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Mathematical Appendix

Complete mathematical derivations, definitions, and academic citations for all algorithms used in the BRISMA Example 1 pipeline.

Key Equations

Empirical Covariance: $$ \hat{\Sigma}_{\text{emp}} = \sum_{t=1}^{T} w_t \, r_t r_t^\top $$
Weighted Covariance: $$ Q^{(k+1)}_{\text{emp}} = \sum_{t=1}^{T} w_t^{(k)} \, r_t r_t^\top $$
Time Decay Weights: $$ w_t^{(0)} = \frac{t}{\sum_{s=1}^{T} s} = \frac{2t}{T(T+1)} $$
Garch Weight Update: $$ w_t^{(k+1)} \propto \left(\frac{t}{T}\right)^{1-\gamma} \cdot \left(\frac{1}{\hat{\sigma}_t^{(k)}}\right)^{\gamma} $$
Garch Weight Strength: $$ \gamma = \left(\frac{(\sum w_t)^2 / \sum w_t^2}{T}\right)^2 $$
Eigendecomposition: $$ Q_{\text{rm}} = V \Lambda V^\top $$
Component Rotation: $$ Q_{\text{comp}} = R^\top Q_{\text{rm}} R $$
Scaling Rotation: $$ R_{i,\text{scaled}} = \frac{0.01}{\sqrt{Q_{\text{rm}}[i,i]}} $$
Beta Estimation: $$ \beta = Q_{\text{port,comp}} \cdot Q_{\text{comp,comp}}^{-1} $$
Fitted Covariance: $$ Q_{\text{fit}} = \beta^\top Q_{\text{comp}} \beta $$
Residual Covariance: $$ Q_{\text{res}} = \text{diag}\left(\text{diag}(Q_{\text{emp}} - Q_{\text{fit}})\right) $$
Shrinkage Covariance: $$ Q_{\text{shrink}} = Q_{\text{fit}} + Q_{\text{res}} = \beta^\top Q_{\text{comp}} \beta + \text{diag}(\sigma_{\epsilon}^2) $$
Variance Threshold: $$ n_{\text{comp}} = \min\left\{ k : \frac{\sum_{i=1}^{k} \lambda_i}{\sum_{i=1}^{n} \lambda_i} \geq 0.95 \right\} $$
Garch Model: $$ \sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 $$
Garch Forecast: $$ \sigma_{t+h}^2 = \omega \cdot \frac{1 - (\alpha + \beta)^h}{1 - (\alpha + \beta)} + (\alpha + \beta)^h \sigma_t^2 $$
Garch Long Run: $$ \bar{\sigma}^2 = \frac{\omega}{1 - \alpha - \beta} $$
Garch Covariance: $$ Q_{\text{garch}} = \beta^\top \text{diag}(\sigma_{\text{comp}}^2) \beta + \text{diag}(\sigma_{\text{resid}}^2) $$
Log Returns: $$ r_t = \ln\left(\frac{P_t}{P_{t-1}}\right) $$
Rolling Returns: $$ r_{t,22} = \sum_{s=0}^{21} r_{t-s} = \ln\left(\frac{P_t}{P_{t-22}}\right) $$
Annualization: $$ \sigma_{\text{annual}} = \sigma_{\text{daily}} \cdot \sqrt{252} $$
Currency Conversion: $$ P_{\text{base},t} = P_{\text{local},t} \cdot S_t $$
Excess Return: $$ P_{\text{excess},t} = \frac{P_{\text{base},t}}{P_{\text{cash},t}} $$
Condition Number: $$ \kappa(Q) = \frac{\lambda_{\max}}{\lambda_{\min}} $$
Symmetry Check: $$ \|Q - Q^\top\|_{\max} < \epsilon $$
Psd Check: $$ \lambda_i \geq 0 \quad \forall i $$
Portfolio Variance: $$ \sigma_p^2 = w^\top Q w $$
Portfolio Volatility: $$ \sigma_p = \sqrt{w^\top Q w} $$

Definitions

Covariance Matrix $\Sigma$ or $Q$

A symmetric positive semi-definite matrix where element $\Sigma_{ij}$ represents the covariance between assets $i$ and $j$. Diagonal elements are variances.

Properties:

Eigenvalue Decomposition $Q = V \Lambda V^\top$

Factorization of a symmetric matrix into its eigenvectors $V$ and diagonal eigenvalue matrix $\Lambda$. For covariance matrices, eigenvalues represent variance explained by each principal component.

Properties:

GARCH(1,1) $\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2$

Generalized Autoregressive Conditional Heteroskedasticity model that captures volatility clustering in financial returns. The conditional variance depends on past squared returns and past variances.

Properties:

Shrinkage Covariance Estimator $Q_{\text{shrink}}$

A regularized covariance estimator that combines a structured estimator (factor model) with the sample covariance to reduce estimation error. Shrinks sample covariance toward a structured target.

Properties:

Factor Model $r = \beta f + \epsilon$

A model where asset returns are explained by exposure (beta) to common factors $f$ plus idiosyncratic noise $\epsilon$. Reduces dimensionality of covariance estimation.

Properties:

Principal Components $PC_k = V_k^\top r$

Linear combinations of original variables that capture maximum variance. The k-th principal component is the projection onto the k-th eigenvector. PCA is used to identify latent risk factors.

Properties:

Algorithms

Iterative Covariance Estimation

An iterative algorithm that combines time-decay weighting with GARCH-based volatility adjustments to estimate a more responsive covariance matrix.

Complexity: $O(T \cdot n^2)$ per iteration

Factor Model Estimation

Estimate a factor model covariance matrix by decomposing returns into systematic (factor) and idiosyncratic components.

Complexity: $O(n^2 k + k^3)$

GARCH(1,1) Volatility Forecasting

Forecast conditional volatility using GARCH(1,1) model, which captures volatility clustering and mean-reversion in financial returns.

Complexity: $O(T)$ for estimation, $O(H)$ for forecasting

References

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