9 Notation and definitions

Here we list some of the notations used in this book.

9.1 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.

  • \(\mu_i\): 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 \(\mathbf{x}\).

  • \(\mu_\mathbf{x}\): 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.

9.2 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.

9.3 Abbreviations

  • MVO: Mean–variance optimization

  • LO: Linear optimization

  • QO: Quadratic optimization

  • QCQO: Quadratically constrained quadratic optimization

  • SOCO: Second-order cone optimization

  • SDO: Semidefinite optimization