Chapter 2 — Risk and Risk Aversion
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Portfolio Selection Under Exponential and Quadratic Utility
Portfolio Selection Under Exponential and Quadratic Utility Steven T. Buccola The production or marketing portfolio that is optimal under the assumption of quadratic utility may or may not be optimal under the assumption of exponential utility. In certain cases, the necessary and sufficient condition for an identical solution is that absolute risk aversion coefficients associated with the two utility functions be the same. In other cases, equality of risk aversion coefficients is a sufficient condition only. A comparison is made between use of exponential and quadratic utility in the analysis of a California farmer's marketing problem. A large number of utility functional forms difference. Under other conditions, the have been proposed for use in selecting opti- choice has a significant impact on optimal mal portfolios of risky prospects [Tsaing]. For behavior. purposes of both theoretical and applied eco- Suppose an analyst is studying a continu- nomic research, attention has primarily been ously divisible set of alternative production directed to the quadratic and exponential or marketing options offering approximately forms. After enjoying widespread use in the normally-distributed returns, and that he 1950s and 1960s, the quadratic form came wishes to select the portfolio of these options under criticism by Pratt and others and lost with highest expected utility. The analyst is much of its reputability. In contrast the expo- considering using an exponential utility func- nential form has become increasingly popu- tion U = -exp(-Xx), X > 0, or a quadratic lar, due in large measure to its mathematical utility function U = x -vx,v, > 0, x < /2v, tractability [Attanasi and Karlinger; O'Con- where U is utility and x is wealth. -
Utility with Decreasing Risk Aversion
144 UTILITY WITH DECREASING RISK AVERSION GARY G. VENTER Abstract Utility theory is discussed as a basis for premium calculation. Desirable features of utility functions are enumerated, including decreasing absolute risk aversion. Examples are given of functions meeting this requirement. Calculating premiums for simplified risk situations is advanced as a step towards selecting a specific utility function. An example of a more typical portfolio pricing problem is included. “The large rattling dice exhilarate me as torrents borne on a precipice flowing in a desert. To the winning player they are tipped with honey, slaying hirri in return by taking away the gambler’s all. Giving serious attention to my advice, play not with dice: pursue agriculture: delight in wealth so acquired.” KAVASHA Rig Veda X.3:5 Avoidance of risk situations has been regarded as prudent throughout history, but individuals with a preference for risk are also known. For many decision makers, the value of different potential levels of wealth is apparently not strictly proportional to the wealth level itself. A mathematical device to treat this is the utility function, which assigns a value to each wealth level. Thus, a 50-50 chance at double or nothing on your wealth level may or may not be felt equivalent to maintaining your present level; however, a 50-50 chance at nothing or the value of wealth that would double your utility (if such a value existed) would be equivalent to maintaining the present level, assuming that the utility of zero wealth is zero. This is more or less by definition, as the utility function is set up to make such comparisons possible. -
Stochastic Dominance Under Independent Noise
Stochastic Dominance Under Independent Noise∗ § Luciano Pomatto† Philipp Strack‡ Omer Tamuz May 21, 2019 Abstract Stochastic dominance is a crucial tool for the analysis of choice under risk. It is typically analyzed as a property of two gambles that are taken in isolation. We study how additional independent sources of risk (e.g. uninsurable labor risk, house price risk, etc.) can affect the ordering of gambles. We show that, perhaps surprisingly, background risk can be strong enough to render lotteries that are ranked by their expectation ranked in terms of first-order stochastic dominance. We extend our re- sults to second order stochastic dominance, and show how they lead to a novel, and elementary, axiomatization of mean-variance preferences. 1 Introduction A choice between risky prospects is often difficult to make. It is perhaps even harder to predict, since decision makers vary in their preferences. One exception is when the prospects are ordered in terms of stochastic dominance. In this case, a standard prediction is that the dominant option is chosen. For this reason, stochastic dominance has long been seen as a central concept in decision theory and the economics of information, and remains an area of active research (see, e.g. Müller et al., 2016; Cerreia-Vioglio et al., 2016, among others). More generally, stochastic dominance is an important tool for non-parametric comparison of distributions. arXiv:1807.06927v5 [math.PR] 20 May 2019 In the typical analysis of choices under risk, stochastic dominance is studied as a property of two prospects X and Y that are taken in isolation, without considering other risks the decision maker might be facing. -
Statistical Risk Estimation for Communication System Design: Results of the HETE-2 Test Case
IPN Progress Report 42-197 • May 15, 2014 Statistical Risk Estimation for Communication System Design: Results of the HETE-2 Test Case Alessandra Babuscia* and Kar-Ming Cheung* ABSTRACT. — The Statistical Risk Estimation (SRE) technique described in this article is a methodology to quantify the likelihood that the major design drivers of mass and power of a space system meet the spacecraft and mission requirements and constraints through the design and development lifecycle. The SRE approach addresses the long-standing challenges of small sample size and unclear evaluation path of a space system, and uses a combina- tion of historical data and expert opinions to estimate risk. Although the methodology is applicable to the entire spacecraft, this article is focused on a specific subsystem: the com- munication subsystem. Using this approach, the communication system designers will be able to evaluate and to compare different communication architectures in a risk trade-off perspective. SRE was introduced in two previous papers. This article aims to present addi- tional results of the methodology by adding a new test case from a university mission, the High-Energy Transient Experiment (HETE)-2. The results illustrate the application of SRE to estimate the risks of exceeding constraints in mass and power, hence providing crucial risk information to support a project’s decision on requirements rescope and/or system redesign. I. Introduction The Statistical Risk Estimation (SRE) technique described in this article is a methodology to quantify the likelihood that the major design drivers of mass and power of a space system meet the spacecraft and mission requirements and constraints through the design and development lifecycle. -
The Case Against Power Utility and a Suggested Alternative: Resurrecting Exponential Utility
Munich Personal RePEc Archive The Case Against Power Utility and a Suggested Alternative: Resurrecting Exponential Utility Alpanda, Sami and Woglom, Geoffrey Amherst College, Amherst College July 2007 Online at https://mpra.ub.uni-muenchen.de/5897/ MPRA Paper No. 5897, posted 23 Nov 2007 06:09 UTC The Case against Power Utility and a Suggested Alternative: Resurrecting Exponential Utility Sami Alpanda Geoffrey Woglom Amherst College Amherst College Abstract Utility modeled as a power function is commonly used in the literature despite the fact that it is unbounded and generates asset pricing puzzles. The unboundedness property leads to St. Petersburg paradox issues and indifference to compound gambles, but these problems have largely been ignored. The asset pricing puzzles have been solved by introducing habit formation to the usual power utility. Given these issues, we believe it is time re-examine exponential utility. Exponential utility was abandoned largely because it implies increasing relative risk aversion in a cross-section of individuals and non- stationarity of the aggregate consumption to wealth ratio, contradicting macroeconomic data. We propose an alternative preference specification with exponential utility and relative habit formation. We show that this utility function is bounded, consistent with asset pricing facts, generates near-constant relative risk aversion in a cross-section of individuals and a stationary ratio of aggregate consumption to wealth. Keywords: unbounded utility, asset pricing puzzles, habit formation JEL Classification: D810, G110 1 1. Introduction Exponential utility functions have been mostly abandoned from economic theory despite their analytical convenience. On the other hand, power utility, which is also highly tractable, has become the workhorse of modern macroeconomics and asset pricing. -
Pareto Utility
Theory Dec. (2013) 75:43–57 DOI 10.1007/s11238-012-9293-8 Pareto utility Masako Ikefuji · Roger J. A. Laeven · Jan R. Magnus · Chris Muris Received: 2 September 2011 / Accepted: 3 January 2012 / Published online: 26 January 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com Abstract In searching for an appropriate utility function in the expected utility framework, we formulate four properties that we want the utility function to satisfy. We conduct a search for such a function, and we identify Pareto utility as a func- tion satisfying all four desired properties. Pareto utility is a flexible yet simple and parsimonious two-parameter family. It exhibits decreasing absolute risk aversion and increasing but bounded relative risk aversion. It is applicable irrespective of the prob- ability distribution relevant to the prospect to be evaluated. Pareto utility is therefore particularly suited for catastrophic risk analysis. A new and related class of generalized exponential (gexpo) utility functions is also studied. This class is particularly relevant in situations where absolute risk tolerance is thought to be concave rather than linear. Keywords Parametric utility · Hyperbolic absolute risk aversion (HARA) · Exponential utility · Power utility M. Ikefuji Institute of Social and Economic Research, Osaka University, Osaka, Japan M. Ikefuji Department of Environmental and Business Economics, University of Southern Denmark, Esbjerg, Denmark e-mail: [email protected] R. J. A. Laeven (B) · J. R. Magnus Department of Econometrics & Operations Research, Tilburg University, Tilburg, The Netherlands e-mail: [email protected] J. R. Magnus e-mail: [email protected] C. -
Risk Aversion and Stochastic Dominance: a Revealed Preference Approach
Risk Aversion and Stochastic Dominance: A Revealed Preference Approach David M. Bruner† This version: December 2007 Abstract Theoretically, given a choice over two risky assets with equivalent expected returns, a risk averse expected utility maximizer should choose the second-order stochasti- cally dominant asset. We develop a theoretical framework that allows for decision error, which should decrease in risk aversion. We conduct an experiment using a risk preference elicitation mechanism to identify risk averse individuals and eval- uate the frequency that they choose the stochastically dominant of two lotteries. 75.76% of risk averse and 96.15% of very risk averse subjects chose at least 7 out of 10 dominant lotteries. Estimates of the effect of elicited risk aversion on the number of stochastically dominant lotteries chosen are positive and highly significant across specifications. The results suggest risk averse individuals make choices that satisfy stochastic dominance and violations are due, in large part, to decision error, which is decreasing in risk aversion. Keywords: stochastic dominance, risk, uncertainty, experiments JEL classification: C91, D81 †Department of Economics, University of Calgary, 2500 University Drive NW, SS 409, Calgary Alberta T2N 1N4, Canada. E-mail address: [email protected] I would like to thank Christopher Auld, John Boyce, Michael McKee, Rob Oxoby, and Nathaniel Wilcox for their many helpful comments and suggestions. This research was undertaken at the University of Calgary Behavioural and Experimental Economics Laboratory (CBEEL). Risk Aversion and Stochastic Dominance Bruner 1 Introduction This paper presents the results of an experiment intended to determine the frequency that risk averse individuals make choices that satisfy second-order stochastic dominance (SSD). -
Chapter 4 How Do We Measure Risk?
1 CHAPTER 4 HOW DO WE MEASURE RISK? If you accept the argument that risk matters and that it affects how managers and investors make decisions, it follows logically that measuring risk is a critical first step towards managing it. In this chapter, we look at how risk measures have evolved over time, from a fatalistic acceptance of bad outcomes to probabilistic measures that allow us to begin getting a handle on risk, and the logical extension of these measures into insurance. We then consider how the advent and growth of markets for financial assets has influenced the development of risk measures. Finally, we build on modern portfolio theory to derive unique measures of risk and explain why they might be not in accordance with probabilistic risk measures. Fate and Divine Providence Risk and uncertainty have been part and parcel of human activity since its beginnings, but they have not always been labeled as such. For much of recorded time, events with negative consequences were attributed to divine providence or to the supernatural. The responses to risk under these circumstances were prayer, sacrifice (often of innocents) and an acceptance of whatever fate meted out. If the Gods intervened on our behalf, we got positive outcomes and if they did not, we suffered; sacrifice, on the other hand, appeased the spirits that caused bad outcomes. No measure of risk was therefore considered necessary because everything that happened was pre-destined and driven by forces outside our control. This is not to suggest that the ancient civilizations, be they Greek, Roman or Chinese, were completely unaware of probabilities and the quantification of risk. -
Probability Density Function of Steady State Concentration in Two-Dimensional Heterogeneous Porous Media Olaf A
WATER RESOURCES RESEARCH, VOL. 47, W11523, doi:10.1029/2011WR010750, 2011 Probability density function of steady state concentration in two-dimensional heterogeneous porous media Olaf A. Cirpka,1 Felipe P. J. de Barros,2 Gabriele Chiogna,3 and Wolfgang Nowak4 Received 4 April 2011; revised 27 September 2011; accepted 8 October 2011; published 23 November 2011. [1] Spatial variability of hydraulic aquifer parameters causes meandering, squeezing, stretching, and enhanced mixing of steady state plumes in concentrated hot-spots of mixing. Because the exact spatial distribution of hydraulic parameters is uncertain, the spatial distribution of enhanced mixing rates is also uncertain. We discuss how relevant the resulting uncertainty of mixing rates is for predicting concentrations. We develop analytical solutions for the full statistical distribution of steady state concentration in two-dimensional, statistically uniform domains with log-hydraulic conductivity following an isotropic exponential model. In particular, we analyze concentration statistics at the fringes of wide plumes, conceptually represented by a solute introduced over half the width of the domain. Our framework explicitly accounts for uncertainty of streamline meandering and uncertainty of effective transverse mixing (defined at the Darcy scale). We make use of existing low-order closed-form expressions that lead to analytical expressions for the statistical distribution of local concentration values. Along the expected position of the plume fringe, the concentration distribution strongly clusters at its extreme values. This behavior extends over travel distances of up to tens of integral scales for the parameters tested in our study. In this regime, the uncertainty of effective transverse mixing is substantial enough to have noticeable effects on the concentration probability density function. -
Estimating Exponential Utility Functions
,~ j,;' I) ':;l) . G rio"". 0 • 0 (? _.0 o I') Q () l% a 9(;' o Q estimatiqg "ExPC?ttenti~1 ·:U····:t·····~·I·····I:·I·+u·· ;F;";;u::·:n6',.t5"'·;~:n·'.' ·s.. ·, .' ..., .'.: :','. ..•. ".. '..'. ;. ~':,,;'V~ ..' .'. o c:::::::> ' ,;", o o (b~s. ) EconOm:ics, .Statistics'·Oftd (OOfI,ati,tS /} o i_JS~~n;'~~t'~C:'T" " o .j o i) " (> C. "French " State Un I vers" I ty, and - ~~ ~ from Agric~ltur;al Economic Research, Janucrry 1978" \/01. 30, No. 1 ~~'~....' .. ' .. exponential.utility fun~tion for money has 10ng att:racted attentic;m fromAtheorists~~ because it exhibits\~oni"creasing absolute, risk aversion. j\lso, under certain con- ~ ditions, it generates an expected utility functi()n that is maximizable in a quadratic 1 program. However, "this "'functiona,l form presents 'estimation problems. Logarithmic • transformation of an exponential uti 1i,t,y, function does not conform to the Von Neumann Horgenste~ axioms. Hence', it cannot be used as ,.a basis for best fit in statistical analysis. A criterion is described that can be used to select a best-flt exponential uti 1ity.fun.ction, and its appl.ica,ti~!!c"",L~I.J~~~er utJ1ity an~,,~J~!s. is (~en~~~.~~r.~~ed ,~-».,.....'~\ ~ ....,.,." .,-~~ '1\, ~.... jt-~ .. -~..,q,·""·"~.""""11!"~""""'"""'\.'<'..".tft4._ ... 1' ,,''-,. < ~ i\ ,c,( ~--~""..-~,."".... .... ,~. '" ~. "~:"''''.jq''K>''''~~''l;:' ", " & ' (J 11' 1 i ,~ r.~ .... ' t ' .'~ >':J ~rr 1 C >J '¥ ~r ~1 f' if! 1 t::' ~ t,n t; ~,l IG " Canputerized simulation o Computers c Economic analysis Est imates 1\, Exponential functions " ~ogarlthm functions Risk ~Statistical analysis Utility routines 1711. IdenEifier./Open-Eaded Titnu (;:::; Absolute risk aversion Compute r app,l i cat i ens' Logari thmi c traosformatian Honey (/ Von Neumann-Morgenstern axioms )) ~ 12-A, 12-8 PB . -
Statistical Machine Learning: Introduction
Statistical Machine Learning: Introduction Dino Sejdinovic Department of Statistics University of Oxford 22-24 June 2015, Novi Sad slides available at: http://www.stats.ox.ac.uk/~sejdinov/talks.html Tom Mitchell, 1997 Any computer program that improves its performance at some task through experience. Kevin Murphy, 2012 To develop methods that can automatically detect patterns in data, and then to use the uncovered patterns to predict future data or other outcomes of interest. Introduction Introduction What is Machine Learning? Arthur Samuel, 1959 Field of study that gives computers the ability to learn without being explicitly programmed. Kevin Murphy, 2012 To develop methods that can automatically detect patterns in data, and then to use the uncovered patterns to predict future data or other outcomes of interest. Introduction Introduction What is Machine Learning? Arthur Samuel, 1959 Field of study that gives computers the ability to learn without being explicitly programmed. Tom Mitchell, 1997 Any computer program that improves its performance at some task through experience. Introduction Introduction What is Machine Learning? Arthur Samuel, 1959 Field of study that gives computers the ability to learn without being explicitly programmed. Tom Mitchell, 1997 Any computer program that improves its performance at some task through experience. Kevin Murphy, 2012 To develop methods that can automatically detect patterns in data, and then to use the uncovered patterns to predict future data or other outcomes of interest. Introduction Introduction -
Loss-Based Risk Measures Rama Cont, Romain Deguest, Xuedong He
Loss-Based Risk Measures Rama Cont, Romain Deguest, Xuedong He To cite this version: Rama Cont, Romain Deguest, Xuedong He. Loss-Based Risk Measures. 2011. hal-00629929 HAL Id: hal-00629929 https://hal.archives-ouvertes.fr/hal-00629929 Submitted on 7 Oct 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Loss-based risk measures Rama CONT1,3, Romain DEGUEST 2 and Xue Dong HE3 1) Laboratoire de Probabilit´es et Mod`eles Al´eatoires CNRS- Universit´ePierre et Marie Curie, France. 2) EDHEC Risk Institute, Nice (France). 3) IEOR Dept, Columbia University, New York. 2011 Abstract Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We charac- terize loss-based risk measures by a representation theorem and give examples of such risk measures. We then discuss the statistical robustness of estimators of loss-based risk measures: we provide a general criterion for qualitative ro- bustness of risk estimators and compare this criterion with sensitivity analysis of estimators based on influence functions.