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## Abstract

This article analyzes the benefits of adding hedge fund replication products (clones) to an existing portfolio in comparison to adding actual hedge funds as represented by hedge fund indexes. The authors employ the marginal Sharpe methodology to evaluate the benefit of adding an investment to an existing long-only portfolio. The marginal Sharpe is decomposed into a return component and a diversification component, and the authors conduct separate tests on the two components and compare clones with hedge fund indexes. Hedge fund clones, which are liquid trading strategies, seem to be able to replicate the benefit stemming from the diversification component. With respect to the return component, the benefits of adding a hedge fund index to an existing portfolio are significantly higher than those obtained by the clones. However, these results are mitigated after accounting for fees and hedge fund premiums.

Since the beginning of the 21st century, various hedge fund replication products (clones) have been developed and sold to investors.^{1} The purpose of these instruments is to provide quasi-hedge fund exposure while limiting trading to a few liquid assets. Some developers employ statistical models to identify a hedge fund index’s exposure to various market risk factors and create a portfolio with similar exposures. Others claim that they are able to systematically mimic the trading patterns of real hedge funds. The aim of these clones is to offer investors hedge fund exposure without many of the downside risks associated with true hedge fund allocation: lack of liquidity, long lock-ups, lack of transparency, operational risks, fraud risk, reputational risks, and more.^{2}

This article focuses on analyzing the relative benefits of adding clones to an existing portfolio and adding actual hedge funds (represented by hedge fund indexes).^{3} We distinguish between *trackers*, which are investment formulas or algorithms, and *clones*, which are actual investments that implement the trackers’ formulas (i.e., they trade [long or short] the different assets included in the trackers and incur expenses such as bid/offer spreads, market impact costs, and fees). We argue that clones should not be assessed by their ability to approximate the target returns of hedge funds (represented by hedge fund indexes) but rather they should be evaluated according to the overall benefits they provide to investors.

We compare the benefits of clones to the benefits of actual hedge fund exposure in order to evaluate the success of the replication process and to assess the value of clones as an investment alternative. Most previous studies that examined hedge fund cloning focused on two questions: First, is replication possible; and second, what is the best method to carry out the replication process? Some of these studies suggest algorithms for replication and implement them *on paper*, thus proving that replication is feasible (see, e.g., Hasanhodzic and Lo [2007]).^{4} This article is similar to that of Wallerstein, Tuchschmid, and Zaker [2010]: We do not attempt to replicate hedge funds but rather evaluate the merits of the final replications.

We employ a model to evaluate the marginal benefit of adding an investment to an existing portfolio. This benefit is measured in terms of the marginal contribution of the investment to the Sharpe ratio of an existing portfolio, which is called the *marginal Sharpe*. We break down the marginal Sharpe into two testable components, the *return component* and the *diversification component*, which enables us to conduct conventional statistical tests on each component separately and to compare clones with hedge funds. The advantage of using this tool is that it provides a theoretical justification for the statistical tests employed in this article.

The early literature on asset valuation measured the beta component of asset returns as exposure to equities (Sharpe [1964], Lintner [1965]). Any statistically significant residual not explained by the linear model was called *alpha*. Another set of articles dealing with asset performance attribution expanded the number of factors that explain the returns of traditional assets (King [1966], Meyers [1973]), thus generalizing the definition of alpha. More recently, a series of articles, pioneered by Fung and Hsieh [1999, 2001], Mitchell and Pulvino [2001], Agarwal and Naik [2004], and others, has recognized that, while there may be some alpha inherent in hedge fund portfolios as a result of managers’ skills or active management of positions and exposures, returns can also be attributed to other risk factors that are not easily accessible via a more traditional long-only bond and equity portfolio and can therefore be misinterpreted as alpha.^{5} These authors have captured a variety of risk factors that explain the returns of a typical hedge fund portfolio. The factors vary from study to study, but the main ones that have been identified are equity, credit spreads, volatility, commodities, and currencies. Attempts have also been made to capture some of the more difficult factors, such as the illiquidity premium or nonlinearity of returns (Getmansky, Lo, and Makarov [2004]). These factors are sometimes referred to as hedge fund betas.

**HEDGE FUND CLONES AND HEDGE FUND INDEXES**

Hedge fund clones seek to replicate hedge fund returns by using exchange-traded instruments and over-the-counter liquid instruments, thus avoiding the complexities associated with investing in actual hedge funds. These clones are offered to institutional investors as well as to retail investors (in the form of mutual funds or exchange-traded funds).

The main justification for investing in a clone portfolio stems from the argument that in a well-diversified portfolio of hedge funds, diversifiable risk is expected to wash out and returns should be determined by the exposure to nondiversifiable risk factors and alpha. If one can identify and invest in those factors, then the factor portfolio should serve as a good proxy for a real hedge fund portfolio. There are also additional merits to buying a clone portfolio: fee savings, liquidity,^{6} simplicity of positions, full transparency of positions, and the minimization of reputational risk.^{7}

Within the universe of hedge fund clones, the aim of the product can range from replicating the returns of a composite hedge fund index to replicating a particular hedge fund strategy or trading style (e.g., a discretionary macro or long-short equity strategy) to creating vehicles that mimic the risk behavior of market-neutral hedge funds (e.g., clones that are marketed as absolute return funds). This article only considers clones from the first group, as our analysis focuses on the marginal contribution of hedge fund clones from the perspective of a single, consolidated asset class rather than that of a particular hedge fund strategy.

In the realm of hedge fund clones that replicate the returns of a hedge fund index, the actual replication approach can vary. One method of replication involves running regressions on the returns of the index against various investable market factors (equity indexes, interest rates, credit indexes, foreign exchange, various options, etc.) to create a liquid trading strategy based on the index’s exposure to these factors. This method can be categorized as *factor-based replication*. There is another approach, which also employs the factor-based method but makes additional use of *bottom up* techniques. These bottom up techniques seek to identify the trading strategies and market exposures of each hedge fund substrategy in the index (e.g., long-short equity, macro) by studying regulatory filings, industry reports, hedge fund databases, and other data. Such techniques also involve creating specific trading strategies that aim to explain strategy risks that regressions may not pick up. For instance, in the case of a merger arbitrage strategy, a trading strategy involving going long the target company and short the acquirer is thought to be more suitable for capturing the more idiosyncratic risk and return characteristics of the strategy. Once the various market exposures and trading strategies have been identified, quantitative models are used to generate weights for each in a manner that best represents the risk and return characteristics of the substrategy. In some cases, a qualitative overlay can also be applied to the process. Subsequently, appropriate weights are designated to each substrategy in order to replicate the overall hedge fund index.

The hedge fund indexes employed in this study are not easily investable as they typically aggregate thousands of funds (across hedge fund strategies), many of which could be closed to new investors or demand investment conditions that are not suitable for every investor. This means that if an investor perceives that a hedge fund index fits his or her portfolio better than the alternative clone, it could be difficult for him or her to allocate money to that index specifically.

Nonetheless, an important assumption in this study is that each of the hedge fund indexes in the sample represents a typical hedge fund portfolio. The logic behind this assumption is that a well-diversified hedge fund portfolio would converge to an index portfolio. In other words, although an investor would not buy the exact index, a typical hedge fund portfolio, which consists of numerous hedge funds of different strategies, is expected to have return characteristics similar to those of the indexes. The validity of this assumption is strengthened by the high correlations between the six hedge fund indexes in the sample (Exhibit 1).

**DATA**

The database for this study includes monthly returns of six hedge fund indexes and five hedge fund index clones. Data were collected from Bloomberg.^{8}

The study analyzes a series of index and clone returns. We conducted the statistical tests on data starting from April 2010 through March 2015.

Exhibit 2 presents summary statistics for all indexes and clones for the complete sample period, April 2010-March 2015. Exhibit 3 presents a scatter diagram of the returns and standard deviations of the various indexes and clones, and Exhibits 1 and 4 present the correlations among the hedge fund indexes and clones. Exhibit 5 summarizes the characteristics of each of the individual clones selected for this study.

The correlations between the various hedge fund indexes (Exhibit 1) range between 0.92 and 0.99, indicating that the various indexes are representative of the same asset class, in spite of variations in returns and volatilities. Out of the five hedge fund index clones, three utilize the factor replication method described in the previous section, while the remaining two combine factor analysis with the bottom up approach outlined in the same section. Each clone seeks to replicate a specific composite hedge fund index, except for the Goldman Sachs clone that aims to “approximate the return and risk patterns of a diversified universe of hedge funds” (Goldman Sachs [2016]). This in itself is not problematic as the hedge fund indexes used in this study represent a large and diversified set of hedge funds. Furthermore, the correlation statistics in Exhibit 5 indicate that the Goldman Sachs clone exhibits correlations to hedge fund indexes that are similar to those of the other clones.

As can be seen from Exhibit 4, correlations among clones are marginally lower than those of the indexes, ranging from 0.88 to 0.93. Exhibit 3 reveals that there is a higher variability in the returns of the clones but not in their volatility.

**METHODOLOGY**

We will review the merits of adding asset *i* (a hedge fund portfolio) to a long-only portfolio, denoted *m,* where the criterion for inclusion is the hedge fund portfolio’s marginal contribution to the Sharpe ratio of the total portfolio, denoted as marginal Sharpe or *MS*
^{9}:

where µ_{i} and µ_{m} are the expected return of the hedge fund portfolio *i* and the long-only portfolio *m*, s_{m} is the standard deviation of portfolio *m*, *S _{m}
* is the Sharpe ratio of portfolio

*m*, and ß

_{i}is the beta of portfolio

*i*with respect to portfolio m (see the Appendix for the derivation of all equations).

The first term on the right-hand side of Equation (1) is the return component, and the second term is the diversification component. Both terms are additive and are equal to the total marginal Sharpe.

To test the benefits of clones versus hedge funds, we will employ Equation (2) (also found in the Appendix as Equation (A8)):

2where *MS _{i}
* -

*MS*denotes the difference between the marginal Sharpe of two portfolios,

_{j}*i*and

*j*, and ß

_{j}, ß

_{i}, are the betas of portfolio

*j*and portfolio

*i*, respectively, against the long-only portfolio

*m*. The two terms on the right-hand side of Equation (2) are the difference in the return component and the difference in the diversification component of asset

*i*versus asset

*j*.

The advantage of employing Equation (2) is that it enables separate statistical testing of the two components that contribute to the overall Sharpe ratio of the portfolio. The first test examines different measures of the benefit to the investor in a cross-sectional analysis.^{10} The null hypothesis is that the various clone products are able to attain similar rates of return, risk adjusted returns, and marginal Sharpe as real hedge fund indexes. The second group of tests uses time series analysis. For these tests, we employ Equation (2). We create two equally weighted portfolios, one consisting of all the hedge fund indexes in the sample (equally weighted portfolio of indexes [EWI]) and another consisting of all the clones in the sample (equally weighted portfolio of clones [EWC]). The test is conducted separately for the difference in the return component and in the diversification component between clones and indexes. The difference in the returns component is examined by a *t*-test on the difference in expected returns µ_{i} - µ_{j}; the difference in the diversification component is examined by a *t*-test on ß_{i} - ß_{j}.

To test possible test biases due to autocorrelation, we will run the following two regressions:

3 4Tests on a in Equation (3) and on µ_{i} - µ_{j} in Equation (4) are identical to tests on the return component and on the diversification component appearing in Equation (2) and will enable the use of standard statistical methods to test and correct autocorrelation phenomena in the returns data.

Finally, Equation (2) is affected by leverage: Leverage increases both the expected return and the beta of a portfolio (assuming they are both positive). If necessary, we will correct for this possible influence on the statistical results by equating the standard deviation of returns of the clones and the hedge fund indexes (Equations (A9-A11)).

**RESULTS: ANALYSIS OF THE CONTRIBUTION OF CLONES**

The results for the cross-sectional tests comparing the different measures of performance of both indexes and clones are presented in Exhibit 6. Mean returns, Sharpe ratio, and marginal Sharpe to a 50/50 stock-bond portfolio are reported for each of the six hedge fund indexes and five clones described in the previous section, as well as for the equally weighted portfolios of indexes and clones.

Panel A, which covers the period from April 2010 to March 2015, shows that, although highly correlated, there are noticeable differences in performance, even between the widely available hedge fund indexes. Panel B compares the performance of the five clones. Panel C reports *p*-value test statistics for the null hypothesis to determine whether the returns of indexes and the returns of clones follow the same distribution. The hypothesis is tested using both the nonparametric Mann-Whitney-Wilcoxon test^{11} (MWW) and the parametric *t*-test. The *p*-value statistics of these tests are presented for each of the three performance measures, including marginal Sharpe.

The statistical results indicate that the performance of hedge fund indexes was superior to that of the clones and that the differences are statistically significant. The marginal Sharpe is higher on average for the indexes in comparison to the clones (-0.052 versus -0.183), and the dominance of indexes over clones is significant (*p*-value = 0.0022 in the case of the nonparametric MWW test). Overall, these results indicate that the performance of clones is inferior to that of hedge fund indexes. We will discuss the implications of this result in another section.

Exhibit 7 focuses on the marginal Sharpe of adding clones or indexes to a traditional stock and bond portfolio. The marginal Sharpe is broken up into its two components as presented in Equation (1). While the return component of indexes is significantly higher than that of the clones (*p*-value = 0.0022 for the nonparametric test), no significant difference in the diversification components of the two groups was found (*p*-value = 0.3961 for the nonparametric test).

The two right-most columns of Exhibit 7 present the marginal Sharpe for alternative reference portfolios. We examine two extreme cases: a 90%-stock portfolio and a 90%-bond portfolio. The tests suggest that previous results using the total marginal Sharpe measure are robust to changes in the reference portfolio and thus may be valid for various types of portfolios whose composition of bonds and equities varies.

Exhibit 8 presents the results of the second type of tests: time series *t*-tests on the return component and on the diversification component as portrayed in Equation (2). The *t-*tests are based on time series returns of the equally weighted portfolio of indexes versus the equally weighted portfolio of clones. By decomposing the marginal Sharpe into its two components, we are able to analyze the sources of contribution of the clones versus the indexes.

The results indicate that the return component of the equally weighted portfolio of indexes is significantly higher than the return component of the equally weighted portfolio of clones (*p*-value = 0.0047 for the one-sided *t*-test). However, as presented in the second part of Exhibit 7, we cannot reject the null hypothesis that the indexes’ diversification component is greater than that of the clones (*p*-value = 0.2574 for the one-sided *t*-test). These results are consistent with results of the nonparametric MWW test presented in Exhibit 7. In the sample period tested, clones were able to provide a diversification contribution to the marginal Sharpe similar to that of hedge fund indexes but were inferior in their contribution to the return.

Hedge fund returns suffer from autocorrelations, which can affect the results of the statistical tests. Getmansky, Lo, and Makarov [2004] claimed that the holding of illiquid securities by hedge funds will create a positive serial correlation and a downward bias of the ordinary least squares standard error estimates. However, the Durbin-Watson statistic presented in Exhibit 8 for the return component is not statistically significant at the 5% level and therefore does not necessitate a correction. There is no need to correct for autocorrelations in the case of the diversification component because the null hypothesis was not rejected (any correction would increase the standard error of the estimator and would strengthen the statistical result).

The portfolio of clones was originally run at a slightly lower standard deviation than that of the portfolio of indexes, which can affect the results of the statistical tests. To overcome this concern, we repeat the results of Exhibit 8 but this time scale up the standard deviation of the portfolio of clones to that of the portfolio of indexes. The scaling up is achieved through leverage, as defined in Equations (A9-A11).^{12} We end up with a comparison of two series that have an equal standard deviation. Exhibit 9 presents the results of this test conducted on the scaled portfolio of clones.

Exhibit 9 reveals that adding leverage to the portfolio of clones does not change the results regarding the return component: The return component of the portfolio of indexes remains significantly higher than the return component of the portfolio of clones (with a *p*-value of 0.0060). As in Exhibit 8, the Durbin-Watson statistic for this component is not statistically significant at the 5% level.

As for the diversification component, we still cannot reject the null hypothesis that the portfolio of indexes’ diversification component is equal to that of the portfolio of clones (although the *p*-value for the test is lower, 0.1552 for the one-sided *t*-test). As stated in the discussion of Exhibit 8, there is no need to correct for autocorrelations in this case, as the null hypothesis cannot be rejected and increasing the standard error estimates will intensify this result.

**CONCLUDING REMARKS**

This article tests the relative benefit of adding hedge fund clones to a typical long-only portfolio in comparison to adding actual hedge funds (represented by hedge fund indexes). For this purpose, we utilized the marginal Sharpe methodology. We deconstructed the marginal Sharpe into two testable components: a return component and a diversification component, which are two measurable benefits that a new investment can bring to an existing portfolio. The methodology allows for the comparison of clones and indexes by testing the two components separately.

The conclusion from both cross-sectional tests and time-series analysis is that the benefits of adding clones to a traditional long-only portfolio are significantly lower than those of adding an actual hedge fund portfolio as represented by an index of hedge funds. While clones were able to replicate the benefit stemming from the diversification component, they were not able to match the return component benefit derived from indexes. These results are sustained even after correcting for the slight standard deviation difference in the sample between clones and indexes.

There are several possible explanations for these results, which are not necessarily mutually exclusive. The first explanation is that in a large and well-diversified hedge fund index, which is a representation of an extremely diversified hedge fund portfolio, the idiosyncratic risks are washed out through diversification and what remains are hedge fund betas, market betas, and alpha. Apparently, these betas can be captured through the trading algorithms of the various clones, but aggregate alpha is more difficult to capture. This can explain why the return component of the clones is significantly lower than that of the indexes, while the diversification component is not significantly different. The second explanation is related to fees. Investment in a diversified hedge fund portfolio requires either investment through a fund of funds or paying direct costs to cover the due diligence process and ongoing maintenance of the investment. These costs vary and in the case of fund of funds can be quite substantial^{13} and much higher than the costs of investing in a clone portfolio. The third explanation is related to the additional premium that investors require from direct hedge fund investments due to illiquidity, lack of transparency, and other shortcomings of hedge funds, which are not reflected in a simple mean-variance model and are translated into a premium of hedge funds over a direct investment in a clone. As a result, one cannot necessarily conclude that clones are inferior to hedge funds as the returns shortfall in comparison to a real hedge fund portfolio could very well reflect the additional costs, risk, and illiquidity premiums required by investors investing in hedge funds. Also, as reflected in Exhibits 2 and 3, there are differences between the various hedge fund clones of which investors should be aware, as some clones have been able to better deliver replication properties than others.

**APPENDIX**

**Derivation of Testing Models**

Assume that the investor is holding portfolio *m* (long only) whose return (a random variable) is denoted by
and is adding to the portfolio asset *i* (hedge fund portfolio) whose return is denoted by
. The addition is carried out as a zero investment strategy (see Sharpe [1964]): Proportion *x* will be invested in asset *i* and proportion (1 - *x*) will be invested in portfolio *m*.

The return on the combined portfolio p, is given by

A1The expected return of the combined portfolio *p*, µ_{p} is

where µ_{i} and µ_{m} are the expected returns of asset *i* and portfolio *m*, respectively.

The standard deviation of the return on the combined portfolio p, s_{p} is

where s_{i} and s_{m} are the standard deviations of the returns of asset *i* and portfolio *m*, respectively, and µ_{im} is the covariance of the returns of asset *i* and portfolio *m*.

We denote the Sharpe ratio of portfolio *p* by *S _{p}
*, and it is given by

where *R _{f}
* is the risk free rate.

Taking the first derivative of the right-hand side of Equation (A4) with respect to *x* will give

Evaluating Equation (A5) at *x* = 0 and substituting *S _{m}
* = (µ

_{m}-

*R*)/s

_{f}_{m}, we get the following equation for the marginal contribution of asset

*i*to the Sharpe ratio of portfolio

*m*, which is the marginal Sharpe of asset

*i*with respect to portfolio

*m*

^{14}:

Please note that *S _{m}
* is the Sharpe ratio of the existing (long-only) portfolio

*m*and ß

_{i}is the beta of asset

*i*(the hedge fund portfolio) with respect to portfolio

*m*(the long-only portfolio).

As can be seen from Equation (A6), the marginal Sharpe (MS) of asset *i* with respect to portfolio *m* can be divided into two additive components.

The first component,
, is the (risk-adjusted) return component and will be positive as long as the return of asset *i* (hedge funds) is expected to be greater than the return on portfolio *m* (long only). The second component, *S _{m}
*(1 - ß

_{i}), is the diversification component, and it will be positive as long as the beta of asset

*i*with respect to portfolio

*m*(the existing portfolio) is less than 1.

The implication of Equation (A6) is that, for any (sufficiently small) addition of size ?*x* of asset *i* to portfolio *m*, the Sharpe ratio of the portfolio will change by^{15}

Equation (A7) is valid as an approximation for a relatively small ?*x*. For a more accurate evaluation of the marginal Sharpe, one would need to take the second term of a Taylor approximation.

The optimal allocation between the benchmark portfolio *m* and the additional portfolio *i* is obtained from the well-known mean-variance (M-V) optimization formula. This allocation has changed over time, especially in the aftermath of the 2008 financial crisis. Normally, investors would underallocate to hedge funds, well below what the simple M-V model would indicate, for a variety of reasons, including risks not incorporated in the model and the relative illiquidity of hedge funds. The optimal M-V allocation to portfolio *i* (hedge funds) is obtained by equating Equation (A5) to zero.

From Equation (A6) we can also compute the difference between the marginal Sharpe of any two assets, *i* and *j*:

Equation (A8) presents a breakdown of the difference between the marginal Sharpe of two assets into two components: the returns component and the diversification component. This breakdown will enable us to conduct statistical tests on the relative efficiency of asset *i* in comparison to asset *j* and to assess the sources of the relative efficiency of the two assets.

Equation (A8) is affected by the relative leverage of each of the assets. To clean up this effect, we will equate the volatility of the two assets *i* and *j* by creating the following leveraged (or deleveraged) version of asset *j*:

where s_{i} and s_{j} are the standard deviations of assets *i* and asset *j*, respectively; *R _{j}
* is the rate of return of asset

*j*(a random variable); and

*R*is the risk free rate.

_{f}From Equation (A9) we derive the expected return (
) and beta (
) of asset *i* vis-à-vis the reference portfolio:

Finally, empirical evidence suggests that hedge fund returns are serially autocorrelated (Getmansky, Lo, and Makarov [2004]). To test possible test biases due to this property, we will run the following two regressions:

A12 A13A test on a in Equation (A12) is a test on the return component (first part of the right-hand side of Equation (A8)), and a test on ß_{i} - ß_{j} in Equation (A13) is a test on the diversification component (second part of the right-hand side of Equation (A8)). We will employ standard autocorrelation tests to determine the existence of autocorrelation and make the necessary corrections if needed.

## ENDNOTES

The authors would like to thank the anonymous referee of this journal for insightful and helpful comments. The views expressed are those of the authors alone and do not reflect the position of the World Bank.

↵

^{1}Hedge funds as an asset class underperformed during the 2008 financial crisis and in its aftermath. Nevertheless, hedge funds continue to attract large pools of capital from institutional investors, and the analysis of this asset class and its replicators continues to be very important (Summers [2015]).↵

^{2}The downside of hedge fund investing is more pronounced during times of stress, such as the credit crisis of 2007-2008, a time when many hedge funds faced large redemptions and margin calls and were forced to impose special restrictions such as gating and suspension.↵

^{3}The benefits of hedge funds and the role they play in an institutional investor’s portfolio are well known (Lamm [1999], Agarwal and Naik [2000]). However, unlike some of the more traditional asset classes, quantifying the benefits of hedge funds is difficult for a variety of reasons, such as biases in the measurement of index returns, exposure to nonlinear risk factors, short history of returns, and nonstability of exposures to risk factors.↵

^{4}Wallerstein, Tuchschmid, and Zaker [2010] reviewed the different approaches to hedge fund replication.↵

^{5}The authors recognize that the definition of alpha is problematic, as it depends upon the specifications of the model and on the statistical significance of the factors employed within the sample period. In addition, capturing nonlinear relationships is difficult and can result in misinterpreting complex factor sensitivity as alpha.↵

^{6}An investment in a hedge fund usually requires an initial lockup as well as a fairly long redemption notification period. In addition, at times of crisis the hedge fund manager has the option to suspend or limit redemptions.↵

^{7}*Reputational risk*is defined as the risk stemming from being invested in a hedge fund that has either suffered heavy losses or has received negative media coverage relating to allegations of fraud or violation of security laws. This risk may be independent of the size of the actual exposure to the fund or to the size of the losses suffered.↵

^{8}Indexes: Barclays-Barclays Hedge Fund Hedge Fund (BGHSHEDG), CISDM-CASAM/CISDM Equal Weighted Hedge Fund Index (CISDMEW), DJCS-Dow Jones Credit Suisse Hedge Fund Index (HEDGNAV), Eurekahedge-Eurekahedge Hedge Fund Index (EHFI251), Greenwich-Greenwich Global Hedge Fund Index (GVAINDX), and HFRI-HFRI Fund Weighted Composite Index (HFRIFWI).Products: Credit Suisse-Credit Suisse Liquid Alternative Beta Fund Luxembourg USD (LABINDX), Goldman-Goldman Sachs Absolute Return Tracker Fund Class A USD (GARTX), Lombard Odier-LO Funds Alternative Beta USD (LOALBIA), Merrill-Merrill Lynch Factor Model Fund Luxembourg USD (MLFM02 March 2010-April 2014 and MLFM01 May 2014-March 2015), and NN-NN Alternative Beta Luxembourg USD (INALTUI).

↵

^{9}The analysis can be extended to other measures of risk and return (e.g., measures that take into account more extreme tail events and are not captured by the more conventional Sharpe ratio).↵

^{10}In a cross-sectional analysis, each hedge fund index or clone is a single observation.↵

^{11}The rationale for employing the MWW test stems from the assumption that clones are constructed to stochastically replicate a hedge fund index. The MWW methodology tests the null hypothesis that clones mimic the distribution of hedge funds. If this is the case, then the ranking of the various tested parameters (performance, marginal Sharpe, etc.) between the two groups is expected to be random. If the null hypothesis cannot be rejected, we would conclude that clones indeed deliver what they are expected to offer: synthetic hedge fund exposure. We also ran a parametric*t*-test for robustness.↵

^{12}The scaling is conducted ex post (i.e., with full knowledge of the observed standard deviations).↵

^{13}Brown, Goetzmann, and Liang [2004] estimated the median management fee and incentive fee of a fund of funds in their sample to be 1.5% and 10%, respectively. There is anecdotal evidence that these costs decreased substantially following the 2008 financial crisis.↵

^{14}Equation (A6) can also be presented as follows:where,

*r*is the correlation coefficient between and ._{i,m}The advantage of the above equation is that the term in brackets is independent of leverage. We will conduct statistical tests on Equation (1), which portrays the true properties of the clones and indexes, and also scale the returns to clean up the effect of leverage.

↵

^{15}Equation (A7) is the first order Taylor approximation of the change in the portfolio’s Sharpe ratio due to an increase of magnitude ?*x*in the investment in asset*i*. This approximation will hold reasonably well for ?*x*< 0.2. Investors will normally not invest in hedge funds up to the optimal point (where the marginal Sharpe is zero) due to other risks not embedded in the simple mean-variance model: illiquidity, nontransparency of positions, etc.

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