $S$ satisfy $\theta=\frac{1-R}{1-S}>1$. In this sense, this paper is complementary to Herdegen, Hobson and Jerome [arXiv:2107.06593]. The main novelty of the case $\theta>1$ (as opposed to $\theta\in(0,1)$) is that there is an infinite family of utility processes associated to every nonzero consumption stream. To deal with this issue, we introduce the economically motivated notion of a proper utility process, where, roughly speaking, a utility process is proper if it is nonzero whenever future consumption is nonzero. We then proceed to show that for a very wide class of consumption streams $C$, there exists a proper utility process $V$ associated to $C$. Furthermore, for a wide class of consumption streams $C$, the proper utility process $V$ is unique. Finally, we solve the optimal investment-consumption problem in a constant parameter financial market, where we optimise over the right-continuous attainable consumption streams that have a unique proper utility process associated to them.

Estimating and Testing for Smooth Structural Changes in Moment Condition Models

Numerous studies have been devoted to estimating and testing for moment condition models. Most of the existing studies assume that structural parameters are either fixed or changed abruptly over time. This study considers estimation of and testing for smooth structural changes in moment condition models where the data generating process is assumed to be locally stationary. A novel local generalized method of moment estimator and its boundary-corrected counterpart are proposed to estimate the smoothly changing structural parameters. Consistency and asymptotic normality are established, and an optimal weighting matrix and its consistent estimator are obtained. In particular, a consistent nonparametric test is proposed to check both smooth changes and abrupt breaks in structural parameters. The test is asymptotically pivotal and does not require prior information about the alternative. A Monte Carlo study is performed to illustrate the merits of the proposed test. In an empirical application, we document the time-varying features of the risk aversion parameter in an asset pricing model, which are consistent with business cycles and financial crisis.

Multiscale Intertemporal Capital Asset Pricing Model

This study investigates the multiscale intertemporal capital asset pricing model. We focus upon differences across timescales since they represent heterogeneities of investors in markets. This study employs a wavelet approach to decompose return data into multiple timescales. Furthermore, we impose a same risk-aversion parameter constraint into all portfolios, which is proposed by Engel and Bali (2010) who show that the constraint provides a reasonable equity risk premium at a daily frequency. We observe positive relations between the expected returns on portfolios and the covariance of the market at a daily frequency, while these relations change as timescales increase. We find that a negative risk-return relation, which might be related to a correction process of overreaction at an approximately weekly frequency (2 days to 16 days). The strongest positive relation is observed at an approximately monthly frequency (16 to 32 days). Monthly portfolio rebalances are widely used and might impact stock market return patterns. The equity risk premium in the longer frequency ranges from 8.64% to 11.10%. Our results are robust after controlling for macroeconomic variables, market implied volatility and test portfolios. Moreover, we investigate size and value factors and reveal that the risk premia disappear in the longer frequency, which suggests that Intertemporal CAPM is satisfied.

A loss-averse retailer–supplier supply chain model under trade credit in a supplier-Stackelberg game

At present, the research on relevant decision-making considering the loss risk/ default risk in trade credit is on the rise This paper presents a supply chain model with a risk-averse retailer and a supplier offering loss sharing and trade credit in a supplier-Stackelberg game where the decision variables are the optimal order quantity and loss sharing ratio. The model is analyzed and compared to a newsvendor model without loss sharing and trade credit. Meanwhile, we provide a numerical example and sensitivity analysis to obtain managerial insights. We determine the trend and sensitivity hierarchies between the parameters and the optimal results. The main novelty and contribution of the paper can be demonstrated in two aspects. First, the risk tolerance largely affects the profit of supply chain members, therefore, we incorporate risk-averse parameter into trade credit policy. Second, we incorporate loss sharing into the supplier-Stackelberg game with trade credit for a loss-averse retailer. The findings highlight that for a risk-averse retailer, the supplier can improve the retailer’s utility and its own profits through trade credit and loss sharing.

Economic Evaluation of Cryptocurrency Investment

This study proposes a method to enhance cryptocurrency portfolios constructed by forecast models. This study forecasts returns on four liquid cryptocurrencies (Bitcoin, Litecoin, Ripple, and Dash) and determines the weights on the cryptocurrencies based upon a dynamic allocation framework. We assess the performances of the portfolios using the performance fee measure. Our results present that the proposed portfolios outperform the benchmark portfolio with the conventional level of the risk aversion parameter. The economic gain for an investor is equivalent to 12% per week. The economic gain is sensitive to a change in the risk aversion parameter, which contrasts with the studies of exchange rates which is due to the high volatility on the cryptocurrencies. Our predictors are related to the price momentum effects and they outperform widely used network factors.

The Time-Varying Nature of Risk Aversion: Evidence from 60 Years of U.S. Stock Market Data

We investigate the time variations of the relative risk aversion parameter of a U.S. representative agent using 60 years of stock market data. We develop a methodology to identify the variables that explain the variations of risk aversion, based on an asset pricing model without valuation (or preference) risk. In this framework, the variables that predict the excess return of a market index (but not the second moments) also explain the variations of risk aversion. To wit, the variables include the price-dividend ratio and the short-term interest rate. A shock on the dividend-price ratio exerts a positive, highly persistent, though modest, effect on risk aversion, while a shock on the short-term interest rate exerts a highly negative, less persistent effect. The resulting measure of risk aversion follows a macroeconomically and financially countercyclical pattern.

Comparing bio-inspired heuristic algorithm for the mean-CVaR portfolio optimization

Risk aversion parameter is a coefficient that denotes the trade-off between the risk and the return in an optimal investment. This coefficient had widely used to modify the mean-variance portfolio optimization procedure. In this study, we develop become a mean-CVaR optimization problem with risk aversion. We investigate the usage of several biological-based heuristic algorithms such as genetic algorithm, grasshopper optimization, firefly optimization, moth flame optimization, particle swarm optimization, grey-wolf optimization, and dragonfly optimization to solve this portfolio optimization procedure. Empirical study with Indonesian Stock data show that the Grey-Wolf Optimization yields better performance than the others.

Unemployment-based Capital Asset Pricing Model

In this paper, we wish to examine how prospects of unemployment might change predictions of consumption based CAPM with regards to equity premium and term structure of interest. In particular, instead of agents having varied consumption growth, we study agents with probability of losing significant portion of their consumption. Thus, volatility of unemployment determines risk in the economy, rather than volatility of the consumption. Within framework of Epstein-Zin utility function, our model correctly predicts term structure premium and equity premium. In particular, the model predicts one-year risk-free rate of 1.71%, 30-year risk-free rate of 3.17% and equity rate of 7.33% given parameter of risk aversion of 1,2, EIS of 1,54, and time discount factor of 0.9578. Additionally, we study importance of third moments in our model and discover new pricing factor that is missing in current skewness models. Empirical tests under various conditions confirm statistical significance of our factor. Finally, we utilize our model to predict market changes based on shifts in term structure and then empirically verify model predictions. This paper demonstrates that standard C-based CAPM is more than enough to predict observed term structure and equity premium and no model modification is required if proper consumption variation statistics is used.

Do we really need heterogeneous agent models?

We investigate the existence of representative agent under various heterogeneities in a recursive utility framework. We provide the analytical solution of household allocations. We numerically explore whether we can find a representative agent whose income is the aggregate income of the society and whose allocations are the aggregate allocations of the society under heterogeneity in the parameter of risk aversion and/or parameter of intertemporal substitution, or discount rate or survival probability. We find that there is no representative agent when the heterogeneity is in survival probability. From the data, we know that there is heterogeneity among people in terms of survival probability. For instance, women tend to live longer than men, rich people tend to live longer than poor people. We conclude that it may be better to use heterogeneous agents models than the representative agent models under the recursive utility framework. Our results support the idea that we do need the heterogeneous agent models indeed.

Dynamic Surplus Optimization with Performance- and Index-Linked Liabilities

The increasing importance of liability-driven investment strategies and the shift towards retirement products with lower guarantees and more performance participation provide challenges for the development of portfolio optimization frameworks which cover these aspects. To this end, we establish a general and flexible terminal-surplus optimization framework in continuous time, allowing for dynamic investment strategies and stochastic liabilities, which can be linked to the performance of an index or the asset portfolio of the insurance company. Besides optimality results in a fairly general surplus optimization setting, we obtain closed-form solutions for the optimal investment strategy for various specific liability models, which include the cases of index-linked and performance-linked liabilities and liabilities which are completely or only partly hedge-able. We compare the results in numerical examples and study the impact of the performance participation, un-hedge-able risk components, different ways of modeling the liabilities and the relative risk aversion parameter. We find that performance- or index-linked liabilities, which provide a close link between the wealth of the insurance company and its liabilities, allow for a higher allocation in the risky investment. On the other hand, un-hedge-able risks reduce the allocation in the risky investment. We conclude that, aiming at a high expected return for the policy holder, insurance companies should try to connect the performance of insurance products closely to the wealth and minimize un-hedge-able risks.

Assessment of the Sustainability of Budget Revenues in the Regions of the Russian Federation

The paper proposes new approaches to assessing the sustainability of budget revenues in a region, which have been tested on the example of 83 subjects of the Russian Federation in 2010-2017. The sustainability of budget revenues is estimated on the basis of time series of two indicators: the ratio of sub-federal budget revenues to GRP and real income of sub-federal budgets per capita. To exclude economies of scale and eliminate system-wide risk, both indicators are calculated as a ratio to average Russian values. Using the construction of temporary linear regressions, a steady growth rate, absolute risk and relative risk of budget revenues are determined. For a comprehensive assessment of the sustainability of budget revenues in a region, we proposed the Arrow-Pratt function with the incorporated risk aversion parameter of 1.5. Using the proposed technique, we evaluated the absolute and relative risk, as well as the sustainability of budget revenues in the Russian regions. Assessments based on the two alternative indicators showed a high correlation. Our study revealed that the least stable budget systems are typical for some border regions, including the backward North Caucasian republics and Far Eastern regions with an unstable economy, and Tyumen Oblast, which is quite developed, but depends on the world market. At the same time, some extractive regions (Republic of Sakha), on the contrary, showed a high stability of budget revenues

A two-level model for the participation of microgrids in energy and reserve markets using hybrid stochastic-IGDT approach

In this paper, the decision making problem of microgrids (MGs) in simultaneous participation in the energy and reserve markets under uncertainty is investigated using a two-level framework. In the first level, the uncertainties of wind speed, solar radiation, probability of calling reserve, energy and reserve market prices, and demand are modeled using their probability distribution functions regarding which the operation problem of MG operator (MGO) is formulated as a two-stage stochastic optimization approach. The expected total cost and the amount of the provided reserve by the MGO for the reserve market are the output decisions of this problem which are considered as the parameters in the second level problem. The uncertainty of the accepted reserve by the market regarding the required reserve of the system and the behavior of the market players are modeled using the information gap decision theory (IGDT) approach as the second level problem. Therefore, the risk of the MGO is controlled using the conditional value at risk (CVaR) and IGDT risk-aversion parameters. Applying the proposed model on the 15-bus modified MG and 40-bus real test system shows the optimal decisions of the MGO in both markets to manage its uncertainties. Moreover, the sensitivity analysis is done to investigate the behavior of the MGO with changing the risk aversion parameters.

The implied Sharpe ratio

In an incomplete market, including liquidly traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky assets, there is no such measure available to help investors choose among the different European options. We introduce a new concept—the implied Sharpe ratio—which allows investors to make such a comparison in an incomplete financial market. Specifically, when comparing various European options, it is the option with the highest implied Sharpe ratio that, if included in an investor's portfolio, will improve his expected utility the most. Through the method of Taylor series expansion of the state-dependent coefficients in a nonlinear partial differential equation, we also establish the behaviour of the implied Sharpe ratio with respect to an investor's risk-aversion parameter. In a series of numerical studies, we compare the investment attractiveness of different European options by studying their implied Sharpe ratio.

Empirical test of employee incentive in supply chain network based on asymmetric information game analysis and fuzzy model

The construction of the evaluation index system of employee satisfaction level is the basis for an enterprise to measure the level of employee satisfaction. In this paper, the author analyzes the employee incentive in supply chain network based on asymmetric information game analysis and fuzzy model. Through the correction of network weight and threshold value, the error function decreases along the gradient direction. The construction and training of the network can be realized by MATLAB. The newff function in the software is used to construct the network, and the train function is used to train. This evaluation index system is a multi-level target evaluation system based on asymmetric information game, and its basic principle is to simplify complex problems. Long term effort is bigger when the salesperson’s risk averse parameter is more than a certain value or risk averse parameter is less but discount factor is more than a threshold; short term effort is bigger when risk averse parameter and discount factor is both less. For both information scenarios, the compensation contracts are designed and comparison analyses are conducted.

Stochastic Up to Congestion Bidding Strategy in the Nodal Electricity Markets Considering Risk Management

Up to congestion (UTC) is a type of financial product available in the nodal electricity markets of the United States, based on which a financial participant can earn profits by utilizing the different congestion and loss components of the electricity prices in the day-ahead (DA) and real-time (RT) markets. This paper proposes the UTC bidding strategy by using stochastic optimization technique, where the uncertain electricity prices on the UTC transaction paths are represented via scenario sets. In the established stochastic model, the total expected profit and the Conditional Value at Risk (CVaR) of the UTC bidding strategy are maximized simultaneously considering risk management, where the risk preference of financial participant is characterized by using a risk aversion parameter. By solving the proposed stochastic model, non-increasing DA UTC bidding curves can be generated for all the time periods of the next operating day, where the credit requirements for UTC transactions are taken into account in detail. Finally, to verify the effectiveness of the proposed strategy, case studies are carried out based on the historical data and trading policies of the Pennsylvania-New Jersey-Maryland (PJM) electricity market, and the UTC bidding strategies generated by different models are analyzed. The numeral results indicate that, compared to the deterministic UTC bidding strategy, the proposed stochastic strategy can bring much higher expected profit and lower potential risks for the financial participant. Moreover, by adjusting the risk aversion parameter in the proposed model, the risks can be managed efficiently according to the financial participant’s preference.

Option Pricing with Random Risk Aversion

Based on a standard general equilibrium economy, we develop a framework for pricing European options where the risk aversion parameter is state dependent and aggregate wealth and the underlying asset have a bi-variate transformed-normal distribution. Our results show that the pricing kernel may become non-monotonic for high levels of volatility and low levels of skewness of the risk aversion parameter. Also, as the volatility of the risk aversion parameter increases, the (Black and Scholes) implied volatility shifts upwards but its shape remains the same, which implies that the volatility of the risk aversion parameter does not change the shape of the risk neutral distribution. Finally, an empirical example shows that the estimated volatility of the risk aversion parameter tends to be low in periods of high market volatility and vice-versa.

Insurance With Heterogeneous Preferences

This paper studies an optimal insurance problem with finitely many potential policyholders. A monopolistic, risk-neutral insurer offers an insurance contract, and exponential utility maximizing individuals accept the offer or not. We allow for heterogeneity in the preferences of the individuals, while the insurer cannot discriminate in the insurance premium. We show that it is optimal for the insurer to offer only a full insurance contract, and the price optimization problem is reduced to a discrete problem, where the premium is an indifference premium for one individual in the market. Moreover, if individuals can self-select their insurance coverage given the market premium rate, then we find that partial insurance is generally optimal. Since the risk aversion parameters of individuals is generally unobserved, we also present a simulation-based framework. We show its convergence, and provide numerical examples.