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Financial Analysis, Planning and Forecasting Theory and Application Chapter 8 Risk and Return Trade-off Analysis By Alice C. Lee San Francisco State University John C. Lee J.P. Morgan Chase Cheng F. Lee Rutgers University 1 Outline 8.1 Introduction 8.2 Capital Market Line, Efficient-market hypothesis, and capital asset pricing model 8.3 The market model and beta estimation 8.4 Empirical evidence for the risk-return relationship 8.5 Why beta is important in financial management 8.6 Systematic risk determination 8.7 Some applications and implications of the capital asset pricing model 8.8 Arbitrage pricing theory 8.9 Summary Appendix 8A. Mathematical derivation of the capital asset pricing model Appendix 8B. Arbitrage Pricing Model 2 8.2 Capital Market Line, Efficient-market hypothesis, and capital asset pricing model Lending, borrowing, and the market portfolio The capital market line The efficient-market hypothesis - Weak-form efficient-market hypothesis - Semistrong-form efficient-market hypothesis - Strong-form efficient-market hypothesis The capital asset pricing model 3 8.2 Capital Market Line, Efficient-market hypothesis, and capital asset pricing model From Figure 8.1, we can derive the capital market line as follows. Step 1. Since PRf E M p Rf D therefore, E ( Rp ) R f p E( Rm ) R f m Figure 8.1 The Capital Market Line then we can obtain Eq. 8.2. Eq(8.2) P E ( RP ) R f [ E ( Rm ) R f ] m 4 8.2 Capital Market Line, Efficient-market hypothesis, and capital asset pricing model market value of an individual asset x market value of all asset (8.3) Figure 8.2 The Capital Asset Pricing Model (SML) E(Ri ) Rf i [E(Rm ) Rf ] (8.4) Derivation of Equation (8.4) can be found in Appendix 8A 5 8.2 Capital Market Line, Efficient-market hypothesis, and capital asset pricing model COV ( Ri , Rm ) i ,m i m VAR ( Rm ) m2 i ,m i m i E ( Ri ) R f [ E ( Rm ) R f ] m (8.5) (8.6) 6 8.3 The market model and beta estimation Ri, t = αi + i Rm, t + εi, t (8.7) Ri, t – Rf, t = αi + i (Rm, t – Rf, t) + εi, t (8.8) ˆi (Ri Rf ) i (Rm Rf ) (8.9) 7 8.3 The market model and beta estimation Example 8-1 i JNJ cov( Ri , Rm ) m2 cov( RJNJ , Rm ) m2 MRK PG .000407 .3246 .00126 cov( RMRK , Rm ) m2 cov( RPG , Rm ) m2 .000974 .7763 .00126 .000244 .1943 .00126 8 8.3 The market model and beta estimation Table 8-1 Average Return ( Ri ) Variance (2i) Covariance [COV (Ri, Rm)] JNJ .0042 .00162 .000407 MRK .0021 .00610 .000974 PG .0105 .00124 .000244 Market .0043 .00126 ˆ JNJ (.0042 .004) (0.3255)(.0043 .004) .0001 ˆ MRK (.0021 .004) (0.7790)(.0043 .004) .0021 ˆ PG (.0105 .004) (0.1941)(.0043 .004) .0064 9 8.3 The market model and beta estimation Figure 8-3 Mean-Variance Comparison Mean-Variance Comparison 0.007 MRK Mean Return 0.006 0.005 0.004 0.003 JNJ 0.002 Market PG 0.001 0 0 0.002 0.004 0.006 0.008 0.01 0.012 Variance 10 8.3 The market model and beta estimation Figure 8-4 SML Comparison SML Comparison Mean Return 0.012 0.01 PG 0.008 0.006 JNJ 0.004 Market MRK 0.002 0 0 0.2 0.4 0.6 0.8 1 1.2 Beta 11 8.4 Empirical evidence for the risk-return relationship E(Ri) – Rf = [E(Rm) – Rf] i (8.10) im i ,m i i 2 m m (8.5) Rp Rf r0 r1 p ep (8.11) 12 8.5 Why beta is important in financial management The use of beta is of great importance to the financial manager because it is a key component in the estimation of the cost of capital. A firm’s expected cost of capital can be derived from E(Ri), where E(Ri) = Rf + βi [E (Rm) – Rf]. This assumes that the manager has access to the other parameters, such as the risk-free rate and the market rate of return. (This is explored in more detail in chapter 11.) Ri, t = αi + βi Rm, t + єi, t (8.7) Beta is also an important variable because of its usefulness in security analysis. In this type of analysis, beta is used to measure a security’s response to a change in the market and so can be used to structure portfolios that have certain risk characteristics. 13 8.6 Systematic risk determination Business risk Financial risk Other financial variables Capital labor ratio Fixed costs and variable costs Market-based versus accounting-based beta forecasting 14 8.6 Systematic risk determination Business risk and financial risk risk of financial leverage ROE ROA Other Financial Variables Salest Salest 1 gs 100% Salest 1 gTA total assetst total assetst 1 100% total assetst 1 15 8.6 Systematic risk determination EPSi,t a0 a1EPSm,t Ei,t (8.12) N 2 ( EBIT ) 2 ( EBIT EBIT ) t t 1 N 1 Capital labor ratio Q=f(K, L) a0 a1 X1 a2 X 2 a3 X 3 ... an X n (8.13) (8.14) 16 8.7 Some applications and implications of the capital asset pricing model Table 8-2 Regression Results for the Accounting Beta a0 a1 X1 a2 X 2 ... an X n Coefficients t Values Intercept ( a0 ) .911 11.92 Financial leverage ( a1 ) .704 5.31 Dividend payout ( a2 ) -.175 -3.50 Sales beta( a3) .03 3.02 Operating income beta ( a4 ) .011 2.18 17 8.7 Some applications and implications of the capital asset pricing model E ( P1 ) P0 E ( Ri ) P0 E ( P1 ) P0 R f [ E ( Rm ) R f ]i P0 E ( P1 ) P0 1 R f [ E ( Rm ) R f ]( i ) (8.15) (8.16) (8.17) 18 8.7 Some applications and implications of the capital asset pricing model FIGURE 8-5 Capital Budgeting and Business Strategy Matrix 19 8.8 Liquidity and Capital Asset Pricing Model Acharya and Pedersen (2005) proposed the following model for liquidity considering the impact of liquidity risk on security pricing [E(Rit )] a bj1 (factor 1) bj 2 (factor 2) ... bjk (factor k ) (8.18) The overall risk of a security account for three kinds of liquidity risk defined as follows L1 Cov(Ci , CM ) Var ( RM CM ) L2 Cov( Ri , CM ) Var ( RM CM ) L3 Cov(Ci , RM ) Var ( RM CM ) First, L1 measures the sensitivity of security illiquidity to market illiquidity. In general, investors demand higher premium for holding an illiquid security when the overall market liquidity is low. Further, L 2 measures the sensitivity of security’s return to market illiquidity. Investors are willing to accept a lower average return on security that will provide higher returns when market L3 illiquidity is higher. Finally measures the sensitivity of security illiquidity to the market rate of return. In general, investors are willing to accept a lower average return on security that can be20 sold more easily. 8.9 Arbitrage Pricing Theory Ross (1976, 1977) has derived a generalized capital asset pricing relationship called arbitrage pricing theory (APT). To derive the APT, Ross assumed the expected rate of return on ith security be explained by k independent influences (or factors) as [E(Rit )] a bj1 (factor 1) bj 2 (factor 2) ... bjk (factor k ) (8.19) Using Equation 8-18, Ross has shown that the risk premium of jth security can be defined as E(Ri ) Rf [E(Fi ) Rf ]bi1 ... [E(Fk ) Rf ]bik (8.20) By comparing Equation 8-19 with the CAPM equation, we can conclude that the APT is a generalized capital asset pricing model. Therefore, it is one of the important models for students of finance to understand. This model is generally discussed in upper-level financial management or investment courses. 21 8.10 Intertemporal CAPM Merton (1973,1992) derived an Intertemporal CAPM which allows the change of investment opportunity set as defined in Eq. (8.21). This Intertemporal CAPM will reduce to the one-period CAPM defined in Eq. (8.4). i r i iM in nM 2 M (1 nM ) ( M r ) i in iM nM 2 M (1 Mn ) ( n r ) (8.21) 22 8.8 Summary In Chapter 8 we have discussed the basic concepts of risk and diversification and how they pertain to the CAPM, as well as the procedure for deriving the CAPM itself. It was shown that the CAPM is an extension of the capital market line theory. The possible uses of the CAPM in financial analysis and planning were also indicated. The concept of beta and its importance to the financial manager were introduced. Beta represents the firm’s systematic risk and is a comparative measure between a firm’s security or portfolio risk in comparison with the market risk. Systematic risk was further discussed by investigating the relationship between the beta coefficient and other important financial variables. 23 Appendix 8A. Mathematical derivation of the capital asset pricing model E( RP ) wi E( Ri ) (1 wi ) E( Rm ) P [w (1 wi ) 2(1 wi )wi im ] 2 i 2 i 2 2 m E ( RP ) E ( Ri ) E ( Rm ) wi (8A-1) 1 2 (8A-2) (8A-3) 1 ( P ) 1 2[wi2 i2 (1 wi )2 m2 2wi (1 wi )] 2 (2wi i2 2 m2 2wi m2 2 im 4wi im ) wi 24 Appendix 8A. Mathematical derivation of the capital asset pricing model When wi 0 E ( RP ) E ( Ri ) E ( Rm ) wi im m2 E ( RP ) ( ) ( P ) m P wi CML Slop: E ( Rm ) R f m (8A-4) (8A-5) im E ( Ri ) R f [ E ( Rm ) R f ] 2 m (8A-6) E( Ri ) Rf [ E( Rm ) R f ]i (8A-7) 25 Appendix 8A. Mathematical derivation of the capital asset pricing model FIGURE 8-A1 The Opportunity Set Provided by Combinations of Risky Asset / and Market Portfolio, M 26 Appendix 8B. Arbitrage Pricing Model Rj E(Rj ) b1 j1 b2 j 2 bkj X j 0 bkj k j (8B-1) (8B-2) n Xj 0 j 1 X j Rj 0 (8B-3) (8B-4) 27 Appendix 8B. Arbitrage Pricing Model E(Rj ) 0 1bj1 2bj 2 k bjk (8B-5) i Ei E0 E(Rj ) E0 (E1 E0 )bj1 Rjt a0 a1R1t (Ek E0 )bjk (8B-6) ai 1Ri 1t ai 1Ri 1t ak Rk it (8B-7) 28