Capital Asset Pricing Model - UNSW

1 Capital Asset Pricing ModelEcon 487 Outline CAPM Assumptions and Implications CAPM and the Market Model Testing the CAPM Conditional CAPMCAPM Readings Zivot, Ch. 8 (pp. 185-191) (page # s at top of page) Benninga, Ch. 10 (pp. 221-228) Perold (2004) (pp. 288-289)What is the CAPM? Theory of Asset price determination for firms Based on portfolio theory and Market Model The only thing that matters is Beta (co-movement with the market) Alternative to valuation theory for individual firmsCAPM Assumption #1 Many investors who are all price takers , financial markets are competitive Returns provide full summary of investment opportunitiesCAPM Assumption #2 All investors plan to invest over the same time horizon Abstracts from heterogeneity in investors ( , risk averse have different time preferences than the risk tolerant) Helps address any deviations from CER modelCAPM Assumption #3 No distortionary taxes or transaction costs Clearly a false assumption (debt vs.)

2 Equity)CAPM Assumption #4 All investors can borrow/lend at same risk-free rate Again, clearly false But we can consider Zero-Beta version of CAPM with short-salesCAPM Assumption #5 Preferences: Investors only care about expected return (like) and variance (dislike) Consistent with portfolio theory and CER Model under Normality False if assets have different co-movements with state of the economy ( , recession vs. boom)CAPM Assumption #6 All investors have same information and beliefs about distribution of returns , CER Model , with same beliefs about parametersCAPM Assumption #7 Market portfolio that determines Beta consists of all publicly traded assetsImplication #1 Investors will use Markowitz algorithm to determine same set of efficient portfoliosImplication #2 Risk-averse investors will put most of their wealth in risk-free Asset Risk-tolerant investors will put most of their wealth in risky assets In equilibrium, no net borrowing=> risk-free rate and tangency portfolioImplication #3 Tangency portfolio = market portfolio Note.

3 Implies positive weights on all assets in tangency portfolio, even allowing for short salesImplication #4 Market portfolio is mean-variance efficient , highest Sharpe RatioImplication #5 Security Market Line (SML) Pricing holds for all assets and portfolios , expected return on Asset i is fully determined by three things: risk for Asset iSecurity Market Line (SML) ! E[Rit]=rf+"i,M(E[RMt]#rf) where i refers to an individual Asset or portfolio Beta-Portfolio ! E[Rpt]=(1"xm)rf+xmE[RMt]=rf+xm(E[RMt]"rf ) Set ! xm="i,M for a given Asset i to compare ! E[Rpt] on Beta-Portfolio to ! E[Rit]. Log-Linear Present Value Relationship ! pt=c1"#+Et#j"1(1"#)dt+jj=1$%& ' ( ( ) * + + "Et#j"1rt+jj=1$%& ' ( ( ) * + + Market Model ! Rit="i+#i,MRMt+$it ! "it~iidN(0,#"i2) ! cov(RMt,"it)=0 Subtract ! rf from both sides ! Rit"rf=#i"rf+$i,MRMt+%it Add and subtract !))

4 "i,Mrf from right-hand-side: ! Rit"rf=#i"(1"$i,M)rf+$i,M(RMt"rf)+%it IstheCAPMU seful? ,wecanexaminewhetherrealworldassetprices andinvestorportfoliosconformtothepredict ionsofthemodel,ifnotalwaysinastrictquant itativesense, ,evenifthemodeldoesnotdescribeourcurrent worldparticularlywell,itmightpredictfutu reinvestorbehavior forexample,asaconse-quenceofcapitalmarke tfrictionsbeinglessenedthroughfinanciali nnovation, , (market) , ,optimalmanagementofcapitalgainstaxesinv olvesearlyrealizationoflossesanddeferral ofcapitalgains,andsotaxableinvestorsmigh treactverydifferentlytochangesinassetval uesdependingonwhentheypurchasedtheasset( Constantinides,1983).Nevertheless, , , (SML)Beta of market ! returnMarketportfolioIn equilibrium, allassets plot on the SMLEM " rf ! slope of SMLEMrf 18 JournalofEconomicPerspectivesIntuition for CAPM Investors should not be compensated for diversifiable , Beta=0 If expected return > risk-free, borrow at risk-free to buy zero-beta Asset If expected return < risk-free, sell zero-beta Asset short and buy more risk-free Both cases imply higher portfolio return without higher risk (at the margin)Equilibrium for Beta=0 Risk-free rate and price of zero-beta Asset adjust to equate expected return and risk-free rate , if expected return < risk-free rate, price falls today to make future expected returns higher recall log-linear present-value relationship between price and expected returnsSecurity Market Line (SML) !

5 E[Rit]=rf+"i,M(E[RMt]#rf) where i refers to an individual Asset or portfolio Beta-Portfolio ! E[Rpt]=(1"xm)rf+xmE[RMt]=rf+xm(E[RMt]"rf ) Set ! xm="i,M for a given Asset i to compare ! E[Rpt] on Beta-Portfolio to ! E[Rit]. Log-Linear Present Value Relationship ! pt=c1"#+Et#j"1(1"#)dt+jj=1$%& ' ( ( ) * + + "Et#j"1rt+jj=1$%& ' ( ( ) * + + Market Model ! Rit="i+#i,MRMt+$it ! "it~iidN(0,#"i2) ! cov(RMt,"it)=0 Subtract ! rf from both sides ! Rit"rf=#i"rf+$i,MRMt+%it Add and subtract ! "i,Mrf from right-hand-side: ! Rit"rf=#i"(1"$i,M)rf+$i,M(RMt"rf)+%it , Beta=1 If expected return > market, sell other assets to buy high-return Asset If expected return < market, sell Asset to buy more of market portfolio Both cases imply higher portfolio return without higher risk (at the margin) Prices adjust to bring about equilibriumIn general Investors can choose mix of risk-free Asset and market portfolio to achieve any desired expected return => Beta Portfolio Weight on market portfolio is Beta in SML If expected return on Asset i is different than SML, prices will adjust as investors buy/sell Beta portfolio and Asset Market Line (SML) !))

6 E[Rit]=rf+"i,M(E[RMt]#rf) where i refers to an individual Asset or portfolio Beta-Portfolio ! E[Rpt]=(1"xm)rf+xmE[RMt]=rf+xm(E[RMt]"rf ) Set ! xm="i,M for a given Asset i to compare ! E[Rpt] on Beta-Portfolio to ! E[Rit]. Log-Linear Present Value Relationship ! pt=c1"#+Et#j"1(1"#)dt+jj=1$%& ' ( ( ) * + + "Et#j"1rt+jj=1$%& ' ( ( ) * + + Market Model ! Rit="i+#i,MRMt+$it ! "it~iidN(0,#"i2) ! cov(RMt,"it)=0 Subtract ! rf from both sides ! Rit"rf=#i"rf+$i,MRMt+%it Add and subtract ! "i,Mrf from right-hand-side: ! Rit"rf=#i"(1"$i,M)rf+$i,M(RMt"rf)+%it IstheCAPMU seful? ,wecanexaminewhetherrealworldassetprices andinvestorportfoliosconformtothepredict ionsofthemodel,ifnotalwaysinastrictquant itativesense, ,evenifthemodeldoesnotdescribeourcurrent worldparticularlywell,itmightpredictfutu reinvestorbehavior forexample,asaconse-quenceofcapitalmarke tfrictionsbeinglessenedthroughfinanciali nnovation, , (market) , ,optimalmanagementofcapitalgainstaxesinv olvesearlyrealizationoflossesanddeferral ofcapitalgains,andsotaxableinvestorsmigh treactverydifferentlytochangesinassetval uesdependingonwhentheypurchasedtheasset( Constantinides,1983).))

7 Nevertheless, , , (SML)Beta of market ! returnMarketportfolioIn equilibrium, allassets plot on the SMLEM " rf ! slope of SMLEMrf 18 JournalofEconomicPerspectivesKey Point Even if there is an implicit present-value Model of stock price determination, there is no need to forecast future dividends for firm i Given SML, all that matters for Pricing firm i is Beta ( , responsiveness to market return) Easier to estimate Beta than to forecast future dividendsCAPM and the Market Model Market Model is a statistical Model CAPM is a theory that places parameter restrictions on Market Model Consider excess return version of Market ModelSecurity Market Line (SML) ! E[Rit]=rf+"i,M(E[RMt]#rf) where i refers to an individual Asset or portfolio Beta-Portfolio ! E[Rpt]=(1"xm)rf+xmE[RMt]=rf+xm(E[RMt]"rf ) Set ! xm="i,M for a given Asset i to compare !

8 E[Rpt] on Beta-Portfolio to ! E[Rit]. Log-Linear Present Value Relationship ! pt=c1"#+Et#j"1(1"#)dt+jj=1$%& ' ( ( ) * + + "Et#j"1rt+jj=1$%& ' ( ( ) * + + Market Model ! Rit="i+#i,MRMt+$it ! "it~iidN(0,#"i2) ! cov(RMt,"it)=0 Subtract ! rf from both sides ! Rit"rf=#i"rf+$i,MRMt+%it Add and subtract ! "i,Mrf from right-hand-side: ! Rit"rf=#i"(1"$i,M)rf+$i,M(RMt"rf)+%it Excess-Return Market Model ! Rit"rf=#i$+%i,M(RMt"rf)+&it, where ! "i#="i$(1$%i,M)rf. Take expectations of both sides and take ! rf to right-hand-side: ! E[Rit]=rf+"i#+$i,M(E[RMt]%rf) The SML for CAPM implies ! "i#=0 or, equivalently, ! "i=(1#$i,M)rf. Testing the H0: *=0 for excess-return Market Model (t-test for one Asset or F-test for a joint test for a set of assets) if market portfolio is efficient and equal to tangency portfolio for assets in market predictions for expected returns based on Beta and SMLWhy might tests reject CAPM in practice?))

9 (I) CAPM does not hold restrictions on short sales heterogeneous probability assessmentsWhy might tests reject CAPM in practice? (II) Maybe CAPM only holds for portfolios, not individual assets liquidity issues for individual assetsWhy might tests reject CAPM in practice? (III) Set of assets in market portfolio proxy incomplete foreign assets, bonds, real estate explains if market portfolio appears less efficient than tangency portfolio Roll critique (only test is if true market portfolio is mean-variance efficient)Why might tests reject CAPM in practice? (IV) Non-tradable assets matter human Capital co-movement of traded assets with business cycle (multiple factor Model ) given same Beta, prefer less correlation with labour incomeEarly Studies Black, Jensen, Scholes (72), Fama and MacBeth (73), Blume and Friend (73) support mean-variance efficiency of market portfolio But estimated mean return on zero-beta portfolio > risk-free rateAnomalies Literature Portfolios based on firm characteristics appear to have higher ex-post Sharpe ratios than market proxy Portfolios emphasizing smaller size, lower P/E ratios, and higher book/market ratios appear to be betters"#$%&Ra *&ph Ih0&%1#&%&2$%3404I5&214632*& r01;$&H=&"#*2 "6@6%RQ4"B&H46%2&>426CD?

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