# Basic Utility Theory for Portfolio Selection

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Basic Utility Theory for Portfolio Selection In economics and finance, the most popular approach to the problem of choice under uncertainty is the expected utility (EU) hypothesis. The reason for this to be the preferred paradigm is that, as a general approach to decision making under risk, it has a sound theoretical basis. Suppose an individual at time 0 has to decide about the composition of her portfolio to be held until period 1, and that there are N assets which can be purchased, with (random) returns R i, i = 1,..., N. If the initial wealth to be invested is W 0, she will have wealth W 1 = 1 + N i=1 x i R i W 0 = (1 + r p )W 0 For an introduction to utility theory, see Eeckhoudt, Gollier, and Schlesinger (2005): Economic and Financial Decisions under Risk. Princeton University Press, Princeton.

2 in period 1, where r p = i x i R i is the portfolio return, and the x i, i = 1,..., N, satisfying i x i = 1, are the portfolio weights. In period 1, the individual will extract utility from consuming goods that can be purchased with this wealth. The relationship between wealth and the utility of consuming this wealth is described by a utility function, U( ). In general, each investor will have a different U( ). The expected utility hypothesis states that the individual will choose the portfolio weights such that the expected value of utility is maximized, i.e., the portfolio problem is max x 1,...,x N E[U(W 1 )] = E subject to i x i = 1. U 1 + N i=1 x i R i W 0 U( ) is also called expected utility function, or von Neumann Morgenstern utility function.

3 Properties of the Expected Utility Function An important property of an expected utility function is that it is unique up to affine transformations. That is, if U( ) describes the preferences of an investor, then so does U ( ) = c 1 U( ) + c 2, where c 1 > 0. While the above is a purely mathematical result, we can restrict the range of reasonable utility functions by economic reasoning. Positive marginal utility. That is, U (W ) > 0 for all W. Risk Aversion. Definition 1 An agent is risk averse if, at any wealth level W, he or she dislikes every lottery with an expected payoff of zero, i.e., E[U(W + ɛ)] < U(W )

4 for all W and every zero-mean random variable ɛ. Note that any random outcome Z can be written as Z = E(Z) + ɛ, where ɛ = Z E(Z) is a zero-mean random variable. Thus, a risk averse agent always prefers receiving the expected outcome of a lottery with certainty, rather than the lottery itself. Using Jensen s inequality, it can readily be shown that a necessary and sufficient condition for risk aversion is that the expected utility function is concave, i.e., U (W ) < 0 for all W.

5 In the context of applications to portfolio choice, it is important to note that risk aversion is closely related to portfolio diversification. As a simple example, consider the case of two assets, the returns of which are identically and independently distributed. Then the problem is to solve max x E{U[(1 + xr 1 + (1 x)r 2 )W 0 ]}. The first order condition is E{U [(1 + xr 1 + (1 x)r 2 )W 0 ](R 1 R 2 )} = 0. (1) By the assumptions about the joint distribution of R 1 and R 2, (1) will hold exactly if x = 1/2. The second order condition, E{U [(1 + xr 1 + (1 x)r 2 )W 0 ](R 1 R 2 ) 2 } < 0, is satisfied for a risk averse investor.

6 Measuring and Comparing Risk Aversion The Risk Premium and the Arrow Pratt Measure Risk averters dislike zero mean risks. Thus, a natural way to measure risk aversion is to ask how much an investor is ready to pay to get rid of a zero mean risk ɛ. This is called the risk premium, π, and is defined implicitly by E[U(W + ɛ)] = U(W π). (2) In general, the risk premium is a complex function of the distribution of ɛ, initial wealth W, and U( ). However, let us consider a small risk.

7 A second order and a first order Taylor approximation of the left hand and the right hand side of (2) gives E[U(W + ɛ)] E[U(W ) + ɛu (W ) + ɛ2 2 U (W )] and = U(W ) + σ2 ɛ 2 U (W ), U(W π) U(W ) πu (W ), respectively, where σɛ 2 of ɛ. := E(ɛ2 ) is the variance Substituting back into (2), we get where π σ2 ɛ 2 A(W ), A(W ) := U (W ) U (W ) (3) is the Arrow Pratt measure of absolute risk aversion, which can be viewed as a measure of the degree of concavity of the utility function.

8 The division by U (W ) can be interpreted in the sense that it makes A(W ) independent of affine transformations of U( ), which do not alter the preference ordering. In view of the above, it is reasonable to say that Agent 1 is locally more risk averse than Agent 2 if both have the same initial wealth W and π 1 > π 2, or, equivalently, A 1 (W ) > A 2 (W ). We can say more, however. Namely, Pratt s theorem on global risk aversion states that the following three conditions are equivalent: 1) π 1 > π 2 for any zero mean risk and any wealth level W. 2) A 1 (W ) > A 2 (W ) for all W. 3) U 1 (W ) = G(U 2 (W )) for some increasing strictly concave function G, where U 1 and U 2 are the utility functions of Agents 1 and 2, respectively. Pratt (1964): Risk Aversion in the Small and in the Large. Econometrica, 32:

9 A related question is what happens with the degree of risk aversion as a function of W. Since Arrow (1971), it is usually argued that absolute risk aversion should be a decreasing function of wealth. For example, a lottery to gain or loose 100 is potentially life threatening for an agent with initial wealth W = 101, whereas it is negligible for an agent with wealth W = The former person should be willing to pay more than the latter for the elimination of such a risk. Thus, we may require that the risk premium associated with any risk is decreasing in wealth. It can be shown that this holds if and only if the Arrow Pratt measure of absolute risk aversion is decreasing in wealth, i.e., dπ dw < 0 da(w ) dw < 0. Essays in the Theory of Rsik Bearing. Markham, Chicago.

10 Thus, we can equivalently require that A (W ) < 0 for all W. Note that this requirement means A (W ) = U (W )U (W ) U (W ) 2 U (W ) 2 < 0. A necessary condition for this to hold is U (W ) > 0, so that we can also sign the third derivative of the utility function.

11 Mean Variance Analysis and Expected Utility Theory Mean variance portfolio theory (or µ σ analysis) assumes that investor s preferences can be described by a preference function, V (µ, σ), over the mean (µ) and the standard deviation (σ) of the portfolio return. Standard assumptions are V µ := V (µ, σ) µ > 0, and V (µ, σ) V σ := < 0, σ which may be interpreted as risk aversion, if the variance is an appropriate measure of risk.

12 The existence of such a preference function would greatly simplify things, because, then, the class of potentially optimal portfolios are those with the greatest expected return for a given level of variance and, simultaneously, the smallest variance for a given expected return. Moreover, portfolio means and variances are easily computed once the mean vector and the covariance matrix of the asset returns are given, and efficient sets can be worked out straightforwardly. However, in general, mean variance analysis and the expected utility approach are not necessarily equivalent. Thus, because the expected utility paradigm, as a general approach to decision making under risk, has a sound theoretical basis (in contrast to the mean variance criterion), we are interested in situations where both approaches are equivalent.

13 Example Consider random variable X with probability function p(x) = 0.8 if x = if x = 100 with E(X) = 20.8 and Var(X) = Let random variable Y have probability function p(y) = 0.99 if y = if x = 1000 with E(Y ) = 19.9 and Var(Y ) = Thus E(Y ) < E(X) and Var(Y ) > Var(X). Let U(W ) = log W. Then, E[U(X)] = 0.8 log log 100 = < = 0.99 log log 1000 = E[U(Y )]. Taken from Hanoch and Levy (1969): The Efficiency Analysis of Choices Involving Risk. Review of Economic Studies, 36:

14 To make µ σ analysis reconcilable with expected utility theory, we have to make assumptions about either 1) the utility function of the decision maker, or 2) the return distribution. 1) Restricting the Utility Function: Quadratic Utility Assume that the investor s expected utility function is given by U(W ) = W b 2 W 2, b > 0. Note that this is the most general form of quadratic utility, because expected utility functions are unique only up to an affine transformation. Marginal utility is U (W ) = 1 bw > 0 for W < 1/b. As all concave quadratic functions are decreasing after a certain point, care must be taken to make sure that the outcome remains in the lower, relevant range of utility.

15 Furthermore, U (W ) = b < 0, so b > 0 guarantees risk aversion. Expected utility is E[U(W )] = E(W ) b 2 E(W 2 ) = E(W ) b 2 [Var(W ) + E2 (W )] = µ b 2 (σ2 + µ 2 ) = : V (µ, σ), a function of µ and σ. V (µ, σ) is called a µ σ preference function. We have V µ : = V µ V σ : = V σ = 1 bµ > 0, as max{w } < 1/b = bσ < 0. In this framework, variance is the appropriate measure of risk.

16 For the analysis of portfolio selection, it will be useful to introduce the concept of an indifference curve in (σ, µ) space. Definition 2 The indifference curve in (σ, µ) space, relative to a given utility level V, is the locus of points (σ, µ) along which expected utility is constant, i.e., equal to V. For quadratic utility, indifference curves can be derived from µ b 2 (σ2 + µ 2 ) V. Multiply both sides by 2/b and add 1/b 2, to get σ 2 + µ 2 2 b µ + 1 b 2 2 ( ) 1 b 2b V, or ( σ 2 + µ 1 ) 2 const. b Note that 1/(2b) V > 0, as max{u} = 1/(2b). Thus, indifference curves are semicircles centered at (0, 1/b).

17 In particular, in the economically relevant range (µ < 1/b), they are convex in (σ, µ) space, i.e., d 2 µ dσ 2 > 0. V (µ,σ)=v Moving in a westward or northern direction in (the economically relevant part of) (σ, µ) space, the indifference curves will correspond to higher levels of expected utility, V, as the same mean is associated with a lower standard deviation, or the same standard deviation is associated with a higher mean. Example Let b = 0.25, and let V 1 = 1, and V 2 = 1.5, so that the indifference curve associated with V 2 reflects a higher expected utility level than that associated with V 1.

18 7 Indifference curves for quadratic utility (b=0.25) 6 5 1/b mean, µ 4 3 V 2 = V 1 = standard deviation, σ

19 Drawbacks of quadratic utility Marginal utility becomes negative for W > 1/b. Quadratic utility implies globally increasing absolute risk aversion, given by A(W ) = U (W ) U (W ) = b 1 bw. This is increasing in b, which is reasonable. But, ( ) A b 2 (W ) = > 0. 1 bw In a portfolio context, Arrow (1971) has shown that this implies that wealthier people invest less in risky assets, which contradicts both intuition and fact. In view of its consequences, Arrow (1971) has characterized the quadratic utility assumption as absurd.

20 1) Restricting the Return Distribution: Normality If the returns have a multivariate normal distribution, then the portfolio return (and wealth) will likewise be normal. Thus, portfolio return (and wealth) distributions will differ only by means and variances. In case of normally distributed wealth, we can write { } E[U(W )] = U(W ) 1 (W µ)2 exp 2πσ 2σ 2 dw = = V (µ, σ), U(σW + µ)φ(w )dw where φ( ) is the standard normal density. We have V µ = U (σw + µ)φ(w )dw > 0

21 by the positivity of marginal utility, and V σ = = W U (σw + µ)φ(w )dw 0 W U (σw + µ)φ(w )dw + 0 W U (σw + µ)φ(w )dw = W [U (σw + µ) U ( σw + µ)]φ(w )dw, 0 where the last line follows from the symmetry of φ(w ). By risk aversion, i.e., U (W ) < 0 for all W, we have U (σw +µ) < U ( σw +µ) for W > 0, thus V σ < 0, i.e., investors like higher expected returns and dislike return variance. To derive indifference curves, set V (µ, σ) V. Differentiating implicitly, dµ = V σ > 0. dσ V (µ,σ)=v V µ

22 Not surprisingly, indifference curves are upward sloping in (σ, µ) space. To figure out their shape, we follow Tobin (1958) and consider pairs (µ, σ) and (µ, σ ) being on the same indifference locus. By concavity of the utility function, 1 2 U(σW + µ) U(σ W + µ ) ( σ + σ < U W + µ + ) µ. 2 2 Taking expectations, we have ( µ + µ V, σ + ) σ > V (µ, σ) = V (µ, σ ), 2 2 so that the indifference curves are again convex in (σ, µ)-space (see the picture). Liquidity Preference as Behavior Towards Risk. Review of Economic Studies, 25:

23 V[(µ+µ*)/2,(σ+σ*)/2] µ* (µ + µ*)/2 µ V(µ,σ) = V(µ*,σ*) σ σ* (σ + σ*)/2

24 Example When normality of returns is assumed, a convenient choice of U( ) is the constant absolute risk aversion (CARA) utility function, given by U(W ) = exp{ cw }, c > 0, where c = A(W ) = U (W )/U (W ) is the constant coefficient of absolute risk aversion. Then, E[U(W )] = exp { cµ + c22 σ2 }, and the µ σ preference function may be written as V (µ, σ) = µ c 2 σ2. Indifference curves relative to a preference level V are defined by µ V + c 2 σ2, where V = log{ E[U(W )]}/c.

25 Note that µ σ analysis does not necessarily require normality of returns but holds within a more general class of distributions, namely, the elliptical distributions, see, e.g., Ingersoll (1987): Theory of Financial Decision Making. Roman and Littlefield Publishing, Savage. Finally, even if µ σ analysis does not hold exactly, it will often serve as a reasonable approximation to the true, expected utility maximizing solution, with the advantage of greatly simplifying the decision problem. In fact, a number of studies (using real data and a range of utility functions) have shown that the exact solutions are often rather close to or even economically indistinguishable from µ σ efficient portfolios (to be defined below).

### Choice under Uncertainty

Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory

### 2. Mean-variance portfolio theory

2. Mean-variance portfolio theory (2.1) Markowitz s mean-variance formulation (2.2) Two-fund theorem (2.3) Inclusion of the riskfree asset 1 2.1 Markowitz mean-variance formulation Suppose there are N

### An Overview of Asset Pricing Models

An Overview of Asset Pricing Models Andreas Krause University of Bath School of Management Phone: +44-1225-323771 Fax: +44-1225-323902 E-Mail: a.krause@bath.ac.uk Preliminary Version. Cross-references

### Fund Manager s Portfolio Choice

Fund Manager s Portfolio Choice Zhiqing Zhang Advised by: Gu Wang September 5, 2014 Abstract Fund manager is allowed to invest the fund s assets and his personal wealth in two separate risky assets, modeled

### C2922 Economics Utility Functions

C2922 Economics Utility Functions T.C. Johnson October 30, 2007 1 Introduction Utility refers to the perceived value of a good and utility theory spans mathematics, economics and psychology. For example,

### Lecture 13: Risk Aversion and Expected Utility

Lecture 13: Risk Aversion and Expected Utility Uncertainty over monetary outcomes Let x denote a monetary outcome. C is a subset of the real line, i.e. [a, b]. A lottery L is a cumulative distribution

### 1 Portfolio mean and variance

Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a one-period investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring

### Financial Services [Applications]

Financial Services [Applications] Tomáš Sedliačik Institute o Finance University o Vienna tomas.sedliacik@univie.ac.at 1 Organization Overall there will be 14 units (12 regular units + 2 exams) Course

### Covariance and Correlation

Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such

### Managerial Economics

Managerial Economics Unit 9: Risk Analysis Rudolf Winter-Ebmer Johannes Kepler University Linz Winter Term 2012 Managerial Economics: Unit 9 - Risk Analysis 1 / 1 Objectives Explain how managers should

### Economics 1011a: Intermediate Microeconomics

Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore

### Lecture 1: Asset Allocation

Lecture 1: Asset Allocation Investments FIN460-Papanikolaou Asset Allocation I 1/ 62 Overview 1. Introduction 2. Investor s Risk Tolerance 3. Allocating Capital Between a Risky and riskless asset 4. Allocating

### Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10?

Return, Risk, and Risk Aversion Holding Period Return Ending Price - Beginning Price + Intermediate Income Return = Beginning Price R P t+ t+ = Pt + Dt P t An Example You bought IBM stock at \$40 last month.

### Lecture 11 Uncertainty

Lecture 11 Uncertainty 1. Contingent Claims and the State-Preference Model 1) Contingent Commodities and Contingent Claims Using the simple two-good model we have developed throughout this course, think

### MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

### Analyzing the Demand for Deductible Insurance

Journal of Risk and Uncertainty, 18:3 3 1999 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Analyzing the emand for eductible Insurance JACK MEYER epartment of Economics, Michigan State

### The Effect of Ambiguity Aversion on Insurance and Self-protection

The Effect of Ambiguity Aversion on Insurance and Self-protection David Alary Toulouse School of Economics (LERNA) Christian Gollier Toulouse School of Economics (LERNA and IDEI) Nicolas Treich Toulouse

### Mean Variance Analysis

Finance 400 A. Penati - G. Pennacchi Mean Variance Analysis Consider the utility of an individual who invests her beginning-of-period wealth, W 0,ina particular portfolio of assets. Let r p betherandomrateofreturnonthisportfolio,sothatthe

### 3. The Economics of Insurance

3. The Economics of Insurance Insurance is designed to protect against serious financial reversals that result from random evens intruding on the plan of individuals. Limitations on Insurance Protection

### On Compulsory Per-Claim Deductibles in Automobile Insurance

The Geneva Papers on Risk and Insurance Theory, 28: 25 32, 2003 c 2003 The Geneva Association On Compulsory Per-Claim Deductibles in Automobile Insurance CHU-SHIU LI Department of Economics, Feng Chia

### Understanding the Impact of Weights Constraints in Portfolio Theory

Understanding the Impact of Weights Constraints in Portfolio Theory Thierry Roncalli Research & Development Lyxor Asset Management, Paris thierry.roncalli@lyxor.com January 2010 Abstract In this article,

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### Financial Market Microstructure Theory

The Microstructure of Financial Markets, de Jong and Rindi (2009) Financial Market Microstructure Theory Based on de Jong and Rindi, Chapters 2 5 Frank de Jong Tilburg University 1 Determinants of the

### We have discussed the notion of probabilistic dependence above and indicated that dependence is

1 CHAPTER 7 Online Supplement Covariance and Correlation for Measuring Dependence We have discussed the notion of probabilistic dependence above and indicated that dependence is defined in terms of conditional

### Decision & Risk Analysis Lecture 6. Risk and Utility

Risk and Utility Risk - Introduction Payoff Game 1 \$14.50 0.5 0.5 \$30 - \$1 EMV 30*0.5+(-1)*0.5= 14.5 Game 2 Which game will you play? Which game is risky? \$50.00 Figure 13.1 0.5 0.5 \$2,000 - \$1,900 EMV

### A Portfolio Model of Insurance Demand. April 2005. Kalamazoo, MI 49008 East Lansing, MI 48824

A Portfolio Model of Insurance Demand April 2005 Donald J. Meyer Jack Meyer Department of Economics Department of Economics Western Michigan University Michigan State University Kalamazoo, MI 49008 East

### Topic 4: Multivariate random variables. Multiple random variables

Topic 4: Multivariate random variables Joint, marginal, and conditional pmf Joint, marginal, and conditional pdf and cdf Independence Expectation, covariance, correlation Conditional expectation Two jointly

### Lecture 10 - Risk and Insurance

Lecture 10 - Risk 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 about how risk affects choice

### Portfolio Allocation and Asset Demand with Mean-Variance Preferences

Portfolio Allocation and Asset Demand with Mean-Variance Preferences Thomas Eichner a and Andreas Wagener b a) Department of Economics, University of Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany.

### How to assess the risk of a large portfolio? How to estimate a large covariance matrix?

Chapter 3 Sparse Portfolio Allocation This chapter touches some practical aspects of portfolio allocation and risk assessment from a large pool of financial assets (e.g. stocks) How to assess the risk

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### A Generalization of the Mean-Variance Analysis

A Generalization of the Mean-Variance Analysis Valeri Zakamouline and Steen Koekebakker This revision: May 30, 2008 Abstract In this paper we consider a decision maker whose utility function has a kink

### Uncertainty. BEE2017 Microeconomics

Uncertainty BEE2017 Microeconomics Uncertainty: The share prices of Amazon and the difficulty of investment decisions Contingent consumption 1. What consumption or wealth will you get in each possible

### SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline

### Lesson 5. Risky assets

Lesson 5. Risky assets Prof. Beatriz de Blas May 2006 5. Risky assets 2 Introduction How stock markets serve to allocate risk. Plan of the lesson: 8 >< >: 1. Risk and risk aversion 2. Portfolio risk 3.

### Saving and the Demand for Protection Against Risk

Saving and the Demand for Protection Against Risk David Crainich 1, Richard Peter 2 Abstract: We study individual saving decisions in the presence of an endogenous future consumption risk. The endogeneity

### 4. Joint Distributions of Two Random Variables

4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint

### FOUNDATIONS OF PORTFOLIO THEORY

FOUNDATIONS OF PORTFOLIO THEORY Nobel Lecture, December 7, 1990 by HARRY M. MARKOWITZ Baruch College, The City University of New York, New York, USA When I studied microeconomics forty years ago, I was

### Economics of Insurance

Economics of Insurance In this last lecture, we cover most topics of Economics of Information within a single application. Through this, you will see how the differential informational assumptions allow

### Microeconomics Tutoral 5: Choice under Uncertainty and Revealed Preferences October, 21th

Université Paris 1 Panthéon Sorbonne Microeconomics: PSME - ngelo Secchi 1 Master I, cademic year 2015-2016 Microeconomics tutorials supervised by Elisa Baku 2 and ntoine Marsaudon 3. Microeconomics Tutoral

### Risk and Insurance. Vani Borooah University of Ulster

Risk and Insurance Vani Borooah University of Ulster Gambles An action with more than one possible outcome, such that with each outcome there is an associated probability of that outcome occurring. If

### 2. Information Economics

2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### Combining decision analysis and portfolio management to improve project selection in the exploration and production firm

Journal of Petroleum Science and Engineering 44 (2004) 55 65 www.elsevier.com/locate/petrol Combining decision analysis and portfolio management to improve project selection in the exploration and production

### P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )

Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =

### Asset Pricing. Chapter IV. Measuring Risk and Risk Aversion. June 20, 2006

Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Measuring Risk Aversion Utility function Indifference Curves U(Y) tangent lines U(Y + h) U[0.5(Y + h) + 0.5(Y h)] 0.5U(Y + h) + 0.5U(Y h) U(Y

### CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS

CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e). (b) A higher borrowing is a consequence of the risk of the borrowers default. In perfect markets with no additional

### Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

### Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents

Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents William H. Sandholm January 6, 22 O.. Imitative protocols, mean dynamics, and equilibrium selection In this section, we consider

### .4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.

Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, \$100, for one month, and is considering

### 1 Uncertainty and Preferences

In this chapter, we present the theory of consumer preferences on risky outcomes. The theory is then applied to study the demand for insurance. Consider the following story. John wants to mail a package

### Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

### The Values of Relative Risk Aversion Degrees

The Values of Relative Risk Aversion Degrees Johanna Etner CERSES CNRS and University of Paris Descartes 45 rue des Saints-Peres, F-75006 Paris, France johanna.etner@parisdescartes.fr Abstract This article

### Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*:

Problem 1. Consider a risky asset. Suppose the expected rate of return on the risky asset is 15%, the standard deviation of the asset return is 22%, and the risk-free rate is 6%. What is your optimal position

### Regret and Rejoicing Effects on Mixed Insurance *

Regret and Rejoicing Effects on Mixed Insurance * Yoichiro Fujii, Osaka Sangyo University Mahito Okura, Doshisha Women s College of Liberal Arts Yusuke Osaki, Osaka Sangyo University + Abstract This papers

### The Bivariate Normal Distribution

The Bivariate Normal Distribution This is Section 4.7 of the st edition (2002) of the book Introduction to Probability, by D. P. Bertsekas and J. N. Tsitsiklis. The material in this section was not included

### Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

### Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

### An Introduction to Portfolio Theory

An Introduction to Portfolio Theory John Norstad j-norstad@northwestern.edu http://www.norstad.org April 10, 1999 Updated: November 3, 2011 Abstract We introduce the basic concepts of portfolio theory,

### Some probability and statistics

Appendix A Some probability and statistics A Probabilities, random variables and their distribution We summarize a few of the basic concepts of random variables, usually denoted by capital letters, X,Y,

### Choice Under Uncertainty

Decision Making Under Uncertainty Choice Under Uncertainty Econ 422: Investment, Capital & Finance University of ashington Summer 2006 August 15, 2006 Course Chronology: 1. Intertemporal Choice: Exchange

### Optimization under uncertainty: modeling and solution methods

Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino e-mail: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte

Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a risk-free and risky asset, with returns given by r f and E(r p ), respectively. Let y be the

### 5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

### FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

### Christfried Webers. Canberra February June 2015

c Statistical Group and College of Engineering and Computer Science Canberra February June (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 829 c Part VIII Linear Classification 2 Logistic

### Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier

Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier Prof. Massimo Guidolin Portfolio Management Spring 2016 Overview The

### How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

### Chapter 2 Portfolio Management and the Capital Asset Pricing Model

Chapter 2 Portfolio Management and the Capital Asset Pricing Model In this chapter, we explore the issue of risk management in a portfolio of assets. The main issue is how to balance a portfolio, that

### Chapter 25: Exchange in Insurance Markets

Chapter 25: Exchange in Insurance Markets 25.1: Introduction In this chapter we use the techniques that we have been developing in the previous 2 chapters to discuss the trade of risk. Insurance markets

### Constrained Bayes and Empirical Bayes Estimator Applications in Insurance Pricing

Communications for Statistical Applications and Methods 2013, Vol 20, No 4, 321 327 DOI: http://dxdoiorg/105351/csam2013204321 Constrained Bayes and Empirical Bayes Estimator Applications in Insurance

### Principle of Data Reduction

Chapter 6 Principle of Data Reduction 6.1 Introduction An experimenter uses the information in a sample X 1,..., X n to make inferences about an unknown parameter θ. If the sample size n is large, then

### Compulsory insurance and voluntary self-insurance: substitutes or complements? A matter of risk attitudes

Compulsory insurance and voluntary self-insurance: substitutes or complements? A matter of risk attitudes François Pannequin, Ecole Normale Supérieure de Cachan and Centre d Economie de la Sorbonne pannequin@ecogest.ens-cachan.fr

### INFERIOR GOOD AND GIFFEN BEHAVIOR FOR INVESTING AND BORROWING

INFERIOR GOOD AND GIFFEN BEHAVIOR FOR INVESTING AND BORROWING Felix Kubler University of Zurich Larry Selden Columbia University University of Pennsylvania August 14, 211 Xiao Wei Columbia University Abstract

### Lecture 15. Ranking Payoff Distributions: Stochastic Dominance. First-Order Stochastic Dominance: higher distribution

Lecture 15 Ranking Payoff Distributions: Stochastic Dominance First-Order Stochastic Dominance: higher distribution Definition 6.D.1: The distribution F( ) first-order stochastically dominates G( ) if

### Black-Litterman Return Forecasts in. Tom Idzorek and Jill Adrogue Zephyr Associates, Inc. September 9, 2003

Black-Litterman Return Forecasts in Tom Idzorek and Jill Adrogue Zephyr Associates, Inc. September 9, 2003 Using Black-Litterman Return Forecasts for Asset Allocation Results in Diversified Portfolios

### Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

### Lecture Notes on Elasticity of Substitution

Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before

### Random Vectors and the Variance Covariance Matrix

Random Vectors and the Variance Covariance Matrix Definition 1. A random vector X is a vector (X 1, X 2,..., X p ) of jointly distributed random variables. As is customary in linear algebra, we will write

### General Sampling Methods

General Sampling Methods Reference: Glasserman, 2.2 and 2.3 Claudio Pacati academic year 2016 17 1 Inverse Transform Method Assume U U(0, 1) and let F be the cumulative distribution function of a distribution

### 14.16. a. Again, this question will be answered on the basis of subjective judgment. Here are some possible types :

14.11. If the two agree on the bet, then for both EU(et) > U(). a. If P(Rain tomorrow) =.1, and they agree on this probability, For,.1 U(4).9 U(-1) > U(), and E(Payoff) = -5. For,.1 U(-4).9 U(1) > U(),

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### The Capital Asset Pricing Model

Finance 400 A. Penati - G. Pennacchi The Capital Asset Pricing Model Let us revisit the problem of an investor who maximizes expected utility that depends only on the expected return and variance (or standard

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### Multi-variable Calculus and Optimization

Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus

### Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

### Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model

Finance 400 A. Penati - G. Pennacchi Market Micro-Structure: Notes on the Kyle Model These notes consider the single-period model in Kyle (1985) Continuous Auctions and Insider Trading, Econometrica 15,

### Financial Markets. Itay Goldstein. Wharton School, University of Pennsylvania

Financial Markets Itay Goldstein Wharton School, University of Pennsylvania 1 Trading and Price Formation This line of the literature analyzes the formation of prices in financial markets in a setting

### CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between r P and

5. Capital Asset Pricing Model and Factor Models Capital market line (CML) CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between

### SCREENING IN INSURANCE MARKETS WITH ADVERSE SELECTION AND BACKGROUND RISK. Keith J. Crocker

SCREENING IN INSURANCE MARKETS WIT ADVERSE SEECTION AND BACKGROUND RISK Keith J. Crocker University of Michigan Business School 70 Tappan Street Ann Arbor, MI 4809 and Arthur Snow Department of Economics

### Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS

Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS 5-1 a. Stand-alone risk is only a part of total risk and pertains to the risk an investor takes by holding only one asset. Risk is

### Preferences. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Preferences 1 / 20

Preferences M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Preferences 1 / 20 Preference Relations Given any two consumption bundles x = (x 1, x 2 ) and y = (y 1, y 2 ), the

### An Introduction to Utility Theory

An Introduction to Utility Theory John Norstad j-norstad@northwestern.edu http://www.norstad.org March 29, 1999 Updated: November 3, 2011 Abstract A gentle but reasonably rigorous introduction to utility

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### 3 Random vectors and multivariate normal distribution

3 Random vectors and multivariate normal distribution As we saw in Chapter 1, a natural way to think about repeated measurement data is as a series of random vectors, one vector corresponding to each unit.

### WHY THE LONG TERM REDUCES THE RISK OF INVESTING IN SHARES. A D Wilkie, United Kingdom. Summary and Conclusions

WHY THE LONG TERM REDUCES THE RISK OF INVESTING IN SHARES A D Wilkie, United Kingdom Summary and Conclusions The question of whether a risk averse investor might be the more willing to hold shares rather

### Exam Introduction Mathematical Finance and Insurance

Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closed-book exam. The exam does not use scrap cards. Simple calculators are allowed. The questions

### The Simple Linear Regression Model: Specification and Estimation

Chapter 3 The Simple Linear Regression Model: Specification and Estimation 3.1 An Economic Model Suppose that we are interested in studying the relationship between household income and expenditure on