General Notation. Return and Risk: The Capital Asset Pricing Model
|
|
- Kory Price
- 6 years ago
- Views:
Transcription
1 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 efficient set with two assets diversification with many assets efficient set with many assets the capital market line the capital asset pricing model AFM Return and Risk: The Capital Asset Pricing Model Slide 1 General Notation E(R j ) = expected return on a security/portfolio j σ 2 j = variance of some security/portfolio j σ AB = covariance between two variables A and B ρ AB = correlation between two variables A and B Ω = the number of possible future states of the economy ω i = a possible future state of the economy, i {1,...,Ω} p i = probability of occurrence of state ω i, i {1,...,Ω} R i, j = return of security j in state ω i, i {1,...,Ω} R t, j = return of security j in period t, t {1,...,T } β j = beta of some security/portfolio j AFM Return and Risk: The Capital Asset Pricing Model Slide 2
2 Single Security Statistics expected return historical sample: R j = T 1 T t=1 R t, j population: E(R j ) = Ω i=1 p i R i, j example: ω i p i R A - auto stock R B - gold stock recession % 20% normal % 3% boom % 20% calculate the expected return for each stock: AFM Return and Risk: The Capital Asset Pricing Model Slide 3 variance and standard deviation historical sample: σ 2 j = T 1 1 T t=1 (R t, j R j ) 2 population: σ 2 j = Ω i=1 p i (R i, j E(R j )) 2 standard deviation: s.d.(r j ) = σ j a measure of risk: a probability weighted average of squared deviations of a security s return from its expected return calculate the variances and standard deviations for A and B AFM Return and Risk: The Capital Asset Pricing Model Slide 4
3 Y Y Covariance and Correlation variance (s.d.) measures variability of a single variable (e.g. stock) correlation and covariance measure the statistical relationship between two variables (e.g. 2 stocks) covariance (σ AB ) - direction of the relationship correlation (ρ AB ) - strength and direction of relationship ( 1 ρ AB 1) examples: negative covariance: interest rates and bond prices, housing starts and interest rates, exchange rates and exports, etc. positive covariance: profits and stock prices, exchange rates and imports, dividends and stock prices, etc. AFM Return and Risk: The Capital Asset Pricing Model Slide 5 positive correlation/covariance two variables X and Y tend to move together when X is above (below) its mean, Y tends to be above (below) its mean 120 Perfect Positive Correlation 90 Perfect Positive Correlation X Y Value Time X 120 Correlation = Correlation = 0.85 X Y Value Time X AFM Return and Risk: The Capital Asset Pricing Model Slide 6
4 Y Y negative correlation/covariance two variables X and Y tend to move in opposite directions when X is above (below) its mean, Y tends to be below (above) its mean 120 Perfect Negative Correlation 90 Perfect Negative Correlation X Y Value Time X 120 Correlation = Correlation = 0.85 X Y Value Time X AFM Return and Risk: The Capital Asset Pricing Model Slide 7 zero correlation/covariance two variables X and Y are not (linearly) related 120 Zero Correlation 90 Zero Correlation X Y Value 90 Y Time X AFM Return and Risk: The Capital Asset Pricing Model Slide 8
5 covariance formulas historical sample: σ AB = cov(r A,R B ) = 1 T 1 population: σ AB = cov(r A,R B ) = Ω i=1 T t=1 [(R t,a R A ) (R t,b R B )] [p i (R i,a E(R A )) (R i,b E(R B ))] calculate the covariance between stock A and stock B: AFM Return and Risk: The Capital Asset Pricing Model Slide 9 correlation formula historical sample/population: ρ AB = corr(r A,R B ) = cov(r A,R B ) σ A σ B = σ AB σ A σ B calculate the correlation between stock A and stock B: graphically: R i, j R A 20% 20% 10% 10% 0% 10% R N B state 20% 10% 10% 20% 10% R B 20% 20% AFM Return and Risk: The Capital Asset Pricing Model Slide 10
6 correlation and covariance - further observations σ AB = σ BA ρ AB = ρ BA even if two variables are actually uncorrelated, a historical sample will not yield a zero covariance due to measurement errors, sampling error, etc. covariance σ AB can take any value, and is in squared deviation units indicates direction of (linear) relationship correlation coefficient ρ AB can take on values between -1 and 1 (inclusive) indicates direction and strength of (linear) relationship AFM Return and Risk: The Capital Asset Pricing Model Slide 11 Return and Risk for a Portfolio let Y,Z be random variables, and a,b be real numbers, recall E(aY + bz) = ae(y ) + be(z) var(ay + bz) = a 2 var(y ) + b 2 var(z) + 2abcov(Y,Z) the expected return of a portfolio P is a weighted average of the expected returns of the individual securities: E(R P ) = N X j E(R j ) j=1 where N is the number of securities in P, X j is the percentage of funds invested in security j ( N j=1 X j = 1), and E(R j ) is the expected return of security j e.g. find the expected return on a portfolio consisting of 75% in stock A and 25% in stock B: AFM Return and Risk: The Capital Asset Pricing Model Slide 12
7 to see why the E(R P ) formula holds, consider the N = 2 case and let n i be the number of shares held of stock i (i = 1,2) letting S i (t) be the price of stock i at time t, the portfolio value at time t = 0 is V (0) = n 1 S 1 (0) + n 2 S 2 (0) note that X 1 = n 1 S 1 (0)/V (0), X 2 = n 2 S 2 (0)/V (0) since S i (1) = S i (0)(1 + R i ), we have V(1) = n 1 S 1 (1) + n 2 S 2 (1) = n 1 S 1 (0)(1 + R 1 ) + n 2 S 2 (0)(1 + R 2 ) V (1) V (0) 1 = R p = n 1S 1 (0)(1 + R 1 ) + n 2 S 2 (0)(1 + R 2 ) 1 n 1 S 1 (0) + n 2 S 2 (0) = X 1 (1 + R 1 ) + X 2 (1 + R 2 ) 1 = X 1 R 1 + X 2 R 2 E(R p ) = X 1 E(R 1 ) + X 2 E(R 2 ) AFM Return and Risk: The Capital Asset Pricing Model Slide 13 variance of a portfolio P: σ 2 P = N j=1 N k=1 X jx k σ jk where j and k each represent a security, σ jk = cov(r j,r k ), and σ j j = σ 2 j = var(r j ) for N = 2 σ 2 P = 2 2 j=1 k=1 X j X k σ jk 2 ( ) = Xj X 1 σ j1 + X j X 2 σ j2 j=1 = X 1 X 1 σ 11 + X 1 X 2 σ 12 + X 2 X 1 σ 21 + X 2 X 2 σ 22 = X 2 1 σ X 2 2 σ X 1 X 2 σ 12 = X 2 1 σ X 2 2 σ X 1 X 2 σ 1 σ 2 ρ 12 AFM Return and Risk: The Capital Asset Pricing Model Slide 14
8 s.d. of a portfolio P: σ P = var P e.g. find the variance and standard deviation of a portfolio 75% in stock A and 25% in stock B observations: AFM Return and Risk: The Capital Asset Pricing Model Slide 15 Diversification combining stocks into a portfolio reduces risk: how does this happen? claim: as long as ρ AB < 1, we get diversification (i.e. the s.d. of the portfolio of 2 securities is less than the weighted average of the s.d. s of the individual securities). Proof: σp 2 = X1 2 σ1 2 + X2 2 σ X 1 X 2 σ 1 σ 2 ρ 12 < X1 2 σ1 2 + X2 2 σ X 1 X 2 σ 1 σ 2 (if ρ 12 < 1) = (X 1 σ 1 + X 2 σ 2 ) 2 σ P < X 1 σ 1 + X 2 σ 2 thus portfolio risk is not a weighted average of the risks of the stocks in the portfolio; we get diversification whenever correlation is less than +1 AFM Return and Risk: The Capital Asset Pricing Model Slide 16
9 Efficient Set With Two Assets suppose E(R A ) = 0.20, E(R B ) = 0.15, σ A = , σ B = , ρ AB = 0.5 some possible risk/return combinations: X A E(R P ) σ P Feasible set with 0.5 correlation E(Rp) s.d. AFM Return and Risk: The Capital Asset Pricing Model Slide 17 opportunity set: (a.k.a. feasible set) set of all attainable portfolios (attained via various combinations of two stocks) domination: P 1 dominates P 2 if E(R P1 ) = E(R P2 ) and σ P1 < σ P2, or σ P1 = σ P2 and E(R P1 ) > E(R P2 ) efficient set: set of attainable portfolios which result in maximum expected return for a given s.d. (or alternatively, minimum s.d. for a given expected return) note that the efficient set does not contain any portfolios which are dominated minimum variance portfolio: portfolio having the lowest s.d. of all the portfolios in the feasible set AFM Return and Risk: The Capital Asset Pricing Model Slide 18
10 finding the minimum variance portfolio: σ 2 P = X 2 Aσ 2 A + (1 X A ) 2 σ 2 B + 2X A (1 X A )σ AB this is a differentiable concave function, so a minimum exists. To find it, differentiate w.r.t. X A and set the result to zero: so dσ 2 P dx A = 0 X A = dσ 2 P dx A = 2X A σ 2 A + 2(1 X A )( 1)σ 2 B + 2(1 2X A )σ AB = 2 [ X A σ 2 A (1 X A )σ 2 B + (1 2X A )σ AB ] = 2 [ X A (σ 2 A + σ 2 B 2σ AB ) + (σ AB σ 2 B) ] σ 2 B σ AB σ 2 A + σ 2 B 2σ AB AFM Return and Risk: The Capital Asset Pricing Model Slide 19 what happens for different values of ρ AB? Portfolio Standard Deviation σ P X A E(R P ) ρ AB = 1.0 ρ AB = 0.5 ρ AB = 0.0 ρ AB = 0.5 ρ AB = % % % % % % Feasible set with varying correlations E(Rp) s.d. AFM Return and Risk: The Capital Asset Pricing Model Slide 20
11 observations: lower ρ AB : more bend in the curve lower s.d. (see table) greater diversification if ρ AB = 1, we can find a risk free portfolio (i.e. having σ P = 0) recall that this is extremely unlikely with two stocks, but can happen in a portfolio consisting of a stock and a derivative (e.g. stock and a put on that stock) over short periods in reality, only one curve is possible, since ρ AB is unique; other curves in our graph are hypothetical AFM Return and Risk: The Capital Asset Pricing Model Slide 21 Diversification With Many Assets the risk an asset adds to a portfolio must be measured with reference to its relationship to other securities in the portfolio the variance of the return on a portfolio with many securities is more dependent on the covariances between the individual securities than on the variances of the individual securities proof: AFM Return and Risk: The Capital Asset Pricing Model Slide 22
12 proof cont d: AFM Return and Risk: The Capital Asset Pricing Model Slide 23 σ 2 P = total risk = diversifiable risk + undiversifiable risk σ 2 P about 30 stocks required for optimal diversification diversifiable risk: risk relating to uncorrelated events that gets eliminated when we hold many securities undiversifiable risk: risk relating to correlated events that is not eliminated by owning many securities N AFM Return and Risk: The Capital Asset Pricing Model Slide 24
13 Efficient Set With Many Assets feasible set becomes an area efficient frontier is still a curve E(R P ) how do we find the efficient frontier? σ P AFM Return and Risk: The Capital Asset Pricing Model Slide 25 Summarizing to Here investors are risk-averse; try to increase expected return and reduce risk of their portfolios investors can reduce risk by choosing stocks which are not perfectly correlated the risk added by a security to a portfolio has to be measured with reference to its relationship to other securities in the portfolio (variances and covariances determine risk) a rational risk-averse investor chooses investments that are efficient the optimal investment lies on the efficient frontier investors are not compensated for bearing diversifiable risk, and only undiversifiable risk is priced in the market AFM Return and Risk: The Capital Asset Pricing Model Slide 26
14 The Capital Market Line introduce risk-free borrowing/lending asset f with return R f : σ f = 0 by definition σ A f = 0 by definition for any risky asset A portfolio combinations of risky portfolio B and risk free asset f : E(R P ) = (1 X B )R f + X B E(R B ) σ P = [ (1 X B ) 2 σ 2 f + X 2 Bσ 2 B + 2(1 X B )X B σ f σ B ρ B f ] 1/2 = XB σ B X B = σ P σ B ( 1 σ P ) R f + σ P E(R P ) = E(R B ) σ B σ B [ ] E(RB ) R f = R f + σ P σ B AFM Return and Risk: The Capital Asset Pricing Model Slide 27 E(R P ) σ P choose point of tangency A to get capital market line (CML) separation principle: 1. determine point A, independent of personal preferences; and 2. based on degree of personal risk aversion, pick a point on CML as your investment point AFM Return and Risk: The Capital Asset Pricing Model Slide 28
15 The Capital Asset Pricing Model so far we have considered decisions of one investor now consider many investors: homogeneous expectations assumption: all individuals have same expectations regarding returns, variances, and covariances under homogeneous expectations, all investors choose the same portfolio of risky assets represented by A in equilibrium, A must be the market portfolio - a value-weighted portfolio of all existing securities (often proxied by indexes (e.g. S&P/TSX)) one implication: index investing (pick a broad stock index such as S&P/TSX, then divide your investment between this index and the risk free asset) AFM Return and Risk: The Capital Asset Pricing Model Slide 29 β measures the sensitivity of the change in return of an individual security to the change in return of the market portfolio M: β j = cov(r j,r M ) σ 2 M sign of β depends on sign of covariance/correlation example: β C = 2 E(R C ) E(R M ) AFM Return and Risk: The Capital Asset Pricing Model Slide 30
16 β can be estimated by linear regression: R t, j = α j + β j R t,m + ε t, j, where α j = intercept, β j = security β, and ε t, j = regression error term characteristics of β β M = 1 β R f = 0 β j and E(R j ) have a positive, linear relationship β j > 1 unusually sensitive to market movements β j < 1 unusually insensitive to market movements AFM Return and Risk: The Capital Asset Pricing Model Slide 31 Aside: Linear Regression linear regression is a widely used tool in statistical analysis consider a simple case with a dependent variable y and a single independent variable x: y x AFM Return and Risk: The Capital Asset Pricing Model Slide 32
17 the simple linear regression model is y t = α + βx t + ε t basic idea is to minimize the variance of the error term the best fit line is given by ˆα = ȳ ˆβ x ˆβ = cov(x,y) var(x) simple e.g.: let x be monthly returns on TSE 300 and y be monthly returns on RBC (last half of 2002): y x AFM Return and Risk: The Capital Asset Pricing Model Slide 33 we can calculate: x = 1.02 ȳ = 1.97 cov(x,y) = var(x) = ˆα = 2.61 ˆβ = 0.63 y x ŷ ε since var(y) = = ˆβ 2 var(x) + var(ε) = , the regression explains 9.51/23.18 = 41% of the variation in y the implication here is that of RBC s total risk, 41% is market risk and 59% is unique risk AFM Return and Risk: The Capital Asset Pricing Model Slide 34
18 R RBC R M AFM Return and Risk: The Capital Asset Pricing Model Slide 35 Back to CAPM there is positive linear relationship between risk and expected return E(R j ) = R f + β j [ E(R M ) R f ] basic idea: investors expect a reward for waiting (R f ) and worrying (risk premium) CAPM conclusions: expected return on a security depends on security s risk relative to the risk of the market portfolio, i.e. undiversifiable risk β is the only reason that expected returns differ between securities AFM Return and Risk: The Capital Asset Pricing Model Slide 36
19 CAPM applies to portfolios as well as to individual securities for portfolios: E(R P ) = R f + β P [ E(R M ) R f ] β P = N X j β j j=1 for well-diversified portfolios: var(r P ) = β 2 P σ 2 M s.d.(r P ) = σ P = β P σ M AFM Return and Risk: The Capital Asset Pricing Model Slide 37 the security market line (SML): E(R j ) E(R M ) R f 0 1 there is a positive linear relationship between E(R j ) and β j : β j AFM Return and Risk: The Capital Asset Pricing Model Slide 38
20 in equilibrium, all securities must line on the SML, e.g.: comparison of capital market line and security market line: CML SML - traces efficient set of portfolios - describes the return-β relationship formed from risky assets and riskless asset - relates return to total risk - relates expected return to systematic/undiversifiable risk - holds only for efficient portfolios - holds for all individual securities and all possible portfolios AFM Return and Risk: The Capital Asset Pricing Model Slide 39 overall summary: the stock market is risky and investors want a reward for risk a measure of risk for a single security is σ or σ 2 in a portfolio, do not look at the risk of a security in isolation risk consists of diversifiable and undiversifiable risk only undiversifiable/systematic risk is rewarded a security s contribution to the total risk of a portfolio is measured by β, which represents sensitivity of the security to market changes (i.e. the systematic risk) CAPM/SML - positive linear relationship between expected returns and systematic risk in equilibrium, all stocks must lie on the SML CAPM is the best known model of risk and return AFM Return and Risk: The Capital Asset Pricing Model Slide 40
Return 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 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 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 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 informationFINC 430 TA Session 7 Risk and Return Solutions. Marco Sammon
FINC 430 TA Session 7 Risk and Return Solutions Marco Sammon Formulas for return and risk The expected return of a portfolio of two risky assets, i and j, is Expected return of asset - the percentage of
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 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 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 informationLecture 5. Return and Risk: The Capital Asset Pricing Model
Lecture 5 Return and Risk: The Capital Asset Pricing Model Outline 1 Individual Securities 2 Expected Return, Variance, and Covariance 3 The Return and Risk for Portfolios 4 The Efficient Set for Two Assets
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 informationChapter 11. Return and Risk: The Capital Asset Pricing Model (CAPM) Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 11 Return and Risk: The Capital Asset Pricing Model (CAPM) McGraw-Hill/Irwin Copyright 2013 by The McGraw-Hill Companies, Inc. All rights reserved. 11-0 Know how to calculate expected returns Know
More informationRisk and Return. CA Final Paper 2 Strategic Financial Management Chapter 7. Dr. Amit Bagga Phd.,FCA,AICWA,Mcom.
Risk and Return CA Final Paper 2 Strategic Financial Management Chapter 7 Dr. Amit Bagga Phd.,FCA,AICWA,Mcom. Learning Objectives Discuss the objectives of portfolio Management -Risk and Return Phases
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 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 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 informationChapter 13 Return, Risk, and Security Market Line
1 Chapter 13 Return, Risk, and Security Market Line Konan Chan Financial Management, Spring 2018 Topics Covered Expected Return and Variance Portfolio Risk and Return Risk & Diversification Systematic
More informationFinancial Markets. Laurent Calvet. John Lewis Topic 13: Capital Asset Pricing Model (CAPM)
Financial Markets Laurent Calvet calvet@hec.fr John Lewis john.lewis04@imperial.ac.uk Topic 13: Capital Asset Pricing Model (CAPM) HEC MBA Financial Markets Risk-Adjusted Discount Rate Method We need a
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 informationApplication to Portfolio Theory and the Capital Asset Pricing Model
Appendix C Application to Portfolio Theory and the Capital Asset Pricing Model Exercise Solutions C.1 The random variables X and Y are net returns with the following bivariate distribution. y x 0 1 2 3
More informationPortfolio models - Podgorica
Outline Holding period return Suppose you invest in a stock-index fund over the next period (e.g. 1 year). The current price is 100$ per share. At the end of the period you receive a dividend of 5$; the
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 informationPrinciples 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 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 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 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 informationP2.T8. Risk Management & Investment Management. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition.
P2.T8. Risk Management & Investment Management Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju
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 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 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 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 informationFoundations of Finance
Lecture 5: CAPM. I. Reading II. Market Portfolio. III. CAPM World: Assumptions. IV. Portfolio Choice in a CAPM World. V. Individual Assets in a CAPM World. VI. Intuition for the SML (E[R p ] depending
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 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 informationRisk and Return. Return. Risk. M. En C. Eduardo Bustos Farías
Risk and Return Return M. En C. Eduardo Bustos Farías Risk 1 Inflation, Rates of Return, and the Fisher Effect Interest Rates Conceptually: Interest Rates Nominal risk-free Interest Rate krf = Real risk-free
More informationChapter 5. Asset Allocation - 1. Modern Portfolio Concepts
Asset Allocation - 1 Asset Allocation: Portfolio choice among broad investment classes. Chapter 5 Modern Portfolio Concepts Asset Allocation between risky and risk-free assets Asset Allocation with Two
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 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 informationAdjusting discount rate for Uncertainty
Page 1 Adjusting discount rate for Uncertainty The Issue A simple approach: WACC Weighted average Cost of Capital A better approach: CAPM Capital Asset Pricing Model Massachusetts Institute of Technology
More informationUniversity 18 Lessons Financial Management. Unit 12: Return, Risk and Shareholder Value
University 18 Lessons Financial Management Unit 12: Return, Risk and Shareholder Value Risk and Return Risk and Return Security analysis is built around the idea that investors are concerned with two principal
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 informationWhen we model expected returns, we implicitly model expected prices
Week 1: Risk and Return Securities: why do we buy them? To take advantage of future cash flows (in the form of dividends or selling a security for a higher price). How much should we pay for this, considering
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 informationIn March 2010, GameStop, Cintas, and United Natural Foods, Inc., joined a host of other companies
CHAPTER Return and Risk: The Capital 11 Asset Pricing Model (CAPM) OPENING CASE In March 2010, GameStop, Cintas, and United Natural Foods, Inc., joined a host of other companies in announcing operating
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 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 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 informationSession 8: The Markowitz problem p. 1
Session 8: The Markowitz problem Susan Thomas http://www.igidr.ac.in/ susant susant@mayin.org IGIDR Bombay Session 8: The Markowitz problem p. 1 Portfolio optimisation Session 8: The Markowitz problem
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 informationPowerPoint. to accompany. Chapter 11. Systematic Risk and the Equity Risk Premium
PowerPoint to accompany Chapter 11 Systematic Risk and the Equity Risk Premium 11.1 The Expected Return of a Portfolio While for large portfolios investors should expect to experience higher returns for
More informationPort(A,B) is a combination of two stocks, A and B, with standard deviations A and B. A,B = correlation (A,B) = 0.
Corporate Finance, Module 6: Risk, Return, and Cost of Capital Practice Problems (The attached PDF file has better formatting.) Updated: July 19, 2007 Exercise 6.1: Minimum Variance Portfolio Port(A,B)
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 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 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 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 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 informationChapter. Diversification and Risky Asset Allocation. McGraw-Hill/Irwin. Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Diversification and Risky Asset Allocation McGraw-Hill/Irwin Copyright 008 by The McGraw-Hill Companies, Inc. All rights reserved. Diversification Intuitively, we all know that if you hold many
More informationAnswer FOUR questions out of the following FIVE. Each question carries 25 Marks.
UNIVERSITY OF EAST ANGLIA School of Economics Main Series PGT Examination 2017-18 FINANCIAL MARKETS ECO-7012A Time allowed: 2 hours Answer FOUR questions out of the following FIVE. Each question carries
More informationEstimating Betas in Thinner Markets: The Case of the Athens Stock Exchange
Estimating Betas in Thinner Markets: The Case of the Athens Stock Exchange Thanasis Lampousis Department of Financial Management and Banking University of Piraeus, Greece E-mail: thanosbush@gmail.com Abstract
More informationEfficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9
Efficient Portfolio and Introduction to Capital Market Line Benninga Chapter 9 Optimal Investment with Risky Assets There are N risky assets, named 1, 2,, N, but no risk-free asset. With fixed total dollar
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 informationCOPYRIGHTED MATERIAL. Portfolio Selection CHAPTER 1. JWPR026-Fabozzi c01 June 22, :54
CHAPTER 1 Portfolio Selection FRANK J. FABOZZI, PhD, CFA, CPA Professor in the Practice of Finance, Yale School of Management HARRY M. MARKOWITZ, PhD Consultant FRANCIS GUPTA, PhD Director, Research, Dow
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 informationLecture #2. YTM / YTC / YTW IRR concept VOLATILITY Vs RETURN Relationship. Risk Premium over the Standard Deviation of portfolio excess return
REVIEW Lecture #2 YTM / YTC / YTW IRR concept VOLATILITY Vs RETURN Relationship Sharpe Ratio: Risk Premium over the Standard Deviation of portfolio excess return (E(r p) r f ) / σ 8% / 20% = 0.4x. A higher
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 informationChapter 10. Chapter 10 Topics. What is Risk? The big picture. Introduction to Risk, Return, and the Opportunity Cost of Capital
1 Chapter 10 Introduction to Risk, Return, and the Opportunity Cost of Capital Chapter 10 Topics Risk: The Big Picture Rates of Return Risk Premiums Expected Return Stand Alone Risk Portfolio Return 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 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 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 informationCHAPTER 2 RISK AND RETURN: Part I
CHAPTER 2 RISK AND RETURN: Part I (Difficulty Levels: Easy, Easy/Medium, Medium, Medium/Hard, and Hard) Please see the preface for information on the AACSB letter indicators (F, M, etc.) on the subject
More informationRisk and Return: From Securities to Portfolios
FIN 614 Risk and Return 2: Portfolios Professor Robert B.H. Hauswald Kogod School of Business, AU Risk and Return: From Securities to Portfolios From securities individual risk and return characteristics
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 informationCHAPTER 8 Risk and Rates of Return
CHAPTER 8 Risk and Rates of Return Stand-alone risk Portfolio risk Risk & return: CAPM The basic goal of the firm is to: maximize shareholder wealth! 1 Investment returns The rate of return on an investment
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 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 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 informationFoundations of Finance. Lecture 8: Portfolio Management-2 Risky Assets and a Riskless Asset.
Lecture 8: Portfolio Management-2 Risky Assets and a Riskless Asset. I. Reading. A. BKM, Chapter 8: read Sections 8.1 to 8.3. II. Standard Deviation of Portfolio Return: Two Risky Assets. A. Formula: σ
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 informationDerivation Of The Capital Asset Pricing Model Part I - A Single Source Of Uncertainty
Derivation Of The Capital Asset Pricing Model Part I - A Single Source Of Uncertainty Gary Schurman MB, CFA August, 2012 The Capital Asset Pricing Model CAPM is used to estimate the required rate of return
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 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 informationMacroeconomics Sequence, Block I. Introduction to Consumption Asset Pricing
Macroeconomics Sequence, Block I Introduction to Consumption Asset Pricing Nicola Pavoni October 21, 2016 The Lucas Tree Model This is a general equilibrium model where instead of deriving properties of
More informationCHAPTER 2 RISK AND RETURN: PART I
1. The tighter the probability distribution of its expected future returns, the greater the risk of a given investment as measured by its standard deviation. False Difficulty: Easy LEARNING OBJECTIVES:
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 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 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 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 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 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 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 informationDefine risk, risk aversion, and riskreturn
Risk and 1 Learning Objectives Define risk, risk aversion, and riskreturn tradeoff. Measure risk. Identify different types of risk. Explain methods of risk reduction. Describe how firms compensate for
More informationRisk, return, and diversification
Risk, return, and diversification A reading prepared by Pamela Peterson Drake O U T L I N E 1. Introduction 2. Diversification and risk 3. Modern portfolio theory 4. Asset pricing models 5. Summary 1.
More informationRisk Reduction Potential
Risk Reduction Potential Research Paper 006 February, 015 015 Northstar Risk Corp. All rights reserved. info@northstarrisk.com Risk Reduction Potential In this paper we introduce the concept of risk reduction
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 informationMean-Variance Model for Portfolio Selection
Mean-Variance Model for Portfolio Selection FRANK J. FABOZZI, PhD, CFA, CPA Professor of Finance, EDHEC Business School HARRY M. MARKOWITZ, PhD Consultant PETTER N. KOLM, PhD Director of the Mathematics
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 informationCHAPTER 5: ANSWERS TO CONCEPTS IN REVIEW
CHAPTER 5: ANSWERS TO CONCEPTS IN REVIEW 5.1 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 informationAsset Pricing Model 2
Outline Note 6 Return, Risk, and the Capital Risk Aversion Portfolio Returns and Risk Portfolio and Diversification Systematic Risk: Beta (β) The Capital Asset Pricing Model and the Security Market Line
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 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 information