Principles of GARCH
How to Generate Covariance Matrices
The generation of large, positive semi-definite covariance matrices that properly reflect market conditions has been a challenge for finance practitioners for several years. Since the 1996 Amendment to the 1988 Basle accord, where the principals of internal models for the calculation of market risk capital were outlined, it has been a major problem to generate the covariance matrices that are necessary to calculate firm-wide Value-at-Risk estimates. In very large portfolios a risk factor model may be employed, but it is still necessary to have a covariance matrix for all the risk factors of the portfolio.
Of the three methods that are in standard use (the covariance method for linear portfolios, and Monte Carlo and historical simulation) only the historical simulation method requires no covariance matrix. And even with historical simulation, covariance matrices are normally used in stress testing and scenario analysis. The need for large covariance matrices is not just confined to the middle office of a large investment bank. Traders in the front office also require these matrices to price and hedge complex option portfolios. Large covariance matrices also have a major role to play in investment analysis, because portfolio risk is determined by the covariance matrix of all the assets in the portfolio.
The RiskMetrics Methodology
JP Morgan and Reuters have risen to this challenge by providing the RiskMetricsTM data sets. JP Morgan launched the first version of RiskMetrics in October 1994, and over the course of the next two years have made several improvements to the methodology and the data. The data consist of three large covariance matrices of the returns to major FX rates, money market rates, equity indices, bonds and some key commodities. The statistical methodology for calculating these covariance matrices, and a description of how the data should be implemented in Value-at-Risk models is given in the highly informative RiskMetrics Technical Document (4th ed. December 1996).
There are some disadvantages with these data. One problem with the two exponentially weighted matrices is that the same smoothing constant has to be used for all assets, otherwise the matrix would not be positive semi-definite. The smoothing constant in an exponentially weighted moving average corresponds to the persistence in volatility and correlation estimates; they have been set at 0.94 for the daily matrix and 0.97 for the monthly matrix for every market return. However the volatility and correlation persistence characteristics are quite different for different markets; they also change over time. For example, in some of the major equity markets GARCH models indicate that volatility has become less persistent during the last few years and current daily data on the S&P500 indicates a volatility persistence of less than 0.9.
A problem with the equally weighted matrix is that it suffers from 'ghost effects' of extreme events that occurred any time during the past 250 days. Thus a market crash, such as that occurring after September 11th 2001, will have exactly the same effect on volatility and correlation estimates at the beginning of September 2002 as it did at the end of September 2001.
The Orthogonal Alternative
Principal component analysis (PCA) may be used to construct large covariance matrices that have many desirable characteristics. The method is computationally very simple: it takes the univariate volatilities of the first few principal components of a system of risk factors and the factor weights matrix of the principal components representation to produce a full covariance matrix for the original system. A schematic of the method is shown in the figure below. Here the matrix A is a matrix of re-scaled factor weights and the diagonal matrix D is a matrix of either generalized autoregressive conditional heteroscedasticity (GARCH) or exponentially weighted moving average (EWMA) variances of the principal components.
The key behind the success of the orthogonal method is to choose a reduced set of principal components; if there are n returns in the full covariance matrix, only k << n principal components are used. The higher principal components are not included because they are assumed to be picking up only the unnecessary noise in the system; thus correlation estimates in particular become more stable over time. Another key to successful calibration of the model is to divide the data into relatively highly correlated blocks before the principal component analysis is applied. This would normally entail division of the risk factors into country and product categories, and/or industry or credit categories. Full description of the method is beyond the scope of this article: further details may be found in the references.
There are many:
- The full covariance matrix is always positive semi-definite
- Very few constraints are imposed on the movements in volatility and correlation. In particular, it is not necessary to impose the constraint that all volatility and correlation estimates have the same reaction and persistence to market shocks. Instead, these characteristics of an asset or risk factor are determined by its correlation in the system.
- The computational burden is much lighter; all of the n(n+1)/2 variances and covariances are simple linear transformations of just k EWMA or GARCH variances, and in many systems k will be of the order of 2 or 3
- The method allows one to implement multivariate GARCH for large systems, something that has previously been beyond the scope of any multivariate GARCH calibration.
- If the method is applied with GARCH, rather than EWMA variances for the principal components, one can generate term structures of covariance matrix term structures that are mean reverting. That is, there is no necessity to apply the 'square-root-of-time' rule and assume that volatilities and correlations are constant over all risk horizons. Instead the usual GARCH analytic formulae for computing the term structure of volatility and correlation are applied so that the h-day covariance matrix forecast converges to the long term average as h increases
- Data may be difficult to obtain directly, particularly on new issues or on financial assets that are not heavily traded. When data are sparse or unreliable on some of the variables in the system a direct estimation of volatilities and correlation may be very difficult. However, provided there is sufficient information to infer their factor weights in the principal component representation, their volatilities and correlations may be obtained using the orthogonal method
- By using only the first few principal components to represent the system, the correlations estimates become more stable and less influenced by variation that would be better ascribed to 'noise' in the data.
I predict that this method will soon become the industry standard for generating large, positive semi-definite covariance matrices. The model can be implemented very easily and commercial software for orthogonal GARCH has been available from Algorithmics Inc. since 1997.
Alexander, C. and A. Chibumba (1996) "Multivariate Orthogonal Factor GARCH" University of Sussex Discussion Papers in Mathematics
Alexander, C. (2000) "Orthogonal Methods for Generating Large Positive Semi-Definite Covariance Matrices" ISMA Centre Discussion Papers in Finance 2000-06 available from www.ismacentre.rdg.ac.uk
Alexander, C. (2001) Market Models: a Guide to Financial Data Analysis, John Wiley and Sons
Alexander, C. (2001) Orthogonal GARCH, in C. Alexander (ed.), Mastering Risk Volume 2.Financial Times – Prentice Hall, pp 21-38