1. Copyright © 2000 by Harcourt, Inc. All rights reserved. Introduction The next two chapters (together with Chs. 2 – 5 of Haugen) will briefly examine the following aspects of quantitative investment management: Modeling risk and return – CAPM & APT – theory, testing, and extensions Estimating risk and return – the Single-Index Model (SIM) and multiple-factor models for risk and expected return

2. Version 1.2 Copyright © 2000 by Harcourt, Inc. All rights reserved. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department Harcourt, Inc. 6277 Sea Harbor Drive Orlando, Florida 32887-6777 Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Sixth Edition by Frank K. Reilly & Keith C. Brown Chapters 9 & 10

3. Copyright © 2000 by Harcourt, Inc. All rights reserved. Modeling Risk & Return Part One: The Risk-Free Asset, Portfolio Separation, and The Capital Asset Pricing Model (CAPM)

4. THE RISK-FREE ASSET WHAT IS A RISK FREE-ASSET? – DEFINITION: an asset whose terminal value is certain variance of returns = 0, covariance with other assets = 0 If then

5. THE RISK-FREE ASSET WHAT IS A RISK FREE-ASSET? – DEFINITION: an asset whose terminal value is certain An investment with NO risk An asset with zero variance Zero correlation with all other risky assets Provides the risk-free rate of return (RFR) Will lie on the vertical axis of a portfolio graph

6. THE RISK-FREE ASSET DOES A RISK-FREE ASSET EXIST? – CONDITIONS FOR EXISTENCE: Fixed-income security No possibility of default No interest-rate risk no reinvestment risk

7. THE RISK-FREE ASSET DOES A RISK-FREE ASSET EXIST? – Given the conditions, what qualifies? a U.S. Treasury security with a maturity matching the investor’s horizon

8. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining the Risk-Free Asset with a Risky Portfolio Portfolio expected return is a linear relationship  the weighted average of the two returns

9. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining the Risk-Free Asset with a Risky Portfolio Portfolio standard deviation is also a linear relationship, equal to the weighted average of the two standard deviations (zero for the risk- free asset and  i for the risky portfolio)

10. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Standard deviation The expected variance for a two-asset portfolio is Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this formula would become Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

11. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Given the variance formula the standard deviation is Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.

12. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Example Assume: –E(R F ) = 7%, –E(R S&P ) = 12%, –  S&P = 20% Expected Return on Combined Portfolio: Standard Deviation on Combined Portfolio:

13. Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining a Risk-Free Asset with a Risky Portfolio Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets. Thus, the existence of a risk-free asset adds value to investors by expanding the set of portfolios available to them.

14. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier Figure 9.1 RFR P* C A B D

15. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Risk-Free Asset and Portfolio Separation Theory Assuming the investor can both lend (by buying Treasury bonds) and borrow (by shorting the bonds with full use of the proceeds) at the risk- free rate, this means that the investor now faces a linear (rather than convex) efficient frontier:

16. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Risk-Free Asset and Portfolio Separation Theory This linear efficient frontier, comprising various combinations of the risk-free asset and the risky portfolio P*, dominates all other possible risky portfolios within the original (Markowitz) efficient frontier. This fact led to the development of the Portfolio Separation Theory (cf., James Tobin).

17. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Separation Theory Under the portfolio separation theory, the ideal risky portfolio in which an investor should invest is the same (P*), regardless of how aggressive or risk averse the investor is. –I.e., the point on the Markowitz efficient frontier at which the investor will invest is independent of the investor’s risk preferences. Where risk preferences are reflected is in terms of how much of his or her portfolio is allocated to P* and how much is invested in the risk-free asset.

18. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Separation Theory Thus, in order to obtain his or her optimal portfolio, there are two separate decisions for the investor to make: 1.The investment decision Which portfolio on the Markowitz efficient frontier to choose? This is determined by the point of tangency between the Markowitz efficient frontier and a line extending from the risk-free rate This leads to the choice of portfolio P* as the optimal risky portfolio for the investor, regardless of the investor’s risk preferences

19. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Separation Theory 2.The financing decision This is where risk preferences come into the picture If the investor is more risk averse, he or she will put part of his or her money in P* and the rest in Treasury bonds (this is known as a lending portfolio, because the rest of the investor’s money is lent to the federal government) If the investor is more aggressive, he or she will leverage up his or her holdings and invest in P* on margin by borrowing at the risk-free rate (this is known as a borrowing portfolio)

20. Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier RFR P* Borrowing Lending

21. Copyright © 2000 by Harcourt, Inc. All rights reserved. Capital Market Theory: An Overview Question: What are the general implications for security prices if investors act the way Markowitz portfolio theory and portfolio separation theory say they should? If such theories hold, what would equilibrium in the capital markets entail? Capital market theory extends portfolio theory and develops a model for pricing all risky assets The capital asset pricing model (CAPM) will allow you to determine the required rate of return (for use in discounting future cash flows) for any risky asset

22. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 1. All investors are Markowitz mean-variance efficient investors who want to target points on the efficient frontier. –Also, they include all investable assets in their estimation of the efficient frontier –Not necessarily a realistic assumption! Most investors do not use Markowitz optimization Of those who do, they typically optimize w.r.t. alpha and tracking error rather than mean and variance

23. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). –This means that the conditions of portfolio separation theory will hold, at least at the individual level. –Note: it is always possible to lend money at the risk- free rate by buying securities such as T-bills, but (unless you’re the government) it is not usually possible to borrow at this risk-free rate. –However, assuming a higher borrowing rate does not change the general results.

24. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 3. All investors have homogeneous expectations; that is, investors have identical estimates for the probability distributions of future rates of return. –This implies that all investors will estimate the efficient frontier to be in the exact same location, and the optimal portfolio P* (i.e., the investment decision from portfolio separation theory) will be the same for all investors. –This assumption can be relaxed, and as long as the differences in expectations are not vast their effects will be minor.

25. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 4. All investors have the same one-period time horizon such as one-month, six months, or one year. –Markowitz portfolio theory is a single-period model; making the model dynamic requires additional constraints, such as on portfolio turnover, in calculating the efficient frontier. –With regard to capital market theory, differences in investors’ time horizons would require investors to derive risk measures and risk-free assets that are consistent with their time horizons.

26. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 5. Capital markets are “frictionless,” i.e.: –No taxes – true for many classes of investors –No transactions costs – becoming more true over time, but still can be an impediment –Fixed supply of stocks – i.e., don’t have to worry about incorporating IPO shares into the analysis –Infinitely divisible supply of stocks – this assumption allows us to discuss investment alternatives as continuous curves. Changing it would have little impact on the theory, and it is also becoming more true over time. –Information is costless and available to all investors

27. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory 6. Capital markets are in equilibrium. –This means that we begin with all investments properly priced in line with their risk levels.

28. Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Capital Market Theory Note that some of these assumptions are unrealistic, But relaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions; Moreover, a theory can be useful for helping to explain and predict behavior, even if not all of its assumptions hold true (e.g., many useful models in physics assume the absence of any friction).

29. Copyright © 2000 by Harcourt, Inc. All rights reserved. Derivation of the Capital Market Line a)Homogeneous expectations (together with the same investment horizon) means that investors all face the same estimated efficient frontier. b)Existence of a risk-free asset means that each investor can mix the riskless asset with a risky portfolio. c)(a) and (b) imply that all investors choose the same risky portfolio to hold in combination with the risk-free asset. Call this portfolio P*

30. Copyright © 2000 by Harcourt, Inc. All rights reserved. Derivation of the Capital Market Line d)In order to have equilibrium (supply = demand), all risky assets must be included in P*. If this were not the case, then some assets would not be held at all. e)In view of (d), the optimal portfolio P* is called the Market Portfolio (M) Value-weighted portfolio, with E(R M ) and  M f)The line connecting RFR with M now represents the market-wide opportunities for expected return and risk. Thus, this line is called the: Capital Market Line (CML)

31. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Capital Market Line (CML) Figure 9.2 RFR M CML Borrowing Lending

32. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio Because portfolio M lies at the point of tangency, it has the highest portfolio possibility line Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML Therefore this portfolio must include ALL RISKY ASSETS

33. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio Because the market is in equilibrium, all risky assets are included in this portfolio in proportion to their market value

34. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio Because it contains all risky assets, it is a completely diversified portfolio (once you already own everything, you can’t diversify any more!), which means that all the unique risk of individual assets (unsystematic risk) is diversified away (all the risk that’s left over is, by definition, systematic risk)

35. Copyright © 2000 by Harcourt, Inc. All rights reserved. Systematic Risk Only systematic risk remains in the market portfolio Systematic risk is the variability in all risky assets caused by macroeconomic variables Systematic risk can be measured by the standard deviation of returns of the market portfolio and can (and does) change over time

36. Copyright © 2000 by Harcourt, Inc. All rights reserved. Examples of Macroeconomic Factors that Affect Systematic Risk Variability in growth of money supply Interest rate volatility Variability in: industrial production corporate earnings and cash flow

37. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Market Portfolio and How to Measure Diversification All portfolios on the CML are perfectly positively correlated with each other and with the completely diversified market Portfolio M A completely diversified portfolio would have a correlation with the market portfolio of +1.00 Thus, can use regression R 2 of portfolio’s returns regressed on the “market” portfolio’s returns as a measure of the extent of diversification

38. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Capital Market Line (CML) Describes the risk / return relationship for well- diversified portfolios (idiosyncratic risk has been diversified away). Portfolio standard deviation (  Q ) is the relevant measure of risk, and the portfolio’s expected return (E(R Q )) will be a direct linear function of its risk: To obtain higher expected returns, must accept higher risk.

39. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Security Market Line (SML) Key Question: What is the relevant measure of risk for an individual security when it is held as part of a well diversified portfolio (i.e., the Market portfolio, M)? The Security Market Line describes the risk / return relationship for an individual security. –Also applies to non-diversified portfolios or any other holding for which the total risk may include some diversifiable or idiosyncratic risk.

40. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Security Market Line (SML) The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Cov i,m ) This is the risk measure for the SML, which describes the relationship between risk and expected return for all portfolios, whether well- diversified or not, as well as for all securities The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance:

41. Copyright © 2000 by Harcourt, Inc. All rights reserved. Graph of Security Market Line (SML) Figure 9.5 RFR SML

42. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Security Market Line (SML) The equation for the risk-return line is We then redefine as beta

43. Copyright © 2000 by Harcourt, Inc. All rights reserved. Graph of SML with Normalized Systematic Risk Figure 9.6 SML Negative Beta RFR

44. Copyright © 2000 by Harcourt, Inc. All rights reserved. Determining the Expected Rate of Return for a Risky Asset The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset The risk premium is determined by the systematic risk of the asset (  ) and the prevailing market risk premium (R M -RFR) In equilibrium, to obtain higher expected returns, investors must accept higher “covariance” risk In equilibrium, investors receive no compensation for diversifiable (non-systematic or idiosyncratic) risk Q: What is market is not in equilibrium?

45. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Capital Asset Pricing Model: Expected Return and Risk CAPM indicates what should be the expected or required rates of return on risky assets This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models Conversely, you can compare an estimated rate of return to the required rate of return implied by CAPM – over / under valued ?

46. Copyright © 2000 by Harcourt, Inc. All rights reserved. Determining the Required Rate of Return for a Risky Asset Assume: RFR = 6% (0.06) R M = 12% (0.12) Implied market risk premium = 6% (0.06) E(R A ) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2% E(R B ) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0% E(R C ) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9% E(R D ) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4% E(R E ) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%

47. Copyright © 2000 by Harcourt, Inc. All rights reserved. Determining the Required Rate of Return for a Risky Asset In equilibrium, all assets and all portfolios of assets should plot on the SML Any security with an estimated return that plots above the SML is underpriced Any security with an estimated return that plots below the SML is overpriced A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor

48. Copyright © 2000 by Harcourt, Inc. All rights reserved. Identifying Undervalued and Overvalued Assets Compare the required rate of return to the expected rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment Independent estimates of return for the securities provide price and dividend outlooks

49. Copyright © 2000 by Harcourt, Inc. All rights reserved. Price, Dividend, and Rate of Return Estimates Table 9.1

50. Copyright © 2000 by Harcourt, Inc. All rights reserved. Comparison of Required Rate of Return to Estimated Rate of Return Table 9.2

51. Copyright © 2000 by Harcourt, Inc. All rights reserved. Plot of Estimated Returns on SML Graph Figure 9.7 SML.20.40.60.801.20 1.40 1.60 1.80 -.40 -.20.22.20.18.16.14.12 R m.10.08.06.04.02 A B C D E

52. Copyright © 2000 by Harcourt, Inc. All rights reserved.

53. Modeling Risk & Return Part Two: Extensions, Testing, and The Arbitrage Pricing Theory (APT)

54. 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

55. 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

56. 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

57. 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

58. 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 0.01.0 E(R z ) E(R m ) - E(R z )

59. 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

60. 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 0.01.0 E(R z ) E(RFR) or

61. 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!

62. 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

63. 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?

64. 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)

65. 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

66. 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

67. 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

68. 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)

69. 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

70. 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)

71. 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

72. 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

73. 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

74. 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

75. 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

76. 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

77. 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

78. 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….

79. 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.

80. 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

81. 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

82. 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:

83. 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

84. 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

85. 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

86. 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%

87. 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

88. 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

89. 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

90. 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

91. 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

92. 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

93. 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

94. 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

95. 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  0.127 + 0.879  VL Securities market value affects the size and direction of the interval affect Trading volume also affects the beta estimates

96. 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:

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

98. 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

99. ESTIMATING FACTOR MODELS THREE METHODS – TIME-SERIES APPROACH – CROSS-SECTIONAL APPROACH – FACTOR-ANALYTIC APPROACH

100. ESTIMATING FACTOR MODELS TIME-SERIES APPROACH – BEGINNING ASSUMPTIONS: investor knows in advance of 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

101. 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

102. 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

103. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Internet Investments Online www.valueline.com www.barra.com www.stanford.edu/~wfsharpe.com

104. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Internet Investments Online www.barra.com www.wsharpe.com www.cob.ohio-state.edu/~fin/journal.jof.htm www3.oup.co.uk/revfin/scope

105. Copyright © 2000 by Harcourt, Inc. All rights reserved. Future topics The Efficient Markets Hypotheses – basic theory and future directions: –The Market as a Complex Adaptive System “Shift Happens” - Mauboussin –The New Finance “The Wrong 20-Yard Line” - Haugen Introduction to Behavioral Finance

106. Copyright © 2000 by Harcourt, Inc. All rights reserved.