Downloading . . . If the document you requested does not automatically load, please click here. PITFALLS IN USING THE S&P BOGEY FOR FINANCIAL ANALYSIS AND PORTFOLIO MANAGEMENT Somnath Basu and Ron Rizzuto Abstract The S&P 500 Index is a benchmark that is widely accepted and used though much maligned. In this study we add a chapter to the index's detractors by investigating and revealing an industry bias built-in to the index. By the nature of the construction of the S&P there exists sectoral imbalances. By constructing 'S&P Comparable' indexes we establish first that such an industry bias does exist and then we probe deeper to find the sectors that are favored or disfavored. We find that the problem is not insevere and neither does it go away with time. INTRODUCTION The S&P 500 is one of the most widely used stock indexes and, invariably, the return of the S&P 500 is included in reports that summarize the performance of institutional money managers. Some portfolio managers use it as their primary "bogey" or benchmark for performance. Other equity managers, who may use different indexes for benchmarks, still use it as a quick reference common denominator or yard stick of performance. The firms included in the S&P 500 are large in size. Inherently, some industries are mainly populated by large firms (e.g., petro-chemicals, auto manufacturing) while others are characterized by smaller sized firms (e.g., textile, furniture). Hence, constructing a portfolio composed of large firms may not only induce a large firm bias but also a non-random industry bias. The problems associated with benchmark construction have been studied. For example Fridson [1992] considers the effects of high-yield indexes while Bailey [1992] notes some of the issues concerning the validity of benchmarks. Gould [1991] considers the impact of weighting procedures in the construction of indexes. In this study, the particular nature of the industry bias, as well as the size bias in the S&P and problems associated with them are established. The consequences of a biased benchmark is especially painful when it is used to evaluate the performance of portfolio managers. The risk of such consequences is even greater for managers who may specialize in certain industries or sectors and whose performance may be understated or overstated due to the built-in industry bias in the benchmark itself. Another undesirable outcome is potential if managers make investment decisions of asset allocation, stock selection or risk management based on the benchmark since such decisions will be suboptimal or even erroneous. Although the S&P 500 is an invaluable tool in the management of portfolios, our study indicates there are substantial risks in dependence on a biased index. Biases in benchmarks and performance evaluation are also addressed by Allen [1991] and Feinstein [1990]. Over the years, several writers have criticized the S&P 500 on a number of points. Roll [1977, 1978, 1980, 1981] raised concerns about this index because: 1) it, like all other conceptual stock indices, is not "the" true market index; 2) the lack of a "true" or "optimized" index introduces benchmark error into performance evaluations; and 3) the S&P 500 is not a market value weighted index as suggested by CAPM theory. Nagorniak [1982] and Etzioni [1992] conclude that benchmarks are easy to "game" against and that it may not be appropriate to use any benchmark in evaluating a portfolio manager's individual security or industry selection skills without first considering the benchmark quality to the portfolio that is evaluated. The main problem with a biased benchmark is the problem of inefficiency. From the theory of finance we understand that the benchmark portfolio (e.g. the market) must be an efficient portfolio composed of risky securities. Thus, if the benchmark is not an appropriate proxy for the true market portfolio then decisions on risk and return will be erroneously made; since risk-return decisions are fundamental to many other related financial decisions, the errors will be transformed widely. Hence, the need for well-constructed benchmarks. This paper expands on prior research and provides further indictments against the S&P 500. The S&P 500's tilt towards large capitalization stocks has been well documented (see, for example, Haugen and Baker [1991], Collins and Fabozzi [1990], Banz [1981], Blume and Stambaugh [1983], Chan et al. [1985], Christie and Hertzel [1981], Keim [1983], Reinganum [1981] and Roll [1981]). These authors note that the risk (betas) measures of small firms may be understated. Other authors (see, for example, Dumas and Sengupta [1991], Sengupta [1990] and Gibbons, Ross and Shanken [1989]) have questioned the efficiency of index portfolios. Multi (industry)-index models too have been studied extensively (see, for example, Cohen and Pogue [1967], Elton and Gruber [1970], Farrell [1974], Fertuck [1975] and King [1966]), but the industry-bias in the S&P and its effects have not been investigated. We first reveal the nature of the industry and size bias by constructing for comparison three 'S&P comparable' indexes. After establishing the nature, we investigate deeper the extent of the bias by comparing industry groups to the S&P. This latter comparison allows us to segregate the sectoral departures built into the S&P which, in turn, can provide the necessary guidelines for the proper use of the benchmark. The first section of this paper describes the data and methodology used to construct the 'comparable' indexes while section two illustrates the nature of the biases. Section three explores the sectoral imbalance built into the S&P and section four concludes. DATA AND METHODOLOGY The data used in the construction of the indexes were extracted from the financial data of the 7,200 firms included in the Compustat tapes. Since the main purpose of this study is to critically evaluate the S&P 500 index, constructing indexes strictly comparable to the S&P index was integral to this study. We constructed three indexes and calculated monthly returns on those indexes for the period 1972-1990. The Compustat tape for 1991 contains monthly financial data for 7,200 firms over the previous 20 year period. The limitation from data in our study is an extension of the limitation of data available in the Compustat tapes. For a more complete description on the Compustat tapes, see McElreath and Wiggins [1984]. These indexes were created so that at the beginning of each month, the values of each index was exactly equal to the S&P's market value. Note that the Compustat tapes specially code the firms included in the S&P 500. Any additions and deletions from the index is updated monthly as are stock splits, stock dividends, treasury stocks and new stock issues. This provision allows a monthly reconstruction of the index from the data included in the tapes. The following is a detailed description of the steps involved in the construction of the indexes. Step I: Determination Of The Inputs Of The S&P Index The S&P index is constructed as follows: where Pi = price per share, Qi = # of shares outstanding, for the ith S&P 500 firm. The Base Market Value (BMV) is the (P0Q0 (1941-1943), an average value for all the industry groups and is computed for the period 1941-19431. As the base period value is relatively constant, the index number (the S&P index) reflects only fluctuations in current market value. In this relative form an index number attains its maximum usefulness for statistical purposes. At the beginning and end of each month, the aggregate market value of the S&P index (S&P MV) was determined from all the firms included in the index. Using the S&P index value, the BMV at the beginning and end of each month was then derived. Step II: Calculation Of Industry Aggregates We determined the following at the beginning of each month: a) The Total Market Value (TMV) for all the firms on the Compustat tapes (7,200 firms) as: where P = price per share, Q = # of shares outstanding. b) The Aggregate Market Value (IMV) for 270 industry groups (3-digit SIC code). where n = # of firms in each 3-digit SIC industry group. c) The Industry Weights, xj: xj = IMVj/TMV d) The Representative Industry Value (RIV): RIVj = xj ( S&P MV such that (j RIVj = S&P MV, j = 1,.., 270. Step III: Construction Of Industry Value-Weighted Indexes. a) Large Firm Index: At the beginning of each month the market values for firms in each industry group was sorted by size. Starting with the largest firm, the market values of the firms were summed until the aggregate exactly equaled the RIV for the industry. Within this scheme of construction, it was necessary to use a fraction of the last firm included in the index such that the aggregate market value of the firms in the large-firm index exactly equalled the RIV of that industry. For a more detailed description of the construction of the large, random and small firm index, see Appendix. Using the Base Market Value (BMV) of the S&P index and using a construct similar to the S&P index, an index is derived for the large-firm index. Thus, this index allowed comparison with the S&P index and provided a measure of the extent of industry-wise large-firm bias inherent in using the largest capitalization stocks. This comparison is valid since each industry is represented in the index in proportion to its share in the total market value, only the largest capitalization firms are included in the index and the Base Market Value and constant are identical to the S&P. b) Small Firm Index: This index was built similarly to the large firm index except that the firms included in the index are the smallest firms in each industry group. This index, along with the large firm index, provides a comprehensive measure of the large firm bias in the S&P index. c) Random Index: This index was constructed similarly to the other two indexes, but the firms included from each industry group were chosen randomly, by a random number generator. As in the other two indexes, the number of firms representing each industry group depended on the RIV of each group though, ex ante, we may expect the number of firms in the random index to be lesser than the small firm index; by design the random index could include firms of all sizes. Since an industry bias in the S&P is essentially non-random, this index provides a measure of the extent of non-randomness in the S&P index. Step IV: Returns And Other Calculations. Using market values from the end of each month, index values were recalculated at the end of each month. Again, for comparisons, each end-of-month index value was calculated using the constant and the BMV of the S&P 500. Such a scheme ensured the validity of the comparisons. The percentage returns for the indexes were then computed from the index values at the beginning and end of each month. The returns took into consideration that indexes contain fractions of one firm in each index in every month. For a detailed numerical example of the index construction and measurement of returns see Appendix. By utilizing the three "S&P comparable" indexes, we now compare and contrast these different indexes and very quickly discover some of the problems inherent in the use of the S&P 500 index. S&P 500 PROBLEM AREAS The market value of the S&P 500 firms (S&P MV) constitutes a very consistent seventy percent (small variance) of the total market value (TMV) of the 7,200 firms, in each month. We can appreciate the largeness of the firms included in the S&P when we note that 6.9 percent of the firms (500/7,200) constitute seventy percent of the total value. Moreover, these large firms have grown larger over the years as the TMV itself has increased over the years of the study. Figure 1 charts the number of firms required by each of the constructed indexes to equal in value the S&P MV, while Table 1 enumerates the same results. There are a few interesting points to note from Table 1 and Figure 1. First, the number of small or randomly chosen firms required to replicate the S&P MV has monotonically increased over the years. In 1972, only 1,597 firms were required by the small index and 1,215 firms by the random index to replicate the S&P MV. In 1990, 5,366 and 3,184 firms, respectively, were required for replication. This indicates that the firms included in the S&P index themselves have grown significantly over the time period. It should be noted that the requirements of a firm to be included in the Compustat Tapes have not changed significantly over the same time period. Second, the number of firms required by the large cap index to replicate the S&P MV fluctuates considerably during the study period (1972-1990). Since the large cap index allows us to establish the existence of a size and industry-bias, the fluctuations in the replication number of firms may imply that the bias itself may fluctuate. This may, therefore, pose a problem if adjustments are to be made. Table 2 shows the mean monthly and the annualized returns for each of the indexes over the entire period. Note that the small index has the highest return, while the S&P and the random indexes have approximately equal returns. The large index shows the lowest return for the entire period. The small firm index returns have the largest variance while the S&P 500 has the smallest variance. The risk-returns of the indexes are fairly consistent with other tabulations, including Ibbotson and Sinquefield [1982] and with data published by the U.S. Department of Commerce in their Survey of Current Business series as well as the data published in the CRSP tapes. Table 3 shows the mean and standard deviations of monthly returns on each of the indexes. We observe that the index mean returns were similar for each index (and higher than returns for the entire period) and that the S&P index return was the smallest of the four indexes. Table 4 shows the result of correlation analysis between index returns. It should be noted that correlation analysis is a test of the closeness of association between different series. The high correlations reported in Table 4 are expected, a priori. Each of the indexes represent 70% of the Total Market Value and hence, market-wide movements would similarly affect all indexes. Moreover, the smallest correlations are between the small firm index returns and both the large firm and S&P index returns. This result is not unexpected either because of the size difference of component firms in the indexes. SECTORAL TESTS From the discussion so far we conclude that there may exist a serious size and industry bias in the S&P index, and a biased benchmark is detrimental in evaluating either the performance of managers or even the performance of industry sectors. Therefore, to extend the analysis we try to establish the specifics of this bias, i.e. for which industries does the S&P significantly depart as a benchmark, the nature of these departures over time, etc. However, it is not feasible for comparison to include 270 different industry groupings but rather to aggregate the firms under a broader framework. We aggregated the 7200 firms into 28 industry groups using the broad headings from the compustat SIC listings guide as a basis for segregation. The summary statistics of the market value for each of the 28 groups are shown in Tables 5A and 5B. The results in these tables for both absolute and changes in market value were computed for the period 1981-1990. For every S&P firm there are on average 14 firms in an industry. While the S&P does not have a single firm in group 1, its firms in group 18 are nearly 33 times greater than the size of average firms in this group. In 8 of the 28 groups, S&P firms are greater than the size of an average 14 firms in that sector. From the correlations between industry and S&P market value over the time period we can observe how stable this size relationship has been. In 19 of the 28 groups, the correlation in absolute market value is greater than 0.80. Earlier, in Figure 1 and Table 1 we observed that the number of firms needed to construct the large firm index fluctuated in relation to the S&P. The correlations below 0.80 in absolute value indicate the possibility of sectoral accounting of such fluctuations. Further, the correlation in changes in market value (returns) between all firms in an industry and the S&P firms in that industry are significantly smaller than those for absolute market values. This indicates that the developments in a sector may lead or lag the developments in large firms in those sectors. In sum, the results in Table 5A reveal the direction of the bias and the direction for further investigation. In Table 5B we extend the comparison to include time patterns. We observe that for each year the correlation between industry and S&P in mean market values is around 0.80 whereas the correlation in the mean changes in market value is significantly lower. This reinforces the observation from Table 5A that general changes in firms in an industry may not be related in time to the changes in the large firms in that industry. The pairwise T-statistic for changes in market value shows significant differences (99%) in how the value of S&P firms in an industry group changed as compared to all the firms in that industry for the years 1982, 1983, 1987 and 1988. The overall difference (pairwise over 252 observations, i.e. 28 groups for 9 years) is also significant at the 95% level. In general we observe that changes in value for S&P firms, by industries, have not reflected very well the changes in value for all firms in those industrial sectors. Table 6 shows the results from a variance analysis (ANOVA) test of the data; the results are a composite of the results in Tables 5A and 5B. The ANOVA procedure allows us to observe whether changes in market values are dependent on the industries' firms may belong to and whether this dependence persists in time. The procedure also allows us to measure (F-statistic) the strength of any dependence that may exist. The F-test also allows us to statistically qualify our observations. In the ANOVA test we use F-statistics to determine how significant the differences were between market values over the time period under study. The analysis is conducted in each case for both absolute and change in market values. The model F-statistic denotes the significance in the relationship between the dependent variable (market value) and the independent variables (industry, year). In Table 6 we observe from the significant statistics that the market value of all firms in an industry is significantly different for sectors, i.e., some sectors are much bigger than others or that the industry bias is built into market values. Further, the overall market value of sectors change in different ways in time. For S&P firms, large firm market values are different by sectors as well but on the whole (model) these differences are not so significant. On the other hand, changes in value for S&P firms show a very different picture. These results indicate that all large firms changed in value over the time period significantly but similarly. So far the observations from Table 5A, 5B and 6 have further clarified the nature of the industry and size bias. One point that is strongly indicated is that the largest firms are in a world of their own; their differences with their universe is of extreme magnitude. Therefore, we seek to explore deeper into the particulars of these differences. We begin with a scatterplot (Figure 2) of the mean market values of S&P firms and all firms by industry group. From the figure we observe that: a) The industry groups which contain the smallest of the 500 S&P firms have market values in consonance with market values for firms in that industry; b) Industry groups which contain medium sized S&P firms are underrepresented in the S&P, i.e. such industry groups have more value than what their S&P firms contain; and c) Industry groups with the largest S&P firms are overrepresented in the S&P, i.e. such industry groups have less value than what their S&P firms contain. The above specifications imply that managers and analysts of industry groups included in observation a) above may suitably use the S&P benchmark whereas for the other two subsets, corrections are required. We now seek to determine exactly the industries falling in each of these categories. To do so we conduct a principal components analysis. Principal components analysis is a multivariate technique for examining relationships among several quantitative variables. The technique enables us to summarize data and detect linear relationships. The technique is also useful in exploratory data analysis, in reducing the number of independent variables and reducing the effects of clustering. Principal components analysis allows us to infer the number of important factors (components) affecting and explaining changes in a dependent variable. However, care should be taken in applying this technique; for example, if the components are calculated from unstandardized covariance matrices rather than correlation matrices, then the components may provide biased (biased factor loadings) results. In our analysis we took all precautions necessary to provide impartial results. By the same token we also benefit from the validity of our results as the output of a very powerful and robust technique. The principal components analysis reveals that the first two components account for 97% of the variations in the market value for both firms in the S&P and the industry and that the first component accounts for 89% of all variations. Therefore, an analysis of the first two components is considered sufficient. The first component measures the industry groups that dominate the market in value. Given that the factor loading for each of the years are about equal, the second component may be used to measure which industries have dominated other industries over different time intervals. In the analysis of the components (Table 7 shows the principal component analysis for market value while Table 8 exhibits the same analysis but for changes in market value) we break the 28 groups into 3 categories: the nine most dominating groups, the middle 10 groups and the 9 least dominating groups. This should provide accord with our earlier categorization from the scatterplot. Consider the first component in Table 7. There are two areas of concern. Groups that are important when considered against all other industries in market value (e.e. 4, 7, 20) are not given the same importance in the S&P index whereas for other groups (e.g. 14, 13, 17, 18) the S&P considers them of more value than how the entire market values them. The second area of concern are differences in valuation between moderately and least dominating market value groups. For example, the S&P index favors some groups (12, 5, 26, 25) much more than what their proportionate industry value is and the reverse is true for other groups (i.e. 1, 10, 11). In general, we may say that if a group is much higher (much lower down) up in dominance in the S&P than the industry, that industry is overrepresented (underrepresented) by its large firms in the S&P. The second principal component in Table 7 shows a near perfect fit between the S & P and the industry. This result suggests the existence of an industry life-cycle and in which firm size is irrelevant. Thus, we can say that firms of all sizes in the services industry (eg. 28.27,26,25) had more value in the latter years of this study than in the earlier years. This observation is consistent with the history of the large growth in the services sector during the mid-eighties onwards. The principal components analysis in Table 8 concerns changes in market value. Since percentage changes in market value are essentially market returns, the analysis in Table 8 is important as this allows us to observe the mismatching of returns and therefore, be able to determine the exact industries for which the S&P benchmark was inappropriate and the magnitude of comparison errors. The analysis of the first component in Table 8 shows a tremendous mismatch of returns for an industry and for the returns of the S&P firms in that industry. We will point out a few examples of the above here. Consider the returns from all the firms in the industry groups 28, 21, 25, 27. The largest firms in these industry groups had very high returns and for the S & P, these groups show up right at the top. However, most of the other firms in these groups did not fare as well and subsequently these industry returns are far down the industry list. The opposite is also true for other groups (e.g. 5, 7, 12, 17). Such drastic differences have obviously serious implications. The second component indicates which groups' returns were highest in the latter years as compared to earlier years, i.e., lead-lag between industry and large firm. One striking observation is that the S&P very inadequately represents the changing values in industry sectors over time. For example, most industries which changed in value greatly in the latter years were considerably under-represented in the S&P. The reverse is also true. The inflexibility of the S&P structure (its inability to easily change the component firms) is clearly detrimental to its role as a benchmark. However, the consequences of such a tainted benchmark are much more damaging. To summarize the results of the principal components analysis, we tabulate its main inferences in Table 9. These troubling inferences can be divided primarily into three observations: a) industry groups that are under or overrepresented by the S&P; b) inability of the S&P to accurately represent changes in sectoral values; and c) the combined effect of a) and b) which shows up in leads-lags in growth of industries to the S&P firms. CONCLUSION The S&P index is a widely used benchmark for reasons of liquidity, investability (index funds) and its acceptance by plan sponsors. It is also a benchmark that will probably survive all its detractors. Therefore, we consider it more appropriate to interpret the results of this study more in the spirit of a guide in the benchmark's use. Managers measuring performance vis-a-vis the S&P 500 should be wary of the structure of their portfolios. Caution is especially warranted for managers of medium to small cap funds, sectoral funds and growth funds. The results of this study point to the need for increased understanding of appropriate benchmarks. Active managers, too, must take into consideration the S&P biases. Selection and valuation methods that incorporate a market index (a S&P proxy) or factor models that incorporate both market and industry indexes would not be appropriate for all industrial sectors. We conclude this study by noting that much remains to be done in the area of performance benchmarks and the construction of benchmarks appropriate for different purposes promises to provide a rich avenue for further research. FIGURE 1 Number Of Firms Required To Replicate S&P 500 Index (January 1972 Through January 1991) FIGURE 2 Scatterplot Of Market Values Of S&P Firms To All Firms, By Industry Groups TABLE 1 Number Of Firms Required To Replicate The S&P 500 Index (January 1972 Through January 1991) TABLE 2 Monthly Rates Of Return And Betas For The S&P 500 Index, Large Cap Index, Small Cap Index, And Random Index (January 1972 To January 1991) TABLE 3 Monthly Rates Of Return And Betas For The S&P 500 Index, Large Cap Index, Small Cap Index, And Random Index (March 1986 Through January 1991, 59 Months) TABLE 4 Correlation Analysis (Monthly Returns And Betas, March 1986 To January 1991) TABLE 5A Mean Market Value Comparisons And Correlations Between The S&P Firms And Average Firms By Industry TABLE 5B Comparison Of Absolute And Changes In Market Value Yearly Summary Statistics TABLE 6 Analysis Of Variance Significance Of F-Statistic Test TABLE 7 Principal Components Analysis Market Value TABLE 8 Principal Components Analysis Percentage Change in Market Value TABLE 9 Sector Discrepanciesa ENDNOTE 1. The BMV is occasionally adjusted when companies are substituted in the S&P Index. Since we back-out the BMV in our returns calculations, all changes in the BMV are reflected in our index return calculations. REFERENCES [1] Allen, G., "Performance Attribution for Global Equity Portfolios," The Journal of Portfolio Management, Fall 1991, pp. 59-65. [2] Bailey, J., "Evaluating Benchmark Quality," Financial Analysts Journal, May-June 1992, pp. 33-39. [3] Banz, Rolf W., "The Relationship Between Return and Market Value of Common Stock," Journal of Financial Economics 9, 1981, pp. 3-18. [4] Blume, Marshall E. and Robert F. Stambaugh, "Biases in Computed Returns," Journal of Financial Economics 12, 1983, pp. 387-404. [5] Chan, K.C., Nai-fu Chen, and David Hsieh, "An Explanatory Investigation of the Firm Size Effect," Journal of Financial Economics 14, September 1985, pp. 451-471. 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APPENDIX In this section we exemplify in some detail how the comparative indexes were constructed and returns calculated. Table A1 shows the firms in the pharmaceutical industry (SIC 283X) along with their market capitalization values for the months of July and August, 1989. The firms are sorted by size. TABLE A1 Market Values Of Firms In The Pharmaceutical Industry (SIC 283X) In constructing the large firm index and calculating the returns, we will use RIV (July). We begin by adding the market values of the firms in Table A1 for the month of July until the sum equals RIV283X (July). Table A2 shows the firms included from this industry group for the month of July in the large firm index. Note 1) only 79% of Warner-Lambert Company's market value is included in the index for this month, 2) that this industry's large firms are definitely overrepresented in the S&P 500, and 3) that for this industry, all the large firms are also S&P firms. TABLE A2 Index Firms, July 1989: SIC 283X This procedure is repeated for each of the 270 industry groups. Next, the market values for all the firms included in the large firm index for the month of July is summed and the value of the large firm index is calculated as follows: Equation 1 Large Firm Index (July, '89) = and for the same firms included in the July large firm index, the index value for August, '89 is calculated as: Equation 2 Large Firm Index (July firms, August value) = where N in both Equations 1 and 2 indicate the same firms included in the index for July '89. Now the large firm index returns are calculated as: Large Firm Index Returns (July) = Observe, first that the inclusion of a percentage of the nth. firms for each 4-digit SIC group (e.g. SIC 283X -- 79% of Warner-Lambert Company) implies that by construction design the value of the large firm index for July '89 must equal the value of the S&P Index for July '89, since the sum PiQi = S&P MV. However, the large firm index for August '89 will be different, very generally, from the S&P Index August Value.1 Also note that maintaining the fractional values of firms is not difficult but important for comparison. In each of the 270 industry groups this fractional firm may exist and the sum of their market values may imply that at the beginning of each month, the large firm index may differ substantially from the S&P Index Value. This in turn may defeat the purpose of this study by making the returns comparisons somewhat invalid. Second, the S&P 500 firms do not include ADR's in their composition even though these firms may be of considerable value. In Table A1 note that Glaxo Holdings U.S. market capitalization makes it the second largest in this industry group, but is ignored in the S&P 500. Further, Allegran, Inc. is larger than Genentech, Inc. but not included either for some unknown reason. In the indexes we construct, these flaws do not exist. Finally, the procedure applied to construct the large firm index is repeated for each of the 229 months in the period under study. The same procedure is followed for the small firm and the random firm index as a well. For the small firm index, we sum up market values for this industry for the smallest firm upward till the sum equals the RIV for the industry. For the industry group SIC 283X it is obvious that the industry is characterized by a few gigantic firms and many small firms. We chose this group intentionally to highlight the industry bias problem. Because this industry is overrepresented by its large firms and due to the industry's character, small firms in this industry are grossly underrepresented. For this industry, the small firm index in July contains all firms in this group except Merck and Glaxo. Thus, if small firm behavior departs significantly from large firms, then the small firm index returns will also be significantly different from large firm and S&P returns. The same argument will also hold for firms randomly chosen. For the random index, the firms are chosen by using a random generator. The return calculations follow in a similar manner.