Unformatted text preview: Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory 7.1 The Capital Asset Pricing Model Capital Asset Pricing Model (CAPM) • Equilibrium model that predicts the rela3onship between risk and expected returns on risky assets. CAPM underlies all modern ﬁnancial theory • Derived using principles of diversiﬁca3on, but with other simplifying assump3ons • Treynor, Sharpe, Lintner and Mossin are researchers credited with its development A basic version of CAPM • Is developed under the idea of – All individuals are alike, except ini3al wealth and risk aversion • Detailed assump3ons are … (see next 2 slides) Simplifying AssumpAons 1. 2. 3. Individual investors are price takers. They cannot aﬀect prices by their individual trades. Single
period investment horizon: All investors plan for one iden3cal holding period. Investors form porRolios from a common universe of publicly traded ﬁnancial assets, and have access to unlimited risk
free borrowing or lending opportuni3es. No taxes and no transac3on costs. Investors are ra3onal mean
variance op3mizers: all investors attempt to construct efficient frontier portfolios Homogeneous expecta3ons: all investors analyze securi3es in the same way and share the same economic view of the world. As a consequence, they use the same es3mates on expected returns, std dev., and correla3ons to generate the eﬃcient fron3er and the unique op3mal risky porRolio. 4. 5. 6. ResulAng Equilibrium CondiAons 1. All investors will hold the same porRolio for risky assets; the “market porRolio” 2. The market porRolio will be the op3mal risky porRolio – – Market porRolio will be on the eﬃcient fron3er. Capital market line (CML) is the best a\ainable CAL. 3. Risk premium on market porRolio will be propor3onal to the variance of the market porRolio and investors’ typical degree of risk aversion. – E(rM) –rf = A*σM2 4. Risk premium on individual assets will be propor3onal to the risk premium on the market porRolio and to the beta coeﬃcient of th e security on the market porRolio. i.e. market return is the single risk factor of the security market. Known Tangency PorGolio of CML • Equilibrium condi3ons: All investors will hold the same porRolio for risky assets; the “market porRolio” Capital Market Line M = The value weighted “Market” PorGolio of all risky assets. Eﬃcient FronAer CAPM: M is the op3mal risky porRolio. What is the slope of CML? Risk Premium of the Market PorGolio • E(rM) –rf = A*σM2 – The equilibrium risk premium of the market porRolio is propor3onal to the risk of the market (σM2) and the the degree of risk aversion of the average investor (A*). – Risk premium of the market porRolio should be high enough to a\ract investors to hold the available supply of stocks. Example: risk premium of market porGolio Ex: Suppose the risk
free rate is 5%, the average investor has a risk
aversion coeﬃcient is 2. and the standard devia3on of the market porRolio is 20%. Then, what is the equilibrium value of the market risk premium? What is the expected rate of return on the market? ANS: rf=5%, A*=2, σM=20%, • E(rM) –rf = A*σM2 = 2(20%)2=0.08 • So E(rM) = rf + 0.08= 5% + 8%=13% • What if the average degree of risk aversion were lower? CAPM: Risk Premium of Individual SecuriAes in a diversiﬁed porGolio • An individual security’s total risk (σ2i) can be par33oned into systema3c and unsystema3c risk – σ2i = βi2 σM2 + σ2(ei) – M = market porRolio of all risky securi3es – Systema3c risk is measured by individual security’s Beta βi, • Investors require risk premium as compensa3on for bearing systema3c risk, NOT ﬁrm
speciﬁc risk. Why? – Risk premium of an individual asset [E(ri) – rf ] is propor3onal to its Beta. E(ri) – rf = βi [E(rM) – rf ] • Ra3o of risk premium to systema3c risk is the same for any two assets or porRolios. – [E(ri) – rf ]/βi = [E(rM) – rf ]/1 = [E(rj) – rf ]/βj Expected Returns on Individual SecuriAes • CAPM’s expected return and beta relaAonship: – E(ri)= rf + βi [E(rM)
rf] – Risk premium of any asset equals to the asset’s systemaAc risk (measured by βi) Ames the risk premium of the market porGolio (E(rM)
rf). – Individual security contributes to the risk of the market porGolio via βi • βi = [COV(ri,rM)] / σ2M Example: expected return and risk premium Ex: Suppose the risk premium of the market porRolio is 9%, the T
bill rate is 5%. We es3mate the beta of Dell to be 1.3. What is the risk premium for the Dell’s stock? What is its expected rate of return? • • • • ANS: E(rM) –rf = 9%, rf =5%, βD =1.3 E(rD)
rf = βi [E(rM)
rf]=1.3(9%)=11.7% E(rD) = rf + 11.7%= 5%+11.7%=16.7% What if the systema3c risk of Dell is higher? PorRolio Beta and PorRolio Risk Premium • The Beta of a porRolio is the weighted average of the Betas of the individual assets in the porRolio. • The risk premium of a porRolio is the weighted average of the risk premiums of the the individual assets in the porRolio. Example: PorGolio Beta and Risk Premium Asset Beta Risk Premium Weight in the porGolio 0.5 0.3 0.2 1.0 Microsos American Electric Power Gold PorRolio 1.2 0.8 0.0 ? 9.0% 6.0% 0.0% ? Base on CAPM: What is the risk premium on the porRolio? What is the porRolio beta? Can you ﬁnd the market risk premium? Another example on porGolio beta Ex: If you put half your money in a stock with a beta of 1.5 and 30% of your money in a stock with a beta of 0.9 and the rest in T
bills, what is the porGolio beta? ANS: Because βP = ΣWi βi, βP = 0.50(1.5) + 0.30(0.9) + 0.20(0) = 1.02 Individual Stocks: Security Market Line Slope SML = [E(rM) – rf ]/ 1 = risk premium of market Equa3on of the SML (CAPM) E(ri) = rf + βi[E(rM)
rf] Sample Calcula3ons for SML Equa3on of the SML E(ri) = rf + βi[E(rM)
rf] • E(rM)
rf =0.08, rf = 0.03 • Consider asset x and asset y: βx = 1.25 – E(rx) = 0.03 + 1.25(.08) = .13 or 13% βy = .6 – E(ry) = 0.03 + 0.6(0.08) = 0.078 or 7.8% • Higher systema3c risk demands higher expected return! CML vs SML • BOTH relates expected return to risk. • CML graphs the risk premiums of (complete) eﬃcient porRolios as a func3on of porRolio risk, – Where porRolio risk is measured by standard devia3on of the porRolio. • SML graphs individual asset risk premiums as a func3on of asset risk, – Where asset risk is measured by the asset’s beta. – Where individual asset is part of a diversiﬁed porRolio. SML: expected return
beta rela3onship Whenever the CAPM is correct, (1) all securi3es are fairly priced, (2) all securi3es must lie on SML in market equilibrium, i.e. (E(ri) – rf) / βi should be the same for all securi3es Suppose a security with a β of 1.25 is oﬀering an expected return of 15% 13% According to the SML, the E(r) should be 13%, because E(r) = 0.03 + 1.25(.08) = 13% Is the security under or overpriced? Underpriced: It is oﬀering too high of a rate of return for its level of risk The diﬀerence between the return required for the risk level as measured by the CAPM in this case and the actual return is called the stock’s alpha denoted by α What is the α in this case? α = +2% PosiAve α is good, negaAve α is bad + α gives the buyer a + abnormal return More example on alpha and beta E(rM)= 14% βS = 1.5 rf = 5% Required return = rf + β S [E(rM) – rf] = 5% + 1.5 [14% – 5%] = 18.5% If you believe the stock will actually provide a return of 17%, what is the implied alpha? α = 17%
18.5% =
1.5% A stock with a negaAve alpha plots below the SML & gives the buyer a negaAve abnormal return 7.2 The CAPM and Index Models CAPM and Index Model • CAPM deals with expected returns while index model is about realized (historical) returns – E(ri) and E(rM) vs. rit and rMt in period t. • CAPM talks about theore3cal market porRolio while index model uses proxy (e.g. S&P200) for market porRolio. – Proxy may not be representa3ve for the broad market. Index Model • Interpreted as a regression rela3onship rit – rs = αi + βi (rMt –rs) + eit (7.3) – Where rit is the holding
period return (HPR) on asset i in period t, – rMt is the HPR of proxy porRolio for market index in the same period. – eit is ﬁrm
speciﬁc eﬀects, it measures the devia3on of security i’s realized HPR from the regression line. • (7.3) expressed in terms of expecta3on: E(rit) – rs = αi + βi [E(rMt)–rs] (7.4) – If CAPM is true, αi in (7.4) is zero. Es3ma3ng Index Model • Denote excess return of Google in each month as RGt=rGt
rs, denote excess return of market proxy in each month as RMt=rMt
rs – RGt = αG + βG RMt + eGt – The excess return of Google is aﬀected by both market factor and ﬁrm
speciﬁc factor. – Residual = Actual excess return RGt – predicted excess return of Google based on market return – Std dev of residuals σe measures the magnitude of Google’s ﬁrm
speciﬁc risk. Processing Data • Data: Monthly T
bill rates, Monthly prices of Google and monthly prices of S&P500 • Period: Jan 2006
Dec 2008 Summary Sta3s3cs of Monthly returns Time Series Plots of Monthly Returns SCL for Google Dispersion of the points around the line measures unsystema3c risk. The sta3s3c is called σe RGt = αG + ßGRMt + eGt SCL Slope = β Es3ma3on Results Implica3on of Index model es3ma3on Applica3ons of CAPM • Investment management – Is Google a cyclical stock? What is the es3mated systema3c risk of Google? Is Google under
priced? • Capital budge3ng decision – The required rate of return for an investment with similar risk as Google’s equity is required rate= rf + ßG (rM
rf) = 2.75%+1.65x5.5%=11.83% • Assess the performance of a porRolio manager who invested heavily in Google stock during this period. Adjusted Betas • Calculated betas are adjusted to account for the empirical ﬁnding that betas tend to “regress toward the mean”. • A ﬁrm with a beta >1 will tend to have a lower beta (closer to 1) in the future. A ﬁrm with a beta <1 will tend to have a higher beta (closer to 1) in the future. • Adjusted β = (2/3)(Calculated β) + (1/3) (1) = 2/3 (1.276) + 1/3 (1) =1.184 • Alterna3vely, use 3me
varying beta, and augment beta with ﬁrm informa3on such as P/E 7.3 The CAPM and the Real World Evalua3ng the CAPM • The principles of CAPM are well accepted – Relate expected return to systema3c risk – Separate systema3c risk and ﬁrm
speciﬁc risk – Beta is widely used as a measure of systema3c risk • Trading strategies involving beta etc. • However … CAPM in real applica3on • True market porRolio cannot be observed. So, CAPM is untestable (Roll, 1977). • Empirically, average returns were higher for higher
beta porRolio. However, the reward for (market) beta risk was less than what CAPM predicted (Black, Jensen, and Scholes, 1972; Fama and MacBeth, 1973). • Systema3c risk comes from mul3ple sources. 7.4 Mul3factor Models and the CAPM Mul3factor Models • Systema3c risk comes from more than one sources, e.g. business
cycle risk, interest rate risk, inﬂa3on rate risk, energy price risk, etc. • Individual security/porRolio could display diﬀerent sensi3vity to diﬀerent risk factors. • Mul3factor model reﬁne CAPM, – It relates security returns to mul3ple risk factors. – CAPM is a single
factor model, with market risk being the sole risk factor. Two Factors • Two factor model: – Rit=αi + βiM RMt + βiTB RTBt + eit (7.5) – Excess return of market porRolio RMt: proxy for market risk, – excess return of T
bond porRolio : proxy for interest rate risk – Betas: sensi3vity to diﬀerent systema3c risk. • A two
factor security market line for security i: – E(ri) =rf + βiM [E(rM) – rf] + βiTB [E(rTB) – rf] (7.6) – Total compensa3on comes from risk exposures to both market risk and interest rate risk. A Two
Factor SML Example Northeast Airlines has a market beta of 1.2 and T
bond beta of 0.7. Suppose the risk premium of the market index is 6%, while that of the T
bond porRolio is 3%. Also suppose the risk
free rate is 4%. Assuming a two
factor model, what is the overall risk premium on Northeast Airlines stock? What would be the equilibrium expected rate of return? ANS: total risk premium = 1.2(6%)+0.7(3%) Fama
French (FF) 3 factor Model • Fama and French noted that stocks of smaller ﬁrms and stocks of ﬁrms with a higher ra3o of book value of equity to market value of equity (B/ M) have had higher stock returns than predicted by single factor models. • Three factors: – Excess return of broad market porRolio: proxy for market risk – Diﬀerence in returns between small ﬁrms and big ﬁrms: proxy for ﬁrm
size risk – Diﬀerence in returns between high B/M ﬁrms and low B/M ﬁrms Apply FF model to Google • rG
rf = αG+ βM (rM – rf) + βHML rHML + βSMB rSMB + eG • What did Table 7.4 tell us? Apply FF model to Google • Assume T
bill rate of 2.75%, market risk premium of 5.5%. We forecast that SMB porRolio has rate of return 2.5%, HML porRolio has rate of return 4%. Given results from Table 7.4, what is the overall risk premium for Google stock? What is the required rate for an investment with the same risk ask Google’s equity? Arbitrage Pricing Theory (APT) • Arbitrage: Crea3on of riskless proﬁts by exploi3ng mispricing of two or more securi3es. • APT provides an alterna3ve path to a security market line. • With eﬃcient markets, proﬁtable arbitrage opportuni3es will quickly disappear. APT in one
factor security market In a single
factor security market, • The excess rate of return on security i, Ri=ri –rf, can be represented as • Construct a highly diversiﬁes porRolio with a given beta beta βp, this portolﬁo’s excess return Rp can be wri\en as • Under no
arbitrage condi3on, αp=0 – Rp = αp+ βpRM – Ri = αi+ βiRM+ ei APT in one
factor market Arbitrage Pricing Model The result: For a well diversiﬁed porRolio Rp = βpRS (Excess returns) RS is the excess return and for an individual security (rp,i – rf) = βp(rS,i – rf) + ei on a porRolio with beta of 1 rela3ve to systema3c factor “S” Advantage of the APT over the CAPM: • No par3cular role for the “Market PorRolio,” which can’t be measured anyway • Easily extended to mul3ple systema3c factors, for example – (rp,i – rf) = βp,1(r1,i – rf) + βp,2(r2,i – rf) + βp,3(r3,i – rf) + ei A porRolio manager is using the capital asset pricing model for making recommenda3ons to her clients. Her research department has developed the informa3on: Forecasted Return Stock X Stock Y Market index Risk
free rate 14.0% 17.0 14.0 5.0 Standard DeviaAon 36% 25 15 Beta 0.8 1.5 1.0 Problem 1 Calculate expected return and alpha for each stock. Which stock to pick for (a) adding to a well
diversiﬁed equity porRolio? (b) holding it as a single
stock porRolio. Problem 2 What is the beta of a porRolio with expected return of 20%? Assume the risk
free rate is 5% and the expected return of market porRolio is 15%? Problem 3 Are the following statements true or false? Explain. 1. The CAPM implies that investors require a higher return to hold highly vola3le securi3es. 2. You can construct a porRolio with a beta of 0.75 by inves3ng 75% of the budget in T
bills and the remainder in the market porRolio. 3. Stocks with a beta of zero oﬀers an expected rate of return of zero. Problem 4 If CAPM is valid, which of the situa3ons is possible? PorGolio A B Expected Return 20% 25 Beta 1.4 1.2 PorGolio C D Expected Return 30% 40 VolaAlity 35% 25 Problem 5 If CAPM is valid, Are the following situa3ons possible? PorGolio Risk
free Market A PorGolio Risk
free Market B Expected Return 10% 18 16 Expected Return 10% 18 16 Standard DeviaAon 0% 24 12 Beta 0% 1.0 1.5 ...
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 Spring '11
 Wang
 Arbitrage, the00, risk00

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