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of the European Equity Model
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Estimation of the European Equity ModelBy Gregory Connor and Nick Herbert This report introduces the European Equity Model and describes the results from the estimation of this new model. Section 2 discusses the construction of the risk indices and the estimation of the factor model. Section 3 presents the specific risk model. Section 4 shows the results from performance testing of the risk forecasts from the model. Section 5 analyzes the predicted betas from the model and compares them to historically estimated betas. Section 6 concludes the report. 1. Introduction The integration movement in Western Europe is one of the most important political-economic developments of our time. Its basic objective is the creation of an integrated super-state in which the member states retain some individual economic and political identity. It is not surprising that Western European equity markets have responded over recent decades by becoming increasingly homogeneous (see for example Beckers, Connor and Curds (1996) or Freimann (1998)). As various analysts have noted, the entity being created in Western Europe, particularly within the single currency region, is in many ways unprecedented. BARRA has, accordingly, created a new type of equity risk model, the European Equity Model (EEM). The new model sits halfway between BARRAıs Global Equity Model (GEM) and its family of single country equity models. Like the GEM, the European Equity Model contains country, industry and risk index factors. Unlike the GEM, all the factors are estimated simultaneously rather than country factors first. This allows the regional influences to fully exert their explanatory power. When we started the research for this new model, we expected to find a significant distinction between the ³Ins² (those countries adopting the Euro currency in 1999) and the "Outs." Empirical evidence, however, indicated otherwise - many Outs, such as Switzerland, for example, are well integrated into the equity factor structure of continental Europe. Rather, we discovered that the key distinction was between the UK and everyone else. For this reason the European equity model includes separate industry factors for the UK, and makes no modeling distinction between the Ins and Outs except the obvious one in their currency covariances. 2. Construction of the Risk Indices and the Factor Model Regressions The European Equity Model has six risk indices: Value, Size, Momentum, Volatility, Yield, and Blue Chip Membership. The first five of these are standard BARRA risk indices that have been applied successfully in a large number of our equity risk models. The Blue Chip Membership risk index is designed to capture the common movement of the top-tier equities. It is a dummy variable whose value is one if the stock is among the top 100 capitalization stocks in the universe at the beginning of the month, and zero otherwise. table 1 shows the risk indices and the descriptors contained in them. All descriptors are filtered for errors using the skipped Huber method, that is, values which are greater than 5.2 median absolute deviations from the median are set to this limit value. The risk index exposures are standardized to have a capitalization weighted mean of zero and an equally weighted standard deveation 1 of 1.0 each month. TABLE 1: The Risk Indices and their Underlying Descriptors
We have chosen to use the Stoxx2 industry classifications for the model. TABLE 2 lists the industry classifications and their coverage across the sixteen countries, both in terms of number of issues and percentage capitalization. The model also includes country dummies for each of the sixteen countries in the model. The countries are listed in table 3 along with number of issues and percentage capitalization in each. The UK market is by far the largest by either criterion. TABLE 2: The Industry Categories Used in EUE1
TABLE 2: The Countries Used in EUE1
Unlike the GEM, the EEM estimates all the factor returns including the country factors simultaneously, rather than estimating the country factor returns first and then all other factor returns on the first-stage residuals. The simultaneous presence of both industry dummies and country dummies creates a singularity in the matrix of independent variables. To adjust for this, we impose a linear restriction on the country factor returns. The weighted sum of country factor returns is constrained to equal zero. The weight for each security is the square root of its market capitalization. Details are shown in the Appendix. In our preliminary empirical work we found that the industry factors estimated Europe-wide fit poorly on the UK subset. This lack of integration between UK and continental equities cannot be attributed solely to the UKıs opt-out from the single currency. Sweden, Denmark, Norway and Switzerland are not joining the single currency either and yet their equity returns are well explained by the Europe-wide industry factor returns. It does however conform to anecdotal and empirical findings that UK equities behave distinctly differently from continental equities.3 To account for the lack of integration of UK and continental markets, we include a second set of industry factor for the UK only. All continental4 stocks have zero exposure to the UK industries and all UK stocks have zero exposure to the continental industries. The UK and continental industry dummies (2 x 19 industries) plus sixteen country dummies plus six risk index exposures of all assets at time t constitute a 60 x nt matrix Xt where nt is the number of assets in the cross-section at time t. Let Rt denote the nt vector of asset excess returns at time t. The factor model regressions are performed using excess returns in local currency for each security. This means that the factor returns are measured from a ³fully hedged² perspective. We estimate the 60-vector of factor returns by cross-sectional weighted least squares regression:
Ft is the 60-vector of industry, country and risk index factor
returns and TABLE 4 shows the square-root-cap-weighted adjusted R2s for each of the cross-sectional regressions (1). The cross-sectional regressions are run over the sixty-nine month period April 1992 to December 1997. table 4 also shows the number of country, industry, and risk index factors that are significant each month. Since these are risk factors, we do not expect them all to have a significant effect on the cross-section of returns every month. Each month, some of the factors have significant returns. Each factor has a significant return at least occasionally. The adjusted R2 of the regression varies from 9.6% to 56.3% with a time-series average of 30.2%. This means that in a typical month, 30.2% of the return to a typical stock comes from common return and the rest from asset-specific return. By the nature of a risk model, the R2 varies widely between a low value in very quiet months to a high value in months with large market-wide moves. Note that the proportion of factor return in a broad-based portfolio will be much higher than in a typical stock, due to the effect of diversification on asset-specific return. TABLE 4: Adjusted R-squareds
TABLE 5 shows the percentage of months for which the t-statistic for each factor return from the cross-sectional regression (1) is significant. This indicates which factor returns are most frequently important in explaining cross-sectional returns. All six risk factors have good-to-excellent explanatory power by this measure. The same applies to the industry and country factors. table 5 also shows the market betas of each of the factors and t-statistics for these betas. These measure the extent to which the different risk factors have market risk exposure or are purely extra-market sources of risk. The size factor has a large positive beta of 0.39. The betas of the country factors are generally near zero and often negative. This reflects the presence of the industry factors and the linear constraint placed upon the country factors. The "general market move" is by construction placed in the industry factors rather than in the country factors. TABLE 5: Individual factors: Percentage of months significant and market betas
The currency covariance matrix is constructed from a Euro-perspective using the Deutschmark as historical proxy for the Euro. The ten Euro In countries in the model have zero currency volatility from this perspective. The covariance matrix has nonzero variances and covariances for the other six currencies, those for Denmark, Greece, Norway, Sweden, Switzerland and the UK. There are also non-zero covariances of these currency returns with the 60 risk factors. The factor covariance matrix is estimated using exponential smoothing with a 48-month half-life. 3. The Specific Risk Model The specific risk model uses the standard BARRA methodology. Let eit denote the asset-specific return to security i in month t from the factor model regressions. Let 1/4it denote the relative absolute specific return of asset i in month t:
where St is the square-root-cap-weighted average of |
where b are the estimated regression coefficients and uit are the regression residuals. St is forecast using an equally-weighted moving average of the past
six months' realized St. The product of the one-month-ahead forecasts
for TABEL 6 shows the empirical ratio for cap-ranked decile portfolios (each portfolio contains 10% of total capitalisation). TABLE 6: Emperical Ratio for Capitalisation-Ranked Decile Portfolios
4. Risk Forecasting Performance of the Model The performance of the equity model is tested by generating risk forecasts for a variety of portfolios and then observing whether the magnitude of realized returns on the portfolios is consistent with the risk forecasts. Define a standardized outcome as the realized return on a portfolio divided by its ex-ante predicted risk. If the risk forecast is accurate, then the time series of standardized outcomes should have a sample standard deviation close to 1.0. TABLE 7 shows the sample standard deviations of standardized outcomes for a variety of portfolios. For these tests, we use fully hedged returns and all the portfolios are cap-weighted. We use the fifty-six month test period January 1994 to August 1998 for table 7. Table 7: Bias Test Results
The first portfolio consists of the top 10 stocks by capitalization, and the second portfolio the next 40 (stocks 11 - 50). The next 11 portfolios consist of all securities in the top and bottom deciles (as percent of capitalization) of exposure for each risk index exposure. The exception to this rule is the Blue Chip Membership risk index, where we use a portfolio of all assets with unit exposure. The next 38 portfolios consist of all stocks in each of the industries (19 continental and 19 UK). The final sixteen portfolios consist of all securities in each country. The risk forecasting bias test is performed for both total and active risk forecasts. The active risks use the cap-weighted universe portfolio as benchmark. The results are generally excellent, with the test values tightly clustered around 1.0. 5. Analysis of the Factor Model Predicted Betas The next two tables compare the performance of time-series estimated betas (called historical betas) and betas estimated using the BARRA factor model (called predicted betas). Let
TABLE 8 shows the cross-sectional distribution statistics for the historic and predicted betas for December 1997. Note the too-wide range for the historic betas. It is not credible that any European equities have betas as low as -3.447 or as high as 4.02. This shows the weakness of historic betas - they tend to have large measurement errors. The predicted betas all lie in a more reasonable interval: a maximum of 1.684 and a minimum of -0.123. Table 8: Distribution of Market Betas
TABLE 9 sorts securities using each type of beta and displays the top ten and bottom ten securities with their associated betas. TABLE 9:
6. Conclusion The European Equity Model is an innovation for BARRA in that it takes a regional rather than global or single-country perspective in modeling equity returns. It was developed in response to the integration movement in western Europe, in particular the Economic and Monetary Union (EMU) program of the European Union, and the growing empirical evidence for equity market integration in the region. The development of the model is partly motivated by the adoption of a single currency in eleven European countries. However, we find that the single-currency component of EMU is not definitive in terms of European capital market integration. Some countries which are not joining the currency are well integrated into the regional capital market. For this reason, we include all the Outs in the model’s estimation universe. A significantly lower level of integration than the continental markets distinguishes the UK equity market, which is why we include separate industry factors for the UK. The model has very good fit both in terms of explanatory power and risk forecasting performance. Appendix The Linear Constraints on Country and Industry Factors The factor model includes 16 country dummies and 38 industry dummies. Each asset has a unit exposure to one country dummy and one industry dummy, and this creates a singularity in the matrix of independent variables. To see the problem intuitively, consider adding 10% (or any arbitrary amount) to each of the 16 country factor returns, and subtracting the same amount from each of the 38 industry factor returns. Since every asset has unit exposure to exactly one industry factor and one country factor, every asset has 10% added and 10% subtracted from its explained return - leaving every asset’s explained return unchanged. We can, therefore, make these arbitrary changes without affecting the fit of the model - that is the nature of an indeterminacy. We need to put a constraint on the factor returns to "identify" them, that is, to properly separate the country factor returns from the industry factor returns. We resolve the indeterminacy by placing a linear constraint on the country factor returns. The weighted average country factor return is constrained to equal zero. We use the square root of equity capitalization of each security as weights. The linear constraint is therefore:
where CAPi is the market capitalization of security i, dijc equals 1 if security i is in country j and zero otherwise, fjc is country factor return j, and n is the number of securities in the model. If the model had only one set of industry factors then (A1) would fully resolve the indeterminacy. However the presence of two full sets of industry factors, one for the UK and one for all other countries, means that (A1) alone is not enough. We need to ensure that the UK and continental industry factor returns capture the same overall market move, so that the constraint (A1) is binding. To do this, we constrain the weighted sum of the differences between the UK and continental industry factors to equal zero:
where References Beckers, Stan, Gregory Connor and Ross Curds (1996) "National versus Global Influences on Equity Returns," Financial Analysts Journal, vol. 52, no. 2, 31-39. Chaumeton, Lucie, Gregory Connor and Ross Curds (1996) "A Global Stock and Bond Model," Financial Analysts Journal, vol. 52, no. 6, 65-74. Freimann, Eck | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
![[ Equity Analytics ]](../images/NLEquityAn.gif){kind=small}
is the nt-vector of
asset-specific returns.
it we use a set of descriptors which
includes all the industry and country dummies. Let Zit-1 denote the
vector of descriptors for asset i at time t-1. We stack the
time-series/cross-sectional sample of 1/4it and regress them against Zit-1:
TS
denote the n x n time-series sample covariance matrix of
asset excess returns. Let SFM denote the n x n covariance matrix
estimated from BARRA’s factor model. Let m denote the n-vector of
market portfolio weights (the cap-weighted portfolio of all stocks in the model). The
linear algebraic formulas for the n-vectors of historic and predicted betas are:
(A1)
ihUK
equals 1 if security i is in UK industry h and zero otherwise, fhUK
is UK industry factor return h, and