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【正文】 ves 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。 Return Summary ? Total risk: variance (or standard deviation of an asset’s return ? ? Total return: expected return + unexpected return ? Systematic risk: unanticipated events that affect almost all assets to some degree ? Unsystematic risk: unanticipated events that affect single assets or small groups of assets ? Effect of diversification: elimination of unsystematic risk via the bination of assets into a portfolio ? Systematic risk principle amp。 u: risk of specific firm (portfolio) – Method: ?Define a riskfree asset such as TBills ?Select a market index or benchmark such as the Samp。 xij: the proportion invested in security j by investor i. m: the numbers of investors n: numbers of securities xij= xi+1,j = xi+2,j =… =kj (assumption and MV approach) Pj=?mWi xij ? Pj=kj?Wi In equilibrium, ?mWi =?nPj? kj=Pj/ ?nPj 14/02/2022 44 – The first order equation – The slope of the curved line iM at the endpoint M. where – Security market line ( ) MiiP RRdwRd ??2/1222 2PiMiiMMiMiiiP dwd??????? ?????? ?2MiMMMiPP RRdRd???? ???0?iwS M LC M L ?? ?? ?MfMMiMMMi RRRR???? ????2 iMMfMfiRRRR ?? ????214/02/2022 45 Portfolio Separation and Risk references M Lender Borrower Capital Market Line M?)( MREfR)(RE?14/02/2022 46 The Security Market Line Bet aE x p ect ed Ret u rnR i s k f r e e R e tu r nB e ta of the M ar k e t = 1. 00E ( M ar k e t R e tur n)Sec u r it y M ar k e t L in eOv e r pr i c e d S e c ur i t i e sUn de r pr i c e d S e c ur i t i e sAB+ al pha al phai M 14/02/2022 47 (3) implication: The CAPM shows that the expected return for an asset depends on: – Pure time value of money: riskfree rate – Reward for bearing systematic risk: market risk premium – Amount of systematic risk: beta coefficient – Prices (Payoff) and CAPM where ? Disequilibrium example – Suppose a security with a “b” of is offering expected return of 15% – According to SML, it should be 13% – Underpriced: offering too high of a rate of return for its level of risk ? ?fMfMPiMiiRPRPEPEP?????1)1()()( 1221b)(),(222MMiPiM PV arPPCov?b14/02/2022 48 ? Estimation of Beta (Market Model) where ER: excess return。 Risk Aversion E(r) Efficient frontier of risky assets More riskaverse investor U’’’ U’’ U’ Q P S St. Dev Less riskaverse investor 14/02/2022 37 Optimal Allocation ? We have shown that different investors will choose different positions in the risky asset. In particular, the more risk averse investors will choose to hold less of the risky asset and more of the riskfree asset. ? How do we quantify this? ? We start from the utility function of the investor: ? ? ?ArEU ??14/02/2022 38 Optimal Allocation ? The investor attempts to maximize her utility level, by choosing the best allocation to the risky asset, w. ? Taking the first order derivatives of U with respect to w and set it to zero. ? ?? ?? ? 222PfPf AyrrEwrArEU????????? ?? ?22*0PfPPfPArrEwwArrE????????14/02/2022 39 Optimal Allocation ? If In other
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