## Abstract

Stock sector investing has become increasingly popular since the creation of the Global Industry Classification System in 1999, yet little academic research has explored sector behavior. This study begins to fill the void by examining how each sector contributes to the optimal risky portfolio over time. Using a much longer time series than previous studies, a crisp definition of sectors, and the perspective of mean Markowitz optimization, this study provides evidence that investors mostly price return and risk information efficiently. But they find it difficult to price sector-level correlation information efficiently into the optimal risky portfolio.

**TOPICS:** Fundamental equity analysis, portfolio theory, statistical methods

Academic studies tend to view the US stock market as efficient, although many studies document evidence to the contrary.^{1} However, even if individual stocks are inefficient enough to reward those who seek the inefficiencies, building the optimal portfolio of these inefficient stocks is costly given the size of the investment set. An alternative that mitigates the cost of building portfolios of individual stocks is to build a portfolio of industry-concentrated mutual funds/ETFs. Yet such an approach can still be challenging due to the large number of industry funds available. So an even more parsimonious and less costly alternative is to build a portfolio of funds that targets specific industry sectors.

Sector investing has become increasingly popular since the creation of the Global Industry Classification System (GICS) in 1999 by MSCI and Standard & Poors. Eakins and Stansell (2007) report that a few sector-concentrated investments were available in the late 1980s and a full spectrum was available by 1995. These funds invest in market sectors, which are distinguishable from industries in that they represent a broader market partition. In particular and from the MSCI website, the GICS originally consisted of 10 market sectors^{2} made up of 24 industry groups, 67 industries, and 156 subindustries.

Arguably, the leader in the innovation of sector-investing vehicles has been Fidelity Investments, which offers both sector mutual funds and exchange-traded funds (ETFs) constructed according to the GICS. But surprisingly little research in the academic literature has documented the behavior of the sectors’ return, risk, and correlation characteristics. While many studies employ individual sectors as laboratories for other research issues, few have studied the behavior of the sectors in cross-section, providing the motivation for this study.

Notable exceptions to the void in sector-level research, however, include O’Neal (2000) who followed Moskowitz and Grinblatt (1999) by studying the extent to which momentum in stock returns reflected industry-level information. He found that industry/sector momentum over the intermediate term is clearly evident. Sassetti and Tani (2006) simulated rotational strategies that invest in sector funds based on various criteria, finding that their strategies seem to consistently outperform the S&P 500 Index. Jacobsen and Visaltanachoti (2009) examined the Halloween effect, also known as the “sell in May” effect, in US stock market sectors. They followed a line of research started by Bouman and Jacobsen (2002) documenting the idea that investors should move their portfolios down the capital allocation line during the summer months (May through October) and up the capital allocation line during the winter months (November through April). Their study finds that the Halloween effect persists but varies widely across sectors.

To examine sector rebalancing strategies, Wyatt and Kee (2014) documented whether or not an upper capitalization limit on any one GICS sector as a percentage of a total portfolio allocation is beneficial for investors. Consistent with the model of overreaction, they found that investors may benefit from rebalancing their portfolios across sectors whenever a particular sector’s weight exceeds 10%. Their study was similar to a study by Eakins and Stansell (2007), who also studied rebalancing sector-allocated portfolios. They found that rebalancing based on a 9% trigger was optimal, but inferior to annual rebalancing. These results were similar to Sturm (2010) who studied State Street’s Select Sector ETFs based on the GICS. He found that an equally weighted portfolio of the nine Select Sector ETFs reliably outperforms the S&P 500 ETF, implying that investors should hold a rebalanced version of the market portfolio.

In summary, documented evidence from prior studies provides mixed interpretations as to the efficient pricing of sector-level information. Adding to the literature and in the spirit of Bouman and Jacobsen (2002), Jacobsen and Visaltanachoti (2009), Eakins and Stansell (2007), Sturm (2010), and Wyatt and Kee (2014), who effectively studied trading strategies using sectors, the primary purpose of this study is to examine the extent to which investors price sector-level information into the optimal risky portfolio (ORP). Unlike previous studies, however, we approach the analysis from the perspective of Markowitz mean–variance optimization and use a much larger sample size.

An immediate problem in studying sectors, and one of the most important differences across prior studies, is the non-uniform definition of sectors. For example, O’Neal (2000) used 31 sector/industry mutual funds, Sassetti and Tani (2006) ran simulations on 41 Fidelity sector funds, Jacobsen and Visaltanachoti (2009) focused on 17 sector portfolios’ value-weighted data from Kenneth French’s website, Wyatt and Kee (2014) used the 10 GICS-defined sectors, Eakins and Stansell (2007) used 19 sector funds, and Sturm (2010) used the State Street 9 sector funds. Additionally, many studies tend to use the labels “sector” and “industry” interchangeably.

Therefore, for clarity and consistent with Wyatt and Kee (2014), this study uses the 10 sectors^{3} defined by the GICS classification for three reasons. First, the GICS provides a crisp distinction between sectors and industries. Second, sector vehicles consistent with the GICS definitions are currently and easily available to investors, especially through Fidelity. Finally, and as pointed out by O’Neal (2000), using sector funds rather than individual stocks as investment vehicles has the advantages of known transaction costs and a realistically manageable set of assets.

Another contribution of this study to the literature is the construction of a much larger sample size. Most of the prior studies only go back to the 1980s or 1990s, because sector investing is a fairly new innovation. For example, O’Neal (2000) studied 10 years from mid-1989 to mid-1999; Sassetti and Tani (2006) studied the 1998–2003 period (6 years); Wyatt and Kee (2014) used data over the period September 1989–February 2009 (almost 20 years); Eakins and Stansell (2007) studied the period December 1995–December 2002 (about 7 years); Jacobsen and Visaltanachoti (2009) examined the May 1998–April 2007 (almost 9 years) period; and Sturm used the period January 1999–December 2007 (almost 9 years). The sample size used in this study encompasses 89.5 years, July 1926–December 2015, by using proxies chosen from Kenneth French’s website.

But the primary contribution of this study is to gather evidence on whether or not investors efficiently price sector-level information into the ORP. To obtain the evidence, we employ Markowitz optimization due to its time-tested importance to the portfolio formation problem and efficient pricing as judged from the following perspectives:

1.

*The frequency with which the sectors are priced into the ORP*. The model of market efficiency predicts that sectors should be priced efficiently in at least 50% of the subsamples, because investors should learn from mispricings. For example, if forecasting error causes actual returns to vary from expected returns in year*t*, then investors should adjust prices in year*t*+ 1 to eliminate the error. Accordingly, even in the presence of forecasting errors, sectors should be priced into the ORP at least 50% of the time. Lower frequencies indicate an inability by investors to learn from their mistakes and price sectors efficiently.2.

*The importance of each sector to the ORP*. In a state of equilibrium, the portfolio weight of all 10 sectors on average across time should be equal. This would result from the increase and decrease in demand based on each sector’s diversification benefits as measured by correlation. If a sector did not contribute high diversification benefits relative to the others, then demand would drop, prices would drop, and expected returns would increase. Accordingly, the sector would become a more attractive investment, increasing demand for the sector resulting in a higher portfolio weight.3.

*The performance of the ORP to various benchmarks*. In an efficient market, the performance of the*ORP*should not be reliably different from the market portfolio. Any test of market efficiency is subject to the joint hypothesis problem (Campbell, Lo, and MacKinlay 1997). But in this study, the sectors studied are partitions of the S&P 500, so using the S&P 500 as a benchmark effectively eliminates the error that would result from using an incorrect equilibrium model to form expected returns. Therefore, differences between expected and actual returns are likely evidence of inefficiencies. The result, on average over time, is equal portfolio weights for all sectors.4.

*The tendency for sectors that do not lie on the efficient frontier in year t to move toward the efficient frontier in year t + 1*. While frictions may cause an ex post failure by investors to efficiently price a sector, the model of market efficiency predicts that on average, sectors should become more efficiently priced and therefore outperform sectors already efficiently priced. Similar to the literature on overreaction,^{4}the testable evidence is the presence of abnormal returns of these sectors in year*t*+ 1.

In addition and for completion as juxtaposed against prior literature, the performance of portfolios forecasted out-of-sample using Markowitz optimization is compared to various benchmarks.

The evidence together shows that investors price sector-level return and risk information efficiently, but it appears difficult for investors to efficiently price sector-level information for all sectors into the ORP. In addition, Markowitz optimization does not appear valuable for forecasting sector prices.

This article proceeds as follows. The second section explains how the proxies used in this study were chosen, the third section provides descriptive statistics, and the fourth section analyzes the efficient pricing of sectors from the perspective of Markowitz optimization. The fifth section investigates the implications of this study for investors, and the sixth section provides concluding remarks.

## PROXY SELECTION

Data for Fidelity’s sector products go back to the early 1980s, although not for every sector, but Kenneth French provides volumes of data for a variety of industry portfolios going back to 1926. The challenge with French’s data, however, is that none appear to be directly available to investors, either historically or presently. Therefore, to capture a very long time series while at the same time identifying series that are closely matched to current choices in the investment set, we calculate correlations for the concurrent periods between data from Kenneth French’s website and Fidelity’s GICS sectors. Then to proxy for current investment choices, we chose the highest correlating data from Kenneth French’s website as proxies for the GICS sectors currently available in the investment set.

Specifically, monthly data are obtained for the 10 GICS sector mutual funds available from Fidelity over the period March 1997 through December 2015. We chose mutual funds over the Fidelity ETFs because they have a much longer time series; we chose March 1997 because data for every Fidelity fund is available starting at that point in time; we chose December 2015 because it is the most recent data available on Kenneth French’s website.

We calculated correlations between these data and the monthly industry portfolio data from Kenneth French’s website to find the series that most highly correlate with Fidelity’s funds over the available data for the 10 GICS sector mutual funds. Correlations between Kenneth French’s data and Fidelity’s funds (*P*_{(KT)}) are calculated as follows:

where *K* and κ are the observations and average from Kenneth French’s data, and *T* and τ are the observations and average from Fidelity’s fund data. Then, we used the entire data series from Kenneth French’s website going back to 1926 as a proxy for the GICS defined sectors that would have been available to investors, thereby providing a much longer time series of data than in previous studies.

Kenneth French has available eight sets of industry portfolios that partition the market into 5, 10, 12 17, 30, 38, 48, and 49 industry portfolios. To begin the search for highly correlated series, we selected French’s value-weighted 10-industry portfolio set. As a result, 8 of the 10 portfolios are highly correlated with Fidelity’s funds, with an average correlation of 0.92.^{5} The two least correlated are materials/shops, with a correlation of 0.67, and surprisingly, utilities, with a correlation of 0.60. From Kenneth French’s data, we substituted chemicals for shops, yielding a correlation between the materials sector from Fidelity and the chemicals industry from French of 0.89.

For the utilities sector, we tested all the other time series available from French and found no reasonable alternatives. Therefore, we used French’s utilities series, with a correlation of 0.60, as the best available proxy for Fidelity’s utilities mutual funds. The results should not be significantly biased, because 0.60 is still a high correlation, and both series have the same directional sign for 80% of the observations of monthly returns.^{6} Moreover, using French’s data increases the sample size for all sectors by 273% (1926–2015 versus 1997–2015), so the trade-off is beneficial. Finally, and to provide a benchmark, we used the market returns from French’s website as a proxy for the market portfolio.

## DESCRIPTIVE STATISTICS

To begin analyzing whether investors price sectors into the ORP, we compiled descriptive statistics to gain insight into how the various sectors differ in their behavior and the contribution of each sector to market portfolio returns. Since the GICS sectors simply divide the market portfolio into 10 subsets, the model of market efficiency predicts that each sector’s risk-adjusted returns should not be significantly different, because investors should discount sector-level information and demand more favorable sectors to the exclusion of less favorable sectors. Hence, prices in the sectors should adjust to realign the expected risk-adjusted returns of each sector.

Exhibit 1 shows that over the sample period 1926–2015, the market portfolio averaged monthly returns (δ_{M}) of 0.93%, and all the sectors were within 0.17% of the market (Columns 2 and 3). To test the difference between each sector’s average monthly returns (δ_{S}) and the market’s average monthly returns (δ_{M}), we conducted the difference of means test as follows:

where σ_{S} and σ_{M} are the standard deviation of returns for each sector and the market, respectively, and n_{S} and n_{M} are the sample size for each sector and the market, respectively. None of the sectors’ returns were significantly different from the market portfolio (Row 2 in Column 3), although the information technology and health sectors were almost significantly different (p-values of 0.105 and 0.111, respectively).

Preliminarily, this result seems to indicate that all sectors are efficiently priced and all make an important contribution to market portfolio returns. The healthcare sector was the largest contributor to market returns, at 1.10%, and telecommunications contributed the lowest, with average monthly returns of 0.86%. Interestingly, 7 of the 10 sectors contributed higher average returns than the market portfolio, but with the exception of consumer staples, they all came with higher standard deviations of returns (σ) as well (Column 4). Moreover, two of the three with lower returns also had higher risk than the market—a violation of the risk/return relationship. The exception was telecommunications, with both lower returns and risk.

Similarly, the market variability of monthly returns averaged 5.38% and all sectors were within 2.36% of the market (Column 4). The more risky sectors were the consumer discretionary sector, with an average standard deviation of 7.73%, and the information technology sector, with a standard deviation of 7.30%. The telecommunications sector was the least risky at 4.61%, and both the utilities and heath sectors had standard deviations very close to the market portfolio (differences of 0.19% and 0.22% respectively).

Combining the return and risk characteristics reveals that the consumer staples sector dominated the market portfolio over the period 1926–2015, contributing higher returns with lower risk as measured by the standard deviation of returns. Conversely, the utilities and financial sectors were dominated by the market portfolio proxy, with both lower average returns and higher risk than the market portfolio proxy. However, none of the differences were significantly reliable (Column 3), but both the information technology and heath sectors were only slightly insignificant (p-values of 0.11).

These results are additionally interesting because consumer staples and utilities are both considered defense sectors, yet they were priced very differently. Since rotating to defensive sectors during recessionary environments is prominent in many smart beta strategies, the effect of incorporating economic indicators on the portfolios would be interesting. Such an analysis will be deferred to future studies.

So from this study, and in sum from Columns 2, 3, and 4, investors appear to efficiently price sector-level return and risk information. The last column in Exhibit 1 provides preliminary statistics describing the relation between the various sectors and the market—and not surprisingly, the returns of all sectors were highly correlated (ρ) with the market portfolio (0.01 level). The industrials sector was the most correlated with the market (0.96), the utilities sector was the least correlated (0.76), and the sectors together had an average correlation with the market of 0.85, so they would be expected to be the most demanded sectors by investors seeking diversification. The opposite would be expected for the most highly correlated sectors—industrials, financials, and consumer staples and materials.

In sum from Exhibit 1, none of the sectors’ risk-adjusted returns were reliably different from the market portfolio, indicating preliminarily that investors appear to efficiently price sector-level information. To investigate the stability of the results from Exhibit 1 and to continue gaining a feel for the efficiency with which sector-level information is priced, we examined the return–risk characteristics of each sector across various subperiods. More specifically, we partitioned the sample into quartiles. The results are presented in Exhibit 2.

Market efficiency predicts that no sectors should consistently outperform the market portfolio. Panel A shows that the only two sectors to consistently provide higher returns than the market portfolio were the health and energy sectors. But in Panel B, the energy sector always came with higher risk, and the health sector was more risky in two of the four quartiles examined. No sectors were less risky for all four quartiles, but the telecommunications and utilities sectors were less risky over three of the four subperiods.

Panel C in Exhibit 2 presents the risk-adjusted returns (returns/standard deviation) across all four subperiods. None of the sectors experienced consistently higher returns across the periods, but the best performing sector was the health sector, which generated higher risk-adjusted returns over three of the four subperiods. No more than 3 of the 10 sectors produced higher risk-adjusted returns than the market portfolio over the most recent three periods, and the energy and financial sectors were the only sectors to produce lower risk-adjusted returns than the market portfolio across all four subperiods. The model of market efficiency predicts that 5 of the 10 sectors would outperform the market portfolio in each of the periods, but not the same 5 across periods. While the sample size is too small for statistical reliability, the results from Exhibit 2 suggest that investors may have had difficulty efficiently pricing the sectors, especially the energy and financial sectors.

However, to increase the sample size, partitioning the sample into years reveals that for each year of the sample period, 53.03% of the sectors, on average, outperformed the market portfolio in any given year.^{7} Conducting the binomial difference of proportions test yielded a test statistic of only 0.57, indicating that investors appear to price sectors efficiently. Moreover, with the exception of the consumer staples sector, none of the sectors outperformed or underperformed the market annually reliably in 50% of the years. The lone exception, consumer staples, outperformed the market in 60% of the years (0.10 level).

In summary from Exhibits 1 and 2, we observe several characteristics of sector behavior that together imply that investors efficiently price sector-level information. The highest returns over the sample period came from the health sector, which also had higher returns than the market portfolio over each quartile of the sample period. Conversely, the lowest returns were from the telecommunications sector, which experienced lower returns than the market portfolio for three of the four quartiles. The utilities sector was the only sector with lower returns than the market over all four quartiles. The most risky sector was the consumer discretionary sector, which also was more risky than the market portfolio for the last three quartiles. While the risk-adjusted returns of each sector were not reliably different from the market portfolio, the sector offering the greatest diversification benefits from the market portfolio was the utilities sector. The least benefits came from the industrials sector. Finally, and as predicted by the model of market efficiency, 53.03% of the sectors outperformed the market, on average, each year of the sample.

## THE EFFICIENT FRONTIER OF SECTORS

The descriptive statistics from Exhibits 1 and 2 suggest that while some sectors have performed better than others, they all seemed to be efficiently priced, as evidenced by the return and risk characteristics being insignificantly different from the market portfolio proxy, the inconsistency with which they produced higher risk-adjusted returns than the market portfolio, and the fact that, on average, approximately half of the sectors out/underperform the market in any given year. Thus, investors appear to successfully discount return and variability information at the sector level. However, an equally important driver of portfolio performance is how each asset in a portfolio behaves relative to others and contributes to the risk-adjusted returns of the total portfolio (Markowitz 1952). Hence, this section explores how efficiently investors discount sector-level correlation information into prices, by examining the extent to which investors price the various sectors in the opportunity set as represented by the efficient frontier and especially the optimal risky portfolio.

The model of market efficiency predicts that all sectors should contribute to the efficient frontier, because the discounting of information into prices by investors drives returns to the optimal risk/return relation in the presence of each other. To do otherwise would be irrational. However, the model makes several assumptions. For example, it assumes that investors understand the information’s importance, that they gather the information, and that they act upon it, thereby moving the supply and demand equilibrium. The violation of these and other assumptions can drive ex post results that differ from expectations, causing a breakdown in market efficiency and resulting in suboptimal investments, at least ex post. Yet even if such inefficiencies appear, the model of market efficiency also predicts that the mispricing will be temporary, because investments not contributing to the efficient set should subsequently move towards the efficient set. Accordingly and consistent with the analysis of reversals in the overreaction literature, those sectors becoming more efficiently priced would outperform those already efficiently priced. Also, as investors arbitrage away mispricings, sectors should be priced efficiently at least 50% of the time. Hence, this section continues documenting the ex post behavior of sectors by incorporating correlation information into the price discovery process, a key variable in the ORP.

If news events are identically important across sectors, then the correlation among sector returns will be perfectly positive regardless of the news events’ correlation with each other.^{8} Moreover, if news events are perfectly positively correlated, the correlation between sector returns still will be perfectly positive regardless of their importance across sectors. Therefore, less than perfectly positive correlations between sector returns require: 1) news events that differ in importance to each sector, and 2) less than perfectly positively correlated news events.^{9} For purposes of this study, such news events will collectively be referred to as *differing news*.

To begin understanding how investors discount differing sector-level news events, we calculate the correlations among sector returns. Exhibit 3 shows the lowest correlations were between the energy and telecommuncations (0.52), energy and health (0.57), and information technology and utilities (0.61) sectors. The highest correlations were between the materials and industrials sector (0.92), followed by the financials and industrials (0.90) and consumer discretionary and industrials (0.84). From Exhibits 1 and 3, the sectors with the lowest correlations with the market portfolio and each other appear to be the utilities, telecommunications, and energy sectors. Therefore, if the return–risk characteristics of all sectors were equal, the demand for these sectors in portfolios would be higher because of their superior diversification benefits relative to the other sectors. Such demand would drive returns lower for the sectors with low correlations and higher for sectors with high correlations until all sectors were priced equally. This theoretical state of equilibrium among returns, variability, and correlations is the perspective of market efficiency throughout the remainder of this study.

Therefore, to begin examining how efficiently investors incorporate sector-level correlation information into prices with return and risk information, the efficient frontier is constructed from monthly returns over the entire sample (Exhibit 1). Two observations are immediately noticeable. First, the market portfolio has not been the ORP as predicted by the well-known modern portfolio theory, and second, an equally weighted portfolio of sectors has been more efficient than the market portfolio.

More specifically, and starting with the market portfolio, the 10 sectors plot in every direction around the market portfolio (Exhibit 4). An investor’s goal of Markowitz optimization is to choose assets that move the risky portfolio in the most “northwest” direction; such a portfolio is the ORP. Over the 1926–2015 time period with the proxies chosen for the 10 GCIS sectors, creating an equally weighted portfolio of sectors appears more efficient than the market portfolio proxy because it moves the return/risk relationship northwest from the market portfolio proxy. This result is consistent with the findings of Sturm (2010), Eakins and Stansell (2007), and Wyatt and Kee (2014), who approximate an equally weighted portfolio with their rebalancing.

Continuing northwest (Exhibit 4) and inconsistent with expectations, the efficient frontier consists of just four sectors: consumer staples, energy, telecommunications, and health.^{10} More specifically and as reported in Exhibit 1, the minimum-return portfolio would have been a 100% allocation to the telecommunications sector and the maximum-return portfolio would have been a 100% allocation to the health sector. With the inclusion of correlation information, the minimum-variance portfolio would have been a weighting across the four sectors of 41.3%, 14.1%, 17.4%, and 27.2%, respectively. Finally, the ORP would have been constructed with weightings of 43%, 13%, 27%, and 18%, respectively. Moving up the efficient frontier, these weights are mostly adjusted down as the portfolios are allocated to the maximum-return asset.

Toward the top, there would have been a small and brief allocation to the consumer discretionary and materials sectors with portfolio weights of only 0.80% and 1.49%, respectively.

So over the period 1926–2015, the preliminary results indicate that the market has not efficiently priced all sectors into the ORP or the efficient frontier. Only 4 of the 10 market sectors contribute to the ORP, and only 6 of the 10 to the efficient set. Of course, this result could merely be the result of sample selection bias. Moreover, these results are limited by the unavoidable fact that the 10 ETF managers decide the portfolio weights of their own sectors. If the sectors were rebalanced otherwise, the efficient region would change. Studying the relation between the ORP and this re-sampled efficient region may yield different results, but such an examination will be left to future studies.

In this study, to examine more closely how efficiently sectors are priced by increasing the sample size, we calculated the ex post ORP each year using monthly returns. Then we analyzed the results for the frequency of each sector’s inclusion in the ORP and the average portfolio weight when included. The results are presented in Exhibit 5.

The second column in Exhibit 5 shows how frequently over the 89.5-year sample period each sector was included in the ORP, together with its test statistics from two one-sided binomial difference of proportions tests. The null hypothesis for the first test is that in an efficient market, on average, each sector should be priced efficiently at least 50% of the years. This follows from the notion that due to forecasting errors of each sector’s expected returns, there will be some years during which the sector is not priced efficiently ex post. However, as explained previously, investors in aggregate should learn from the error in year *t* and adjust sector returns toward the ORP in year *t* + 1. The test statistic (*t*) is calculated using the binomial test of proportions as follows:

For rigor, the second test statistic in the first row is calculated identically, but the null hypothesis follows from the idea that even if investors fail to price sectors into the ORP 50% of the time, the frequency with which sectors are priced in the ORP should be stable. Hence, the null equals the average of all the frequencies in Column 1 (31.25%).

All of the sectors were priced in the ORP during some of the years, but none of the sectors were priced into the ORP for more than half of the years studied (except for consumer staples). When they were, the more important sectors appeared to be the health sector, with an average portfolio weight of 51.1%; the telecommunications sectors, with an average weight of 39.4%; and the information technology sector, with an average weight of 37.9%.

More specifically, from the perspective of the null hypothesis that each sector should be priced into the ORP at least 50% of the time, the first test statistic in Column 2 of Exhibit 5 provides evidence that with the exception of consumer staples, all of the sectors were priced in the ORP significantly less than 50% of the time. The consumer staples sector was priced the most frequently, while the financial sector was priced the least frequently in the ORP. If the null hypothesis follows from the assumption that the average frequency of all sectors represents efficient pricing, then the expectation becomes that each sector should be priced efficiently 31.25% of the years observed. Yet, the second test statistic shows that three sectors are still not priced in the ORP as frequently as expected: the industrials, materials, and financials sectors.

Thus, Column 2 suggests that investors had difficulty pricing sector-level information efficiently. All sectors except consumer staples were priced in the ORP much less than 50% of the time, and three sectors were priced significantly less consistently than the average of all sectors. While the second column shows the results of testing how consistently sectors were priced in the ORP, the third column shows the results of testing how important each sector’s contribution was to the ORP. That is, the third column shows the results of testing the average annual portfolio weight of each sector, together with the test statistics calculated from Equation 3. Similar to Column 2, the first null hypothesis assumes that all 10 sectors should be weighted equally at 10% on average, and the second tests the difference between the individual sector weightings and the average portfolio weighting of all sectors in the ORP (31.09%).

From Column 3 of Exhibit 5, clearly the first null hypothesis is rejected for all sectors. While none of the sectors are included in the ORP as frequently as expected (Column 2), when they are included in the ORP, they are all weighted, on average, significantly higher than 10%. Moreover, as suggested by the second test statistic in Column 3, the consistency of the weightings across sectors varies more than expected. The average weight of all sectors included in the ORP is 31.09%, but the average ORP weighting of 5 of the 10 sectors is significantly different than the average.

The testing of consistency and importance of pricing presented in Exhibit 5 suggests that investors have difficulty pricing sector-level information consistently into the ORP. For rigor, the fourth column in Exhibit 5 presents each sector’s best contribution to the ORP. Most sectors were priced exclusively, or almost exclusively, in the ORP during at least one year of the sample period except for one sector. The financial sector is the clear outlier given that its maximum contribution to the ORP has only been 54.8%. Thus, Column 4 provides evidence that during some years over the 89.5-year sample period, 9 of the 10 sectors were not included in the ORP.

In addition to the results reported in Exhibit 5, the average number of sectors included in the ORP each year was 3.13, indicating that, on average, between six and seven sectors were not efficiently priced (standard deviation = 1.41). Moreover, of those not priced efficiently at year *t*, only 29% were priced efficiently in year *t* + 1 (results not reported).

Together, the evidence presented in Exhibit 5 shows that investors often do not efficiently price sector-level information into the ORP. Again, market efficiency predicts that all of the sectors should contribute to the ORP at least half of the years studied, yet the results in Column 2 show that sectors often are not priced in the ORP. Additionally, the average portfolio weight of inclusion in the ORP for the 10 sectors should be unreliably different from 10%, but the average weight of most sectors is significantly higher (Column 3). Moreover, the average portfolio weight of several sectors is reliably different from the average weight of all sectors. Finally, from Column 4, most sectors stood alone (or mostly alone) in the ORP during at least one year of the sample period, except for financials.

In addition to the results presented in Exhibits 1–3 and 5, additional tests revealed the following.^{11} Only approximately 31% of the sectors were included in the ORP in any given year, and during many years of the sample, the optimal risky portfolio contained only 1 of the 10 sectors. The one sector that appeared to be clearly inferior to the others was the financial sector. It was only priced efficiently 10.2% of the time, and had the third lowest average portfolio weight and the lowest maximum contribution to the optimal risky portfolio. The sector that appeared superior was consumer staples, which was clearly priced efficiently more consistently than the other sectors.

Intuition forms the expectation that some sectors (e.g., financials) would be more sensitive to some macro-economic variables (e.g., interest rates) than other sectors. Hence, the expected variability in returns to some sectors would be higher than others, but investors should price the variability so that the return–risk relationship is in equilibrium across sectors. The evidence should therefore show that the return–risk relationship of the sectors should not be statistically different. The evidence presented in Exhibits 1 and 2 confirms these expectations.

Continuing, because of the varying reactions among sectors to differing news, the correlations among sectors will vary, driving varying demand for each sector conditional upon its value to a well-diversified portfolio. Therefore, prices in an efficient market should discount both macroeconomic news and the value of each sector to a portfolio as measured by the correlations of reactions to differing news. The result of discounting the correlations should be twofold. First, each sector should be included in the ORP because each sector should contribute the benefits of diversification. If a sector were not valuable to the ORP, then market efficiency predicts that there would be no demand for the sector, so returns would adjust in order for it to become valuable to the ORP. Second, each sector should be equally weighted in the ORP because more (less) valuable sectors would be more (less) in demand, thereby driving returns. That is, if a sector were priced in the ORP but with a very low weight, there would be less demand for that sector. Again, market efficiency predicts that prices would drop due to the low demand increasing expected returns. The evidence presented in Exhibit 5 does not confirm these expectations.

## IMPLICATIONS FOR INVESTORS

The state of equilibrium should therefore be characterized by insignificantly different risk-adjusted returns among the sectors and sectors that, on average, contribute equally and consistently to the ORP. Exhibit 5 shows that investors have difficulty finding this state of equilibrium. While the implications of these findings are clear for investors, this study is not intended to design a trading strategy like other important and useful studies. However, much of the prior literature, such as Wyatt and Kee (2014), Eakins and Stansell (2007), and Sturm (2010), studied investment strategies using sectors with the common implication that an equal weighting of the sectors was usually optimal. Therefore, to gain a feel for this study’s implications for investors, we examine an ORP-based trading strategy and compare it to an equally weighted strategy as well as the market portfolio to determine if investors are able to use Markowitz optimization to forecast the ORP.

More specifically, we test two strategies to determine if either outperforms the benchmarks. The first strategy holds a portfolio formed using the ORP weights from year *t*, to speculate whether investors are able to use information at year *t* to forecast the ORP’s composition at year *t* + 1. The second strategy holds an equal weighting of all of the sectors in year *t* + 1 that were not included in the ORP during year *t*, to test if this “inefficient portfolio” from year *t* moves towards the ORP in year *t* + 1. The model of market efficiency predicts that the inefficient portfolio should generate higher risk-adjusted returns than the forecasted ORP. Additionally for completion with Exhibit 5, the ex post ORP is tested for a reliable difference from the market and an equally weighted portfolio. The results are presented in Exhibit 6.

The first two rows in Exhibit 6 present the average (δ) and standard deviation (σ) of monthly returns for the benchmarks and the various portfolios, and the third row presents the risk-adjusted returns (returns/standard deviation). Rows 4 and 5 present the difference of mean monthly returns, together with the p-values from the various portfolios and the benchmarks (the market portfolio and equally weighted portfolio, respectively). From Columns 2 and 3, and consistent with prior studies, an equally weighted portfolio of proxies for the GICS sectors does appear to significantly differ from market portfolio returns. Over the 1926–2015 period, monthly returns for the market averaged 0.91% and monthly returns for the sectors averaged only 0.97%, a difference of 0.06% per month (0.01 level). While this piece of evidence is consistent with prior literature, it is especially compelling given the substantially longer sample period than in previous studies.

Column 4 in Exhibit 6 presents the results of testing the difference in average monthly returns between the ORP from Exhibit 4 and the two benchmarks. While the results presented in Exhibit 5 suggested that investors had trouble pricing sectors efficiently, the results in Column 4 of Exhibit 6 show no reliable difference in returns from either the market portfolio or an equally weighted portfolio of the sectors. These results suggested that at least over the entire sample period, investors were able to price sector-level information efficiently.

Columns 5 and 6 of Exhibit 6 show the results of testing a trading strategy using Markowitz optimization. From Column 5, the ORP portfolio at time *t* + 1 constructed from the optimal weights calculated at time *t* appears to reliably underperform both the market portfolio and an equally weighted portfolio (0.01 level). Indeed, such a strategy appears to yield returns that are lower and more risky than both benchmarks. Similarly, Column 6 presents the results of forming an equally weighted portfolio at time *t* + 1 of sectors not included in the ORP at time *t*. Similar to testing for abnormal returns in reversals to market overreaction, market efficiency predicts that this portfolio should produce abnormal risk-adjusted returns as investors recognize the inefficient pricing and demand the sectors. The results show that this *inefficient portfolio* provided higher returns with lower risk than the ex post ORP portfolio, but unreliably.^{12}

The inefficient portfolio also had lower returns than the equally weighted portfolio, but with higher risk, resulting in nearly identical risk-adjusted returns. However, the higher risk-adjusted returns (Row 3) in the inefficient portfolio over the ORP suggest that investors may be trying to more efficiently price inferior sectors. But, the inefficient portfolio’s performance is significantly lower than both the market and equally weighted portfolios. Moreover, only 29% of the sectors, on average, were included in the ORP following a year during which they were not included.^{13}

In summary, from Exhibit 6, the ORP constructed ex post did not reliably outperform the benchmarks, and it appears that a simple Markowitz optimization trading strategy would not only underperform, but would actually be a detrimental strategy for investors to pursue. However, and consistent with prior literature, an equally weighted portfolio of sectors did appear to reliably outperform the market portfolio over the time period studied. But pursuing such a strategy would involve relatively high transaction costs, since 10 funds must be managed. A potential alternative available to investors would be to simply allocate to an equal-weight ETF such as the Invesco S&P 500 Equal Weight ETF (RSP). Therefore, the RSP ETF is analyzed to gain a feel for the value of avoiding the costs of rebalancing 10 ETFs.

A direct comparison between the RSP and the previous tests in this study is not practical for at least two reasons. First, the RSP fund only has data back to 2003, a common limitation of prior studies. Also, the portfolio weights in the RSP will not exactly match those of equally weighting the 10 sectors. However, an advantage is the lower costs of allocating to just one fund rather than across 10 funds. So we repeated the tests conducted over the 1926–2015 sample period in Exhibit 6 over the 2003–2015 sample period to include the RSP fund. The results are presented in Exhibit 7.

More specifically, data for the RSP is available only back to May 2003, so we compare the market portfolio and the equally weighted portfolio to the RSP over the time period May 2003–December 2015. From Exhibit 7, the descriptive statistics of the market portfolio and an equally weighted sector portfolio over the shorter sample periods is consistent with Exhibit 6, except that there is not a statistically reliable difference between the returns. The insignificant difference is probably due to the much smaller sample size, implying that investors seeking to capitalize on the additional benefits of an equally weighted portfolio, as documented in Exhibit 6, need to hold the portfolio over a long time interval.

However, the purpose of the tests in Exhibit 7 is to compare the equally weighted sector portfolio to the RSP ETF for differences. If the same value can be generated from the RSP as from managing 10 sector ETFs, the RSP would be a viable alternative for investors. From Exhibit 7, none of the results are statistically significant. In particular, the average monthly returns of the RSP is 0.01% lower than the value-weighted market portfolio and 0.09% lower than an equally weighted sector portfolio. Although the differences are not statistically significant (p-values of 0.938 and 0.349, respectively), the fact that both differences are negative suggests consistency of the results.

Moreover, since the insignificant difference of market versus the equally weighted portfolio appears to be driven by a smaller sample size, it is reasonable to speculate that over a longer time interval, the RSP would probably reliably underperform the equally weighted portfolio—especially given the much lower p-value relative to the value-weighted portfolio. Consequently, incurring the additional cost of rebalancing the 10 sector ETFs appears to be worth the effort to investors, assuming that transaction costs would average less than 0.09% per month. If transaction costs exceed this percentage, the RSP would be a valuable alternative.

## CONCLUSIONS

This study has made three contributions to the literature. First, it gathers evidence using a more crisp and easily reproducible definition of “sectors” than many previous studies. Second, the conclusions of this study are supported by evidence from a much longer time series than most previous studies. Finally, this study documents investors’ apparent struggle to efficiently price sector-level correlation information from the perspective of Markowitz mean–variance optimization.

Over the entire sample, the consumer staples sector convincingly produced higher risk-adjusted returns than the market portfolio, and the utilities and financial sectors produced lower risk-adjusted returns. Most of the risk-adjusted returns are not reliably different than market portfolio returns, indicating that investors efficiently price return and risk information. However, the consumer staples sector did reliably outperform the market portfolio over 60% of the years studied, suggesting that investors were not able to efficiently price this sector. But investors appear to find it more difficult to discount correlation information between sectors, as evidenced by the inconsistency with which sectors are priced in the ORP.

In particular, the ORP over the period 1926–2015 consisted of just four sectors: consumer staples, energy, telecommunications, and health. When the ORP was recalculated annually, its composition appeared unstable and the more important sectors were health, telecommunications, and information technology. None of the sectors were priced in the ORP as frequently as expected, and when they were, their average portfolio weights were significantly higher than expected. Further, of those not included in the ORP during any given year, only 29% were included in the following year’s ORP, and their risk-adjusted returns were not significantly different than those priced in the previous year’s ORP. This indicates that investors were not able to efficiently correct sector-level information mispricing. Finally, an ORP investing strategy revealed that investors would not have been able to build portfolios from the prior year’s ORP information to outperform the market.

One common theme was present throughout all of the tests: The consumer staples sector was the superior sector, and the financial sector was noticeably inferior, across most of the tests.

Based on the evidence gathered in this study, it seems clear that it is difficult for investors to completely price all sector-level information efficiently into the ORP. Investors appear to discount and efficiently price sector-level information regarding returns and volatility, but pricing correlation information appears more difficult. Correlation information may not be discounted because investors either are not aware of the importance of the information, or because of forecasting error between expected and actual correlations. Regardless, and consistent with prior literature, an equally weighted portfolio of the sectors appears to be a superior long-term investing strategy.

## ADDITIONAL READING

**Select Sector SPDRs and the S&P 500: Is the Sum of the Parts Greater than the Whole?**

Ray R. Sturm

*The Journal of Wealth Management*

**https://jwm.pm-research.com/content/13/1/62**

**ABSTRACT:** *In December 1998, Select-Sector ETFs, which subdivide the S&P 500 SPDR (SPY) into nine distinct industry/sector groupings, were introduced into the market. In the study presented in this article, the author compares the returns from actively managing a portfolio of the nine Select-Sector funds using various techniques from the academic literature to the benchmark SPY. To test the funds’ performance, the author forms portfolios based on an equal weighting of the nine funds, the Markowitz mean–variance selection process, and momentum and overreaction price behaviors. His findings indicate that an equal-weighted portfolio of the sector funds reliably outperforms the SPY over all measures of performance and are robust across time.*

## ENDNOTES

↵

^{1}See entire lines of researching beginning with DeBondt and Thaler (1985), Jegadeesh and Titman (1993), and many others.↵

^{2}On September 1, 2016, real estate was added as an 11th sector to the GICS.↵

^{3}Excluding the relatively new real estate sector.↵

^{4}See DeBondt and Thaler (1985) and many others who followed.↵

^{5}Results not reported.↵

^{6}Results not reported.↵

^{7}Results not reported.↵

^{8}Assuming the correlation is non-negative. Negatively correlated news events are not considered in this study.↵

^{9}Proof omitted.↵

^{10}Results not reported.↵

^{11}Results not reported.↵

^{12}Results not reported.↵

^{13}Results not reported.

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