Principles of Finance Lecture 9&10: Market Equilibrium PDF

Introduction CAPM Factor models & APT Principles of Finance Lecture 9&10: Market Equilibrium CAPM and APT (CWS ch. 6 & Fama and French (1992)) Rikke Sejer Nielsen 1 / 47  Introduction CAPM Fac...

Introduction CAPM Factor models & APT Principles of Finance Lecture 9&10: Market Equilibrium CAPM and APT (CWS ch. 6 & Fama and French (1992)) Rikke Sejer Nielsen 1 / 47  Introduction CAPM Factor models & APT Introduction Last time: Mean-variance analysis by Markowitz (1952, 1959) Today: Market equilibrium asset pricing models ⇒ Optimal portfolio/investment decisions under uncertainty ⇒ How to price a single asset. ⇒ How to determine market price of risk. ⇒ Two models: 1 Capital asset pricing model (CAPM) introduced by Sharpe [1963, 1964] and Treynor 2 Arbitrage pricing theory (APT) introduced by Ross 3 / 47  Introduction CAPM Factor models & APT Settings One-period model Capital market is perfect (no frictions or imperfections) and complete. ▶ N risky assets, with return of R e = (R e 1 ,..., R eN ) e σ 2 ), e ∼ N(E(R), R ▶ A risk free asset, with the return of Rf ▶ All assets are marketable and perfectly divisible. Investors: ▶ Greedy and risk averse (allow for difference in levels of risk aversion). ▶ With homogeneous expectations about asset returns. Investment decision: Find optimal portfolio that maximizes expected utility of end-of-period wealth. ⇒ Optimal defined by the portfolio weights, w = (w1 ,..., wN ) and wf. 4 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Mean-variance efficient PFs in equilibrium with a risk-free asset Two funds separation: Investors’ optimal portfolio consists of two portfolios: 1 Risk-free asset 2 Tangency portfolio 7 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Mean-variance efficient PFs in equilibrium with a risk-free asset In equilibrium, prices are established such that Supply = Demand for any asset Meaning that Supply (market value) of asset j = Demand for asset j Total supply (market value) of assets = Total demand for assets Thus, Market value of asset j Demand for asset j Total market value of assets = total demand of assets ⇔ Market portfolio = Tangency portfolio 8 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Mean-variance efficient PFs in equilibrium with a risk-free asset ⇒ Referred to as Capital Market Line (CML) e M ) − Rf E(R E(R e p ) = Rf + σp σM 9 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Mean-variance efficient PFs in equilibrium with a risk-free asset Optimal portfolio for different investors: E(R eM )−Rf ⇒ MRSi = MRSj ⇒ MPR(market price of risk) =. σ(R eM ) ⇒ Implication for corporate policy: MRSi = MRSj = MPR = MRT. 10 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Capital asset pricing model (CAPM) An equilibrium model, where: Tangency portfolio = Market portfolio. Interested in pricing all asset and portfolios ⇒ New risk-measure accounting for only market risk ▶ Idiosyncratic (asset-specific) risk e.g. Changes in oil-prices. ▶ Systematic (market) risk e.g. Changes in interest rates, recessions, etc. ⇒ Only idiosyncratic risk is diversifiable. ⇒ Investors need to be compensated only for market risk. 11 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Capital asset pricing model (CAPM) New risk-measure accounting for only market risk Cov (R ei , R eM ) σi,M βi = = 2 Var (RM ) e σM Security Market Line (SML):
e M ) − Rf e i ) = Rf + βi E(R E(R Proof on the board 12 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Security Market Line 13 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM First property of CAPM: Systematic risk and Idiosyncratic risk Total risk = systematic risk + idiosyncratic risk σi = σi,sys + σi,unsys We know σi , but how do we determine σi,sys and σi,unsys ? To find the systematic risk, σi,sys , for assets/portfolios i, we compare: CML pricing well-diversified portfolios (with no idiosyncratic risk) ⇒ pricing based on systematic risk, so σp = σi,sys , SML pricing all assets/portfolios based on its systematic risk only. 14 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM First property of CAPM: Systematic risk and Idiosyncratic risk Thus, for assets/portfolios i, we know from SML that:
E(R e i ) =Rf + βi E(Re M ) − Rf (SML) E(Re M ) − Rf =Rf + βi σ M σM If βi σM = σp for well-diversified portfolio, e M ) − Rf E(R E(R e i ) =Rf + σp (CML) σM ⇒ SML = CML if σp = βi σM , So ⇒ Systematic risk: σi,sys = βi σM ⇒ Idiosyncratic risk: σi,unsys = σi − σi,sys 14 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM First property of CAPM: Systematic risk and Idiosyncratic risk 15 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Second property of CAPM: Portfolio betas - measure the systematic risk of portfolios Cov (R ep, R eM ) βp = Var (ReM ) PN Cov ( i=1 wi Rei , R eM ) = Var (ReM ) Since Cov (aXe , bYe ) = abCov (Xe , Ye ) and Cov (Xe + Ye , Ze ) = Cov (Xe , Ze ) + Cov (Ye , Ze ) N X Cov (R ei , R eM ) ⇔ βp = wi i=1 Var (R eM ) N X = wi β i = w ⊤ β i=1 ⇒ Portfolio betas = weighted average of the betas of the securities. 16 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM CAPM - usage for investment decision for firms Simple example for a firm without debt. CAPM can be used to determine the cost of equity/required rate of return on equity. ⇒ When no debt: required rate of return on equity = required rate of return on project Invest in project if E(R e project ) > E(R e CAPM ) ⇒ if project’s expected rate of return > required rate of return. 17 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Exercise: Assume a market in equilibrium, such that the market portfolio is equal to the tangent portfolio based on 8 Danish stocks and a risk-free asset with a rate of 0.2% (from the exercise in lecture 8). ⇒ E(R e tan ) = E(R e M ) = 1.99% and σtan = σM = 4.72% Assume a project is funded using 100% equity, expected to generate a return of 2.25%, and with a Cov (R e project , R e M ) = 0.003. Should the firm invest in this project? 18 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Exercise: 18 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Empirical test of CAPM Expectations cannot be measured ⇒ Transform theoretical CAPM to ex ante form to apply on observed data. Transformation: Assume rate of return on any asset is a fair gamble, defined as Rjt =E(Rjt ) + βj δmt + ϵjt , where δmt =Rmt − E(Rmt ), with E(δmt ) = 0, ϵjt =random-error term, with E(ϵjt ) = 0, Cov (ϵjt , δmt ) =0, Cov (Rjt , Rmt ) βj = Var (Rmt ) Note: Fair gamble because   E[Rjt ] =E E(Rjt ) + βj δmt + ϵjt ⇔ E[Rjt ] = E[Rjt ] 20 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Empirical test of CAPM Transformation, cont.: Rjt =E(Rjt ) + βj δmt + ϵjt   =E(Rjt ) + βj Rmt − E(Rmt ) + ϵjt     =Rft + E(Rmt ) − Rft βj + βj Rmt − E(Rmt ) + ϵjt =Rft + (Rmt − Rft )βj + ϵjt ⇔ Rjt − Rft =(Rmt − Rft )βj + ϵjt ⇒ An empirical version of the CAPM that is expressed in terms of ex post observations of return data instead of ex ante expectations. ⇒ Ex post form of the CAPM. 21 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Empirical test of CAPM Ex post form of the CAPM. Rjt − Rft =(Rmt − Rft )βj + ϵjt Time-series regression: Testing the variation in returns over time for a given asset j: Rjt − Rft =αj + (Rmt − Rft )βj + ϵjt , Cross-sectional analysis: Understanding average returns in relation to betas in a given time period t: E[Rjt − Rft ] =αj + λβj , For CAPM (E(Rp ) − Rf = (E(RM ) − Rf )βj ) to hold: ⇒ λ = E(Rmt ) − Rft ⇒ αj = 0. 22 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Frequently used technique: 1 For every security j, estimate βj during a five-year pre-holding period based on a time-series regression: Rit − Rft = aj + βj (Rmt − Rft ) + ϵjt ⇒ Obtain β̂j 2 Construct N portfolios based on pre-ranking β̂j ’s (Typically N = 10, 12, or 20.) 3a Estimate β̂p and calculate E(Rp ) over a post-period 4a Run one cross-sectional regression for the post-period: E(Rp ) − Rf = (γ) + λβ̂p + αp ⇒ Obtain α̂p and λ̂ Or use Fama-MacBeth cross-sectional regression approach: 4b Run a cross-sectional regression at each time period, t: Rit − Rft = (γt ) + λt β̂p + αpt ⇒ Obtain α̂pt and λ̂t The estimates of λ and αp are the averages across time: T T 1 X 1 X λ̂ = λ̂t α̂p = α̂pt T t=1 T t=1 23 / 47  Introduction CAPM Factor models & APT Theoretical framework Empirical test of CAPM Empirical findings For CAPM to hold α̂ should not be significantly different from zero ⇒ General empirical finding: α̂ is significantly different from 0. Coefficient of β̂ (λ̂) should be equal to E(RM ) − Rf ⇒ General empirical finding: λ̂ < E(RM ) − Rf. Over long periods of time, the average excess return on the market portfolio should be greater than the risk-free rate. ⇒ General empirical finding: Over long periods of time, λ̂ > 0 Linear relationship between excess return on assets and beta. ⇒ General empirical finding: The model is linear in beta. Beta should be the only factor that explains the rate of return on the risky asset. ⇒ General empirical finding: Beta dominates squared-beta and unsystematic risk as a measure of risk. ⇒ General empirical finding: Other factors also explains security returns, not captured by β. 24 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Factor models Asset returns are affected by Common factors Firm specific noise Asset return for asset i with K factors R e1 + · · · + biK F e i = αi + bi1 F eK + εei , where R e i =return on asset i, E(R e i ) =expected return on asset i, bik =sensitivity of ith asset’s returns to the k th factor, F ek =k th factor common to the returns on all assets, εei =Noise term for asset i. E(F ek ) = 0 & E(e ϵi ) = 0 ⇒ αi = E(R e i ), Cov (e εi , εej ) = 0 & Cov (eεi , F ek ) = 0. 27 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Factor model for portfolios For portfolio with N assets, the return is N X N X
R ep = wi R ei = e1 + · · · + biK F wi αi + bi1 F eK + εei i=1 i=1 N X N X N X N X = wi αi + e1 + · · · + wi bi1 F wi biK F eK + wi εei i=1 i=1 i=1 i=1 So N X N X αp = wi αi = wi E(R e i ) = E(R ep) i=1 i=1 N X K X bpk = wi bik → Systematic risk = bpk F ek i=1 k=1 N X εep = wi εei = unsystematic risk i=1 28 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Factor model for portfolios For portfolio with N assets, the variance of the return is Var (R e1 + · · · + bpK F e p ) =Var (αp + bp1 F eK + εep ) Remember αp = E(R e p ) = constant and εep = PN wi εei , so since i=1 Cov (e εi , εej ) = 0 and Cov (eεi , F ek ) = 0 for k = 1,...K , we have K X X K ⇔ Var (R ep) = bpk bpm Cov (F ek , F em ) + Var (e εp ) k =1 m=1 For factor models with uncorrelated factors (Cov (F ek , F em ) = 0 for k ̸= m), we have K X 2 ⇔ Var (R ep) = bpk Var (F ek ) + Var (e εp ) k =1 Note: Uncorrelated factors are usually assumed for factor models. For a well-diversified portfolio (usually assumed): Var (e εp ) = 0. 29 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Tracking portfolios Definition of Tracking portfolio: Perfect tracking: A portfolio replicating the return of a benchmark portfolio in all scenarios Imperfect tracking: A portfolio with the same systematic risk as the benchmark portfolio ⇒ Factor betas for the tracking portfolio (bpk ) = factor beta for the benchmark portfolio (bk∗ ) for k = 1,..., K. 31 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Use of Tracking portfolios What do we use tracking portfolios for: 1 Performance evaluation 2 Corporate Hedging 3 Capital allocating 32 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Use of Tracking portfolios Example - Corporate Hedging BMW: Two relevant factors for BMW’s return: Exchange rate risk and growth (BNP) e1 : EUR ↑ by 5% ⇒ German-return ↓ by 3% : b∗ F ≈ −0.6 USD BMW ,1 e2 : Unexpected decrease in BNP in US of 10% ⇒ German-return ↓ F ∗ by 3% : bBMW ,2 ≈ 0.3 Hedging (tracking) portfolio: ∗ For factor 1: bp1 = −bBMW ,1 ∗ For factor 2: bp2 = −bBMW ,2 ∗ ⇒ no risk for BMW after hedging: bHp1 + bBMW ,1 = 0 33 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Portfolio weights for tracking portfolio Determining portfolio weights for tracking portfolio: For factor 1:bp1 =w1 b11 + · · · + wN bN1 = b1∗... For factor K:bpK =w1 b1K + · · · + wN bNK = bK∗ PN and i=1 wi = 1. ⇒ Solve system of equations to find portfolio weights for tracking portfolio. ▶ Note, need N = K+1 assets to solve the system of equations. 34 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Pure factor portfolios Definition of a pure factor portfolio (pm) wrt. factor m: ⇒ bpm,m = 1 ⇒ bpm,k = 0 for k ̸= m ⇒ no firm-specific risk Thus, the return of the pure factor portfolio is defined as R e pm = αpm + F em The factor risk-premiums can therefore be defined as: e pm ) − Rf = αpm − Rf λm = E(R Example on the board 35 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Pure factor portfolios A tracking portfolio for security i with no firm-specific risk: K X R e i =αi + bim F e1 + · · · + biK F em = αi + bi1 F eK m=1 Tracking pf = weighted avg. of pure factor pfs and risk-free asset K X  K X  ⇔R ei = e pm + 1 − bim R bim Rf m=1 m=1 With the expected return of K X  K X  E(R ei ) = e pm ) + 1 − bim E(R bim Rf m=1 m=1 K X  K X  = bim (λm + Rf ) + 1 − bim Rf m=1 m=1 K X = bim λm + Rf = bi1 λ1 + · · · + biK λK + Rf m=1 36 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Asset pricing Theory (APT) Assuming: Return defined by a factor model No firm-specific risk (Eliminated by diversification) Perfect capital market Market in equilibrium, so no arbitrage ⇒ Asset i’s expected return is determined by the tracking portfolio e i ) = bi1 λ1 + · · · + biK λK + Rf E(R where λm is factor m’s risk-premium. 38 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Asset pricing Theory (APT) Comments about APT: No assumptions about empirical distribution of asset returns No assumptions about individuals’ utility functions Allow equilibrium returns of assets to be dependent on many factors, not just one (e.g., beta) No special role for the market portfolio in the APT, whereas the CAPM requires that the market portfolio be efficient. Roll’s critique (Roll, 1977): ▶ Market portfolio contains all assets ⇒ impossible to observe ⇒ Impossible to test whether the true market portfolio is efficient ⇒ CAPM cannot be tested! ⇒ APT is more general than CAPM! ⇒ APT easier to test than CAPM! 39 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Empirical test of Asset pricing Theory (APT) Generally, the same test as for CAPM. Addition to empirical test of CAPM ⇒ Identify relevant factors that explains financial risk (the variability in the returns ⇒ covariance structure). Three methods: 1 Factor-analysis: A statistical method identifying relevant factors based on how much of the variability in the returns that is described by the factors 2 Macro-economic variables that do the best job explaining observed return-patterns 3 Firm-characteristics that affect returns. Note: When factors are measured precisely for individual stocks ⇒ No need of portfolios in empirical testing. 40 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Fama and French (1992) An example of empirical testing of factor models/APT Test the cross-sectional relationship between return and risk β, ▶ estimated using the value-weighted portfolio of NYSE, AMEX, and, NASDAQ stocks. Firm size, ln(ME) ▶ the natural logarithm of the equity market capitalization of a firm book-to-market equity, ln(BE/ME), ▶ where BE = book value of common equity plus balance-sheet deferred taxes etc. Databases used: Daily individual stock returns 1963 – 1990: all NYSE and AMEX listed stocks 1973 – 1990: all NASDAQ-listed stocks market equity, ME (a stock’s price times shares outstanding) 41 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Fama and French (1992) An example of empirical testing of factor models/APT Table II: Properties of Porfolios Formed on Size or Pre-ranking β: 1963-1990. 42 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Fama and French (1992) An example of empirical testing of factor models/APT 43 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Fama and French (1992) An example of empirical testing of factor models/APT Use cross-sectional regression approach by Fama-MacBeth: Table III: Average Slopes (t-Statistics) from Month-by-Month Regressions of Stock Returns on β , Size, Book-to-Market Equity,...: July 1963 to December 1990. (Small part of Table III from paper): 44 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Fama and French’s three factor model Introduce three-factor model: E(Ri ) − Rf =bi [E(RM ) − Rf ] + si E(SMB) + hi E(HML) where SMB =size-premium =Diff. in returns between small and big firms HML =value-premium =Diff. in returns between firms w. high and low book-to-market value ratio 45 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT Comments regarding empirical testing of Factor models Debate about empirical validity of factor models continues - three key directions in the research of empirical testing of CAPM: 1 Errors in the execution and design of the empirical tests ⇒ Many different designs! Only presented one here! (see more in Cochrane (2009)) 2 Misspecification of model ⇒ What factors explain expected excess returns? ⇒ Lack of intuitive explanation behind factors (problem: Fishing for factors (see more in Harvey, Liu and Zhu (2016) or Harvey and Liu (2021)). 3 The possibility that the market risk premium and betas change over time 46 / 47  Introduction CAPM Factor models & APT Factor models Tracking portfolios APT References CWS, ch. 5 Cochrane, J. H. (2009). Asset pricing: Revised edition. Princeton university press. Fama, E. F., & French, K. R. (1992). The cross-section of expected stock returns. Journal of Finance, 47(2), 427-465. Harvey, C. R., Liu, Y., & Zhu, H. (2016).... and the cross-section of expected returns. The Review of Financial Studies, 29(1), 5-68. Harvey, C. R., & Liu, Y. (2021). Lucky factors. Journal of Financial Economics, 141(2), 413-435. Roll, R. (1977). A critique of the asset pricing theory’s tests Part I: On past and potential testability of the theory. Journal of financial economics, 4(2), 129-176. 47 / 47