Bias¶

The objective of a bias test is to examine the forecasting accuracy of volatility on a time series of observed portfolio returns.

First, standardardized returns \(b_t\) are defined as the observed returns \(r_t\) normalised by the volatility forecast \(\sigma_t\) deduced from the risk model

\[ b_t = \frac{r_t}{\sigma_t} \]

where \(r_t\) is the return obtained from the price change in the interval \([t, t+1]\) and \(\sigma_t\) is the volatility forecast of which its risk model is trained / fitted until time \(t\).

Given a rolling window \(T\), the bias statistic is defined as

\[ B_T(t) = \sqrt{\frac{1}{T-1} \sum_{\tau=t-T+1}^{t} (b_{\tau} - \bar{b})^2} \]

The expected bias statistic would be equal to one, while the risk model

overestimates the risk if it is below one, and
underestimates the risk otherwise

Assuming the portfolio returns are normally distributed, the 95% confidence bounds for \(B_T(t)\) are between \(1-\sqrt{\frac{2}{T}}\) and \(1+\sqrt{\frac{2}{T}}\).

Usage¶

Assume that you have market returns returns in a DataFrame, portfolio weights weights in a DataFrame (it could be equally weighted, or market cap weighted), and rolling risk models, which can be a RollingRiskModel object, or just a dictionary of covariances. To compute the bias statistic in a 21-day rolling window, you can use the following code snippet.

from fpm_risk_model.accuracy.bias import compute_bias_statistics

compute_bias_statistics(
  X=returns,
  weights=weights,
  rolling_risk_model=rolling_risk_model,
  window=30,
)

In the meantime, if you have the forecast portfolio volatility, named forecast_vols, from vendor, you can directly pass it into the function as well.

compute_bias_statistics(
  X=returns,
  weights=weights,
  forecast_vols=forecast_vols,
  window=30,
)

Please refer to the below section for more information.

Reference¶

Alexander, Carol (2009). Market risk analysis, value at risk models. John Wiley & Sons.

Module¶

fpm_risk_model.accuracy.bias.compute_standardized_returns(X: DataFrame, weights: DataFrame, rolling_risk_model: Optional[Union[RollingFactorRiskModel, Dict[Any, Any]]] = None, forecast_vols: Optional[Series] = None, cov_halflife: Optional[float] = None) → Series¶

Compute the standardized returns given the rolling risk model.

Standardized return is defined as

\[b_t = \frac{r_t}{\sigma_t}\]

Parameters¶

X: ndarray: The instrument forecast returns.
weights: ndarray: Weights of the instruments.
rolling_risk_model: Union[RollingFactorRiskModel, Dict[Any, Any]]: A rolling risk model object or dictionary of covariances of which the keys and values are dates and covariances.
forecast_vols: Series: The forecast volatility.
cov_halflife: Optional[float]: Halflife in computing covariances.

Returns¶

Series: A timeseries of standardized returns.

fpm_risk_model.accuracy.bias.compute_bias_statistics(X: DataFrame, weights: DataFrame, window: int, rolling_risk_model: Optional[Union[RollingFactorRiskModel, Dict[Any, Any]]] = None, forecast_vols: Optional[Series] = None, min_periods: Optional[int] = None, cov_halflife: Optional[float] = None) → Series¶

Compute the bias statistics.

Standardized return is defined as

\[b_t = \frac{r_t}{\sigma_t}\]

and the bias statistic is expressed as

\[B_T(t) = \sqrt{\frac{1}{T} \sigma_{\tau}(b_{\tau} - \bar{b})^2}\]

Parameters¶

X: ndarray: The instrument forecast returns.
weights: ndarray: Weights of the instruments.
forecast_vols: Optional[Series]: The forecast volatility.
rolling_risk_model: Union[RollingFactorRiskModel, Dict[Any, Any]]: A rolling risk model object or dictionary of covariances of which the keys and values are dates and covariances.
cov_halflife: Optional[float]: Halflife in computing covariances.

Returns¶

Series: A timeseries of bias statistic.