UEH - Bài giảng Môn quản lý danh mục đầu tư Chương 07 Các mô hình định giá - Sách ReillyBrown. Bài giảng tham khảo của đại học kinh tế TPHCM

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CHAPTER 7

Asset Pricing Models

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7.1 The Capital Asset Pricing Model

•The capital asset pricing model (CAPM) extends capital market theory in a way that allows

investors to evaluate the risk–return trade-off for both diversified portfolios and individual

• The risk measure is called the beta coefficient and

calculates the level of a security’s systematic risk compared to that of the market portfolio

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7.1.1 A Conceptual Development of the CAPM (slide 1 of 2)

•The existence of a risk-free asset resulted in

deriving a capital market line (CML) that became the relevant frontier

•However, CML cannot be used to measure the expected return on an individual asset

•For individual asset (or any portfolio), the relevant risk measure is the asset’s covariance with the

market portfolio

•That is, for an individual asset i, the relevant risk is not σi, but rather σiriM, where riMis the correlation coefficient between the asset and the market

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•Inserting this product into the CML and adapting the

notation for the ith individual asset:

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7.1.2 The Security Market Line(slide 1 of 18)

•The SML

•Is a graphical form of the CAPM

•Shows the trade-off between risk and expected return as a straight line intersecting the vertical axis (zero-risk point) at the risk-free rate

•Considers only the systematic component of an investment’s volatility

•Can be applied to any individual asset or collection of assets

•Exhibit 7.1

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•Determining the Expected Rate of Return for a Risky Asset

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•Risk-free rate is 5 percent and the market return is 9 percent

•This implies a market risk premium of 4 percent

0.05 1.00 0.09 0.050.099.00%

0.05 1.15 0.09 0.050.0969.60%

0.05 1.40 0.09 0.050.10610.60%

0.050.30 0.09 0.050.05 0.012

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•Identifying Undervalued and Overvalued Assets

• 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 adjusted rates of return than the average investor

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• Compare the required rate of return to the

estimated rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment

•Exhibits 7.2, 7.3, 7.4

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•Calculating Systematic Risk

•A beta coefficient for Security i can be calculated directly from the following formula:

•Security betas can also be estimated as the slope coefficient in a regression equation between the

returns to the security (Rit) over time and the returns (RMt) to the market portfolio (the security’s

characteristic line):

( )M

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•The Impact of the Time Interval

•The number of observations and time interval used in the calculation of beta vary widely,

causing beta to vary

•There is no “correct” interval for analysis

• Morningstar uses monthly returns over five years• Reuters Analytics uses daily returns over two years• Bloomberg uses weekly returns over two years

although the system allows users to change the time intervals

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•The Effect of the Market Proxy

•The Standard & Poor’s 500 Composite Index is often used as the proxy because:

• It contains large proportion of the total market value of U.S stocks

• It is a value weighted index

•Theoretically, the market portfolio should include all 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

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•Computing a Characteristic Line: An Example

•The example shows how to estimate a

characteristic line for Microsoft Corp (MSFT) using monthly return data from January 2016 to December 2016

•Betas for MSFT are calculated using:

• The S&P 500 (SPX)

• The MSCI World Equity (MXWO) index•Exhibits 7.5, 7.6, 7.7

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•Industry Characteristic Lines

•The characteristic line used to estimate beta value can be computed for sector indexes

•Exhibit 7.8

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7.2 Empirical Tests of the CAPM

•When testing the CAPM, there are two major questions

1.How stable is the measure of systematic risk (beta)?

2.Is there a positive linear relationship as

hypothesized between beta and the rate of return on risky assets?

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7.2.1 Stability of Beta

stability of beta and generally concluded that the risk measure was not stable for individual stocks but was stable for portfolios of stocks

•The larger the portfolio and the longer the period, the more stable the beta estimate

•The betas tended to regress toward the mean

•High-beta portfolios tended to decline over time toward 1.00, whereas low beta portfolios tended to increase over time toward unity

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7.2.2 Relationship Between Systematic Risk and Return (slide 1 of 5)

•The ultimate question regarding the CAPM is whether it is useful in explaining the return on risky assets

•Specifically, is there a positive linear relationship between the systematic risk and the rates of

return on these risky assets?

•Study (Jensen) shows that:

•Most of the measured SMLs had a positive slope

•The slopes change between periods

•The intercepts are not zero

•The intercepts change between periods

•Exhibit 7.9

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•Effect of a Zero-Beta Portfolio

•The characteristic line using a zero-beta

portfolio instead of RFR should have a higher intercept and a lower slope coefficient

•Several studies have tested this model with its higher intercept and flatter slope and found

conflicting results

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•Effect of Size, P/E, and Leverage

•Size and P/E are additional risk factors that need to be considered along with beta

•Expected returns are a positive function of

beta, but investors also require higher returns from relatively small firms and for stocks with relatively low P/E ratios

•Bhandari (1988) found that financial leverage also helps explain the cross section of

average returns after both beta and size are considered

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•Effect of Book-to-Market Value

•Fama and French (1992) concluded that size and book-to-market equity capture the cross-sectional variation in average stock returns associated with size, E/P, book-to-market equity, and leverage

•Two variables, BE/ME, appear to subsume E/P and leverage

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7.2.3 Additional Issues

•Effect of Transaction Costs

•With transactions costs, the SML will be a band of securities, rather than a straight line

•Effect of Taxes

•Differential tax rates could cause major differences in the CML and SML among investors

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7.2.4 Summary of Empirical Results for the CAPM

•Early evidence supported the CAPM; there was evidence that the intercepts were generally higher than implied by the RFR that prevailed, which is

either consistent with a zero-beta model or the existence of higher borrowing rates

•To explain unusual returns, size, the P/E ratio, financial leverage, and the book-to-market value ratio are found to have explanatory power regarding returns beyond beta

•Further studies;

•Kothari, Shanken, and Sloan (1995) measured beta with annual returns and found substantial compensation for beta risk, which suggested that the results obtained by Fama and French may have been time-period specific

•Jagannathan and Wang (1996) employed a conditional CAPM that allows for changes in betas and in the market risk premium and found that this model performed well in explaining the cross section of returns

•Reilly and Wright (2004) examined the performance of 31 different asset classes with betas computed using a broad market portfolio proxy; the risk–return

relationship was significant and as expected by theory

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•The true market portfolio should

•Included all the risky assets in the world

•In equilibrium, the assets would be included in the portfolio in proportion to their market value

•Using U.S Index as a market proxy

•Most studies use an U.S index

•The U.S stocks constitutes less than 15% of a truly global risky asset portfolio

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7.3 The Market Portfolio: Theory versus Practice (slide 2 of 4)

•The beta intercept of the SML will differ if

•There is an error in selecting the risk-free asset

•There is an error in selecting the market portfolio

•Using the incorrect SML may lead to incorrect evaluation of a portfolio

•Exhibits 7.10, 7.11

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7.3 The Market Portfolio: Theory versus Practice (slide 4 of 4)

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7.4 Arbitrage Pricing Theory (slide 1 of 5)

•CAPM is criticized because of

•The many unrealistic assumptions

•The difficulties in selecting a proxy for the market portfolio as a benchmark

•An alternative pricing theory with fewer assumptions was developed: Arbitrage Pricing Theory (APT)

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•Three major assumptions:

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 expressed as a linear function of a set of K

factors or indexes

•In contrast to CAPM, APT does not assume:

1.Normally distributed security returns

2.Quadratic utility function

3.A mean-variance efficient market portfolio

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•Theory assumes that the return-generating

process can be represented as a K factor model of the form:

R=E R+b+b+ +b+i=n

Ri= actual return on Asset i during a specified time period, i= 1, 2, 3, … n

E(Ri) = expected return for Asset i if all the risk factors have zero changes

bij= reaction in Asset i’s returns to movements in a common risk factor j

δk= set of common factors or indexes with a zero mean that influences the returns on all assets

εi= unique effect on Asset i’s return (a random error term that, by assumption, is

completely diversifiable in large portfolios and has a mean of zero)n = number of assets

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•The APT requires that in equilibrium the return on a investment, zero-systematic-risk portfolio is zero when the unique effects are fully diversified

zero-•This assumption implies that the expected return on any Asset i can be expressed as:

bij= pricing relationship between the risk premium and the asset; that is, how

responsive Asset i is to the jth common factor (These are called factor betas or

factor loadings.)

•Exhibit 7.12

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7.4.1 Using the APT (slide 1 of 3)

•Two-stock and a two-factor model example:

•Assume that there are two common factors: one

related to unexpected changes in the level of inflation and another related to unanticipated changes in the real level of GDP

•Risk factor definitions and sensitivities:

δ1= unanticipated changes in the rate of inflation The risk premium related to this factor is 2 percent for every 1 percent change in the rate (λ = 0.02).

δ2= unexpected changes in the growth rate of real GDP The average risk premium related to this factor is 3 percent for every 1 percent change in the rate growth (λ2= 0.03).

λ0= rate of return on a zero-systematic risk asset (zero-beta) is 4 percent (λ0= 0.04).

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•Assume also that there are two assets (x and y) that have the following sensitivities to these common risk factors:

bx1= response of Asset x to changes in the inflation factor is 0.50

bx2= response of Asset x to changes in the GDP factor is 1.50

by1= response of Asset y to changes in the inflation factor is 2.00

by2= response of Asset y to changes in the GDP factor is 1.75

•Exhibit 7.13

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•Suppose that three stocks (A, B, and C) and two common

systematic risk factors (1 and 2) have the following relationship (for simplicity, it is assumed that the zero-beta return [λ0] equals zero):

E(RA) = (0.80)λ1+ (0.90)λ2E(RB) = (−0.20)λ1+ (1.30)λ2E(RC) = (1.80)λ1+ (0.50)λ2

•If λ1= 4 percent and λ2= 5 percent, then the returns expected by the market over the next year can be expressed as:

E(RA) = (0.80)(4%) + (0.90)(5%) = 7.7%

E(RB) = (−0.20)(4%) + (1.30)(5%) = 5.7%

E(RC) = (1.80)(4%) + (0.50)(5%) = 9.7%

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7.4.2 Security Valuation with the APT: An Example (slide 2 of 3)

•Assuming that all three stocks are currently priced at $35 and do not pay a dividend, the following are the expected prices a year from now:

E(PA) = $35(1.077) = $37.70

E(PB) = $35(1.057) = $37.00

E(PC) = $35(1.097) = $38.40

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PA= ($37.20) ÷ (1.077) = $34.54

PB= ($37.80) ÷ (1.057) = $35.76

PC= ($38.50) ÷ (1.097) = $35.10

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7.4.3 Empirical Tests of the APT(slide 1 of 5)

•Roll-Ross Study (1980)

•Methodology followed a two-step procedure:

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

•The authors concluded that the evidence generally supported the APT but

acknowledged that their tests were not conclusive

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•Extensions of the Roll–Ross Tests

•Cho, Elton, and Gruber (1984) examined the number of factors in the return-generating process that were priced

•Dhrymes, Friend, and Gultekin (1984)

reexamined techniques and their limitations and found the number of factors varies with the size of the portfolio

•Roll and Ross (1984) pointed out that the number of factors is a secondary issue compared to how well the model can explain the expected return

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7.4.3 Empirical Tests of the APT(slide 3 of 5)

•Connor and Korajczyk (1993) developed a test that identifies the number of factors in a model that allows the unsystematic

components of risk to be correlated across assets

•Harding (2008) also showed the connection between systematic and unsystematic risk factors

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•The APT and Stock Market Anomalies

•An alternative set of tests of the APT considers how well the theory explains pricing anomalies: the small-firm effect and the January effect

•APT Tests of the Small-Firm Effect

• Reinganum: Results inconsistent with the APT• Chen: Supported the APT model over CAPM

•APT Tests of the January Effect

• Gultekin and Gultekin: APT not better than CAPM

• Burmeister and McElroy: Effect not captured by model but still rejected CAPM in favor of APT