Risk Aversion Function Research Articles

Estimating Risk Attitudes in Conventional and Artefactual Lab Experiments: The Importance of the Underlying Assumptions

In this paper the authors assess the importance of sample type in the estimation of risk preferences. The authors elicit and compare risk preferences from student subjects and subjects drawn from the general population, using the multiple price list method devised by Holt and Laury (Risk aversion and incentive effects, 2002). The authors find evidence suggesting that under Rank Dependent Utility and an expo-power function, students exhibit similar risk attitudes to subjects drawn from the general population. However, when the authors assume an incorrect characterization of risk preferences, in particular they adopt the framework of Expected Utility theory and a Constant Relative Risk Aversion function, their estimation results lead to erroneous inferences. In this case, students are on average risk averse, while subjects drawn from the general population exhibit risk loving preferences. The results have implications for economic policy making under uncertainty.

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%.

Coherent risk measures in real estate investment

PurposeThe purpose of this paper is to propose a new way to measure risk in real estate investment, which departs from traditional statistical methodology borrowed from finance.Design/methodology/approachAn argument is advanced for the use of so‐called coherent risk measures for real estate investment decisions. Central to this class of measure is the computation of spectral risk, a measure that covers the spectrum of an individual's risk aversion function. Intuition being hard to observe, empirical data from two large databases are presented as support.FindingsAt the heart of the controversy is a discussion of the nature of risk and how it should be measured. This paper seeks common ground where peace may be made between these two warring factions. Scenarios are tested wherein different risk aversion functions are used to compute spectral risk for different sectors. Ex‐post analysis shows that reasoning of this nature can lead to improved risk‐adjusted investment results.Practical implicationsThe route to finding an appropriate risk measure for real estate investment has been tortuous. It is not certain that the destination has been reached. Complicating the task is a considerable gap between academic and practitioner methodology, the former relying on the mathematics of objective probability, the latter dwelling quite successfully in a habitat of subjective risk measures.Social implicationsIt is widely accepted that risk represents a cost to society. Real estate, as a repository of roughly half the world's wealth, can be viewed as having risk of a structurally different nature. The better understanding of this risk reduces the cost to society.Originality/valueFor practitioners, spectral measures offer formal support for something they have been doing their entire careers: evaluating risk subjectively. The simple dot product of weights and ordered outcomes is an extension of the widely used “Best case – Worse case – Most likely” methodology that has served professionals well for decades. Perhaps a by‐product of spectral measures is to bring academics and field gladiators closer together. If there is merit to a new way of thinking about risk independent of its implementation, real estate investment could benefit from that thinking. Among those who doubt that everything is “normal” are those who believe that previous attempts to explain risk in real estate have fallen far short of the mark. The challenge to those still not satisfied by the spectral approach is to offer an alternative that represents both a departure from and an improvement of the old methods.

Option Pricing with Realistic Arch Processes

Option pricing should be 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. This paper presents a realistic and complete, first principles,'' computation of option prices. The underlying is modeled by 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 P measure) and allows to reproduce all empirical statistics, from 1 day to 1 year. For a given risk aversion function, the change of measure from P to the risk-neutral measure Q can be derived rigorously along each path drawn from the process, namely the Radon-Nikodym derivative dQ/dP can be constructed. A small DT expansion allows to compute explicitly the change of measure. Finally, the European option price is obtained as the expectation in P of the discounted payoff with a weight given by the change of measure dQ/dP. This procedure is implemented in a Monte Carlo simulation, and allows to compute the option prices, without further adjustable parameters. The empirical study uses European put and call options on the SP500 from 1996 to 2010. The computed implied volatility surfaces are compared with the empirical surfaces, in particular with respect to the level, smile, smirk and term structure. All the main characteristics of the implied volatility surfaces are correctly reproduced. Further points concern the respective role of the P and Q measures, the distribution of the terminal prices in both measures, the negligible effect of the risk aversion and drift premium, and the empirical validity of the no-arbitrage argument. Finally, simplifications of the present exact scheme are suggested.

Development and Validation of a Utility Weighting Function for the Patient-Oriented Prostate Utility Scale (PORPUS)

Previously, we developed a prostate cancer (PC)-specific health state classification system, the Patient Oriented Prostate Utility Scale (PORPUS). In this study, we developed a scoring system to allow indirect calculation of utilities from the PORPUS. We interviewed 234 PC outpatients, including those with newly diagnosed and metastatic disease, to obtain rating scale (RS) values on 4 to 6 levels of each of the 10 attributes of the PORPUS, and on 10 corner states (worst level on 1 attribute, best on 9). Patients also completed standard gamble (SG) and RS tasks on 4 multiattribute states (impotence and pain corner states, mild and severe PC symptoms). We used the RS and SG scores for multiattribute states to determine a risk aversion function for mapping values to utilities. We then tested 15 different strategies to estimate the multiattribute utility function (MAUF), using the single attribute disutilities for each level of the 10 PORPUS attributes, and the disutilities for the corner states. The root mean squared error (RMSE) of prediction of the SG on the 4 multiattribute states was used to identify the optimal strategy and scoring system. The optimal strategy gave an RMSE of 0.06. Comparison of mean MAUF-predicted utilities to directly elicited SG utilities for the 2 multiattribute states from patients in 2 previously published studies (n = 248 and n = 141) supported the validity of the MAUF. The scoring system together with the PORPUS comprise an indirect utility instrument, the PORPUS-U, which can be used in clinical and research settings.

Decomposing the Cross Derivatives of a Multiattribute Utility Function into Risk Attitude and Value

The cross derivatives of a multiattribute utility function play an important role in the choice between multivariate lotteries and in multiattribute Taylor expansions of the utility function. This paper decomposes the cross derivatives into two components: the derivatives of a single-attribute utility function over value and the cross derivatives of the value function. This approach provides a simple method for reasoning about the signs of the cross derivatives of a multiattribute utility function using derivatives of a univariate utility function and the properties of the value function. To illustrate the approach, we relate the multivariate risk aversion concept, which involves the mixed partial derivative of the utility function, to the Arrow–Pratt risk aversion function. We show that for additive value functions, a decision maker is multivariate risk averse if and only if he is risk averse over value in the Arrow–Pratt sense. For other value functions, however, a decision maker can be risk averse or risk seeking over value and still exhibit multivariate risk aversion. The approach also derives the conditions on the value function that relate two important classes of utility functions: single attribute utility functions whose derivatives alternate in sign and multiattribute utility functions whose cross derivatives alternate in sign. These two classes are widely used in practice and form the basis of univariate and multivariate stochastic dominance. Several examples illustrate the approach.

A Moment Expansion of Downside Risk Measures

A decomposition of a sub-class of spectral risk measures is introduced in terms of L-moments accounting for geometric characteristics of the return distribution similar to the ones described by the ordinary moments. The decomposition characterizes completely the spectral risk measures with square-integrable risk aversion functions and can be regarded as a link between higher-order moment risk and downside risk measures. Coherent approximations based on only a few L-moments can be successfully constructed for continuous risk aversion functions and can be applied to problems in portfolio theory to analyze the incremental impact of higher order moments on optimal allocations.

What Can the Option-Implied Risk Aversion Really Tell Us?

In this paper we raise a question on the reliability of option implied risk aversion. We prove that given any number of options, there exist numerous risk aversion functions which are piecewise constant, have only two values, either a lower bound or an upper bound on the true risk aversion function, and are consistent with the prices of all these options. Similar results are proved with respect to the true risk aversion function's derivatives. These results show that the errors in the estimation of risk aversion and its derivatives caused purely by market incompleteness can be infinitely large and it is extremely difficult to ensure that the risk aversion function we extract from option prices is the true one.

Endogenous Risk Measures

We present a methodology that allows endogenous derivation of the moments of the probability distributions. In doing so, we, present an alternative objective function and alternative concept of risk aversion. In addition, we show that the risk measure depends on the preferences. Moreover, we show that a higher level of risk aversion yields higher values of the risk measure.

ENDOGENOUS INEQUALITY OF NATIONS THROUGH FINANCIAL ASSET MARKET INTEGRATION

The paper analyzes an endogenous mechanism leading perfectly symmetric economies to diverge in the long run after unifying their financial asset markets. The standard growth model with overlapping generations of consumers (OLG) is extended to include uncertainty and a financial asset. In the absence of an international asset market, the two autarkic economies converge to the same globally attracting steady state under rational expectations dynamics. When the two asset markets are unified internationally, additional asymmetric steady states appear, implying that the steady state with equal levels of capital becomes unstable, causing symmetry breaking. The paper derives general sufficient conditions for a saddle node bifurcation of the symmetric steady state. A numerical example shows that these effects occur, in particular when the production function and the function of absolute risk aversion are isoelastic.

A method for determining risk aversion functions from uncertain market prices of risk

In Gzyl and Mayoral (2008) we developed a technique to solve the following type of problems: How to determine a risk aversion function equivalent to pricing a risk with a load, or equivalent to pricing different risks by means of the same risk distortion function. The information on which the procedure is based consists of the market prices of the risk. Here we extend that method to cover the case in which there may be uncertainties in the market prices of the risks.

The limits to risk aversion: Part 1. The point of indiscriminate decision

The paper uses utility theory to investigate how much should be spent to avert all costs from an industrial accident apart from direct human harm. These “environmental costs” will include those of evacuation, clean-up and business disruption. Assuming the organisation responsible will need to pay such costs, the difference between its expected utility with and without an environmental protection system constitutes a rational decision variable for whether or not the scheme should be installed. The value of utility is dependent on the coefficient of relative risk aversion, “risk-aversion” for short. A model of an organisation's decision-making process has been developed using the ABCD model, linking the organisation's assets, A, the cost of the protection scheme, B, the cost of consequences, C, and the expected utility difference with and without the scheme, D. Increasing the organisation's risk-aversion parameter will tend to make it less reluctant to invest in a protection system, but can bring about such investment only when the scheme is relatively close to financial break-even. For such borderline schemes, the amount the organisation is prepared to spend on the protection system will rise as the risk-aversion increases. The ratio of this sum to the break-even cost is named the “Limiting Risk Multiplier”, the maximum value of which is governed by the maximum feasible value of risk-aversion. However, the mathematical model shows that increasing the risk-aversion will reduce the clarity of decision making generally. Although the reluctance to invest in a protection scheme may change sign and turn into a positive desire to invest as the risk-aversion increases, the absolute value of this parameter is a continuously decreasing function of risk-aversion, tending asymptotically to zero. As a result, discrimination will gradually diminish, being lost altogether at the “point of indiscriminate decision”. Here the decision maker will be able to distinguish neither advantage in installing the scheme nor disadvantage in installing its inverse. There is a close correspondence between this mathematically predicted state and that of panic, where an individual has become so fearful that his actions become random. The point of indiscriminate decision provides a natural upper bound for the value of risk-aversion. This bounds the Limiting Risk Multiplier in turn, and so sets an objective upper limit on the amount that it is rational to spend on an environmental protection system.

Reply to Comment on "Significance of "high probability/low damage" versus "low probability/high damage" flood events" by C. M. Rheinberger (2009)

Abstract. In a comment to our recently published paper on the "Significance of "high probability/low damage" versus "low probability/high damage" flood events" (Merz et al., 2009), C. M. Rheinberger questions the use of relative damage as a suitable indicator for risk aversion and the use of the resulting risk aversion functions in judging flood mitigation measures. While the points of criticism are important and should be accounted for, most of these points are considered in our original paper. More importantly, we do not agree with the conclusion that the use of relative damage as indicator for risk aversion is generally not appropriate in decision making about flood mitigation measures.

An Enjoyable Life Puzzling Over Modern Finance Theory

This is a terse account of group creation of modern finance theory; and a sampling of my prosaic autobiographical investing and consulting for nonprofit academies. Eschewing 1900 Bachelier and 1905 Einstein white noise randomness, my martingale version of market micro efficiency invoked no violation of economic law. My attempts to establish pricing theory for options fell a bit short of the Black-Scholes-Merton Holy Grail. For life cycle investing, mathematicians’ maximum growth Kelly criterion was debunked, as were vulgar notions that necessarily riskiness is averaged downward for long-term investors. Popular Markowitz-Tobin quadratic programming was shown to hold generically only for smallest price variations or for unrealistic risk-aversion functions. Because economic history at best obeys only quasi-stationary probabilities, no sure-thing formulas will ever be definable. Excess returns—excess “alphas”—can result only from early new “insider” knowledge, however acquired—legally or illegally. Boo hoo.

Target Reliability Levels for Design and Assessment of Onshore Natural Gas Pipelines

This paper proposes a set of reliability targets that can be used in the design and assessment of onshore natural gas pipelines. The targets were developed as part of a PRCI-sponsored project that aims to establish reliability-based methods as a viable alternative for pipeline design and assessment. The proposed targets are calibrated to meet risk levels that are considered widely acceptable. The proposed criteria are based on a detailed consideration of both societal and individual risk criteria. Two societal risk criteria were considered: the first based on a fixed expectation of the number of fatalities and the second based on a risk aversion function as characterized by a F/N relationship. Societal risk criteria were calibrated to match or exceed the average safety levels implied by current codes. Individual risk criteria were based on published tolerable levels. The target reliability levels corresponding to the three criteria are presented and a recommended set of targets is presented.

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.

Implied Risk Preferences from Option Prices: Evidence from KOSPI 200 Index Options

In this paper, we investigated the risk averse ness of KOSPI 200 option investors with very flexible risk preference structure. Contrary to the most of previous research either assuming a time-invariant underlying asset return distribution or assuming a well-known functional form for the underlying utility functions. we directly assume functional forms for Investors’risk aversion functions. With the direct specification on the risk aversion functions themselves. we can avoid the possibility 이 suffering from Internal inconsistency and of obtaining misleading risk aversion functions. From our empirical results using KOSPI 200 Index option prices from 1997 through 2006. we discovered that the investors' relative risk aversions exhibit ‘sharply decreasing' across wealth. In addition, our Implied subjective PDFs are found to more accurately forecast the distribution of realization than both the risk neutral PDFs and implied subjective PDFs from previous methods. For the robustness of our empirical results, we test the effects of estimation errors In the expected risk premium, and of financial crisis in the late of 1990s.

Pareto Optimality and Equilibrium in an Insurance Market

The concept of economic equilibrium under uncertainty is applied to a model of insurance market where, in distinction to the classic Borch’s model of a reinsurance market, risk exchanges are allowed between the insurer and each insured only, not among insureds themselves. Conditions characterizing an equilibrium are found. A variant of the conditions, based on the Pareto optimality notion and involving risk aversion functions of the agents, is derived. An existence theorem is proved. Computation of the market premiums and optimal indemnities is illustrated by an example with exponential utility functions.

Pareto Optimality and Equilibrium in an Insurance Market

The concept of economic equilibrium under uncertainty is applied to a model of insurance market where, in distinction to the classic Borch’s model of a reinsurance market, risk exchanges are allowed between the insurer and each insured only, not among insureds themselves. Conditions characterizing an equilibrium are found. A variant of the conditions, based on the Pareto optimality notion and involving risk aversion functions of the agents, is derived. An existence theorem is proved. Computation of the market premiums and optimal indemnities is illustrated by an example with exponential utility functions.