Factor Models

FactorModel provides a low-rank decomposition of the asset covariance matrix and serves as the covariance estimator in sliding-window mode (SlidingWindowConfig). It is also available for standalone low-rank covariance inspection.

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Mathematical background

The factor risk model represents a covariance matrix as:

Σ = B · F · Bᵀ + D

where:

Term	Shape	Meaning
B	n × k	Factor loading matrix — column j gives each asset's sensitivity to factor j
F	k × k	Factor covariance matrix (positive definite)
D	n × n (diagonal)	Per-asset idiosyncratic variance (strictly positive)

Fitting proceeds by truncated SVD of the vol-adjusted return matrix R ∈ ℝ^(T×n):

Ĉ = (1/T) · V_k · Σ_k² · V_kᵀ + D̂

where V_k (n×k) contains the top-k right singular vectors and Σ_k is the diagonal of the top-k singular values. D̂ is set so that Ĉ has unit diagonal.

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Quick example

from basanos.math import FactorModel
import numpy as np

# Fit from a return matrix (T×n) using truncated SVD
returns = np.random.default_rng(0).normal(size=(200, 5))
fm = FactorModel.from_returns(returns, k=2)

fm.n_assets    # 5
fm.n_factors   # 2
fm.covariance  # reconstructed (5, 5) covariance matrix

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FactorModel reference

Attributes

Attribute	Shape	Description
factor_loadings	(n, k)	Factor loading matrix B; column j gives each asset's sensitivity to factor j
factor_covariance	(k, k)	Factor covariance matrix F (positive definite)
idiosyncratic_var	(n,)	Per-asset idiosyncratic variance (strictly positive)

Properties and methods

Name	Returns	Description
n_assets	int	Number of assets n
n_factors	int	Number of factors k
covariance	np.ndarray (n, n)	Reconstructed full covariance matrix Σ = B·F·Bᵀ + D
from_returns(returns, k)	FactorModel	Class method; fits the model from a (T, n) return matrix via truncated SVD

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Using FactorModel in BasanosEngine

Pass a SlidingWindowConfig to switch the engine to factor-model covariance:

from basanos.math import BasanosConfig, BasanosEngine, SlidingWindowConfig

cfg = BasanosConfig(
    vola=16,
    corr=32,    # unused in sliding_window mode but still required
    clip=3.5,
    shrink=0.5, # unused in sliding_window mode
    aum=1e6,
    covariance_config=SlidingWindowConfig(
        window=60,    # rolling window length W; rule of thumb: W >= 2 * n_assets
        n_factors=2,  # number of latent factors k; fewer = stronger regularisation
    ),
)

engine = BasanosEngine(prices=prices, mu=mu, cfg=cfg)

At each timestamp the engine fits a rolling FactorModel over the last window rows and solves Ĉ · x = μ via the Woodbury identity, reducing the per-step cost from O(n³) to O(k³ + kn).

For guidance on choosing window and n_factors see Concepts — Mode 2.

For the full API documentation see Risk Models & Helpers.