Risk aversion and choice under uncertainty
|
|
- Charlotte Young
- 5 years ago
- Views:
Transcription
1 Risk aversion and choice under uncertainty Pierre Chaigneau June 14, 2011
2 Finance: the economics of risk and uncertainty In financial markets, claims associated with random future payoffs are traded. What should be their price? (asset pricing) In a frictionless market, it depends on the economic fundamentals, and on the time preferences and the risk preferences of economic agents. Price of time, price of risk. Once you understand these time and risk preferences, you can understand changes in asset prices.
3 A simple example Why intuitions only get you so far... Consider a lottery whose payoffs are $1,000 with probability 0.4, and 0 with probability 0.6. What would you be willing pay to participate in this lottery? $400? $300? Intuitively, it should depend on your preferences: are you adventurous or cautious? It will also depend on your wealth to your exposure to other sources of risk and to the number of opportunities you have of playing the lottery (casinos usually have rules against playing too much ). Bottom line: even this simple example is too complicated to be solved intuitively. We need a model!
4 Another simple example The gamble is the following: someone offers you to flip a fair coin. If it s tails you receive $1 and that s it. If it s heads the coin is flipped again. If it s tails you receive $2 and that s it. If it s heads the coin is flipped again. At every round, the amount paid if tails obtains doubles. How much would be willing to pay to play this gamble?
5 Another simple example A shocking result Suppose that you are risk neutral. Then the price of this gamble is given by its expected payoff. With probability 1 2 you get $1. With probability 1 4 you get $2. With probability 1 8 you get $4. The expected payoff is ( 1 ) i 2 i 1 = 2 i=1 i=1 1 2 =
6 Another simple example This gamble has an infinite value! Yet with a 93% probability, it pays off $8 or less. The probability to get more than $64 is only 1%... It seems that economic agents either discount these very high payoffs (this is the idea behind the utility function), or underweight these extreme probabilities (this is the idea behind risk neutral probabilities). Solution to this puzzle: risk aversion.
7 Lotteries We assume that the outcome is unidimensional: money! The set of possible outcomes is X {x s } s=1,...,s. s denotes the state of the world. We denote the random outcome by x. A lottery L is described by a vector a probabilities (p 1,..., p S ). Example 1: Suppose that S = 3, and the possible outcomes are x 1 = 1, x 2 = 10, x 3 = 16. Lottery La is defined by p 1 = 0.1, p 2 = 0.8 (p 3 = 0.1). Lottery L b is defined by p 1 = 0.2, p 2 = 0.3 (p 3 = 0.5). Example 2: S =. The outcomes are x 1 = 0, x 2 = 1, x 3 = 2, etc. Any lottery is defined by the set of probabilities {p 1, p 2, p 3,... }, which must be such that s=1 p s = 1. Given a set of possible outcomes X, the set of all lotteries is L {(p 1,..., p S ) R+ p S p S = 1}. A degenerated lottery takes one value x s X with probability 1.
8 Preferences In consumer theory, preferences are defined over baskets of goods. In financial economics, preferences are defined over lotteries. Axioms on preferences: Completeness: for any two lotteries La and L b, either L a L b or L b L a. Transitivity: If La L b and L b L c then L a L c. Continuity: If La L b L c, there exists a scalar α [0, 1] such that L b αl a + (1 α)l c. NB: a preference relation which is both complete and transitive is rational.
9 Preferences Theorem: Given a preference relation which satisfies these axioms, there exists a preference functional U which maps any lottery L L into R such that L a L b U(L a ) U(L b ) NB: U is not unique. Consider any increasing function g. Then V ( ) g(u( )) represents the same preferences as U, in the sense that it will yield the same ordering of lotteries (put differently, an individual with preference functional U would always choose the same lotteries as an individual with preference functional V : they would behave in the same way).
10 Preferences We may want to impose other axioms on preferences: Time consistency: preferences are time consistent if a choice (possibly contingent upon the realization of some random variables) which is optimal at any time t is also optimal at any time T > t. Violation of time inconsistency: hyperbolic discounting, monetary policy, provision of incentives, liquidation of a firm, etc. Anytime there is a conflict between ex-ante and ex-post optimality. Resolution mechanisms: commitment or renegotiation (very important in corporate finance). Independence: for all α [0, 1], L a L b αl a + (1 α)l c αl b + (1 α)l c. If the lottery L a is preferred to the lottery L b, then a mix of L a with a lottery L c is also preferred to the same mix of L b with L c, for any L c.
11 Choice criterion: expected utility Theorem(von Neumann and Morgenstern): If the preference relation satisfies the axioms above, then it can be represented by a preference functional which is linear in probabilities. This means that, for each outcome x s, u s s.t. S S L a L b ps a u s ps b u s s=1 s=1 Or, with outcomes continuously distributed in [0, ), there exists a function u such that: L a L b 0 u(x s )dφ a (x s ) 0 u(x s )dφ b (x s ) where Φ a is the cumulative distribution function (c.d.f.) of x with the lottery a. In any case, we also write that E a [u( x)] E b [u( x)]. NB: The utility function u is not unique. Any increasing linear transformation of u will give the same ranking of lotteries.
12 Risk aversion Definition: An agent is risk averse if he dislikes all zero-mean risks at all wealth levels. This implies that, for a risk averse agent, E[u(w + ɛ)] < E[u(w)] = u(w) = u(e[w]) = u(e[w + ɛ]), where E[ ɛ] = 0. Proposition: An agent is risk averse if and only if his utility function u is concave. Jensen inequality: E[u( x)] u(e[ x]) if and only if u is concave What are the implications of concavity for marginal utility? What is the form of the utility function of a risk neutral agent? Risk aversion explains the behavior of economic agents: purchase of insurance, purchase of risky assets. It is a crucial element in GE asset pricing models. Why doesn t it matter in derivatives pricing?
13 To summarize Some perspective, and expected utility theory in practice The class of models that combine linearity in probabilities with a nonlinear utility function remains the workhorse in much of modern economics. The seminal contributions of Kenneth J. Arrow (1965) and John W. Pratt (1964) provided the foundations for measuring individual risk attitudes using the curvature of the utility function. von Gaudecker et al., AER Only assume preferences over lotteries. Expected utility theorem. Existence of a utility function. u is usually given: Functional form. Calibration: risk aversion.
14 Applying the expected utility criterion The possible outcomes are x 1 = 1, x 2 = 10, x 3 = 16. Lottery L a is defined by p 1 = 0.1, p 2 = 0.8 (p 3 = 0.1). Lottery L b is defined by p 1 = 0.2, p 2 = 0.3 (p 3 = 0.5). Consider the utility function such that u (x) = x γ. 1 γ = 0 γ = 0.5 γ = 1 γ = 2 E a [u( x)] E b [u( x)] It can be represented by u(x) = ln(x) for γ = 1, and u(x) = x1 γ 1 γ otherwise.
15 The risk premium Given the utility function, can we quantify the tradeoff between risk and return? In other words, can we find the maximum amount of money that the agent would be willing to pay to eliminate a pure risk? This amount is called the risk premium, Π. In the case of a risk ɛ which is additive in wealth, it solves: E[u(w + ɛ)] = u(w Π) where E[ ɛ] = 0 This amount Π is positive if the agent is risk averse: E[u(w + ɛ)] < u(w) if u is concave Another similar concept is the certainty equivalent. Given a lottery x (whose expected payoff is not necessarily zero), the certainty equivalent CE is the value of the lottery: E[u(w + x)] = u(w + CE)
16 The Arrow-Pratt approximation (1) A closed form expression for the tradeoff between risk and return Consider a pure risk ɛ additive in wealth: end-of-period wealth = w + ɛ. Then the absolute risk premium (expressed as the amount of wealth) associated with this risk is Π 1 2 A(w)σ2 where var[ ɛ] = σ 2 and A(w) u (w) u (w) > 0. Important limitation: this approximation is only approximately valid for small risks. The risk premium is increasing in the variance of the risk. The risk premium is increasing in the concavity of the utility function (which is a measure of risk aversion) at w: the coefficient of absolute risk aversion A(w) is a measure of the tradeoff between risk and return at w. The coefficient of absolute risk aversion measures the agent s preferences regarding additive risks (with pure & small risks).
17 The Arrow-Pratt approximation (2) A closed form expression for the tradeoff between risk and return Consider a pure risk ɛ multiplicative in wealth, so that the end-of-period wealth is w(1 + ɛ). Then the relative risk premium (expressed as a share of wealth) associated with this risk is ˆΠ 1 2 R(w)σ2 where var[ ɛ] = σ 2 and R(w) w u (w) u (w) > 0. Important limitation: this approximation is only approximately valid for small risks. The risk premium is increasing in the variance of the risk. The risk premium is increasing in the concavity of the utility function at w: the coefficient of relative risk aversion R(w) is a measure of the tradeoff between risk and return at w. The coefficient of relative risk aversion measures the agent s preferences regarding pure and small multiplicative risks.
18 Differences in risk aversion Who is more risk averse? Who is willing to pay more to escape a given risk? Consider a linear transformation of the utility function: ν(x) = a + bu(x). Does that change the risk premium Π? Proposition: the following statements are equivalent: The Arrow-Pratt risk premium is larger for an individual with utility function v than for an individual with utility function u. For all w, A v (w) > A u (w). v is a concave transformation of u.
19 Wealth and risk aversion Intuitively, a millionaire will not be as averse to winning or losing $100 as a struggling student. Definition: Preferences are characterized by decreasing absolute risk aversion (DARA) if the risk premium associated to any pure additive risk is decreasing in wealth. Preferences are DARA if and only if A(w) is decreasing in w. A necessary condition is u (w) > 0 (this is called prudence), i.e., a convex marginal utility.
20 Constant absolute risk aversion CARA The coefficient of absolute risk aversion A(w) is a constant function of wealth with the following utility function (also known as negative exponential): As you can check, A(w) = ρ. u(w) = exp{ ρw} With these preferences, an economic agent is as averse to an additive risk if he owns $10 or $1,000,000: constant absolute risk aversion. Unrealistic... but these utility functions are tractable (especially with a normally distributed risk) and are used in stylized models. Increasing relative risk aversion.
21 Quadratic utility Vintage: popular in the 1950s Simple form: u(w) = aw bw 2 Advantage: preferences are fully described by the mean and the variance of the payoff. Drawbacks: Increasing absolute risk aversion. Only defined on the interval of wealth where marginal utility is increasing. Not prudent: u (w) = 0.
22 Constant relative risk aversion CRRA, or power utility The coefficient of relative risk aversion A(w) is a constant function of wealth with the following utility function: { w 1 γ u(w) = 1 γ if γ 1 ln(w) if γ = 1 As you can check, R(w) = γ. Constant RRA. With these preferences, an economic agent is as averse to a multiplicative risk whether he owns $10 or $1,000,000. It is also known as the isoelatic utility function: the elasticity of substitution between consumption at any two points in time is equal to 1 γ. Decreasing absolute risk aversion. Widely used in macroeconomics and finance, especially in calibration exercises. Usual assumption: γ [1, 4].
23 What is your coefficient of relative risk aversion? Which proportion of your wealth would you be willing to pay to escape the risk of your wealth either increasing or decreasing by 10% with equal probabilities? Suppose you own $50,000. If you participate in the lottery, your wealth will be $45,000 with probability 0.5, or $55,000 with probability 0.5. What percentage of $50,000 would you be willing to pay to escape this risk? Answer the same question with 30% instead of 10%. Of course, your answer will also depend on personal circumstances, notably to what extent you are exposed to other sources of risk. More on this later on.
24 What is your coefficient of relative risk aversion? coef RRA k = 10% k = 30% γ = % 2.3% γ = 1 0.5% 4.6% γ = 4 2.0% 16.0% γ = % 24.4% γ = % 28.7% A RRA less than one may be considered a low risk aversion, whereas a RRA higher than four may be considered a high risk aversion.
25 Caveats of the expected utility criterion The Allais paradox. Your wealth is $6,000. You have the option to pay $2,000 to get $8,000 with probability 0.5 or $0 with probability 0.5. Your wealth is $12,000. You can either choose to lose $8,000 with probability 0.5 or $0 with probability 0.5, or pay $6,000 to escape this lottery. Framing: preferences over lotteries depend on whether they are presented as a gamble or as insurance. Loss aversion.
26 Caveats of the expected utility criterion Prospect theory (Kahnman and Tversky). Habit formation (Campbell and Cochrane). Recursive utility.
27 Takeaway If a preference relation satisfies certain axioms, the expected utility criterion can be used to rank lotteries. A risk averse agent has a concave utility function. We can derive the price of a small risk, whether it is additive or multiplicative in wealth. We can use different functional forms for the utility function, with different implications for ARA and RRA as a function of wealth.
28 Exercises Homework will be solved next week Consider an investor with a relative risk aversion which is equal to 4 at any level of wealth. His current wealth is $100. What is his certainty equivalent for an asset whose payoff is equal to $100 with probability 0.5 and $0 otherwise? (think about a bond with a risk of default)
29 Acknowledgements: Some sources for this series of slides include: The slides of Martin Boyer, for the same course at HEC Montreal. Asset Pricing, by John H. Cochrane. Finance and the Economics of Uncertainty, by Gabrielle Demange and Guy Laroque. The Economics of Risk and Time, by Christian Gollier.
Risk preferences and stochastic dominance
Risk preferences and stochastic dominance Pierre Chaigneau pierre.chaigneau@hec.ca September 5, 2011 Preferences and utility functions The expected utility criterion Future income of an agent: x. Random
More informationCHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION
CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Choice Theory Investments 1 / 65 Outline 1 An Introduction
More informationECON 581. Decision making under risk. Instructor: Dmytro Hryshko
ECON 581. Decision making under risk Instructor: Dmytro Hryshko 1 / 36 Outline Expected utility Risk aversion Certainty equivalence and risk premium The canonical portfolio allocation problem 2 / 36 Suggested
More informationExpected Utility and Risk Aversion
Expected Utility and Risk Aversion Expected utility and risk aversion 1/ 58 Introduction Expected utility is the standard framework for modeling investor choices. The following topics will be covered:
More informationUtility and Choice Under Uncertainty
Introduction to Microeconomics Utility and Choice Under Uncertainty The Five Axioms of Choice Under Uncertainty We can use the axioms of preference to show how preferences can be mapped into measurable
More informationChoice under risk and uncertainty
Choice under risk and uncertainty Introduction Up until now, we have thought of the objects that our decision makers are choosing as being physical items However, we can also think of cases where the outcomes
More informationChoice under Uncertainty
Chapter 7 Choice under Uncertainty 1. Expected Utility Theory. 2. Risk Aversion. 3. Applications: demand for insurance, portfolio choice 4. Violations of Expected Utility Theory. 7.1 Expected Utility Theory
More informationChapter 1. Utility Theory. 1.1 Introduction
Chapter 1 Utility Theory 1.1 Introduction St. Petersburg Paradox (gambling paradox) the birth to the utility function http://policonomics.com/saint-petersburg-paradox/ The St. Petersburg paradox, is a
More informationLecture 3: Utility-Based Portfolio Choice
Lecture 3: Utility-Based Portfolio Choice Prof. Massimo Guidolin Portfolio Management Spring 2017 Outline and objectives Choice under uncertainty: dominance o Guidolin-Pedio, chapter 1, sec. 2 Choice under
More informationModels and Decision with Financial Applications UNIT 1: Elements of Decision under Uncertainty
Models and Decision with Financial Applications UNIT 1: Elements of Decision under Uncertainty We always need to make a decision (or select from among actions, options or moves) even when there exists
More informationInvestment and Portfolio Management. Lecture 1: Managed funds fall into a number of categories that pool investors funds
Lecture 1: Managed funds fall into a number of categories that pool investors funds Types of managed funds: Unit trusts Investors funds are pooled, usually into specific types of assets Investors are assigned
More informationMICROECONOMIC THEROY CONSUMER THEORY
LECTURE 5 MICROECONOMIC THEROY CONSUMER THEORY Choice under Uncertainty (MWG chapter 6, sections A-C, and Cowell chapter 8) Lecturer: Andreas Papandreou 1 Introduction p Contents n Expected utility theory
More informationExpected value is basically the average payoff from some sort of lottery, gamble or other situation with a randomly determined outcome.
Economics 352: Intermediate Microeconomics Notes and Sample Questions Chapter 18: Uncertainty and Risk Aversion Expected Value The chapter starts out by explaining what expected value is and how to calculate
More informationComparison of Payoff Distributions in Terms of Return and Risk
Comparison of Payoff Distributions in Terms of Return and Risk Preliminaries We treat, for convenience, money as a continuous variable when dealing with monetary outcomes. Strictly speaking, the derivation
More informationExpected utility theory; Expected Utility Theory; risk aversion and utility functions
; Expected Utility Theory; risk aversion and utility functions Prof. Massimo Guidolin Portfolio Management Spring 2016 Outline and objectives Utility functions The expected utility theorem and the axioms
More informationMicro Theory I Assignment #5 - Answer key
Micro Theory I Assignment #5 - Answer key 1. Exercises from MWG (Chapter 6): (a) Exercise 6.B.1 from MWG: Show that if the preferences % over L satisfy the independence axiom, then for all 2 (0; 1) and
More informationBEEM109 Experimental Economics and Finance
University of Exeter Recap Last class we looked at the axioms of expected utility, which defined a rational agent as proposed by von Neumann and Morgenstern. We then proceeded to look at empirical evidence
More informationSTOCHASTIC CONSUMPTION-SAVINGS MODEL: CANONICAL APPLICATIONS FEBRUARY 19, 2013
STOCHASTIC CONSUMPTION-SAVINGS MODEL: CANONICAL APPLICATIONS FEBRUARY 19, 2013 Model Structure EXPECTED UTILITY Preferences v(c 1, c 2 ) with all the usual properties Lifetime expected utility function
More informationChapter 18: Risky Choice and Risk
Chapter 18: Risky Choice and Risk Risky Choice Probability States of Nature Expected Utility Function Interval Measure Violations Risk Preference State Dependent Utility Risk-Aversion Coefficient Actuarially
More informationFoundations of Financial Economics Choice under uncertainty
Foundations of Financial Economics Choice under uncertainty Paulo Brito 1 pbrito@iseg.ulisboa.pt University of Lisbon March 9, 2018 Topics covered Contingent goods Comparing contingent goods Decision under
More informationModels & Decision with Financial Applications Unit 3: Utility Function and Risk Attitude
Models & Decision with Financial Applications Unit 3: Utility Function and Risk Attitude Duan LI Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong http://www.se.cuhk.edu.hk/
More informationUTILITY ANALYSIS HANDOUTS
UTILITY ANALYSIS HANDOUTS 1 2 UTILITY ANALYSIS Motivating Example: Your total net worth = $400K = W 0. You own a home worth $250K. Probability of a fire each yr = 0.001. Insurance cost = $1K. Question:
More informationMock Examination 2010
[EC7086] Mock Examination 2010 No. of Pages: [7] No. of Questions: [6] Subject [Economics] Title of Paper [EC7086: Microeconomic Theory] Time Allowed [Two (2) hours] Instructions to candidates Please answer
More informationCopyright (C) 2001 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of version 1 of the
Copyright (C) 2001 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of version 1 of the open text license amendment to version 2 of the GNU General
More informationE&G, Chap 10 - Utility Analysis; the Preference Structure, Uncertainty - Developing Indifference Curves in {E(R),σ(R)} Space.
1 E&G, Chap 10 - Utility Analysis; the Preference Structure, Uncertainty - Developing Indifference Curves in {E(R),σ(R)} Space. A. Overview. c 2 1. With Certainty, objects of choice (c 1, c 2 ) 2. With
More informationFinancial Economics. A Concise Introduction to Classical and Behavioral Finance Chapter 2. Thorsten Hens and Marc Oliver Rieger
Financial Economics A Concise Introduction to Classical and Behavioral Finance Chapter 2 Thorsten Hens and Marc Oliver Rieger Swiss Banking Institute, University of Zurich / BWL, University of Trier July
More informationMicroeconomic Theory III Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 14.123 Microeconomic Theory III Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIT 14.123 (2009) by
More informationAttitudes Toward Risk. Joseph Tao-yi Wang 2013/10/16. (Lecture 11, Micro Theory I)
Joseph Tao-yi Wang 2013/10/16 (Lecture 11, Micro Theory I) Dealing with Uncertainty 2 Preferences over risky choices (Section 7.1) One simple model: Expected Utility How can old tools be applied to analyze
More informationSTOCHASTIC CONSUMPTION-SAVINGS MODEL: CANONICAL APPLICATIONS SEPTEMBER 13, 2010 BASICS. Introduction
STOCASTIC CONSUMPTION-SAVINGS MODE: CANONICA APPICATIONS SEPTEMBER 3, 00 Introduction BASICS Consumption-Savings Framework So far only a deterministic analysis now introduce uncertainty Still an application
More informationRational theories of finance tell us how people should behave and often do not reflect reality.
FINC3023 Behavioral Finance TOPIC 1: Expected Utility Rational theories of finance tell us how people should behave and often do not reflect reality. A normative theory based on rational utility maximizers
More informationThe stochastic discount factor and the CAPM
The stochastic discount factor and the CAPM Pierre Chaigneau pierre.chaigneau@hec.ca November 8, 2011 Can we price all assets by appropriately discounting their future cash flows? What determines the risk
More informationSAC 304: Financial Mathematics II
SAC 304: Financial Mathematics II Portfolio theory, Risk and Return,Investment risk, CAPM Philip Ngare, Ph.D April 25, 2013 P. Ngare (University Of Nairobi) SAC 304: Financial Mathematics II April 25,
More informationCONVENTIONAL FINANCE, PROSPECT THEORY, AND MARKET EFFICIENCY
CONVENTIONAL FINANCE, PROSPECT THEORY, AND MARKET EFFICIENCY PART ± I CHAPTER 1 CHAPTER 2 CHAPTER 3 Foundations of Finance I: Expected Utility Theory Foundations of Finance II: Asset Pricing, Market Efficiency,
More informationEconomic of Uncertainty
Economic of Uncertainty Risk Aversion Based on ECO 317, Princeton UC3M April 2012 (UC3M) Economics of Uncertainty. April 2012 1 / 16 Introduction 1 Space of Lotteries (UC3M) Economics of Uncertainty. April
More informationCONSUMPTION-SAVINGS MODEL JANUARY 19, 2018
CONSUMPTION-SAVINGS MODEL JANUARY 19, 018 Stochastic Consumption-Savings Model APPLICATIONS Use (solution to) stochastic two-period model to illustrate some basic results and ideas in Consumption research
More informationAdvanced Risk Management
Winter 2014/2015 Advanced Risk Management Part I: Decision Theory and Risk Management Motives Lecture 1: Introduction and Expected Utility Your Instructors for Part I: Prof. Dr. Andreas Richter Email:
More informationModule 1: Decision Making Under Uncertainty
Module 1: Decision Making Under Uncertainty Information Economics (Ec 515) George Georgiadis Today, we will study settings in which decision makers face uncertain outcomes. Natural when dealing with asymmetric
More informationLecture 11 - Risk Aversion, Expected Utility Theory and Insurance
Lecture 11 - Risk Aversion, Expected Utility Theory and Insurance 14.03, Spring 2003 1 Risk Aversion and Insurance: Introduction To have a passably usable model of choice, we need to be able to say something
More informationStat 6863-Handout 1 Economics of Insurance and Risk June 2008, Maurice A. Geraghty
A. The Psychology of Risk Aversion Stat 6863-Handout 1 Economics of Insurance and Risk June 2008, Maurice A. Geraghty Suppose a decision maker has an asset worth $100,000 that has a 1% chance of being
More informationAnalysing risk preferences among insurance customers
Norwegian School of Economics Bergen, spring 2016 Analysing risk preferences among insurance customers Expected utility theory versus disappointment aversion theory Emil Haga and André Waage Rivenæs Supervisor:
More informationECMC49F Midterm. Instructor: Travis NG Date: Oct 26, 2005 Duration: 1 hour 50 mins Total Marks: 100. [1] [25 marks] Decision-making under certainty
ECMC49F Midterm Instructor: Travis NG Date: Oct 26, 2005 Duration: 1 hour 50 mins Total Marks: 100 [1] [25 marks] Decision-making under certainty (a) [5 marks] Graphically demonstrate the Fisher Separation
More informationExpected Utility And Risk Aversion
Expected Utility And Risk Aversion Econ 2100 Fall 2017 Lecture 12, October 4 Outline 1 Risk Aversion 2 Certainty Equivalent 3 Risk Premium 4 Relative Risk Aversion 5 Stochastic Dominance Notation From
More informationCharacterization of the Optimum
ECO 317 Economics of Uncertainty Fall Term 2009 Notes for lectures 5. Portfolio Allocation with One Riskless, One Risky Asset Characterization of the Optimum Consider a risk-averse, expected-utility-maximizing
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics May 29, 2014 Instructions This exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationEffects of Wealth and Its Distribution on the Moral Hazard Problem
Effects of Wealth and Its Distribution on the Moral Hazard Problem Jin Yong Jung We analyze how the wealth of an agent and its distribution affect the profit of the principal by considering the simple
More informationLecture 6 Introduction to Utility Theory under Certainty and Uncertainty
Lecture 6 Introduction to Utility Theory under Certainty and Uncertainty Prof. Massimo Guidolin Prep Course in Quant Methods for Finance August-September 2017 Outline and objectives Axioms of choice under
More informationFinancial Decisions and Markets: A Course in Asset Pricing. John Y. Campbell. Princeton University Press Princeton and Oxford
Financial Decisions and Markets: A Course in Asset Pricing John Y. Campbell Princeton University Press Princeton and Oxford Figures Tables Preface xiii xv xvii Part I Stade Portfolio Choice and Asset Pricing
More informationECON Financial Economics
ECON 8 - Financial Economics Michael Bar August, 0 San Francisco State University, department of economics. ii Contents Decision Theory under Uncertainty. Introduction.....................................
More informationOutline. Simple, Compound, and Reduced Lotteries Independence Axiom Expected Utility Theory Money Lotteries Risk Aversion
Uncertainty Outline Simple, Compound, and Reduced Lotteries Independence Axiom Expected Utility Theory Money Lotteries Risk Aversion 2 Simple Lotteries 3 Simple Lotteries Advanced Microeconomic Theory
More informationAREC 815: Experimental and Behavioral Economics. Measuring Risk Preferences. Professor: Pamela Jakiela
AREC 815: Experimental and Behavioral Economics Measuring Risk Preferences Professor: Pamela Jakiela Department of Agricultural and Resource Economics University of Maryland, College Park Expected Utility
More informationName. Final Exam, Economics 210A, December 2014 Answer any 7 of these 8 questions Good luck!
Name Final Exam, Economics 210A, December 2014 Answer any 7 of these 8 questions Good luck! 1) For each of the following statements, state whether it is true or false. If it is true, prove that it is true.
More informationFinancial Economics: Making Choices in Risky Situations
Financial Economics: Making Choices in Risky Situations Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY March, 2015 1 / 57 Questions to Answer How financial risk is defined and measured How an investor
More information3. Prove Lemma 1 of the handout Risk Aversion.
IDEA Economics of Risk and Uncertainty List of Exercises Expected Utility, Risk Aversion, and Stochastic Dominance. 1. Prove that, for every pair of Bernouilli utility functions, u 1 ( ) and u 2 ( ), and
More informationPart 4: Market Failure II - Asymmetric Information - Uncertainty
Part 4: Market Failure II - Asymmetric Information - Uncertainty Expected Utility, Risk Aversion, Risk Neutrality, Risk Pooling, Insurance July 2016 - Asymmetric Information - Uncertainty July 2016 1 /
More informationIf U is linear, then U[E(Ỹ )] = E[U(Ỹ )], and one is indifferent between lottery and its expectation. One is called risk neutral.
Risk aversion For those preference orderings which (i.e., for those individuals who) satisfy the seven axioms, define risk aversion. Compare a lottery Ỹ = L(a, b, π) (where a, b are fixed monetary outcomes)
More informationOptimizing Portfolios
Optimizing Portfolios An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Investors may wish to adjust the allocation of financial resources including a mixture
More informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College April 3, 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationMicroeconomics 3200/4200:
Microeconomics 3200/4200: Part 1 P. Piacquadio p.g.piacquadio@econ.uio.no September 25, 2017 P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 September 25, 2017 1 / 23 Example (1) Suppose I take
More informationFINC3017: Investment and Portfolio Management
FINC3017: Investment and Portfolio Management Investment Funds Topic 1: Introduction Unit Trusts: investor s funds are pooled, usually into specific types of assets. o Investors are assigned tradeable
More informationMaking Hard Decision. ENCE 627 Decision Analysis for Engineering. Identify the decision situation and understand objectives. Identify alternatives
CHAPTER Duxbury Thomson Learning Making Hard Decision Third Edition RISK ATTITUDES A. J. Clark School of Engineering Department of Civil and Environmental Engineering 13 FALL 2003 By Dr. Ibrahim. Assakkaf
More informationChoice Under Uncertainty
Chapter 6 Choice Under Uncertainty Up until now, we have been concerned with choice under certainty. A consumer chooses which commodity bundle to consume. A producer chooses how much output to produce
More informationMicroeconomic Theory III Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 14.123 Microeconomic Theory III Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIT 14.123 (2009) by
More informationPeriod State of the world: n/a A B n/a A B Endowment ( income, output ) Y 0 Y1 A Y1 B Y0 Y1 A Y1. p A 1+r. 1 0 p B.
ECONOMICS 7344, Spring 2 Bent E. Sørensen April 28, 2 NOTE. Obstfeld-Rogoff (OR). Simplified notation. Assume that agents (initially we will consider just one) live for 2 periods in an economy with uncertainty
More informationEconomic & Financial Decisions under Risk (Chapters 1&2) Eeckhoudt, Gollier & Schlesinger (Princeton Univ Press 2005)
Economic & Financial Decisions under Risk (Chapters &2) Eeckhoudt, Gollier & Schlesinger (Princeton Univ Press 2005) Risk Aversion This chapter looks at a basic concept behind modeling individual preferences
More informationECON4510 Finance Theory Lecture 1
ECON4510 Finance Theory Lecture 1 Kjetil Storesletten Department of Economics University of Oslo 15 January 2018 Kjetil Storesletten, Dept. of Economics, UiO ECON4510 Finance Theory Lecture 1 15 January
More informationRepresenting Risk Preferences in Expected Utility Based Decision Models
Representing Risk Preferences in Expected Utility Based Decision Models Jack Meyer Department of Economics Michigan State University East Lansing, MI 48824 jmeyer@msu.edu SCC-76: Economics and Management
More informationFinancial Economics: Risk Aversion and Investment Decisions
Financial Economics: Risk Aversion and Investment Decisions Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY March, 2015 1 / 50 Outline Risk Aversion and Portfolio Allocation Portfolios, Risk Aversion,
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program August 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program August 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More information3.1 The Marschak-Machina triangle and risk aversion
Chapter 3 Risk aversion 3.1 The Marschak-Machina triangle and risk aversion One of the earliest, and most useful, graphical tools used to analyse choice under uncertainty was a triangular graph that was
More informationAndreas Wagener University of Vienna. Abstract
Linear risk tolerance and mean variance preferences Andreas Wagener University of Vienna Abstract We translate the property of linear risk tolerance (hyperbolical Arrow Pratt index of risk aversion) from
More informationUnit 4.3: Uncertainty
Unit 4.: Uncertainty Michael Malcolm June 8, 20 Up until now, we have been considering consumer choice problems where the consumer chooses over outcomes that are known. However, many choices in economics
More informationMicroeconomics of Banking: Lecture 2
Microeconomics of Banking: Lecture 2 Prof. Ronaldo CARPIO September 25, 2015 A Brief Look at General Equilibrium Asset Pricing Last week, we saw a general equilibrium model in which banks were irrelevant.
More informationPrevention and risk perception : theory and experiments
Prevention and risk perception : theory and experiments Meglena Jeleva (EconomiX, University Paris Nanterre) Insurance, Actuarial Science, Data and Models June, 11-12, 2018 Meglena Jeleva Prevention and
More informationManagerial Economics
Managerial Economics Unit 9: Risk Analysis Rudolf Winter-Ebmer Johannes Kepler University Linz Winter Term 2015 Managerial Economics: Unit 9 - Risk Analysis 1 / 49 Objectives Explain how managers should
More information1. Expected utility, risk aversion and stochastic dominance
. Epected utility, risk aversion and stochastic dominance. Epected utility.. Description o risky alternatives.. Preerences over lotteries..3 The epected utility theorem. Monetary lotteries and risk aversion..
More informationReview Session. Prof. Manuela Pedio Theory of Finance
Review Session Prof. Manuela Pedio 20135 Theory of Finance 12 October 2018 Three most common utility functions (1/3) We typically assume that investors are non satiated (they always prefer more to less)
More informationThis paper addresses the situation when marketable gambles are restricted to be small. It is easily shown that the necessary conditions for local" Sta
Basic Risk Aversion Mark Freeman 1 School of Business and Economics, University of Exeter It is demonstrated that small marketable gambles that are unattractive to a Standard Risk Averse investor cannot
More informationProblem Set 2. Theory of Banking - Academic Year Maria Bachelet March 2, 2017
Problem Set Theory of Banking - Academic Year 06-7 Maria Bachelet maria.jua.bachelet@gmai.com March, 07 Exercise Consider an agency relationship in which the principal contracts the agent, whose effort
More informationLecture 11: Critiques of Expected Utility
Lecture 11: Critiques of Expected Utility Alexander Wolitzky MIT 14.121 1 Expected Utility and Its Discontents Expected utility (EU) is the workhorse model of choice under uncertainty. From very early
More informationIntroduction to Economics I: Consumer Theory
Introduction to Economics I: Consumer Theory Leslie Reinhorn Durham University Business School October 2014 What is Economics? Typical De nitions: "Economics is the social science that deals with the production,
More informationSWITCHING, MEAN-SEEKING, AND RELATIVE RISK
SWITCHING, MEAN-SEEKING, AND RELATIVE RISK WITH TWO OR MORE RISKY ASSETS 1. Introduction Ever since the seminal work of Arrow (1965) and Pratt (1964), researchers have recognized the importance of understanding
More informationMaximizing the expected net future value as an alternative strategy to gamma discounting
Maximizing the expected net future value as an alternative strategy to gamma discounting Christian Gollier University of Toulouse September 1, 2003 Abstract We examine the problem of selecting the discount
More informationAdvanced Financial Economics Homework 2 Due on April 14th before class
Advanced Financial Economics Homework 2 Due on April 14th before class March 30, 2015 1. (20 points) An agent has Y 0 = 1 to invest. On the market two financial assets exist. The first one is riskless.
More informationPrincipes de choix de portefeuille
Principes de choix de portefeuille 7 e édition Christophe Boucher christophe.boucher@u-paris10.fr 1 Chapitre 3 7 e édition La théorie du choix en incertitude 2 Part 3. The Theory of Choice under Uncertainty
More information05/05/2011. Degree of Risk. Degree of Risk. BUSA 4800/4810 May 5, Uncertainty
BUSA 4800/4810 May 5, 2011 Uncertainty We must believe in luck. For how else can we explain the success of those we don t like? Jean Cocteau Degree of Risk We incorporate risk and uncertainty into our
More informationAsset Pricing. Teaching Notes. João Pedro Pereira
Asset Pricing Teaching Notes João Pedro Pereira Nova School of Business and Economics Universidade Nova de Lisboa joao.pereira@novasbe.pt http://docentes.fe.unl.pt/ jpereira/ June 18, 2015 Contents 1 Introduction
More informationSlides III - Complete Markets
Slides III - Complete Markets Julio Garín University of Georgia Macroeconomic Theory II (Ph.D.) Spring 2017 Macroeconomic Theory II Slides III - Complete Markets Spring 2017 1 / 33 Outline 1. Risk, Uncertainty,
More informationLECTURE NOTES 10 ARIEL M. VIALE
LECTURE NOTES 10 ARIEL M VIALE 1 Behavioral Asset Pricing 11 Prospect theory based asset pricing model Barberis, Huang, and Santos (2001) assume a Lucas pure-exchange economy with three types of assets:
More informationAMS Portfolio Theory and Capital Markets
AMS 69.0 - Portfolio Theory and Capital Markets I Class 5 - Utility and Pricing Theory Robert J. Frey Research Professor Stony Brook University, Applied Mathematics and Statistics frey@ams.sunysb.edu This
More informationPricing and hedging in incomplete markets
Pricing and hedging in incomplete markets Chapter 10 From Chapter 9: Pricing Rules: Market complete+nonarbitrage= Asset prices The idea is based on perfect hedge: H = V 0 + T 0 φ t ds t + T 0 φ 0 t ds
More informationChapter 6: Risky Securities and Utility Theory
Chapter 6: Risky Securities and Utility Theory Topics 1. Principle of Expected Return 2. St. Petersburg Paradox 3. Utility Theory 4. Principle of Expected Utility 5. The Certainty Equivalent 6. Utility
More informationConsumption and Asset Pricing
Consumption and Asset Pricing Yin-Chi Wang The Chinese University of Hong Kong November, 2012 References: Williamson s lecture notes (2006) ch5 and ch 6 Further references: Stochastic dynamic programming:
More informationExercises for Chapter 8
Exercises for Chapter 8 Exercise 8. Consider the following functions: f (x)= e x, (8.) g(x)=ln(x+), (8.2) h(x)= x 2, (8.3) u(x)= x 2, (8.4) v(x)= x, (8.5) w(x)=sin(x). (8.6) In all cases take x>0. (a)
More informationMORAL HAZARD AND BACKGROUND RISK IN COMPETITIVE INSURANCE MARKETS: THE DISCRETE EFFORT CASE. James A. Ligon * University of Alabama.
mhbri-discrete 7/5/06 MORAL HAZARD AND BACKGROUND RISK IN COMPETITIVE INSURANCE MARKETS: THE DISCRETE EFFORT CASE James A. Ligon * University of Alabama and Paul D. Thistle University of Nevada Las Vegas
More informationMaximization of utility and portfolio selection models
Maximization of utility and portfolio selection models J. F. NEVES P. N. DA SILVA C. F. VASCONCELLOS Abstract Modern portfolio theory deals with the combination of assets into a portfolio. It has diversification
More informationUncertainty in Equilibrium
Uncertainty in Equilibrium Larry Blume May 1, 2007 1 Introduction The state-preference approach to uncertainty of Kenneth J. Arrow (1953) and Gérard Debreu (1959) lends itself rather easily to Walrasian
More information8/28/2017. ECON4260 Behavioral Economics. 2 nd lecture. Expected utility. What is a lottery?
ECON4260 Behavioral Economics 2 nd lecture Cumulative Prospect Theory Expected utility This is a theory for ranking lotteries Can be seen as normative: This is how I wish my preferences looked like Or
More informationStandard Risk Aversion and Efficient Risk Sharing
MPRA Munich Personal RePEc Archive Standard Risk Aversion and Efficient Risk Sharing Richard M. H. Suen University of Leicester 29 March 2018 Online at https://mpra.ub.uni-muenchen.de/86499/ MPRA Paper
More informationChoice under Uncertainty
1 Choice under Uncertainty ASSET PRICING THEORY aims to describe the equilibrium in financial markets, where economic agents interact to trade claims to uncertain future payoffs. Both adjectives, uncertain
More informationUse (solution to) stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research
TOCATIC CONUMPTION-AVING MODE: CANONICA APPICATION EPTEMBER 4, 0 s APPICATION Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing
More information