Risk model covariance¶

For the factor risk model, the covariance is computed by

\[ Q = B \Sigma B^T + \Delta ^ 2 \]

where \(B\) is the factor exposure, \(\Sigma\) is the factor covariance and \(\Delta\) is the specific variance (sample variance of residual returns).

Unmodified¶

The unmodified covariances can be returned from the factor model directly by method cov, and same for correlations with corr.

For example, covariances and correlation on a single factor model can be retrieved as follows.

from fpm_risk_model import FactorModel
factor_model = FactorModel(...)
cov = factor_model.cov()
corr = factor_model.corr()

For rolling factor model, the returns are a dict of covariances with dates as keys.

Covariance estimator¶

Covariance estimators provide advanced features in transforming the factor exposures and returns into pairwise covariances and correlations.

Covariance shrinkage¶

The usual approach to unbiased estimator is a maximum likelihood estimator and the approach converges to the population variance with Law of large numbers.

However, it is not a good estimator of the eigenvalues of the covariance matrix, and especially its inverse. Covariance estimation requires a large sample of data to converge to its true population. Due to curse of dimensionality, the estimate on a smaller set of observations yields to a noisy sample covariance, and even its noisier inverse. Also, it is a general phenomenon of a finite sample data with extreme and possibly noisy returns in financial data.

Covariance shrinkage is a technique to produce a greater signal-to-noise ratio on covariance estimation. Primarily, covariance shrinkage takes a parameter \(\delta\) to suppress its off-diagonal elements on covariance (or actually its correlation) matrix.

\[ \hat{Q} = (1 - \delta) * Q + \delta * \mu * I \]

where \(\mu\) is the mean of all diagonal entries (variances) of \(Q\).

Covariance shrinkage can be specified in the covariance estimator creation. For example, to specify a constant shrinkage with delta equal to 0.2,

from fpm_risk_model import RollingCovarianceEstimator

estimator = RollingCovarianceEstimator(
  rolling_risk_model,
  shrinkage_method="constant",
  delta=0.2
)
estimator.cov()

Volatility adjustments¶

Unlike covariance estimation, volatility estimation does not require as much data and more options, like GARCH, in volatility estimation are available to improve the short-term forecast performance. So it is a common practice to generate the covariance with medium / long-term observations and then adjust the variances (diagonal entries) with estimations from short-term observations.

For example, GARCH volatility estimation is computed as garch_est, and the volatilities can be adjusted with argument volatility in cov method,

estimator.cov(volatility=garch_est)

Reference¶

Ledoit, O., & Wolf, M. (2004). Honey, I Shrunk the Sample Covariance Matrix. The Journal of Portfolio Management, 30(4), 110–119.

Module¶

class fpm_risk_model.cov_estimator.CovarianceEstimator(risk_model: RiskModel, shrinkage_method: Optional[str] = None, delta: Optional[float] = None)¶

Covariance estimator.

__init__(risk_model: RiskModel, shrinkage_method: Optional[str] = None, delta: Optional[float] = None)¶

Constructor.

Parameters¶

risk_modelRiskModel: Risk model object.
shrinkage_methodOptional[str]: Shrinkage method. Options are “constant” and “ledoit_wolf_constant_variance”.
deltaOptional[float]: Delta, only used in constant shrinkage.

corr() → DataFrame¶: Correlation

cov(volatility: Optional[Series] = None, strict: bool = True) → DataFrame¶

Correlation

Parameters¶

volatilitypd.Series: Volaility series to convert from correlation to covariance. Optional.
strictbool: Indicates to throw exception if volatility series does not align with correlation matrix.

static constant_shrinkage(cov: DataFrame, delta: float)¶: Constant shrinkage.

class fpm_risk_model.cov_estimator.RollingCovarianceEstimator(rolling_risk_model: RollingFactorRiskModel, shrinkage_method: Optional[str] = None, delta: Optional[float] = None)¶

Rolling covariance estimator.

__init__(rolling_risk_model: RollingFactorRiskModel, shrinkage_method: Optional[str] = None, delta: Optional[float] = None)¶

Constructor.

Parameters¶

rolling_risk_modelRollingFactorRiskModel: Rolling risk model object.
shrinkage_methodOptional[str]: Shrinkage method. Options are “constant” and “ledoit_wolf_constant_variance”.
deltaOptional[float]: Delta, only used in constant shrinkage.

cov(volatility: Optional[DataFrame] = None)¶

Correlation

Parameters¶

volatilityDataFrame: Volaility series to convert from correlation to covariance. Optional.