(2.1) Markowitz’s mean-variance formulation (2.2) Two …

2. mean-variance portfolio theory( ) markowitz s mean-variance formulation ( ) Two-fund theorem( ) Inclusion of the riskfree markowitz mean-variance formulationSuppose there areNrisky assets, whose rates of returns are given by the randomvariablesR1, , RN, whereRn=Sn(1) Sn(0)Sn(0), n= 1,2, , (w1 wN)T, wndenotes the proportion of wealth invested in assetn,withNXn=1wn= 1. The rate of return of the portfolio isRP=NXn= There does not exist any asset that is a combination of other assets in theportfolio, that is, non-existence of redundant = (R1R2 RN) and1= (1 1 1) are linearly independent, otherwiseRPis a constant irrespective of any choice of portfolio first two moments ofRPare P=E[RP] =NXn=1E[wnRn] =NXn=1wn n,where n=Rn,and 2P= var(RP) =NXi=1 NXj=1wiwjcov(Ri, Rj) =NXi=1 NXj=1wi denote the covariance matrix so that 2P=wT example whenn= 2, we have(w1w2) 11 12 21 22!

To find the global minimum variance portfolio, we set dσ2 P dµP = 2aµP − 2b ∆ = 0 so that µP = b/a and σ2 P = 1/a. Correspondingly, λ1 = 1/a and λ2 = 0. The weight vector that gives the global minimum variance is found to be wg = Ω−11 a = Ω−11 1TΩ−11. 15

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