Fixed Income Risk Modeling with the Global Industry Classification Standard (GICS )

Transcription

1 Fixed Income Risk Modeling with the Global Industry Classification Standard (GICS ) BIM301 Fixed Income Credit Model Enhancements Carl Gold John Fox

2 BIM301 Adds More Detailed Credit Spread Factors In this paper, we introduce a new version of the Barra Fixed Income Model for developed markets, showing how the new model adds more detailed credit spread factors. This new model is part of the Barra Integrated Model version 301 (BIM301). This document first discusses the model and credit spread factors. Then we describe fixed income corporate and non-corporate sectors and the use of GICS, the global industry classification standard jointly developed by MSCI and Standard & Poor s, to identify corporate sub-sector factors. We present a procedure for ranking factors to determine how much sector granularity to include, concluding with model performance statistics. The new GICS-based models significantly improve the performance over the previous versions. All versions of the Barra Fixed Income Model include common risk factors for credit spreads in developed markets. 1 Until BIM301, the credit spread models were based on a strict sector-by-rating grid of the bond universe: each sector and rating combination had one factor that was a weighted average of the change in spread for the bonds with that sector and rating. There were a few exceptions to the rule: high yield sectors were merged for a given rating to form a rating factor that was not sector specific; at the same time, the US Agency factor only applied to the AAA-rated US agencies. The investible bond market is not as evenly distributed across sectors and ratings as the model factors: in some sectors there are no issues with certain ratings (for example, supranational organizations do not have high yield ratings); for other sectors, there are enough issues that the sector designation was broad and could be made more granular if there are important differences between market segments within a sector. Additionally, there have been some changes in the composition of the market over time: e.g., there are now virtually no companies in the industrial sector with AAA ratings; the financial sector became much larger and could be divided between banks and insurance companies. Starting with the new Barra Integrated Model version 301 (BIM301), fixed income models will take a more nuanced approach to credit spread factor construction: sector and rating combinations with relatively few issues will be merged with other similar categories, while sectors and ratings with a large number of issues will be further decomposed. BIM301 also introduces GICS as the basis for corporate sub-sector decomposition. The underlying grid is based on sub-sectors; neighboring sectors and ratings may be merged to form more general categories as appropriate. The Factor Hierarchy and GICS To allow a consistent definition of factors that provides more flexibility, BIM301 uses a hierarchical decomposition of the bond universe (see Figure 1). At the top of the hierarchy, the most general classification used is the rating, with each rating as the root of a tree of factors. For any given rating, bonds are sub-divided into corporate versus non-corporate. For corporate bonds, the model divides between financial and non-financial issuers. Among financial, non-financial, and non-corporate bonds, the model may further sub-divide the universe into sectors and sub-sectors. Factors may be defined at each level of the hierarchy. 1 For a summary of the structure of Barra Fixed Income Model, see Appendix D: Overview of Barra Fixed Income Model. 2of 27

3 This structured approach allows the creation of highly specific factors when it is appropriate, and a logical fall back on more general factors when it is not. For example, in one market we may choose to define a factor based on the automotive industry. In another market, we may not take this approach and use the parent sector, Consumer Discretionary, instead. In a smaller market, there may not be enough bonds from Consumer Discretionary sector issuers to create such a factor; in that case, bonds from those issuers are covered by a Corporate Non-Financial factor. It is important to understand that factors calculated at a more general level of the hierarchy include the assets in the more specific levels. For example, a Consumer Discretionary factor is calculated based on all Consumer Discretionary assets, including Auto and any other more specific sub-sector factors that may be defined inside it (e.g., media, or retail). Put another way, factors at a higher level in the hierarchy represent all assets below them, but not other assets left over after more specific categories are carved out. This ensures that factors based on higher levels of the hierarchy are truly general. Rating Corporate Non-Corporate Non-Financial Non-Financial Sectors Financial Sectors Non-Corporate Sectors GICS Non-Financial Sub- Sectors Financial Sub- Sectors Non-Corporate Sub-Sectors Figure 1: Illustration of Factor hierarchy and the use of GICS in factor definition. We chose the GICS to define sectors and sub-sectors for corporate bonds. GICS is a four-level hierarchy consisting of 10 sectors, 24 industry groups, 68 industries and 154 sub-industries. While it is better known in equity markets, GICS was chosen for the classification of corporate bonds because it provides a consistent framework for analyzing fixed income sector risk across global markets. In contrast, widely used fixed income indices are produced by different vendors in major markets and use slightly different sector schemes. Also, GICS is used to define the Barra equity industry factors and use of GICS in fixed income facilitates the comparison of risk across asset classes. In comparison to most fixed income corporate classification schemes, GICS has a larger number of sectors, along with a more restricted definition of the Industrial sector. Some major indices (and the previous versions of the Barra models) define the Industrial sector to mean roughly all Corporate Non- Financial firms, including such diverse activities as food production, computer technology, mining and retailing. GICS restricts the Industrial sector to heavy industry and production of capital goods, including additional categories at the sector level for Consumer Discretionary, Consumer Staples, Healthcare, 3of 27

4 Technology, Materials, and so forth. In choosing GICS as our sector framework we adopt this more specialized approach to classifying corporate non-financials. Note that this decision to create factors for all the GICS level 1 classifications creates a larger number of fixed income sectors when compared with a risk model based on the traditional fixed income approach (even if sub-sectors are not used for factors at all). Given its specificity, GICS had to be adapted for the analysis of fixed income risk. Although we adopted the 10 GICS sectors, we chose from among the GICS industry groups (level 2) and industries (level 3) the most relevant 39 categories for fixed income markets. GICS sub-industries were always found to be overly specific for use in fixed income risk modeling. We refer to the GICS industry groups and industries generically as sub-sectors. As described in the next section, we ultimately chose only a small fraction of the sub-sectors to use as the basis for model factors. In Appendix A: Sectors for the Barra Fixed Income Model, Table 2 contains a list of the corporate sectors and sub-sectors used in the fixed income models, including which sub-sectors have factors based on them in one or more of the model markets. In addition to corporate sectors, the Barra Fixed Income Model requires non-corporate sectors, which are not a part of GICS. The non-corporate issuer sectors are the same as in the previous iterations of the Barra Fixed Income Model: government agencies, local/provincial governments ( Munis ), covered bonds (Pfandbriefe), sovereigns (national governments issuing debt denominated in a foreign currency), and supranational organizations. In the new model we have also defined sub-sectors that differentiate segments of the European Covered Bond market. The new Euro model includes covered bond sub-sector factors for German Jumbo Pfandbriefe and German Non-Jumbo Pfandbriefe, as well as factors for covered bonds from French and Spanish issuers, Obligations Foncières and Cédulas Hipotecarias. The new model also includes covered bond sector factors for covered bonds that do not fall into one of the four major subsectors. However, in contrast to the method for other sector factors, we exclude German Pfandbriefe from the covered bond sector factor. This prevents the large German market from dominating the sector factor, resulting in a better model of covered bonds that are issued in smaller Euro member countries. Method of Factor Selection Our goal is to improve the performance of the factor model by adding differentiated factors. However, we are careful to preserve the parsimonious nature of the factor model by not adding too many factors. Too many factors reduce the interpretability of the model, and too many factors in the covariance matrix can lead to degeneracy and the erroneous prediction of riskless portfolios. 2 More subtly, portfolio optimization with a risk model biases the risk forecast, and this bias increases with the ratio of the number of factors to the effective number of observations (becoming infinite when the two are equal). 3 For all these reasons, we apply rigorous criteria when adding factors. 2 See e.g. RC Grinold and RN Kahn, Active Portfolio Management,McGraw-Hill, New York, 2000, pages P Shepard, Second Order Risk, Quantitative Finance Papers , August of 27

5 As noted above, we began with all level 1 GICS sectors (10) and our additional fixed income noncorporate sectors, then we evaluated sub-sector factors based on GICS levels 2 and 3 classifications. The major criterion for adding a sub-sector factor to the model is that it confers a significant improvement in the cross sectional R-squared of the model. If we find that addition of a factor does not improve R- squared, then we do not include the factor in the model; instead, we expose its bonds to the factor s parent in the hierarchy. Because we are particularly interested in the performance of the credit factors, we use the R-squared of the credit model after interest rate return has been taken into account. That is, we calculate the R-squared of the credit model on the residual returns from the interest rate model, and this R-squared is lower than the overall R-squared of the model. The credit model R-squared is typically around 50%-75% of the R-squared of the full model, including interest rate, swap and credit factors. We started by analyzing the mean cross-sectional credit R-squared improvement for the bonds in all GICS sub-sectors covered by our estimation universe, as well as for GICS corporate non-financial sectors, and for different categories of covered bonds as described above. We compared the cross-sectional R- squared of those bonds when exposed to a sub-sector factor versus those bonds being exposed to the sub-sector s parent in the hierarchy. For GICS corporate non-financial sectors, the comparison is with the corporate non-financial factor, roughly the same as the Industrial factor in the previous iterations of the Barra models. An example of the comparison is shown in Figure 2 for the European Auto BBB factor. The sub-sector factor increases the average cross-sectional R-squared of the credit factor from 0.16 to The improvement is greatest around the time of the General Motors downgrade in Using this test on all sub-sectors, we identified which sub-sector factors had potential to improve the model and which did not. 0.4 Euro Auto BBB Factor (Sub-sector) Euro Auto BBB exposed to Consumer Discretionary (Sector) Improvement Cross Sectional R Figure 2: Illustration of Cross Sectional Credit R-squared Improvement using Sub-sector factors. At the model level, we must also account for how large a sub-sector is in the market. It may be preferable to add a factor for a large sub-sector with a modest R-squared improvement instead of a factor for a very small sub-sector that had a greater improvement. But choosing an optimal set of factors for the entire bond universe is an exponentially complex problem. To make the problem tractable, we ordered the factors by the market-weighted improvement in the sub-sector R-squared, 5of 27

6 and then added the factors to the model in order of most-to-least improvement. As each factor was added, we re-calculated the estimation universe cross-sectional credit R-squared (the R-squared for the complete model, not the sub-sectors in isolation). We observed that most of the improvement in cross sectional R-squared accrues from adding only a minority of the sub-sectors in the estimation universe. The results of this test for the Euro zone and the US are shown in Figure 3. Cross Sectional R- squared Cross Sectional R- squared Euro Estimation Universe Credit R-squared # Sub-Sector Factors US Estimation Universe Credit R-squared # Sub-Sector Factors Figure 3: Improvement to Estimation Universe Credit R-squared with addition of sub-sector credit factors. (This is the overall power of the model to explain bond return that is residual to interest rate factors; total R-squared including interest rate factors is substantially greater.) For the US, we analyzed 62 sub-sectors and chose 15 that conferred 61% of the total R-squared improvement, and for the Euro zone we analyzed 35 factors and chose 15, resulting in 84% of the total possible R-squared improvement. Note that the choice of 15 for both the US and Euro is a coincidence. In fact, we chose relatively more sub-sector factors in the Euro zone (15 of 35) because the model began with somewhat lower R-squared, and the US model already had a larger number of sector factors. The number of sub-sectors analyzed, and the number of sub-sector factors in each market, is summarized below in Table 1. In each case we chose the final factor list to optimize the model R-squared in relation to the total number of factors and the overall model performance. Market # Sub-Sectors Analyzed # Sub-Sector Selected Total # Credit Factors US Euro UK Canada Switzerland Australia Table 1: Number of sub-sector factors and total number of credit factors per market. Total # of credit factors includes sector factors (GICS level 1 and non-corporate sectors) as well as sub-sector factors. 6of 27

7 One of our most interesting findings is which GICS sub-sector factors gave the most improvement in the model explanatory power. As summarized in Appendix A: Sectors for the Barra Fixed Income Model, Table 2 and Table 3, sub-sector factors were added almost exclusively within the Financial and Consumer Discretionary sectors (the exceptions being distinctions between covered bonds in the Euro zone and Utility sub-sectors in the Canadian model). This suggests, from the point of view of fixed income market risk, that the Financial and Consumer Discretionary sectors contain a significantly more varied set of sub-sectors than other GICS sectors. For a complete description of the factors added in each market model, see Appendix C: BIM301 Credit Model Factor Tables. In selecting factors, we also observed measures of the conditioning of the covariance matrix and the bias that would result for a portfolio optimized with the risk model. We found that the criterion of R- squared improvements restricted the factor set well enough that matrix conditioning and optimized portfolio bias did not become problems. Therefore, we did not apply strict metrics based on that type of criteria when choosing how many factors to add to the models. Factor Selection and Model Performance We use the explanatory power measured by R-squared to construct the model rather than risk forecasting performance, in order to avoid overfitting the model to a particular backtesting metric. In general, performance in backtesting is highly dependent on the half life used for the covariance estimate. At the same time, our ultimate goal is to deliver better risk forecasts to model users, and we need to confirm that the model constructed using cross-sectional statistics confers a benefit in time series forecasting. Our primary metric of the model risk forecast is the bias statistic: the standard deviation of returns normalized by volatility forecasts. When the bias statistic is close to 1, the model risk forecast is accurate because realized volatility matches forecasts. To summarize bias performance on a portfolio, we use the Mean Rolling Absolute Deviation (MRAD) of the bias statistic: the mean deviation of the bias from 1 over the back-test simulation. For example, an MRAD of 0.2 indicates that the bias statistic averages 0.2 away from a perfect score of 1 (without indicating whether it is over- or under-forecast), while a lower MRAD indicates less bias in the model. Unless otherwise noted, when we refer to the bias statistic we mean the MRAD of the bias statistic. 4 To analyze improvement in bias statistic displayed by the new factor model, we used a similar approach to that used in analyzing cross-sectional R-squared: we compare the bias on a portfolio with (and without) the addition of one of the factors. This is illustrated in Figure 4 for the Merrill European Automotive sub-index, and the Auto BBB factor that was previously shown in Figure 2. Figure 4 shows both the total risk of the portfolio (top) and the active risk of the Auto sub-index relative to the European All Corporate benchmark (bottom). We also show the realized portfolio volatility calculated with an exponentially weighted moving average (EWMA) with the same half life as the factor covariance matrix; note that the realized volatility also has some degree of bias due to changes in the volatility regime. We consider the bias of the EWMA volatility as a benchmark against which to compare the factor model: for a portfolio without turnover it is not generally possible for a model to have a less biased forecast than the realized portfolio volatility itself (although this is the not the case when there is turnover, since factor models can outperform the realized volatility if there is any significant portfolio turnover). 4 See Appendix E: Model Performance Statistics for details of the model performance statistics. 7of 27

8 In general, we find that active risk relative to a parent benchmark is more challenging to forecast than total risk, as illustrated in Table 2, on the next page. For total risk we find that EWMA and the factor model with the Auto factor included both have the same MRAD, while the factor model without the Auto factor has a bias that is greater by only On the other hand, for active risk we find that all biases are higher and that the Factor model with the Auto Factor outperforms EWMA by 0.04 (probably due to turnover in the index), and outperforms the factor model without the auto factor by Annualized Volatility, bps European Auto Sub-Index, Total Risk Rolling Bias Statistic European Auto Sub-Index, Total Risk Bias Statistic EWMA 2002 Factor Model 2004 w/ Auto Sub-Sector Factor Model w/out Sub-Sector Annualized Volatility, bps European Auto Sub-Index, Active Risk European Auto Sub-Index, Active Risk Bias Statistic Rolling Bias Statistic Figure 4: Total Risk and Active Risk relative to the All Corporate Benchmark (ticker ER00), for Merrill European Auto Index (ticker EJAU). 8of 27

9 Total Risk Active Risk EWMA Factor model w/ Auto Factor Factor model w/out Auto Table 1: Mean Rolling Absolute Deviation (MRAD) of the bias statistic for the Merrill European Automotive Sub-Index (EJAU). The higher MRAD score indicates that the Active Risk forecast relative to a parent benchmark is substantially more difficult. We performed similar tests for all of the Merrill sub-indices covered by our estimation universe, as well as some additional portfolios that we introduced in market segments where there are no differentiated Merrill sub-indices. Figure 4 shows the average active risk bias (market weighted) over all sub-indices tested in the Euro and US with addition of each sub-sector factor (added in the order of marketweighted R-squared improvement, as in Figure 3). The figure also shows the baseline bias level determined by the EWMA calculation. As in the test of model explanatory power, we find that most of the improvement conferred by sub-sector factors comes from only a fraction of the possible sub-sectors covered by the estimation universe Average Euro Merrill Sub-Index Active Risk MRAD Model MRAD EWMA MRAD # Sub-Sector Factors Average US Merill Sub-Index Active Risk MRAD Model MRAD EWMA MRAD # Sub-Sector factors Figure 4: Bias improvement (MRAD) in Euro and US with addition of Sub-sector factors. Model Performance Comparison To compare the performance of the new models used in BIM301 with the previous version, BIM207, we constructed 99 different portfolios in the six markets, as summarized in Appendix B: Model Performance Comparison, Table 12. We then compared the model bias statistics from 2003 through We used broad market segment portfolios (e.g., Corporate, Non-Corporate, High Yield) as the active risk 9of 27

10 benchmarks for sector and sub-sector portfolios, and we also analyzed the total risk forecast on the benchmarks. For active risk tests the portfolios were weighted to match the duration of the benchmark; otherwise, the test portfolios were weighted by amount outstanding. We compared both the Short Horizon version of new models and the Long Horizon version ( BIM301S and BIM301L respectively; for details see The Barra Integrated Model (BIM301) Research Notes, Shepard [2011]). The results of these tests are presented in Appendix B: Model Performance Comparison, Table 4 through Table 13. The results for the rolling absolute deviation of the bias statistic (RAD) are shown in Table 4 through Table 7. This measure clearly shows the improvement due to the new models. The deviations of the bias statistic (from ideal value of 1) are significantly reduced compared to the old models. BIM207 is a longer-horizon model, so the most direct comparison is with BIM301L, where we see most of the improvement. The short horizon model (BIM301S) improves the bias statistics further. These results clearly show the greater difficulty of active risk forecasting, and the significant improvement made by the new models. The year-by-year RAD statistics (Table 6 and Table 7) illustrate the much greater difficulty of accurate forecasting during the period of the financial crisis. Longer horizon models are designed for more stable risk forecasts, trading off stability against responsiveness, thus they tend to under-forecast during the start of the crisis in late 2008 and then over-forecast beginning in The short horizon model shows a significant improvement in backtests during this period. Careful examination of the year-by-year RAD statistics also shows that there are some markets and years early in the test when the old model outperforms the new model (or at least the Long Horizon version of the new model). The reason for this apparently weaker performance is that the new models have not only added sub-sector factors, but also pruned factors for which significant markets no longer exist, particularly high-grade corporate non-financial categories like Industrial AAA. These factors can no longer be supported in ongoing models due to a paucity of assets in the estimation universe, and they must be filled by proxies. We believe this is a long term secular shift in the bond markets and, in the interest of parsimony, we do not include such factors in the new models. However, in the earlier years of the backtest (when the factors were not filled by proxies) these factors confer a benefit to the old model which includes them. Table 8 though Table 11 show the percentage of portfolio bias statistics that are within the 95% confidence region, and divide the result by calendar year. We see a similar trend to that presented in the RAD statistics: while the new models outperform the old models in the most recent years, there are several cases where they underperform in the early years of the analysis. And during all years we find that for active risk the percentage of portfolios within the confidence bounds is actually slightly lower for the Long horizon version of the new model than the old model. It is worth noting that the percentage within confidence says nothing about how far out of confidence portfolios with poor forecasts actually are, and also within a given year overforecasting in one part of the year can be offset by underforecasting in another part. The RAD statistic actually measures how far off the forecasts are, and allows no offsetting of the results for these reasons we consider RAD to be the more reliable measure of overall model performance. The overall RAD statistic shows that the occasionally poorer performance of the new models in earlier years is more than offset by improved performance in the most recent period. 10 of 27

11 Conclusion We found that GICS provides an effective decomposition of sectors and industries for constructing a fixed income factor model of credit risk. The use of hierarchical factor decomposition of the estimation universe allows coverage of all sectors and ratings and provides a consistent framework for choosing alternative factors. The method of R-squared comparison between candidate factors and their alternatives using market-weighted rankings is an effective method for making the factor selection problem tractable. In practice, we find very good model performance with only a fraction of the subsectors that could be chosen. The new fixed income credit models deliver a substantial improvement in risk forecast performance, particularly for the active risk of industry-specific portfolios relative to their parent benchmarks. 11 of 27

12 Appendices Appendix A: Sectors for the Barra Fixed Income Model Sector Description Sub-Sector Description Factor? Sector Description Sub-Sector Description Factor? Consumer Discretionary Auto Y Materials Chemicals N Consumer Discretionary Durables & Apparel Y Materials Construction & Building Materials N Consumer Discretionary Media Y Materials Forest & Paper Products N Consumer Discretionary Retail Y Materials Metals & Mining N Consumer Discretionary Services Y Materials Packaging N Consumer Staples Consumer Products N Technology Hardware N Consumer Staples Food N Technology Semiconductor N Consumer Staples Retail N Technology Software N Energy Equipment & Services N Telecommunications Diversified Telecommunication Services Energy Oil & Gas N Telecommunications Wireless N Financial Bank Y Transportation Air Transport N Financial Capital Markets Y Transportation Ground Transport N Financial Consumer Finances N Transportation Infrastructure N Diversified Financial Financial Services Y Transportation Maritime Transport N Financial Insurance Y Utility Electric Utility Y Financial Real Estate Y Utility Gas Utility Y Health Care Equipment & Services N Utility Independent Energy N Health Care Pharmacy N Utility Multi Utility N Industrial Capital Goods N Utility Water Utility N Industrial Services N Table 2: GICS (Corporate) Sectors and Sub-sectors defined in the Barra Fixed Income Models. The third column ( Factor? ) indicates whether there is a factor for the sector and sub-sector in any model market. See Appendix C: BIM301 Credit Model Factor Tables for a detailed presentation of all the factors. N Sector Description Agency Covered Bonds Covered Bonds Covered Bonds Covered Bonds Municipal Municipal Sovereign Supranational Sub-Sector Description French Obligations Foncières German Jumbo Pfandbriefe German Non-Jumbo Pfandbriefe Spanish Cédulas Hipotecarias Provincial Table 3: Non-Corporate Sectors, including Covered Bonds. 12 of 27

13 Appendix B: Model Performance Comparison Total Risk RAD Robust Total Risk RAD Active Risk RAD Market BIM207 BIM301L BIM301S BIM207 BIM301L BIM301S BIM207 BIM301L BIM301S U.S.A Euro zone Canada U.K Switzerland Australia Overall Table 4: Rolling Absolute Deviation of the bias statistic for the test portfolios in different markets for the sample period. All Markets, BIM 207 All Markets, BIM 301L All Markets, BIM 301S RAD RAD RAD Total Total Total Robust Robust Robust Active Active Active Table 5: Rolling Absolute Deviation of the bias statistic, average over portfolios by year, all markets combined. US, BIM 207 US, BIM 301L US, BIM 301S RAD RAD RAD Total Total Total Robust Robust Robust Active Active Active Euro zone, BIM 207 Euro zone, BIM 301L Euro zone, BIM 301S RAD RAD RAD Total Total Total Robust Robust Robust Active Active Active Table 6: Rolling Absolute Deviation of the bias statistic, average over portfolios by year and market (part 1). For RAD, a lower statistic indicates a more reliable risk forecast. We find that for nearly every market, year, and test, BIM301S outperforms BIM301L, which in turn outperforms BIM207. For a detailed explanation of the statistics see Appendix E: Model Performance Statistics. For a detailed discussion of the model results see the section Model Performance Comparison. 13 of 27

14 Canada, BIM 207 Canada, BIM 301L Canada, BIM 301S RAD RAD RAD Total Total Total Robust Robust Robust Active Active Active UK, BIM 207 UK, BIM 301L UK, BIM 301S RAD RAD RAD Total Total Total Robust Robust Robust Active Active Active Switzerland, BIM 207 Switzerland, BIM 301L Switzerland, BIM 301S RAD RAD RAD Total Total Total Robust Robust Robust Active Active Active Australia, BIM 207 Australia, BIM 301L Australia, BIM 301S RAD RAD RAD Total Total Total Robust Robust Robust Active Active Active Table 7: Rolling Absolute Deviation of the bias statistic, average over portfolios by year and market (part 2). For RAD, a lower statistic indicates a more reliable risk forecast. We find that for nearly every market, year, and test, BIM301S outperforms BIM301L, which in turn outperforms BIM207. For a detailed explanation of the statistics, see Appendix E: Model Performance Statistics. For a detailed discussion of the model results, see the section Model Performance Comparison. 14 of 27

15 Total Risk % In idence Robust Total Risk % In idence Active Risk % In idence Market BIM207 BIM301L BIM301S BIM207 BIM301L BIM301S BIM207 BIM301L BIM301S U.S.A Euro zone Canada U.K Switzerland Australia Overall Table 8: Total of % of portfolios within confidence bounds for the test portfolios in different markets for the sample period. All Markets, BIM 207 All Markets, BIM 301L All Markets, BIM 301S Total Total Total Robust Robust Robust Active Active Active Table 9: Percentage of portfolio bias statistics within confidence bounds by year, All Markets combined. US, BIM 207 US, BIM 301L US, BIM 301S % IN CONF Total Total Total Robust Robust Robust Active Active Active Euro zone, BIM 207 Euro zone, BIM 301L Euro zone, BIM 301S Total Total Total Robust Robust Robust Active Active Active Table 10: Percentage of portfolio bias statistics within confidence bounds, by market (part 1). For bias tests, a higher percentage of portfolios within the confidence bounds indicates a more reliable risk forecast. We find that for nearly every market, year, and test, BIM301S outperforms BIM301L, which in turn outperforms BIM207. For a detailed explanation of the statistics, see Appendix E: Model Performance Statistics. For a detailed discussion of the model results, see the section Model Performance Comparison. 15 of 27

16 Canada, BIM 207 Canada, BIM 301L Canada, BIM 301S Total Total Total Robust Robust Robust Active Active Active UK, BIM 207 UK, BIM 301L UK, BIM 301S Total Total Total Robust Robust Robust Active Active Active Switzerland, BIM 207 Switzerland, BIM 301L Switzerland, BIM 301S Total Total Total Robust Robust Robust Active Active Active Australia, BIM 207 Australia, BIM 301L Australia, BIM 301S Total Total Total Robust Robust Robust Active Active Active Table 11: Percentage of portfolio bias statistics within confidence bounds, by market (part 2). For bias tests, a higher percentage of portfolios within the confidence bounds indicates a more reliable risk forecast. We find that for nearly every market, year, and test, BIM301S outperforms BIM301L, which in turn outperforms BIM207. For a detailed explanation of the statistics, see Appendix E: Model Performance Statistics. For a detailed discussion of the model results, see the section Model Performance Comparison. 16 of 27

17 United States Canada Portfolio Benchmark Portfolio Benchmark Corporate NA Corporate NA Auto Corporate Banks Corporate Banks Corporate Corporate Non-Financial Corporate Consumer Discretionary excl. Auto Corporate Diversified Financial Services Corporate Consumer Staples Corporate Energy Corporate Corporate Non-Financial Corporate Financial Corporate Diversified Financial Services Corporate Industrial Corporate Energy Corporate Transportation Corporate Financial Corporate Utility Corporate Health Corporate Non-Corporate NA Industrial Corporate Muni Non-Corporate Materials Corporate Provincial Non-Corporate Technology Corporate United Kingdom Telecommunications Corporate Portfolio Benchmark Transportation Corporate Corporate NA Utility Corporate Banks Corporate Non-Corporate NA Consumer Discretionary Corporate Agency Non-Corporate Consumer Staples Corporate Foreign Agency Non-Corporate Corporate Non-Financial Corporate Sovereign Non-Corporate Diversified Financial Services Corporate Supranational Non-Corporate Energy Corporate High Yield NA Financial Corporate High Yield Consumer Disc. High Yield Industrial Corporate High Yield Consumer Staples High Yield Telecommunications Corporate High Yield Energy High Yield Transportation Corporate High Yield Finance High Yield Utility Corporate High Yield Health High Yield Non-Corporate NA High Yield Industrial High Yield Sovereign Non-Corporate High Yield Materials High Yield Supranational Non-Corporate High Yield Telecom. High Yield Covered Bonds NA High Yield Utility High Yield High Yield NA Euro zone Switzerland Portfolio Benchmark Portfolio Benchmark Corporate NA Corporate NA Auto Corporate Banks Corporate Banks Corporate Corporate Non-Financial Corporate Consumer Discretionary excl. Auto Corporate Diversified Financial Services Corporate Consumer Staples Corporate Financial Corporate Corporate Non-Financial Corporate Non-Corporate NA Diversified Financial Services Corporate Muni Non-Corporate Financial Corporate Covered Bonds NA Health Corporate Australia Industrial Corporate Portfolio Benchmark Materials Corporate Corporate NA Telecommunications Corporate Banks Corporate Transportation Corporate Corporate Non-Financial Corporate Utility Corporate Energy Corporate Covered Bonds NA Financial Corporate NonPfandCovered Covered Bonds Non-Corporate NA Pfandbrief Covered Bonds Muni Non-Corporate High Yield NA High Yield Finance High Yield Non-Corporate NA Agency Non-Corporate Muni Non-Corporate Sovereign Non-Corporate Supranational Non-Corporate Table 12: Test Portfolios, by market. 17 of 27

18 Appendix C: BIM301 Credit Model Factor Tables US Model Factors Category Sector Sub-Sector AAA AA A BBB BB B CCC Auto Auto A Auto BBB Cons. Disc. BB Durables & Apparel Dur. App. BBB Consumer Cons. Disc. Media Media BBB Media BB Cons. Disc. B Discretionary Cons. Disc. A CCC Retail Retail BBB Cons. Disc. BB Services Cons. Disc. BBB Services BB Corporate Cons. Stap. Consumer Staples Corporate Cons. Stap. A Cons. Stap. BBB Cons. Stap. B-BB Non- CCC Non-Financial Industrial Financial Industrial A Industrial BBB Industrial BB Industrial B Industrial CCC AA Materials AAA Materials A Materials BBB Materials BB Materials B Materials CCC Telecommunication Telco. A Telco. BBB Telco. BB Telco. B Telco. CCC Technology Technology Technology A Technology BBB Technology B Corporate BB Non-Financial Transportation Transportation Transportation Transportation B-BB CCC A BBB Energy Energy AA-AAA Energy A Energy BBB Energy BB Energy B Energy CCC Healthcare Health AA-AAA Health A Health BBB Health BB Health B Health CCC Banking Bank AAA Bank AA Bank A Corporate Non-Financial Financial Non-Corporate & Regulated Markets Diversified Financial Divers. Fin. Divers. Fin. Services Servc. AA Servc. A Capital Markets Financial BBB Financial Financial Financial BB Financial B CCC Capital Markets A AAA Consumer Finance Financial AA Insurance Financial A Real Estate Real Estate BBB Agency (Domestic) Agency NA Local & Provincial (Domestic) NA Muni AA Muni A Muni BBB NA NA NA Agency (Foreign) Foreign Foreign Local & Provincial Agency & Agency & A BBB BB B (Foreign) Local AAA Local AA CCC Covered Bonds AAA AA Utility Utility A Utility BBB Utility BB Utility B Sovereign (NA) Sovereign & Supranational Supranational (NA) AA-AAA Sovereign & Supranational BBB-A Sovereign & Supranational CCC-BB 18 of 27

19 Euro zone Model Factors Category Sector Sub-Sector AAA AA A BBB BB B CCC Auto Auto BBB Consumer Media Cons. Disc. A Media BBB Discretionary (Other) Cons. Disc. BBB Cons. Disc. BB Cons. Disc. CCC-B Industrial Corporate Industrial A Industrial BBB Industrial BB Industrial CCC-B Materials AAA Non- Materials A Materials BBB Materials CCC-BB Telecommunication Financial AA Telco. A Telco. BBB Telco. CCC-BB Technology Corp. Non-Fin. A Corp. Non-Fin. A Corporate Consumer Staples Cons. Stap. A Cons. Stap. BBB Corporate Non- Non-Financial Healthcare Health BBB-A Financial B BB Energy Energy AA-AAA Energy BBB-A CCC Transportation Transport AA-AAA Transport BBB-A Transport CCC-BB Diversified Financial Divers. Fin. Divers. Fin. Divers. Fin. Servc. Divers. Fin. Services Servc. AAA Servc. AA A Servc. BBB Financial Capital Financial BB Financial CCC-B Capital Markets A Capital Markets Financial AAA Markets AA Financial BBB (Other) Financial AA Financial A Agency Agency AAA Agency AA Agency A Sovereign Sovereign AA-AAA Sovereign A France French Cov. Bonds AAA Spain Spanish Cov. Covered Bonds AAA Bonds AA Covered Bonds (Other) Covered Bonds AAA BBB Jumbo German Jumbo Jumbo A BB B CCC Pfand.AAA Pfand. AA Non-Jumbo German Non-Jumbo Non-Jumbo Pfand. AAA Pfand. AA Local & Provincial Muni AAA Muni AA Supranational Supranational AAA AA Utility AAA Utility A Utility BBB Corporate Non-Financial Financial Non-Corporate & Regulated Markets 19 of 27

20 United Kingdom Model Factors Model Insight Category Sector Sub-Sector AAA AA A BBB BB B CCC Consumer Discretionary Cons. Disc. BBB-A Cons. Disc.BB- CCC Consumer Staples Cons. Staples Cons. Staples A Corporate BBB Industrial AAA Non- Industrial BBB-A Materials Financial AA Materials BBB-A Telecommunication Telco. A Telco. BBB BB B CCC Healthcare Corporate Non- Corporate Non- Technology Financial A Financial BBB Transportation Transport AA-AAA Energy Energy AA-AAA Energy BBB-A Banking Bank AAA Bank AA Bank A Financial Diversified Financial Divers. Fin. Services Financial AAA Servc. AA Financial A Financial BBB Financial CCC-BB (Other) Financial A Sovereign Sovereign Sovereign AAA AA Supranational Supranational AAA A BBB Covered BB B CCC Covered Bonds Bonds AAA AA Agency Agency AAA Local & Provincial AAA Utility Utility A Utility BBB Corporate Non-Financial Financial Non-Corporate & Regulated Markets 20 of 27

21 Canada Model Factors Model Insight Category Sector Sub-Sector AAA AA A BBB BB Consumer Discretionary Consumer Disc. BBB-A Energy Energy A Energy BBB Industrial Industrial BBB-A Telecommunication Corporate Non-Financial AA- Telco. BBB-A Transportation AAA Transportation BBB-A Consumer Staples Healthcare Corporate Non- Corporate Non- Materials Financial A Financial BBB Technology Banking Bank AA Bank A Financial Diversified Financial Divers. Fin. Financial AAA Financial BBB Services Servc. AA Financial A BB (Other) Financial AA Provincial Provincial Local & Provincial Provincial AAA AA Provincial A Muni Muni AAA Muni AA Agency (NA) Agency AAA BBB Supranational Supranational A (NA) AAA Covered Bonds Sovereign (NA) AA Electric AAA Electric Utility A Utility Gas Gas Utility A Utility BBB (Other) Utility A Corporate Non-Financial Financial Non-Corporate & Regulated Markets 21 of 27

22 Switzerland Model Factors Category Sector Sub-Sector AAA AA A BBB Consumer Staples Consumer Staples. BBB-A Materials Materials BBB-A Telecommunication Telco. BBB-A Consumer Corporate Corporate Discretionary Non-Financial Non- Healthcare AAA Financial AA Corporate Non- Corporate Non- Industrial Financial A Financial BBB Technology Transportation Energy Energy AA-AAA Banking Bank AAA Bank AA Bank A Financial Divers. Fin. Divers. Fin. Diversified Fin. Serv. Servc. AAA Servc. AA Financial A Financial BBB (Other) Financial AAA Financial AA Local & Provincial Muni AAA Muni AA Muni BBB-A Agency Agency AA-AAA Pfandbriefe Covered Bonds AAA A Supranational AA BBB Supranational AAA Sovereign AAA Utility Utility A-AA Corporate Non-Financial Financial Non-Corporate & Regulated Markets Australia Model Factors Category Sector Sub-Sector AAA AA A BBB Energy Energy AA Consumer Discretionary Consumer Staples Healthcare Corporate Non- Corporate Non- Industrial AAA Corporate Non- Financial A Financial BBB Materials Financial AA Technology Telecommunication Transportation Financial Banking Bank AAA Bank AA Bank A (Non-Bank) Financial AAA Financial AA Financial A Financial BBB Agency (NA) Agency AAA Covered Bonds AAA AA Local & Provincial Muni AAA Muni AA Sovereign (NA) Sovereign AAA A BBB Corporate Non-Financial Financ ial Non-Corporate & Regulated Markets Suprnational Supranational (NA) AAA Utility AAA AA 22 of 27

23 Appendix D: Overview of Barra Fixed Income Model Factor models decompose individual asset returns into components that are common to all assets, plus a residual, idiosyncratic component for each asset. This approach allows intuitive attribution of return and risk to the common systematic components, as well as robust calculation of correlations between a relatively small number of factors. 5 The Barra Fixed Income Model includes common factors such as interest rates and credit spreads, and a residual return component that is due to non-systematic idiosyncratic asset characteristics, such as asset-specific liquidity. The attributed asset return is the excess return after subtracting the predictable return component (due to the passage of time) from the price return. The following diagram shows the hierarchical decomposition of asset excess return. Every fixed income asset is exposed to factors in a single market determined by its currency. The common factor return of a bond is equal to the return to the term structure of interest rates multiplied by exposure to the interest rates, plus the returns to the credit spread factors multiplied by the exposure to credit spreads. That is, the excess return of a fixed income asset is expressed by: Markets have up to three nominal interest rate risk factors (. These three factors capture the principal movements of term structure: shift (parallel movements), twist (steepening or flattening), and butterfly (curvature). Shift, twist and butterfly are commonly abbreviated as STB. The Euro market is distinct: it has three term structure factors for each country, and three that describe average changes in rates across the Euro zone. Exposures to STB factors are analogous to duration, where the shock is a unit change in the level, steepness, or curvature of the term structure. STB returns are calculated in a regression of the excess returns onto the exposures. (Some smaller markets have only shift and twist, or only shift factors.) Specific risk for government bonds is modeled by an average of the cross-sectional spread volatility of bonds in the estimation universe. Every market has either a swap spread factor, or three swap shift, twist and butterfly factors ( -- for details see BIM301 Enhanced Developed Market Term Structure Models, Demond et al. [2011]. This represents a common baseline credit risk across the entire market. The only instruments not exposed to swap spread are sovereign issues. The US dollar, Euro, Sterling, Swiss franc, Canadian dollar and Australian dollar markets have additional detailed credit spread factors ( ) that are measured over the swap curve. These factors represent the spread risk of specific market segments (e.g., financial, industrial, etc.). The credit factor exposure of a bond is its spread duration. Credit factor returns are 5 For more details on factor modeling, see the Barra Risk Model Handbook and The Barra Integrated Model (BIM301) Research Notes, Shepard, [2011] 23 of 27

24 averages of bond spread returns within market segments. For markets with detailed credit models, the specific risk model captures the price risk due to rating change and defaults, using rating transition matrices and spread levels from the estimation universe. For markets with no detailed credit model, the specific risk is the risk of a treasury of the same duration plus a term proportional to the bond s spread over the swap curve. Since, by construction, factor and specific returns are uncorrelated, and since specific returns are also uncorrelated with each other (except for correlation of one for bonds from a common issuer), the portfolio risk is modeled as: with where h = the vector of portfolio holdings = the covariance matrix of asset returns = the covariance matrix of factor returns = the diagonal matrix of specific variances Σ Σ Φ Δ 24 of 27

25 Appendix E: Model Performance Statistics Bias Tests Bias statistics are a convenient metric to test the forecasting performance and consistency of a model. Bias tests can be performed on portfolios, single assets, or single-factor return series; they compare risk forecasts with realized, out-of-sample returns. First, out-of-sample z-scores,,, of the entities (assets, factors, portfolios) under investigation are calculated:, Here, is the risk forecast at time,, and is the realized return of the entity at, where is the forecast horizon. If forecasts are perfect, then the standard deviation of the z-scores,,, is 1. A standard deviation below 1 is indicative of over-forecasting, and a standard deviation above 1 indicates under-forecasting. The bias statistic,,, is defined as the sample standard deviation of the z-scores over a lookback horizon,, where denotes the mean z-score for each window:, 1 1, The finite size of the lookback horizon implies that, even for perfect forecasts, the bias statistic will deviate from one. Under the simplifying assumption that the returns,, are normally distributed, a 95% confidence interval can be given for the bias statistic: 1 2/,1 2/ All bias test results presented in this report use a short lookback horizon of 12 months. Though a short lookback horizon implies a broader confidence interval,, it would be incorrect to assume that longer lookback horizons automatically increase the accuracy of the test. Instead, individual bias statistics,,, that span over longer periods can be deceptively close to 1 if they average over a period that contains phases when the model both under-forecasts and over-forecasts. The percentage of all bias statistics within the confidence interval,,is denoted by. This indicates the forecast accuracy of the model for portfolio P,,,, Robust Bias Tests The main disadvantage of bias statistics with a short lookback horizon is a high sensitivity to outliers. Assuming 12 and 0, the upper limit in corresponds to a maximum of 21.8 for the sum of the squared z-scores. If the sample contains a single outlier with 5, then the 12-month bias statistic will always be out-of-confidence, regardless of the other 11 z-scores used. This means that a single outlier masks the forecasting performance over two years of rolling 12-month tests. To control for the influence of outliers, we calculate robust bias statistics by using winsorized z-scores: / 25 of 27