# Factor Model. Arbitrage Pricing Theory. Systematic Versus Non-Systematic Risk. Intuitive Argument

Save this PDF as:

Size: px
Start display at page:

Download "Factor Model. Arbitrage Pricing Theory. Systematic Versus Non-Systematic Risk. Intuitive Argument"

## Transcription

1 Ross [1],[]) presents the aritrage pricing theory. The idea is that the structure of asset returns leads naturally to a odel of risk preia, for otherwise there would exist an opportunity for aritrage profit. Factor Model Assue that there exists a risk-free asset, and consider a factor odel for the excess return ξ on a set of assets: ξ = + B f + e. The ean excess return is the vector of risk preia. 1 Here and Ef)=0 Varf)=I, Ee)=0 Var e)=d. Here D is diagonal, and f and e are uncorrelated. One refers to f as factors. The eta coefficients B are also called factor loadings. 3 Systeatic Versus Non-Systeatic Risk Assue that ost of the coponents of B are not near zero. The diagonal eleents of D are not too large, and the nuer of assets n is large. Then the ter B f represents ost of the variation in the returns. Interpret B f as the systeatic risk, and e as the non-systeatic risk. One can argue that the non-systeatic risk can e eliinated y diversification, so the eta coefficients B should deterine the risk preiu. 4 Intuitive Arguent Ross gives the following intuitive arguent. Consider a portfolio x. Each coponent denotes the fraction of wealth invested in that asset, and 1 1 x is the fraction invested in the risk-free asset. The excess return on the portfolio is ξ x = x + f Bx + e x. Suppose that the portfolio is well-diversified: ost of the coponents of x are non-zero. By the law of large nuers, e x 0; diversification eliinates the non-systeatic risk. If the portfolio is chosen to eliinate the systeatic risk Bx = 0), then the resulting portfolio is nearly risk-free. Then the law of one price iplies x =

2 Ross suarizes his arguent y the following: Bx = 0 x = 0. 1) Of course this arguent is not valid for an aritrary portfolio ut only for a well-diversified portfolio.) Exact Factor Model Consider first an exact factor odel, in which e = 0 so D = 0). Reark 1 In the exact factor odel, the law of one price is equivalent to the condition 1). 7 8 Law of One Price For the exact factor odel, the law of one price 1) says that is orthogonal to NB). By the fundaental theore of linear algera, ust lie in R B ). Thus we otain the following theore. Theore ) In the exact factor odel, the law of one price holds if only if the ean excess return is a linear coination of the eta coefficients, for soe. = B, ) 9 10 The Versus the Capital-Asset Pricing Model Like the capital-asset pricing odel, the systeatic risk eodied in the eta coefficients deterines the risk preia. However the reasoning is different. The capital-asset pricing odel is derived fro arket equiliriu, the equality of asset deand and supply. This equality iplies that the arket portfolio ust e efficient, and a typical investor holds the arket portfolio. In contrast, the aritrage pricing theory is derived fro an aritrage arguent, not a arket equiliriu arguent. The risk preia ) follow fro the factor structure of the asset returns. Asset supply is irrelevant to the arguent. If soe set of asset returns has the factor structure, then the conclusion follows for this set. 11 1

3 Weighted Least Squares We next suppose that the factor odel is not exact, that e 0. Then any value for is consistent with the law of one price the only portfolio with a constant excess return is x = 0). Nevertheless we put forward a duality arguent that ) isa good approxiation. We choose y weighted least squares. Prole 3 Prial) in [ 1 B ) D B ) ]. The prial is a weighted regression of the ean on the eta coefficients. The arguent is that the value is sall, for the optiu Dual An alternate axiization prole is dual to the prial. Prole 4 Dual) sup 1 D δ B =0 ). By definition, the indicator function δ is zero if the elongs to the set such that B = 0, and is otherwise. The prial and the dual are equivalent proles, in that either one can e calculated fro the other, and we explain their relationship. 15 The ojective function of the prial is jointly convex in and, and it follows that the value function V ) is convex. The choice variale is, and the perturation variale is. 16 Conjugate Definition 5 Conjugate) The conjugate of V ) is V ) :[, V )]. The conjugate is a convex function. Conjugate Duality Proposition 6 Conjugate Duality) Under general conditions, the conjugate of the conjugate is the original function, V )=V ). In atheatics, conjugate has any definitions, ut always the conjugate of the conjugate is the original; an exaple is the coplex conjugate

4 Dual Definition 7 Dual) For a prial with value function V ), the dual is the axiization prole sup [, V )]. By conjugate duality, the optiu value in the dual is V ). Theore 8 No Duality Gap) The iniu value in the prial is the axiu value in the dual. 19 Calculation of the Dual fro the Prial Let us derive the dual prole 4) y calculating the conjugate: V ), [, V )] { [ 1, in { [, + sup 1 [, 1 B ) ) ]} D B B ) ) ]} D B B ) D B 0 ) ] Sustituting c := B separates the axiization into two parts: V ),c c,c + B 1 ) c D c,c 1 ) c D c = 1 D + δ B =0, + sup B, Prial Greater than or Equal to the Dual For this prole, let us verify directly the asic duality properties. Always the value of the prial is greater than or equal to the value of the dual. If B 0, then δ B =0 =, so the value of the dual is. Necessarily the value of the prial is greater than or equal to the value of the dual. otaining the dual prole 4). 1 If B = 0, then the prial less the dual is 1 B ) ) D B [, 1 ] D = 1 [ ) D B ] [ ) D D B ] 0 3) a su of squares. Here,B = B, = 0, = 0. Again, the value of the prial is greater than or equal to the value of the dual. 3 Prial Equal to the Dual In the prial, the first-order condition for a iniu is ) BD B = 0. Given the solution to the prial, solve the dual y setting ) = D B. By the first-order condition, B = 0, so the dual constraint is satisfied. Furtherore, the quadratic for 3) is zero. That the value of the prial equals the value of the dual proves that indeed we have the solution to oth proles. 4

5 Envelope Theore Applying the envelope theore to the prial yields ) V )/ = D B =, in which is the solution to the prial and is the solution to the dual. This relationship is a general duality result: the solution to the dual shows how the perturation variale affects the optiu value. The solution to the dual is a Lagrange ultiplier. First-Order Condition Osolete Even though this verification akes use of the first-order condition, a thee of duality theory is that the first-order condition is osolete. Because there is no duality gap, one can solve the prial and the dual siultaneously, y setting the prial equal to the dual. A systeatic procedure then finds the optiu values for the choice variales in the prial and the dual, to achieve this equality. 5 6 Econoic Interpretation of the Dual The dual has an econoic interpretation. The choice variale is a portfolio of investents in the risky assets, with 1 1 as the investent in the risk-free asset. The constraint B = 0 says that the portfolio is chosen to e uncorrelated with the factors; the variaility of the return arises solely fro the non-systeatic risk. Thus is the excess return on the portfolio, and D is the variance of the return. 7 The dual reseles the derivation of the separation theore with sall risks, in which the ojective function is a linear function of the ean excess return and the variance. Just as for the separation theore, the solution to the dual axiizes the ratio of the ean excess return to the standard deviation, D, 4) here suject to the constraint that the portfolio return is uncorrelated with the factors. Furtherore, the square of the axiu value of this ratio is the optiu value of the dual. 8 Efficient Frontier Define s as the axiu value of the ratio 4). It is the slope of an efficient frontier, suject to the constraint that the portfolio return is uncorrelated with the factors. The slope s of the efficient frontier is of course greater than or equal to s. Upper Bound to the Weighted Su of Squares As there is no duality gap, in [ 1 B ) D B ) ] = s s. 5) Thus the slope s of the efficient frontier provides an upper ound to the weighted su of squares. The slope provides an upper ound to how far the predicted ean B can deviate fro the actual ean. 9 30

6 Many Assets A key property is that this upper ound is independent of the nuer n of assets. Conclusion 9 ) If the nuer of assets is large, it follows that for ost assets. B 6) Otherwise the upper ound would e violated. The approxiation ay e poor for a few assets, ut for ost assets the approxiation ust e excellent. 31 Trivial Case Note that this conclusion holds even for the trivial case B = 0, for which B = 0. Then the duality relation 5) says that so 1 D = s s, 0 is a good approxiation. For ost assets, the ean excess return is near zero. 3 Irrelevance of Non-Systeatic Risk? Ross s point of view is that the error e is non-systeatic risk, and this risk should e eliinated y portfolio diversification. Hence the non-systeatic risk should have no effect on ean returns. If this point of view is true, then s should e sall, sall even if s is large. By the duality relation 5), it would then follow that the approxiation 6) would e extreely good. Diagonal Variance That the variance of the error e is diagonal is iportant, and allows one to see the error as non-systeatic risk. The duality relation 5) holds regardless of whether D is in fact diagonal. If the coponents of e were highly correlated, then the approxiation 6) ight e poor for any assets. For exaple, for the trivial case B = 0, there is no presuption that ost of the coponents of should e near zero References [1] S. Ross. Return, risk, and aritrage. In I. Friend and J. L. Bicksler, editors, Risk and Return in Finance, pages Ballinger, Caridge, MA, HG4539R57. [] S. A. Ross. The aritrage theory of capital asset pricing. Journal of Econoic Theory, 133): , Deceer HB1J

### IV Approximation of Rational Functions 1. IV.C Bounding (Rational) Functions on Intervals... 4

Contents IV Approxiation of Rational Functions 1 IV.A Constant Approxiation..................................... 1 IV.B Linear Approxiation....................................... 3 IV.C Bounding (Rational)

Advanced Microeconoics (ES3005) Advanced Microeconoics (ES3005) Matheatics Review : The Lagrange Multiplier Outline: I. Introduction II. Duality Theory: Co Douglas Exaple III. Final Coents I. Introduction

### Quality evaluation of the model-based forecasts of implied volatility index

Quality evaluation of the odel-based forecasts of iplied volatility index Katarzyna Łęczycka 1 Abstract Influence of volatility on financial arket forecasts is very high. It appears as a specific factor

### Page 1. Algebra 1 Unit 5 Working with Exponents Per

Algera 1 Unit 5 Working with Exponents Nae Per Ojective 1: Siplifying Expressions with Exponents (Powers INSIDE parentheses) Exponents are a shortcut. They are a quicker way of writing repeated ultiplication.

### Use of extrapolation to forecast the working capital in the mechanical engineering companies

ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2014. Vol. 1. No. 1. 23 28 Use of extrapolation to forecast the working capital in the echanical engineering copanies A. Cherep, Y. Shvets Departent of finance

### Fixed-Income Securities and Interest Rates

Chapter 2 Fixed-Incoe Securities and Interest Rates We now begin a systeatic study of fixed-incoe securities and interest rates. The literal definition of a fixed-incoe security is a financial instruent

### Financial Risk: Credit Risk, Lecture 1

Financial Risk: Credit Risk, Lecture 1 Alexander Herbertsson Centre For Finance/Departent of Econoics School of Econoics, Business and Law, University of Gothenburg E-ail: Alexander.Herbertsson@econoics.gu.se

### Analysis of the purchase option of computers

Analysis of the of coputers N. Ahituv and I. Borovits Faculty of Manageent, The Leon Recanati Graduate School of Business Adinistration, Tel-Aviv University, University Capus, Raat-Aviv, Tel-Aviv, Israel

### SOME APPLICATIONS OF FORECASTING Prof. Thomas B. Fomby Department of Economics Southern Methodist University May 2008

SOME APPLCATONS OF FORECASTNG Prof. Thoas B. Foby Departent of Econoics Southern Methodist University May 8 To deonstrate the usefulness of forecasting ethods this note discusses four applications of forecasting

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### Reliability Constrained Packet-sizing for Linear Multi-hop Wireless Networks

Reliability Constrained acket-sizing for inear Multi-hop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois

### 1 Interest rates and bonds

1 Interest rates and bonds 1.1 Copounding There are different ways of easuring interest rates Exaple 1 The interest rate on a one-year deposit is 10% per annu. This stateent eans different things depending

### Note on a generalized wage rigidity result. Abstract

Note on a generalized age rigidity result Ariit Mukheree University of Nottingha Abstract Considering Cournot copetition, this note shos that, if the firs differ in labor productivities, the equilibriu

### Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing Models ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing odels ANSWERS TO END-OF-CHAPTER QUESTIONS 7-1 a. A portfolio is made up of a group of individual assets held in combination. An asset that

### arxiv:0805.1434v1 [math.pr] 9 May 2008

Degree-distribution stability of scale-free networs Zhenting Hou, Xiangxing Kong, Dinghua Shi,2, and Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of

### Is Pay-as-You-Drive Insurance a Better Way to Reduce Gasoline than Gasoline Taxes?

Is Pay-as-You-Drive Insurance a Better Way to Reduce Gasoline than Gasoline Taxes? By Ian W.H. Parry Despite concerns about US dependence on a volatile world oil arket, greenhouse gases fro fuel cobustion,

### International Journal of Management & Information Systems First Quarter 2012 Volume 16, Number 1

International Journal of Manageent & Inforation Systes First Quarter 2012 Volue 16, Nuber 1 Proposal And Effectiveness Of A Highly Copelling Direct Mail Method - Establishent And Deployent Of PMOS-DM Hisatoshi

### Physics 211: Lab Oscillations. Simple Harmonic Motion.

Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.

Advanced Volatility Add a little wind and we get a little increase in volatility. Add a hurricane and we get a huge increase in volatility. (c) 006-0, Gary R. Evans. For educational uses only. May not

### Meadowlark Optics LCPM-3000 Liquid Crystal Polarimeter Application Note: Determination of Retardance by Polarimetry Tommy Drouillard

Meadowlark Optics LCPM- Liquid Crystal Polarieter Application Note: Deterination of Retardance by Polarietry Toy Drouillard 5 Meadowlark Optics, Inc.. Introduction: The iediate purpose of a polarieter

### RECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION. Henrik Kure

RECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION Henrik Kure Dina, Danish Inforatics Network In the Agricultural Sciences Royal Veterinary and Agricultural University

### 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

### Semi-invariants IMOTC 2013 Simple semi-invariants

Sei-invariants IMOTC 2013 Siple sei-invariants These are soe notes (written by Tejaswi Navilarekallu) used at the International Matheatical Olypiad Training Cap (IMOTC) 2013 held in Mubai during April-May,

### Online Appendix I: A Model of Household Bargaining with Violence. In this appendix I develop a simple model of household bargaining that

Online Appendix I: A Model of Household Bargaining ith Violence In this appendix I develop a siple odel of household bargaining that incorporates violence and shos under hat assuptions an increase in oen

### LEASING, LEMONS, AND MORAL HAZARD

LEASING, LEMONS, AND MORAL HAZARD by Justin P. Johnson Johnson Graduate School of Manageent Cornell University Sage Hall Ithaca, NY 14853 jpj25@cornell.edu and Michael Waldan Johnson Graduate School of

### HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES?

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Abstract. We copare the option pricing forulas of Louis Bachelier and Black-Merton-Scholes

### The Velocities of Gas Molecules

he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes

### Lecture L9 - Linear Impulse and Momentum. Collisions

J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9 - Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,

### This paper studies a rental firm that offers reusable products to price- and quality-of-service sensitive

MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol., No. 3, Suer 28, pp. 429 447 issn 523-464 eissn 526-5498 8 3 429 infors doi.287/so.7.8 28 INFORMS INFORMS holds copyright to this article and distributed

### A CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS

A CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS Isaac Zafrany and Sa BenYaakov Departent of Electrical and Coputer Engineering BenGurion University of the Negev P. O. Box

### MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS

MINIMUM VERTEX DEGREE THRESHOLD FOR LOOSE HAMILTON CYCLES IN 3-UNIFORM HYPERGRAPHS JIE HAN AND YI ZHAO Abstract. We show that for sufficiently large n, every 3-unifor hypergraph on n vertices with iniu

### Chapter 7 Portfolio Theory and Other Asset Pricing Models

Chapter 7 Portfolio Theory and Other sset Pricing Models NSWERS TO END-OF-CHPTER QUESTIONS 7-1 a. portfolio is made up of a group of individual assets held in combination. n asset that would be relatively

### Modified Latin Hypercube Sampling Monte Carlo (MLHSMC) Estimation for Average Quality Index

Analog Integrated Circuits and Signal Processing, vol. 9, no., April 999. Abstract Modified Latin Hypercube Sapling Monte Carlo (MLHSMC) Estiation for Average Quality Index Mansour Keraat and Richard Kielbasa

### Data Set Generation for Rectangular Placement Problems

Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George

### Preference-based Search and Multi-criteria Optimization

Fro: AAAI-02 Proceedings. Copyright 2002, AAAI (www.aaai.org). All rights reserved. Preference-based Search and Multi-criteria Optiization Ulrich Junker ILOG 1681, route des Dolines F-06560 Valbonne ujunker@ilog.fr

### Coloring Relatives of Intervals on the Plane, I: Chromatic Number Versus Girth

Europ. J. Cobinatorics (1998) 19, 103 110 Coloring Relatives of Intervals on the Plane, I: Chroatic Nuber Versus Girth A. V. KOSTOCHKA AND J. NEŠETŘIL For the intersection graphs of intervals, rays and

### ESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET

ESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET Francisco Alonso, Roberto Blanco, Ana del Río and Alicia Sanchis Banco de España Banco de España Servicio de Estudios Docuento de

### Algorithmica 2001 Springer-Verlag New York Inc.

Algorithica 2001) 30: 101 139 DOI: 101007/s00453-001-0003-0 Algorithica 2001 Springer-Verlag New York Inc Optial Search and One-Way Trading Online Algoriths R El-Yaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin

### Chapter 8: Newton s Third law

Warning: My approach is a soewhat abbreviated and siplified version of what is in the text, yet just as coplete. Both y treatent and the text s will prepare you to solve the sae probles. Restating Newton

### CHAPTER 11: ARBITRAGE PRICING THEORY

CHAPTER 11: ARBITRAGE PRICING THEORY 1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times

### OpenGamma Documentation Bond Pricing

OpenGaa Docuentation Bond Pricing Marc Henrard arc@opengaa.co OpenGaa Docuentation n. 5 Version 2.0 - May 2013 Abstract The details of the ipleentation of pricing for fixed coupon bonds and floating rate

### The public private partnership paradox

The public private partnership paradox Stephen Gray * UQ Business School, University of Queensland Jason Hall UQ Business School, University of Queensland Grant Pollard Value Decisions ABSTRACT A public

### ON SELF-ROUTING IN CLOS CONNECTION NETWORKS. BARRY G. DOUGLASS Electrical Engineering Department Texas A&M University College Station, TX 77843-3128

ON SELF-ROUTING IN CLOS CONNECTION NETWORKS BARRY G. DOUGLASS Electrical Engineering Departent Texas A&M University College Station, TX 778-8 A. YAVUZ ORUÇ Electrical Engineering Departent and Institute

### Lecture L26-3D Rigid Body Dynamics: The Inertia Tensor

J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L6-3D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall

### INTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE SYSTEMS

Artificial Intelligence Methods and Techniques for Business and Engineering Applications 210 INTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE

### Construction of Graeco Sudoku Square Designs of Odd Orders

onfring International Journal of Data Mining, Vol, No, June 0 37 Construction of Graeco Sudoku Square Designs of Odd Orders J Subraani Abstract--- The Sudoku puzzle typically consists of a nineby-nine

### THREE-PHASE DIODE BRIDGE RECTIFIER

Chapter THREE-PHASE DIODE BRIDGE RECTIFIER The subject of this book is reduction of total haronic distortion (THD) of input currents in three-phase diode bridge rectifiers. Besides the reduction of the

### 10.1 Systems of Linear Equations: Substitution and Elimination

726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations

### SAMPLING METHODS LEARNING OBJECTIVES

6 SAMPLING METHODS 6 Using Statistics 6-6 2 Nonprobability Sapling and Bias 6-6 Stratified Rando Sapling 6-2 6 4 Cluster Sapling 6-4 6 5 Systeatic Sapling 6-9 6 6 Nonresponse 6-2 6 7 Suary and Review of

### AUTOMATIC SATELLITE IMAGE REGISTRATION BY COMBINATION OF STEREO MATCHING AND RANDOM SAMPLE CONSENSUS

AUTOATIC SATELLITE IAGE REGISTRATION BY COBINATION OF STEREO ATCHING AND RANDO SAPLE CONSENSUS Taejung Ki* Yong-Jo I** *Satellite Technology Research Center Korea Advanced Institute of Science and Technology

### 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

### 6. Time (or Space) Series Analysis

ATM 55 otes: Tie Series Analysis - Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction,

### A Strategic Approach to Software Protection U

Â Â Strategic pproach to Software Protection U OZ SHY University of Haifa, Israel and Stockhol School of Econoics, Sweden ozshy@econ.haifa.ac.il JCQUES-FRNCË OIS THISSE CORE, Universite Catholique de Louvain,

### Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

### F=ma From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.edu

Chapter 4 F=a Fro Probles and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, orin@physics.harvard.edu 4.1 Introduction Newton s laws In the preceding two chapters, we dealt

### Mutual fund flight-to-liquidity

Mutual fund flight-to-liquidity Aleksandra Rzeźnik February 2015 Abstract This paper shows that active utual fund anagers actively anage the liquidity of their portfolio. Specifically, I use onthly holdings

### PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW

PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW ABSTRACT: by Douglas J. Reineann, Ph.D. Assistant Professor of Agricultural Engineering and Graee A. Mein, Ph.D. Visiting Professor

### 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

### Calculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance

Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 45-3167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21

### Example: Suppose that we deposit \$1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?

Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,

### Method of supply chain optimization in E-commerce

MPRA Munich Personal RePEc Archive Method of supply chain optiization in E-coerce Petr Suchánek and Robert Bucki Silesian University - School of Business Adinistration, The College of Inforatics and Manageent

### Plane Trusses. Section 7: Flexibility Method - Trusses. A plane truss is defined as a twodimensional

lane Trusses A plane truss is defined as a twodiensional fraework of straight prisatic ebers connected at their ends by frictionless hinged joints, and subjected to loads and reactions that act only at

### 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

### Construction Economics & Finance. Module 3 Lecture-1

Depreciation:- Construction Econoics & Finance Module 3 Lecture- It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset

### CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

### WEB APPENDIX. Calculating Beta Coefficients. b Beta Rise Run Y 7.1 1 8.92 X 10.0 0.0 16.0 10.0 1.6

WEB APPENDIX 8A Calculating Beta Coefficients The CAPM is an ex ante model, which means that all of the variables represent before-thefact, expected values. In particular, the beta coefficient used in

### On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information

Finance 400 A. Penati - G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information

### The Mathematics of Pumping Water

The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role

### PERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO

Bulletin of the Transilvania University of Braşov Series I: Engineering Sciences Vol. 4 (53) No. - 0 PERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO V. CAZACU I. SZÉKELY F. SANDU 3 T. BĂLAN Abstract:

### Endogenous Market Structure and the Cooperative Firm

Endogenous Market Structure and the Cooperative Fir Brent Hueth and GianCarlo Moschini Working Paper 14-WP 547 May 2014 Center for Agricultural and Rural Developent Iowa State University Aes, Iowa 50011-1070

### 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

### A magnetic Rotor to convert vacuum-energy into mechanical energy

A agnetic Rotor to convert vacuu-energy into echanical energy Claus W. Turtur, University of Applied Sciences Braunschweig-Wolfenbüttel Abstract Wolfenbüttel, Mai 21 2008 In previous work it was deonstrated,

### A Log-Robust Optimization Approach to Portfolio Management

A Log-Robust Optimization Approach to Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Ban Kawas Research partially supported by the National Science Foundation Grant CMMI-0757983

### Machine Learning Applications in Grid Computing

Machine Learning Applications in Grid Coputing George Cybenko, Guofei Jiang and Daniel Bilar Thayer School of Engineering Dartouth College Hanover, NH 03755, USA gvc@dartouth.edu, guofei.jiang@dartouth.edu

### The Capital Asset Pricing Model. Capital Budgeting and Corporate Objectives

The Capital Asset Pricing odel Capital Budgeting and Corporate Objectives Professor Ron Kaniel Simon School of Business University of Rochester 1 Overview Utility and risk aversion» Choosing efficient

### Problem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)

Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x

### Financial Risk Management Exam Sample Questions/Answers

Financial Risk Management Exam Sample Questions/Answers Prepared by Daniel HERLEMONT 1 2 3 4 5 6 Chapter 3 Fundamentals of Statistics FRM-99, Question 4 Random walk assumes that returns from one time period

### Reconnect 04 Solving Integer Programs with Branch and Bound (and Branch and Cut)

Sandia is a ultiprogra laboratory operated by Sandia Corporation, a Lockheed Martin Copany, Reconnect 04 Solving Integer Progras with Branch and Bound (and Branch and Cut) Cynthia Phillips (Sandia National

### Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM)

Lecture 05: Mean-Variance Analysis & Capital Asset Pricing Model (CAPM) Prof. Markus K. Brunnermeier Slide 05-1 Overview Simple CAPM with quadratic utility functions (derived from state-price beta model)

### Extended-Horizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona Network

2013 European Control Conference (ECC) July 17-19, 2013, Zürich, Switzerland. Extended-Horizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona

### 6 Hedging Using Futures

ECG590I Asset Pricing. Lecture 6: Hedging Using Futures 1 6 Hedging Using Futures 6.1 Types of hedges using futures Two types of hedge: short and long. ECG590I Asset Pricing. Lecture 6: Hedging Using Futures

### Fuzzy Sets in HR Management

Acta Polytechnica Hungarica Vol. 8, No. 3, 2011 Fuzzy Sets in HR Manageent Blanka Zeková AXIOM SW, s.r.o., 760 01 Zlín, Czech Republic blanka.zekova@sezna.cz Jana Talašová Faculty of Science, Palacký Univerzity,

### On Computing Nearest Neighbors with Applications to Decoding of Binary Linear Codes

On Coputing Nearest Neighbors with Applications to Decoding of Binary Linear Codes Alexander May and Ilya Ozerov Horst Görtz Institute for IT-Security Ruhr-University Bochu, Gerany Faculty of Matheatics

### The Lagrangian Method

Chapter 6 The Lagrangian Method Copyright 2007 by David Morin, orin@physics.harvard.edu (draft version In this chapter, we re going to learn about a whole new way of looking at things. Consider the syste

### Insurance Spirals and the Lloyd s Market

Insurance Spirals and the Lloyd s Market Andrew Bain University of Glasgow Abstract This paper presents a odel of reinsurance arket spirals, and applies it to the situation that existed in the Lloyd s

ANSWER KEY 1 BUDGETS W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE (1) Draw the budget set for the following paraeters. =, p beer = 1, p pizza =. The units for the two goods are pints and

### Adaptive Modulation and Coding for Unmanned Aerial Vehicle (UAV) Radio Channel

Recent Advances in Counications Adaptive odulation and Coding for Unanned Aerial Vehicle (UAV) Radio Channel Airhossein Fereidountabar,Gian Carlo Cardarilli, Rocco Fazzolari,Luca Di Nunzio Abstract In

### Calculation Method for evaluating Solar Assisted Heat Pump Systems in SAP 2009. 15 July 2013

Calculation Method for evaluating Solar Assisted Heat Pup Systes in SAP 2009 15 July 2013 Page 1 of 17 1 Introduction This docuent describes how Solar Assisted Heat Pup Systes are recognised in the National

### No. 2004/12. Daniel Schmidt

No. 2004/12 Private equity-, stock- and ixed asset-portfolios: A bootstrap approach to deterine perforance characteristics, diversification benefits and optial portfolio allocations Daniel Schidt Center

### 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

### Several Views of Support Vector Machines

Several Views of Support Vector Machines Ryan M. Rifkin Honda Research Institute USA, Inc. Human Intention Understanding Group 2007 Tikhonov Regularization We are considering algorithms of the form min

### ABSTRACT KEYWORDS. Comonotonicity, dependence, correlation, concordance, copula, multivariate. 1. INTRODUCTION

MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS BY INGE KOCH AND ANN DE SCHEPPER ABSTRACT In this contribution, a new easure of coonotonicity for -diensional vectors is introduced, with values between

### 1 Capital Asset Pricing Model (CAPM)

Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available

### Int. J. Production Economics

Int. J. Production Econoics 18 (1) 175 187 Contents lists available at ScienceDirect Int. J. Production Econoics journal hoepage www.elsevier.co/locate/ijpe Outsourcing structures and inforation flow in

### 1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset

Department of Economics Financial Economics University of California, Berkeley Economics 136 November 9, 2003 Fall 2006 Economics 136: Financial Economics Section Notes for Week 11 1 Capital Allocation

### REQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES

REQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES Charles Reynolds Christopher Fox reynolds @cs.ju.edu fox@cs.ju.edu Departent of Coputer

### 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

### Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, In which we introduce the theory of duality in linear programming.

Stanford University CS261: Optimization Handout 6 Luca Trevisan January 20, 2011 Lecture 6 In which we introduce the theory of duality in linear programming 1 The Dual of Linear Program Suppose that we