Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Sixth Edition by Frank K. Reilly & Keith C. Brown Chapters 8 & 9

Copyright © 2000 by Harcourt, Inc. All rights reserved. Modeling Risk & Return Part Two: Extensions, Testing, and The Arbitrage Pricing Theory (APT)

Copyright © 2000 by Harcourt, Inc. All rights reserved. Relaxing the Assumptions of the CAPM CAPM assumption: all investors can borrow or lend at the risk-free rate – unrealistic Two possible alternatives: 1.Differential borrowing and lending rates Unlimited lending at risk-free rate Borrowing at higher rate Leads to “bent” Capital Market Line 2.Zero-Beta CAPM Eliminates theoretical need for risk-free asset Leads to same form for SML but with a shallower slope

Copyright © 2000 by Harcourt, Inc. All rights reserved. Differential Borrowing and Lending Rates (Cost of Borrowing higher than Cost of Lending) Figure 10.1 E(R) RbRb RFR Risk (standard deviation  ) F G K

Copyright © 2000 by Harcourt, Inc. All rights reserved. Zero-Beta CAPM Zero-beta portfolio: create a portfolio that is uncorrelated to the market (beta 0) –The return of the zero-beta portfolio may differ from the risk-free rate Any combination of portfolios on the efficient frontier will be on the frontier Any efficient portfolio will have associated with it a zero-beta portfolio

Copyright © 2000 by Harcourt, Inc. All rights reserved. Implications of Black’s Zero-beta model The expected return of any security can be expressed as a linear relationship of any two efficient portfolios E(R i ) = E(R z ) +  i [E(R m ) - E(R z )] If original CAPM defines the relationship between risk and return, then the return on the zero-beta portfolio should equal RF –Typically, in real world, RFR < E(R Z ), so the zero-beta SML would be less steep than the original SML –Consistent with empirical results of tests of original CAPM To test directly - identify a market portfolio and solve for the return of a zero-beta portfolio –Leads to less consistent results

Copyright © 2000 by Harcourt, Inc. All rights reserved. Security Market Line With A Zero-Beta Portfolio Figure 10.2 E(R) E(R m ) i i SML M E(R z ) E(R m ) - E(R z )

Copyright © 2000 by Harcourt, Inc. All rights reserved. Relaxing the Assumptions of the CAPM Another assumption of CAPM – zero transactions costs Existence of transaction costs: –affect mispricing corrections –affect diversification –Leads to a “security market ‘band’” in place of the security market line

Copyright © 2000 by Harcourt, Inc. All rights reserved. Security Market Line With Transaction Costs Figure 10.3 E(R) E(R m ) i i SML E(R z ) E(RFR) or

Copyright © 2000 by Harcourt, Inc. All rights reserved. Relaxing the Assumptions of the CAPM Heterogenous expectations –If all investors have different expectations about risk and return, each investor would have a different idea about the position and composition of the efficient frontier, hence would have a different idea about the location and composition of the tangency portfolio, M –Hence, each would have a unique CML and/or SML, and the composite graph would be a band of lines with a breadth determined by the divergence of expectations –Since each investor would have a different idea about where the SML lies, each would also have unique conclusions about which securities are under- and which are over-valued –Also note that small differences in initial expectations can lead to vastly different conclusions in this regard!

Copyright © 2000 by Harcourt, Inc. All rights reserved. Relaxing the Assumptions of the CAPM Planning periods –CAPM is a one period model, and the period employed should be the planning period for the individual investor, which will vary by individual, affecting both the CML and the SML Taxes –Tax rates affect returns –Tax rates differ between individuals and institutions

Copyright © 2000 by Harcourt, Inc. All rights reserved. Empirical Testing of CAPM Key questions asked: How stable is the measure of systematic risk (beta)? Is there a positive linear relationship as hypothesized between beta and the rate of return on risky assets? How well do returns conform to the SML equation?

Copyright © 2000 by Harcourt, Inc. All rights reserved. Empirical Testing of CAPM Beta is not stable for individual stocks over short periods of time (52 weeks or less) –Need to estimate over 3 or more years (5 typically used) Stability increases significantly for portfolios The larger the portfolio and the longer the period, the more stable the beta of the portfolio Betas tend to regress toward the mean ( = 1.0)

Copyright © 2000 by Harcourt, Inc. All rights reserved. Empirical Testing of CAPM In general, the empirical evidence regarding CAPM has been mixed. Empirically, the most serious challenge to CAPM was provided by Fama and French (discussed in the Introductory lecture) Conceptually, the most serious challenge is provided by Roll’s Critique

Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio: Theory Versus Practice Impossible to test full market Portfolio used as market proxy may be correlated to true market portfolio Benchmark error – 2 possible effects: –Beta will be wrong –SML will be wrong

Copyright © 2000 by Harcourt, Inc. All rights reserved. Criticism of CAPM by Richard Roll Key limit on potential tests of CAPM: –Ultimately, the only testable implication from CAPM is whether the market portfolio is efficient (i.e., whether it lies on the efficient frontier) Range of SML’s - infinite number of possible SML’s, each of which produces a unique estimate of beta

Copyright © 2000 by Harcourt, Inc. All rights reserved. Criticism of CAPM by Richard Roll Market efficiency effects - substituting a proxy, such as the S&P 500, creates two problems –Proxy does not represent the true market portfolio –Even if the proxy is not efficient, the market portfolio might be (or vice versa)

Copyright © 2000 by Harcourt, Inc. All rights reserved. Criticism of CAPM by Richard Roll Conflicts between proxies - different substitutes may be highly correlated even though some may be efficient and others are not, which can lead to different conclusions regarding beta risk/return relationships So, ultimately, CAPM is not testable and cannot be verified, so it must be used with great caution Stephen Ross devised an alternative way to look at asset pricing that uses fewer assumptions – the Arbitrage Pricing Theory, or APT

Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Arbitrage Pricing Theory (APT) 1. Capital markets are perfectly competitive 2. Investors always prefer more wealth to less wealth with certainty 3. The stochastic process generating asset returns can be presented as K factor model (to be described)

Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of CAPM That Were Not Required by APT APT does not assume: A market portfolio that contains all risky assets, and is mean-variance efficient Normally distributed security returns Quadratic utility function

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period RiRi

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period = expected return for asset i RiEiRiEi

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period = expected return for asset i = reaction in asset i’s returns to movements in a common factor R i E i b ik

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period = expected return for asset i = reaction in asset i’s returns to movements in a common factor = a common factor with a zero mean that influences the returns on all assets R i E i b ik

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period = expected return for asset i = reaction in asset i’s returns to movements in a common factor = a common factor with a zero mean that influences the returns on all assets = a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero R i E i b ik

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period = expected return for asset i = reaction in asset i’s returns to movements in a common factor = a common factor with a zero mean that influences the returns on all assets = a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero = number of assets R i E i b ik N

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) Economic factors expected to have an impact on all assets: –Growth rate of GDP –The level of interest rates –Inflation –Various yield spreads –Changes in oil prices –Major political upheavals –Etc. Or, alternatively….

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) Market plus Sector factors expected to have an impact on all assets: –General market factor, plus –Sector factors, such as Utilities Transportation Financial Etc. And potentially many more alternatives, In contrast with CAPM insistence that only beta and the market portfolio factor are relevant.

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: –Inflation –Growth in GNP –Major political upheavals –Changes in interest rates –And potentially many more…. Contrast with CAPM insistence that only beta is relevant

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) b ik determine how each asset reacts to this common factor Each asset may be affected by growth in GNP, but the effects will differ In application of the theory, the factors are not identified Similar to the CAPM, the unique effects (the  i ’s) are independent and will be diversified away in a large portfolio Caveat: impossible to completely diversify away unique risk if all the relevant systematic risk factors are not correctly identified

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) APT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away The expected return on any asset i (E i ) can be expressed as:

Copyright © 2000 by Harcourt, Inc. All rights reserved. Arbitrage Pricing Theory (APT) where: = the expected return on an asset with zero systematic risk where = the risk premium related to each of the common factors - for example the risk premium related to interest rate risk b ij = the pricing relationship between the risk premium and asset i - that is how responsive asset i is to this common factor j

Copyright © 2000 by Harcourt, Inc. All rights reserved. Example of Two Stocks and a Two-Factor Model = changes in the rate of inflation. The risk premium related to this factor is 1 percent for every 1 percent change in the rate = percent growth in real GNP. The average risk premium related to this factor is 2 percent for every 1 percent change in the rate = the rate of return on a zero-systematic-risk asset (zero beta: b oj =0) is 3 percent

Copyright © 2000 by Harcourt, Inc. All rights reserved. Example of Two Stocks and a Two-Factor Model = the response of asset X to changes in the rate of inflation is 0.50 = the response of asset Y to changes in the rate of inflation is 2.00 = the response of asset X to changes in the growth rate of real GNP is 1.50 = the response of asset Y to changes in the growth rate of real GNP is 1.75

Copyright © 2000 by Harcourt, Inc. All rights reserved. Example of Two Stocks and a Two-Factor Model =.03 + (.01)b i1 + (.02)b i2 E x =.03 + (.01)(0.50) + (.02)(1.50) =.065 = 6.5% E y =.03 + (.01)(2.00) + (.02)(1.75) =.085 = 8.5%

Copyright © 2000 by Harcourt, Inc. All rights reserved. Roll-Ross Study of APT 1. Estimate the expected returns and the factor coefficients from time-series data on individual asset returns 2. Use these estimates to test the basic cross- sectional pricing conclusion implied by the APT

Copyright © 2000 by Harcourt, Inc. All rights reserved. Empirical Tests of the APT Studies by Roll and Ross and by Chen support APT by explaining different rates of return with some better results than CAPM But, Dhrymes and Shanken question the usefulness of APT because it was not possible to identify the factors

Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimating Risk & Return The Market Model, the Characteristic Line, and Beta Estimating Multiple Factor Models Haugen’s comments

Copyright © 2000 by Harcourt, Inc. All rights reserved. Calculating Systematic Risk: The Characteristic Line The systematic risk input of an individual asset is derived from a regression model, referred to as the asset’s characteristic line with the model portfolio: where: R i,t = the rate of return for asset i during period t R M,t = the rate of return for the market portfolio M during t

Copyright © 2000 by Harcourt, Inc. All rights reserved. Scatter Plot of Rates of Return Figure 9.8 RMRM RiRi The characteristic line is the regression line of the best fit through a scatter plot of rates of return

Copyright © 2000 by Harcourt, Inc. All rights reserved. The Impact of the Time Interval Number of observations and time interval used in the regression vary Value Line Investment Services (VL) uses weekly rates of return over five years (260 obs.) Merrill Lynch, Pierce, Fenner & Smith (ML) uses monthly return over five years (60 obs.) There is no “correct” interval for analysis Weak relationship between VL & ML betas due to difference in intervals used Interval effect impacts smaller firms more

Copyright © 2000 by Harcourt, Inc. All rights reserved. The Effect of the Market Proxy The market portfolio of all risky assets must be represented in computing an asset’s characteristic line Standard & Poor’s 500 Composite Index is most often used –Large proportion of the total market value of U.S. stocks –Value weighted series

Copyright © 2000 by Harcourt, Inc. All rights reserved. Weaknesses of Using S&P 500 as the Market Proxy –Includes only U.S. stocks –The theoretical market portfolio should include U.S. and non-U.S. stocks and bonds, real estate, coins, stamps, art, antiques, and any other marketable risky asset from around the world

Copyright © 2000 by Harcourt, Inc. All rights reserved. Comparisons of Beta Estimates Different estimates of beta for a stock vary typically in data used Value Line estimates use 260 weekly observations and compare to the NYSE Composite Index Merrill Lynch estimates use 60 monthly observations and compare to the S&P 500  ML   VL Securities market value affects the size and direction of the interval affect Trading volume also affects the beta estimates

Copyright © 2000 by Harcourt, Inc. All rights reserved. Comparing Market Proxies Calculating Beta for Coca-Cola using Morgan Stanley (M-S) World Equity Index and S&P 500 as market proxies results in a 1.27 beta when compared with the M-S index, but a 1.01 beta compared to the S&P 500 The difference is exaggerated by the small sample size (12 months) used, but selecting the market proxy can make a significant difference Here are the computations from page 303:

Copyright © 2000 by Harcourt, Inc. All rights reserved. Computation of Beta of Coca-Cola with Selected Indexes Table 9.3

Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimating Risk & Return The complications and range of choices make things difficult with a single-index model For multiple-factor models, there is even greater complexity

ESTIMATING FACTOR MODELS THREE METHODS – TIME-SERIES APPROACH Macro-economic-based factors – Chen, Roll, and Ross model – CROSS-SECTIONAL APPROACH Micro-economic-based factors – Fama and French three-factor model Extensions of characteristic-based factors – Haugen model – FACTOR-ANALYTIC APPROACH

ESTIMATING FACTOR MODELS TIME-SERIES APPROACH – BEGINNING ASSUMPTIONS: investor knows in advance the factors that influence a security's returns – e.g., the return on the S&P 500 Index, or – the growth rate in GDP Regress the stocks’ historical returns against the historical values of these time series the information may be gained from an economic analysis of the firm

ESTIMATING FACTOR MODELS CROSS-SECTIONAL APPROACH – BEGINNING ASSUMPTION Identify Attributes: estimates of a securities sensitivities to certain factors – e.g., the firm’s size, – beta, or – M/B ratio estimate attributes in a particular period of time repeat over multiple time periods to estimate the factor’s standard deviations and correlations

ESTIMATING FACTOR MODELS FACTOR-ANALYTIC APPROACH – BEGINNING ASSUMPTIONS: neither factor values nor securities attributes are know uses factor analysis approach take the returns over many time periods from a sample to identify one or more significant factors generating covariances

Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimating Risk in a Multifactor Setting: Examples Estimating Expected Returns for Individual Stocks Comparing Mutual Fund Risk Exposures –Style Analysis

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