portfolio risk and return, and the mean-variance analysis developed by Markovitz (1959) is a common tool for asset allocations (§7.2). But whereas factor models attempt to explain individual asset returns by returns to market indices or other factors, tracking models turn the problem around, seeking to explain a market index as a linear function of its constituents, or more generally to track the performance of a portfolio to another benchmark return.

This chapter examines how the volatility of a linear portfolio depends on the volatility of the underlying risk factors, the factor sensitivities and the specific risks. After describing some common specifications for linear factor models in §8.1, it looks at different modelling perspectives and how these relate to methods for estimating factor sensitivities. A number of classical statistical methods are described in §8.2, and then §8.3 discusses how Bayesian methods have been applied to the estimation of factor models. The chapter closes with general guidelines for specifying and estimating factor models and some quite cautionary conclusions.

8.1 Decomposing Risk in Factor Models 8.1.1 The Capital Asset Pricing Model

One of the most common applications of a simple linear regression model in financial markets is the capital asset pricing model (CAPM).1 The model is written

j>, = a + pz, + e, {t - \, . . ., T), (8.1)

where y, denotes the return to a stock, X, the return to a market index and e, the stock-specific return, all measured at time t? In the original derivation of the model, based on the mean-variance analysis of Markovitz (1959), investors have access to risk-free lending and borrowing and so the returns X and are excess over some risk-free rate (Sharpe, 1964; Lintner, 1965). Without this assumption, the Black (1972) version of the model is based on real returns. Linear regression provides a method, though not necessarily the best method, for estimating:

5* the mispricing of the stock relative to the market, a;

the stock sensitivity to the market risk factor, P; >» the residual return. 8.

Active portfolio managers seek to gain incremental returns with a positive alpha, but if markets are efficient and the Sharpe-Lintner version of the

This is not a text on portfolio theory and it is not intended to describe the fundamental assumptions and developments of the CAPM here. There are already many excellent texts that cover this topic - for example. Campbell et at. (1997).

2 The subscript 1 is used throughout, assuming time series data are used to fit the model. However cross-sectional or panel data might also be used.

CAPM is the correct model, alpha should be zero. Statistical procedures such as those described in Appendix 2 may be used to test the hypothesis that a = 0; in fact these form the basis of many empirical tests of the validity of different versions of the CAPM.

Beta provides a measure of how the stock responds to changes in the index: If P is insignificantly different from 0 the index has no statistical effect on the stock, and if it is insignificantly different from 1 then changes in the index are matched by changes in the stock. The residual returns are assumed to be diversifiable and therefore unimportant: if the portfolio has many assets the stock-specific returns will tend to cancel each other out. The market represents the undiversifiable risk this is common to all stocks and therefore the stock beta reflects the riskiness of the stock relative to the market: stocks that have a beta significantly less than 1 are regarded as low risk investments, those with beta significantly greater than 1 are categorized as high risk. Assuming the true value of beta is a constant, it is defined as

The residual returns are assumed to be diversifiable and therefore unimportant: if the portfolio has many assets the stock-specific returns will tend to cancel each other out

Pv = 0"A>/oi,

(8.2)

where aXy is the unconditional covariance of the return of stock with the return of the market X, and crl is the unconditional variance of the return of the market.

The estimation of beta in a CAPM can be very sensitive to the definitions of the market portfolio and the risk-free rate. The market portfolio should include all risky assets, but most indices only contain a subset. So betas with respect to the Dow Jones 30 can be very different from betas with respect to the S&P 500, for example. It is not always easy to know what is the appropriate risk-free rate either. Short-term treasury bills are often taken as risk-free, thus ignoring uncertainties about inflation and assuming bills are held to maturity. However, the beta may equally well be measured in terms of total returns if the risk-free rate really does have zero variance.3

Portfolio betas may be measured directly by using return data at the portfolio level. An artificial history of portfolios returns is constructed using the current portfolio weights and historic data on each asset. Alternatively, the individual stock betas may be weighted by the proportion of the fund that is invested in stock with return y, denoted wy. Then summation gives the net beta of the portfolio as

= -

When ordinary least squares is used to estimate the betas the two methods will give the same results (§8.2.2).

The portfolio beta provides a very simple framework for predicting portfolio

1 Denote by r the risk-free rate, so the estimate of beta is oov{X-r,j-r)/V(X- r) = [cov(X,y)~cov{A\r)-cov(v,r) + V(r)] I [ V(X)-2cov(X,r) + V(r)} = cov(X.y)IV(X) if F(r) = 0.

returns and modelling portfolio risk. A 1% fall in the market is expected to be matched by a (3% fall in the portfolio, so a portfolio with beta greater than 1 is considered more risky than a portfolio with beta less than 1. A portfolio manager seeking only to track an index will diversify the portfolio to achieve a beta of 1, an alpha of 0 and residual returns as small as possible to minimize the tracking error. On the other hand, active portfolio managers that seek positive alpha may have betas that are somewhat greater than 1 if they are willing to accept an increased risk for the incremental return above the index.

The CAPM attributes risk to three sources: assuming cov(X„ £,) = 0 for all t, taking variances of (8.1) gives

V{y) = tfV{X)+V{z).

More generally, recall that for n assets with returns r, (i - 1, . . ., n) in a portfolio with weights w = (wb . . ., w„) where T,Wj = 1, the portfolio return is RP = wlrl+...+ w„r„-wr. The portfolio variance V(P) is given by the quadratic form wVw, where V is the covariance matrix of individual asset returns (§7.1.1). For equities, which are modelled by the CAPM and it is assumed that cov(A",, £,,) = 0 for all i and /, the diagonal elements of V are V{X) + V{Zi) and the off-diagonal elements of V are given by P,P/F(A) + cov(£„ e;). Thus

V = PKWP + VE, (8.3)

where p = (pb . . ., Pn) is the vector of equity betas and VE is the n x n specific risk covariance matrix. Portfolio risk, as measured by the portfolio variance, is given by

V(P) = wPFTAPw + wVew. (8.4)

This illustrates the three sources of risk in a factor model:

5* factor sensitivities (the beta vector, P);

market risk (the variance of the market risk factor, specific risks (the residual covariances, VE).

V{X));

An example of the application of (8.4) will be given in §8.2.4, after the methods used to estimate factor sensitivities have been described. See also the risk decomposition spreadsheet on the CD.

If relevant risk factors are omitted from the model the assumption cov(Z,, e,) = 0 will not hold

This risk decomposition is totally dependent on the assumption co\(X„ e,) = 0 and this in turn depends very much on the model specification (§8.5). If relevant risk factors are omitted from the model, the variation from these factors can only be attributed to the specific risks, and if these omitted factors are correlated with the market risk factor then the assumption \{ „ ,) = 0 will not hold. This is one of the reasons why it is important to include all possible risk factors in the model for a linear portfolio, and often there are too many sources of risk to capture with the simple CAPM alone.