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