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.

• $\mathrm{\Sigma }$: 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 $\mathbf{x}$.

• $\mathit{\mu }$: Estimate of expected security return vector $\mu$.

• $\mathbf{\Sigma }$: Estimate of security return covariance matrix $\mathrm{\Sigma }$.

• $T$: 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.

• ${\mathbb{R}}^n$: Set of $n$-dimensional real vectors.

• ${\mathbb{Z}}^n$: Set of $n$-dimensional integer vectors.

• $⟨\mathbf{a},\mathbf{b}⟩$: 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