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.

Spreading a portfolio across assets can reduce risk that is not shared by them. For centered asset risks with covariance matrix $\Gamma$, the variance of the weighted risk is $w^T\Gamma w$. Equal weights across $n$ independent risks with common variance $s^2$ give variance $s^2/n$, whereas equal weights across risks that are all the same random variable leave variance $s^2$. Thus small individual weights do not by themselves remove common risk; the covariance criterion for diversification of factor residuals identifies sufficient dependence and weight conditions.

Selling a borrowed asset and later buying it back to return it. A negative portfolio weight represents a short position. The position benefits from falling prices and loses from rising prices; its sale proceeds may be invested in the bank account. Unrestricted short positions are a modelling assumption, not a guarantee of unlimited practical borrowing.

The expected value of a random financial return $R$. For asset return vector $R$ with mean $\mu$, a portfolio of weights $w$ has expected return $w^T\mu$ by linearity of expectation. This measures average payoff growth rather than a guaranteed realized gain.

The value of an investment portfolio, obtained by summing each holding times its current asset price. Consumption removes money from portfolio wealth; a self-financing portfolio changes its value only through asset gains. Nonnegative portfolio wealth is a common admissibility restriction in an investment-consumption problem.

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.

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.