14 Notation and definitions¶
Here we list some of the notations used in this book.
14.1 Financial notations¶
\(N\): Number of securities in the security universe considered.
\(p_{0,i}\): The known price of security \(i\) at the beginning of the investment period.
\(\mathbf{p}_0\): The known vector of security prices at the beginning of the investment period.
\(P_{h,i}\): The random price of security \(i\) at the end of the investment period.
\(P_h\): The random vector of security prices at the end of the investment period.
\(R_i\): The random rate of return of security \(i\).
\(R\): The random vector of security returns.
\(\mathbf{r}\): The known vector of security returns.
\(\mathbf{R}\): The sample return data matrix consisting of security return samples as columns.
\(\mu_i\): The expected rate of return of security \(i\).
\(\mu\): The vector of expected security returns.
\(\Sigma\): The covariance matrix of security returns.
\(x_i\): The fraction of funds invested into security \(i\).
\(\mathbf{x}\): Portfolio vector.
\(\mathbf{x}_0\): Initial portfolio vector at the beginning of the investment period.
\(\tilde{\mathbf{x}}\): The change in the portfolio vector compared to \(\mathbf{x}_0\).
\(x^\mathrm{f}\): The fraction of funds invested into the risk-free security.
\(x_0^\mathrm{f}\): The fraction of initial funds invested into the risk-free security.
\(\tilde{x}^\mathrm{f}\): The change in the risk-free investment compared to \(x_0^\mathrm{f}\).
\(R_\mathbf{x}\): Portfolio return computed from portfolio \(\mathbf{x}\).
\(\mathbf{R}_\mathbf{x}\): Sample portfolio return computed from data matrix \(\mathbf{R}\) and portfolio \(\mathbf{x}\).
\(\mu_\mathbf{x}\): Expected portfolio return computed from portfolio \(\mathbf{x}\).
\(\sigma_\mathbf{x}\): Expected portfolio variance computed from portfolio \(\mathbf{x}\).
\(\EMean\): Estimate of expected security return vector \(\mu\).
\(\ECov\): Estimate of security return covariance matrix \(\Sigma\).
\(T\): Number of data samples or scenarios.
\(h\): Time period of investment.
\(\tau\): Time period of estimation.
14.2 Mathematical notations¶
\(\mathbf{x}^\mathsf{T}\): Transpose of vector \(\mathbf{x}\). Vectors are all column vectors, so their transpose is always a row vector.
\(\mathbb{E}(R)\): Expected value of \(R\).
\(\mathrm{Var}(R)\): Variance of \(R\).
\(\mathrm{Cov}(R_i, R_j)\): Covariance of \(R_i\) and \(R_j\).
\(\mathbf{1}\): Vector of ones.
\(\mathcal{F}\): Feasible region generated by a set of constraints.
\(\R^n\): Set of \(n\)-dimensional real vectors.
\(\integral^n\): Set of \(n\)-dimensional integer vectors.
\(\langle \mathbf{a}, \mathbf{b} \rangle\): Inner product of vectors \(\mathbf{a}\) and \(\mathbf{b}\). Sometimes used instead of notation \(\mathbf{a}^\mathsf{T}\mathbf{b}\).
\(\mathrm{diag}(\mathbf{S})\): Vector formed by taking the main diagonal of matrix \(\mathbf{S}\).
\(\mathrm{Diag}(\mathbf{x})\): Diagonal matrix with vector \(\mathbf{x}\) in the main diagonal.
\(\mathrm{Diag}(\mathbf{S})\): Diagonal matrix formed by taking the main diagonal of matrix \(\mathbf{S}\).
\(\mathcal{F}\): Part of the feasible set of an optimization problem. Indicates that the problem can be extended with further constraints.
\(e\): Euler’s number.
14.3 Abbreviations¶
MVO: Mean–variance optimization
LO: Linear optimization
QO: Quadratic optimization
QCQO: Quadratically constrained quadratic optimization
SOCO: Second-order cone optimization
SDO: Semidefinite optimization
