Variance of a 3-Asset Portfolio
What is the variance of a portfolio with 3 assets, A, B and C, with weights xA x_A xA , xB x_B xB , and xC x_C xC ?
xAσA2+xBσB2+xCσC2+2xAxBCov(rA,rB)+2xAxCCov(rA,rC)+2xBxCCov(rB,rC) x_Aσ_A^2 + x_Bσ_B^2 + x_Cσ_C^2 + 2x_Ax_B\mathrm{Cov}(r_A, r_B) + 2x_Ax_C\mathrm{Cov}(r_A, r_C) + 2x_Bx_C\mathrm{Cov}(r_B, r_C) xAσA2+xBσB2+xCσC2+2xAxBCov(rA,rB)+2xAxCCov(rA,rC)+2xBxCCov(rB,rC)
xA2σA2+xB2σB2+xC2σC2+2xAxBCov(rA,rB)+2xAxCCov(rA,rC)+2xBxCCov(rB,rC) x_A^2σ_A^2 + x_B^2σ_B^2 + x_C^2σ_C^2 + 2x_Ax_B\mathrm{Cov}(r_A, r_B) + 2x_Ax_C\mathrm{Cov}(r_A, r_C) + 2x_Bx_C\mathrm{Cov}(r_B, r_C) xA2σA2+xB2σB2+xC2σC2+2xAxBCov(rA,rB)+2xAxCCov(rA,rC)+2xBxCCov(rB,rC)
xA2σA2+xB2σB2+xC2σC2+2xAxBCov(rA,rB)+2xAxCCov(rA,rC)+2xBxCCov(rB,rC)+8xAxBxCCov(rA,rB)Cov(rB,rC)Cov(rA,rC) {\tiny x_A^2σ_A^2 + x_B^2σ_B^2 + x_C^2σ_C^2 + 2x_Ax_B\mathrm{Cov}(r_A, r_B) + 2x_Ax_C\mathrm{Cov}(r_A, r_C) + 2x_Bx_C\mathrm{Cov}(r_B, r_C) + 8x_Ax_Bx_C\mathrm{Cov}(r_A, r_B)\mathrm{Cov}(r_B, r_C)\mathrm{Cov}(r_A, r_C)} xA2σA2+xB2σB2+xC2σC2+2xAxBCov(rA,rB)+2xAxCCov(rA,rC)+2xBxCCov(rB,rC)+8xAxBxCCov(rA,rB)Cov(rB,rC)Cov(rA,rC)
1+xA2σA2+xB2σB2+xC2σC2+xAxBCov(rA,rB)+xAxCCov(rA,rC)+xBxCCov(rB,rC) {1 + x_A^2σ_A^2 + x_B^2σ_B^2 + x_C^2σ_C^2 + x_Ax_B\mathrm{Cov}(r_A, r_B) + x_Ax_C\mathrm{Cov}(r_A, r_C) + x_Bx_C\mathrm{Cov}(r_B, r_C)} 1+xA2σA2+xB2σB2+xC2σC2+xAxBCov(rA,rB)+xAxCCov(rA,rC)+xBxCCov(rB,rC)
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