Hedge Portfolios, the No Arbitrage Condition & Arbitrage Pricing Theory
|
|
- Sharlene Walters
- 6 years ago
- Views:
Transcription
1 Hedge Portfolios, the No Arbitrage Condition & Arbitrage Pricing Theory Hedge Portfolios A portfolio that has zero risk is said to be "perfectly hedged" or, in the jargon of Economics and Finance, is referred to as a hedge portfolio. An investor who invests 100% of wealth in risk-free debt has obviously procured a hedge portfolio but this is not the only way to achieve this result. Consider, for example, a portfolio made up of some number k of risky securities where α i is used to denote the share of the portfolio invested in the risky asset with rate of return r i. Then the rate of return on the portfolio is r p such that r p = Σ α i r i and Σ α i = 1, where both summations are from i = 1 to k. If the investor chooses values for the α i such that Var[r p ] = 0, the portfolio will be a hedge portfolio with a rate of return E[r p ] = Σ α i E[ r i ] with absolute certainty. The No Arbitrage Condition A necessary condition for financial markets to be in equilibrium is something economists have termed the no arbitrage condition. In words it says that any investor who incurs zero risk and invests zero wealth must earn zero profits. A few examples will suffice to convince students that financial markets cannot be in equilibrium if the no arbitrage condition is not satisfied. Example 1 -- the foreign exchange market. Suppose the $US/$CAD exchange rate is simultaneously 0.75 in New York City and 0.76 in Toronto. This means that an investor can instruct her broker to purchase 1 Canadian dollar in New York for a price of $0.75 US and to simultaneously sell 1 Canadian dollar in Toronto for $0.76. The investor faces no risk and invests none of her own wealth and yet earns a profit of $0.01 US for every Canadian dollar simultaneously bought and sold. Clearly the no arbitrage condition is not satisfied here and just as clearly the foreign exchange market is not in equilibrium. What will happen is that large numbers of investors will try to buy Canadian dollars in New York and sell them in Toronto. This will cause exchange rate to swiftly rise in New York and fall in Toronto until both exchange rates are identical. The market will then be in equilibrium and the no arbitrage condition will be satisfied. Example 2 -- a hedge portfolio Consider the hedge portfolio of risky assets described in the previous subsection and suppose that the dollar cost of acquiring this risk-free portfolio is W. Instead of purchasing the portfolio outright, the investor could acquire it by borrowing W dollars at the risk-free rate of interest r f. If the investor does borrow, she will have invested zero of her own wealth in a portfolio that has zero risk. She will receive the following profit: 1
2 Profit = W (1+ E[r p ] ) - W (1+r f ). = W (E[r p ] - r f ). The first term on the RHS of the upper equation represents the gross return on the W dollars invested in the hedge portfolio and is known with certainty. The second term on the RHS represents the repayment of interest and principal on the investor's borrowing of W dollars and is also known with certainty. For the no arbitrage condition to hold, it must be the case that the certain rate of return on the hedge portfolio be equal to the risk free rate; i.e. that E[r p ] = r f. This makes intuitive sense; a portfolio with no risk should earn the risk-free rate of interest. But suppose that the no arbitrage condition did not hold here and that E[r p ] > r f. Then the security market can not be in equilibrium. A great many investors would wish to borrow to buy the hedge portfolio and this would bid up the prices of the securities that comprise it, causing the expected rates of return on these securities to decline. Since E[r p ] = Σ α i E[ r i ], the value of E[r p ] must also decline until E[r p ] = r f, at which point security prices would stabilize and the market would reach equilibrium. Example 3 -- a problem for students to solve Consider a setting with 1 risk-free asset and 2 risky securities, A and B. There are only 2 possible states of the world in the future as described by the following table. State 1 State 2 Probability of occurrence 1/2 1/2 Payoff of Security A $2.00 $0.00 Payoff of Security B $1.50 $0.50 In words: If State 1 occurs next time period, Security A will pay $2 and Security B will pay $1.50. If, instead, State 2 occurs, Security A will pay nothing and Security B will pay $0.50. Each State has an equal probability of occurring. Observe from their respective payoffs that a portfolio composed of +2 units of Security B and -1 unit of Security A is a hedge portfolio that will deliver a payoff of $1.00 in each of the 2 states. Currently the risk-free rate of interest is r f = 0.10 and the price of Security A is P A = $0.80. You are asked to use the no arbitrage condition to determine the value of P B (the current price of Security B). {The solution will be revealed at a later date.) 2
3 Observe that in each of the examples the no arbitrage condition imposes some restrictions on the relationships that exist among various asset prices and/or rates of return. In the foreign exchange example there are 2 asset prices -- the US dollar price of 1 Canadian dollar in New York and in Toronto. The no arbitrage condition requires that these 2 asset prices be identical. In Example 2, the no arbitrage condition requires that the rate of return on a hedge portfolio be equal to the risk-free rate. This imposes some restrictions on the expected rates of return of the risky assets that make up a hedge portfolio, though it is unclear from our example what form these restrictions might take. Arbitrage Pricing Theory (APT) spells out the nature of these restrictions and it is to that theory that we now turn. A Short Introduction to Arbitrage Pricing Theory APT is the impressive creation of Steve Ross. It is a much more general theory of the pricing of risky securities than the CAPM. (Indeed, the CAPM can be shown to be a special case of APT). APT requires no assumptions about investor's preferences other than that they are risk averse and does not require special assumptions (e.g. normality) about the probability distributions of rates of return. The rate of return on the market portfolio plays no special role in the APT. In addition, APT readily generalizes to multiple time periods. What follows is a derivation of APT in as simple a setting as possible. APT begins by trying to identify the underlying sources of uncertainty that make securities risky. Any source of uncertainty that creates risk among many securities is called a factor. There may be many factors in the real world (more on this later), but for simplicity I will derive the APT under the assumption that there is a single factor affecting the returns on risky securities and that factor is uncertainty about the overall performance of the economy. (A not totally implausible assumption). Let the variable G denote the rate of growth of real GDP during any time period. Then we will use as a measure the performance of the economy the factor F 1 defined as F 1 = G - E[G]. Observe that F 1 is the difference between the actual rate of growth in any time period and the rate of growth that had been expected to occur a priori. In other words, F 1 here is the unexpected rate of growth in real GDP. If F 1 > 0 in some time period, this means that the economy grew faster than expected; if F 1 < 0, then growth was less than expected. Finally, observe that E[F 1 ] = 0. We can express the relationship between the rate of return on any security j and the performance of the economy via the following equation: 3
4 (1) r j = E[r j ] + b j F 1 + ε j, where b j = Cov(r j,f 1 ) / Var(F 1 ) and Cov( ε j, F 1 ) = 0. Equation (1) says that there is a systematic relationship between the return on any security and the performance of the economy. If b j > 0, then the return on security j tends to be above / below its expected value as F 1 is above / below zero. [We might expect the value of b to be positive for most securities but there will be securities for which b is 0 or negative.] Uncertainty about the value of F 1 gives rise to systematic risk in all securities for which the value of b j 0. The equation also says that r j has an unsystematic component in the form of the random ε j, which gives rise to unsystematic (diversifiable) risk.. Var[r j ] = b 2 j Var[F 1 ] + Var[ε j ] systematic risk unsystematic risk We know that the unsystematic risk can be eliminated in a portfolio involving a large number k of risky assets. r p = Σ α j E[r j ] + F 1 Σ α j b j + Σ α j ε j, with the summation from j = 1 to k. When k is large, Σ α j ε j will equal zero and the portfolio return can be expressed as r p = E[r p ] + b p F 1, where E[r p ] = Σ α j E[r j ] and b p = Σ α j b j. Now consider 2 large portfolios that we shall refer to as portfolios A and B. r A = E[r A ] + b A F 1 r B = E[r B ] + b B F 1 Let's combine these 2 portfolios into a hedge portfolio. Let w denote the share of A in the hedge portfolio and (1-w) denote the share of B. We choose w so that w b A + (1-w) b B = 0 (zero systematic risk); hence w = b B / (b B -b A ). Then the expected return on the hedge portfolio is (2) E[r H ] = [b B / (b B -b A )]E[r A ] + [-b A / (b B -b A )] E[r B ] with 100% certainty. The no arbitrage condition (from which APT derives its name) requires that r H = r f, the risk-free rate and this, along with Equation (2) implies 4
5 (3) (E[r A ]- r f ) / b A = (E[r B ]- r f ) / b B. Equation (3) was derived directly for portfolios in which unsystematic risk is zero but it applies as well to all securities because the unsystematic risk of a security can have no influence on its expected return; in other words, (E[r j ]- r f ) / b j must have the same constant value for all securities. [If b j = 0 for some security, the value of (E[r j ]- r f ) / b j is undefined, but any security with b j = 0 has no systematic risk and must, therefor, have an expected rate of return equal to the risk-free rate r f. Let E[R 1 ] denote the expected rate of return on a security that has a value of b=1. Then APT implies that the following holds for all securities. (4) E[r j ]- r f = b j (E[R 1 ] - r f ). Observe that although our measure of overall economic performance F 1 does not appear directly in Equation (4) it does appear indirectly since b j = Cov(r j,f 1 ) / Var(F 1 ). Now let's step back and take an intuitive look at what Equation (4) says. It says that in a world in which the only source of overall uncertainty is unexpected growth in real GDP (as measured by the factor F 1 ) the risk of any security is proportional to the Covariance between the rate of return on the security and F 1. The larger is this Covariance, the higher will be the expected rate of return on the security. [An implication of the Equation is that if we could find a way to predict rates of growth of real GDP with 100% accuracy, there would be no systematic risk and the expected return on all securities would be equal to r f.] It is easy to see from this result that CAPM is just a special case of APT. Had we not chosen overall economic performance as our explanatory factor, but had instead chosen the return on the market portfolio, then F 1 would have been (r M - E[r M ] ) and Equation (4) would reduce to the familiar CAPM relationship. Our derivation was purposely based on only one factor for simplicity, but APT readily generalizes to any number of factors. Remember that a factor is an aggregate source of uncertainty, and it is easy to think of things other than uncertainty about the rate of growth of real GDP that might affect security returns. Empirical investigations using APT have identified a total of 4 factors that appear to be important sources of systematic risk for US securities. These are: F 1 -- unexpected growth in real GDP (as in our 1-factor derivation) F 2 -- unexpected price inflation F 3 -- differences between the yields on AAA versus Baa corporate bonds F 4 -- differences between the yields on long- versus short-term government bonds. 5
6 The 4-factor counterpart to Equation (1) is (5) r j = E[r i ] + b j1 F 1 + b j2 F 2 + b j3 F 3 + b j4 F 4 + ε j, where b jk = Cov( r j, F k ) / Var(F k ) for k = 1,2,3,4 and the factors are measured in such a way that Cov( F i, F k ) = 0 for all i,k pairings. Here there are four types of systematic risk -- one for each of the factors: Var[r j ] = b j1 2 Var[F 1 ] + b j2 2 Var[F 2 ] + b j3 2 Var[F 3 ] + b j4 2 Var[F 4 ] + Var[ε j ]. systematic risk 1 systematic risk 2 systematic risk 3 systematic risk 4 unsystematic risk We can create several large portfolios that each eliminate unsystematic risk, then combine these into a hedge portfolio that eliminates all 4 types of systematic risk. Applying the no arbitrage condition to the hedge portfolio yiels the following counterpart to Equation 4. (6) E[r j ]- r f = b j1 (E[R 1 ] - r f ) + b j2 (E[R 2 ] - r f )+ b j3 (E[R 3 ] - r f )+ b j4 (E[R 4 ] - r f ), where E[R i ] = the expected rate of return on a security that has b=1 for the i th b = 0 for the other 3 factors. factor and For a couple of fairly simple examples of numerical applications of APT, students should complete Problems 7.20 and 7.21 at the end of Chapter 7 in our textbook. 6
Principles of Finance Risk and Return. Instructor: Xiaomeng Lu
Principles of Finance Risk and Return Instructor: Xiaomeng Lu 1 Course Outline Course Introduction Time Value of Money DCF Valuation Security Analysis: Bond, Stock Capital Budgeting (Fundamentals) Portfolio
More informationPrinciples of Finance
Principles of Finance Grzegorz Trojanowski Lecture 7: Arbitrage Pricing Theory Principles of Finance - Lecture 7 1 Lecture 7 material Required reading: Elton et al., Chapter 16 Supplementary reading: Luenberger,
More informationChapter 8: CAPM. 1. Single Index Model. 2. Adding a Riskless Asset. 3. The Capital Market Line 4. CAPM. 5. The One-Fund Theorem
Chapter 8: CAPM 1. Single Index Model 2. Adding a Riskless Asset 3. The Capital Market Line 4. CAPM 5. The One-Fund Theorem 6. The Characteristic Line 7. The Pricing Model Single Index Model 1 1. Covariance
More information1.1 Interest rates Time value of money
Lecture 1 Pre- Derivatives Basics Stocks and bonds are referred to as underlying basic assets in financial markets. Nowadays, more and more derivatives are constructed and traded whose payoffs depend on
More informationAnswers to Concepts in Review
Answers to Concepts in Review 1. A portfolio is simply a collection of investment vehicles assembled to meet a common investment goal. An efficient portfolio is a portfolio offering the highest expected
More informationUse partial derivatives just found, evaluate at a = 0: This slope of small hyperbola must equal slope of CML:
Derivation of CAPM formula, contd. Use the formula: dµ σ dσ a = µ a µ dµ dσ = a σ. Use partial derivatives just found, evaluate at a = 0: Plug in and find: dµ dσ σ = σ jm σm 2. a a=0 σ M = a=0 a µ j µ
More informationECMC49S Midterm. Instructor: Travis NG Date: Feb 27, 2007 Duration: From 3:05pm to 5:00pm Total Marks: 100
ECMC49S Midterm Instructor: Travis NG Date: Feb 27, 2007 Duration: From 3:05pm to 5:00pm Total Marks: 100 [1] [25 marks] Decision-making under certainty (a) [10 marks] (i) State the Fisher Separation Theorem
More informationFinal Exam Suggested Solutions
University of Washington Fall 003 Department of Economics Eric Zivot Economics 483 Final Exam Suggested Solutions This is a closed book and closed note exam. However, you are allowed one page of handwritten
More informationConsumption- Savings, Portfolio Choice, and Asset Pricing
Finance 400 A. Penati - G. Pennacchi Consumption- Savings, Portfolio Choice, and Asset Pricing I. The Consumption - Portfolio Choice Problem We have studied the portfolio choice problem of an individual
More informationModels of Asset Pricing
appendix1 to chapter 5 Models of Asset Pricing In Chapter 4, we saw that the return on an asset (such as a bond) measures how much we gain from holding that asset. When we make a decision to buy an asset,
More information1 Asset Pricing: Replicating portfolios
Alberto Bisin Corporate Finance: Lecture Notes Class 1: Valuation updated November 17th, 2002 1 Asset Pricing: Replicating portfolios Consider an economy with two states of nature {s 1, s 2 } and with
More informationArchana Khetan 05/09/ MAFA (CA Final) - Portfolio Management
Archana Khetan 05/09/2010 +91-9930812722 Archana090@hotmail.com MAFA (CA Final) - Portfolio Management 1 Portfolio Management Portfolio is a collection of assets. By investing in a portfolio or combination
More informationGMM Estimation. 1 Introduction. 2 Consumption-CAPM
GMM Estimation 1 Introduction Modern macroeconomic models are typically based on the intertemporal optimization and rational expectations. The Generalized Method of Moments (GMM) is an econometric framework
More informationMonetary Economics Risk and Return, Part 2. Gerald P. Dwyer Fall 2015
Monetary Economics Risk and Return, Part 2 Gerald P. Dwyer Fall 2015 Reading Malkiel, Part 2, Part 3 Malkiel, Part 3 Outline Returns and risk Overall market risk reduced over longer periods Individual
More informationLecture 3: Return vs Risk: Mean-Variance Analysis
Lecture 3: Return vs Risk: Mean-Variance Analysis 3.1 Basics We will discuss an important trade-off between return (or reward) as measured by expected return or mean of the return and risk as measured
More informationB6302 Sample Placement Exam Academic Year
Revised June 011 B630 Sample Placement Exam Academic Year 011-01 Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized units). Fund
More informationLecture 4: Return vs Risk: Mean-Variance Analysis
Lecture 4: Return vs Risk: Mean-Variance Analysis 4.1 Basics Given a cool of many different stocks, you want to decide, for each stock in the pool, whether you include it in your portfolio and (if yes)
More informationP1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes
P1.T1. Foundations of Risk Management Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments, 10th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com BODIE, CHAPTER
More informationProblem Set 3. Thomas Philippon. April 19, Human Wealth, Financial Wealth and Consumption
Problem Set 3 Thomas Philippon April 19, 2002 1 Human Wealth, Financial Wealth and Consumption The goal of the question is to derive the formulas on p13 of Topic 2. This is a partial equilibrium analysis
More informationLecture 5 Theory of Finance 1
Lecture 5 Theory of Finance 1 Simon Hubbert s.hubbert@bbk.ac.uk January 24, 2007 1 Introduction In the previous lecture we derived the famous Capital Asset Pricing Model (CAPM) for expected asset returns,
More informationDerivation of zero-beta CAPM: Efficient portfolios
Derivation of zero-beta CAPM: Efficient portfolios AssumptionsasCAPM,exceptR f does not exist. Argument which leads to Capital Market Line is invalid. (No straight line through R f, tilted up as far as
More informationFIN 6160 Investment Theory. Lecture 7-10
FIN 6160 Investment Theory Lecture 7-10 Optimal Asset Allocation Minimum Variance Portfolio is the portfolio with lowest possible variance. To find the optimal asset allocation for the efficient frontier
More informationOne-Period Valuation Theory
One-Period Valuation Theory Part 2: Chris Telmer March, 2013 1 / 44 1. Pricing kernel and financial risk 2. Linking state prices to portfolio choice Euler equation 3. Application: Corporate financial leverage
More informationECO 317 Economics of Uncertainty Fall Term 2009 Tuesday October 6 Portfolio Allocation Mean-Variance Approach
ECO 317 Economics of Uncertainty Fall Term 2009 Tuesday October 6 ortfolio Allocation Mean-Variance Approach Validity of the Mean-Variance Approach Constant absolute risk aversion (CARA): u(w ) = exp(
More informationRisk and Return and Portfolio Theory
Risk and Return and Portfolio Theory Intro: Last week we learned how to calculate cash flows, now we want to learn how to discount these cash flows. This will take the next several weeks. We know 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 informationGeneral Notation. Return and Risk: The Capital Asset Pricing Model
Return and Risk: The Capital Asset Pricing Model (Text reference: Chapter 10) Topics general notation single security statistics covariance and correlation return and risk for a portfolio diversification
More informationConsumption-Savings Decisions and State Pricing
Consumption-Savings Decisions and State Pricing Consumption-Savings, State Pricing 1/ 40 Introduction We now consider a consumption-savings decision along with the previous portfolio choice decision. These
More informationCHAPTER III RISK MANAGEMENT
CHAPTER III RISK MANAGEMENT Concept of Risk Risk is the quantified amount which arises due to the likelihood of the occurrence of a future outcome which one does not expect to happen. If one is participating
More informationModule 3: Factor Models
Module 3: Factor Models (BUSFIN 4221 - Investments) Andrei S. Gonçalves 1 1 Finance Department The Ohio State University Fall 2016 1 Module 1 - The Demand for Capital 2 Module 1 - The Supply of Capital
More informationDiversification. Finance 100
Diversification Finance 100 Prof. Michael R. Roberts 1 Topic Overview How to measure risk and return» Sample risk measures for some classes of securities Brief Statistics Review» Realized and Expected
More informationArbitrage Pricing Theory (APT)
Arbitrage Pricing Theory (APT) (Text reference: Chapter 11) Topics arbitrage factor models pure factor portfolios expected returns on individual securities comparison with CAPM a different approach 1 Arbitrage
More informationINTERTEMPORAL ASSET ALLOCATION: THEORY
INTERTEMPORAL ASSET ALLOCATION: THEORY Multi-Period Model The agent acts as a price-taker in asset markets and then chooses today s consumption and asset shares to maximise lifetime utility. This multi-period
More informationMATH 4512 Fundamentals of Mathematical Finance
MATH 451 Fundamentals of Mathematical Finance Solution to Homework Three Course Instructor: Prof. Y.K. Kwok 1. The market portfolio consists of n uncorrelated assets with weight vector (x 1 x n T. Since
More informationEcon 422 Eric Zivot Fall 2005 Final Exam
Econ 422 Eric Zivot Fall 2005 Final Exam This is a closed book exam. However, you are allowed one page of notes (double-sided). Answer all questions. For the numerical problems, if you make a computational
More informationMicroéconomie de la finance
Microéconomie de la finance 7 e édition Christophe Boucher christophe.boucher@univ-lorraine.fr 1 Chapitre 6 7 e édition Les modèles d évaluation d actifs 2 Introduction The Single-Index Model - Simplifying
More informationCapital Asset Pricing Model
Topic 5 Capital Asset Pricing Model LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Explain Capital Asset Pricing Model (CAPM) and its assumptions; 2. Compute Security Market Line
More informationFinancial Economics: Capital Asset Pricing Model
Financial Economics: Capital Asset Pricing Model Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY April, 2015 1 / 66 Outline Outline MPT and the CAPM Deriving the CAPM Application of CAPM Strengths and
More informationEconomics 430 Handout on Rational Expectations: Part I. Review of Statistics: Notation and Definitions
Economics 430 Chris Georges Handout on Rational Expectations: Part I Review of Statistics: Notation and Definitions Consider two random variables X and Y defined over m distinct possible events. Event
More informationChapter 23: Choice under Risk
Chapter 23: Choice under Risk 23.1: Introduction We consider in this chapter optimal behaviour in conditions of risk. By this we mean that, when the individual takes a decision, he or she does not know
More informationMean-Variance Portfolio Theory
Mean-Variance Portfolio Theory Lakehead University Winter 2005 Outline Measures of Location Risk of a Single Asset Risk and Return of Financial Securities Risk of a Portfolio The Capital Asset Pricing
More informationLecture 3: Factor models in modern portfolio choice
Lecture 3: Factor models in modern portfolio choice Prof. Massimo Guidolin Portfolio Management Spring 2016 Overview The inputs of portfolio problems Using the single index model Multi-index models Portfolio
More informationLimits to Arbitrage. George Pennacchi. Finance 591 Asset Pricing Theory
Limits to Arbitrage George Pennacchi Finance 591 Asset Pricing Theory I.Example: CARA Utility and Normal Asset Returns I Several single-period portfolio choice models assume constant absolute risk-aversion
More informationChapter 8. Markowitz Portfolio Theory. 8.1 Expected Returns and Covariance
Chapter 8 Markowitz Portfolio Theory 8.1 Expected Returns and Covariance The main question in portfolio theory is the following: Given an initial capital V (0), and opportunities (buy or sell) in N securities
More informationPortfolio Management
Portfolio Management Risk & Return Return Income received on an investment (Dividend) plus any change in market price( Capital gain), usually expressed as a percent of the beginning market price of the
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 informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Fall 2017 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Spring 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationu (x) < 0. and if you believe in diminishing return of the wealth, then you would require
Chapter 8 Markowitz Portfolio Theory 8.7 Investor Utility Functions People are always asked the question: would more money make you happier? The answer is usually yes. The next question is how much more
More informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Fall 2017 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationForward and Futures Contracts
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Forward and Futures Contracts These notes explore forward and futures contracts, what they are and how they are used. We will learn how to price forward contracts
More informationMicroeconomics 3. Economics Programme, University of Copenhagen. Spring semester Lars Peter Østerdal. Week 17
Microeconomics 3 Economics Programme, University of Copenhagen Spring semester 2006 Week 17 Lars Peter Østerdal 1 Today s programme General equilibrium over time and under uncertainty (slides from week
More informationCHAPTER 11 RETURN AND RISK: THE CAPITAL ASSET PRICING MODEL (CAPM)
CHAPTER 11 RETURN AND RISK: THE CAPITAL ASSET PRICING MODEL (CAPM) Answers to Concept Questions 1. Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of
More informationFinancial Mathematics III Theory summary
Financial Mathematics III Theory summary Table of Contents Lecture 1... 7 1. State the objective of modern portfolio theory... 7 2. Define the return of an asset... 7 3. How is expected return defined?...
More informationPortfolio Theory and Diversification
Topic 3 Portfolio Theoryand Diversification LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Explain the concept of portfolio formation;. Discuss the idea of diversification; 3. Calculate
More informationIndex Models and APT
Index Models and APT (Text reference: Chapter 8) Index models Parameter estimation Multifactor models Arbitrage Single factor APT Multifactor APT Index models predate CAPM, originally proposed as a simplification
More informationCHAPTER 10. Arbitrage Pricing Theory and Multifactor Models of Risk and Return INVESTMENTS BODIE, KANE, MARCUS
CHAPTER 10 Arbitrage Pricing Theory and Multifactor Models of Risk and Return INVESTMENTS BODIE, KANE, MARCUS McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. INVESTMENTS
More information3. Capital asset pricing model and factor models
3. Capital asset pricing model and factor models (3.1) Capital asset pricing model and beta values (3.2) Interpretation and uses of the capital asset pricing model (3.3) Factor models (3.4) Performance
More informationEG, Ch. 12: International Diversification
1 EG, Ch. 12: International Diversification I. Overview. International Diversification: A. Reduces Risk. B. Increases or Decreases Expected Return? C. Performance is affected by Exchange Rates. D. How
More informationOPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS. BKM Ch 7
OPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS BKM Ch 7 ASSET ALLOCATION Idea from bank account to diversified portfolio Discussion principles are the same for any number of stocks A. bonds and stocks B.
More informationAn Analysis of Theories on Stock Returns
An Analysis of Theories on Stock Returns Ahmet Sekreter 1 1 Faculty of Administrative Sciences and Economics, Ishik University, Erbil, Iraq Correspondence: Ahmet Sekreter, Ishik University, Erbil, Iraq.
More informationProblem set 5. Asset pricing. Markus Roth. Chair for Macroeconomics Johannes Gutenberg Universität Mainz. Juli 5, 2010
Problem set 5 Asset pricing Markus Roth Chair for Macroeconomics Johannes Gutenberg Universität Mainz Juli 5, 200 Markus Roth (Macroeconomics 2) Problem set 5 Juli 5, 200 / 40 Contents Problem 5 of problem
More informationE(r) The Capital Market Line (CML)
The Capital Asset Pricing Model (CAPM) B. Espen Eckbo 2011 We have so far studied the relevant portfolio opportunity set (mean- variance efficient portfolios) We now study more specifically portfolio demand,
More informationChapter 5: Answers to Concepts in Review
Chapter 5: Answers to Concepts in Review 1. A portfolio is simply a collection of investment vehicles assembled to meet a common investment goal. An efficient portfolio is a portfolio offering the highest
More informationElements of Economic Analysis II Lecture XI: Oligopoly: Cournot and Bertrand Competition
Elements of Economic Analysis II Lecture XI: Oligopoly: Cournot and Bertrand Competition Kai Hao Yang /2/207 In this lecture, we will apply the concepts in game theory to study oligopoly. In short, unlike
More informationECO 100Y INTRODUCTION TO ECONOMICS
Prof. Gustavo Indart Department of Economics University of Toronto ECO 100Y INTRODUCTION TO ECONOMICS Lecture 16. THE DEMAND FOR MONEY AND EQUILIBRIUM IN THE MONEY MARKET We will assume that there are
More informationProblem Set. Solutions to the problems appear at the end of this document.
Problem Set Solutions to the problems appear at the end of this document. Unless otherwise stated, any coupon payments, cash dividends, or other cash payouts delivered by a security in the following problems
More informationMarkowitz portfolio theory
Markowitz portfolio theory Farhad Amu, Marcus Millegård February 9, 2009 1 Introduction Optimizing a portfolio is a major area in nance. The objective is to maximize the yield and simultaneously minimize
More informationB. Arbitrage Arguments support CAPM.
1 E&G, Ch. 16: APT I. Background. A. CAPM shows that, under many assumptions, equilibrium expected returns are linearly related to β im, the relation between R ii and a single factor, R m. (i.e., equilibrium
More informationLabor Economics Field Exam Spring 2011
Labor Economics Field Exam Spring 2011 Instructions You have 4 hours to complete this exam. This is a closed book examination. No written materials are allowed. You can use a calculator. THE EXAM IS COMPOSED
More informationAnswers to chapter 3 review questions
Answers to chapter 3 review questions 3.1 Explain why the indifference curves in a probability triangle diagram are straight lines if preferences satisfy expected utility theory. The expected utility of
More informationTwo Equivalent Conditions
Two Equivalent Conditions The traditional theory of present value puts forward two equivalent conditions for asset-market equilibrium: Rate of Return The expected rate of return on an asset equals the
More informationFinance 100: Corporate Finance. Professor Michael R. Roberts Quiz 3 November 8, 2006
Finance 100: Corporate Finance Professor Michael R. Roberts Quiz 3 November 8, 006 Name: Solutions Section ( Points...no joke!): Question Maximum Student Score 1 30 5 3 5 4 0 Total 100 Instructions: Please
More informationCorporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005
Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate
More informationEconomics 424/Applied Mathematics 540. Final Exam Solutions
University of Washington Summer 01 Department of Economics Eric Zivot Economics 44/Applied Mathematics 540 Final Exam Solutions I. Matrix Algebra and Portfolio Math (30 points, 5 points each) Let R i denote
More informationAn Intertemporal Capital Asset Pricing Model
I. Assumptions Finance 400 A. Penati - G. Pennacchi Notes on An Intertemporal Capital Asset Pricing Model These notes are based on the article Robert C. Merton (1973) An Intertemporal Capital Asset Pricing
More informationMarket Liquidity and Performance Monitoring The main idea The sequence of events: Technology and information
Market Liquidity and Performance Monitoring Holmstrom and Tirole (JPE, 1993) The main idea A firm would like to issue shares in the capital market because once these shares are publicly traded, speculators
More informationCapital Asset Pricing Model and Arbitrage Pricing Theory
Capital Asset Pricing Model and Nico van der Wijst 1 D. van der Wijst TIØ4146 Finance for science and technology students 1 Capital Asset Pricing Model 2 3 2 D. van der Wijst TIØ4146 Finance for science
More informationMBA 203 Executive Summary
MBA 203 Executive Summary Professor Fedyk and Sraer Class 1. Present and Future Value Class 2. Putting Present Value to Work Class 3. Decision Rules Class 4. Capital Budgeting Class 6. Stock Valuation
More informationA. Huang Date of Exam December 20, 2011 Duration of Exam. Instructor. 2.5 hours Exam Type. Special Materials Additional Materials Allowed
Instructor A. Huang Date of Exam December 20, 2011 Duration of Exam 2.5 hours Exam Type Special Materials Additional Materials Allowed Calculator Marking Scheme: Question Score Question Score 1 /20 5 /9
More informationSession 10: Lessons from the Markowitz framework p. 1
Session 10: Lessons from the Markowitz framework Susan Thomas http://www.igidr.ac.in/ susant susant@mayin.org IGIDR Bombay Session 10: Lessons from the Markowitz framework p. 1 Recap The Markowitz question:
More informationThe Capital Asset Pricing Model as a corollary of the Black Scholes model
he Capital Asset Pricing Model as a corollary of the Black Scholes model Vladimir Vovk he Game-heoretic Probability and Finance Project Working Paper #39 September 6, 011 Project web site: http://www.probabilityandfinance.com
More informationJ B GUPTA CLASSES , Copyright: Dr JB Gupta. Chapter 4 RISK AND RETURN.
J B GUPTA CLASSES 98184931932, drjaibhagwan@gmail.com, www.jbguptaclasses.com Copyright: Dr JB Gupta Chapter 4 RISK AND RETURN Chapter Index Systematic and Unsystematic Risk Capital Asset Pricing Model
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 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 informationSolutions to the problems in the supplement are found at the end of the supplement
www.liontutors.com FIN 301 Exam 2 Chapter 12 Supplement Solutions to the problems in the supplement are found at the end of the supplement Chapter 12 The Capital Asset Pricing Model Risk and Return Higher
More informationArbitrage Pricing Theory and Multifactor Models of Risk and Return
Arbitrage Pricing Theory and Multifactor Models of Risk and Return Recap : CAPM Is a form of single factor model (one market risk premium) Based on a set of assumptions. Many of which are unrealistic One
More informationThe Capital Assets Pricing Model & Arbitrage Pricing Theory: Properties and Applications in Jordan
Modern Applied Science; Vol. 12, No. 11; 2018 ISSN 1913-1844E-ISSN 1913-1852 Published by Canadian Center of Science and Education The Capital Assets Pricing Model & Arbitrage Pricing Theory: Properties
More informationChilton Investment Seminar
Chilton Investment Seminar Palm Beach, Florida - March 30, 2006 Applied Mathematics and Statistics, Stony Brook University Robert J. Frey, Ph.D. Director, Program in Quantitative Finance Objectives Be
More informationQR43, Introduction to Investments Class Notes, Fall 2003 IV. Portfolio Choice
QR43, Introduction to Investments Class Notes, Fall 2003 IV. Portfolio Choice A. Mean-Variance Analysis 1. Thevarianceofaportfolio. Consider the choice between two risky assets with returns R 1 and R 2.
More informationNote on Cost of Capital
DUKE UNIVERSITY, FUQUA SCHOOL OF BUSINESS ACCOUNTG 512F: FUNDAMENTALS OF FINANCIAL ANALYSIS Note on Cost of Capital For the course, you should concentrate on the CAPM and the weighted average cost of capital.
More informationMacroeconomics I Chapter 3. Consumption
Toulouse School of Economics Notes written by Ernesto Pasten (epasten@cict.fr) Slightly re-edited by Frank Portier (fportier@cict.fr) M-TSE. Macro I. 200-20. Chapter 3: Consumption Macroeconomics I Chapter
More informationEFFICIENT MARKETS HYPOTHESIS
EFFICIENT MARKETS HYPOTHESIS when economists speak of capital markets as being efficient, they usually consider asset prices and returns as being determined as the outcome of supply and demand in a competitive
More informationSolution Guide to Exercises for Chapter 4 Decision making under uncertainty
THE ECONOMICS OF FINANCIAL MARKETS R. E. BAILEY Solution Guide to Exercises for Chapter 4 Decision making under uncertainty 1. Consider an investor who makes decisions according to a mean-variance objective.
More informationWe examine the impact of risk aversion on bidding behavior in first-price auctions.
Risk Aversion We examine the impact of risk aversion on bidding behavior in first-price auctions. Assume there is no entry fee or reserve. Note: Risk aversion does not affect bidding in SPA because there,
More informationOverview of Concepts and Notation
Overview of Concepts and Notation (BUSFIN 4221: Investments) - Fall 2016 1 Main Concepts This section provides a list of questions you should be able to answer. The main concepts you need to know are embedded
More informationPortfolio theory and risk management Homework set 2
Portfolio theory and risk management Homework set Filip Lindskog General information The homework set gives at most 3 points which are added to your result on the exam. You may work individually or in
More informationTechniques for Calculating the Efficient Frontier
Techniques for Calculating the Efficient Frontier Weerachart Kilenthong RIPED, UTCC c Kilenthong 2017 Tee (Riped) Introduction 1 / 43 Two Fund Theorem The Two-Fund Theorem states that we can reach any
More informationFINANCE 402 Capital Budgeting and Corporate Objectives. Syllabus
FINANCE 402 Capital Budgeting and Corporate Objectives Course Description: Syllabus The objective of this course is to provide a rigorous introduction to the fundamental principles of asset valuation and
More information1 Asset Pricing: Bonds vs Stocks
Asset Pricing: Bonds vs Stocks The historical data on financial asset returns show that one dollar invested in the Dow- Jones yields 6 times more than one dollar invested in U.S. Treasury bonds. The return
More information