7.1 THE CAPITAL ASSET PRICING MODEL CHAPTER 7 Capital Asset Pricing and Arbitrage Pricing Theory
Capital Asset Pricing Model (CAPM) Assumptions
Equilibrium model that underlies all modern Individual investors are price takers financial theory Single -period investment horizon Derived using principles of diversification Investments are limited to traded financial with simplified assumptions assets Markowitz, Sharpe, Lintner and Mossin are No taxes nor transaction costs researchers credited with its development
Assumptions (cont.) Resulting Equilibrium Conditions
Information is costless and available to all All investors will hold the same portfolio investors for risky assets – market portfolio Investors are rational mean -variance Market portfolio contains all securities optimizers and the proportion of each security is its Homogeneous expectations market value as a percentage of total market value
7-1 Resulting Equilibrium Conditions (cont.) Figure 7.1 The Efficient Frontier and the Capital Market Line Risk premium on the market depends on the average risk aversion of all market participants Risk premium on an individual security is a function of its covariance with the market
The Risk Premium of the Market Portfolio Expected Returns On Individual Securities The risk premium on individual M = Market portfolio securities is a function of the individual r = Risk free rate f security ’s contribution to the risk of the E(r ) - r = Market risk premium M f market portfolio Individual security ’s risk premium is a E(r M) - rf = Market price of risk = M f function of the covariance of returns σ M with the assets that make up the market = Slope of the CAPM portfolio
Expected Returns On Individual Securities: Figure 7.2 The Security Market Line an Example and Positive Alpha Stock Using the Dell example: Er− r Er − r ()Mf= () Df
1 βD Rearranging gives us the CAPM ’s expected return -beta relationship
Er= r + Er − r ()D fβ D () M f
7-2 SML Relationships Sample Calculations for SML
E(r m) - rf = .08 rf = .03 2 β = [COV(r i,r m)] / σm
E(r m) – rf = market risk premium m f βx = 1.25
E(r x) = .03 + 1.25(.08) = .13 or 13% SML = r f + β[E(r m) - rf]
βy = .6
e(r y) = .03 + .6(.08) = .078 or 7.8%
Graph of Sample Calculations
E(r) SML 7.2 THE CAPM AND INDEX MODELS Rx=13% .08 Rm=11% Ry=7.8% 3% ß .6 1.0 1.25 ßyy ßm ßxx
Estimating the Index Model Table 7.1 Monthly Return Statistics for T-bills, S&P 500 and General Motors Using historical data on T -bills, S&P 500 and individual securities Regress risk premiums for individual stocks against the risk premiums for the S&P 500 Slope is the beta for the individual stock
7-3 Figure 7.3 Cumulative Returns for Figure 7.4 Characteristic Line for GM T-bills, S&P 500 and GM Stock
Table 7.2 Security Characteristic GM Regression: What We Can Learn Line for GM: Summary Output
GM is a cyclical stock Required Return:
rrr+ −=+ x = fβ( M f ) 2.75 1.24 5.5 9.57% Next compute betas of other firms in the industry
Predicting Betas
The beta from the regression equation is an estimate based on past history THE CAPM AND THE REAL WORLD Betas exhibit a statistical property – Regression toward the mean
7-4 CAPM and the Real World
The CAPM was first published by Sharpe 7.4 MULTIFACTOR MODELS AND THE Journal of Finance in the in 1964 CAPM Many tests of the theory have since followed including Roll ’s critique in 1977 and the Fama and French study in 1992
Multifactor Models Fama French Three-Factor Model
Limitations for CAPM Returns are related to factors other than Market Portfolio is not directly observable market returns Research shows that other factors affect Size returns Book value relative to market value Three factor model better describes returns
Table 7.3 Summary Statistics for Rates of Table 7.4 Regression Statistics for the Return Series Single-index and FF Three-factor Model
Arbitrage - arises if an investor can 7.5 FACTOR MODELS AND THE construct a zero beta investment portfolio ARBITRAGE PRICING THEORY with a return greater than the risk -free rate If two portfolios are mispriced, the investor could buy the low -priced portfolio and sell the high -priced portfolio In efficient markets, profitable arbitrage opportunities will quickly disappear
Figure 7.5 Security APT and CAPM Compared Line Characteristics APT applies to well diversified portfolios and not necessarily to individual stocks With APT it is possible for some individual stocks to be mispriced - not lie on the SML APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio APT can be extended to multifactor models
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