 Var（rp）= w1 2 Var（r1）+ w2 2 Var（r2）+ 2w1w2 Cov（r1r2）

（1）式中的投资组合ω是独立的风险最优均值 – 方差有效率，其中n是有风险证券的回报向量R-r的预期超额回报率，n是有风险证券收益率之间的非奇异协方差矩阵Ω。当且仅当ω=Ω-1（R-r）/ [eTΩ-1（R-r）]时，均方差框架“

“在给定风险证券收益之间的期望收益率向量R-r和非奇异协方差矩阵Ω的情况下，当且仅当市场投资组合ωm=Ω-1（R-r）/ [eTΩ-1 R-R）]”

Define and describe the efficient frontier in CAPM. Discuss proposition 1 and 3 on efficient portfolios and CAPM. List basic assumptions, main results and intuitions of the model
In the risk-return space each possible combination of assets can be plotted and the collection of plotted portfolios is defined by the region in this space. Efficient frontier is referred to the line along upper edge of the region that includes plot of portfolios. All the combinations that lie along this line are known as portfolios that exclude those having risk free alternative. These portfolios offer lowest risk for a fixed level of return. In other words it can be said that efficient frontier offers best possible return for a given level of risk (Jagannathan & Wang, 1996).
In mathematical terms the efficient frontier is obtained by the intersection of set of portfolios having a minimum variance that represents a set of portfolio having maximum return.
Var(rp) = w1 2 Var(r1) + w2 2 Var(r2) + 2w1w2 Cov(r1r2)
Proposition 1
“Given the expected excess rate of return vector R–r on n risky securities and the non–singular covariance matrix Ω between n risky securities rate of returns, the portfolio ω in equation (1) is the unique risky optimal mean–variance efficient within the mean–variance framework if and only if ω = Ω–1(R–r)/[eTΩ–1(R–r)]”
Proposition 2
“Given the expected excess rate of return vector R–r and non–singular covariance matrix Ω between the risky securities returns, CAPM holds if and only if the market portfolio ωm = Ω–1(R–r)/[eTΩ–1(R–r)]”