8.1.2 Multi-factor Fundamental Models

Suppose a portfolio consists of two groups of stocks, a low-beta group in industry sector A and a high-beta group in industry sector B, and suppose that the net portfolio beta is 1, so that the CAPM predicts that the portfolio should move in line with the market. Any differences between the portfolio return and the market return are attributed to stock-specific risks, which in a sufficiently large portfolio should be diversified away. However, if the market index contains stocks from other industry sectors that have different characteristics to sector A and sector B, the portfolio will not necessarily move in line with the market.

The arbitrage pricing theory (APT), introduced by Ross (1976), resolves this problem by extending the CAPM to a more general linear model. In the APT model the returns from each asset are represented as a linear sum of several economic risk factors that are common to all assets. The no-arbitrage principle implies that the expected returns to portfolios that have the same net exposure to these common risk factors will be equalized provided the number of assets in the portfolios is large in relation to the number of risk factors and that specific risks have been diversified away.

More general factor models that are based on APT represent the returns to each asset in a portfolio by many risk factors:

Rj, = P„-*i, + 2/ 2, + ... + $ 1 1 + e,, (8.5)

where R denotes the returns to the jth asset in the portfolio (./ = 1, . . ., ), X, denotes the return to the /th risk factor ( = 2, . . ., , and Xx = 1), (3,:/ denotes the sensitivity of the jth asset to the /th risk factor, and e7 denotes the residual of the yth asset. The model (8.5) is a system of n general linear regression equations. A matrix form of this model is4

where r, is the T x 1 vector of returns to asset j, X is the T x matrix whose /th column is the T x 1 vector X, of data on the /th risk factor returns, P; is the x 1 vector (Piy, . . ., PA/) and £, is the T x 1 vector ( 1, . . ., e/T).

How should the risk factors in an APT model be specified? Economic theory would imply that factors are at levels of increasing granularity: international equity market indices; industrial sector indices within each market; and the input-output criteria used for national accounts within each sector (wage rates, exchange rates, interest rates, taxes, commodity prices and other variables in the Standard Industrial Classification code). However, an enormous quantity of data are required to estimate such models with a complete specification, and it is more usual to determine potential factors by a consensus view from market

4 Note that an alternative matrix form, r - Bx + c, was used in §7.2.2.

Certain factors such as prices/earnings ratio, book-to-price ratio, debt/ equity ratio and market capitalization have emerged as standard fundamental factors

analysts. Certain factors such as prices/earnings ratio, book-to-price ratio, debt/equity ratio and market capitalization have emerged as standard fundamental factors:

The S&P/BARRA Growth and Value Indexes are constructed by dividing the stocks in an index according to a single attribute: book-to-price ratio. This splits the index into two mutually exclusive groups designed to track two of the predominant investment styles in the U.S. equity market. The value index contains firms with higher book-to-price ratios; conversely, the growth index has firms with lower book-to-price ratios. Each company in the index is assigned to either the value or growth index so that the two style indices add up to the full index. Like the full S&P indexes, the value and growth indexes are capitalization-weighted, meaning that each stock is weighted in proportion to its market value. The design of the indexes is an outgrowth of research into investment styles in the U.S. equity market performed by 1990 Nobel Laureate William F. Sharpe. Sharpe found that the value/growth dimension (as represented by price-to-book ratios), along with the large/small dimension (as represented by market capitalization), appears to explain many of the differences in returns to U.S. equity mutual funds.

(www.barra.com, 1999)

The point to note is that a vast number of decisions must be made in order to apply the APT framework

For international equity markets, data analysis firms such as Barra use certain common fundamental factors such as: volatility, momentum, size, trading activity, growth, earnings yield, value, earnings variability, leverage, labour intensity and foreign currency exposure. But some country models may require quite idiosyncratic factors (e.g. the family control indicator in Thailand, or foreign exports in South Africa). The point to note is that a vast number of decisions must be made about the nature of the variables and the data that are relevant for analysis in order to apply the APT framework to equity markets. We shall return to this problem at the end of the chapter.

The risk decomposition (8.4) for portfolios of assets modelled by the simple CAPM may be generalized to portfolios where assets are modelled by multi-factor models. Stacking the estimates of the betas in (8.5) in an n x estimated sensitivity matrix gives an estimated portfolio beta with respect to each risk factor as follows: for an n x 1 vector of portfolio holdings w = ( >,, . . ., w„), where Eiv,- = 1, the net portfolio beta vector is x 1 vector Bw, where the ith element is the estimated net portfolio sensitivity with respect to the rth risk factor. With this notation it can be seen that multi-factor models allow the decomposition of portfolio variance into two terms corresponding to the risks due to fundamental and specific factors:

VP = w BVVB w + wVEw,

(8.6)

where Vv is the x covariance matrix of risk factor returns and Ve is the n x n specific risk covariance matrix.

The residuals eb . . ., e„ from the estimation of (8.5) are used to estimate their covariance matrix VE. This specific risk covariance matrix has a number of applications in addition to the risk decomposition (8.6), that is used by portfolio risk analysts. For example, if the choice of risk factors differs for different stocks in the portfolio, the most efficient parameter estimates will be obtained using generalized least squares (§A.3.3) which requires the matrix Ve. The specific risk covariance matrix is also necessary to measure the contribution of the specific risks to the total value-at-risk of a portfolio. This can reduce market risk capital charges considerably compared to the standard methods (§9.3.3).

8.1.3 Statistical Factor Models

Statistical factor models are a common alternative to fundamental factor models. They have no need for data on possible explanatory variables and no problems with multicollinearity, but there is no economic interpretation of the variables. One of the most popular statistical techniques for developing multi-factor models for asset risk management is principal component analysis. Although the first principal component will still represent a simultaneous shift in all assets if they are highly correlated (§6.2), the interpretation of lesser principal components will depend on an ordering in the system. This is possible in term structures and in implied volatilities (according to strike or moneyness), but otherwise an ordering is not normally possible.

If the risk factors are taken as the principal components of a system we have an orthogonal factor model. To see this, suppose that the /th asset return is Xh so that the normalized variables are X* = (X, - u,)/a„ where u, and o, are the mean and standard deviation of Xt for /= 1, . . ., k. Write the principal components representation (6.2) as

X*=w*lPl+w%P2 + ... + w*kPk, (8.7)

where w* is the factor weight on the jth component in the representation for the /th normalized asset return. In terms of the original asset returns this can be written

X, = u, + wnPi + wi2P2 + ... + w,mPm + e„ (8.8)

where w,-,- = w*o, and the error term in (8.8) picks up the approximation from using only m of the principal components. The factors Pu . . ., Pm are orthogonal with each other and with the error term, and the model parameters Wy are easily calculated from the principal component factor weights and the asset return standard deviations.

Risk decomposition in statistical factor models is analogous to (8.6). The factor weights wtj - w*Oj are equivalent to the beta estimates, and to calculate the

The specific risk covariance matrix is also necessary to measure the contribution of the specific risks to the total value-at-risk of a portfolio