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83

specific risk matrix in (8.6) calculate the model errors in (8.8) and take an equally weighted average covariance matrix of these as described in §3.1.5

With statistical factor models the difficult model specification issues that will be discussed in §8.4 are avoided. It is only necessary to decide on the number of principal components and the length of data used to calculate them; the choice will depend on the results of backtesting the model.

8.2 Classical Risk Measurement Techniques

The diversity of multi-factor model implementations is not only due to differences in risk factors or different estimation periods. The choice of estimation methods will also influence results: in particular, it will affect the ability of the model to capture portfolio risk characteristics. Portfolio risk has been ascribed to sensitivities, risk factors and specific components. This section shows how estimates of all these quantities will depend on a number of different covariance matrices: covariances between the risk factors, covariances between the assets and the risk factors, and covariances between residuals of different assets. We discuss the different methods that are available for estimating these covariance matrices and the context where each could be applied.

8.2.1 The Different Perspectives of Risk Managers and Asset Managers

Many asset managers simply take an average beta over a long period of time; typically 5 years of monthly data will produce robust results when fitting a CAPM. In doing so they are taking the economists perspective. Classical economics assumes that there is one true value for each model parameter. In this case OLS is often the most efficient estimator (§A.2.4) and better results are obtained when long periods of data are used, because any differences in estimates are only due to sampling error and the more data are used the lower the sampling variation (§5.2.1). It is fine to estimate betas using long-term equally weighted averages when considering long-term buy-hold strategies. However, conditions in a firm or an industry can change quite abruptly, so covariances may be quite unstable. Risk managers are concerned by the accurate estimation of factor sensitivities from day to day, since these factor sensitivities are crucial inputs to the risk model. For modelling short-term risk exposures such as 1-day VaR, rather than assuming there is one true value for the underlying parameter it is preferable to assume that the parameters themselves change over time. So from a risk management perspective it is better to assume that the true betas vary over time. In this case OLS is not the appropriate modelling methodology. For accurate daily measures of

5 Since principal components are based on the equally weighted correlation matrix, it is consistent to apply equal weighting for the specific covariance matrix also.

Classical economics assumes that there is one true value for each model parameter. However, from a risk management perspective it is better to assume that the true betas vary over time



sensitivities to be used in risk models such as the covariance VaR model of §9.3, we require a theoretical model that is based on the time-varying parameter assumption.

8.2.2 Methods Relevant for Constant Parameter Assumptions

The ordinary least squares method of estimating the covariance matrices that determine portfolio risk in factor models is described in Appendix 2. It gives equal weighting to all observations in the sample. Thus OLS betas are related to the unconditional correlation p between the risk factor return and the asset return. For example, in the simple CAPM defined by (8.1) and (8.2) the unconditional correlation p and the OLS beta are related by

P=pv,

(8.9)

where v denotes the relative volatility of the asset relative to the market. Thus if the asset is perfectly correlated with the market, so that there is no specific return, the beta is just the relative volatility. Stocks that tend to move in the opposite direction to the market may have negative betas; and when stocks are relatively volatile, betas may be much greater than one.

In multi-factor models the OLS sensitivity estimates, which are determined by the covariance and variances of stock and market excess returns, are obtained by applying the formulae in §A. 1.2. The OLS estimator of is the inverse of the covariance matrix between risk factors, times the vector of covariances between the /th asset returns and the risk factors:

the asset is perfectly correlated with the market, so that there is no specific return, the beta is just the relative volatility

.- ,

(8.10)

These estimates are x 1 vectors in which the /th element is the estimated asset sensitivity to the /th risk factor. To obtain the OLS specific risk covariance matrix first obtain the OLS estimates from (8.10) and then calculate the OLS residuals as e; = r- - Xb;. Then efej/T is the estimate of the (/, j)th element of

Note that fitting the model (8.10) for each stock separately makes some over-simplistic assumptions about the behaviour of stock returns. In particular, it should not be assumed that the movements of some stocks in a portfolio will be unrelated to the behaviour of other stocks in a portfolio. Instead, if the return on one asset is unusually high, this should give some useful information about the returns on some of the other assets in the portfolio. However, this type of useful information will be overlooked if each parameter vector is estimated on an individual basis using OLS applied to each equation in (8.10) separately.

Generalized least squares is a more general method of estimation that takes account of the relations between different asset returns by allowing the error



If different assets in the portfolio have a different set of risk factors there will be considerable efficiency gains when generalized least squares is used to estimate factor sensitivities

processes £b . . ., £„ to be correlated with each other. In §A.3.3 it is shown that if all the risk factors are identical in each equation, generalized least squares is equivalent to ordinary least squares. But if different assets in the portfolio have a different set of risk factors there will be considerable efficiency gains when generalized least squares is used to estimate factor sensitivities.

8.2.3 Methods Relevant for Time-Varying Parameter Assumptions

The OLS method gives equal weights to all observations in the sample used to estimate the beta. Therefore, it does not emphasize the importance of the most recent market conditions. If a stock and the index have even just one large synchronous movement during the sample period, this single event will have a great effect on the beta estimate; and this effect will not diminish with time - until it drops out of the average. Similarly, if the stock experiences an exceptional return at any point in the data period, this will also increase the current beta estimate for as long as that return remains in the averaging period. In fact OLS beta estimates have exactly the same problems of ghost features and artificial stability as the equally weighted measures of volatility and correlation (§3.1).

For the purpose of measuring short-term risk exposures, therefore, it may be better to use a model that does not assume that the beta is constant over time. Instead one can assume that the true parameter beta takes different values at different points in time. This time-varying parameter assumption is an extension of the time-varying conditional correlation parameters introduced in §1.4. To see this, note that one can extend the definition (8.2) of a CAPM market beta to conditional covariances and variances in the natural way:

(8.11)

OLS beta estimates have exactly the same problems of ghost features and artificial stability as the equally weighted measures of volatility and correlation

where aXr, is the conditional covariance, ax , is the conditional variance of the market, p, is the conditional correlation and v, is the relative conditional volatility at time t. Similarly, in a multi-factor model, where in (8.10) the beta estimate is given by the inverse risk factor covariance matrix multiplied by the vector of covariances between the stock and the risk factors, the use of conditional covariance matrices in place of unconditional covariance matrices will extend the constant parameter estimate to an estimate of a time-varying beta vector. In this way the estimates of time-varying market betas can be obtained from time-varying volatility and correlation estimates. An example of the estimation of a time-varying beta was given in §4.5.1.

8.2.4 Index Stripping

Having estimated a stock beta and found the residuals, a useful technique for the risk analysis of a stock is to use (8.3) to decompose the total volatility of the stock into market and specific components. This can also be done at the



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