Mean Variance Analysis
|
|
- Harvey Morgan
- 7 years ago
- Views:
Transcription
1 Mean Variance Analysis Karl B. Diether Fisher College of Business Karl B. Diether (Fisher College of Business) Mean Variance Analysis 1 / 36 A Portfolio of Three Risky Assets Not a two risky asset world We need to be able to deal with portfolios that have many assets. The return on a three-asset portfolio r p = w a r a + w b r b + w c r c r a is the return and w a is the weight on asset A. r b is the return and w b is the weight on asset B. r c is the return and w c is the weight on asset C. w a + w b + w c = 1. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 2 / 36
2 A Portfolio of Three Risky Assets Three asset portfolio: E(r p ) and σ(r p ) If A, B, and C are assets, w a, w b, and w c are the portfolio weights on each asset, and w a + w b + w c = 1, then the expected return of the portfolio is E(r p ) = w a E(r a ) + w b E(r b ) + w c E(r c ) the variance of the portfolio is σ 2 (r p ) = w 2 a σ 2 (r a ) + w 2 b σ2 (r b ) + w 2 c σ 2 (r c ) + 2w a w b cov(r a, r b ) + 2w a w c cov(r a, r c ) + 2w b w c cov(r b, r c ), and the standard deviation of the portfolio is σ(r p ) = σ 2 (r p ). Karl B. Diether (Fisher College of Business) Mean Variance Analysis 3 / 36 A Portfolio of Three Risky Assets Expected returns The expected return of a portfolio is just a weighted sum of the individual security expected returns. Variance is a bit more complicated Covariances are more influential than with a 2-asset portfolio. Three covariances now instead of one. As we add more assets, covariances become a more important determinate of the portfolio s variance (diversification). Karl B. Diether (Fisher College of Business) Mean Variance Analysis 4 / 36
3 Example: Three Risky Assets You hate Microsoft so you invest in three competitors Covariance Matrix E(r i ) Apple Sun Micro Red Hat Apple Sun Micro Red Hat The covaraince matrix The diagonal of the covariance matrix holds variances. cov(r apple, r apple ) = σ 2 (r apple ) The covariance matrix is symmetric (row 1, column 2 = row 2, column 1): cov(r apple, r sun ) = cov(r sun, r apple ) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 5 / 36 Example: Three Risky Assets Computing E(r p ) and σ(r p ) E(r p ) = w a E(r a ) + w s E(r s ) + w h E(r h ) = 1 3 (0.2) (0.12) (0.15) = 15.7% σ 2 (r p ) = w 2 a σ 2 (r a ) + w 2 s σ 2 (r s ) + w 2 h σ2 (r h ) + 2w a w s cov(r a, r s ) + 2w a w h cov(r a, r h ) + 2w s w h cov(r s, r h ) = 1 9 (0.09) (0.07) (0.06) + 2( )(0.045) + 2( )(0.05) + 2( )(0.04) = σ(r p ) = = 23.3% Karl B. Diether (Fisher College of Business) Mean Variance Analysis 6 / 36
4 A Portfolio of Many Risky Assets An N asset portfolio Consider a portfolio that has n assets (n can be a very large number like 1000, 7000, or even more than that). The return on an N asset portfolio r p = w 1 r 1 + w 2 r w n r n n = w i r i i=1 r i is the return on the ith asset in the portfolio. w i is the portfolio weight for the ith asset. The sum of the weights equals one: n i=1 w i = 1 Karl B. Diether (Fisher College of Business) Mean Variance Analysis 7 / 36 Many Risky Assets: Expected Return Expected return still a weighted sum The expected return is a weighted average of the expected returns of the individual securities in the portfolio: ) E(r p ) = E (w 1 r 1 + w 2 r w n r n = w 1 E(r 1 ) + w 2 E(r 2 ) + + w n E(r n ) n = w i E(r i ) i=1 Karl B. Diether (Fisher College of Business) Mean Variance Analysis 8 / 36
5 Many Risky Assets: Variance The variance σ 2 r p = w 2 1 σ 2 (r 1 ) + w 2 2 σ 2 (r 2 ) + + w 2 n σ 2 (r n ) + 2w 1 w 2 cov(r 1, r 2 ) + 2w 1 w 3 cov(r 1, r 3 ) + + 2w 1 w n cov(r 1, r n ) + 2w 2 w 3 cov(r 2, r 3 ) + 2w 2 w 4 cov(r 2, r 4 ) + + 2w 2 w n cov(r 2, r n ).. + 2w n 2 w n 1 cov(r n 2, r n 1 ) + 2w n 2 w n cov(r n 2, r n ) + 2w n 1 w n cov(r n 1, r n ). Writing the variance more compactly n n σ 2 (r p ) = wi 2 σ 2 (r i ) + i=1 n i=1 j=i+1 2w i w j cov(r i, r j ) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 9 / 36 Thinking About Risk Risky Securities What makes a stock risky? Risky Portfolios What makes a portfolio risky? Karl B. Diether (Fisher College of Business) Mean Variance Analysis 10 / 36
6 Many Risky Assets: Variance The importance of covariances If the number of securities is large (and the weights are small), covariances become the most important determinant of a portfolio s variance. For a n asset portfolio there are n(n 1)/2 covariance terms, and n variance terms. For example, a portfolio of a 1000 stocks, has 1000 variance terms and 499,950 unique covariance terms. Compare that to a portfolio of two assets which has 2 variance terms, and 1 covariance term. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 11 / 36 The Power of One How does an asset affect a portfolio s variance? Just find all the terms in the variance equation that involve the asset. Let s do it for asset 1. w 2 1 σ 2 (r 1 ) + 2w 1 w 2 cov(r 1, r 2 ) + + 2w 1 w n cov(r 1, r n ) The risk of security 1 in portfolio p An asset s influence on a portfolio s variance primarily depends on how it covaries with the other assets in the portfolio. Thus, what matters is how an asset covaries with the portfolio. The above expression almost equals, w 1 cov(r 1, r p ) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 12 / 36
7 Minimizing the Portfolio Variance A small thought experiment To illustrate the importance of covariances lets think about how we could find the minimum variance portfolio. Finding the portfolio with the smallest possible variance 1 Find two securities already in the portfolio with different covariances with the portfolio. 2 Add a little weight to the security with a lower cov(r i, r p ), and subtract a little from the security with the higher covariance. 3 The portfolio variance is a little lower. Repeat steps 1 and 2 until the variance cannot be lowered anymore. The variance of the portfolio will be minimized when all the securities have the same covariance with the portfolio: cov(r 1, r p ) = cov(r 2, r p ) = = cov(r n, r p ) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 13 / 36 Mean Variance Analysis Asset allocation with many assets Suppose you can invest in many risky assets. What is the optimal allocation? Clearly it depends on preferences. Can we say anything about the portfolios that should be chosen? What properity will the optimal or best portfolios have? What does the investment opportunity set look like when we have many assets? Karl B. Diether (Fisher College of Business) Mean Variance Analysis 14 / 36
8 Investment Opportunity Set Red curve called the efficient frontier Any point on the curve or to the right is a feasible portfolio. E(r) Feasible investment set is an area. Global Min. Variance portfolio σ(r) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 15 / 36 The Investment Opportunity Set The investment opportunity set When there are many risky assets the feasible investment set is a curve and the area to the right of the curve. The boundary and the frontier The curve in the graph is called the mean-variance boundary or the mean-variance frontier. The portion of the curve above the global minimum variance portfolio is called the mean-variance efficient frontier or just the efficient frontier. An investor will only choose a portfolio on the efficient frontier. Portfolios on the efficient frontier are called mean variance efficient portfolios or efficient portfolios. Adding more assets pushes the curve towards the Northwest corner of the graph. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 16 / 36
9 Mean Variance Frontier: Adding Assets E(r) σ(r) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 17 / 36 The Mean Variance Boundary How is the mean-variance boundary formed? We pick the expected return we want, and then choose the weights of the portfolio so that the variance of the portfolio is minimized. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 18 / 36
10 Two-Fund Seperation Two-Fund Separation The entire mean-variance boundary can be created from any two portfolios on the mean-variance boundary. The entire efficient frontier can be created from any two mean variance efficient portfolios. More implications A portfolio of portfolios on the mean-variance boundary will also be on the boundary. A portfolio of mean variance efficient portfolios will be on the efficient frontier if the weight on each portfolio is positive. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 19 / 36 Adding a Riskless Asset How do you allocate your money between a riskless asset and risky mean variance efficient portfolios? Logically, this is the same as when we only had two risky assets. We can draw a CAL between every efficient portfolio (the only ones we care about) and the riskfree asset. The best or optimal CAL is the one with the highest slope. The optimal risky portfolio is the portfolio that is tangent to the optimal CAL (the tangency portfolio). Karl B. Diether (Fisher College of Business) Mean Variance Analysis 20 / 36
11 Risk Free Asset and Many Risky Assets CAL 3 CAL 2 E(r) CAL 1 riskfree rate σ(r) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 21 / 36 The Optimal CAL E(r) The tangency portfolio riskfree rate σ(r) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 22 / 36
12 The Optimal CAL The optimal CAL: Definition The CAL with the highest slope (Sharpe ratio): The tangency portfolio Slope = E(r p) r f σ(r p ) The optimal CAL is the tangency line between the riskfree rate and the mean variance efficient frontier comprised entirely of risky assets. The tangency portfolio is the portfolio comprised of only risky assets with the highest Sharpe ratio. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 23 / 36 The New Efficient Frontier The new efficient frontier is the optimal CAL If there are many risky assets and a riskfree asset, the efficient frontier is the optimal CAL. Optimal Allocations Investors combine the tangency portfolio with the riskfree asset to form their overall portfolio. The allocation they choose depends on their preferences for risk and expected return. Note: Two-fund separation still holds; we can trace out the entire frontier with two portfolios on the frontier. For example, the tangency portfolio and the riskfree asset. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 24 / 36
13 The Optimal CAL E(r) The tangency portfolio riskfree rate σ(r) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 25 / 36 Finding the Tangency Portfolio How do we find the tangency portfolio? It is the portfolio comprised of only risky assets with the highest Sharpe ratio: An inccorect method SR = E(r p) r f σ(r p ) 1 Form a portfolio using all the risky securities (any portfolio will do). 2 Find two securities already in the portfolio with different Sharpe ratios. Wrong! Step 2 is not quite right. Let s think about the goal. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 26 / 36
14 Finding the Tangency Portfolio What we care about We only care about how a security affects the portfolio s Sharpe ratio. Remember, securities only affect the variance of a portfolio via its covariance with the portfolio. Examine the marginal sharpe ratios We really just need to look for securities with different, E(r i ) r f cov(r i, r p ) Note: the above ratio is called the marginal Sharpe ratio. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 27 / 36 Finding the Tangency Portfolio The correct method 1 Form a portfolio using all the risky securities (any portfolio will work). 2 Find two securities already in the portfolio with different risk premium to covariance ratios: E(r i ) r f cov(r i, r p ) 3 Add a little weight to the security with a higher ratio, and subtract a little from the security with the lower ratio. 4 Keep repeating steps 1 3 until, marginal Sharpe ratios are equal across all securities. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 28 / 36
15 Finding the Tangency Portfolio All the marginal Sharpe ratios are equal For the tangency portfolio (T), the following is true: E(r 1 ) r f cov(r 1, r T ) = E(r 2) r f cov(r 2, r T ) = = E(r n) r f cov(r n, r T ) This includes all individual securities and other portfolios. Rewriting compactly We can write the tangency condition as the following: E(r i ) r f cov(r i, r T ) = E(r j) r f cov(r j, r T ) for all i and j Karl B. Diether (Fisher College of Business) Mean Variance Analysis 29 / 36 Example: The Tangency Portfolio Suppose you can invest in two risky assets and a riskfree asset E(r) σ Stock 1 11% 20% Stock 2 16% 25% Riskfree 6% cov(r 1, r 2 ) = The tangency condition Remember, the following is true for the tangency portfolio (T): E(r 1 ) r f cov(r 1, r T ) = E(r 2) r f cov(r 2, r T ) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 30 / 36
16 Example: The Tangency Portfolio Computing covariance of a security with a portfolio If x, y, and z are random variables and A and B are constants, cov(z, Ax + By) = A cov(z, x) + B cov(z, y) If portfolio P is comprised of two assets called A and B, w is the weight on A and 1 w is the weight on B, then the covariance of a security I with portfolio P is the following: cov(r i, r p ) = w cov(r i, r a ) + (1 w) cov(r i, r b ) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 31 / 36 Example: The Tangency Portfolio The rule applied to the example cov(r 1, r T ) = w cov(r 1, r 1 ) + (1 w) cov(r 1, r 2 ) = wσ 2 (r 1 ) + (1 w) cov(r 1, r 2 ) cov(r 2, r T ) = w cov(r 2, r 1 ) + (1 w) cov(r 2, r 2 ) = w cov(r 1, r 2 ) + (1 w)σ 2 (r 2 ) Now we are ready to find the tangency portfolio We can solve for the optimal w by using the tangency portfolio condition: E(r 1 ) r f cov(r 1, r T ) = E(r 2) r f cov(r 2, r T ) Karl B. Diether (Fisher College of Business) Mean Variance Analysis 32 / 36
17 Example: The Tangency Portfolio E(r 1 ) r f cov(r 1, r T ) = E(r 2) r f cov(r 2, r T ) ( )[ E(r1 ) r f cov(r2, r T ) ] = ( )[ E(r 2 ) r f cov(r1, r T ) ] 5 [ wcov(r 1, r 2 ) + (1 w)σ 2 (r 2 ) ] = 10 [ wσ 2 (r 1 ) + (1 w)cov(r 1, r 2 ) ] 5[0.025w + (1 w)0.0625] = 10[0.04w + (1 w)0.025] 5[ w ] = 10[0.015w ] w = 0.15w w = w = The optimal portfolio is 18.5% in stock 1 and 81.5% in stock 2. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 33 / 36 Example: The Tangency Portfolio What is the Sharpe Ratio of the tangency portfolio? E(r T ) = 0.185(0.11) (0.16) = 15.08% σ 2 (r t ) = (0.2 2 ) ( ) + 2(0.185)(0.815)(0.025) SR T = = = Compare that to the Sharpe ratios of stock 1 and 2: SR 1 = SR 2 = = 0.25 = 0.40 Karl B. Diether (Fisher College of Business) Mean Variance Analysis 34 / 36
18 A Few Things to Think About Problems with mean variance analysis Can you think of any limitations with respect to mean variance analysis? What are some of the practical problems people face when they try to use mean variance analysis? How might people manage some of these practical problems? Warren Buffet Warren Buffet claims he doesn t engage in mean variance analysis. Is that a mistake or can you think of reasons why it doesn t make sense for him to use mean variance analysis? Karl B. Diether (Fisher College of Business) Mean Variance Analysis 35 / 36 Summary An asset s influence on a portfolio s variance primarily depends on how it covaries with the other assets in the portfolio. A rational and risk averse investor will only invest in efficient portfolios. Efficient portfolios have the highest expected return for a given standard deviation If there is a risk free asset then the efficient frontier is a CAL line between the riskfree asset and the tangency portfolio. The tangency portfolio is composed of only risky assets. The tangency portfolio has the highest Sharpe ratio of any portfolio composed of risky assets. In the tangency portfolio all the marginal Sharpe ratios are equal. Karl B. Diether (Fisher College of Business) Mean Variance Analysis 36 / 36
Lecture 1: Asset Allocation
Lecture 1: Asset Allocation Investments FIN460-Papanikolaou Asset Allocation I 1/ 62 Overview 1. Introduction 2. Investor s Risk Tolerance 3. Allocating Capital Between a Risky and riskless asset 4. Allocating
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
Capital Allocation Between The Risky And The Risk- Free Asset. Chapter 7
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
Lesson 5. Risky assets
Lesson 5. Risky assets Prof. Beatriz de Blas May 2006 5. Risky assets 2 Introduction How stock markets serve to allocate risk. Plan of the lesson: 8 >< >: 1. Risk and risk aversion 2. Portfolio risk 3.
15.401 Finance Theory
Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lecture 13 14 14: : Risk Analytics and Critical Concepts Motivation Measuring Risk and Reward Mean-Variance
CHAPTER 7: OPTIMAL RISKY PORTFOLIOS
CHAPTER 7: OPTIMAL RIKY PORTFOLIO PROLEM ET 1. (a) and (e).. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash and real estate.
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
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
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
Review for Exam 2. Instructions: Please read carefully
Review for Exam 2 Instructions: Please read carefully The exam will have 25 multiple choice questions and 5 work problems You are not responsible for any topics that are not covered in the lecture note
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
Instructor s Manual Chapter 12 Page 144
Chapter 12 1. Suppose that your 58-year-old father works for the Ruffy Stuffed Toy Company and has contributed regularly to his company-matched savings plan for the past 15 years. Ruffy contributes $0.50
Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10?
Return, Risk, and Risk Aversion Holding Period Return Ending Price - Beginning Price + Intermediate Income Return = Beginning Price R P t+ t+ = Pt + Dt P t An Example You bought IBM stock at $40 last month.
SAMPLE MID-TERM QUESTIONS
SAMPLE MID-TERM QUESTIONS William L. Silber HOW TO PREPARE FOR THE MID- TERM: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below,
Optimal Risky Portfolios Chapter 7 Investments Bodie, Kane and Marcus
Optimal Risky ortfolios Section escription 7.0 Introduction 7.1 iversification and ortfolio Risk 7. ortfolios of Two Risky Assets 7.3 Asset Allocation with Stocks, Bonds and Bills 7.4 The Markowitz ortfolio
1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises.
1. Solutions to PS 1: 1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises. 7. The bill has a maturity of one-half year, and an annualized
Using the Solver add-in in MS Excel 2007
First version: April 21, 2008 Last revision: February 22, 2011 ANDREI JIRNYI, KELLOGG OFFICE OF RESEARCH Using the Solver add-in in MS Excel 2007 The Excel Solver add-in is a tool that allows you to perform
15.433 Investments. Assignment 1: Securities, Markets & Capital Market Theory. Each question is worth 0.2 points, the max points is 3 points
Assignment 1: Securities, Markets & Capital Market Theory Each question is worth 0.2 points, the max points is 3 points 1. The interest rate charged by banks with excess reserves at a Federal Reserve Bank
AFM 472. Midterm Examination. Monday Oct. 24, 2011. A. Huang
AFM 472 Midterm Examination Monday Oct. 24, 2011 A. Huang Name: Answer Key Student Number: Section (circle one): 10:00am 1:00pm 2:30pm Instructions: 1. Answer all questions in the space provided. If space
Chapter 13 Composition of the Market Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z,
Chapter 13 Composition of the arket Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z, and a riskless government security. Evaluated at current prices in
CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM)
CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM) Answers to Concepts Review and Critical Thinking Questions 1. Some of the risk in holding any asset is unique to the asset in question.
CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS
CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e). (b) A higher borrowing is a consequence of the risk of the borrowers default. In perfect markets with no additional
Note: There are fewer problems in the actual Final Exam!
HEC Paris Practice Final Exam Questions Version with Solutions Financial Markets Fall 2013 Note: There are fewer problems in the actual Final Exam! Problem 1. Are the following statements True, False or
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
Practice Set #4 and Solutions.
FIN-469 Investments Analysis Professor Michel A. Robe Practice Set #4 and Solutions. What to do with this practice set? To help students prepare for the assignment and the exams, practice sets with solutions
Chapter 2 Portfolio Management and the Capital Asset Pricing Model
Chapter 2 Portfolio Management and the Capital Asset Pricing Model In this chapter, we explore the issue of risk management in a portfolio of assets. The main issue is how to balance a portfolio, that
Chapter 1. Introduction to Portfolio Theory. 1.1 Portfolios of Two Risky Assets
Chapter 1 Introduction to Portfolio Theory Updated: August 9, 2013. This chapter introduces modern portfolio theory in a simplified setting where there are only two risky assets and a single risk-free
Portfolio Optimization Part 1 Unconstrained Portfolios
Portfolio Optimization Part 1 Unconstrained Portfolios John Norstad j-norstad@northwestern.edu http://www.norstad.org September 11, 2002 Updated: November 3, 2011 Abstract We recapitulate the single-period
END OF CHAPTER EXERCISES - ANSWERS. Chapter 14 : Stock Valuation and the EMH
1 EN OF CHAPTER EXERCISES - ANSWERS Chapter 14 : Stock Valuation and the EMH Q1 oes the dividend discount model ignore the mass of investors who have bought their shares with the intention of selling them
Portfolio Performance Measures
Portfolio Performance Measures Objective: Evaluation of active portfolio management. A performance measure is useful, for example, in ranking the performance of mutual funds. Active portfolio managers
Risk and Return Models: Equity and Debt. Aswath Damodaran 1
Risk and Return Models: Equity and Debt Aswath Damodaran 1 First Principles Invest in projects that yield a return greater than the minimum acceptable hurdle rate. The hurdle rate should be higher for
Lecture 3: CAPM in practice
Lecture 3: CAPM in practice Investments FIN460-Papanikolaou CAPM in practice 1/ 59 Overview 1. The Markowitz model and active portfolio management. 2. A Note on Estimating β 3. Using the single-index model
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)
M.I.T. Spring 1999 Sloan School of Management 15.415. First Half Summary
M.I.T. Spring 1999 Sloan School of Management 15.415 First Half Summary Present Values Basic Idea: We should discount future cash flows. The appropriate discount rate is the opportunity cost of capital.
Models of Risk and Return
Models of Risk and Return Aswath Damodaran Aswath Damodaran 1 First Principles Invest in projects that yield a return greater than the minimum acceptable hurdle rate. The hurdle rate should be higher for
Review for Exam 2. Instructions: Please read carefully
Review for Exam Instructions: Please read carefully The exam will have 1 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation questions.
CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between r P and
5. Capital Asset Pricing Model and Factor Models Capital market line (CML) CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between
Chapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver
FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver Question: How do you create a diversified stock portfolio? Advice given by most financial advisors
The Capital Asset Pricing Model (CAPM)
Prof. Alex Shapiro Lecture Notes 9 The Capital Asset Pricing Model (CAPM) I. Readings and Suggested Practice Problems II. III. IV. Introduction: from Assumptions to Implications The Market Portfolio Assumptions
Independent samples t-test. Dr. Tom Pierce Radford University
Independent samples t-test Dr. Tom Pierce Radford University The logic behind drawing causal conclusions from experiments The sampling distribution of the difference between means The standard error of
Executive Summary of Finance 430 Professor Vissing-Jørgensen Finance 430-62/63/64, Winter 2011
Executive Summary of Finance 430 Professor Vissing-Jørgensen Finance 430-62/63/64, Winter 2011 Weekly Topics: 1. Present and Future Values, Annuities and Perpetuities 2. More on NPV 3. Capital Budgeting
15 CAPM and portfolio management
ECG590I Asset Pricing. Lecture 15: CAPM and portfolio management 1 15 CAPM and portfolio management 15.1 Theoretical foundation for mean-variance analysis We assume that investors try to maximize the expected
Optimal Risky Portfolios
Otimal Risky Portolio Otimal Risky Portolios When choosing the otimal allocation between a risk-ree asset and a risky ortolio, we have assumed that we have already selected the otimal risky ortolio In
The fundamental question in economics is 2. Consumer Preferences
A Theory of Consumer Behavior Preliminaries 1. Introduction The fundamental question in economics is 2. Consumer Preferences Given limited resources, how are goods and service allocated? 1 3. Indifference
Chap 3 CAPM, Arbitrage, and Linear Factor Models
Chap 3 CAPM, Arbitrage, and Linear Factor Models 1 Asset Pricing Model a logical extension of portfolio selection theory is to consider the equilibrium asset pricing consequences of investors individually
Simple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
The CAPM (Capital Asset Pricing Model) NPV Dependent on Discount Rate Schedule
The CAPM (Capital Asset Pricing Model) Massachusetts Institute of Technology CAPM Slide 1 of NPV Dependent on Discount Rate Schedule Discussed NPV and time value of money Choice of discount rate influences
DERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
1. Portfolio Returns and Portfolio Risk
Chapter 8 Risk and Return: Capital Market Theory Chapter 8 Contents Learning Objectives 1. Portfolio Returns and Portfolio Risk 1. Calculate the expected rate of return and volatility for a portfolio of
Understanding the Impact of Weights Constraints in Portfolio Theory
Understanding the Impact of Weights Constraints in Portfolio Theory Thierry Roncalli Research & Development Lyxor Asset Management, Paris thierry.roncalli@lyxor.com January 2010 Abstract In this article,
Multivariate Analysis of Variance (MANOVA): I. Theory
Gregory Carey, 1998 MANOVA: I - 1 Multivariate Analysis of Variance (MANOVA): I. Theory Introduction The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the
Introduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
Enhancing the Teaching of Statistics: Portfolio Theory, an Application of Statistics in Finance
Page 1 of 11 Enhancing the Teaching of Statistics: Portfolio Theory, an Application of Statistics in Finance Nicolas Christou University of California, Los Angeles Journal of Statistics Education Volume
Standard Form of a Linear Programming Problem
494 CHAPTER 9 LINEAR PROGRAMMING 9. THE SIMPLEX METHOD: MAXIMIZATION For linear programming problems involving two variables, the graphical solution method introduced in Section 9. is convenient. However,
CAS Exam 8 Notes - Parts A&B Portfolio Theory and Equilibrium in Capital Markets Fixed Income Securities
CAS Exam 8 Notes - Parts A&B Portfolio Theory and Equilibrium in Capital Markets Fixed Income Securities Part I Table of Contents A Portfolio Theory and Equilibrium in Capital Markets 1 BKM - Ch. 6: Risk
2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
10 Binomial Trees. 10.1 One-step model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1
ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 10 Binomial Trees 10.1 One-step model 1. Model structure ECG590I Asset Pricing. Lecture 10: Binomial Trees 2 There is only one time interval (t 0, t
Midterm Exam:Answer Sheet
Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a risk-free and risky asset, with returns given by r f and E(r p ), respectively. Let y be the
Chapter 5. Risk and Return. Copyright 2009 Pearson Prentice Hall. All rights reserved.
Chapter 5 Risk and Return Learning Goals 1. Understand the meaning and fundamentals of risk, return, and risk aversion. 2. Describe procedures for assessing and measuring the risk of a single asset. 3.
Chapter 5 Uncertainty and Consumer Behavior
Chapter 5 Uncertainty and Consumer Behavior Questions for Review 1. What does it mean to say that a person is risk averse? Why are some people likely to be risk averse while others are risk lovers? A risk-averse
Because the slope is, a slope of 5 would mean that for every 1cm increase in diameter, the circumference would increase by 5cm.
Measurement Lab You will be graphing circumference (cm) vs. diameter (cm) for several different circular objects, and finding the slope of the line of best fit using the CapStone program. Write out or
How to Win the Stock Market Game
How to Win the Stock Market Game 1 Developing Short-Term Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.
Years after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540
To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. You could do this with any two columns of data, but for demonstration
Distinction Between Interest Rates and Returns
Distinction Between Interest Rates and Returns Rate of Return RET = C + P t+1 P t =i c + g P t C where: i c = = current yield P t g = P t+1 P t P t = capital gain Key Facts about Relationship Between Interest
CHAPTER 15 INTERNATIONAL PORTFOLIO INVESTMENT SUGGESTED ANSWERS AND SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS
CHAPTER 15 INTERNATIONAL PORTFOLIO INVESTMENT SUGGESTED ANSWERS AND SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS QUESTIONS 1. What factors are responsible for the recent surge in international portfolio
A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study
A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study But I will offer a review, with a focus on issues which arise in finance 1 TYPES OF FINANCIAL
Risk Decomposition of Investment Portfolios. Dan dibartolomeo Northfield Webinar January 2014
Risk Decomposition of Investment Portfolios Dan dibartolomeo Northfield Webinar January 2014 Main Concepts for Today Investment practitioners rely on a decomposition of portfolio risk into factors to guide
ATHENS UNIVERSITY OF ECONOMICS AND BUSINESS
ATHENS UNIVERSITY OF ECONOMICS AND BUSINESS Masters in Business Administration (MBA) Offered by the Departments of: Business Administration & Marketing and Communication PORTFOLIO ANALYSIS AND MANAGEMENT
Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture - 13 Consumer Behaviour (Contd )
(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture - 13 Consumer Behaviour (Contd ) We will continue our discussion
Finance Homework p. 65 (3, 4), p. 66-69 (1, 2, 3, 4, 5, 12, 14), p. 107 (2), p. 109 (3,4)
Finance Homework p. 65 (3, 4), p. 66-69 (1, 2, 3, 4, 5, 12, 14), p. 107 (2), p. 109 (3,4) Julian Vu 2-3: Given: Security A Security B r = 7% r = 12% σ (standard deviation) = 35% σ (standard deviation)
15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
Circuit Analysis using the Node and Mesh Methods
Circuit Analysis using the Node and Mesh Methods We have seen that using Kirchhoff s laws and Ohm s law we can analyze any circuit to determine the operating conditions (the currents and voltages). The
A Mean-Variance Framework for Tests of Asset Pricing Models
A Mean-Variance Framework for Tests of Asset Pricing Models Shmuel Kandel University of Chicago Tel-Aviv, University Robert F. Stambaugh University of Pennsylvania This article presents a mean-variance
Linear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
Econ 422 Summer 2006 Final Exam Solutions
Econ 422 Summer 2006 Final Exam Solutions 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
The Tangent or Efficient Portfolio
The Tangent or Efficient Portfolio 1 2 Identifying the Tangent Portfolio Sharpe Ratio: Measures the ratio of reward-to-volatility provided by a portfolio Sharpe Ratio Portfolio Excess Return E[ RP ] r
FOUNDATIONS OF PORTFOLIO THEORY
FOUNDATIONS OF PORTFOLIO THEORY Nobel Lecture, December 7, 1990 by HARRY M. MARKOWITZ Baruch College, The City University of New York, New York, USA When I studied microeconomics forty years ago, I was
CHAPTER 5 HISTORY OF INTEREST RATES AND RISK PREMIUMS
CHAPTER 5 HISTORY OF INTEREST RATES AND RISK PREMIUMS HISTORY OF INTEREST RATES AND RISK PREMIUMS Determinants of the Level of Interest Rates Risk and Risk Premium The Historical Record Real Versus Nominal
CHAPTER 1: SPREADSHEET BASICS. AMZN Stock Prices Date Price 2003 54.43 2004 34.13 2005 39.86 2006 38.09 2007 89.15 2008 69.58
1. Suppose that at the beginning of October 2003 you purchased shares in Amazon.com (NASDAQ: AMZN). It is now five years later and you decide to evaluate your holdings to see if you have done well with
Chapter 11, Risk and Return
Chapter 11, Risk and Return 1. A portfolio is. A) a group of assets, such as stocks and bonds, held as a collective unit by an investor B) the expected return on a risky asset C) the expected return on
Decision Analysis. Here is the statement of the problem:
Decision Analysis Formal decision analysis is often used when a decision must be made under conditions of significant uncertainty. SmartDrill can assist management with any of a variety of decision analysis
t = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3
MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate
Achievement of Market-Friendly Initiatives and Results Program (AMIR 2.0 Program) Funded by U.S. Agency for International Development
Achievement of Market-Friendly Initiatives and Results Program (AMIR 2.0 Program) Funded by U.S. Agency for International Development Equity, Portfolio Management, Real Estate Practice Exam 2 and Solutions
An Oracle White Paper October 2012. Optimizing Loan Portfolios
An Oracle White Paper October 2012 Optimizing Loan Portfolios Introduction... 1 The Loan Portfolio... 3 Correlation... 3 Portfolio Risk... 4 Using Oracle Crystal Ball... 6 The Effects of Diversification...
1. CFI Holdings is a conglomerate listed on the Zimbabwe Stock Exchange (ZSE) and has three operating divisions as follows:
NATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY FACULTY OF COMMERCE DEPARTMENT OF FINANCE BACHELOR OF COMMERCE HONOURS DEGREE IN FINANCE PART II 2 ND SEMESTER FINAL EXAMINATION MAY 2005 CORPORATE FINANCE
TRADING SYSTEM EVALUATION By John Ehlers and Mike Barna 1
TRADING SYSTEM EVALUATION By John Ehlers and Mike Barna 1 INTRODUCTION There are basically two ways to trade using technical analysis Discretionarily and Systematically. Discretionary traders can, and
Makeup Exam MØA 155 Financial Economics February 2010 Permitted Material: Calculator, Norwegian/English Dictionary
University of Stavanger (UiS) Stavanger Masters Program Makeup Exam MØA 155 Financial Economics February 2010 Permitted Material: Calculator, Norwegian/English Dictionary The number in brackets is the
CHAPTER 6 RISK AND RISK AVERSION
CHAPTER 6 RISK AND RISK AVERSION RISK AND RISK AVERSION Risk with Simple Prospects Risk, Speculation, and Gambling Risk Aversion and Utility Values Risk with Simple Prospects The presence of risk means
Prot Maximization and Cost Minimization
Simon Fraser University Prof. Karaivanov Department of Economics Econ 0 COST MINIMIZATION Prot Maximization and Cost Minimization Remember that the rm's problem is maximizing prots by choosing the optimal
How Many Days Equal A Year? Non-trivial on the Mean-Variance Model
How Many Days Equal A Year? Non-trivial on the Mean-Variance Model George L. Ye, Dr. Sobey School of Business Saint Mary s University Halifax, Nova Scotia, Canada Christine Panasian, Dr. Sobey School of
Pigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
Linear Programming Problems
Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number
Use the table for the questions 18 and 19 below.
Use the table for the questions 18 and 19 below. The following table summarizes prices of various default-free zero-coupon bonds (expressed as a percentage of face value): Maturity (years) 1 3 4 5 Price
WHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
CHAPTER 14 NONPARAMETRIC TESTS
CHAPTER 14 NONPARAMETRIC TESTS Everything that we have done up until now in statistics has relied heavily on one major fact: that our data is normally distributed. We have been able to make inferences
To give it a definition, an implicit function of x and y is simply any relationship that takes the form:
2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to
PowerScore Test Preparation (800) 545-1750
Question 1 Test 1, Second QR Section (version 1) List A: 0, 5,, 15, 20... QA: Standard deviation of list A QB: Standard deviation of list B Statistics: Standard Deviation Answer: The two quantities are
OA3-10 Patterns in Addition Tables
OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20