Models of evaluation of the effectiveness of hedge funds
Hedge fund research. Multifactor assetpricing models. Hedge funds factor models. Data on asset pricing factors and descriptive statistics. Carhart fourfactor model. Betting against beta. Quality minus junk. Analysis of regional focuses, asset pricing.
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Models of evaluation of the effectiveness of hedge funds
Abstract
Hedge funds today are among the most active institutions in the modern financial system and experience consistent growth in their number and assets under management. For investors the main criteria for choosing a hedge fund to invest in is its profitability. That is why understanding hedge funds' return factors is important. This paper studies the possibility of application of extended assetpricing models for hedge funds exposure and efficiency analysis. We say a hedge fund is efficient if its Alpha is above sample average and Rsquared for asset pricing models is low. The research includes the usage of multifactor models for the return analysis of over 3600 funds during years 20052015. The contribution of this paper is comparative analysis of different hedge fund strategies, which allows to track whether a particular strategy beats the market or not. We track the relationship between hedge funds' returns and exposure to commonly applied models between different strategies, regions and time.
Keywords: hedgefunds, Alpha, risk exposure, asset pricing models, performance
Contents
Introduction
1. Hedge funds efficiency and exposure
1.1 Hedge fund research
1.2 Multifactor assetpricing models
1.3 Hedge funds factor models
1.4 Literature drawbacks
2. Data
2.1 Data set
2.2 Hedge fund strategy classification
2.3 Data on asset pricing factors and descriptive statistics
3. Framework of the research
3.1 Capital Asset Pricing Model
3.2 FamaFrench threefactor model
3.3 Carhart fourfactor model
3.4 Betting against beta
3.5 FamaFrench fivefactor model
3.6 Quality minus junk
4. Outcomes
4.1 Analysis of strategies and regions
4.2 Analysis of strategies and asset pricing models
4.3 Analysis of regional focuses and asset pricing models
4.4 Analysis of timerelated dependencies
5. Contribution, limitations and opportunities for further research
Conclusion
References
Appendix 1
Appendix 2
Appendix 3
Appendix 4
Introduction
The evaluation of hedge fund's effectiveness is one of the most important research problems of modern theory and practice of financial management. The increase in the number of hedge funds is a result of the introduction of Volcker rule, restricting investment banks in trading on financial markets, and the growing public attention attracted by extremely high profits that some hedge funds obtained even in the periods of crises. Although the multimillion dollar payouts that hedge fund managers receive are quite rare, the mere existence of this possibility is of great interest for both private investors and professional market participants.
Hedge funds have been experiencing consistent growth in their number and assets under management during last decades. Their popularity has reached its peaks during financial crises, and today hedge funds still show excessive returns compared to other asset classes. Due to the increasing demand, it is obvious that hedge funds will continue to be a distinguished investment vehicle among institutional and wealthy investors. However, the industry starts to experience tough times since the latest financial crises, so the propriety on high commissions and management fees has to be justified by the efficiency of investment strategy employed by a fund. The ability to profit in any market condition though still causes great interest among investors of hedge funds. The investors start to experience the problem of choosing a fund to invest. The main criterion of a hedge fund is its yield (or efficiency).
The novelty of hedge funds as a collective investment instrument is accompanied with the infancy of hedge funds' theoretical background. Assetpricing models, which are the core of financial management theory, can be applied to the hedgefunds as well in order to find out their return factors. More conservative investment vehicles like mutual funds are usually analyzed via CAPM, with the Markowitz portfolio theory behind it. However, for hedge funds it is necessary to move on from solely theoretical models and employ empirical models, due to the opacity of hedge funds' returns, so that additional difficulty is created. The models that explain hedge fund returns are relevant to the hedge fund investors, which want to find out the factors that are driving the profit of a particular fund. In order to do so, a vast amount of research has been done in order to reveal the factors of hedge fund returns.
The presence of high management and performance fees though creates additional necessity in having an efficient way to distinguish truly efficient funds that create value above benchmarks. The way hedge funds are organized and managed makes investors significantly exposed to the skills of hedge fund managers. Stating that, investors are not likely to invest in a hedge funds that follows the strategy of following market trends, going long "blue chips" even if these strategies bring them positive Alpha considering fees. This is the reason an investor would be eager to invest in a hedge fund that uses its own strategy and using profitmaking opportunities not identified by its competitors.
Hedge funds that not only get positive returns, but also do it in a way most other financial institutions and asset managers can not, are to be desired by many investors. In order to spot this kind of funds, we create a framework that will allow to track the performance, novelty of investment strategy and the relationship between them. We state that the fund is efficient if it has high Alpha and low exposure to classic asset pricing models. We use CAPM and extended CAPMbased models in order to track the exposure of different strategies' hedge funds to such factors and market risk, size or value investing, as well as a few other factors commonly accepted in asset pricing research.
The purpose of this paper is to measure hedge funds' returns' exposure to the asset pricing models (Rsquared) and produce comparative analysis of Alpha  Rsquared relationship between strategies, regional focuses and time.
The following hypotheses will be tested throughout the research:
1. There is a large difference between performance and exposure of hedge funds, which focus on different geographical regions;
2.1. Hedge funds with lower exposure (model Rsquared) to the risk factors on average provide higher returns;
2.2. The relationship between past exposure and future Alpha is negative;
3. It is possible to outline specific return factors of hedge funds using classic assetpricing models;
The structure of the research is the following: In Section 1 we discuss most relevant research papers devoted to hedge fund and asset pricing analysis. Section 2 is dedicated to the review of the data used in our sample, as well as description of hedge fund strategies outlined in our research. In Section 3 we discuss our methodology and the framework of analysis, stating asset pricing models employed in the research. In Section 4 we state the results and outcomes, outlining criteria for hedge funds' efficiency. In Section 5 we conclude the contribution of this paper, state the limitations and opportunities for further research.
1. Hedge funds efficiency and exposure
The research positions itself in between the hedge funds performance research, which derives profitability, survivorship and other factors based on the intrinsic characteristics of a particular fund, and the assetpricing research, which employs risk exposure factors to explain a funds returns as the objects of a fund's investment strategy.
1.1 Hedge fund research
A significant number of papers was devoted to the problem of assessing the performance of the hedge fund during the last 1520 years (e.g., Gruber, 1996; Carhart, 1997; Agarwal, Naik, 2000; Boyson, 2003; Bollen, Busse, 2004). Many papers addressed the problem of hedge funds' performance persistence. Some works point to the presence of shortterm (13 months) persistence of profits among some hedge funds (Agarwal, Naik, 1999, 2000; Bares, Gibson, Gyger, 2002; Baquero, Ter Horst, Verbeek, 2002). However, at large time intervals (from 6 months) stability of yield observed very rarely (Brown, Goetzmann, Ibbotson, 1999; Brown, Goetzmann, 2001). A large proportion of the work uses regression analysis of the stability of funds' efficiency in order to reveal return factors. The most important factors affecting the profitability of hedge funds typically are the fund's investment strategy and the personal characteristics of the portfolio manager. Thus, the persistence of returns is detected among funds with "young" (relating to the tenure) managers (Boyson, 2003). Managers also increase the likelihood of obtaining a positive return, while reducing the likelihood of establishing a new high watermark (the profitability measure, above which the premium will be paid to the manager) (Motaze, 2009). Other work find a positive correlation between the rate of return of funds and the commission, the size of the fund and the period of inability to withdraw funds (Liang, 1998). The relationship between the index of strategy definition (SDI) and the longterm sustainability of return also stands out (Amman, Huber, Schmid, 2010).
1.2 Multifactor assetpricing models
Among the more fundamental studies are papers on the CAPM models application to the estimation of the effectiveness of funds. Ever since Treynor, Sharpe, Lintner introduced Capital Asset Pricing Model, many have been trying to improve its quality by using additional riskfactors with the model. A significant step was done by Jensen (1968), who introduced such a factor as Alpha for the calculation of riskadjusted returns excess of CAPM. The Alpha concept is still used by many asset managers as a measure of their effectiveness. CAPM was also expanded in Fama and French's research (1993). The authors include into the model parameters such as the capitalization of the portfolio (SMB) and the ratio of book value to market value (HML). Carhart (1997) upgraded the FamaFrench model. He adds to the threefactor model another factor  oneyear momentum. Under momentum he meant the average difference in yield of best and worstperforming funds over the previous 12 months.
The idea of extended CAPMbased models is still supported by various authors. Frazzini and Pedersen (2013) constructed a model that takes into account leverage and margin constrains, therefore introducing BAB (Betting Against Beta) factor, which is long leveraged lowbeta assets and short highbeta assets. The authors found that the factor produces significant positive riskadjusted returns.
Asness, Frazzini and Pedersen (2013) developed the concept of quality securities, accounting for stocks that are safe, profitable, growing, and well managed. They employed QMJ (Quality Minus Junk) factor that goes long highquality stocks and shorts lowquality stocks earns significant riskadjusted returns in the U.S. and globally. Assness and Frazzini (2013) also took another approach at calculating HML factor, considering the momentum factor and reducing the lags.
Fama and French (2014) expanded their classic threefactor model by including profitability and investments factors into the model. The RMW is the difference between the returns on diversified portfolios of stocks with robust and weak profitability, while CMA is the difference between the returns on diversified portfolios of the stocks of low and high investment firms, which are called conservative and aggressive.
1.3 Hedge funds factor models
Much research has been done on the subject of linear models that may explain hedge fund returns. Most of the papers differ by the set of variables, their quantity and interpretation. For the theoretical base of the research we take a few papers that stand out from the masses.
The paper by Fung, Hsieh (2004) offers a sevenfactor linear model that accounts for different type of riskexposure. The authors use the data on hedge funds' returns from 1994 to 2002 and evaluate returns' sensitivity to seven risk types: Bondtrend, Currencytrend, Commoditytrend, Equitymarket, Sizespread, Bondmarket, Creditspread. Applying linear regression model the authors received the following results: The sevenfactor model explains most of monthly return variation. However, the risk characteristics of specific hedge fund strategies are better explained by risk factors that are constructed to fit the particular strategy. In addition, the risk factors in the model are not necessarily unique. Other linear combinations of the seven risk factors can produce substantially similar results.
Getmansky, Lee, Lo (2015) use linear regression model to study the dependence of hedge funds' returns on the equitymarket risk, sizespread factors, bondmarket factor and 2to3 month lagged returns of the S&P 500 on the 20 year data period. They found out that fourfactor model does a better job than sevenfactor model (Fung, Hsieh, 2004) for seven of the 10 hedge fund categories (strategies). This suggests that the best linear factor model forecast of a hedge fund's future risk may be obtained from simpler, investable factors, rather than illiquid factors.
Asness, Krail, Liew (2001) construct a model that only uses market and hedge fundspecific indices. Using the data on 656 funds from 1994 to 2000, they regress hedge funds' returns by S&P 500 returns and CSFB/Tremont hedge fund indices returns. The regressions show only modest market exposure and positive added value. Hedge funds in the aggregate contain significantly more market exposure than simple estimates indicate. After accounting for this increased market exposure, it is found that taken as a whole the broad universe of hedge funds does not add value over this period.
Titman, Tiu (2008) use the data on 8500 hedge funds (the largest sample used in academic research) from 1994 to 2005. The authors construct a linear regression of hedge funds' returns and Equity, Fixed Income, Commodities and Primitive Trend Following Strategies factors. The authors also regress future performance of funds by their previous Rsquares and find out that low Rsquared funds outperform high Rsquared funds. A 10% drop in the Rsquared is associated with an annual increase in returns of about 60 basis points and an increase in the annual Sharpe ratio of about 0.05.
Similar results were found by Aragon (2004). The author found that longer lockup periods generate superior performance, which is consistent with the idea that these funds are able to generate excess returns by taking less liquid positions. Since these funds also tend to have lower Rsquares than other funds there is a possibility that the relation between Rsquare and future Sharpe ratios is generated by the same liquidity effect.
The bottom line of the abovementioned literature is the following: linear risk factors can well explain hedge funds' returns, however this does not necessary mean that these hedge funds add value. Conversely, sometimes exposure to the risk factors may result in the lower Sharpe ration, or lower efficiency. It is therefore reasonable to create the perspective of assessing hedge funds' performance in terms of value added.
1.4 Literature drawbacks
Most of these papers suffer from two types of problems: data biases and linear models drawbacks. There are three types of data bias. The first one is the survivorship bias: dead funds usually have poor performance and are not presented in the calculation, therefore the average estimation of population returns is biased upwards. Backfill bias occurs when new or lowperforming funds "backfill" the database later when they have established a track record of success. Selfselection bias occurs when top or bottom performing funds lack the same incentive as other funds to report to data vendors and thus are excluded from index calculations (Agarwal, Naik, 2000; Boyson, 2003).
Linear models drawbacks also exist in all the papers that use this method of studying hedge funds' returns. Factor model usually explains only part of the hedge funds' returns, even if Rsquared is high. Even if a factor model accurately explains a fund's returns, the risks associated with every hedge fund change over time with the market structure and geopolitical situation. It is also impossible to replicate hedge funds' returns by investing in the factors that may describe its performance, since some of the factors may be noninvestable or illiquid (Getmansky, Lee, Lo, 2015). In this paper we will try to address last type of problems by choosing tradable factors that will allow to replicate hedge funds' returns.
There is also the necessity of tradability and liquidity of hedge funds' return factors, that is determined by the practical value of these models, which are used in order to forecast hedge funds' future performance or to imitate these returns outside the hedge fund. It is not possible to simulate an excess return of a hedge fund, if its return factor is not traded anymore: for example, the bond may expire, or the stock may be delisted. In order to avoid this problem it is needed to find proxies for these factors, that will be both tradable and liquid. It will allow other institutions to obtain hedge fundslike returns regardless of its legal form.
To underline the abovementioned literature we can state that many researchers were able to capture some factors defining hedge funds return and outline relationships between funds' profitability and its characteristics. However, most of the papers are tend to be concentrated on absolute returns. Since hedge funds are quite risky, secretive and expensive (in terms of fees and commissions) way to invest, it is necessary to have a benchmark for returns in order to make weighed investment decisions.
We return to the concept introduced by Jensen to measure the efficiency of investments in terms of excess returns above selected benchmark. We use asset pricing models outlined above in order to derive Alphas, returns factors and exposure of funds' returns' to them, or the Rsquared of the models. We then address findings of Titman, Tiu and expand them on the crosssection of strategies, regional focuses and time horizons.
2. Data
Previously we stated that even though data vendors often lack valuable data on hedge funds that most of other financial institutions are obliged to report it is nevertheless possible to extract information on hedge funds' returns and general characteristics of funds. In this section we discuss our data sample.
2.1 Data set
As hedge funds are not obliged to report to any databases and data vendors, the information available for any particular hedge fund is very scarce. The data on hedge funds' returns is downloaded from Bloomberg Terminal for years 20052015. Besides hedge funds' yield, Bloomberg Terminal also offers data on hedge funds' strategies. There are 10 strategies in Bloomberg classification: MultiStrategy, Macro, Fixed Income Directional, Fixed Income Relative Value, Event Driven, Equity Hedge, CTA/Managed Futures, Global Allocation, Long/Short or Strategy is not available (N.A.). Strategies Long/Short and Global Allocation were excluded from analysis due to the insufficient number of observations. The following data is also available on Bloomberg Terminal for some hedge funds: country of domicile, country of operations, assets under management. Other databases such as Lipper TASS offer additional data, but we do not have access to them. The data on hedge funds' return factors, such as riskfree rate, market returns, etc. was downloaded from public sources: Kenneth French data library, Andrea Frazzini data library.
The distribution of hedge funds across strategies and regions can be seen below (Table 1), as well as abbreviations we use for strategies and regions throughout the research.
Table 1
Distribution of hedge funds in the sample across strategies and geographies
Strategy/Geographical Focus 
Global (G) 
North America (NA) 
Asia & Pacific (P) 
Europe (E) 
United States (US) 
Total 

CTA/Managed Futures (CTA) 
324 
1 
6 
5 
1 
337 

Equity Hedge (EH) 
526 
27 
346 
118 
44 
1061 

Event Driven (ED) 
76 
5 
20 
8 
2 
111 

Fixed Income Directional (FID) 
159 
1 
40 
11 
11 
222 

Fixed Income Relative Value (FIRV) 
57 
1 
12 
0 
3 
73 

Macro (M) 
313 
1 
29 
7 
4 
354 

MultiStrategy (MS) 
373 
3 
58 
12 
3 
449 

N.A. (NA) 
710 
22 
167 
72 
90 
1061 

Total 
2538 
61 
678 
233 
158 
3668 

As it can be seen from the table, most of the funds are oriented at global markets and Asia and Pacific region. The most used strategy is Equity Hedge, which includes such traditional for hedge funds investing styles as Statistical Arbitrage. 1061 funds did not reported their trading strategy and therefore were put under "N.A." mark. The United States were taken as a separate geographical region in order to separate U.S.oriented funds from offshore funds (most of which are located in North America). Relative performance of each strategy to the S&P 500 index can be seen in the chart (Figure 1).
Figure 1. Cumulative return of strategies analyzed and S&P 500 index
It seems that we can easily state profitable and efficient funds based on the performance chart. However, as we find out later, the nature of hedge funds' returns is much more sophisticated than the cumulative return.
2.2 Hedge fund strategy classification
The hedge fund strategy classification was taken in compliance with Bloomberg classification. To each hedge fund strategy there are investing strategies and objectives, which can also interconnect between strategies. We outline the basic concepts of each strategy as it is important to understand strategies' characteristics and nature in our analysis. The cumulative return of each strategy, divided into different regional focuses and compared by volatility and return with S&P 500, which is a commonly accepted benchmark for many asset managers, can be seen in Appendix 1.
Commodity Trading Advisors/Managed futures strategies hedge funds concentrate on the commodity and financial futures markets, specializing on sophisticated algorithmic computerdriven trading programs. Hedge funds of this type tend to use employ precise trading rules to capture price movements and focus on shortterm trading patterns.
Equity Hedge strategies maintain both long and short positions primarily in equity securities and equity derivatives.
These strategies can be broadly diversified, as well as narrowly focused on specific industries or sectors and can range broadly in terms of levels of net exposure, leverage employed, holding period and other characteristics. Equity Hedge fund managers would typically maintain positions with at least 50% exposure to equities  both long and short.
Event Driven hedge fund managers maintain positions in companies awaiting or currently involved in corporate transactions of a wide variety including mergers, acquisitions, restructurings, financial distress, tender offers, shareholder buybacks, as well as other capital structure adjustments. Event Driven exposure includes a combination of sensitivities to equity markets, credit markets and idiosyncratic, company specific risk.
Fixed Income Directional strategy includes trades on realization of a spread between related instruments in which one or multiple components of the spread is a fixed income instrument or derivative backed by physical collateral or other financial obligations.
Strategies employ an investment process designed to isolate attractive opportunities between a range of fixed income instruments specifically securitized by collateral commitments or similar structures.
Fixed Income Relative Value includes strategies on realization of a spread between related instruments in which one or multiple components of the spread is a convertible fixed income instrument. Convertible arbitrage positions maintain characteristic sensitivities to credit quality of the issuer, implied and realized volatility of the underlying instruments, interest rates levels and the valuation of the issuer's equity, trying to capture pricing inefficiencies and other trading opportunities.
Macro fund managers employ a variety of strategies in which the investment process is predicated on movements in underlying economic variables and the impact these would have on equity, fixed income, hard currency and commodity markets. Managers employ a variety of techniques, both discretionary and systematic analysis, combinations of top down and bottom up theses, quantitative and fundamental macroeconomic approaches and long and shortterm holding periods.
Multistrategy funds may refer to a strategy not directly specified, as well as to a combination of other strategies outlined above. Managers of multistrategy funds may employ various strategies and obtain exposure to different types of risks.
Nonavailability of hedge fund strategy information creates additional difficulties to analyze the hedge fund industry. It is therefore up to researchers to understand which risk factors will these funds show exposure to.
2.3 Data on asset pricing factors and descriptive statistics
The parameters for CAPM models were downloaded the following way: the risk factors were taken in compliance with the hedgefunds geographical focus. Therefore, estimating such parameters as riskfree rate or market return, we take into account the market that the fund operates in. Consequently, for the abovementioned factors we take different factors depending on the region of fund's operations.
We employ such factors as the market premium, which is the excess return of market index over riskfree rate (variable MKTRF), size factor (SMB), value factor (HML), profitability (RMW), investment (CMA), Betting against beta factor (BAB), quality factor (QMJ) and momentum (MOM).
One of our research priorities is to track the differences if hedge funds' performance across various geographies. Since the geographical focus is available to us, we split all the hedge funds into five categories as shown in the table. The return factors were also downloaded and used in the models correspondingly with the hedge fund's focus: as to if a fund is Europeoriented, the factors used to calculate the benchmark return are taken for Europe.
Below is descriptive statistics of hedge funds' returns, as well as for the return factors.
These are the values for the average strategies (Table 2) and factors (Table 3), descriptive statistics for the regionally distributed strategies and return factors can be seen in Appendix 2 and Appendix 3 respectively.
Table 2
Descriptive statistics for hedge funds' returns
CTA 
ED 
EH 
FID 
FIRV 
M 
MS 
NA 

Average 
0.33% 
0.24% 
0.50% 
0.27% 
0.29% 
0.28% 
0.48% 
0.09% 

Min 
9.34% 
12.85% 
11.15% 
5.50% 
4.40% 
6.56% 
5.64% 
10.88% 

Max 
8.27% 
7.32% 
9.17% 
3.15% 
4.96% 
6.01% 
9.48% 
13.67% 

Std dev 
3.42% 
2.35% 
3.13% 
1.09% 
0.90% 
1.59% 
1.85% 
2.37% 

Table 3
Descriptive statistics for asset pricing factors
MKTRF 
SMB 
HML 
RMW 
CMA 
RF 
MOM 
BAB 
QMJ 

Average 
0.55% 
0.04% 
0.01% 
0.25% 
0.12% 
0.11% 
0.60% 
0.69% 
0.46% 

Min 
20.96% 
5.87% 
6.45% 
5.08% 
4.73% 
0.00% 
25.40% 
8.61% 
7.84% 

Max 
13.78% 
6.44% 
6.34% 
4.65% 
5.84% 
0.44% 
9.97% 
9.17% 
8.31% 

Std dev 
5.15% 
2.25% 
2.14% 
1.57% 
1.62% 
0.15% 
3.99% 
2.61% 
2.47% 

As it can be seen, hedge funds on average provide 0.31% monthly return, which corresponds with 3.7% annual return, with Equity hedge and Multistrategy being the most profitable strategies at the first sight. We will later discuss profitability of these strategies in more details.
3. Framework of the research
As we target to outline over or underperformance of hedge funds, it is necessary for us to determine a benchmark and criteria for performance. The findings of Titman, Tiu (2008) point us to the idea of the relationship between the performance of the fund, and the return factors associated with its strategy. Considering the vast amount of restrictions and risk associated with investing in hedge funds, as well as high performance fee and commissions, it is sensible that their performance should be measured in more complexity than the simple absolute returns or the returns above benchmark indices.
We therefore develop a framework, consistent with which a fund would be outperforming and a "value for investing", if its Alpha is positive and is obtained by some kind of an innovative strategy. To track the novelty of the investing strategy we are eliminating the strategy for following the market index (since this strategy does not require an investor to be a hedge fund), as well as other commonly known strategies such as "value" or "growth" investing.
To measure the extent to which a particular hedge fund is exposed to the outlined kinds of strategies, we use the Rsquared of the regression models of returns, by the factors corresponding to these strategies. It is therefore would be sensible to mark hedge funds as outperforming if their Alpha is high, while the Rsq of the corresponding model is low.
In the research we employ the most commonly used assetpricing models, as well as the ones extended with additional factors. The specifications of chosen models vary by the number of variables and its nature.
3.1 Capital Asset Pricing Model
The CAPM model is typically addressed as the most simple and therefore most contradictory. One of the simplest investment strategies can be captured if we see significant exposure to this model: following the market. There is not much reason for investors to pay 2% commission and 20% performance fee in order to use a strategy they can implement by simply following the market index. Use this model in order to track "lazy" funds.
The CAPM model can be written as follows:
, (1)
where: r  fund's returns; б  Jensen's alpha; Rf  riskfree rate; Rm  market return;  portfolio (fund's) beta;
3.2 FamaFrench threefactor model
Fama and French have managed to track the outperformance of value and small cap stocks. Although the model proved to be successful in explaining returns during previous century, the model itself is quite conservative. We use this model to capture whether hedge funds follow this concept or not.
The FamaFrench model can be written as follows:
(2)
where: r  fund's returns; б  Jensen's alpha; Rf  riskfree rate; Rm  market return;  portfolio (fund's) beta;  regression coefficients; SMB  historic excess returns of small caps over big caps; HML  historic excess returns of value stocks over growth stocks;
3.3 Carhart fourfactor model
Carhart has tracked the positive relationship between an asset's past and future performance, called as momentum. Given that momentum is usually calculated on 12month basis, it is unlikely that the strategy is frequently implemented by hedge funds, since some of them may exist for half that time.
Carhart model can be written as follows:
(3)
where: r  fund's returns; б  Jensen's alpha; Rf  riskfree rate; Rm  market return;  portfolio (fund's) beta;  regression coefficients; SMB  historic excess returns of small caps over big caps; HML  historic excess returns of value stocks over growth stocks; MOM  momentum factor;
3.4 Betting against beta
Another commonly used strategy, which was outlined in a separate riskfactor is to short highbeta and long lowbeta assets. A not so obvious theory was developed Frazzini and Pedersen, principals of at a large hedge fund, in 2013. Due to its nature the implementation of this model requires a large amount of capital and access to low trading costs to succeed, which is not really suitable for middle and smallcapitalization funds.
The model with betting against beta factor is written as follows:
(4)
where: r  fund's returns; б  Jensen's alpha; Rf  riskfree rate; Rm  market return;  portfolio (fund's) beta;  regression coefficients; SMB  historic excess returns of small caps over big caps; HML  historic excess returns of value stocks over growth stocks; BAB  returns of short highbeta and long lowbeta assets strategy;
3.5 FamaFrench fivefactor model
Fama and French have extended their basic model by including profitability and investment factors. The paper received a lot of critics, however there is a sense of logic behind the two new factors. Although, taking into account that information on companies' profitability and investment prospects appears as rarely as once a quarter (in the best case), so it is not the way for hedge funds to follow.
The FamaFrench fivefactor model can be written as follows:
(5)
where: r  fund's returns; б  Jensen's alpha; Rf  riskfree rate; Rm  market return;  portfolio (fund's) beta;  regression coefficients; SMB  historic excess returns of small caps over big caps; HML  historic excess returns of value stocks over growth stocks; RMW  historic excess returns of high operating profitability over weak; CMA  historic excess returns of conservative over aggressive portfolios;
3.6 Quality minus junk
Another recently introduced paper involves long highquality stocks and shorts lowquality stocks. The concept of high quality assets being more profitable is logical, however it is rarely observable whether a company is of a good "quality" or not, unless there is private information exploitation, which we do not address in our research.
The qualityminusjunk model can be written as follows:
(6)
where: r  fund's returns; б  Jensen's alpha; Rf  riskfree rate; Rm  market return;  portfolio (fund's) beta;  regression coefficients; SMB  historic excess returns of small caps over big caps; HML  historic excess returns of value stocks over growth stocks; RMW  historic excess returns of high operating profitability over weak; CMA  historic excess returns of conservative over aggressive portfolios; QMJ  returns of long highquality stocks and shorts lowquality stocks;
The selection of factors to specify the models was performed according to the relevance of economic research first to introduce them. We have included theoretical (for example, CAPM), as well as empirical models (FamaFrench models, Carhart model) and factors (BAB and QMJ, introduced by Asness, Frazzini and Pedercen, researchers and principals of a multibillion dollars hedgefund) in order to capture many asset pricing aspects. It was possible to include other factors and models in the research, such as marketspecific risk factors (commodity, FX, interest rate, etc.) or other factors subject to asset pricing. However the factors specified above were chosen from the most cited and relevant articles, and inclusion of additional factors would require an indepth economical validation so they were retained for future research.
We use the abovementioned models in order to track the relationship between hedge funds Alpha's and Rsquared for different strategies. In total we construct 205 random effects regressions. We then separate the stated relationship for different regions, strategies and models.
Having outlined the specifications of the models, it is then necessary to choose the toolset. Since we have panel data to analyze and no timeinvariant variables, we get to construct random effects panel regressions. Since fixed and random effects models produce close results in terms of Rsquared (within and overall) and constant (Alpha) values for our sample, there is no reason for us to use fixed effects. We also use reported Rsquared overall for our benchmark for funds' exposure to the factors in the models.
4. Outcomes
Having outlined model specifications (1)  (6), we then perform regression analysis. From the model constructed we a most interested in Rsquared of regressions and constant term, which we interpret as funds' Alpha. In the following section we discuss results of these models. We analyze Alpha, Rsquared and their relationships varying strategies, regional focuses, models and timeframes.
4.1 Analysis of strategies and regions
First of all we are interested in how the returns differ between absolute returns and excess returns, measured as the average Alpha of models constructed. The values for absolute returns (Table 4) and Alphas for different strategies and regions (Table 5) are shown below. The "" exist if a model could not be constructed by statistical package due to the insufficient number of observations or some other reason.
Table 4
Average annual return for different strategies and regions
Return 
E 
G 
P 
NA 
US 
Average 

CTA 
2.17% 
4.23% 
5.86% 
6.06% 
1.73% 
4.01% 

ED 
6.16% 
2.09% 
1.54% 
1.54% 
3.24% 
2.91% 

EH 
6.67% 
6.54% 
6.43% 
5.94% 
4.78% 
6.07% 

FID 
3.41% 
4.44% 
5.01% 
0.47% 
3.84% 
3.25% 

FIRV 
0.00% 
4.20% 
6.13% 
1.84% 
5.21% 
3.48% 

M 
1.92% 
5.15% 
6.23% 
0.86% 
2.67% 
3.37% 

MS 
5.72% 
5.11% 
4.90% 
14.67% 
0.58% 
5.96% 

NA 
1.53% 
5.57% 
2.09% 
10.39% 
0.32% 
1.14% 

Average 
3.06% 
3.27% 
4.77% 
5.10% 
2.65% 
3.77% 

Table 5
Average annual Alpha for different strategies and regions
Alpha 
E 
G 
P 
NA 
US 
Average 

CTA 
2.66% 
0.84% 
13.82% 
 
 
5.22% 

ED 
11.10% 
1.52% 
3.96% 
1.63% 
0.71% 
1.92% 

EH 
0.57% 
1.25% 
2.96% 
2.24% 
1.51% 
0.02% 

FID 
4.48% 
1.15% 
3.09% 
 
2.58% 
2.83% 

FIRV 
 
3.40% 
1.56% 
 
5.31% 
2.38% 

M 
5.96% 
1.82% 
0.65% 
 
9.28% 
4.43% 

MS 
0.25% 
2.24% 
7.06% 
3.02% 
5.29% 
3.57% 

NA 
4.68% 
4.19% 
1.88% 
6.47% 
1.37% 
1.13% 

Average 
0.43% 
0.42% 
3.51% 
1.41% 
2.90% 
1.74% 

To compare the values we calculated the difference between returns and Alphas (Table 6).
Table 6
Excess of funds' returns over Alpha
Delta 
E 
G 
P 
NA 
US 
Average 

CTA 
0.49% 
5.07% 
7.96% 
6.06% 
1.73% 
1.21% 

ED 
17.25% 
3.61% 
2.42% 
3.17% 
2.54% 
4.83% 

EH 
7.23% 
5.29% 
3.47% 
8.19% 
6.29% 
6.09% 

FID 
1.07% 
3.28% 
1.92% 
0.47% 
1.25% 
0.42% 

FIRV 
0.00% 
0.80% 
7.70% 
1.84% 
0.09% 
1.10% 

M 
4.04% 
3.33% 
5.58% 
0.86% 
6.61% 
1.06% 

MS 
5.47% 
2.86% 
2.16% 
11.65% 
5.87% 
2.39% 

NA 
3.15% 
1.38% 
3.96% 
3.92% 
1.69% 
2.27% 

Average 
3.49% 
2.86% 
1.26% 
3.70% 
0.25% 
2.03% 

It can be seen that on average Alphas are 2.4% lower than the absolute return values. In some cases, such as for Event driven European funds or Multistrategy North American funds the difference could be as much as 17% or 11%, which is critical in terms of wellconsidered investment allocation.
We can observe that for Europeoriented funds the average difference between Alpha and returns is among the highest, 3.49%, following only North American funds with 3.7% difference. The lowest value is achieved by Asia and Pacific and USfocused funds, with excess of absolute returns of 1.26% and 0.25% respectively. Global funds' Alphas' tend to be 2.86% lower than returns, which is close to the average value across the sample.
CTA and Macro strategies appeared to be among the most efficient with 1.21% and 1.06% difference between Alpha and returns, followed by Fixed Income Directional strategy. For Equity hedge and Event driven strategy funds returns are on average 6.09% and 4.83% higher than the calculated Alpha. European Event Driven funds have Alpha to be 17.25 lower than the actual average returns according to our models. Fixed Income Relative Value funds, Multistrategy funds and funds with no available information for strategy have returns to be higher than Alphas on the average value of the sample, or 22.5%. Considering that the average management fee for hedge funds is usually around 2%, the purpose of investing into hedge funds with return  Alpha surplus of 2% or higher is questionable. Below is the distribution of performance fee across strategies and regions (Table 7). Considering mandatory management fee of 2%, additional 17% of profits cut off takes another 0.64% off of 3.77% average return, therefore increasing the barrier for Alpha.
Table 7
Average management fees across strategies and regions
Fees 
E 
G 
P 
NA 
US 
Average 

CTA 
20% 
18% 
17% 
10% 
20% 
17% 

ED 
20% 
17% 
19% 
13% 
20% 
18% 

EH 
18% 
17% 
18% 
17% 
15% 
17% 

FID 
15% 
14% 
10% 
 
13% 
13% 

FIRV 
 
16% 
12% 
20% 
22% 
17% 

M 
19% 
16% 
17% 
20% 
23% 
19% 

MS 
15% 
13% 
18% 
18% 
17% 
16% 

NA 
19% 
17% 
19% 
15% 
18% 
18% 

Average 
18% 
16% 
16% 
16% 
18% 
17% 

Having outlined the necessity of further investigation, we then need to consider how the exposure to asset pricing models relates to profitability, and how this relationship changes over time.
We then turn to the analysis of hedge funds' returns' exposure, which we measure via Rsquared of the asset pricing models constructed (Table 8).
Table 8
Exposure of hedge funds' strategies to asset pricing models by region
Rsquared 
E 
G 
P 
NA 
US 
Rsq 
Alpha 1yr 

CTA 
1.34% 
1.26% 
2.66% 
 
 
1.75% 
5.22% 

ED 
7.48% 
29.26% 
12.32% 
27.20% 
19.07% 
1.92% 

EH 
15.91% 
10.47% 
25.69% 
16.79% 
26.41% 
19.06% 
0.02% 

FID 
22.13% 
8.62% 
18.13% 
 
0.48% 
12.34% 
2.83% 

FIRV 
 
4.02% 
6.33% 
 
2.95% 
4.43% 
2.38% 

M 
21.75% 
3.09% 
21.75% 
 
1.64% 
12.05% 
4.43% 

MS 
16.09% 
1.69% 
15.44% 
40.05% 
5.69% 
15.79% 
3.57% 

NA 
18.01% 
7.77% 
14.00% 
8.92% 
8.10% 
11.36% 
1.13% 

Rsq 
14.67% 
5.27% 
16.66% 
19.52% 
10.35% 
12.64% 
1.92% 

Alpha 1yr 
0.43% 
0.42% 
3.51% 
1.41% 
2.90% 
1.56% 
1.74% 

In the table are the overall Rsquared values of the regression models specified by 16 for each strategy and region. The values presented in the table are the average Rsquares across all model specifications. It is good for a hedge fund to have low Rsquared, since that implies it follows a strategy different from what asset pricing models suggest. We can see that the values differ quite significantly, from 0.48% for Fixed Income Directional strategy in the United States to 40% for North American Multistrategy hedge funds. North America tends to be the most exposed among all regions, while globallyoriented funds experience the lowest exposure.
We would call a hedge fund to be efficient if its Alpha is above average (1.74%) and Rsquared is below average (12.64%). In the table below are results of applying the condition to our fund groups (Table 9). It is 1, if the fund meets the criteria, and 0 otherwise.
Table 9
Outcomes after application of efficiency criteria for Alpha and Rsquared
E 
G 
P 
NA 
US 

CTA 
1 
0 
1 
0 
0 

ED 
0 
0 
0 
0 
0 

EH 
0 
0 
0 
0 
0 

FID 
0 
0 
0 
0 
1 

FIRV 
0 
1 
0 
0 
1 

M 
0 
1 
0 
0 
1 

MS 
0 
1 
0 
0 
1 

NA 
0 
0 
0 
1 
0 

From the table we can see that only a few strategies turned out to be not only profitable, but also not exposed to the asset pricing models specified in previous section. Fixed Income Relative Value, Macro, CTA and Multistrategy funds suited the criteria for more than one geographical focus, while funds with no available strategy information perform efficiently only in North America. Most efficient region turned out to be the United States, followed by globally oriented funds.
However, we do not only care about absolute Alpha  Rsquared values. Of a much interest to us is the relationship between these two factors. Since across our model specifications we include additional factors in the asset pricing models, the increase in Rsquared would imply exposure to the added factor (we will assess exposure to particular factors later). The correlation values can be observed below (Table 10).
Table 10
Correlation of funds' Alphas and Rsquared for different strategies and regions
AlphaRsq correlation 
E 
G 
P 
NA 
US 
Alpha 1yr 
Correlation average 

CTA 
25.35% 
91.38% 
71.56% 
 
 
5.22% 
6.89% 

ED 
1.42% 
28.69% 
83.90% 
91.37% 
68.09% 
1.92% 
70.41% 

EH 
21.41% 
74.82% 
29.99% 
69.27% 
75.18% 
0.02% 
10.35% 

FID 
35.93% 
4.16% 
53.84% 
 
74.05% 
2.83% 
52.47% 

FIRV 
 
88.79% 
26.35% 
 
70.19% 
2.38% 
75.51% 

M 
78.70% 
13.46% 
15.41% 
 
88.87% 
4.43% 
32.01% 

MS 
94.86% 
18.11% 
41.66% 
85.34% 
15.62% 
3.57% 
6.25% 

NA 
49.40% 
66.25% 
66.55% 
8.89% 
53.52% 
1.13% 
38.68% 

Alpha 1yr 
0.43% 
0.42% 
3.51% 
1.41% 
2.90% 
1.78% 
3.27% 

Correlation average 
37.76% 
32.56% 
26.34% 
9.58% 
64.16% 
15.14% 
9.20% 

If an increase in exposure is accompanied with an increase of Alpha, that would mean that the Alpha is derived from the return factor added, which is not suitable for efficient funds. In a case a Rsquared increase is followed by a decrease of Alpha, it is even worse since these funds do not even outperform. Ideally, we would expect zero correlation, however, negative correlation is also a positive case given that Alpha remains growing.
As to the strategies, the most profitable strategies in term of Alpha (in descending order) are CTA, Macro, Multistrategy and two Fixed Income strategies. Among those, Multistrategy and CTA funds tend to have the smallest absolute correlation value of Alpha and Rsquared, which means that on average hedge funds that follow these strategies tend to invest in an original, not described by commonly known models way.
On the other hand, hedge funds that have high positive Alphas and high positive (FID) correlations can not be named as efficient, since they follow typical investment strategies and trends. Overall, hedge funds tend to have a small positive Alpha of 1.74% and AlphaRsquared correlation of 9%. This finding can support the vision of hedge fund industry as semiefficient: while providing a positive Alpha, hedge funds tend to be innovative and deviate from typical investment factors. From the table we can also conclude that our Hypothesis 1 has been confirmed, since significant difference between Alphas and correlation values can be observed between funds of different regional focus.
We apply additional criteria to our efficiency condition, so that a fund would be efficient is its Alpha is above average (1.74%), Rsquared is below average (12.64%) and the Alpha  Rsquared correlation is below average (9.20%). In order to account for decreasing positive Alpha  increasing low Rsquared mistake, we also add the condition that fund group's Alpha has to be above the strategyregion average Alpha. The outcomes are presented in Table 11.
Table 11
Outcomes after application of correlation criteria for efficiency
E 
G 
P 
NA 
US 

CTA 
0 
0 
0 
0 
0 

ED 
0 
0 
0 
0 
0 

EH 
0 
0 
0 
0 
0 

FID 
0 
0 
0 
0 
0 

FIRV 
0 
0 
0 
0 
1 

M 
0 
0 
0 
0 
0 

MS 
0 
1 
0 
0 
1 

NA 
0 
0 
0 
0 
0 

By this condition, we can see that very few funds suit the outlined criteria. Multistrategy Global and U.S.oriented funds remained efficient, as well as FI Relative Value U.S. funds. CTA and Macro funds of some regional focuses that we outlined above as efficient, though, were eliminated. Although the correlation constraint may seem too severe, we have to account for strategies that do not follow it anyway.
4.2 Analysis of strategies and asset pricing models
Another thing we want to track with our research is the Alpha  Rsquared relationship between different strategies and model specifications. Following are the Alphas (Table 12), Rsquares (Table 13) and correlations between them (Table 15).
Table 12
Average annual Alpha for different strategies and models
Alpha 
MKTRF 
MKTRF SMB HML 
MKTRF SMB HML BAB 
MKTRF SMB HML MOM 
MKTRF SMB HML RMW CMA 
MKTRF SMB HML RMW CMA QMJ 
Alpha 1yr 

CTA 
4.99% 
5.26% 
5.48% 
4.54% 
5.84% 
5.19% 
5.22% 

ED 
1.65% 
2.08% 
3.14% 
2.80% 
0.72% 
1.11% 
1.92% 

EH 
0.35% 
0.59% 
0.34% 
0.70% 
0.20% 
1.63% 
0.02% 

FID 
2.76% 
2.47% 
2.34% 
2.90% 
3.25% 
3.25% 
2.83% 

FIRV 
2.06% 
2.04% 
1.62% 
2.58% 
3.33% 
2.66% 
2.38% 

M 
3.77% 
3.50% 
4.07% 
4.36% 
4.35% 
6.51% 
4.43% 

MS 
3.58% 
3.12% 
3.26% 
2.95% 
3.73% 
4.80% 
3.57% 

NA 
1.28% 
1.00% 
1.30% 
2.05% 
1.17% 
0.04% 
1.13% 

Alpha 1yr 
1.73% 
1.59% 
1.50% 
1.47% 
2.35% 
2.87% 
1.92% 

Table 13
Rsquared 
MKTRF 
MKTRF SMB HML 
MKTRF SMB HML BAB 
MKTRF SMB HML MOM 
MKTRF SMB HML RMW CMA 
MKTRF SMB HML RMW CMA QMJ 
Rsq average 
Alpha 1yr 

CTA 
0.53% 
1.40% 
1.51% 
1.78% 
2.52% 
2.78% 
1.75% 
5.22% 

ED 
14.41% 
16.70% 
21.14% 
17.18% 
17.87% 
18.58% 
17.65% 
1.92% 

EH 
17.48% 
18.85% 
19.23% 
18.94% 
19.60% 
20.23% 
19.06% 
0.02% 

FID 
11.40% 
13.15% 
13.49% 
8.45% 
13.77% 
13.80% 
12.34% 
2.83% 

FIRV 
3.01% 
4.21% 
4.26% 
4.55% 
5.13% 
5.44% 
4.43% 
2.38% 

M 
10.58% 
11.84% 
11.96% 
12.75% 
12.39% 
12.80% 
12.05% 
4.43% 

MS 
13.58% 
15.51% 
15.97% 
15.88% 
16.45% 
17.36% 
15.79% 
3.57% 

NA 
10.56% 
11.16% 
11.50% 
11.44% 
11.50% 
11.99% 
11.36% 
1.13% 

Rsq average 
10.19% 
11.60% 
12.38% 
11.37% 
12.40% 
12.87% 
11.80% 
x 

Alpha 1yr 
1.43% 
1.27% 
1.16% 
1.10% 
2.00% 
2.63% 
1.76% 
x 

Exposure of hedge funds' strategies to asset pricing models
We can observe that across different strategies Alphas and Rsquares change only slightly, with occasional spikes of 21% exposure of Event driven strategy to Betting against beta factor or low exposure of FI Directional strategy to momentum factor. Alphas also tend to be more or less the same.
We then apply the same conditions outlined above regarding funds exposure and performance. In order to outline efficient funds, we say that their Alphas should be above average (1.92%) and Rsquares should be below average (11.8%). The results can be observed below (Table 14).
Table 14
Outcomes after application of efficiency criteria for Alpha and Rsquared
MKTRF 
MKTRF SMB HML 
MKTRF SMB HML BAB 
MKTRF SMB HML MOM 
MKTRF SMB HML RMW CMA 
MKTRF SMB HML RMW CMA QMJ 

CTA 
1 
1 
1 
1 
1 
1 

ED 
0 
0 
0 
0 
0 
0 

EH 
0 
0 
0 
0 
0 
0 

FID 
1 
0 
0 
1 
0 
0 

FIRV 
1 
1 
0 
1 
1 
1 

M 
1 
0 
0 
0 
0 
0 

MS 
0 
0 
0 
0 
0 
0 

NA 
0 
0 
0 
0 
0 
0 

In the betweenmodels dimension, we see that CTA still can be named as efficient. We also observe that FI Relative Value strategy hedge funds show good results in terms of efficiency. However, we still need to apply correlation condition, so below are the correlation values between Alphas and Rsquared across strategies (Table 15).
Table 15
Correlation of funds' Alphas and Rsquared for different strategies and models
Correl 
MKTRF 
MKTRF SMB HML 
MKTRF SMB HML BAB 
MKTRF SMB HML MOM 
MKTRF SMB HML RMW CMA 
MKTRF SMB HML RMW CMA QMJ 
Average 

CTA 
60.37% 
99.97% 
99.66% 
8.06% 
95.55% 
99.90% 
55.12% 

ED 
94.57% 
85.50% 
60.54% 
79.01% 
88.04% 
75.24% 
70.41% 

EH 
3.73% 
17.10% 
21.60% 
8.38% 
8.87% 
0.64% 
10.35% 

FID 
81.97% 
55.44% 
48.19% 
47.31% 
47.23% 
56.40% 
52.47% 

FIRV 
99.98% 
94.91% 
92.13% 
95.97% 
91.07% 
95.73% 
75.51% 

M 
51.39% 
58.96% 
48.26% 
23.86% 
48.99% 
5.28% 
32.01% 

MS 
25.86% 
12.99% 
10.90% 
7.33% 
22.19% 
30.81% 
6.25% 

NA 
43.18% 
48.83% 
36.22% 
42.49% 
49.41% 
21.69% 
38.68% 

Average 
9.08% 
10.54% 
11.14% 
8.27% 
13.60% 
5.87% 
1.65% 

What is interesting to us is the relationships between Alphas and Rsquares. For instance, it can be seen that the dependence is more or less the same across different models, which is logical since the nature of trading strategy stays the same. However, in some cases we can see a significant drop in exposure with addition of some factors. One of such findings relates to the hedge funds of CTA strategy: we can observe a significant rise in correlation value with addition of size and value factors compared to the market factor in CAPM. With such findings, we can state that these hedge funds actively employ investment principles outlined by Fama, French and their threefactor model. We also can say that these hedge funds have no exposure to momentum investing, as the correlation in Carhart model is close to zero. Event driven hedge funds tend to show more exposure solely to the market premium, as well as almost as high exposureAlpha relationship to FF fivefactor model, which tends to capture more fundamental companies' characteristics. That finding is sensible considering the nature of ED funds strategy. Large exposure to the market risk was shown by Fixed Income funds. However, while Directional funds move according to market trends, Relative Value funds show negative correlation, which is interesting since these strategies show almost identical Alpha. Not much can be said about other types of funds apart from the fact that Macro funds are certainly not exposed to the Quality concept, which is inverse for Multistrategy funds that show upward exposure to this factor. Although we managed to capture some relationship between hedge funds' exposure, performance and factors used in outlined models, we cannot approve our Hypothesis 3 applying it to the whole data sample since the Rsquares are low.
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