Efficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9
|
|
- Joy Williams
- 5 years ago
- Views:
Transcription
1 Efficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9
2 Optimal Investment with Risky Assets There are N risky assets, named 1, 2,, N, but no risk-free asset. With fixed total dollar amount of investment, you choose the proportion of investment in each asset, i.e., portfolio choice. Since you are risk averse, you want to maximize the return of your investment, given a certain level of risk. Alternatively, you want to minimize the risk of your investment for a given level of return. 2
3 A review
4 Optimization The optimal investment can be summarized as: choosing portfolio x, which solves the following minimisation problem Min{Var(r x )=x T Sx} st. E(r x ) = x T E(r) = r g, å = N i 1 xi = 1 r is the return vector of the N assets, S is their covariance matrix, r g (or r target ) is the expected return (target). Note, x i is the proportion of investment in asset i, it can be negative in this topic. 4
5 Minimum-Variance Portfolios When E(r), S and r target are known, Excel Solver can solve the optimal portfolio. The next example provides a case of 4 risk assets and it solves three optimal portfolios for r target = 10%, 12% and 7%. Excel Solver can be found in Data/Analysis (the very right end of your menu ribbon) 5
6
7 7
8 Envelope and Global Minimum-Variance Portfolio (GMV) Portfolios that have minimum variance for a given return are called envelope portfolios Consider a new problem: you want the risk of your investment as low as possible, don t mind the return of your investment. This means your are looking for a global minimum variance portfolio (GMV) Mathematically, this can be solved by: min Var(r x )=x T Sx st. å = N i 1 xi = 1 Note we here drop the constraint of E(r x ) = r g in the previous minimization problem (we care about risk only). 8
9 Feasible Portfolios 11% 10% Infeasible portfolio Portfolio mean return 9% 8% 7% GMV Efficient and envelope Feasible, not efficient 6% 5% Envelope, not efficient 4% 10% 20% 30% 40% 50% 60% 70% 80% 90% Portfolio standard deviation FM3: Chapter 9 Efficient portfolio theorems 9
10 GMV and Efficient Frontier The global minimum variance portfolio (GMV) provides lowest risk for all feasible investment choices. GMV separate the frontier to two parts: 1. The upper segment: efficient frontier and portfolios on this segment are efficient portfolios. An efficient portfolio maximizes return for a given variance level. 2. The Lower part: of course consists inefficient portfolios 2. Similarly, we can use Solver to find the GMV portfolio (See excel example) 10
11 Envelope portfolios We can generate minimum-variance portfolios or envelope portfolios by changing the required (expected) rate of return r target to obtain many portfolios. A curve linking all these portfolios in a mean-standard deviation space is called minimum-variance frontier or the envelope of the feasible investment set All envelope portfolios can be found by repeating the previous exercise using Solver for different r target 11
12 Find Envelope Portfolios: alternative methods The disadvantage of previous method: Have to use Solver many times to find the envelope/efficient frontier. The alternative is using concept/theorem about Black (1972) 2-fund proposition: Use Solver directly or find tangency portfolio Data Table 12
13 Black proposition: The convex combination of any two envelope portfolios is also an envelope portfolio. Thus: if x and y are envelope portfolios, so is lx ( 1 l) ì lx1+ - y1 ü ï ï + ( 1- l) y =í L ý ïlxn + ( 1-l) y ï î Nþ FM3: Chapter 9 Efficient portfolio theorems 13
14 Use proposition one To use proposition one and data table to generate the envelope, we need at least 2 efficient portfolios. How to find? (2 ways) Solver directly apply to mean and variance function Tangency portfolio Use Solver
15
16 Second way to find efficient portfolios: Tangency portfolio r c s x E(r x ) - c Tencency porfolios for a given c s Except for the GMV, any envelope portfolio or minimum variance portfolio is a tangency portfolio of a constant c. Given c, at least one tangency portfolio exists. 16
17 If C=r f, the tangent portfolio is on the CML line (more about this later ) Efficient Frontier with CML Capital market line, Portfolio mean return Risk-free rate, r f Market Portfolio standard deviation
18 Find the Tangency portfolio using constant and Solver Mathematically, we can obtain tangency portfolio x for a given c by solving the following problem Max (or Min) [E(r x ) c]/s x st. å = N i 1 xi = 1 The maximization à a portfolio on the upper part of the envelope (efficient frontier) The minimization à a portfolio on the lower segment (not efficient). 18
19
20 Capital Market Line: Idea about investment with a risk-free Asset Let the risk-free rate is r f, so you will not consider any investment with a return lower than r f as long as you are risk averse. For any target rate of expected return greater than r f, we can still determine the optimal investment proportions to N risky assets + 1 risk free asset by minimizing the portfolio s risk. 20
21 Efficient Portfolios with a Risk-Free Asset and CML Similarly, we can vary the target return to obtain all efficient portfolios and the efficient frontier. The efficient frontier (with the risk-free asset) is a straight line in the standard deviations-mean space. This straight line passes through the risk-free asset and the tangency portfolio of efficient frontier of risky assets. If the portfolio has N assets that include all risky assets in the market, the portfolio is the market portfolio (MP) The straight line is called Capital Market Line (CML) 21
22 Market portfolio (MP) and Capital Market Line (CML) CML is a straight line from the risk-free rate through the market portfolio in an s-r plane. r Capital market line Market portforlio rf Envelope s 22
23 Market portfolio Most important implication of CAPM All investors hold the same optimal portfolio of risky assets The optimal portfolio is at the highest point of tangency between r f and the efficient frontier The portfolio of all risky assets is the optimal risky portfolio Called market portfolio
24 Characteristics of Market portfolio All risky assets must be in portfolio, so it is completely diversified Includes only systematic risk All securities included in proportion to their market value Unobservable but proxied by some market index, e.g., all ordinaries, S&P500
25 CAPM s prediction CAPM prediction: Efficient portfolio = invest a proportion in the risk-free asset and (1 a) in the MP The expected return and risk: E(r p ) = ar f +(1 a)e(r M ) s p =(1 a)s M Note the particularly simple form of the standard deviation here. What are the return and risk on CML when a=0, a=1, a<0, 0<a<1? 25
26 How to Find the CML? Two steps of finding the CML 1. Following tangency approach, obtain the MP by setting c = r f. 2. Use Data Table: Change the weights of risk-free asset and the market portfolio to create all expected returns and standard deviations on the CML. 26
27
28 Find a Portfolio with Desired Return or Risk Since all efficient portfolios are on the CML, an investor with a particular target of mean return or risk in mind can choose an investment portfolio on the CML. But with Excel, we can find the portfolio even without drawing the CML. The tool is Goal Seek or Solver, See example next 28
29
u (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 informationCapital Allocation Between The Risky And The Risk- Free Asset
Capital Allocation Between The Risky And The Risk- Free Asset Chapter 7 Investment Decisions capital allocation decision = choice of proportion to be invested in risk-free versus risky assets asset allocation
More informationMean-Variance Portfolio Choice in Excel
Mean-Variance Portfolio Choice in Excel Prof. Manuela Pedio 20550 Quantitative Methods for Finance August 2018 Let s suppose you can only invest in two assets: a (US) stock index (here represented by the
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 informationLecture 2: Fundamentals of meanvariance
Lecture 2: Fundamentals of meanvariance analysis Prof. Massimo Guidolin Portfolio Management Second Term 2018 Outline and objectives Mean-variance and efficient frontiers: logical meaning o Guidolin-Pedio,
More informationMean Variance Analysis and CAPM
Mean Variance Analysis and CAPM Yan Zeng Version 1.0.2, last revised on 2012-05-30. Abstract A summary of mean variance analysis in portfolio management and capital asset pricing model. 1. Mean-Variance
More informationChapter 7: Portfolio Theory
Chapter 7: Portfolio Theory 1. Introduction 2. Portfolio Basics 3. The Feasible Set 4. Portfolio Selection Rules 5. The Efficient Frontier 6. Indifference Curves 7. The Two-Asset Portfolio 8. Unrestriceted
More informationQuantitative Risk Management
Quantitative Risk Management Asset Allocation and Risk Management Martin B. Haugh Department of Industrial Engineering and Operations Research Columbia University Outline Review of Mean-Variance Analysis
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 informationEfficient Frontier and Asset Allocation
Topic 4 Efficient Frontier and Asset Allocation LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Explain the concept of efficient frontier and Markowitz portfolio theory; 2. Discuss
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 informationNote on Using Excel to Compute Optimal Risky Portfolios. Candie Chang, Hong Kong University of Science and Technology
Candie Chang, Hong Kong University of Science and Technology Andrew Kaplin, Kellogg Graduate School of Management, NU Introduction This document shows how to, (1) Compute the expected return and standard
More informationMean-Variance Analysis
Mean-Variance Analysis Mean-variance analysis 1/ 51 Introduction How does one optimally choose among multiple risky assets? Due to diversi cation, which depends on assets return covariances, the attractiveness
More informationCSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization
CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio Optimization March 9 16, 2018 1 / 19 The portfolio optimization problem How to best allocate our money to n risky assets S 1,..., S n with
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 informationFall 2005: FiSOOO - Ouiz #2 Part I - Open Questions
Fall 2005: FiSOOO - Ouiz #2 Part I - Open Questions 1. Rita's utility of final wealth is defined by u(w)=~w. She is broke but one lucky day she found a lottery ticket with two possible outcomes: $196 with
More informationIntroduction to Computational Finance and Financial Econometrics Introduction to Portfolio Theory
You can t see this text! Introduction to Computational Finance and Financial Econometrics Introduction to Portfolio Theory Eric Zivot Spring 2015 Eric Zivot (Copyright 2015) Introduction to Portfolio Theory
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 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 informationOptimal Portfolio Selection
Optimal Portfolio Selection We have geometrically described characteristics of the optimal portfolio. Now we turn our attention to a methodology for exactly identifying the optimal portfolio given a set
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 informationLecture 10-12: CAPM.
Lecture 10-12: CAPM. I. Reading II. Market Portfolio. III. CAPM World: Assumptions. IV. Portfolio Choice in a CAPM World. V. Minimum Variance Mathematics. VI. Individual Assets in a CAPM World. VII. Intuition
More informationPORTFOLIO THEORY. Master in Finance INVESTMENTS. Szabolcs Sebestyén
PORTFOLIO THEORY Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Portfolio Theory Investments 1 / 60 Outline 1 Modern Portfolio Theory Introduction Mean-Variance
More informationThe Baumol-Tobin and the Tobin Mean-Variance Models of the Demand
Appendix 1 to chapter 19 A p p e n d i x t o c h a p t e r An Overview of the Financial System 1 The Baumol-Tobin and the Tobin Mean-Variance Models of the Demand for Money The Baumol-Tobin Model of Transactions
More information!"#$ 01$ 7.3"กก>E E?D:A 5"7=7 E!<C";E2346 <2H<
กก AEC Portfolio Investment!"#$ 01$ 7.3"กก>E E?D:A 5"7=7 >?@A?2346BC@ก"9D E!
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 informationUnderstand general-equilibrium relationships, such as the relationship between barriers to trade, and the domestic distribution of income.
Review of Production Theory: Chapter 2 1 Why? Understand the determinants of what goods and services a country produces efficiently and which inefficiently. Understand how the processes of a market economy
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 informationSolutions to questions in Chapter 8 except those in PS4. The minimum-variance portfolio is found by applying the formula:
Solutions to questions in Chapter 8 except those in PS4 1. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, σ S = 30%, σ B = 15%, ρ =.10 From the standard deviations and the correlation
More informationThe Markowitz framework
IGIDR, Bombay 4 May, 2011 Goals What is a portfolio? Asset classes that define an Indian portfolio, and their markets. Inputs to portfolio optimisation: measuring returns and risk of a portfolio Optimisation
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 informationQuantitative Portfolio Theory & Performance Analysis
550.447 Quantitative ortfolio Theory & erformance Analysis Week February 18, 2013 Basic Elements of Modern ortfolio Theory Assignment For Week of February 18 th (This Week) Read: A&L, Chapter 3 (Basic
More informationMATH362 Fundamentals of Mathematical Finance. Topic 1 Mean variance portfolio theory. 1.1 Mean and variance of portfolio return
MATH362 Fundamentals of Mathematical Finance Topic 1 Mean variance portfolio theory 1.1 Mean and variance of portfolio return 1.2 Markowitz mean-variance formulation 1.3 Two-fund Theorem 1.4 Inclusion
More informationReturn and Risk: The Capital-Asset Pricing Model (CAPM)
Return and Risk: The Capital-Asset Pricing Model (CAPM) Expected Returns (Single assets & Portfolios), Variance, Diversification, Efficient Set, Market Portfolio, and CAPM Expected Returns and Variances
More informationCHAPTER 6: PORTFOLIO SELECTION
CHAPTER 6: PORTFOLIO SELECTION 6-1 21. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, σ S = 30%, σ B = 15%, ρ =.10 From the standard deviations and the correlation coefficient
More informationFinancial Market Analysis (FMAx) Module 6
Financial Market Analysis (FMAx) Module 6 Asset Allocation and iversification This training material is the property of the International Monetary Fund (IMF) and is intended for use in IMF Institute for
More informationExample. Suppose n = 2 and we wish to invest 200 in the portfolio d = (J, ) to do this
o Wk ^ Stochastic Modelling in Finance - Mean-Variance Analysis Three basic assumptions (i) Agents only care about the mean and variance of returns. (ii) Agents have homogeneous beliefs (that is they can
More informationFinancial Economics: Risk Aversion and Investment Decisions, Modern Portfolio Theory
Financial Economics: Risk Aversion and Investment Decisions, Modern Portfolio Theory Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY April, 2015 1 / 95 Outline Modern portfolio theory The backward induction,
More informationSDMR Finance (2) Olivier Brandouy. University of Paris 1, Panthéon-Sorbonne, IAE (Sorbonne Graduate Business School)
SDMR Finance (2) Olivier Brandouy University of Paris 1, Panthéon-Sorbonne, IAE (Sorbonne Graduate Business School) Outline 1 Formal Approach to QAM : concepts and notations 2 3 Portfolio risk and return
More informationChapter 8. Portfolio Selection. Learning Objectives. INVESTMENTS: Analysis and Management Second Canadian Edition
INVESTMENTS: Analysis and Management Second Canadian Edition W. Sean Cleary Charles P. Jones Chapter 8 Portfolio Selection Learning Objectives State three steps involved in building a portfolio. Apply
More informationPortfolio Management
MCF 17 Advanced Courses Portfolio Management Final Exam Time Allowed: 60 minutes Family Name (Surname) First Name Student Number (Matr.) Please answer all questions by choosing the most appropriate alternative
More informationKey investment insights
Basic Portfolio Theory B. Espen Eckbo 2011 Key investment insights Diversification: Always think in terms of stock portfolios rather than individual stocks But which portfolio? One that is highly diversified
More informationMean Variance Portfolio Theory
Chapter 1 Mean Variance Portfolio Theory This book is about portfolio construction and risk analysis in the real-world context where optimization is done with constraints and penalties specified by the
More informationModeling Portfolios that Contain Risky Assets Optimization II: Model-Based Portfolio Management
Modeling Portfolios that Contain Risky Assets Optimization II: Model-Based Portfolio Management C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling January 26, 2012
More informationMean-Variance Analysis
Mean-Variance Analysis If the investor s objective is to Maximize the Expected Rate of Return for a given level of Risk (or, Minimize Risk for a given level of Expected Rate of Return), and If the investor
More informationCOMM 324 INVESTMENTS AND PORTFOLIO MANAGEMENT ASSIGNMENT 1 Due: October 3
COMM 324 INVESTMENTS AND PORTFOLIO MANAGEMENT ASSIGNMENT 1 Due: October 3 1. The following information is provided for GAP, Incorporated, which is traded on NYSE: Fiscal Yr Ending January 31 Close Price
More informationMS-E2114 Investment Science Lecture 5: Mean-variance portfolio theory
MS-E2114 Investment Science Lecture 5: Mean-variance portfolio theory A. Salo, T. Seeve Systems Analysis Laboratory Department of System Analysis and Mathematics Aalto University, School of Science Overview
More informationRisk and Return. Nicole Höhling, Introduction. Definitions. Types of risk and beta
Risk and Return Nicole Höhling, 2009-09-07 Introduction Every decision regarding investments is based on the relationship between risk and return. Generally the return on an investment should be as high
More informationChapter 6 Efficient Diversification. b. Calculation of mean return and variance for the stock fund: (A) (B) (C) (D) (E) (F) (G)
Chapter 6 Efficient Diversification 1. E(r P ) = 12.1% 3. a. The mean return should be equal to the value computed in the spreadsheet. The fund's return is 3% lower in a recession, but 3% higher in a boom.
More informationCHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS
CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e) 2. (b) A higher borrowing is a consequence of the risk of the borrowers default. In perfect markets with no additional
More informationEliminating Substitution Bias. One eliminate substitution bias by continuously updating the market basket of goods purchased.
Eliminating Substitution Bias One eliminate substitution bias by continuously updating the market basket of goods purchased. 1 Two-Good Model Consider a two-good model. For good i, the price is p i, and
More informationLECTURE NOTES 3 ARIEL M. VIALE
LECTURE NOTES 3 ARIEL M VIALE I Markowitz-Tobin Mean-Variance Portfolio Analysis Assumption Mean-Variance preferences Markowitz 95 Quadratic utility function E [ w b w ] { = E [ w] b V ar w + E [ w] }
More informationCh. 8 Risk and Rates of Return. Return, Risk and Capital Market. Investment returns
Ch. 8 Risk and Rates of Return Topics Measuring Return Measuring Risk Risk & Diversification CAPM Return, Risk and Capital Market Managers must estimate current and future opportunity rates of return for
More informationPortfolios that Contain Risky Assets 10: Limited Portfolios with Risk-Free Assets
Portfolios that Contain Risky Assets 10: Limited Portfolios with Risk-Free Assets C. David Levermore University of Maryland, College Park, MD Math 420: Mathematical Modeling March 21, 2018 version c 2018
More informationFIN Second (Practice) Midterm Exam 04/11/06
FIN 3710 Investment Analysis Zicklin School of Business Baruch College Spring 2006 FIN 3710 Second (Practice) Midterm Exam 04/11/06 NAME: (Please print your name here) PLEDGE: (Sign your name here) SESSION:
More informationEcon 424/CFRM 462 Portfolio Risk Budgeting
Econ 424/CFRM 462 Portfolio Risk Budgeting Eric Zivot August 14, 2014 Portfolio Risk Budgeting Idea: Additively decompose a measure of portfolio risk into contributions from the individual assets in the
More informationPortfolios that Contain Risky Assets Portfolio Models 9. Long Portfolios with a Safe Investment
Portfolios that Contain Risky Assets Portfolio Models 9. Long Portfolios with a Safe Investment C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling March 21, 2016 version
More informationP s =(0,W 0 R) safe; P r =(W 0 σ,w 0 µ) risky; Beyond P r possible if leveraged borrowing OK Objective function Mean a (Std.Dev.
ECO 305 FALL 2003 December 2 ORTFOLIO CHOICE One Riskless, One Risky Asset Safe asset: gross return rate R (1 plus interest rate) Risky asset: random gross return rate r Mean µ = E[r] >R,Varianceσ 2 =
More information15.414: COURSE REVIEW. Main Ideas of the Course. Approach: Discounted Cashflows (i.e. PV, NPV): CF 1 CF 2 P V = (1 + r 1 ) (1 + r 2 ) 2
15.414: COURSE REVIEW JIRO E. KONDO Valuation: Main Ideas of the Course. Approach: Discounted Cashflows (i.e. PV, NPV): and CF 1 CF 2 P V = + +... (1 + r 1 ) (1 + r 2 ) 2 CF 1 CF 2 NP V = CF 0 + + +...
More informationStochastic Programming and Financial Analysis IE447. Midterm Review. Dr. Ted Ralphs
Stochastic Programming and Financial Analysis IE447 Midterm Review Dr. Ted Ralphs IE447 Midterm Review 1 Forming a Mathematical Programming Model The general form of a mathematical programming model is:
More informationFinancial Analysis The Price of Risk. Skema Business School. Portfolio Management 1.
Financial Analysis The Price of Risk bertrand.groslambert@skema.edu Skema Business School Portfolio Management Course Outline Introduction (lecture ) Presentation of portfolio management Chap.2,3,5 Introduction
More informationCopyright 2009 Pearson Education Canada
Operating Cash Flows: Sales $682,500 $771,750 $868,219 $972,405 $957,211 less expenses $477,750 $540,225 $607,753 $680,684 $670,048 Difference $204,750 $231,525 $260,466 $291,722 $287,163 After-tax (1
More informationFirst of all we have to read all the data with an xlsread function and give names to the subsets of data that we are interested in:
First of all we have to read all the data with an xlsread function and give names to the subsets of data that we are interested in: data=xlsread('c:\users\prado\desktop\master\investment\material alumnos\data.xlsx')
More informationChapter 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
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 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 informationCHAPTER 9: THE CAPITAL ASSET PRICING MODEL
CHAPTER 9: THE CAPITAL ASSET PRICING MODEL 1. E(r P ) = r f + β P [E(r M ) r f ] 18 = 6 + β P(14 6) β P = 12/8 = 1.5 2. If the security s correlation coefficient with the market portfolio doubles (with
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 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 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 informationLecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall Financial mathematics
Lecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall 2014 Reduce the risk, one asset Let us warm up by doing an exercise. We consider an investment with σ 1 =
More informationCapital Asset Pricing Model
Capital Asset Pricing Model 1 Introduction In this handout we develop a model that can be used to determine how an investor can choose an optimal asset portfolio in this sense: the investor will earn the
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 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 informationJournal of Computational and Applied Mathematics. The mean-absolute deviation portfolio selection problem with interval-valued returns
Journal of Computational and Applied Mathematics 235 (2011) 4149 4157 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
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 informationPortfolio Risk Management and Linear Factor Models
Chapter 9 Portfolio Risk Management and Linear Factor Models 9.1 Portfolio Risk Measures There are many quantities introduced over the years to measure the level of risk that a portfolio carries, and each
More informationSome useful optimization problems in portfolio theory
Some useful optimization problems in portfolio theory Igor Melicherčík Department of Economic and Financial Modeling, Faculty of Mathematics, Physics and Informatics, Mlynská dolina, 842 48 Bratislava
More informationFinancial Economics 4: Portfolio Theory
Financial Economics 4: Portfolio Theory Stefano Lovo HEC, Paris What is a portfolio? Definition A portfolio is an amount of money invested in a number of financial assets. Example Portfolio A is worth
More informationMATH4512 Fundamentals of Mathematical Finance. Topic Two Mean variance portfolio theory. 2.1 Mean and variance of portfolio return
MATH4512 Fundamentals of Mathematical Finance Topic Two Mean variance portfolio theory 2.1 Mean and variance of portfolio return 2.2 Markowitz mean-variance formulation 2.3 Two-fund Theorem 2.4 Inclusion
More informationRETURN AND RISK: The Capital Asset Pricing Model
RETURN AND RISK: The Capital Asset Pricing Model (BASED ON RWJJ CHAPTER 11) Return and Risk: The Capital Asset Pricing Model (CAPM) Know how to calculate expected returns Understand covariance, correlation,
More informationKevin Dowd, Measuring Market Risk, 2nd Edition
P1.T4. Valuation & Risk Models Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Dowd, Chapter 2: Measures of Financial Risk
More informationEQUITIES & INVESTMENT ANALYSIS MAF307 EXAM SUMMARY
EQUITIES & INVESTMENT ANALYSIS MAF307 EXAM SUMMARY TOPIC 1 INVESTMENT ENVIRONMENT & FINANCIAL INSTRUMENTS 4 FINANCIAL ASSETS - INTANGIBLE 4 BENEFITS OF INVESTING IN FINANCIAL ASSETS 4 REAL ASSETS 4 CLIENTS
More informationCorporate Finance, Module 3: Common Stock Valuation. Illustrative Test Questions and Practice Problems. (The attached PDF file has better formatting.
Corporate Finance, Module 3: Common Stock Valuation Illustrative Test Questions and Practice Problems (The attached PDF file has better formatting.) These problems combine common stock valuation (module
More informationMidterm 1, Financial Economics February 15, 2010
Midterm 1, Financial Economics February 15, 2010 Name: Email: @illinois.edu All questions must be answered on this test form. Question 1: Let S={s1,,s11} be the set of states. Suppose that at t=0 the state
More informationIn terms of covariance the Markowitz portfolio optimisation problem is:
Markowitz portfolio optimisation Solver To use Solver to solve the quadratic program associated with tracing out the efficient frontier (unconstrained efficient frontier UEF) in Markowitz portfolio optimisation
More informationMathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should
Mathematics of Finance Final Preparation December 19 To be thoroughly prepared for the final exam, you should 1. know how to do the homework problems. 2. be able to provide (correct and complete!) definitions
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 informationOutline for today. Stat155 Game Theory Lecture 13: General-Sum Games. General-sum games. General-sum games. Dominated pure strategies
Outline for today Stat155 Game Theory Lecture 13: General-Sum Games Peter Bartlett October 11, 2016 Two-player general-sum games Definitions: payoff matrices, dominant strategies, safety strategies, Nash
More informationEE365: Risk Averse Control
EE365: Risk Averse Control Risk averse optimization Exponential risk aversion Risk averse control 1 Outline Risk averse optimization Exponential risk aversion Risk averse control Risk averse optimization
More informationContinuing Education Course #287 Engineering Methods in Microsoft Excel Part 2: Applied Optimization
1 of 6 Continuing Education Course #287 Engineering Methods in Microsoft Excel Part 2: Applied Optimization 1. Which of the following is NOT an element of an optimization formulation? a. Objective function
More informationCalculating EAR and continuous compounding: Find the EAR in each of the cases below.
Problem Set 1: Time Value of Money and Equity Markets. I-III can be started after Lecture 1. IV-VI can be started after Lecture 2. VII can be started after Lecture 3. VIII and IX can be started after Lecture
More informationAversion to Risk and Optimal Portfolio Selection in the Mean- Variance Framework
Aversion to Risk and Optimal Portfolio Selection in the Mean- Variance Framework Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2017 Outline and objectives Four alternative
More informationThe mean-variance portfolio choice framework and its generalizations
The mean-variance portfolio choice framework and its generalizations Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2014 Outline and objectives The backward, three-step solution
More informationSolutions to Problem Set 1
Solutions to Problem Set Theory of Banking - Academic Year 06-7 Maria Bachelet maria.jua.bachelet@gmail.com February 4, 07 Exercise. An individual consumer has an income stream (Y 0, Y ) and can borrow
More informationAnalysis INTRODUCTION OBJECTIVES
Chapter5 Risk Analysis OBJECTIVES At the end of this chapter, you should be able to: 1. determine the meaning of risk and return; 2. explain the term and usage of statistics in determining risk and return;
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 informationLecture 10: Performance measures
Lecture 10: Performance measures Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe Portfolio and Asset Liability Management Summer Semester 2008 Prof.
More informationKEIR EDUCATIONAL RESOURCES
INVESTMENT PLANNING 2017 Published by: KEIR EDUCATIONAL RESOURCES 4785 Emerald Way Middletown, OH 45044 1-800-795-5347 1-800-859-5347 FAX E-mail customerservice@keirsuccess.com www.keirsuccess.com TABLE
More informationAPPENDIX TO LECTURE NOTES ON ASSET PRICING AND PORTFOLIO MANAGEMENT. Professor B. Espen Eckbo
APPENDIX TO LECTURE NOTES ON ASSET PRICING AND PORTFOLIO MANAGEMENT 2011 Professor B. Espen Eckbo 1. Portfolio analysis in Excel spreadsheet 2. Formula sheet 3. List of Additional Academic Articles 2011
More information