Investment portfolio ($H=(H^1,\dots ,H^n)$)

A collection of holdings in traded assets. Its value is the dot product of the holdings with the asset prices. A self-financing portfolio is an investment portfolio whose trading requires no external funding.

Brownian portfolio exposures are the coefficients $y_j$ of independent Brownian motions in the wealth equation. An invertible asset volatility matrix lets one optimize directly over these exposures. The drift risk premium is then the dot product of exposures with the market price of risk vector.

The expected excess return of a portfolio divided by its return standard deviation. Under the capital asset pricing model, the ratio for one asset equals its correlation with the market times the market's ratio.

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.

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.