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An Introduction to Asset Pricing Models Chapter 8 Chapter Objectives CAPM assumptions risk/return structure CAPM equation beta Security Market Line Empirical use of model time intervals variables Capital Market Theory: An Overview Capital market theory extends portfolio theory and develops a model for pricing all risky assets Capital asset pricing model (CAPM) will allow you to determine the required rate of return for any risky asset Assumptions of CMT 1. All investors are Markowitz efficient investors who want to target points on the efficient frontier. 2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). 3. All investors have homogeneous expectations. 4. All investors have the same one-period time horizon. 5. All investments are infinitely divisible. 6. There are no taxes or transaction costs. 7. There is no inflation or any change in interest rates. 8. Capital markets are in equilibrium. Risk-Free Asset An asset with zero variance Zero correlation with all other risky assets Provides the risk-free rate of return (RFR) Will lie on the vertical axis of a portfolio graph Risk-Free Asset Covariance between two sets of returns is n Cov ij [R i - E(R i )][R j - E(R j )]/n i 1 Because the returns for the risk free asset are certain, RF 0 Thus Ri = E(Ri), and Ri - E(Ri) = 0 Consequently, the covariance of the risk-free asset with any risky asset or portfolio will always equal zero. Similarly the correlation between any risky asset and the risk-free asset would be zero. Market Portfolio Under CAPM, in equilibrium each asset has nonzero proportion in M All assets included in risky portfolio M All investors buy M If M does not involve a security, then nobody is investing in the security If no one is investing, then no demand for securities If no demand, then price falls Falls to point where security is attractive and people buy and so it is in M CML and the Separation Theorem CML represents new EF all investors have the same EF but choose different portfolios based on risk tolerances investor spreads money among risky assets in same relative proportions and then borrows/lends separation theorem optimal combination of risky assets for investor can be determined without knowledge of investor’s preferences toward risk and return investment decision financing decision The CML and the Separation Theorem The decision of both investors is to invest in portfolio M along the CML (the investment decision) E ( R port ) CML B M A PFR port Number of Stocks in a Portfolio and the Standard Deviation of Portfolio Return Standard Deviation of Return Figure 9.3 Unsystematic (diversifiable) Risk Total Risk Standard Deviation of the Market Portfolio (systematic risk) Systematic Risk Number of Stocks in the Portfolio A Risk Measure for the CML Covariance with the M portfolio is the systematic risk of an asset The Markowitz portfolio model considers the average covariance with all other assets in the portfolio The only relevant portfolio is the M portfolio Because all individual risky assets are part of the M portfolio, an asset’s return in relation to the return for the M portfolio may be described as follows: R it a i b i R Mi The Capital Asset Pricing Model: Expected Return and Risk The existence of a risk-free asset resulted in deriving a capital market line (CML) that became the relevant frontier An asset’s covariance with the market portfolio is the relevant risk measure This can be used to determine an appropriate expected rate of return on a risky asset - the capital asset pricing model (CAPM) The Capital Asset Pricing Model: Expected Return and Risk CAPM indicates what should be the expected or required rates of return on risky assets This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models You can compare an estimated rate of return to the required rate of return implied by CAPM - over/ under valued ? The Security Market Line (SML) The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m) This is shown as the risk measure The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance: 2 m Determining the Expected Return for a Risky Asset E(R i ) RFR i (R M - RFR) The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset The risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (RM-RFR) Determining the Expected Return for a Risky Asset Stock Beta Assume: RFR = 5% (0.05) A 0.70 RM = 9% (0.09) B 1.00 C 1.15 Implied market risk premium = 4% (0.04) D E 1.40 -0.30 E(R i ) RFR i (R M - RFR) E(RA) = 0.05 + 0.70 (0.09-0.05) = 0.078 = 7.8% E(RB) = 0.05 + 1.00 (0.09-0.05) = 0.090 = 09.0% E(RC) = 0.05 + 1.15 (0.09-0.05) = 0.096 = 09.6% E(RD) = 0.05 + 1.40 (0.09-0.05) = 0.106 = 10.6% E(RE) = 0.05 + -0.30 (0.09-0.05) = 0.038 = 03.8% Determining the Expected Return for a Risky Asset 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 risk-adjusted rates of return than the average investor Price, Dividend, and Rate of Return Estimates Table 9.1 Current Price Expected Dividend Expected Future Rate Stock (Pi ) Expected Price (Pt+1 ) (Dt+1 ) of Return (Percent) A 25 27 0.50 10.0 % B 40 42 0.50 6.2 C 33 39 1.00 21.2 D 64 65 1.10 3.3 E 50 54 0.00 8.0 Comparison of Required Rate of Return to Estimated Rate of Return Table 9.2 Required Return Estimated Return Stock Beta E(Ri ) Estimated Return Minus E(R i ) Evaluation A 0.70 10.2% 10.0 -0.2 Properly Valued B 1.00 12.0% 6.2 -5.8 Overvalued C 1.15 12.9% 21.2 8.3 Undervalued D 1.40 14.4% 3.3 -11.1 Overvalued E -0.30 4.2% 8.0 3.8 Undervalued Plot of Estimated Returns E(R ) on SML Graph i Figure 9.7 .22 .20 Rm C SML .18 .16 .14 .12 Rm .10 A E .08 .06 B .04 D .02 1.0 Beta -.40 -.20 0 .20 .40 .60 .80 1.20 1.40 1.60 1.80 Calculating Systematic Risk: The Characteristic Line The systematic risk input of an individual asset is derived from a regression model, referred to as the asset’s characteristic line with the model portfolio: where: R i,t R i i M, t Ri,t = the rate of return for asset i during period t RM,t = the rate of return for the market portfolio M during t i R i - i R m i Cov i,M M 2 the random error term Scatter Plot of Rates of Return The characteristic Figure 9.8 Ri line is the regression line of the best fit through a scatter plot of rates of return RM Empirical Tests of the CAPM Stability of Beta Comparability of Published Estimates of Beta Number of observations and time interval used in regression vary Value Line Investment Services (VL) uses weekly rates of return over five years Merrill Lynch (ML) uses monthly return over five years There is no “correct” interval for analysis Weak relationship between VL & ML betas due to difference in intervals used Interval effect impacts smaller firms more Market portfolio The Market Portfolio: Theory versus Practice There is a controversy over the market portfolio. Hence, proxies are used There is no unanimity about which proxy to use Microeconomic-Based Risk Factor Models Specify the risk in microeconomic terms using certain characteristics of the underlying sample of securities ( Rit RFRt ) ai bi1 ( Rm t RFRt ) bi 2 SMBt bi 3 HMLt eit extension of Fama-French 3-factor model includes a fourth factor that that accounts for firms with positive past return to produce positive future return - momentum ( Rit RFRt ) ai bi1 ( Rmt RFRt ) bi 2 SMBt bi 3 HMLt bi 4 PR1YRt eit Summary When you combine the risk-free asset with any risky asset on the Markowitz efficient frontier, you derive a set of straight-line portfolio possibilities Summary The dominant line is tangent to the efficient frontier Referred to as the capital market line (CML) All investors should target points along this line depending on their risk preferences Summary All investors want to invest in the risky portfolio, so this market portfolio must contain all risky assets The investment decision and financing decision can be separated Everyone wants to invest in the market portfolio Investors finance based on risk preferences Summary The relevant risk measure for an individual risky asset is its systematic risk or covariance with the market portfolio Once you have determined this Beta measure and a security market line, you can determine the required return on a security based on its systematic risk Summary Assuming security markets are not always completely efficient, you can identify undervalued and overvalued securities by comparing your estimate of the rate of return on an investment to its required rate of return

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