quantlab.risk.bias#

Bias statistics: whether a risk model’s volatility forecasts are calibrated.

Follows Menchero, Orr and Wang, The Barra US Equity Model (USE4), Methodology Notes (MSCI, 2011), Appendix A. For a portfolio n with return R_nt over the bar after t and volatility forecast sigma_nt made at t, the standardized outcome b_nt = R_nt / sigma_nt (A1) has a standard deviation of 1 when the forecasts are right. The bias statistic is the realized standard deviation of the outcomes over a window of T bars (A2, about their mean, T - 1 in the denominator); for normal returns and right forecasts about 95 percent of bias statistics fall within 1 +/- sqrt(2/T) (A3). Above the band the model underpredicts risk, below it overpredicts. The rolling bias statistic is the same over a window rolled one bar at a time (A4); across portfolios, its mean (A5), 5th and 95th percentiles and mean absolute deviation from 1 (MRAD, A6) summarise each window.

bias_statistics computes these from any realized returns and forecasts. risk_model_bias_statistics reads them from a factor risk model’s stores, over one bar or a horizon of several (USE4 tests monthly), the outcome then the sum of the bars’ returns and the forecast sqrt(h) times the one-bar one:

  • factor: each factor’s return over the next bar, the return of its pure factor portfolio, against the square root of its forecast variance;

  • eigenfactor: the eigenfactors of each bar’s forecast covariance (USE4 Methodology Notes §4.2, Figure 4.1), numbered from the lowest variance: the next bar’s return of each, the factor returns weighted by its unit eigenvector, against the square root of its eigenvalue;

  • specific: each symbol’s specific return over the next bar against its forecast specific risk;

  • random: random active portfolios as in the USE4 Empirical Notes (§5, Figure 5.4): portfolio_size randomly selected estimation-universe stocks, capitalization weighted, less the capitalization-weighted estimation universe.

Functions

bias_statistics

Return the bias statistics of volatility forecasts against realized returns.

risk_model_bias_statistics

Return the bias statistics of a factor risk model's forecasts from its stores.