Rank-one covariance

struct RankOneCovariance{T1<:Real} <: AbstractCovarianceEstimator

Estimates the covariance of a short window as a rank-one matrix along the principal direction of its price relatives.

Lai, Tan, Wu and Fang (2020) built the estimate for short-term portfolio optimisation, where the window holds a few observations of many assets. The matrix is positive semidefinite and singular, so it has no Cholesky factor. A consumer reads it as a quadratic form, as Variance does under QuadRiskExpr.

Mathematical definition

\[\begin{align} \mathbf{X}^\intercal \mathbf{X} \boldsymbol{u}_1 &= \theta_1 \boldsymbol{u}_1\,, \\ \hat{\mathbf{\Sigma}}_{\mathrm{MP}} &= \mathbf{X}^\intercal \left( \mathbf{I} - \frac{1}{w} \boldsymbol{1} \boldsymbol{1}^\intercal \right) \mathbf{X}\,, \\ \zeta_1^\star &= \theta_1 \left( \frac{\operatorname{tr}(\hat{\mathbf{\Sigma}}_{\mathrm{MP}})}{N (w - 1)} \right)^{-1/2}\,, \\ \hat{\mathbf{\Sigma}}_{\mathrm{RO}} &= \zeta_1^\star \boldsymbol{u}_1 \boldsymbol{u}_1^\intercal\,. \end{align}\]

Where:

• $\mathbf{X}$: $w \times N$ window of price relatives, one row per observation, not centred.
• $w$: Number of observations in the window.
• $N$: Number of assets.
• $\theta_1$: Largest eigenvalue of $\mathbf{X}^\intercal \mathbf{X}$, the square of the largest singular value of $\mathbf{X}$.
• $\boldsymbol{u}_1$: Unit eigenvector of $\mathbf{X}^\intercal \mathbf{X}$ at $\theta_1$, the principal right singular vector of $\mathbf{X}$.
• $\hat{\mathbf{\Sigma}}_{\mathrm{MP}}$: Scatter matrix of the window, the sum of the outer products of its centred rows.
• $\mathbf{I}$: Identity matrix.
• $\zeta_1^\star$: Energy of the estimate along $\boldsymbol{u}_1$.
• $\hat{\mathbf{\Sigma}}_{\mathrm{RO}}$: Rank-one covariance estimate of a window, the matrix that RankOneCovariance computes.

This is Algorithm 1 of the paper. Its equations 42 and 49 write the trace of $\hat{\mathbf{\Sigma}}_{\mathrm{MP}}$ in the basis of the singular vectors of $\mathbf{X}$, as $\operatorname{tr}(D)$, which is the same number. The quotient $\operatorname{tr}(\hat{\mathbf{\Sigma}}_{\mathrm{MP}}) / (N (w - 1))$ is the mean of the $N$ sample variances of the window. So $\zeta_1^\star$ is $\theta_1$ over the root of the mean variance. That value minimises the trade-off of equation 48 between the magnitude of $\theta_1 \boldsymbol{u}_1 \boldsymbol{u}_1^\intercal$ and the magnitude of $\hat{\mathbf{\Sigma}}_{\mathrm{MP}}$.

Equation 47 divides the trace of a matrix by its rank, and equation 49 puts the rank of $\hat{\mathbf{\Sigma}}_{\mathrm{MP}}$ at $w - 1$. That is the rank of a short window, $w \leq N + 1$, which is the case the paper studies. On a longer window the rank is at most $N$, but the definition keeps $w - 1$, as Algorithm 1 and the authors' code do. When every row of the window is the same, $\operatorname{tr}(\hat{\mathbf{\Sigma}}_{\mathrm{MP}}) = 0$ and $\zeta_1^\star$ is not defined.

Fields

• shift: Number added to every entry of the returns to form the price relatives $\mathbf{X}$. The default 1 turns returns into the price relatives that the paper reads, and 0 decomposes the rows as given.

Constructors

RankOneCovariance(; shift::Real = 1) -> RankOneCovariance

Keywords correspond to the struct's fields.

Validation

• isfinite(shift). A DomainError is thrown otherwise.

Examples

julia> RankOneCovariance()RankOneCovariance  shift ┴ Int64: 1

Related

• AbstractCovarianceEstimator
• ShortTermLossControlPortfolio: the programme that reads the estimate.
• Variance
• QuadRiskExpr

References

• [64] Z.-R. Lai, L. Tan, X. Wu and L. Fang. Loss control with rank-one covariance estimate for short-term portfolio optimization. Journal of Machine Learning Research 21, 1–37 (2020). Algorithm 1 and equations 42 to 49.

Statistics.cov(ce::RankOneCovariance, X::MatNum; dims::Int = 1, kwargs...)

Computes the rank-one covariance of a window of returns, assets × assets.

RankOneCovariance states the mathematics.

Algorithm

1. Orient X to observations × assets with dims_oriented.
2. Add ce.shift to every entry of X to form the price relatives Xs.
3. Subtract the column means from Xs, and sum the squares of the result. The sum trD is the trace of the scatter matrix.
4. Compute the singular value decomposition of Xs. Take theta, the square of the largest singular value, and u, its right singular vector.
5. Compute zeta = theta * sqrt(N * (w - 1) / trD). When every row of Xs is the same, or when trD is zero, zeta is zero. The method compares the rows, because the column means of equal rows carry round-off, and trD of a constant window is often a tiny positive number.
6. Return zeta .* (u .* transpose(u)). The method forms the outer product entry by entry, so the matrix is symmetric to the bit.

Arguments

• ce: The rank-one covariance estimator.
• X: Data matrix observations × assets if the dims keyword does not exist or dims = 1, assets × observations when dims = 2.
• dims: Dimension along which to perform the computation.
• kwargs...: Additional keyword arguments. The method ignores them, because the estimate reads no mean and no observation weights.

Validation

• dims in (1, 2).
• size(X, 1) >= 2 after orientation. An ArgumentError is thrown otherwise, because one row has no centred energy.

Returns

• sigma::Matrix: The rank-one covariance matrix, positive semidefinite and symmetric. It is the zero matrix when every row of the window is the same.

Related

• RankOneCovariance

Statistics.cor(ce::RankOneCovariance, X::MatNum; dims::Int = 1, kwargs...)

Computes the correlation matrix of the rank-one covariance of a window of returns, assets × assets.

Mathematical definition

\[\begin{align} \rho_{ij} &= \frac{\hat{\Sigma}_{ij}}{\sqrt{\hat{\Sigma}_{ii} \hat{\Sigma}_{jj}}} = \operatorname{sign}(u_{1i} u_{1j})\,. \end{align}\]

Where:

• $\rho_{ij}$: Correlation between assets $i$ and $j$.
• $\hat{\Sigma}_{ij}$: Entry of $\hat{\mathbf{\Sigma}}_{\mathrm{RO}}$ in row $i$ and column $j$.
• $u_{1i}$: Entry $i$ of the principal vector $\boldsymbol{u}_1$ of RankOneCovariance.

The second equality holds when $\zeta_1^\star > 0$ and both assets have a non-zero entry in $\boldsymbol{u}_1$. Otherwise the variance of one of the two assets is zero, and $\rho_{ij}$ is not defined.

Algorithm

1. Compute sigma with the cov method of RankOneCovariance.
2. Take the sign of every entry of sigma.
3. Set the diagonal to one.

Arguments

• ce: The rank-one covariance estimator.
• X: Data matrix observations × assets if the dims keyword does not exist or dims = 1, assets × observations when dims = 2.
• dims: Dimension along which to perform the computation.
• kwargs...: Additional keyword arguments. The method passes them to cov, which ignores them.

Validation

• Every check of the cov method applies.

Returns

• rho::Matrix: The correlation matrix, with ones on the diagonal and $\pm 1$ off it. An entry whose correlation is not defined is zero.

Related

• RankOneCovariance