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72

for some time after a market shock. If an average value of 1 has to be taken for all markets it may be better if it were slightly lower.19 To provide VaR measures of holding periods greater than 1 day it is possible to follow the BIS recommendations and use the square root of time rule (§3.3) and assume that current levels of volatility and correlation persist forever. But the EWMA method is not really suitable for forecasts of more than a few days, so 10-day VaR measures that are obtained using the RiskMetrics daily data can be substantially overestimated (or indeed underestimated, depending on the construction of the portfolio). The 1-day matrix does not conform to the current international regulators quantitative standards for internal VaR models, so it may not be admissible to base market risk capital reserves on VaR measures produced with this matrix. Regulators are requiring that at least one year (about 250 trading days) of historical data be used to construct the covariance matrices. Unfortunately, a value of X - 0.94 in an EWMA means that only a few days of data are used. For example 0.9450 is less than 0.05 and 0.94224 is approximately zero.20

The 1-day matrix does not conform to the current international regulators quantitative standards for internal VaR models

Portfolio market risk estimates relevant for a 1-month (25-day) horizon could be based purely on the 1-day covariance matrix. Assuming returns are independent and identically distributed and using the square-root-of-time rule, a covariance matrix for measuring portfolio risk over the next 25 days is obtained by multiplying every element in the 1-day covariance matrix by 25. But the EWMA methodology is only really applicable to very short-term forecasting, assuming as it does that volatility and correlation are constant.

Thus RiskMetrics have produced different data for the 1-month covariance matrix. If the EWMA methodology is to be applied, one approach would be to apply exponential smoothing to 25-day (monthly) returns. However, there is not enough historical data to base covariance matrix on non-overlapping monthly returns, so instead RiskMetrics apply exponential smoothing to the daily squared and cross-products of returns with a smoothing constant X - 0.97 and multiply the resulting variance and covariance estimates by 25.

The same smoothing constant has to be used to calculate all volatilities and correlations in the RiskMetrics 1-day and 1-month covariance matrices, otherwise they would not necessarily be positive semi-definite. However, the RiskMetrics regulatory matrix is based on equally weighted moving averages over the past year of data and so needs no constraints to be positive definite (§7.1.3). Following the discussion in §3.1, it is clear that one large return will continue to keep the regulatory volatility estimates high for exactly one

19 The persistence parameters from GARCH(1,1) models that are tailored to daily data on each individual risk factor are rarely as high as 0.94.

20 What is zero depends on the tolerance level used for calculations. In RiskMetrics the tolerance level is 10-6, but if a higher tolerance were set then more historic data would be used.



The regulatory correlation estimates will appear too stable for the whole year following a single event that affects both markets

year, and exactly one year after a major market event the equally weighted volatility estimate will jump down again as abruptly as it jumped up. By the same token, the regulatory correlation estimates will appear too stable for the whole year following a single event that affects both markets. These ghost features in the regulatory covariance matrix that will certainly follow large market movements are going to bias VaR measures for a whole year, but the direction of the bias is not always easy to determine as it depends on the portfolio construction.

7.4 Orthogonal Methods for Generating Covariance Matrices

There is an alternative: to apply computations to only a few key market risk factors that capture the most important uncorrelated sources of information in the data, that is, to the principal components

Many covariance matrices, including the RiskMetrics matrices, are obtained by applying computations to the full set of assets (or risk factors). But the dimensions of the problem are normally so large that the problem is intractable. That is why restrictive assumptions often need to be made. However, there is an alternative: to apply computations to only a few key market risk factors that capture the most important uncorrelated sources of information in the data, that is, to the principal components. This section describes how to use the principal component analysis that was explained in Chapter 6 to construct large covariance matrices that have many desirable characteristics:

They are always positive semi-definite.

They have relatively few constraints imposed on the movements in volatility and correlation.

Exponentially weighted moving averages can be used to produce EWMA covariance matrices where the persistence in volatilities and correlation is not the same for all factors. Instead it will be determined by the correlation in the system.

Univariate GARCH can be used to produce covariance matrix term structures that are mean reverting. The usual GARCH analytic formulae for computing the term structure of volatility and correlation are applied so that the «-day covariance matrix converges to the long-term average as n increases.

In orthogonal EWMA there is no need to impose the same value of the smoothing constant on all variables, as there is in the RiskMetrics data sets

The method is computationally very simple: it takes the univariate volatilities of the first few principal components of a system of risk factors, together with the factor weights matrix of the principal components representation, to produce a full covariance matrix for the original system. The method can be used with either GARCH or EWMA volatilities of the principal components. In orthogonal EWMA there is no need to impose the same value of the smoothing constant on all variables, as there is in the RiskMetrics data sets.

There are many advantages with the orthogonal method for generating covariance matrices. First, the computational burden is much lighter when all of the k(k + l)/2 volatilities and correlations are simple matrix transformations



of just a few variances. Second, some data may be difficult to obtain directly, particularly on some financial assets that are not actively traded. When data are sparse or unreliable for some of the variables in the system, a direct estimation of volatilities and correlation may be very difficult. However, if there is sufficient information to infer their factor weights in the principal components representation, their volatilities and correlations may be obtained using the orthogonal method. For example, some bonds or futures may be relatively illiquid for certain maturities, and statistical forecasts of their volatilities may be difficult to generate directly on a daily basis. But if the principal components and factor weights of the term structure are known, the orthogonal method will give a full covariance matrix that generates forecasts of all maturities, including the illiquid ones. Other advantages of the orthogonal method include the ability to reduce the amount of noise in the system so that correlation estimates are more stable. The method may also be used to obtain volatility and correlation forecasts of securities for which only a short price series is available (e.g. new issues - see §6.4.2).

Given these considerable advantages for generating high-dimensional covariance matrices, one would not necessarily expect orthogonal GARCH to perform as well as other multivariate GARCH models when the system has only a few dimensions. Indeed, some multivariate GARCH models are designed for the specific purposes of estimating single correlations, so they are only based on a bivariate GARCH specification. Nevertheless, Engle (2000b) shows that the orthogonal GARCH model performs very well according to three out of the four diagnostics that he has chosen for assessing the accuracy of correlation forecasts.

The orthogonal method will give a full covariance matrix that generates forecasts of all maturities, including the illiquid ones. Other advantages of the orthogonal method include the ability to reduce the amount of noise in the system so that correlation estimates are more stable

7.4.1 Using PCA to Construct Covariance Matrices

This section shows how to obtain the full covariance matrix of a large system from the covariance matrix of the principal components. This method is computationally efficient because the covariance matrix of asset or risk factor returns, which may have hundreds of elements, can be obtained as a simple transformation of the variances of the first few principal components.

Recall the principal components representation (6.2), which may also be written in vector form as

X,- = WjiVx + •• • + wtkVki (7-13)

where x, is the T x 1 vector of normalized data on the /th variable in the system, p; is the T x 1 vector of data on the yth principal component and W = (Wy) is the matrix of factor weights. In terms of the original variables the representation (7.13) is equivalent to

,- = / + W* Pi + • • • + wtnVm + 8/,

(7.14)



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