Risk Aversion Function Research Articles

Continuous time mean-variance portfolio optimization with piecewise state-dependent risk aversion

In this paper, a continuous time mean-variance portfolio optimization problem is considered within a game theoretic framework, where the risk aversion function is assumed to depend on the current wealth level and the discounted (preset) investment target. We derive the explicit time consistent investment policy, and find that if the current wealth level is less (larger) than the discounted investment target, the future wealth level along the time consistent investment policy is always less (larger) than the discounted investment target.

Option-implied risk aversion estimation

In this paper, following the Jackwerth (2000) work, we estimate the risk aversion function on the French market implied by the joint observation of the cross-section of option prices and time-series of underlying asset returns from a high-frequency CAC 40 index options. We recover risk aversion empirically around the 2007 crisis using two different Risk-Neutral densitiy estimation approaches and for different options maturities. We studied simultaneously the impacts of the time-to-maturity, the risk-neutral density estimation model and the time periods of the study on the risk aversion function. Our findings show that the estimated risk-aversion functions that we compute from both mixture of log–normals and jump diffusion models for different maturities and three time periods are unfortunately not positive and monotonically downwards sloping as suggested by the standard assumptions of the economic theory. We note also that the values of the risk aversion are very close to that reported by studies based on consumption data. Moreover, for the three chosen trading days, if we change the time-to-maturity, we note that the risk aversion function shape is not affected for both one and two months maturities which corroborates the findings of Jackwerth (2000) but for the three months maturity, the risk aversion function shape is clearly affected and the U-shaped pattern is more pronounced for the longer maturities. Furthermore, we find that whatever the chosen trading date is, if we have the same time-to-maturity, we keep the same shape of the curve as well as the risk aversion function shape is not substantially affected if we extract the RND from a jump diffusion model or a mixture of log–normals model mainly for the short time-to-maturities. Compared to the time-to-maturity impact on the implied risk aversion, the selected RND model impact is less significant. A noteworthy finding is that the variation interval of the implied risk aversion on the postcrisis, when the market trend is upward, is more remarkable and large than the precrisis and postcrisis for the long-run maturity. Concluding, the risk aversion response is asymmetric depending on the Risk-Neutral density, time to maturity options and the period of study.

MATHEMATICAL MORPHOLOGY APPLIED IN RECOGNITION OF HEART SIGNS

With a generalized version of the endogenous growth model by Romer in [20] it is analysed the dynamical characteristics of the effects of the learning-by-doing (LBD) externalities on the memory-dependent production process. It is regarded as an quasi-homogeneous production function whose factors of substitutability are non-constant. It is assumed that the consumer maximizes the utility of consumption according to a constant relative risk-aversion function. The functional forms for the optimal capital formation trajectory and externalities with two-delayed arguments are solutions obtained by optimality principle. The optimal problem admits a steady state. Taking consumption elasticity as a function of one of the delays, we observe economic fluctuations which can be attenuated by the actions of both the delay effect and damping. But, there is a critical value for consumption elasticity at which economic fluctuations become unstable.

S-Shaped Utility Explains the Subprime Crash

Author propose an S-shaped utility function of consumption which, combined with a heterogeneous agents and external habit setting, it’s well the first order moments of the American financial and macroeconomic time series relevant for the equity premium puzzle in the second half of XX century. The average relative risk aversion of the agents remains in the 0-3 range. The shape of the utility and its relative risk aversion function of consumption suggest an explanation for the 2008 suprime crash.

Why does Income Relate to Depressive Symptoms? Testing the Income Rank Hypothesis Longitudinally.

This paper reports a test of the relative income rank hypothesis of depression, according to which it is the rank position of an individual’s income amongst a comparison group, rather than the individual’s absolute income, that will be associated with depressive symptoms. A new methodology is developed to test between psychosocial and material explanations of why income relates to well-being. This method was used to test the income rank hypothesis as applied to depressive symptoms. We used data from a cohort of 10,317 individuals living in Wisconsin who completed surveys in 1992 and 2003. The utility assumed to arise from income was represented with a constant relative risk aversion function to overcome limitations of previous work in which inadequate specification of the relationship between absolute income and well-being may have inappropriately favoured relative income specifications. We compared models in which current and future depressive symptoms were predicted from: (a) income utility alone, (b) income rank alone, (c) the transformed difference between the individual’s income and the mean income of a comparison group and (d) income utility, income rank and distance from the mean jointly. Model comparison overcomes problems involving multi-collinearity amongst the predictors. A rank-only model was consistently supported. Similar results were obtained for the association between depressive symptoms and wealth and rank of wealth in a cohort of 32,900 British individuals who completed surveys in 2002 and 2008. We conclude that it is the rank of a person’s income or wealth within a social comparison group, rather than income or wealth themselves or their deviations from the mean within a reference group, that is more strongly associated with depressive symptoms.

Inconsistent Investment and Consumption Problems

In a traditional Black---Scholes market we develop a verification theorem for a general class of investment and consumption problems where the standard dynamic programming principle does not hold. The theorem is an extension of the standard Hamilton---Jacobi---Bellman equation in the form of a system of non-linear differential equations. We derive the optimal investment and consumption strategy for a mean-variance investor without pre-commitment endowed with labor income. In the case of constant risk aversion it turns out that the optimal amount of money to invest in stocks is independent of wealth. The optimal consumption strategy is given as a deterministic bang-bang strategy. In order to have a more realistic model we allow the risk aversion to be time and state dependent. Of special interest is the case were the risk aversion is inversely proportional to present wealth plus the financial value of future labor income net of consumption. Using the verification theorem we give a detailed analysis of this problem. It turns out that the optimal amount of money to invest in stocks is given by a linear function of wealth plus the financial value of future labor income net of consumption. The optimal consumption strategy is again given as a deterministic bang-bang strategy. We also calculate, for a general time and state dependent risk aversion function, the optimal investment and consumption strategy for a mean-standard deviation investor without pre-commitment. In that case, it turns out that it is optimal to take no risk at all.

Option‐Implied Preference with Model Uncertainty

We present a theoretical model of option‐implied preferences with model uncertainty. An option‐implied risk aversion function with model uncertainty has a higher and a steeper level of risk aversion than an investor without model uncertainty. Based on the theoretical model, we try to extract empirical option‐implied risk aversion functions with S&P 500 index options. Our empirical option‐implied risk aversion with model uncertainty and option‐implied uncertainty premium show a decreasing and a smirk pattern across wealth. After subprime crisis, the shape of option‐implied risk aversion function with model uncertainty is not quite different, and both the level of option‐implied risk aversion function and the option‐implied uncertainty premium become slightly lowered. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 34:498–515, 2014

Impact of Risk Aversion on Price and Quality Decisions under Demand Uncertainty via the CARA Utility Function

This paper investigates optimal price and quality decisions of a manufacturer-retailer supply chain under demand uncertainty, in which players are both risk-averse decision makers. The manufacturer determines the wholesale price and quality of the product, and the retailer determines the retail price. By means of game theory, we employ the constant absolute risk aversion (CARA) function to analyze two different supply chain structures, that is, manufacturer Stackelberg model (MS) and retailer Stackelberg model (RS). We then analyze the results to explore the effects of risk aversion of the manufacturer and the retailer upon the equilibrium decisions. Our results imply that both the risk aversion of the manufacturer and the retailer play an important role in the price and quality decisions. We find that, in general, in MS and RS models, the optimal wholesale price and quality decrease with the risk aversion of the manufacturer but increase with the risk aversion of the retailer, while the retail price decreases with the risk aversion of the manufacturer as well as the retailer. We also examine the impact of quality cost coefficient on the optimal decisions. Finally, numerical examples are presented to illustrate the different degree of effects of players’ risk aversion on equilibrium results and to compare results in different models considered.

Effects of Risk Aversion on the Value of Information in Two-Action Decision Problems

This paper analyzes the value of information in a two-action decision problem where a decision maker is deciding on whether to accept an uncertain investment. We investigate the effects of the decision maker's risk aversion function on the value of information in this setting. We analyze the value of perfect information and the value of partition information where a decision maker with deterministic initial wealth receives information specifying that the outcome lies within a subset of the domain. We show that if two decision makers reject the investment without the information, then the more risk-averse decision maker will value information (both perfect and partition), less than the less risk averse one. On the other hand, if both decision makers initially accept the investment, or are indifferent to the investment without the information, then (i) the more risk-averse decision maker will value perfect information higher than the less risk-averse one, but (ii) this monotonicity result need not apply for the case of partition information. These results enable us to compare the value of information when decision makers have different risk aversion levels. They also hold when comparison of risk aversion is restricted only to a domain defined by the lottery.

Option pricing with realistic ARCH processes

This paper presents a complete computation of option prices based on a realistic process for the underlying and on the construction of a risk-neutral measure as induced by a no-arbitrage replication strategy. The underlying is modelled with a long-memory ARCH process, with relative returns, fat-tailed innovations and multi-scale leverage. The process parameters are estimated on the SP500 stock index (in the physical measure). The change of measure from to the risk-neutral measure is derived rigorously along each path drawn from the process, yielding a Radon–Nikodym derivative for a given choice of a risk aversion function. A small expansion allows to compute explicitly this change of measure. Finally, a given European option’s price is obtained as the expectation in of the discounted payoff with a weight given by the change of measure . This procedure is implemented in a Monte Carlo simulation, and allows to compute the option prices, without further adjustable parameters. The computed implied volatility surfaces are compared with empirical surfaces based on European put and call options on the SP500 from 1996 to 2010. Our pricing scheme is able to reproduce the level, the smile, the smirk and the term structure of the surfaces, without any calibration on the observed option prices. We discuss the respective roles of the and measures, the distribution of the terminal prices in both measures, the small impact of the risk aversion and drift premium, and finally we suggest simplifications of our pricing scheme for practical purposes.

How Australian farmers deal with risk

PurposeIn Australia broadacre crops can be insured against hail and fire damage and some other perils but not against losses caused by drought, flood or frost. The purpose of this paper is to investigate the private provision of crop insurance in Western Australia.Design/methodology/approachA farm survey was conducted with the cooperation of dryland farmers belonging to Western Australian grower groups. The willingness to pay for hail insurance is modelled as a function of risk aversion, risk of crop failure and government assistance; and the effect of expected crop yield in t/ha on the crop area is investigated.FindingsThe empirical analysis shows that the coefficient of relative risk aversion is 2.7. An increase in the variability of crop yield by 20 per cent, which may be caused by future climate change, would raise the willingness to pay for crop insurance one‐to‐one by 20 per cent. Adverse selection plays a minor role because almost all farmers buy full coverage for hail insurance and associated risks. A future supplier of multi‐peril crop insurance must, however, consider the potential for ex ante moral hazard because the size of the crop area depends on the expected crop yield in t/ha.Social implicationsThe Global Financial Crisis has provided a stark reminder that society crucially depends on the efficient and fair allocation of risk. Climate change threatens the livelihood of farmers and food security. Private multi‐peril crop insurance, which has yet to emerge, would improve the welfare of rural populations and the efficiency of farming.Originality/valueFew empirical studies deal with the private provision of multi‐peril crop insurance because the market for multi‐peril crop insurance fails worldwide and private insurance does not exist. In this study, Australian crop insurance serves as a proxy to gain an understanding of multi‐peril crop insurance.

On Aumann and Serrano’s economic index of risk

We study the risk index of an additive gamble proposed in Aumann and Serrano (J Political Econ 116(5):810–836, 2008). We establish a generalized duality result for this index and use it to prove Yaari (J Econ Theory 1:315–329, 1969) alternative characterization of DARA utilities. A new characterization result for the risk index is obtained through essentially monotonic risk aversion utilities. We also extend the domain of gambles by introducing a price for gambles. We then develop a theory on the risk index for multiplicative gambles. Relative risk aversion functions for multiplicative gambles play the same role as absolute risk aversion functions for additive gambles.

On Aumann and Serrano's Economic Index of Risk

We study the risk index of an additive gamble proposed in Aumann and Serrano (2008). We establish a generalized duality result for this index and use it to prove Yaari's (1969) alternative characterization of DARA utilities. A new characterization result for the risk index is obtained through essentially monotonic risk aversion utilities. We also extend the domain of gambles by introducing a price for gambles. We then develop a theory on the risk index for multiplicative gambles. Relative risk aversion functions for multiplicative gambles play the same role as absolute risk aversion functions for additive gambles.

In This Issue

In This Issue

Shape Invariant Modeling of Pricing Kernels and Risk Aversion

Several empirical studies reported that pricing kernels exhibit a common pattern across different markets. The main interest in pricing kernels lies in validating the presence of the peaks and their variability in location among curves. Motivated by this observation we investigate the problem of estimating pricing kernels based on the shape invariant model, a semi-parametric approach used for multiple curves with shape-related nonlinear variation. This approach allows us to capture the common features contained in the shape of the functions and at the same time characterize the nonlinear variability with a few interpretable parameters. These parameters provide an informative summary of the curves and can be used to make a further analysis with macroeconomic variables. Implied risk aversion function and utility function can also be derived. The method is demonstrated with the European options and returns values of the German stock index DAX.

Probability weighting functions implied in options prices

The empirical pricing kernels estimated from index options are non-monotone (Rosenberg and Engle, 2002; Bakshi, Madan, and Panayotov, 2010) and the corresponding risk-aversion functions can be negative (Aït-Sahalia and Lo, 2000; Jackwerth, 2000). We show theoretically that these and several other properties of empirical pricing kernels are consistent with rank-dependent utility model with probability weighting function, which overweights tail events. We also estimate the pricing kernels nonparametrically from the Standard & Poor's 500 index options and construct empirical probability weighting functions. The estimated probability weights typically have the inverse-S shape, which overweights tail events and is widely supported by the experimental decision theory.

An integrated design of sustainable land use and transportation system with uncertainty in future population

This article proposes a new model for integrated design of a sustainable land use and transportation system with uncertainty in future population. In the proposed model, the future population in the urban area is assumed to be a random variable with a given probability distribution. A set of chance constraints with regard to road capacity expansion, housing and employment supplies and environmental impacts is incorporated to consider the sustainability of urban land development and transportation infrastructure improvement. The proposed model is formulated as a two-stage robust optimisation problem. The first stage of the proposed model (before the future urban population is realised) is to optimise the land use and transportation system by maximising a robust risk-averse objective function subject to various chance constraints for consideration of the system sustainability. The second stage of the proposed model, after the future population has been determined, is a scenario-based stochastic location and route choice equilibrium problem. A heuristic solution algorithm, which is a combination of penalty function method, simulated annealing method and Gauss-Seidel decomposition approach is developed to solve the proposed model. An illustrative example is given to show the application of the proposed model and solution algorithm. The findings show that the integrated design of the sustainable land use and transportation system depends very much on the level of uncertainty in future population, capital budget for urban development, and confidence levels of the chance constraints.

Rare Events and Investor Risk Aversion: Evidence from Crude Oil Options

The effect of four distinct market events on investor risk aversion is evaluated using options data on the WTI crude oil futures contract during the 2007-2011 period. The risk aversion function and the stochastic discount factor (SDF) are estimated using parametric approaches before and after each event in a fifteen-day event window. Events of a more persistent, regime change type lead to flatter risk aversion functions (i.e., less decreasing in wealth) and SDFs. In contrast, events of a transitory nature lead to steeper risk aversion functions (i.e., decreasing more rapidly in wealth) and SDFs. Practical implications are drawn.

Delegated portfolio management with socially responsible investment constraints

We consider the problem of how to establish compensation for a portfolio manager who is required to restrict the investment set, for example, because of socially responsible screening. This is a problem of delegated portfolio management, where the reduction of investment opportunities to the subset of sustainable assets involves a loss in expected earnings for the portfolio manager, compensated by the investor through an extra bonus on the realized return. Under simple assumptions on the investor, manager and market, we compute the optimal bonus as a function of the manager's risk aversion and expertise, and of the impact of portfolio restriction on the mean-variance efficient frontier.

Probability Weighting Functions Implied in Options Prices

The empirical pricing kernels estimated from index options are non-monotone (Rosenberg and Engle, 2002; Bakshi, Madan, and Panayotov, 2010) and the corresponding risk aversion functions can be negative (Ait-Sahalia and Lo, 2000; and Jackwerth, 2000). We show theoretically that these and several other properties of empirical pricing kernels are consistent with rank-dependent utility model with probability weighting function which over-weights tail events. We also estimate the pricing kernels non-parametrically from the S&P 500 index options and construct empirical probability weighting functions. The estimated probability weights typically have the inverse-S shape which overweights tail events and is widely supported by the experimental decision theory.