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## Abstract

Sophisticated institutional investors have a growing interest in factor investing, a disciplined approach to portfolio management that is broadly meant to allow investors to harvest risk premia across and within asset classes through liquid and cost-efficient systematic strategies without having to invest with active managers. Although it is now well accepted that the average long-term performance of active mutual fund managers can to a large extent be replicated through a static exposure to traditional long-only risk premia, an outstanding question remains with respect to the best possible approach for harvesting alternative long–short risk premia.

The focus of this article is to empirically analyze (1) whether systematic rules-based strategies based on investable versions of traditional and alternative factors allow for satisfactory in-sample and out-of-sample replication of hedge fund performance, and more generally (2) whether suitably designed risk allocation strategies may provide a cost-efficient way for investors to get attractive exposure to alternative factors, regardless of whether they can be regarded as proxies for any particular hedge fund strategy. The main findings can be summarized as follows. On one hand, the authors find that replication models for hedge funds strategies generally achieve a relatively low out-of-sample explanatory power, regardless of the set of factors and the methodologies used. On the other hand, they find that heuristic allocation strategies, such as risk parity strategies, applied to alternative risk factors could be a better alternative to hedge fund replication for efficiently harvesting alternative risk premia.

Academic research (see Ang [2014] for a synthetic overview) has highlighted that risk and allocation decisions could be best expressed in terms of rewarded risk factors as opposed to standard asset class decompositions, which can be somewhat arbitrary. For example, convertible bond returns are subject to equity risk, volatility risk, interest rate risk, and credit risk. As a consequence, analyzing the optimal allocation to such hybrid securities as part of a broad bond portfolio is not likely to lead to particularly useful insights. Conversely, a seemingly well-diversified allocation to many asset classes that essentially load on the same risk factor (e.g., equity risk) can eventually generate a portfolio with very concentrated risk exposure. More generally, given that security and asset class returns can be explained by their exposure to pervasive systematic risk factors, looking through the asset class decomposition level to focus on the underlying factor decomposition level appears to be a perfectly legitimate approach that is also supported by standard asset pricing models that rely on equilibrium arguments (the intertemporal capital asset pricing model [CAPM] from Merton [1973]) or arbitrage arguments (the arbitrage pricing theory from Ross [1976]).

In a recent paper, Martellini and Milhau [2015] provided further justification for the factor investing paradigm by formally showing that the most meaningful way of grouping individual securities is to form replicating portfolios for asset pricing factors that can collectively be regarded as linear proxies for the unobservable stochastic discount factor, as opposed to forming arbitrary asset class indexes. Building on this insight and a number of associated formal statistical tests, the authors provided a detailed empirical analysis of the relative efficiency of various ways of implementing the factor investing paradigm and analyzed the robustness of these findings with respect to a number of implementation choices, including the use of long-only versus long–short factor indexes, the use of cap-weighted versus optimized factor indexes, and the use of multiasset factor indexes versus asset class factor indexes.

From a practical perspective, two main benefits can be expected from shifting to a representation expressed in terms of risk factors as opposed to asset classes. On one hand, allocating to risk factors may provide cheaper, as well as more liquid and transparent, access to underlying sources of returns in markets in which the value added by existing active investment vehicles has been put in question. For example, Ang, Goetzmann, and Schaefer [2009] argued in favor of replicating mutual fund returns with suitably designed portfolios of factor exposures such as the value, small cap, and momentum factors. In the same vein, Hasanhodzic and Lo [2007] argued in favor of the passive replication of hedge fund vehicles, although Amenc et al. [2008, 2010] found that the ability of linear factor models to replicate hedge fund performance is modest at best. On the other hand, allocating to risk factors should provide a better risk management mechanism, in that it allows investors to achieve an ex ante control of the factor exposure of their portfolios, as opposed to merely relying on ex post measures of such exposures.

Given the increasing interest in risk premium harvesting and the desire to enhance the diversification of their portfolios, large and sophisticated investors are turning their attention to so-called *alternative risk premia*, loosely defined as risk premia that can be earned above and beyond the reward obtained from standard long-only stock and bond exposure. These alternative risk factors are empirically documented sources of return that can be systematically harvested, typically through dynamic long–short strategies, which have been found to have explanatory power for some hedge fund strategies (see, for example, Fung and Hsieh [1997a, 1997b, 2002, 2004, 2007] and Agarwal and Naik [2004, 2005]).

More precisely, this article aims to determine the best possible approach for harvesting alternative risk premia. To answer this question, we empirically analyze whether systematic rule-based strategies based on investable versions of alternative (and traditional) factors allow for satisfactory in-sample and out-of-sample replication of hedge fund performance, or whether it is instead the case that properly harvesting alternative risk premia, which are more complex to extract and trade compared to traditional risk premia, requires the skills of active managers.

As such, our project is related to the stream of research on hedge fund replication (see Hasanhodzic and Lo [2007] and Amenc et al. [2008, 2010], among many others), which we extend in the following two main directions. In the first step, in contrast to some of the previous research that has analyzed the replication of global hedge fund indexes, which are often dominated by long–short equity strategies that are arguably the easiest to replicate, we will focus on replicating hedge fund strategy indexes (see Asness et al. [2015] for a recent reference). It is in fact one of the goals of this article to identify which strategies are easiest or hardest to replicate using alternative risk premia and possibly conditional models that may capture changes in hedge fund exposures by exploiting information from relatively high-frequency conditioning variables (see Kazemi, Tu, and Li [2008] for an analysis of the conditional properties of hedge fund return distributions). Finally, we consider the possible improvement allowed by the introduction of a specific set of factors for each strategy, as opposed to using a single set of systematic factors for all funds. Given the concern over data mining that would arise from a statistical search for the best factors, we have constrained ourselves to a purely economic selection of factors. In the second step, we shift perspective from hedge fund replication to hedge fund substitution and investigate whether suitably designed risk allocation strategies may provide a cost-efficient way for investors to get an attractive exposure to alternative factors, regardless of whether they can be regarded as proxies for any particular hedge fund strategy.

## TAXONOMY OF ALTERNATIVE RISK PREMIA

There is no well-accepted definition of alternative risk premia. They are in fact best defined in contrast to the so-called *traditional risk premia*, which essentially relate to the long-term rewards earned from a long-only exposure to stocks and bonds. In other words, any factor premium, or documented anomaly, that differs from the long-only equity and bond risk premia can be regarded as an alternative factor premium. If the definition of alternative risk factors involves no a priori restrictions, we only consider in this article alternative factors that have been documented to exhibit significant and persistent premia justified by academic research and economic intuition. Furthermore, we focus on those risk factors that can be harvested with relatively liquid instruments. These restrictions are most easily met in the equity universe, in which the accepted list of alternative risk factors encompasses the standard long–short Fama and French [1992] value and size factors and the momentum factor (Carhart [1997]), the low-volatility factor (Ang et al. [2006, 2009]), and other factors, such as the quality factor (Asness, Frazzini, and Pedersen [2013]) or liquidity factor (Idzorek, Xiong, and Ibbotson [2012]), among others. Overall, and given that we wish to set the analysis in a multiasset context that includes stocks and bonds but also commodities, credit, and currencies, we choose to focus mainly on the following four alternative risk factors (see Asness et al. [2015] for a similar choice of factors): value, momentum, carry, and low-beta. In what follows, we provide an overview of these risk factors, including a definition and a discussion of how to apply the definition in a multiasset context.

### The Value Risk Factor

The value risk factor is defined as a long exposure to assets that are cheap and a short exposure to those that are expensive, according to a valuation measure.

Asness, Moskowitz, and Pedersen [2013] showed that the value risk factor can be extended to other asset classes. They defined for each asset class a robust measure of the cheapness of an asset relative to its asset class: the common ratio of the book value of equity to market value of equity for individual stocks,^{1} the previous month’s ratio of the book value of equity to market value of equity for the MSCI index of the country considered for country equity futures, the 5-year change in the yields of 10-year bonds for government bonds, the 5-year change in purchasing power parity for currencies, and the negative of the spot return over the last 5 years for commodities futures.

For a security *i* within an asset class *A* made of *n _{t}* assets, we denote

*S*

_{i,t}as the corresponding value measure at time

*t*for the security

*i*. We can write the value factor for the asset class

*A*as follows:

where *r*_{i,t} is the monthly return of security *i* at time *t*, and *c _{t}* is a scaling factor for constraining the portfolio to be one dollar long and one dollar short.

^{2}

Then we construct a global average across eight global asset classes (country equity index futures, commodity futures, government bonds, currencies, U.S. stocks, U.K. stocks, European stocks, and Japanese stocks). They define the global multiasset class value factor VAL^{everywhere} as the inverse-volatility-weighted-across-asset-class value factor.

### The Momentum Risk Factor

The momentum risk factor is designed to buy assets that performed well and sell assets that performed poorly over the past 3 to 12 months. The premise of this investment style is that asset returns exhibit positive serial correlations at these horizons.

The methodology used by Asness, Moskowitz, and Pedersen [2013] for the global momentum factor is the same as the one used for the global value factor. Nonetheless, it is more straightforward because of the unique measure for all asset classes considered. For a security *i* within an asset class *A* made of *n*_{t} assets, we denote at time *tr*_{i,[t−12M;t−1M]} as the return of security *i* over the past 12 months, skipping the most recent month. We can write the momentum factor for the asset class *A* as follows:

where *r*_{i,t} is the monthly return of security *i* at time *t*, and *c*_{t} is a scaling factor for constraining the portfolio to be one dollar long and one dollar short.^{3}

Then we construct a global average across eight global asset classes (country equity index futures, commodity futures, government bonds, currencies, U.S. stocks, U.K. stocks, European stocks, and Japanese stocks). We define the global multiasset class momentum factor MOM^{everywhere} as the inverse-volatility-weighted-across-asset-class momentum factor.

### The Carry Risk Factor

The carry risk factor is designed to take advantage of the outperformance of higher-yielding assets over lower-yielding assets.

Carry, historically most well known in currencies, is defined by Koijen et al. [2015] as “the expected return on an asset assuming that market conditions, including its price, stay the same.” They developed a unifying concept of carry as a directly observable quantity independent of any model:

3In their paper, they extended the notion of carry to nine asset classes by considering (synthetic, if need be) futures contracts. These asset classes are currencies, equities, global bonds, commodities, U.S. Treasuries, credit, slope of global yield curves, call index options, and put index options.

They considered at time *t* a future contract that expires at time *t* + 1 with a current futures price *F _{t}*, a spot price

*S*of the underlying security, a risk-free rate noted , and an allocation of capital of

_{t}*X*

_{t}to finance the position and then wrote the return per allocated capital over one period as follows:

The return in excess of the risk-free rate and the carry are, respectively,

5The carry factor thus defined is directly observable from current market instruments. Finally, we can rewrite the excess return as defined by Koijen et al. [2015]:

6Considering the all-assets future-based definition of Koijen et al. [2015], we can define a global carry factor per asset class as the weighted sum of the carry factors on the *n _{t}* individual securities of the asset class available at time

*t*:

where denotes the carry at time *t* of asset *i*.

Koijen et al. [2015] proposed the following weights:

8where *n _{t}* is the number of available securities at time

*t*, and

*z*

_{t}is a scaling factor.

^{4}Like Asness, Moskowitz, and Pedersen [2013], they defined the global multiasset class carry factor GCR

^{everywhere}as the inverse-volatility-weighted-across-single-asset-class carry factor.

### The Low-Beta (Defensive) Risk Factor

The low-beta factor is designed to take advantage of the reported outperformance of low-beta assets over high-beta assets. Empirically, the relation between stock beta and returns has been proven to be flatter than predicted by the CAPM (Black, Jensen, and Scholes [1972]; Haugen and Heins [1975]). Merton [1972] empirically showed in a number of different equity markets and extended time periods that stocks with low beta significantly outperformed high-beta stocks.

Building upon this body of evidence, Frazzini and Pedersen [2014] proposed to build a low-beta factor in several asset classes. They first considered the traditional definition of beta to define the low-beta risk factor for equities and then extended it to other asset classes: equity indexes, country bonds, currencies, U.S. Treasury bonds, credit indexes, credit, and commodities.

The market portfolio against which the pre-ranking betas are computed depends on the asset class considered. The market portfolio includes the Center for Research in Security Prices (CRSP) value-weighted market index for U.S. equities, the corresponding MSCI local market index for international equities, an aggregate Treasury Bond index for U.S. Treasury bonds, a gross domestic product–weighted portfolio for equity indexes, country bonds and currencies, an equally weighted portfolio of all the bonds in the database for credit, and an equal risk weight portfolio across commodities for commodities.

They used the following methodology to define their low-risk factor betting against beta (BAB): for a given asset class, they first estimated the pre-ranking betas of its securities from rolling regressions of excess returns on market excess returns. Excess returns are above the U.S. Treasury bill rate.

For an asset class *A* composed of *n _{t}* securities at time

*t*, they estimated the beta of a security

*i*as follows:

where and are the estimated volatilities at time *t* for the security *i* and the market, and is the estimated correlation between the security and the market portfolio at time *t*. A one-year rolling-window standard deviation was used for estimating volatilities, and a five-year time frame was used for estimating the correlation. Daily returns are preferred to monthly returns for estimations, if available.

They considered for a given asset class *A* composed of *n*_{t} securities at time *t* the *n*_{t} × 1 vector and assigned each security to the low-beta corresponding asset class portfolio if the security’s beta was inferior to the asset class median or the high-beta corresponding asset class portfolio if the security’s beta was superior to the asset class median. They then defined the portfolio weights of the low-beta and high-beta portfolios relative to the *n _{t}* securities universe at time

*t*:

where , is a normalizing constant, and ()+ and ()− indicate, respectively, the positive and the negative elements of a vector.

The returns and ex ante betas of low-beta and high-beta portfolios verify

11where ^{t} denotes the transpose operator.

Finally, they defined a global asset class BAB, which is market-neutral and goes long low-beta securities and short high-beta securities:

12Frazzini and Pedersen [2014] also defined a multiasset global BAB factor (see Exhibit 8 of their paper) by considering a portfolio with an equal risk contribution in each asset class global BAB factor with a 10% ex ante volatility.

### Other Factors

Two other factors are often used in factor investing strategies—namely, the size and quality factors—but their definition is peculiar to stocks and cannot be straightforwardly adapted to other asset classes.

The size factor is designed from a long portfolio of small-cap stocks and a short portfolio of large-cap stocks (see Fama and French [1992, 2012] for further details).

The quality factor is designed to invest in stocks with strong quality characteristics like low debt, stable earnings growth, and management credibility (see Novy-Marx [2013] and Asness, Frazzini, and Pedersen [2013] for more details).

## HEDGE FUND REPLICATION WITH TRADITIONAL AND ALTERNATIVE FACTORS

Benchmarking hedge fund performance is particularly challenging because of the presence of numerous biases in hedge fund return databases, the most important of which are sample selection bias, survivorship bias, and backfill bias. In what follows, we use EDHEC Alternative Indexes, which aggregate monthly returns on competing hedge fund indexes to improve the hedge fund indexes’ lack of representativeness and to mitigate the bias inherent to each database (see Amenc and Martellini [2003]). We consider the following 13 categories: convertible arbitrage, CTA Global, distressed securities, emerging markets, equity market neutral, event driven, fixed-income arbitrage, global macro, long–short equity, merger arbitrage, relative value, short selling, and fund of funds. We also define the set of relevant risk factors and suitable proxies that will be used in the empirical analysis. An overview of the 19 traditional and alternative risk factors considered in our empirical analysis is given in Exhibit 1.

We proxy traditional risk factors by returns of liquid and investable equity, bond, commodity, and currency indexes. For alternative risk factors, we inter alia consider long–short proxies for value, momentum, and low beta for various asset classes using data from Asness, Moskowitz, and Pedersen [2013] and Frazzini and Pedersen [2014]. Data for the carry risk factor were not available, so we decided to not consider it. All alternative factors considered (single and multiasset class) are global. We consider the following equity asset class alternative factors: value (Asness, Moskowitz, and Pedersen [2013]), momentum (Asness, Moskowitz, and Pedersen [2013]), size (Frazzini and Pedersen [2014]), defensive (Frazzini and Pedersen [2014]), and quality (Asness, Frazzini, and Pedersen [2013]).

A key difference between the traditional and alternative factors is that the latter cannot be regarded as directly investable,^{5} which implies that reported performance levels are likely to be overstated. Given the presence of performance biases in both hedge fund returns and alternative factor returns, we do not focus on differences in average performance between hedge fund indexes and their replicating portfolios and instead focus on the quality of replication measured by in-sample and out-of-sample (adjusted) R^{2} and the annualized root-mean-squared error (RMSE).

As the first step, we perform an in-sample linear regression for each hedge fund strategy’s monthly returns against a set of *K* factors over the whole sample period, ranging from January 1997 to October 2015. For each hedge fund strategy, we have

with being the monthly return of the hedge fund strategy at date *t*; β_{k} the exposure of the monthly return on hedge fund strategy to factor *k* (to be estimated); *F*_{k,t} the monthly return at date *t* on factor *k*; and *ϵ*_{t} the specific risk in the monthly return of the hedge fund index at date *t* (to be estimated).

We estimate the explanatory power measured in terms of the linear regression adjusted R^{2} on the sample period in three distinct cases.

**Case 1:**Linear regression on an exhaustive set of factors (“kitchen sink” regression)—that is, the set of 19 factors listed in Exhibit 1.**Case 2:**Linear regression on a subset of traditional factors only (five factors: equity, bond, credit, commodity, and currency).**Case 3:**Linear regression on a bespoke subset of a maximum of eight economically motivated traditional and alternative factors for each hedge fund strategy (see Exhibit 1 for the selection of factors for each hedge fund strategy).

The obtained adjusted R^{2} values, reported in Exhibit 2, suggest that we can explain a substantial fraction of hedge fund strategy return variability with traditional and alternative factors, validating that an important part of hedge fund performance can ex post be explained by their systematic risk exposures. The kitchen sink regression (case 1) confirms that more dynamic and/or less directional strategies such as CTA Global, equity market neutral, fixed-income arbitrage, and merger arbitrage strategies, with respective adjusted R^{2} values of 31%, 32%, 50%, and 39%, are harder to replicate than more static and/or more directional strategies such as long–short equity or short selling, for which we obtain an adjusted R^{2} of 81%.

The results we obtain also show the improvement in explanatory power when an economically motivated subset of factors that includes alternative factors is considered (case 3) compared to a situation in which the same subset of traditional factors is used for all strategies (case 2). For example, the adjusted R^{2} increases from 25% to 50% for the global macro strategy and from 52% to 80% for the emerging market strategy.

In the second step, we perform an out-of-sample hedge fund return replication exercise using for each strategy the bespoke subset of factors (case 3). The objective of this analysis is to assess whether one can capture the dynamic factor exposures of hedge fund strategies by explicitly allowing the betas to vary over time in a statistical model. The out-of-sample window considered is January 1999 to October 2015, which allows us to build a 24-month rolling-window linear clone for each strategy. For each hedge fund strategy, we have

14with being the monthly return of the hedge fund strategy clone at date *t*; β_{k,t} the possibly time-varying exposure of the monthly return on hedge fund strategy to factor *k* on the rolling period [*t* − 24 months; *t* − 1 month] (to estimate); *F*_{k,t} the monthly return at date *t* on factor *k*; and *ϵ*_{t} the specific risk in the monthly return of hedge fund index at date *t* (to estimate).

The hedge fund clone monthly return is

15where is the ordinary least squares estimation of β_{k,t} from Equation (14) on the rolling period [*t* − 24 months; *t* − 1]. Because our focus is on hedge fund replication, we take into account the possible leverage of the strategy by adding a cash component proxied by the U.S. 3-month Treasury bill index monthly returns.

A more sophisticated approach consists of explicitly modeling dynamic risk factor exposures through a linear state-space model and then solving its variables by Kalman filtering (see Harvey [1991] for further details). Broadly speaking, a state-space model is defined by a transition equation and a measurement equation as follows:

16where β_{t} is the vector of (unobservable) factor exposures at time *t* to the risk factors (to estimate via the Kalman filter); *F _{t}* is the vector of factors’ monthly returns at time

*t*; and η

_{t}and

*ϵ*

_{t}are assumed to be normally distributed with a variance assumed to be constant over time (to estimate).

The hedge fund clone monthly return is

17where is the estimation of β_{k,t} via the Kalman filter algorithm. The substantial decrease between in-sample (see Exhibit 2) and out-of-sample (see Exhibit 3) adjusted R^{2} for all strategies suggests that the actual replication power of the clones falls sharply when taken out of the calibration sample. For example, the event driven clones have an out-of-sample adjusted R^{2} below 50%, whereas the event driven hedge fund strategy has a corresponding in-sample adjusted R^{2} of 63%. The equity market neutral clones have negative adjusted R^{2}, whereas the equity market neutral hedge fund strategy has a corresponding in-sample adjusted R^{2} of 16%. The CTA Global rolling-window clone has also a negative out-of-sample adjusted R^{2}, corroborating the lack of robustness of the clones.

To get a better sense of what the out-of-sample replication quality actually is, we compute the annualized RMSE (see Exhibit 3), which can be interpreted as the out-of-sample tracking error of the clone with respect to the corresponding hedge fund strategy. Our results suggest that the use of Kalman filter techniques does not systematically improve the quality of replication with respect to the simple rolling-window approach: Kalman filter clones for distressed securities, emerging markets, event driven, global macro, short selling, and fund of funds have RMSEs greater than the those of their rolling-window counterparts. Strategies such as CTA Global or short selling have clones with the poorest replication quality, with RMSEs higher than 7.5%. Overall, these results do not support the belief that hedge fund returns can be satisfactorily replicated in a passive manner with traditional and alternative risk factors.

## FROM HEDGE FUND REPLICATION TO HEDGE FUND SUBSTITUTION

In this section, we revisit the problem from a different perspective. We move away from hedge fund replication, which is not necessarily a meaningful goal for investors, and analyze whether naively diversified strategies based on systematic exposure to the same alternative risk factors perform better from a risk-adjusted perspective than the corresponding hedge fund clones. Because the same proxies for underlying alternative factor premia will be used in the clones and the diversified portfolios, we can perform a fair comparison in terms of risk-adjusted performance in spite of the presence of performance biases in both hedge fund return and factor proxies.

We apply two popular robust heuristic portfolio construction methodologies, equally weighted and equal risk contribution, for each hedge fund strategy relative to its bespoke subset of economically identified risk factors for the period January 1999 to October 2015. We use 24-month rolling windows to estimate the covariance matrix for the equal risk contribution weighting scheme. We then compare the risk-adjusted performance of rolling-window and Kalman filter clones and the corresponding diversified portfolios of the same selected factors in terms of their Sharpe ratios.

The first two rows of Exhibit 4 show the Sharpe ratios of the rolling-window and Kalman filter clones, and the last two rows show the Sharpe ratios of the corresponding equal risk contribution and equally weighted diversified portfolios. The clones for distressed securities, event driven, global macro, relative value, and fund of funds have been built with the same six risk factors: equity, bond, credit, emerging market, multiclass value, and multiclass momentum. The corresponding equal risk contribution and equally weighted portfolios have respective Sharpe ratios of 0.74 and 0.63, which are higher than all of the previous clones’ Sharpe ratios (see, for example, the global macro and distressed securities Kalman filter clones with respective Sharpe ratios of 0.53 and 0.17). Similarly, the equity market neutral, merger arbitrage, long–short equity, and short-selling clones have been built with the same six risk factors: equity, equity defensive, equity size, equity quality, equity value, and equity momentum. All the clones’ Sharpe ratios are lower (see, for example, the equity market neutral Kalman filter clone with Sharpe ratio of 0.74) than those of the corresponding equal risk contribution and equally weighted portfolios (respectively, 1.02 and 0.96), sometimes substantially lower (see, for example, the merger arbitrage and long–short equity Kalman filter clones with respective Sharpe ratios of 0.39 and 0.26).

Exhibit 5 also displays extreme risks of clones and portfolios. We consider only nondirectional strategies like equity market neutral, merger arbitrage, relative value, convertible arbitrage, and fixed-income arbitrage to compare clones and diversified portfolios with similar volatilities. The clones of equity market neutral (12.4% maximum drawdown for the rolling-window clone and 7.7% for the Kalman clone) and merger arbitrage (20.7% maximum drawdown for the rolling-window clone and 17.2% for the Kalman clone), built with the same risk factors, show higher extreme risk than the corresponding equal risk contribution portfolio (2.6% maximum drawdown). This is also the case for the clones of fixed-income arbitrage (14.7% maximum drawdown for the rolling-window clone and 13.7% for the Kalman clone), which displays higher extreme risk than the corresponding equal risk contribution portfolio (3.7% maximum drawdown).

In Exhibit 6, we can compare for each strategy the diversification of its corresponding clones and diversified portfolios considering the two following diversification measures: the effective number of constituents (ENC) and the effective number of uncorrelated bets (ENUB) (see the Appendix for further details). For a given hedge fund strategy, the clones have, by construction, one more constituent (the risk-free asset) than the corresponding diversified portfolios because of the leverage. Consequently, we consider relative values of the average ENC and average ENUB for the out-of-sample period ranging from January 1999 to October 2015. The values for these two statistics therefore vary from 0% to 100%. The equal risk contribution portfolios have, by construction, an average ENUB near 100%, so we will compare only the clones with the associated equally weighted portfolios.

All clones for equity market neutral, merger arbitrage, long–short equity, and short selling built on the same factors display an inferior average ENUB (see, for example, the long–short equity strategy with 54% for the rolling-window clone and 45% for the Kalman clone, and the merger arbitrage strategy with 47% for the rolling-window clone and 46% for the Kalman clone) compared to the associated equally weighted portfolio (66%). The assessment is still verified for more dynamic or nondirectional strategies like CTA Global (57% for the rolling-window clone, 49% for the Kalman clone, and 61% for the corresponding equally weighted strategy) and fixed-income arbitrage (45% for the rolling-window clone, 42% for the Kalman clone, and 71% for the corresponding equally weighted strategy). On the contrary, all the clones for distressed securities, event driven, global macro, relative value, and fund of funds built on the same factors have a superior (or equal) average ENUB (see, for example, the relative value strategy with 53% for the rolling-window clone and 55% for the Kalman clone, and the fund-of-funds strategy with 54% for the rolling-window clone and 65% for the Kalman clone) compared to the associated equally weighted portfolio (53%). For every hedge fund strategy, the average ENC of clones is systematically inferior to those of the diversified portfolios: Apart from the equity market neutral and the emerging market strategies, the average ENC of the hedge fund clones is inferior or equal to 40%, indicating a high concentration of the clones in a few risk factors.

Overall, diversified portfolios seem a better alternative to hedge fund clones from risk-adjusted performance, diversification, and extreme risk perspectives.

Exhibit 7 contains the results for clones’ and diversified portfolios’ one-way annual turnover. At each rebalancing date *t*, the one-way turnover is computed:

where *x*_{i,t} is the weight of constituent *i* imposed on date *t*, and *x*_{i,t‒} is the effective weight just before the rebalancing. The annual turnover is the average of the θ_{t} multiplied by 12 for conversion to an annual quantity.

We see that in all cases the turnover for the hedge fund clones is much higher than that for the corresponding diversified portfolios. The clones’ turnovers vary from 106% for the long–short equity Kalman clone to 29,935% for the short selling Kalman clone. Comparatively, the diversified portfolios’ turnovers vary from 4% for the fixed-income arbitrage equally weighted portfolio to 79% for the CTA Global equal risk contribution portfolio. These substantial differences in turnover can be partly explained by the unconstrained long–short exposures of the clones on risk factors. On the other hand, except for the short-selling strategy, the Kalman clones have lower turnover than the rolling-window clones. We can also see that for all the hedge fund strategies, the equally weighted portfolios display lower turnover than the equal risk contribution portfolios. These results emphasize that the transaction costs inherent to dynamic trading strategies could be significant in the case of the hedge fund clones.

## CONCLUSION

Although the replication of hedge fund factor exposures appears to be a very attractive concept from a conceptual standpoint, our analysis confirms the previously documented intrinsic difficulty of achieving satisfactory out-of-sample replication power, regardless of the set of factors and the methodologies used. Our results also suggest that risk parity strategies applied to alternative risk factors could be a better alternative than hedge fund replication for harvesting alternative risk premia in an efficient way. In the end, the relevant question may not be whether it is feasible to design accurate hedge fund clones with similar returns and lower fees, for which the answer appears to be clearly negative, but instead “Can suitably designed mechanical trading strategies in a number of investable factors provide a cost-efficient way for investors to harvest traditional but also alternative beta exposures?” With respect to the second question, there are reasons to believe that such low-cost alternatives to hedge funds may prove a fruitful area of investigation for asset managers and asset owners. A key challenge for the alternative investment industry remains the capacity to develop investable, efficient, low-cost proxies for harvesting alternative risk premia not only in the equity market but also in the fixed-income, currencies, and commodity markets. Our study could also be extended in a number of useful directions. In particular, more factors, such as *carry everywhere* or *equity short volatility*, for example, could be included in the empirical analysis. Additionally, introducing regional single- and multiasset factors would also be a worthwhile extension of our analysis.

## APPENDIX

### DIVERSIFICATION STATISTICS

The ENC in a portfolio is given by the entropy of the portfolio weight distribution. This dispersion measure of the dollar contributions could be defined as follows:

A-1where ‖·‖ denotes the Euclidean norm. An application of Cauchy–Schwarz inequality shows that ENC ≤ *K*, and that equality holds if, and only if, all weights are equal.^{6}

This quantity is equal to the nominal number of assets *K* for a well-balanced equally weighted portfolio but would converge to 1 if the allocation to all assets but one converges to zero as in the preceding example, thus confirming the extreme concentration in this portfolio.

The ENUB can be interpreted as a more robust measure of diversification for investors’ portfolios. First, we decompose the portfolio return *r _{P}* as the sum of

*K*correlated asset returns

*r*

_{1}, …,

*r*and also the sum of

_{K}*K*uncorrelated factor returns

*r*

_{F}_{1}, …,

*r*(see also papers by Deguest, Martellini, and Meucci [2013] and Meucci, Santangelo, and Deguest [2015] for more details):

_{FK}where *r* denotes the vector of the original constituents’ returns, *r _{F}* is the vector of uncorrelated factors’ returns,

*w*is the weight vector of correlated components, and

*w*is the weight vector of uncorrelated factors. The main challenge with this approach is to turn correlated asset returns into uncorrelated factor returns.

_{F}The volatility of the portfolio is

where

is the covariance matrix of the *K* uncorrelated factors.

The factor returns can typically be expressed as a linear transformation of the original returns *r _{F}*=

^{t}

*Ar*for some well-chosen transformation

*A*guaranteeing that the covariance matrix of the factors

is a diagonal matrix. *A* is a *K* × *K* transition matrix from the assets to the factors, and it is therefore critical that it be invertible because *w _{F}*=

*A*

^{‒1}

*w*. Then, we define the contribution of each factor to the overall variance of the portfolio :

which leads to the following percentage contributions for each factor:

where .

The ENUB can be written as follows:

A-3ENUB reaches a minimum equal to 1 if the portfolio is loaded in a single risk factor and a maximum equal to *K*, the nominal number of constituents, if the risk is evenly spread among factors. This methodology relies on uncorrelated implicit factors, which are not uniquely defined. In fact, it is easy to see that if *A* is a change of basis matrix from constituents to factors and *Q* is any orthogonal matrix, then the matrix is also a change of basis matrix such that ^{t}*M*𝕍*M* is diagonal.^{7} Thus, one has to specify an orthogonalization procedure. An option is to extract the matrix *A* such that the factor covariance matrix is diagonal while keeping the distance between the factors and the asset returns as small as possible, thus involving the smallest deformation of the original components. In detail, *A* is the solution to the following program:

where diag(𝕍) denotes a diagonal matrix with diagonal elements equal to those of 𝕍. We use in our studies this least intrusive orthogonalization procedure, known as *minimum linear torsion*, to extract uncorrelated factors starting from correlated factor indexes (see Meucci, Santangelo, and Deguest [2015] and Carli, Deguest, and Martellini [2014] for the solution to the optimization problem).

## ENDNOTES

This project is sponsored by Lyxor Asset Management in the context of the “Risk Allocation Solutions” research chair at EDHEC-Risk Institute. We would like to thank Nicolas Gaussel and Thierry Roncalli for very useful comments.

↵

^{1}Book values are lagged by six months for data availability, and market values are the most recent available.↵

^{2}For a given asset class*A*and .↵

^{3}For a given asset class*A*and↵

^{4}For a given asset class*A*and↵

^{5}The main reasons for this are the necessity of short positions and the possible lack of liquidity of the assets considered.↵

^{6}The Cauchy–Schwarz inequality states that for any two vectors (*x*_{1}, …,*x*) and (_{K}*y*_{1}, …,*y*), we have . Equality is achieved if, and only if, one of the two vectors is zero or the two vectors are parallel. Here, we take_{K}*y*_{i}= 1.↵

^{7}A matrix*Q*is said to be orthogonal if it satisfies^{t}*QQ*=*Q*= 𝕀^{t}Q_{k}.

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