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## Abstract

As average hedge fund performance continues to wane, investors are increasingly seeking objective criteria to distinguish talented hedge fund managers from among the herd. Differentiating the contributions of systematic and idiosyncratic factors in a fund’s return stream is one way to accomplish this goal. In this article, the authors combine two related, but distinct, measurements of these quantities. Using the Carhart four-factor model, the Fung–Hsieh seven-factor model, and principal components analysis, they find that the hedge funds with the highest factor-adjusted alpha (a proxy for skill) and lowest *R*^{2} to these factor models (a proxy for factor dependence) produce the strongest subsequent returns and factor-adjusted alphas. They also find that those managers with high trailing factor-adjusted alpha have lower exposure to systematic risk factors in general.

Modern portfolio theory tells us that investment returns, whether on individual stocks or entire portfolios, are widely considered the payoffs to systematic and security-specific risks. When hedge funds posted impressive returns in the late 1990s and early 2000s, investors lacked the motivation and tools to differentiate between the two types of risk. More recently, however, competitive pressures and regulatory changes^{1} have eroded hedge fund returns, and investors now seek informative methods to identify the shrinking percentage of managers with sustainable alpha. Fundamental to this process is determining to what extent systematic risk exposures are responsible for generating returns.

The capital asset pricing model (CAPM) was the first formal model to assume that expected returns are not entirely specific to an individual stock. According to CAPM, part of a stock’s expected return is attributable to its systematic covariance with the market’s return (the stock’s beta). Since CAPM was developed, research has uncovered additional systematic factors that influence expected returns beyond the market’s movements. Specifically, researchers identified systematic excess returns to value companies over growth companies (Shiller [1984]; Fama and French [1992]; Lakonishok, Shleifer, and Vishny [1994]), small stocks over large stocks (Banz [1981]; Reinganum [1981]; Fama and French [1992]), and recent winners (as measured by price change over the last 3, 6, or 12 months) over recent losers (Jegadeesh and Titman [1993]; Carhart [1997]). More recent research has uncovered a set of alternative sources of systematic excess returns that are more heavily exploited by hedge funds. These include various trend-following and bond-oriented style factors (Fung and Hsieh [2004]). The expected returns to these factors, however, arise from taking additional risks. Generally speaking, smaller companies tend to be riskier than larger companies, undervalued stocks tend to be unpopular investments, momentum stocks (or any trend-following indicators) are prone to reversals, and exposure to credit and commodity markets inherently involves additional sources of volatility.

In decomposing hedge fund portfolio returns, a portion will be explained by exposure to these systematic sources, but successful managers should also contribute *skill*, a term we use in the article to represent unique, idiosyncratic sources of alpha. By isolating the effects of these systematic factors when analyzing a fund’s returns, an investor can more effectively distinguish the skill-based, active returns (what we call *factor-adjusted alpha*) from the systematic, passive returns. Because many hedge funds have no benchmark (an effective 0% hurdle), they earn performance fees for any positive return—regardless of the attribution between passive and active sources. Nevertheless, many hedge fund investors seem comfortable with high exposure to cheaply attainable systematic factors (chiefly equity beta and exposure to small caps) in their portfolios. Exhibit 1, Panel A tracks the average exposure to the four Fama–French–Carhart style factors (equity beta, value, size, momentum) over time across equity hedge funds in Hedge Fund Research’s HFRI returns database. Panel B does the same using the seven Fung–Hsieh style factors (bond trend following, currency trend following, commodity trend following, market beta, size, bond market, and credit spread factors), although only the two equity factors are plotted because of the differences in units of the other variables. All numbers plotted represent the average multivariate coefficient using a 24-month rolling regression window. Equity hedge funds have maintained exposures toward equity beta (measured against the MSCI World Index) and toward small caps from 2000 to 2014.

## REVIEW OF PAST RESEARCH

Previous research has explored similar issues of factor dependence and skill among mutual funds and hedge funds. Professors Sheridan Titman and Cristian Tiu hypothesized that specific “hedge funds…will choose greater exposure to priced factors if they have less confidence in their abilities to generate abnormal returns from the active component of their portfolios” (Titman and Tiu [2011]). In other words, those who lack skill rely more heavily on standard factors to produce returns. To test this hypothesis, the authors sorted hedge funds by their *R*^{2} to standard risk factors and found that funds with lower *R*^{2} values have higher alphas, attract more capital, and charge higher fees. Zheng Sun, Ashley Wang, and Lu Zheng used an alternate measure of strategy distinctiveness—a fund’s past correlation to its peer average—to sort hedge funds and found that funds with a lower correlation to their peers had better subsequent performance (Sun, Wang, and Zheng [2012]).

More recently, Professors Yakov Amihud and Ruslan Goyenko sorted the mutual fund universe by two characteristics: They not only sorted mutual funds based on trailing *R*^{2} but also by trailing four-factor alpha. The authors found that funds that sorted into the lowest quintile of *R*^{2} and the highest quintile of four-factor alpha went on to produce the highest subsequent alphas (Amihud and Goyenko [2013]). Additionally, they found that those funds with high *R*^{2} measures were larger, were run by managers with shorter tenures, and had lower fees.

## OUR APPROACH

Assessing funds using both factor-adjusted alpha and *R*^{2} is more revealing than focusing on one dimension alone. Therefore, we sought to apply the same methodology used by Amihud and Goyenko to hedge funds rather than to mutual funds. Conveniently, both of these measures—*R*^{2} and factor-adjusted alpha—are outputs of the same regression. When we regress hedge fund returns against returns to the four Carhart factors or the seven Fung and Hsieh factors, we can derive the *R*^{2} (or coefficient of determination) of the regression, which will tell us the percentage of variation in fund returns that is attributable to variation in the factor returns. Additionally, the regression output will include a constant term, which is the return over and above (or under and below) what would be expected from a fund’s average factor exposures alone. This factor-adjusted alpha will measure the idiosyncratic alpha relative to a benchmark tailored specifically for that fund’s recent exposures. Factor-adjusted alpha, in addition to capturing stock selection skill, should also capture factor timing skill provided that factor exposures are adjusted more frequently than the window size (24 months in our analysis). Exhibit 2 displays the distribution of equity hedge funds along both dimensions (and using both factor models) as of December 2014.

The *R*^{2} highlights a fund’s factor dependence (the higher the *R*^{2}, the more a fund relies on standard factors to produce returns), whereas the factor-adjusted alpha is a proxy for skill. More precisely, factor-adjusted alpha will measure the portion of returns attributable to idiosyncratic sources. These sources include stock selection skill and higher-frequency factor timing skill, but they may also include simple luck and other sources of noise in the return stream. There exists a misguided notion that hedge funds with demonstrated skill also come with high levels of systematic factor exposures—that skill and factor dependence come as a package deal. The relatively steady run-up in equities since the market trough in 2009 has supported this notion because many of the funds with the highest recent returns (and therefore perceived skill) have also had high exposure to equity beta. In theory, however, skill is distinct from factor exposure. We assume that an individual manager has a fixed level of skill (or lack of skill), but factor exposures and timing are well within a manager’s control. In other words, a manager has the choice to layer factor exposures on top of any skill-based returns. In Exhibit 3, we propose a high-level classification framework for hedge funds based on the relative values of these two variables.

We assume that all fund managers seek to deploy their maximum level of skill (and, by extension, factor-adjusted alpha because we believe it to be a good proxy). With regard to factor dependence (we use the *R*^{2} to systematic factors as a proxy), managers may have high factor exposures for a variety of reasons, including:

1. They may simply be unaware of their factor exposures, in which case any dependence/bias is unintentional. Admittedly, low factor exposure can also be inadvertent.

2. They may believe that there are long-term returns to systematic risk factors and are seeking to exploit them.

3. They may believe that they can time these factors (in which case a point-in-time measurement may mischaracterize a variable exposure as static).

4. They may be seeking to offset their (lack of) skill by assuming sizable factor exposures in order to deliver acceptable return levels (which may otherwise be unachievable).

However, none of these explanations aligns with investor interests:

1. Most managers have an idea, even if imprecise, as to their equity beta exposure, but many managers are unaware of their exposure to other style factors. Returns to these factors ebb and flow just like equity market returns, so neglecting to monitor factor exposures puts the portfolio at risk of underperforming when specific factors are out of favor.

2. Incorporating factor exposures will add an additional source of volatility, and the precise degree of exposure to each factor will be unknown to investors ahead of time, making it practically impossible for them to hedge out these exposures. Relatedly, an investor can achieve exposure to these return drivers separately through passive exchange-traded funds or other smart beta products, typically offered at a fraction of the hedge fund fee structure.

3. Our analysis captures the value added (or lost) from higher-frequency changes in factor exposures (any repositioning on a time frame shorter than two years). We are less concerned about lower-frequency factor timing; given the length of equity and style factor cycles (usually many years), skill would be much harder to prove to investors because it would involve multiple correct calls over the course of decades.

4. Consistent with Titman and Tiu’s [2011] hypothesis, managers without skill may choose significant factor exposure to offset or disguise their lack of talent. Perhaps they correctly reason that with enough factor exposure, factor performance will dominate their returns, making skill very difficult for investors to isolate and analyze.

This line of reasoning prompts a question: Why would managers who do have skill choose to layer in factor returns that are presumably outside their control? It is possible that some skilled managers simply do not want to be an outlier and may choose a beta similar to that of their peers. In this case, however, systematic factor performance will potentially overwhelm any skill-based returns because factor volatility and returns are usually higher than expected alpha generation.^{2} Targeting mid- or high-single-digit annual alpha production, for example, the skill-based portion of the return stream will be hidden beneath the much more dominant factor returns. Is it true, then, that the combination of high skill and low factor exposure is preferable? Are the two measures related (i.e., how does factor dependence vary with changes in skill levels)? These are two central questions we seek to answer.

Before we report the results of our analysis, we want to mention a couple caveats to our approach. First, the factors used in this analysis must be relevant to the underlying asset class or the results will suffer from model misspecification. *R*^{2} and factor-adjusted alpha provide two powerful and distinct measures of factor dependence and skill, and they are conveniently derived from the same regression. Because our analysis focuses on equity hedge funds, we defaulted to the Carhart four-factor model, which includes four equity-focused systematic factors, each supported by decades of academic research. However, if we applied this framework to a hedge fund with credit or commodity exposure, for example, the results would likely not be informative or predictive of subsequent performance. This is why we supplemented the Carhart model with the seven-factor Fung and Hsieh model, which includes other nonequity factors (including credit and commodity factors) that hedge funds may be exploiting. Additionally, because correctly defining the systematic risk factors underlying different investment strategies or asset classes is critical to reducing omitted variable bias, we also performed principal components analysis on hedge fund returns.

Second, most hedge fund return databases, including HFR (used in this analysis), depend on self-reporting and therefore may suffer from backfill bias. A hedge fund’s decision to begin reporting probably hinges principally on performance in recent periods. Managers who choose to report their returns are more likely to have a successful recent track record, and when they begin reporting, they are permitted to backfill previous performance into the database. Conversely, managers with poor performance may simply choose not to report to a database. To reduce this potential bias, we exclude the first 24 months of returns for the purpose of out-of-sample analysis, and funds with fewer than 25 months of historical returns at any point in time are excluded entirely. If a new fund begins reporting returns, the first 24 months are only used to sort a fund based on its two dimensions (factor dependence and factor-adjusted alpha). Out-of-sample analysis begins only with the 25th month; therefore, any potential backfilling would likely only affect our sorting process. Because factor dependence and factor-adjusted alpha are likely tracked by few managers, however, our sorting process likely is not significantly affected by backfill bias. After all, a fund that derives all of its returns from systematic factor exposures may have high returns and may choose to report even if it has unimpressive factor-adjusted alpha.

## OUR ANALYSIS

In our analysis, we sought to apply the Amihud and Goyenko [2013] methodology to hedge fund returns instead of mutual fund returns, while expanding it to answer the questions we just raised. We drew from HFR’s HFRI database of over 6,500 USD-reported equity hedge funds, current and closed (to mitigate survivorship bias), and observed their monthly net returns since 1997. Because the traditional single-metric definitions of the Fama–French–Carhart factors place a disproportionate weight on small value stocks, which have done relatively well recently, Cremers, Petajisto, and Zitzewitz [2012] suggested using “common and tradable benchmark indices” instead. Therefore, we use MSCI World indexes to represent the four Carhart factors. We use the MSCI World Index for equity market returns, the MSCI World Value Index—MSCI World Growth Index return spread for returns to value, the MSCI World Small Cap Index—MSCI World Large Cap Index return spread for returns to size, and the float-weighted return of the top half of price performers in the MSCI World Index from the previous 12 months—the float-weighted return of the bottom half of price performers in the MSCI World Index from the previous 12 months return spread for returns to momentum. We also use the same equity market and size factor returns as variations within the Fung and Hsieh seven-factor model.

For each fund, on a rolling 24-month basis, we regress net monthly returns (less the risk-free rate *r _{f}*) against returns to the four Carhart factors and the seven Fung and Hsieh factors. Following the Amihud and Goyenko methodology, we sort funds into quintile baskets based on the

*R*

^{2}to these factors. Then, within each

*R*

^{2}quintile, we sort again based on factor-adjusted alpha. For each month, every fund will be a member of 1 out of a possible 25 baskets based on its

*R*

^{2}and four-factor/seven-factor alpha quintile combination. Because of the dependent sorting, all baskets will contain the same number of funds. For each basket, we record the average return of the portfolio of funds for the 25th month (the month immediately following the 24-month sorting window). We then have a time series of 25th-month returns for each of the 25 quintile combination baskets from January 2000 to December 2014 (180 months = 15 years). We regress each of those time series on the historical four- and seven-factor return streams and multiply the resulting alpha term by 12 to get an annualized factor-adjusted alpha term. Exhibit 4 displays this alpha term and the related

*t*-statistic for each quintile combination using the four-factor model (Panel A) and the seven-factor model (Panel B).

Focusing on the All columns and rows in Exhibit 4, subsequent alphas are higher for funds with lower trailing *R*^{2} and higher trailing factor-adjusted alphas. In other words, funds with less factor dependence and higher factor-adjusted returns go on to perform better than those with opposing characteristics. These relationships are nearly monotonic across all rows and columns, and the majority of the alphas in extreme cells are statistically significant. Looking at the quintile spreads (Low–High column and High–Low row), we also see a consistent story with uniformly positive return spreads along both dimensions, regardless of which factor model we use.

One potential weakness of this analysis is that we sorted the funds into quintiles using a dependent (two-stage) sort, following Amihud and Goyenko: first by *R*^{2}, and then by four-factor alpha within each *R*^{2} quintile. The rationale was to create an equal number of funds/observations underlying each cell. However, any correlation between the two variables may affect the results using this method. Our suspicion that there is a relationship prompted us to rerun the analysis using an independent/simultaneous sort. In Exhibit 5, we demonstrate the same analysis, this time using an independent sort and thus allowing for an uneven distribution of funds across cells.

The results largely stay the same using an independent, or simultaneous, sort. Subsequent alphas are higher for those with lower *R*^{2} figures and higher trailing factor-adjusted alphas. All quintile spreads (Low–High column and High–Low row) are also positive, indicating persistence of these relationships across all quintiles. Similar to Exhibit 4, we find that the trailing four-factor alpha is perhaps more determinative than the trailing *R*^{2} because the quintile spread from low to high factor-adjusted alpha (across all quintiles of *R*^{2}) is about twice that of the spread from high to low *R*^{2}.

One anticipated challenge to this analysis and the conclusions drawn is the potential for omitted variable bias. Might a fund be sorted into a low-*R*^{2} quintile simply because our factor models are poor fits? If this is the case, are the low-*R*^{2} funds taking additional risks that are not captured by the two factor models? Other papers have mentioned the potential that low-*R*^{2} funds may be pursuing strategies with option-like payoff structures that may create significant tail risks. Professor Nicolas Bollen found that tail risks expose market neutral (low-*R*^{2}) hedge funds to increased failure rates despite exhibiting lower ex post volatility (Bollen [2013]). We have taken several steps to account for this. As previously mentioned, we use an HFR database that includes all closed funds to mitigate survivorship bias. We also tested the Fung and Hsieh seven-factor model, which includes a broader array of factors than the equity-only Carhart four-factor model. Are we still missing certain tail risks due to omitted variable bias, though? Any such tail risks should appear in the kurtosis and skewness statistics of the low-*R*^{2} funds. In Exhibit 6, we report the peer-relative statistics for each cell using the seven-factor model. We take the de-meaned trailing 24-month statistics for each fund, record the cross-sectional average for each month, and then average those averages across all months.

Starting with the kurtosis statistics, we do observe a steadily increasing peer-relative kurtosis as *R*^{2} decreases, but we see no obvious link to factor-adjusted alpha. Higher kurtosis reflects greater tail risks. However, when we observe the skewness statistics, we observe that the low *R*^{2} quintile exhibits positive skew. Bollen’s findings would suggest a negative skew from these funds, consistent with greater downside tail risk. Instead, we find that the higher-*R*^{2} funds exhibit negative skew. Perhaps less surprisingly, we also find that higher factor-adjusted-alpha funds also exhibit more positive skewness. Both relationships are nearly monotonic.

To further ensure that our analysis has not omitted any systematic exposures that describe hedge fund returns, we performed a separate analysis of the HFR equity hedge universe using principal component analysis (PCA). The test is to take the same underlying funds as sorted by the four-factor model (we will focus on the four-factor model for comparison because it has a greater number of equity-specific factors than the seven-factor model) and subject them to PCA to determine whether missing factors can be identified and to assess whether including these missing factors has a significant impact on our characterization of the return streams from funds in each of the 25 combinations of quintile groups discussed thus far. Funds that see a big change in their factor-adjusted alpha or *R*^{2} in a PCA framework (compared to the four-factor framework) might be exposed to systematic factors that were omitted from our analysis.

One practical problem with using PCA analysis on our hedge fund returns data is that we have many more hedge funds, *n*, (over 6,500) than we have time periods, *t*, (180 months in this case). A covariance matrix of these returns would consist of well over 20 million distinct elements, and we simply do not have enough data points to estimate and decompose such a large covariance matrix. Fortunately, we can use asymptotic principal component analysis (APCA) instead, based on a 1986 paper by Professors Gregory Connor and Robert Korajczyk (Connor and Korajczyk [1986]). Their key insight was recognizing that if *n* >> *T*, the problem can be reframed. Instead of forming an *n* × *n* covariance matrix with time-series asset variances on the diagonal and asset covariance pairs off-diagonal, we form a covariance matrix with cross-sectional variances for each time period on the diagonal and cross-sectional covariance between different time period pairs off-diagonal. The same authors subsequently improved upon the methodology through a two-pass approach that does a better job of handling the cross-sectional heteroskedasticity of asset returns (Connor and Korajczyk [1988]).

Using this methodology, we can analyze the 25 cells in our hedge fund study by regressing their returns against the APCA factors and examining the *R*^{2} values to determine whether a cell’s returns exhibit sensitivity to a systematic factor that was missed in the four-factor framework. To the extent that a cell exhibits significantly higher *R*^{2} in the APCA models compared to the four-factor model, we can assume the cell has exposure to systematic factors that were not captured in the four-factor model. We use the four-factor model to sort funds to ensure that each of the 25 cells has the same fund membership used earlier. We then observe the APCA *R*^{2} of that cell in Panel B in Exhibit 7 and compare it to the average four-factor model *R*^{2} of the same cell (Panel A) as a baseline.

Comparing the panels in Exhibit 7, we see that the *R*^{2} values go up (unsurprising because we now have 10 factors), but the basic trend is identical to what we find using the four-factor model to calculate *R*^{2}. The lower four-factor *R*^{2} cells are similarly difficult to define relative to the high-*R*^{2} cells. Because this general pattern persists, we believe that the baseline four-factor model does not exclude any systematic factors that would change our results (by disproportionately affecting members of a specific cell). We can retest this result by analyzing the number of APCA components required to achieve a minimum *R*^{2} threshold. Using the same cell constituents as before (defined and sorted using the four-factor model), we investigate this question in Exhibit 8 using a 0.90 minimum *R*^{2}. Again, we find the hedge fund returns represented in the left column (the low four-factor *R*^{2} funds) very difficult to explain, requiring over 50 principal components to achieve a 0.90 APCA *R*^{2}. Equally noteworthy is that many of the cells in the right columns (the high four-factor *R*^{2} funds) hit the 0.90 APCA *R*^{2} threshold using only one factor (which likely represents market beta).

Thus far, we have explored the relationship between trailing *R*^{2} and subsequent factor-adjusted alpha, in addition to that between trailing factor-adjusted alpha and subsequent factor-adjusted alpha. The latter essentially tests the persistence of skill. Do managers who have exhibited skill in the past continue to exhibit skill in the future? We have found that, indeed, there is a strong relationship, consistent with our hypothesis that skill is largely fixed—a manager either has it or does not.

What about predictiveness of actual returns? Is that not the first concern of hedge fund investors? With most developed markets in rally mode between 2009 and 2014, the best-performing hedge funds have undoubtedly had positive exposure to equity market beta, but what happens across market cycles? To reduce the potential of timing bias, we look at the last 180 months (15 years) in an attempt to capture cross-cycle representation. We perform the same independent sort that we did for Exhibit 5 based on trailing 24-month factor-adjusted alpha and *R*^{2} and then observe performance in the 25th month. This time, we record 25th-month returns relative to the universe average hedge fund return for that month. We take a cross-sectional average for each month, then average this number across all months and annualize it. These are the returns that appear in Exhibit 9.

A large difference in returns is apparent between corner cells: 7.8% in annualized return separates the high-*R*^{2}, low-four-factor-alpha cell from the opposite corner (the two cells with thicker borders) using the four-factor model, and 5.9% separates the corner cells using the seven-factor model. Focusing on the All columns, the funds that have demonstrated skill in the past consistently earn better returns going forward. Focusing on the bottom rows of factor-adjusted alpha quintile spreads (High–Low), one can see that all spreads are positive. The *R*^{2} trend is also apparent, though admittedly less monotonic (see All rows). As we discussed, however, the *R*^{2} results are still susceptible to cyclical bias. When the equity market is rising and small cap, value, and momentum stocks are performing well, a high *R*^{2} will naturally lead to higher returns. Nevertheless, the general trend remains: Funds with higher skill and lower factor dependence subsequently outperform their peers.

These results strengthen when we risk-adjust hedge fund returns by volatility. Using each fund’s annualized 25th-month return, we then divide by the annualized volatility of each fund over all 25 months. The results in Exhibit 10 then represent the average subsequent Sharpe ratios of fund peers sorted along the two dimensions. The trends become much more monotonic after adjusting for risk because fund volatilities are observed to increase as *R*^{2} increases. This stands to reason because factor returns contribute additional sources of volatility above and beyond the volatility of factor-adjusted alpha itself. Focusing on the All columns and rows, we observe that Sharpe ratios increase as factor-adjusted alpha increases and as *R*^{2} decreases. Furthermore, the quintile spreads (Low—High column and High—Low row) are positive in all but one instance. Comparing the extreme cells in Exhibit 10 (with thicker borders), we observe a large increase in Sharpe ratio moving from the top right cell to the bottom left.

One anticipated criticism of this analysis involves potential serial correlation in hedge fund returns. A number of academic papers have argued that many hedge funds invest in illiquid securities that trade less frequently, creating an artificial smoothing of returns from month to month (Getmansky, Lo, and Makarov [2004]). This serial correlation can potentially push down measures of volatility and inflate risk-adjusted measures of return such as Sharpe ratios. For the purpose of this analysis, we are most concerned with any asymmetric impact on our *R*^{2} quintiles. Specifically, what if a fund generates a low *R*^{2} because its illiquid investments produce artificially smooth returns? If this were indeed the case, our classification system might confuse exploiting the illiquidity premium for low factor dependence and skill. To test if this is the case, we measure average serial correlation across *R*^{2} quintiles, using both the four-factor and seven-factor models. A fund is sorted into an *R*^{2} quintile using its trailing 24-month returns; on month 25, we measure the correlation in returns between months 1–24 and months 2–25 for every fund on a rolling basis. We take the cross-sectional average and then the average for all months. We also calculate the cross-time average using seven points in time spaced 25 months apart to ensure nonoverlapping measures of serial correlation. The results are shown in Exhibit 11.

As can be seen, the highest measures of serial correlation appear in quintiles 2–4. Quintiles 1 and 5 exhibit lower serial correlation. These results do not support the hypothesis that low-*R*^{2} funds invest more heavily in illiquid assets. In fact, we see serial correlations initially increasing as we move from left to right in the chart, especially in the overlapping period measures. This may be partially explained by the serial correlation in the model factors themselves, the same factors to which higher-*R*^{2} funds are more exposed. Over our full test period from January 2000 to December 2014, for example, the equity factor exhibits a 16.6% serial correlation, the bond market factor exhibits a 17.8% serial correlation, and the credit spread factor exhibits a 44.9% serial correlation. Additionally, we should note that all of the serial correlations in Exhibit 11 are relatively low compared to those in other studies of the entire hedge fund landscape. We believe that this disconnect is due to our universe of equity hedge funds only, which generally invest in more liquid assets than other hedge fund strategies.

How has *R*^{2} and factor-adjusted alpha changed over time among equity hedge funds? Using discrete value cutoffs, we examine the actual distribution of equity hedge funds across both dimensions. We focus only on the four-factor model because it includes more equity-specific factors. Assuming cutoffs of 0.25 for *R*^{2} and 5.0% for annual four-factor alpha, the distributions as of December 2014 and December 2000 appear in Exhibit 12. Although nearly half of hedge funds in 2000 fell into the bottom-right quadrant (high skill/high factor dependence), this large percentage of funds appears to have migrated into the top-right quadrant (low skill/high factor dependence) today. Funds depended on factors to produce returns in 2000, but they supplemented that with meaningful factor-adjusted alpha; today, the majority of funds still have significant factor exposure with much less non-factor-dependent alpha. Interestingly, the percentage of funds in the preferred bottom-left quadrant (high skill/low factor dependence) has stayed roughly constant since 2000. It remains as difficult as ever to find the rare combination of high skill and low factor dependence. Yet, as the line graph in Exhibit 12 illustrates, this is the quadrant that has consistently produced the highest factor-adjusted returns. Conversely, the low skill/high factor dependence quadrant has produced the lowest factor-adjusted returns.

One topic we have largely ignored until now is the actual relationship between factor-adjusted alpha and *R*^{2}. We know the relative distribution of funds: The majority of funds today fall into the least desirable quadrant of both low skill and high factor dependence, but how does factor dependence vary with changes in skill levels? Once again, would a manager with true stock selection skill choose to layer on factor exposures that are beyond his or her control? We think not, and if we are correct, we would observe declining factor exposures as skill increases. The median characteristics of funds sorted by factor-adjusted alpha quintiles as of December 2014 appear in Exhibit 13.

As suspected, the highest-skill quintile of four-factor alpha in Exhibit 13, Panel A also has the lowest factor exposures, whether represented by *v* (to the four factors) or individual factor coefficients. These managers derive the majority of their returns from true alpha sources and therefore do not hide behind the major style factors. As skill declines, median *R*^{2} increases and factor exposures—specifically beta and small-cap, growth, and negative momentum exposures—increase. Consistent with what Exhibit 1 illustrated, however, beta and small-cap exposure appear to be the primary factors driving *R*^{2}. In the lowest quintile of skill—significantly negative four-factor alpha—the median beta is high, at 0.77, and the small-cap coefficient is 0.51. These funds are not only producing negative alpha, but their performance will track that of the market and small-cap stocks and they are unlikely to provide the absolute returns that many hedge fund investors seek.

In Exhibit 13, Panel B, the same analysis using the Fung and Hsieh factor model reveals the same trends with regard to *R*^{2}, beta, and small-cap exposure. The other variables are much less monotonic, with the exception of yield sensitivity: Here, the higher seven-factor alpha funds have a positive relationship with interest rates. However, despite the higher interest rate coefficient for the top factor-adjusted alpha quintile, overall exposure to systematic factors (as measured by *R*^{2}) is the lowest among the quintiles, and the factor-adjusted alpha is obviously correcting for this and other exposures.

## CONCLUSION

It remains difficult to distinguish skilled managers in the growing pool of hedge funds. We propose using two tools to decompose returns into factor-dependent and skill-dependent sources. We combine factor-adjusted alpha, a proxy for skill, with *R*^{2}, a proxy for factor dependence, and we find that the highest-skilled, lowest-factor-dependent funds have the best subsequent performance. We also show that the highest-skilled managers have the lowest factor exposures.

## ENDNOTES

We are grateful to Professor Robert Stambaugh of the Wharton School of the University of Pennsylvania and the National Bureau of Economic Research for his review of an earlier version of this article and helpful comments.

↵

^{1}Though difficult to prove empirically, we observe that regulation fair disclosure (2000), decimalization (2001), the global settlement (2002), and the late trading scandal and subsequent regulation (2003) coincided with the largest decline in equity hedge fund four-factor alpha in the last 15 years.↵

^{2}Over the past 15 years (January 2000–December 2014), the volatility of the equity market premium (MSCI World Index—risk-free rate) has been 16%, that of the small-cap premium (MSCI World Small Cap Index—MSCI Large Cap Index) has been 8%, that of the value premium (MSCI World Value Index—MSCI World Growth Index) has been 7%, and that of the momentum premium (float-weighted return of top half of price performers in the MSCI World Index from previous 12 months—float-weighted return of bottom half of price performers in the MSCI World Index from previous 12 months) has been 12%.**Disclaimer**This article should not be relied on as research or investment advice regarding any investment. There is no guarantee that any forecasts made will come to pass. Causeway Capital Management does not guarantee the accuracy, adequacy or completeness of such information. MSCI has not approved, reviewed or produced this article, makes no express or implied warranties or representations and is not liable whatsoever for any data in the article. You may not redistribute the MSCI data or use it as a basis for other indices or investment products.

- Copyright © 2017 Causeway Capital Management. All right reserved. Not to be reproduced or redistributed without permission.