1.6 Notation
Notation differs depending on the research area and on the personal taste of the author. This book mainly follows the widely accepted notation in statistics, signal processing, and operations research communities.
To differentiate the dimensionality of quantities, we employ lowercase for scalars, boldface lowercase for (column) vectors, and boldface uppercase for matrices, e.g., \(x\), \(\bm{x}\), and \(\bm{X}\), respectively. The \(i\)th entry of vector \(\bm{x}\) is denoted by \(x_i\) and the \((i,j)\)th element of matrix \(\bm{X}\) by \(X_{i,j}\). The elementwise product (also termed Hadamard product) and elementwise division are denoted by \(\odot\) and \(\oslash\), respectively, e.g., \(\bm{x}\odot\bm{y}\) and \(\bm{x}\oslash\bm{y}\) (\(\bm{x}/\bm{y}\) abusing notation); similarly, the Kronecker product is denoted by \(\otimes\). The transpose of a vector \(\bm{x}\) or a matrix \(\bm{X}\) are denoted by \(\bm{x}^\T\) and \(\bm{X}^\T\), respectively. The inverse, trace, and determinant of matrix \(\bm{X}\) are denoted by \(\bm{X}^{-1}\), \(\textm{Tr}(\bm{X})\), and \(|\bm{X}|\) (or \(\textm{det}(\bm{X})\)), respectively. The norm of a vector is written as \(\|\bm{x}\|\), which can be further specified as the \(\ell_2\)-norm \(\|\bm{x}\|_2\) (also termed Euclidean norm), \(\ell_1\)-norm \(\|\bm{x}\|_1\), and \(\ell_\infty\)-norm \(\|\bm{x}\|_\infty\). The operator \((\bm{x})^+\) denotes the projection onto the nonnegative orthant, i.e., \((\bm{x})^+\triangleq\textm{max}(\bm{0},\bm{x})\). We denote by \(\bm{I}\) the identity matrix of appropriate dimensions.
For random variables, \(\textm{Pr}[\cdot]\) denotes probability, and the operators \(\E[\cdot]\), \(\textm{Std}[\cdot]\), \(\textm{Var}[\cdot]\), and \(\textm{Cov}[\cdot]\) denote expected value, standard deviation, variance, and covariance matrix, respectively.
The set of real numbers is denoted by \(\R\) (nonnegative real numbers by \(\R_{+}\) and positive real numbers by \(\R_{++}\)). The set of \(m\times n\) matrices is denoted by \(\R^{m\times n}\); the set of symmetric \(n\times n\) matrices by \(\mathbb{S}^{n}\), and the set of positive semidefinite \(n\times n\) matrices by \(\mathbb{S}^{n}_+\). By \(\bm{a}\geq\bm{b}\) we denote elementwise inequality (i.e., \(a_i \geq b_i\)). The matrix inequalities \(\bm{A}\succeq\bm{B}\) and \(\bm{A}\succ\bm{B}\) denote that \(\bm{A}-\bm{B}\) is positive semidefinite and positive definite, respectively. The indicator function is denoted by \(1\{\cdot\}\) or \(I(\cdot)\).
Table 1.1 lists the most common abbreviations used throughout the book, whereas Table 1.2 provides some key financial mathematical symbols.
| Abbreviation | Meaning |
|---|---|
| AI | Artificial intelligence |
| AR | Autoregressive |
| ARCH | Autoregressive conditional heteroskedasticity |
| ARIMA | Autoregressive integrated moving average |
| ARMA | Autoregressive moving average |
| B&H portfolio | Buy and hold portfolio |
| BCD | Block coordinate descent |
| CAPM | Capital asset pricing model |
| CCC | Constant conditional correlation |
| CP | Conic problem/program |
| CVaR | Conditional value at risk |
| DCC | Dynamic conditional correlation |
| DD | Drawdown |
| DL | Deep learning |
| DR | Downside risk |
| ES | Expected shortfall |
| EWMA | Exponentially weighted moving average |
| EWP | Equally weighted portfolio (a.k.a. \(1/N\) portfolio) |
| FP | Fractional problem/program |
| FX | Foreign exchange |
| GARCH | Generalized autoregressive conditional heteroskedasticity |
| GICS | Global Industry Classification Standard |
| GMRP | Global maximum return portfolio |
| GMVP | Global minimum variance portfolio |
| GP | Geometric problem/program |
| HRP | Hierarchical risk parity |
| i.i.d. | independent and identically distributed |
| IPM | Interior-point method |
| IVarP | Inverse variance portfolio |
| IVolP | Inverse volatility portfolio |
| LFP | Linear fractional problem/program |
| LP | Linear problem/program |
| LPM | Lower partial moment |
| LS | Least squares |
| MA | Moving average |
| MDCP | Maximum decorrelation portfolio |
| MDP | Most diversified portfolio |
| ML | Maximum likelihood or machine learning (depending on context) |
| MM | Majorization-minimization |
| MSRP | Maximum Sharpe ratio portfolio |
| MVolP | Mean volatility portfolio |
| MVP | Mean variance portfolio |
| MVSK | Mean-variance-skewness-kurtosis |
| NAV | Net asset value |
| P&L | Profit and loss |
| QCQP | Quadratically-constrained quadratic problem/program |
| QP | Quadratic problem/program |
| QuintP | Quintile portfolio |
| RPP | Risk parity portfolio |
| S&P 500 | Standard & Poor’s 500 |
| SCA | Successive convex approximation |
| SDP | Semidefinite problem/program |
| SOCP | Second-order cone problem/program |
| SR | Sharpe ratio |
| SV | Stochastic volatility |
| TE | Tracking error |
| VaR | Value at risk |
| VARMA | Vector autoregressive moving average |
| VECM | Vector error correction model |
| Term | Meaning |
|---|---|
| \(\bm{w}\) | Normalized portfolio weight vector. |
| \(\bm{w}^\textm{cap}\) | Portfolio capital allocation vector (e.g., in units of US dollar). |
| \(\bm{w}^\textm{units}\) | Portfolio unit allocation vector (e.g., in units of shares for stocks). |
| \(\bm{p}_t\) | Price vector of assets at time \(t\). |
| \(\bm{y}_t\) | Log-price vector of assets at time \(t\). |
| \(\bm{r}_t\) (\(\bm{x}_t\)) | Return vector of assets at time \(t\) (linear or log-returns, depending on context). |
| \(\bm{r}^{\textm{lin}}_t\) | Linear returns vector of assets at time \(t\). |
| \(\bm{r}^{\textm{log}}_t\) | Log-returns vector of assets at time \(t\). |
| \(\bmu_t\) | Vector of expected value of returns \(\bm{r}_t\). |
| \(\bSigma_t\) | Covariance matrix of returns \(\bm{r}_t\). |
| \(N\) | Number of financial assets in the considered universe. |
| \(T\) | Number of temporal observations \(t=1,\dots,T\). |
| \(\mathcal{N}(\bmu, \bSigma)\) | Normal or Gaussian multivariate distribution with mean \(\bmu\) and covariance \(\bSigma\). |