Folio Lab/Documentation

GARCH-Conditional Beta

This page documents GARCH-based beta as related background theory. FolioLab does not currently compute this metric in the live backend, even though the route is kept for completeness and future expansion.

Status

The metric appears in FolioLab's related-work references because it is a well-known approach to time-varying beta estimation, but the current production code does not expose or calculate it. Treat this page as theory documentation rather than an active metric definition.

Overview

Financial return volatility often clusters in time, mean-reverts, and reacts asymmetrically to shocks. The GARCH family of models captures these dynamics by modeling conditional variance as a function of past shocks and past conditional variances.

In a bivariate setup, GARCH models jointly estimate the time-varying variances of and covariance between asset and benchmark returns. The ratio of conditional covariance to conditional benchmark variance yields a time-varying beta that adapts to changing market conditions.

Mathematical Formulation

Univariate GARCH(1,1)

The foundational GARCH(1,1) model specifies conditional variance as:

Here is the constant term, is the ARCH coefficient, and is the GARCH coefficient. Stationarity requires .

Conditional Covariance Matrix

For time-varying beta, we need the conditional covariance matrix between the asset and the benchmark:

where is the asset conditional variance, is the benchmark conditional variance, and is the conditional covariance.

Time-Varying Beta

The GARCH beta at time is:

This is the conditional analog of the static CAPM beta formula, but both the numerator and denominator adapt over time.

DCC Representation

A common parameterization is Dynamic Conditional Correlation (DCC):

where contains conditional standard deviations and is the time-varying correlation matrix.

Advantages & Limitations

Advantages

  • Volatility clustering: It explicitly models the persistence of market volatility.
  • Full-sample efficiency: It uses the whole sample for parameter estimation rather than dropping data in fixed windows.
  • Smooth time variation: It can produce smoother beta paths than rolling-window methods.
  • Forecasting: It supports one-step-ahead beta forecasts.

Limitations

  • Computational complexity: Multivariate GARCH estimation can be slow and numerically fragile.
  • Model specification: Results depend on the chosen GARCH family and can be sensitive to misspecification.
  • Parameter burden: Rich multivariate models introduce many parameters.
  • Inactive in FolioLab: This methodology is not currently exposed by the production engine.

References

  1. Bollerslev, T. (1986). "Generalized Autoregressive Conditional Heteroskedasticity." Journal of Econometrics, 31(3), 307-327.
  2. Engle, R. F. (2002). "Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models." Journal of Business & Economic Statistics, 20(3), 339-350.
  3. Engle, R. F., & Kroner, K. F. (1995). "Multivariate Simultaneous Generalized ARCH." Econometric Theory, 11(1), 122-150.