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【正文】 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 involves choosing a portfolio such that whatever value is taken on by the factor, F, its effect on the portfolio return is zero. iiinii xpyy ????,01???niiiby10Portfolio amp。Individual Security Comparison F E(r)% Portfolio F E(r)% Individual Security 14/02/2022 57 SingleFactor APT b P E[r ] P E * r f ?1= E r * f * 14/02/2022 58 E(r)% Beta for F 10 7 6 Risk Free 4 A D C .5 Disequilibrium Example 14/02/2022 59 Disequilibrium Example ? Short Portfolio C ? Use funds to construct an equivalent risk higher return Portfolio D – D is prised of A amp。 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 …...
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