【正文】 RiskFree Asset ? Arbitrage profit of 1% 14/02/2022 60 ? 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 APT and CAPM Compared 14/02/2022 61 Portfolio Sensitivities U h h h h B S Z Portfolios S Stocks B – Bonds U – Unit Beta Z – Zero Beta Inflation Beta Productivity Beta 14/02/2022 62 And …... 。j 2222 uiMiMi ??b? ??14/02/2022 50 Security Characteristic line Excess Return on Market Excess Return on Asset Characteristic Line Beta for the firm is the slope of this line. Jensen Alpha for the firm is the intercept of this line. Distance from the line to the points are the error terms .14/02/2022 51 Summary of CAPM ? The CAPM is based on many assumptions, including – All investors have homogeneous beliefs – All investors are meanvariance optimizers – All investors are atomistic – Security returns are normally distributed ? The result is a linear pricing relationship which gives expected returns as a function of systematic risk. ? The single source of systematic risk is identified as the market portfolio. All investors hold this same risky market portfolio because it is meanvariance efficient. ? The CAPM contends that if prices move out of line, all investors will make small adjustments to their portfolios and prices will be brought back to equilibrium. 14/02/2022 52 Arbitrage Pricing Theory ? Ross (1976,JET) developed Arbitrage Pricing Theory (APT) which derives asset prices from arbitrage arguments. ? The APT model is based on the law of one price : two items that are the same cannot sell at different prices. ? Like CAPM it is an equilibrium model. ? It does assume homogenous expectations. ? The APT model does not need to make assumptions about utility functions of investors. ? The meanvariance framework is replaced by an assumption of the process generating security returns. ? This process generating returns is assumed to be a multiindex model. 14/02/2022 53 ? Principle of Arbitrage – Definition of arbitrage ? The process of earning riskless profits by taking advantage of differential pricing for the same physical asset or security. – Conditions on arbitrage portfolio ? Zero initial outlay: some assets are held in positive amounts, some in negative amounts and, perhaps, some in zero amounts. ? Riskfree: the payoff (v) or the return (y) on the portfolio in every state (k) must be either positive or zero (in equilibrium). or ???Niii xp10lkxvNiiik ,...,2,1,01, ???????Niii yr1014/02/2022 54 The effect of Arbitrage Portfolio on an Investor’s Position Old portfolio Arbitrage portfolio New portfolio Weight (1) + (2) = (3) X1 .333 .100 .433 X2 .333 .075 .408 X3 .333 .158 Property rp % .975% % bp .000 ?p small 14/02/2022 55 ? Arbitrage Pricing Theory – Singlefactor APT ? The return on security i is ri = E(ri) + biF + ei. – E(ri) is the expected return. – F is the factor. – bi measures the sensitivity of ri to F. – ei is the firm specific return. APT condition 1: APT condition 2: The elimination of systematic risk invol