Stage 2: Denote by the rxs matrix of covariances of principal components across the two systems, that is, = {cov(P,, Qj). It is possible to estimate each element of the covariance directly, using EWMA or bivariate GARCH. But it is better to estimate all elements of simultaneously, and this can be done by using orthogonal EWMA or orthogonal GARCH a second time, now on a system of the r + s principal components Pb . . ., Pr, Q{, . . ., Qs. The cross-factor covariances of the original system will then be given by ACB and the full covariance matrix of the original system is:

The within-factor covariance matrices AD] A and BD2B will always be positive semi-definite. But it is not always possible to guarantee positive semi-definiteness of the full covariance matrix of the original system, unless the off-diagonal blocks ACB are set to zero.28

7.4.5 Summary

The examples presented in this section have been chosen to illustrate some of the many advantages of the orthogonal approach to generating covariance matrices:

>• Computational difficulties are kept to a minimum:

only univariate GARCH models are necessary for orthogonal GARCH;

computation time is very significantly reduced. >• Market reaction and persistence parameters do not have to be the same for

all assets:

these parameters will be determined by the correlation in the system; therefore orthogonal EWMA is quite different from the RiskMetrics daily matrix.

>• Mean-reverting covariance forecasts can be obtained:

the orthogonal GARCH model covariance matrix forecasts are based on

the usual simple form for GARCH term structure forecasts (§4.4.1); they may have convergent term structure forecasts even when univariate estimation gives IGARCH models. > Block-diagonal covariance matrices will be positive semi-definite:

this is because each block is given by AD;A, where D, is a diagonal matrix of GARCH or EWMA volatilities and A is a matrix of constant factor weights;

however, when splicing together the blocks into a very large covariance matrix, positive semi-definite checks will need to be employed.

28 This may be a sensible thing to do, in the light of the huge instabilities often observed in cross-factor covariances.

AD) A ACB (ACB) BD2B

> Choosing a few principal components to represent the system:

allows one to control the amount of noise and therefore produce more

stable correlation estimates; it also allows one to forecast volatilities and correlations for all variables

in the system, including those illiquid variables for which direct

computation of forecasts is difficult.

-8-

Risk Measurement in Factor Models

Few equity portfolios are so small that they can be analysed using their representation as a weighted sum of the constituent assets. Instead the returns and risks of large equity portfolios are usually represented by a factor model. This is a linear model where portfolio returns are written as a sum of risk factor returns, weighted by sensitivities to these risk factors, plus a specific or idiosyncratic return that is not captured by the risk factors. The success of factor models in predicting returns in financial asset markets and analysing risk depends on both the choice of risk factors and the method for estimating factor sensitivities. Factors may be chosen according to economics (interest rates, inflation, gross domestic product, . . .), finance (market indices, yield curves, exchange rates, . . .), fundamentals (price/earning ratios, dividend yields, . . .) or statistics (principal component analysis, factor analysis, . . .). Depending on the type of factor model, sensitivities may be estimated using cross-sectional regression, time series techniques or eigenvalue methods.

The success of factor models depends on both the choice of risk factors and the method for estimating factor sensitivities

Factor models are normally used to model relationships between underlying assets, rather than between an underlying asset and its derivatives. Traditionally these models are based on an economic perspective. That is, returns are modelled by exogenous variables that are not always considered to be stochastic. And if they are, only their basic properties such as correlation really matter for the model. This is in contrast to the option pricing approach, where the probability distributions of underlying factors are fundamental to the model.

Portfolios that are constructed or analysed by factor models typically have pay-offs that are a linear function of the underlying asset, hence the term linear portfolios. In options portfolios these pay-offs are non-linear. Quite different techniques are used to analyse options because there is a deterministic - not a statistical - relationship between the option price and the underlying price. Also, option pricing models are generally based on continuous time diffusion processes for the underlying risk factors, but a discrete time series approach is normally used for factor models.

The statistical modelling procedures for factor models have much in common with tracking models (§12.5). Both use linear statistical models to analyse