Market portfolio

In a one-period normal-return model with common beliefs and unrestricted exponential utility investors, their risky holdings are proportional to $V^{-1}b$, where $b$ is the expected excess-gain vector and $V$ its positive definite covariance matrix. The aggregate risky holding defines the common market portfolio.

Assume a positive definite covariance matrix and nonzero excess-mean vector $d=\mu -r1$. The Cauchy-Schwarz inequality gives $|d^{T}x|\le \sqrt{d^T{\Sigma }^{-1}d}\sqrt{x^T\Sigma x}$, with the positive-premium efficient direction $x=cz$, $c>0$. If $s=1^{T}z\ne 0$, its normalized risky portfolio has Sharpe ratio $\mathrm{sgn}(s)\sqrt{d^T{\Sigma }^{-1}d}$. Thus negative normalization reverses the efficient orientation. If $s=0$, the efficient risky direction is self-funded and cannot be normalized to a unit risky budget.

In mean-variance optimization with unrestricted borrowing/lending, the upper efficient ray through the risk-free point has slope $\theta =\sqrt{(\mu -r1{)}^T{\Sigma }^{-1}(\mu -r1)}$. If $1^T{\Sigma }^{-1}(\mu -r1)>0$, it touches the risky efficient frontier at the positive-premium normalized market portfolio. The Sharpe ratio there equals the slope. If this normalization is negative, the normalized tangency portfolio instead has negative excess return; the upper ray uses a short position in that portfolio and is not tangent to the upper risky branch.

The expected excess return obeys $\mathbb{E}(R_i-r)={\beta }_i\mathbb{E}(R_M-r)$ in the mean-variance market model. The beta of an asset is a covariance-to-market-variance ratio.

For a nondegenerate market return $R_M$, ${\beta }_i=\mathrm{Cov}(R_i,R_M)/\mathrm{Var}(R_M)$. It measures the asset's market-related exposure and enters the capital asset pricing model.