Absolute Risk Aversion Function Research Articles

Calibration of quantum decision theory: aversion to large losses and predictability of probabilistic choices

We present the first calibration of quantum decision theory (QDT) to a dataset of binary risky choice. We quantitatively account for the fraction of choice reversals between two repetitions of the experiment, using a probabilistic choice formulation in the simplest form without model assumption or adjustable parameters. The prediction of choice reversal is then refined by introducing heterogeneity between decision makers through their differentiation into two groups: ‘majoritarian’ and ‘contrarian’ (in proportion 3:1). This supports the first fundamental tenet of QDT, which models choice as an inherent probabilistic process, where the probability of a prospect can be expressed as the sum of its utility and attraction factors. We propose to parameterize the utility factor with a stochastic version of cumulative prospect theory (logit-CPT), and the attraction factor with a constant absolute risk aversion function. For this dataset, and penalising the larger number of QDT parameters via the Wilks test of nested hypotheses, the QDT model is found to perform significantly better than logit-CPT at both the aggregate and individual levels, and for all considered fit criteria for the first experiment iteration and for predictions (second ‘out-of-sample’ iteration). The distinctive QDT effect captured by the attraction factor is mostly appreciable (i.e. most relevant and strongest in amplitude) for prospects with big losses. Our quantitative analysis of the experimental results supports the existence of an intrinsic limit of predictability, which is associated with the inherent probabilistic nature of choice. The results of the paper can find applications both in the prediction of choice of human decision makers as well as for organizing the operation of artificial intelligence.

Risk investment decisions within the deterministic equivalent income model

Purpose The purpose of this paper is to establish a deterministic equivalent income model (DEIM) based on the risk cost (RC) and risk aversion of investors. The model fully considers both subjective and objective factors that affect risk investment and reasonably evaluates risk investment schemes to choose the correct investment scheme and gain greater investment returns. Design/methodology/approach The utility function is used to measure the extent to which an investor is satisfied by investment returns in various scenarios. Risk aversion expresses subjective attitude of investors to risk. RC represents risk loss in currency. This methodology is based on risk aversion function, utility function and RC theory to establish DEIM. Findings This study shows that investors with different risk preferences have different certainty equivalent returns (CER), so their choices of investment options change accordingly. Practical implications In this paper, the authors use DEIM to test an investment case and conclude that the CER and investment scheme both change with different risk preferences. At the same time, case analysis shows that DEIM is reasonable and stable when evaluating risk investment schemes. Originality/value In this study, the authors innovate by introducing both the RC and risk aversion degree into risk investment schemes evaluation and by deriving a utility function from the absolute risk aversion function to build a utility decision matrix and establish DEIM. The model combines the subjective and objective factors that influence risk investment decisions.

Testing for changes in option-implied risk aversion

Purpose – The purpose of this paper is to investigate and test for changes in investor risk aversion and the stochastic discount factor (SDF) using options data on the West Texas Intermediate crude oil futures contract during the 2007-2011 period. Design/methodology/approach – Risk aversion functions and SDFs are estimated using parametric approaches before and after four specific dates of interest. The dates are: the summer 2008 end of the bull market regime; the late 2008 credit freeze trough; the BP Deepwater Horizon explosion; and the Libyan uprising. Findings – Absolute risk aversion functions and SDFs are significantly flatter (less decreasing in wealth) after the end of the bull market and the credit freeze trough. After these two market reversals, oil market participants were less risk-averse for low levels of wealth but more risk-averse for high wealth levels. Oil market investors also increased their valuation of anticipated future wealth in average states of nature relative to very high or very low-asset return states after reversals. The BP explosion and the Libyan uprising led to steeper risk aversion functions (decreasing more rapidly in wealth) and SDF. Oil market investors were more risk-averse for lower future wealth, but less risk-averse for higher future wealth. Oil market investors increased their valuation of anticipated future wealth in extreme states of nature relative to average states of nature after both dates. Originality/value – Documenting statistically and economically significant changes in oil market investors’ attitude toward risk and inter-temporal appetite for risk in relation to changes in financial and political conditions.

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.

Existence of Capital Market Equilibrium in the Presence of Herding and Feedback Trading

This paper attempts to establish the existence of equilibrium, in an asset market inhabited by two representative investors with different risk aversions. In order to capture heterogeneity in information and wealth, the paper segments the investor population into two: (i) Individual investors and (ii) Institutional investors. Based on prior literature, the present study posits that Institutional investors demonstrate rational intentional herding and positive feedback trading (buy when the markets rise and sell when it falls) and individual investors demonstrate negative feedback trading (vice versa). In other words, when the markets are (monotonically) increasing, institutional investors, expecting the trend to continue would buy more, thus demonstrating decreasing absolute risk aversion. Similarly, when the market is (monotonically) decreasing he will try to stem his loss as soon as possible, demonstrating increasing absolute risk aversion. Such an investment behavior is captured in a power utility function. Further, negative feedback trading by individual investors implies that when market is (monotonically) increasing individual investors, expecting the trend to reverse, would sell. Thus demonstrating increasing absolute risk aversion. And when the markets are (monotonically) decreasing, they would hold on to their investments expecting better times to come, thus depicting decreasing absolute risk aversion. Such investment behavior is captured by a quadratic utility function. Given their wealth and investment behavior, the two investor groups would trade with each other such that the market clears. To the best of our knowledge this is the first paper that proposes a asset pricing model that not only allows for behavioural biases but also for heterogeneous agents who are affected differently by these biases. This paper establishes the bounds for the absolute risk aversion function and the shadow rate of interest at which the two investor groups will lend money to each other to enable trading and market clearing. For reasonable endowments and presumed behavioural biases as implied by the chosen utility function, a numerical example at the end of this paper shows that the market clearing interest rate (at which the investors would lend to and borrow from each other) occurs between 15.5% and 28.05%.

Properties of the utility function: A market-based analysis

Using US market data, this paper sheds new empirical light on properties of the utility function. In particular, employing theoretical relations between Stochastic Discount Factors, state prices, and state probabilities, we are successful in recovering the following four functions: (i) Absolute Risk Aversion (ARA); (ii) Absolute Risk Tolerance (ART); (iii) Absolute Prudence (AP); and (iv) Absolute Temperance (AT). Our statistical analysis points out, unequivocally, that the ARA function is decreasing and convex, the ART function is convex, AT is greater than ARA, and the AP function is not decreasing. These empirical results are analyzed in light of established theory concerning, inter-alia, precautionary saving and prudence as well as the way risk attitudes are affected by the presence of “background risks” and by investors’ investment horizon.

Pricing Kernels with Stochastic Skewness and Volatility Risk

I derive pricing kernels in which the market volatility is endogenously determined. Using the Taylor expansion series of the representative investor's marginal utility, I show that the price of market volatility risk is restricted by the investor's risk aversion and skewness preference. The risk aversion is estimated to be between two and five and is significant. The price of the market volatility is negative. Consistent with economic theory, I find that the pricing kernel decreases in the market index return and increases in market volatility. The projection of the estimated pricing kernel onto a polynomial function of the market return produces puzzling behaviors, which can be observed in the pricing kernel and absolute risk aversion functions. The inclusion of additional terms in the Taylor expansion series of the investor's marginal utility produces a pricing kernel function of market stochastic volatility, stochastic skewness, and stochastic kurtosis. The prices of risk of these moments are restricted by the investor's risk aversion, skewness preference, and kurtosis preference. The prices of risk of these moments should not be confused with the price of risk of powers of the market return, such as coskewness and cokurtosis. This paper was accepted by Wei Xiong, finance.

Pricing Kernels with Stochastic Skewness and Volatility Risk

I derive pricing kernels in which the market volatility is endogenously determined. Using the Taylor expansion series of the representative investor’s marginal utility, I show that the price of market volatility risk is restricted by the investor’s risk aversion and skewness preference. The risk aversion is estimated to be between two and five and is significant. The price of the market volatility is negative. Consistent with economic theory, I find that the pricing kernel decreases in the market index return and increases in market volatility. The projection of the estimated pricing kernel onto a polynomial function of the market return produces puzzling behaviors, which can be observed in the pricing kernel and absolute risk aversion functions. The inclusion of additional terms in the Taylor expansion series of the investor’s marginal utility produces a pricing kernel function of market stochastic volatility, stochastic skewness, and stochastic kurtosis. The prices of risk of these moments are restricted by the investor’s risk aversion, skewness preference, and kurtosis preference. The prices of risk of these moments should not be confused with the price of risk of powers of the market return, such as co-skewness and co-kurtosis.

Risk Aversion Behavior. Relationships Between Risk Aversion, Prudence And Cautiousness

This paper defines decreasing absolute risk aversion in purely behavioral termswithout any assumption of differentiability and shows that a strictly increasing and riskaverse utility function with decreasing absolute risk aversion is necessarily differentiable withan absolutely continuous derivative. A risk averse utility function has decreasing absolute riskaversion if and only if it has a decreasing absolute risk aversion density, and if and only if thecumulative absolute risk aversion function is increasing and concave. This leads to acharacterization of all such utility functions. Analogues of these results also hold forincreasing absolute and for increasing and decreasing relative risk aversion.

Monotone Risk Aversion

This paper defines decreasing absolute risk aversion in purely behavioral terms without any assumption of differentiability and shows that a strictly increasing and risk averse utility function with decreasing absolute risk aversion is necessarily differentiable with an absolutely continuous derivative. A risk averse utility function has decreasing absolute risk aversion if and only if it has a decreasing absolute risk aversion density, and if and only if the cumulative absolute risk aversion function is increasing and concave. This leads to a characterization of all such utility functions. Analogues of these results also hold for increasing absolute and for increasing and decreasing relative risk aversion.

Risky Production Processes and Demand for Preventive Safety Measures Under Uncertainty

This paper analyzes the behavior of a competitive risk-averse firm that has to choose the optimal level of labor and preventive safety measures. If the firm's injury function includes a random component, then the firm is characterized by a lower demand for labor and by a higher demand for safety measures with respect to a firm whose injury risk is completely under its control. The comparative statics show that shifts in the parameters of the risk distribution will have effects that depend on the absolute risk-aversion functions. The introduction of safety standards will prompt a reduction in the demand for labor while a tightening of the compensation system will have ambiguous effects on the demand for the two inputs.

The New Soviet Incentive Model: Comment

Elaborating on the analysis of Weitzman and of Snowberger, this article develops four additional comparative statics results for the Soviet incentive model. The article demonstrates how the optimal change in the firm's self-selected target in response to changes in the central planners' parameters crucially depends on properties of the absolute risk aversion function. Extending the model to include preferences over a proportional discretionary bonus, the article shows how the relative risk aversion becomes a key factor.