+ All Categories
Home > Documents > Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio...

Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio...

Date post: 22-Apr-2020
Category:
Upload: others
View: 6 times
Download: 0 times
Share this document with a friend
13
Hedge fund allocation: Evaluating parametric and nonparametric forecasts using alternative portfolio construction techniques Mohan Subbiah a , Frank J. Fabozzi a,b, a EDHEC Business School, United Kingdom b EDHEC Risk Institute, United States abstract article info Article history: Received 16 August 2015 Received in revised form 17 February 2016 Accepted 7 March 2016 Available online 19 March 2016 We propose a model for constructing Asian funds of hedge funds. We compare the accuracy of forecasts of hedge fund returns using an ordinary least squares (OLS) regression model, a nonparametric regression model, and a nonlinear nonparametric model. We backtest to assess these forecasts using three different portfolio construction processes: an optimizedportfolio, an equally-weighted portfolio, and the Kelly criterion-based portfolio. We nd that the Kelly criterion is a reasonable method for constructing a fund of hedge funds, producing better re- sults than a basic optimization or an equally-weighted portfolio construction method. Our backtests also indicate that the nonparametric forecasts and the OLS forecasts produce similar performance at the hedge fund index level. At the individual fund level, our analysis indicates that the OLS forecasts produce higher directional accu- racy than the nonparametric methods but the nonparametric methods produce more accurate forecasts than OLS. In backtests, the highest information ratio to predict hedge fund returns is obtained from a combination of the OLS regression with the FungHsieh eight-factor variables as predictors using the Kelly criterion portfolio construction method. Similarly, the highest information ratio using forecasts generated from a combination of the nonparametric regression using the FungHsieh eight-factor model variables is achieved using the Kelly cri- terion portfolio construction method. Simulations using risk-adjusted total returns indicate that the nonparamet- ric regression model generates superior information ratios than the analogous backtest results using the OLS. However, the benets of diversication plateau with portfolios of more than 20 hedge funds. These results generally hold with portfolio implementation lags up to 12 months. © 2016 Published by Elsevier Inc. Keywords: Hedge fund allocation Hedge funds Funds of hedge funds 1. Introduction As of September 1, 2014, the hedge fund industry had roughly $2.6 trillion asset under management managed by 11,000 funds by more than 4500 separate companies (Delevingne, 2014). The fund of hedge funds industry includes more than 2000 fund of hedge funds seeking to invest client capital in underlying hedge funds. Their objective is to construct a diverse portfolio of individual hedge funds to provide broad exposure to the hedge fund industry while diversifying the risks associated with individual hedge funds. Lack (2012) suggests hedge fund investments have been poor, con- troversially quoting Shocking but true: if all the money that's ever been invested in hedge funds had been in treasury bills, the results would have been twice as good. Lack also points out Hedge funds could still have a place in portfolios but investors need to be thoughtful about hedge fund allocations. In 2012, HFR quotes Funds of hedge funds have underperformed single manager hedge funds in eight of the past ten years.1 Despite such performance, in 2012, fund of hedge funds managed over $640 billion after experiencing a $184 billion outow fol- lowing the 2008 nancial crisis. This performance suggests that there is room for improvement in al- locating assets among hedge funds. In this paper we propose a model to assist in constructing an Asian funds of hedge funds by using three sta- tistical methodologies the ordinary least squares (OLS) regression, a nonparametric regression, and a nonlinear nonparametric approach (the simplex projection) to forecast hedge fund returns. After compar- ing the accuracy of these forecasts, we backtest to evaluate each model's relative performance using three different portfolio construction pro- cesses: an optimizedportfolio, an equally-weighted portfolio, and a Kelly criterion-based portfolio. The paper is organized as follows. Section 2 provides a review of hedge fund return models and how our methodology and data tie into that used in prior studies. Since there are various methods that can be used to construct a portfolio, in Section 3 we review the three portfolio International Review of Financial Analysis 45 (2016) 189201 Corresponding author at: 858 Tower View Circle, New Hope, PA 18938, United States. E-mail addresses: [email protected] (M. Subbiah), [email protected] (F.J. Fabozzi). 1 Going, Going, Gone?The Economist June 2, 2012. http://dx.doi.org/10.1016/j.irfa.2016.03.003 1057-5219/© 2016 Published by Elsevier Inc. Contents lists available at ScienceDirect International Review of Financial Analysis
Transcript
Page 1: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

International Review of Financial Analysis 45 (2016) 189–201

Contents lists available at ScienceDirect

International Review of Financial Analysis

Hedge fund allocation: Evaluating parametric and nonparametricforecasts using alternative portfolio construction techniques

Mohan Subbiah a, Frank J. Fabozzi a,b,⁎a EDHEC Business School, United Kingdomb EDHEC Risk Institute, United States

⁎ Corresponding author at: 858 Tower View Circle, NewE-mail addresses: [email protected] (M. S

(F.J. Fabozzi).

http://dx.doi.org/10.1016/j.irfa.2016.03.0031057-5219/© 2016 Published by Elsevier Inc.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 16 August 2015Received in revised form 17 February 2016Accepted 7 March 2016Available online 19 March 2016

Wepropose amodel for constructing Asian funds of hedge funds. We compare the accuracy of forecasts of hedgefund returns using an ordinary least squares (OLS) regression model, a nonparametric regression model, and anonlinear nonparametricmodel.Webacktest to assess these forecasts using three different portfolio constructionprocesses: an “optimized” portfolio, an equally-weighted portfolio, and the Kelly criterion-based portfolio. Wefind that the Kelly criterion is a reasonable method for constructing a fund of hedge funds, producing better re-sults than a basic optimization or an equally-weighted portfolio constructionmethod. Our backtests also indicatethat the nonparametric forecasts and the OLS forecasts produce similar performance at the hedge fund indexlevel. At the individual fund level, our analysis indicates that the OLS forecasts produce higher directional accu-racy than the nonparametric methods but the nonparametric methods produce more accurate forecasts thanOLS. In backtests, the highest information ratio to predict hedge fund returns is obtained from a combinationof the OLS regression with the Fung–Hsieh eight-factor variables as predictors using the Kelly criterion portfolioconstruction method. Similarly, the highest information ratio using forecasts generated from a combination ofthe nonparametric regression using the Fung–Hsieh eight-factor model variables is achieved using the Kelly cri-terion portfolio constructionmethod. Simulations using risk-adjusted total returns indicate that the nonparamet-ric regression model generates superior information ratios than the analogous backtest results using the OLS.However, the benefits of diversification plateau with portfolios of more than 20 hedge funds. These resultsgenerally hold with portfolio implementation lags up to 12 months.

© 2016 Published by Elsevier Inc.

Keywords:Hedge fund allocationHedge fundsFunds of hedge funds

1. Introduction

As of September 1, 2014, the hedge fund industry had roughly $2.6trillion asset under management managed by 11,000 funds by morethan 4500 separate companies (Delevingne, 2014). The fund of hedgefunds industry includes more than 2000 fund of hedge funds seekingto invest client capital in underlying hedge funds. Their objective is toconstruct a diverse portfolio of individual hedge funds to providebroad exposure to the hedge fund industry while diversifying the risksassociated with individual hedge funds.

Lack (2012) suggests hedge fund investments have been poor, con-troversially quoting “Shocking but true: if all themoney that's ever beeninvested in hedge funds had been in treasury bills, the results wouldhave been twice as good”. Lack also points out “Hedge funds could stillhave a place in portfolios but investors need to be thoughtful abouthedge fund … allocations”. In 2012, HFR quotes “Funds of hedge funds

Hope, PA 18938, United States.ubbiah), [email protected]

have underperformed single manager hedge funds in eight of the pastten years.”1 Despite such performance, in 2012, fund of hedge fundsmanaged over $640 billion after experiencing a $184 billion outflow fol-lowing the 2008 financial crisis.

This performance suggests that there is room for improvement in al-locating assets among hedge funds. In this paperwe propose amodel toassist in constructing an Asian funds of hedge funds by using three sta-tistical methodologies – the ordinary least squares (OLS) regression, anonparametric regression, and a nonlinear nonparametric approach(the simplex projection) – to forecast hedge fund returns. After compar-ing the accuracy of these forecasts, we backtest to evaluate eachmodel'srelative performance using three different portfolio construction pro-cesses: an “optimized” portfolio, an equally-weighted portfolio, and aKelly criterion-based portfolio.

The paper is organized as follows. Section 2 provides a review ofhedge fund return models and how our methodology and data tie intothat used in prior studies. Since there are various methods that can beused to construct a portfolio, in Section 3 we review the three portfolio

1 “Going, Going, Gone?” The Economist June 2, 2012.

Page 2: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

190 M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

construction methods that we evaluated in this paper. Section 4 de-scribes the data, the procedures for removing any data biases.Section 5 describes how themodel's exogenous data are used as predic-tor variables in the analysis. Section 6 describes (1) the methodologiesfor return forecasts using regression methods and simplex projection,and (2) portfolio construction using three portfolio constructionmethods (an equally-weighted portfolio, an “optimized” portfolio, andtheKelly criterion-basedportfolio). In Section7,we report the(1) returnforecast results from all three methods relative to the observed returnsin individual funds and hedge fund indices, (2) results of the backtestsusing the three portfolio construction methods and each of the returnforecasts, (3) robustness checks which include lagged implementationof the return forecasts, a range of portfolio holding settings in the port-folio construction process, and different portfolio horizons/portfoliorebalancing frequencies. Our conclusions are summarized in Section 8.

2. Hedge fund literature

Sharpe's “style regression” (see Sharpe, 1992) given by

Rt ¼ α þ∑k

bk Fkt þ ut

works well in capturing the styles of open-end mutual funds, whosereturns are highly correlated to those of standard asset classes. Fungand Hsieh (1997) introduce five dominant investment styles in hedgefunds which when added to Sharpe's asset class factor model can pro-vide an integrated framework for style analysis for both buy-and-holdand dynamic trading strategies. Fung, Hsieh, Naik, and Ramadorai(2008) report that a large proportion of the variation in hedge fundreturns can be explained by market-related factors.2 Fung and Hsieh(2004a) propose a seven3 factor “APT-like”model of hedge fund returnswith dynamic risk factor coefficients. They find that their model can ex-plain up to 80% of the variation in global hedge fund returns. Fung andHsieh (2007) extend their model, presenting an eight factor model byadding an emerging market factor. As explained in Section 5, we usethese eight factors as our set of explanatory variables in our regressions.

Teo (2009) investigates the performance of Asian hedge funds usingthe Asiahedge and Eurekahedge databases, the same databases we usein this paper, and the Hedge Fund Research Inc. (HFR) global hedgefund database. One of the few studies that focuses on Asian-basedhedge fund returns, they find that hedge fundswith a physical presencein their investment region outperform other hedge funds by 3.72% peryear. Consistent with Bali, Brown, and Caglayan (2012), Fung andHsieh (2004a) investigate the extent to which market risk, residualrisk, and tail risk explain the cross sectional dispersion in hedge fundreturns. They find that systematic risk is a highly significant factor inexplaining the dispersion of cross-sectional returns while residual riskand tail risk have little explanatory power. Sadka (2010) provides evi-dence of liquidity risk as a contributing factor for hedge fund returns.

After controlling for common risk factors to explain hedge fundreturns, Agarwal, Bakshi, andHuij (2009)find risk premiums for volatil-ity, skewness, and kurtosis of about 6%, 3%, and−3% per annum, respec-tively. Kelly and Jiang (2012) find that a conditional tail risk factor is animportant determinant of hedge fund returns, even after controlling for

2 See, for example, Agarwal and Naik (2004) and Fung and Hsieh (1997, 2001, 2002,2004).

3 S&P: Standard & Poors 500 stock return; SC-LC: Wilshire 1750 Small Cap — Wilshire750 Large Cap return; 10Y: month end-to-month end change in the Federal Reserve'sten year constant maturity yield; Cred spr: month end-to-month end change in the differ-ence between Moody's Baa yield and the Federal Reserve's ten year constant maturityyield; Bd Opt: return of a portfolio of lookback straddles on bond futures; FX Opt: returnof a portfolio of lookback straddles on currency futures; Com Opt: return of a portfolio oflookback straddles on commodity futures.

the Fung–Hsieh factors, option-based risk measures in Agarwal et al.(2009) and a liquidity risk factor by Sadka (2010).

Anand, Kutsarov, Maier, and Storr (2011) show the importance oftactical asset allocation in fund of hedge funds allocation, and presentstatistics on the distribution of returns prior to, during, and after the2008 global financial crisis. They report the presence of large right taildistributions pre- and post-crisis. Using a nonparametric regressionmodel, Anand, Kutsarov, Maier, and Storr (2013) extend their earlierwork by presenting sensitivities of hedge fund indices to a four-factormodel which includes macroeconomic and behavioral factors.

3. Portfolio construction techniques

A number of methods are available for constructing a portfolio ofhedge funds. In this studywe consider three portfolio construction tech-niques: an equally-weighted portfolio, an “optimized” portfolio, and theKelly criterion-based portfolio.

DeMiguel, Garlappi, and Uppal (2009) consider 15 asset allocationmodels and conclude that a simple equally-weighted (1/N) approachis difficult to improve upon. There are many papers on optimizationtechniques for portfolio construction, the most popular being themean–variance framework formulated by Markowitz (1952) based onmeans, variances, and covariances of asset returns for generating effi-cient portfolios. We implement a mean–variance optimizer as one ofour portfolio construction methods and provide further details later(see Section 6.3).

Kelly (1956) introduced the criterion for portfolio construction(now referred to as the “Kelly criterion”). Applying Kelly's criterion,we can allocate a fraction of capital, f, such that f = p/L − q/W, wherep is the probability of “winning”, q the probability of “losing” (1 − p)and W is the amount “won” for each $1 bet, and conversely L theamount “lost” for each $1 bet. There is support for this approach to port-folio construction in the literature. Breiman (1961) proved “that Kelly'sapproach beats any other money management approach” and Ethier(2004) showed that “theKelly criterionmaximizes themedian of termi-nal wealth”. The theory has been extended tomultivariate portfolios byMaslov and Zhang (1998) and Laureti, Medo, and Zhang (2010), amongothers.

4. The hedge fund sample

All hedge fund databases have some problems. Fung and Hsieh(2004a) and Titman and Tiu (2011) provide a detailed explanation ofsome of these problems as they pertain to hedge fund databases.Following Agarwal and Naik (2004) and Titman and Tiu (2011) we ag-gregate multiple databases to provide a more comprehensive dataset.The three databases that we consolidate are Morningstar CISDM,4

Asiahedge, and Eurekahedge. Our access to CISDM data was restricted5

to a fund-level dataset ending in 2009, while Asiahedge fund databaseand Eurekahedge fund database cover through 2012.

Fig. 1 shows the funds coverage.We first filter the global CISDMdatabase of hedge funds from 11,402

funds down to 250 funds, including only funds listed as either Asia/Pa-cific, Asia/Pacific (excluding Japan), and Australia/New Zealand. Similarfilters were applied to the Eurekahedge and Asiahedge fund databases,and then overlapping funds were filtered leaving only the data sourcewith the longest history of a fund.

4 The Morningstar CISDM Database (formerly the MAR Database) is the oldest hedgefund database. Information about thedatabase is available from theCenter for Internation-al Securities and Derivatives Markets (CISDM) at https://www.isenberg.umass.edu/CISDM/Hedge_FundCTA_Database/

5 CISDM data up to 2009, was provided to EDHEC as a one offset of historic data, ratherthan a “current” ongoing subscription.

Page 3: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Fig. 1. Fund coverage.

191M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

Fig. 1 shows the reduction of funds from our initial 13,661 fundsfrom the aggregate of the three data sources down to 2123 uniquefunds to be included for analysis in this paper. We next removed dupli-cate share classes of the same fund by selecting only the share classwiththe longest performance history and excluded fund of funds.

For our analysis, in order to produce the first return forecast we re-quire a two-year history for a fund. Ammann, Schmid, and Huber(2011) generate their return forecasts from a two-year rolling window,noting this requirement may introduce a sampling bias. Fung and Hsieh

Fig. 2. Number of funds through time.

Fig. 3. Annualized returns of funds.

(2000) investigate this bias, which they refer to as “multi-period sam-pling bias”, by comparing the average returns of all funds in the sampleto the average returns of the funds with at least a 24-month history ofreturns. They conclude that this bias is very small and therefore can beignored. This two-year filter and the removal of duplicate share classesreduced our universe to 1565 funds, and the exclusion of fund of fundsreduced our universe to 1243 funds. Fig. 2 shows how our coverage offunds expands through time from an initial 50 funds in 2000, to over600 funds by 2010.

Fig. 3 reports that the range of annualized returns fromour sample offunds is from −75% to 88%. The figure also indicates that 76% of thefunds produce a positive annualized return, and 24% a negative return,which may be explained by a contribution bias of positive returnfunds contributing to the database. A similar issue known as a “backfillbias” is addressed by many researchers such as Titman and Tiu (2011)who address this issue by excluding funds that have assets under man-agement (AUM) of less than $30million. As we have only a single snap-shot of hedge fund data6 from each of the three data providers, we areunable to determine if a fund was below $30 million at some priortime to the snapshot date, so we include all the funds in the analysis ir-respective of the amount of AUM. Another, more frequently usedmeth-od to mitigate the potential impact of backfill bias – described inEdwards and Caglayan (2001), Fung and Hsieh (2000), and Teo(2009) – is to remove the first 12 months of a funds history from the“live” universe. In our analysis, we follow this procedure by removingthe first 12 months of a fund's history and require a subsequent two-year history of performance for a fund to be used as an initial trainingwindow before a fund is included in the live universe.

Another issuewith hedge fund databases is “survivorship bias”. BothAmmann et al. (2011) using the Trading Advisor Selection System(TASS)7 database and Titman and Tiu (2011) using multiple databasesstart their analysis in January 1994, noting that a number of databasevendors did not keep records of defunct funds prior to 1994. TheCISDM and Eurekahedge databases provide a fund inception date. TheAsiahedge fund database does not provide a fund inception date. Al-though our analysis includes both defunct and live funds, all three data-bases may suffer from survivorship bias from funds that terminate

6 Our CISDM dataset is a single snapshot as of December 2009. The snapshots forAsiahedge and Eurekahedge are as of December 2012, so we are unable to obtain a timeseries of AUM to see if a fund is consistently under $30 million throughout its life.

7 The Lipper TASS database is provided by Thomson Reuters.

Page 4: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Table 1Statistics on the universe of funds.

(a) Top performing fund statistics

Top nth fund Per annum % return Top 1:nth fund:average return,%, per annum

Risk, nth fund,% per annum

I.R., nth fund Skewness,a nth fund Kurtosis,b nth fund

1 88.5 88.5 65.96 1.34 1.01 2.8110 49.5 64.5 29.05 1.71 1.42 2.2215 44.4 56.1 9.04 4.91 1.20 1.2020 41.2 50.8 15.88 2.60 0.72 0.4925 38.6 47.2 28.57 1.35 3.77 21.0530 35.8 44.3 16.45 2.18 2.77 11.4135 33.9 41.8 18.36 1.85 −0.09 −0.4940 32.2 39.9 19.16 1.68 −1.20 2.6145 31.0 38.3 10.79 2.87 −0.25 −0.0650 29.5 37.0 17.21 1.71 −0.14 1.31

(b) Hedge fund database universe statistics

IR Decile 1 2 3 4 5 6 7 8 9 10

Ave Ann. Rtn 20.69 15.68 12.84 11.43 9.09 6.65 3.72 0.07 −4.37 −17.27Ave Ann. Risk 8.35 11.65 13.72 16.01 16.87 17.98 19.77 19.31 17.81 19.92Ave IRc 2.86 1.35 0.94 0.71 0.53 0.37 0.19 0.00 −0.25 −1.02

Top performing: ranking by annualized return.a The adjusted Fisher–Pearson standardized moment coefficient is the version found in Excel skew() which we use here.b We use the Kurt() function in excel to produce these numbers on our sample data.c The average IR shown is the average of all fund IR's within each IR decile, rather than the average ann. return/average ann. risk.

8 http://faculty.fuqua.duke.edu/~dah7/DataLibrary/TF-FAC.xls— starts Jan. 1994.9 The followingwere downloaded: SPXT Index(S&P 500 Total Return); RU20INTR Index

(Russell 2000 Total Return); TB3MS (3-month Treasury bill);MOODCBAA Index (Moody'sUS Baa Corporate Bond Index), and; H15T10Y Index (10-year Treasury yield).

192 M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

before being recorded by the database providers. We start our analysisin 1994 in order to mitigate the risk of survivorship bias. We use threeanalytical methods — the OLS regression, nonparametric regressionand simplex projection. Because we have no reason to believe a priorithat our results are affected by survivorship bias under a particular an-alytic method while not likewise affecting the results under the othertwo methods, we therefore overlook any issues associated with survi-vorship bias.

Getmansky, Lo, and Makarov (2004) conclude that hedge fundreturns are often highly serially correlated and the source of this serialcorrelation may be illiquidity in the underlying assets. Asness, Krail,and Liew (2001) argue that variance, Sharpe ratios, and betas are under-stated as a result of the serial correlation of returns. Getmansky et al.(2004) argue that returns are unaffected by serial correlation. In ouranalysis, we choose to focus on forecasting the reported performancefrom the database providers without a serial correlation adjustment asit is easier to interpret the results. Table 1(a) provides statistics for thetop n hedge funds in the database as ranked by annualized return, anda yardstick for the magnitude of returns we might want to achieve inthe absence of due diligence on the funds. The table provides statisticsfor the top 10–50 performing funds based on annualized returns, con-sistent with the 10–50 fund portfolios we construct in our analysis.Table 1(b) also provides statistics for the entire universe of hedgefunds. The table provides return/risk statistics for each informationratio (I.R) decile.

Some funds in our data sample existed for only one month whileothers existed for the complete 21 years in our investigation period.Fig. 4 shows the distribution of our sample of funds with respect tothe funds life (length of existence). We observe that for the periodfrom January 1992 to December 2012, over 20% of the funds have afund life less than two years, 60% of the funds exist for less than fiveyears, 6.5% survive for more than 10 years, and only four funds survivefor 21 years.

We observe that the majority of hedge funds have very short lives(60% of funds exist for less than five years), and posit that this shortlife (lack of observations) may hinder the nonparametric method's abil-ity to produce consistently accurate forecasts. This lack of historic obser-vations for each hedge fund, alongwith the array of forecast variables, is

the “curse of dimensionality” problem discussed in applying nonpara-metric regression by Sain (2002).

5. Predictor variables

In addition to the fund level data from the three database pro-viders, we use hedge fund indices in our optimizer-based portfolioconstruction. For this purpose, we choose the CISDM indices. Al-though the earliest CISDM indices start in January 1980, the Fixed In-come Arbitrage Index is the last index to initiate coverage in January1998. Given our two-year history requirement, our analysis begins in2000. We also use the eight CISDM indices as a dependent set of var-iables and report the results of our methodologies at the hedge fundindex level.

Ammann et al. (2011) compare three alternative factor models —Fung and Hsieh (2004a) seven-factor model, Fung and Hsieh (2007)eight-factor model, and a model that selects relevant risk factors froman initial 23 risk factors for each strategy based on a stepwise regressionapproach. Consistent with Ammann et al.'s (2011) findings, we selectthe eight-factor model (henceforth referred to as FH8) presented inFung and Hsieh (2007) as our independent variables. More specifically,we use the trend following indices from Hsieh's web page,8 which areupdated on the 15th of each month, allowing us to use the data in thefollowing month's forecasting. The remaining FH8 variables we con-struct from data downloaded from Bloomberg9 and apply a one-month lag to allow for the data to be released. Our attempts to findcorresponding Asian equivalent variables for the FH8 variables proveddifficult, mainly due to an absence of history in the time series. The anal-ysis presented in this paper uses the original FH8 variables.

Because the original set of FH8 variables is not Asian focused,we tested an alternative set of independent variables using Asian

Page 5: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Table 3Regression statistics on hedge fund indices.

Statistic Hit rate1 Sum of squared errors (SSE)

Independent variable FH82 Lag 83 FH8 Lag 8

Methodology OLS NPR Simplex OLS NPR Simplex

Average4 0.723 0.735 0.740 0.045 0.052 0.040Convertible arbitrage 0.789 0.784 0.804 0.041 0.046 0.042Distressed debt 0.772 0.807 0.744 0.045 0.051 0.044Long short equity 0.620 0.620 0.643 0.080 0.134 0.065Equity market neutral 0.819 0.848 0.845 0.007 0.007 0.005Fixed income arbitrage 0.865 0.883 0.881 0.027 0.029 0.028Global macro 0.667 0.661 0.643 0.021 0.021 0.018Merger arbitrage 0.754 0.807 0.815 0.016 0.018 0.013CTA 0.497 0.468 0.548 0.120 0.113 0.106

Notes:1 Hit rate indicates the percentage of directionally accurate forecasts.2 In theOLS andNPR regressionswe use FH8 (Fung–Hsieh eight factormodel) variables

as independent predictors of hedge fund returns.3 In the simplex projection we use eight lags of monthly performance to produce a

forecast.4 The average row records the simple average of the eight CISDM index results.

Fig. 4. Fund life length.

Table 2Hedge fund indices statistics.

Index Annualized return Annualized risk⁎ Information ratio

Convertible arbitrage 9.83 5.66 1.74Distressed debt 10.95 5.49 2.00Long short equity 8.92 6.42 1.39Equity market neutral 8.80 2.09 4.22Fixed income arbitrage 8.90 4.54 1.96Global macro 8.71 3.52 2.48Merger arbitrage 8.50 2.87 2.96CTA 9.17 8.46 1.08

⁎ We use standard deviation of monthly total returns as our definition of risk.

193M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

macroeconomic variables. Although not presented here, the empiricalresults were similar to those obtained by using the FH8 variables.10

6. Methodology

Seeking to reconcile the opposing evidence found on tail risk, Cregoand Galvez (2014) show the presence of a common regime betweenhedge funds and the market index. The regime-switching approachesemployed by other researchers require the selection of a specificvariable(s) on which to define a state. One possible approach is theuse of nonparametric regression11 that allows the data to dynamicallydetermine the “appropriate” weight across a wider array of regime de-fining variables with more gradual changes in weights across time.DeFusco, McLeavey, Pinto, and Runkle (2001) suggest nonparametricprocedures can be used in three situations: (1) when the data analyzedare non-normally distributed; (2) when the data represent a series ofranks, and; (3) when the issue investigated does not concern a param-eter. Given the studies on hedge fund returns and tail risks mentionedearlier, we will investigate the value of nonparametric approaches foridentifying the existence of regimes for predicting hedge fund returns.

Nonparametric regression was used as an asset allocation techniquein Beckers and Blair (2002) to predict European stock and bond returnsusing a specific set of macroeconomic variables. They use the followingframework to determine the allocation weights to the assets in theiruniverse,

f tþ1 ¼ ∑Nj¼1 wt� jrt� jþ1

∑Nj¼1 wt� j

wherewt− j are theweights and rt− j+ 1 are returns up to and includingtime t.

We evaluate this approach to allocate first between hedge fund indi-ces and then individual hedge funds to create a fund of hedge funds. In anonparametric regression the value of the bandwidth selected deter-mines the tradeoff between the bias of the forecast, which increases asthe bandwidth increases, and the variancewhich decreases as the band-width increases. This critical issue of bandwidth selection in nonpara-metric regression is explained in Sheather (2004).

As an alternative to evaluating exogenous variables to predict hedgefund returns using OLS and a nonparametric regression, we also evalu-ate a nonparametric approach (simplex projection) using endogenousvariables as predictors of hedge fund returns. Sugihara and May(1990) introduce simplex projection for short-term forecasting of non-linear data. This approach does not rely on exogenous variables as pre-dictors, but instead uses a library of historical patterns in the dependenttime series as predictors. This approach is applied in commodity

10 The results are available from the authors.11 Early papers on nonparametric regression include Rosenblatt (1956), Parzen (1962),Watson (1964), Nadaraya (1964), and Sheather and Jones (1991).

markets in Agnon, Golan, and Shearer (1999) and in FX markets inRodríguez, Sosvilla-Rivero, and Andrada-Félix (2000). We apply thesimplex projection method (on hedge fund data) as an alternative tothe nonparametric regression. For each point we seek to forecast, wefind a simplex (i.e. a series of “nearest” points that surround) that con-tains this point, and use the points in the simplex to forecast the desiredpoint. If such a simplex is not found, we use a combination of anoninclusive simplex and near points to forecast. The forecast is derivedfrom μ observations of the historical time series, where the forecastpoint is an exponentially weighted average of the nearest neighbors,

ytþμ ¼∑jy jtþμe

�d j

∑je�d j

where j = 1,2 … E + 1 and dj is the distance between the points.

6.1. Ordinary least squares and nonparametric regressions

We initially evaluate the FH8 model presented in Fung and Hsieh(2007) as the independent variables in both the OLS and nonparametricregressions. Our dependent variables are the monthly total12 returns ofa filtered set of 1243 Asian hedge funds (henceforth referred to as the“total” return) described in Section 5.

We also record the results using two alternative sets of dependentvariables. Our second set, which we will refer to as an “alpha” return,

12 Hedge fund returns may include a benchmark hurdle rate for performance fees. Ouranalysis examines the monthly total return inclusive of a hurdle rate.

Page 6: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Table4

Performan

ceof

hedg

efund

inde

xpo

rtfolio

s.

Inde

pend

ent

Variables

Weigh

ting

Metho

dFu

ndSe

lection

Criteria⁎

Forecast

Metho

dology

Gross

Performan

ce,

%

Ann

ualiz

edGross

Performan

ce,%

Ann

ualiz

edGross

Risk

Net

Performan

ce,

%

Ann

ualiz

edNet

Performan

ce,%

Ann

ualiz

edNet

Risk

I.RTu

rnov

er,

%Tu

rnov

erPe

rAnn

um,%

Tran

saction

Cost,%

Tran

sactionCo

stPe

rAnn

um,%

FH8

Optim

izer

TOLS

184.3

12.3

4.0

182.6

12.2

4.5

2.75

164.3

18.3

1.6

0.2

Non

parametric

173.6

11.8

4.8

173.1

11.8

5.5

2.15

50.6

5.6

0.5

0.1

AOLS

172.3

11.8

4.1

171.8

11.7

4.6

2.5

54.2

6.0

0.5

0.1

Non

parametric

148.7

10.7

6.0

148.3

10.6

6.1

1.7

41.4

4.6

0.4

0.0

ROLS

191.4

12.6

4.3

189.8

12.5

4.7

2.7

165.9

18.4

1.7

0.2

Non

parametric

181.8

12.2

5.8

181.0

12.2

6.8

1.8

80.3

8.9

0.8

0.1

Kelly

TOLS

177.0

12.0

3.7

177.0

12.0

3.7

3.21

2.6

0.3

0.0

0.0

Non

parametric

177.2

12.0

3.8

177.2

12.0

3.8

3.16

2.3

0.3

0.0

0.0

AOLS

169.2

11.6

3.4

169.2

11.6

3.4

3.4

4.9

0.5

0.0

0.0

Non

parametric

172.5

11.8

3.5

172.4

11.8

3.5

3.3

2.6

0.3

0.0

0.0

ROLS

177.0

12.0

3.7

177.0

12.0

3.7

3.2

2.6

0.3

0.0

0.0

Non

parametric

177.2

12.0

3.8

177.2

12.0

3.8

3.2

2.3

0.3

0.0

0.0

Lag8

Optim

izer

Simplex

166.9

11.5

5.2

165.2

11.4

5.4

2.14

177.1

19.7

1.8

0.2

Kelly

177.2

12.0

3.6

177.1

12.0

3.6

3.34

2.6

0.3

0.0

0.0

Equa

l18

3.3

12.3

3.7

183.3

12.3

3.7

3.33

1.0

0.1

0.0

0.0

Notes:

Intheturnov

erco

lumn,

1.0sugg

ests

100%

turnov

erto

open

allp

ositions

,200

%turnov

erto

turn

portfolio

over

once

(i.e.sella

llpo

sition

san

dreop

enalln

ewpo

sition

s)issh

ownas

2.0in

thetable.

⁎“T”de

notesfund

selectionba

sedon

‘Total’R

eturnrank

ing.

“A”de

notesfund

selectionba

sedon

‘Alpha

’Returnrank

ing.

“R”de

notesfund

selectionba

sedon

“Risk-Adjus

tedTo

talR

eturn”

rank

ing.

Table 6Regression statistics on individual hedge funds using j month forecasts.

Forecast horizon,month

Forecastmethodology

Independentvariable

Hit ratea SSEb

2 OLS FH8c 0.57 0.44Nonparametric 0.54 0.37Simplex Lag 8d 0.56 0.37

3 OLS FH8 0.58 0.68Nonparametric 0.56 0.62Simplex Lag 8 0.57 0.60

6 OLS FH8 0.62 1.55Nonparametric 0.57 1.54Simplex Lag 8 0.61 1.36

12 OLS FH8 0.64 3.32Nonparametric 0.57 3.26Simplex Lag 8 0.64 2.80

24 OLS FH8 0.74 5.64Nonparametric 0.65 5.94Simplex Lag 8 0.74 4.63

a Hit rate indicates the percentage of directionally accurate forecasts.b SSE is the sum of squared errors.c In theOLS and NPR regressionswe use FH8 (Fung–Hsieh eight factormodel) variables

as independent predictors of hedge fund returns.d In the simplex projection we use eight lags of monthly performance to produce a

forecast.

Table 5Regression statistics on individual hedge funds using one-month forecasts.

Independent variable Forecast methodology Hit ratea SSEb

FH8c OLS 0.55 0.19Nonparametric 0.53 0.16

Lag 8d Simplex 0.55 0.16

a Hit rate indicates the percentage of directionally accurate forecasts.b SSE is the Sum of Squared Errors.c In the OLS and NPR regressions we use FH8 (Fung Hsieh eight factor model) variables

as independent predictors of hedge fund returns.d In the simplex projection we use eight lags of monthly performance to produce a

forecast.

194 M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

of dependent variables are the alphas derived from the regression of thehedge fund returns using the FH8 variables as the predictor variables(i.e. the regression intercept). If we consider the FH8 variables as prox-ies for themarket, then amanagermay demonstrate skill (alpha) in fac-tor selection in excess of the market return but not market timing. Our“Total Return” dependent variable captures the managers exposure tothe “market” and the manager's “excess (alpha) return” reflectingboth market timing ability and factor selection skill (or stock pickingability). Conversely the “alpha return” dependent variable seeks to cap-ture themanagers skill in factor selection in excess of themarket return,and hence excludes the manager's ability to market time. The third setof dependent variables is the “risk-adjusted” return where the time se-ries of “total” returns is divided by the standard deviation of “total”returns. This third dependent variable focuses the analysis onmanagersthat account for risk13 in their portfolio construction process rather than“purely” return.

After excluding data from 1994 to address backfill issues in theinitial universe, we then use data from January 1995 to December1996 to produce our first forecast in January 1997 using an OLSregression. We then expand the regression window each monthand produce subsequent out-of-sample OLS forecasts. Simultaneous-ly, for our nonparametric regression forecasts using the sameexpanding window as the OLS regression, we apply the regressionestimator proposed by Nadaraya (1964) and Watson (1964) which

13 Throughout this paper we use the standard deviation of monthly total returns as ourmeasure of risk.

Page 7: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Fig. 5. Graphical representation of simulation parameters.

195M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

we refer to as the un-adaptive14 Nadaraya–Watson estimators. Weevaluate the use of nonparametric regression (and simplex projec-tion) as an alternative to OLS regression to determine if we can pro-duce more accurate return forecasts by assigning heavier weights tomore relevant historic observations, and dynamically weighting theindependent series importance according to the current regime.The nonparametric functions provide a weighting structure to his-toric observations. The weighting structure assigns larger (smaller)weights to historic observations that have closer (further) proximityto the current observation. We evaluate if nonparametric methodscan exploit regime shifts better than OLS regression.

In formulating the nonparametric regression, Sheather (2004) notesthat it is generally accepted that the selection of the kernel function haslittle practical impact, while the selection of the bandwidth value is ofcritical importance. A number of possibilities exist for choosing thekernel function. Turlach (1993) provides a summary of some of thepossible choices. The two popular choices of kernel function areEpanechnikov (1969)15 and Gaussian kernel functions. In this paper,we selected the Gaussian function because it is the simplest and mostcommonly used function.

The value of the bandwidth selected determines the tradeoff be-tween the bias of the forecast, which increases as bandwidth increases,and the variance which decreases as bandwidth increases. Prior re-search to determine the “best”method of bandwidth selection for non-parametric regression forecasting is inconclusive. The four generalapproaches to bandwidth selection are (1) rule of thumb as suggestedby Silverman (1986), (2) plug-in method as suggested by Sheatherand Jones (1991), (3) cross validation method as suggested byBowman (1984), and (4) bootstrap method as suggested by Farawayand Jhun (1990).16 We selected the Silverman (1986) rule of thumb(0.9) method for bandwidth selection as a robust and commonly usedbandwidth selection method.

6.2. Nonlinear nonparametric approach—simplex projection

Both regression approaches use sets of exogenous independent var-iables as predictors of our hedge fund performance. As an alternative toboth approaches, we also implement a nonlinear nonparametric ap-proach (simplex projection) introduced by Sugihara and May (1990).Their approach captures a similar concept as the nonparametric regres-sion, relying on the principle that a period of time in the past is similar tothe pattern of returns in the current period, and therefore can be used as

14 In this analysis, we do not investigate adaptive estimators because of the intensiveprocessing power required to compute these forms of forecast, and are therefore unableto address the issue of the “curse of dimensionality” as noted by Sain (2002).15 Epanechnikov (1969) provides the following kernel: 3

�4ð1� z2Þwhere Ɨ(|z| ≤ 1); Ɨ(•)

is an indicator function, and z is a nonparametric estimator such as Nadaraya Watson(1964).16 A description of some of these methods and their implementation in statistical pack-ages is provided in Sheather (2004).

a predictor for the next period. Closely related to the nearest neighborapproach, we use eight17 lags for each fund to select the “minimal” vol-ume simplex18 that “contains” the point under consideration, and thenuse the points in the simplex to project a forecast of the desired point. Asbefore, we initiate our forecasts with the initial 24 months of observa-tions, and then produce subsequent forecasts with an expandingwindow of observations..19

6.3. Portfolio construction

After creating a time-series of return forecasts per fund using each ofthe forecastingmethods, we then evaluate themethodologies in a port-folio context. For each of our (total, “alpha”, or risk-adjusted) returnmethods, using the corresponding return we select the highest N fore-casts each period. In this analysis, we present results for a range ofvalues for N from 10 to 50 in incremental steps of 5. For each returnmethod,we use the correspondingN forecasts, and construct a portfoliousing threemethods— an optimized based portfolio, an equallyweight-ed portfolio, and a Kelly criterion-based portfolio.

For our Kelly criterion-based portfolio we estimate our forecast ac-curacy over a two-year window to generate our first Kelly score foreach fund in the top N forecasts. We set negative Kelly scores to zeroto avoid the implication of shorting a particular hedge fund. We thenscale the cross-sectional Kelly scores proportionately to the sum of thepositive Kelly scores to produce a portfolio holding. We then use anexpanding window to generate subsequent Kelly scores out of sample.

For the equally weighted portfolio, we simply set the portfolioweight to be 1/N for each of theN funds selectedwith the topN forecaststhat period.

For the optimized portfolio, we construct a portfolio using the Nforecasts. We construct a factor covariance matrix using the eightCISDM hedge fund indices (for the Fixed Income Arbitrage Index thestart date is January 1998) as risk factors, and our exposure of eachfund to the risk factors determined from a prior two-year OLS regres-sion, and set the target return to be the mean of the N forecasts (i.e.the same as the equally weighted portfolio). We maintain a rollingtwo-year window for determining a fund's exposure to the indices aswe move through the simulation. We constrain the optimization to en-sure a fully invested portfolio, but no other constraints such as holdingsize, or turnover are applied.

Given the requirement for a two-year window to determine theKelly scores, our first portfolio weight is therefore only generated in

17 We select eight lags based on Agnon et al. (1999) which suggests E-dimension (num-ber of lags) = four for quarterly data, and eight for monthly data.18 Aminimal ‘volume’ simplex is defined byAgnon et al. (1999) as the simplexwhere theratio of the ‘volume’ of the E + 1 simplex to the sum of all the other ‘volumes’ is exactlyone. Their approach differs slightly fromSugihara andMay (1990)who search for themin-imal diameter simplex.19 For a graphical representation of the simplex projection approach, see Hsieh, Glaser,Lucas, and Sugihara (2005).

Page 8: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Table 7Initial simulations.

Independentvariables

Forecastmethodology

Fundselectioncriteriaa

Weightingmethodology

Grossperformance,%

Annualized grossperformance, %

Annualizedgross Risk

Netperformance,%

Annualized netperformance, %

Annualizednet risk

I.R. Turnover,%

Turnover perannum, %

Transactioncosts, %

Transaction costsper annum,%

FH8 OLS T Optimizer 771 27.2 25.2 769 27.2 25.2 1.08 195.0 21.7 2.0 0.22Nonparametric 172 11.8 25.4 171 11.7 25.5 0.46 133.3 14.8 1.3 0.15OLS A 517 22.4 21.8 516 22.4 21.7 1.03 133.8 14.9 1.3 0.15Nonparametric 208 13.3 14.3 207 13.3 14.4 0.92 78.1 8.7 0.8 0.09OLS R 306 16.8 15.7 304 16.8 15.8 1.06 197.8 22.0 2.0 0.22Nonparametric 228 14.1 12.3 227 14.1 12.4 1.13 158.0 17.6 1.6 0.18OLS T Equal 1315 34.2 20.7 1313 34.2 20.7 1.65 145.6 16.2 1.5 0.16Nonparametric 375 18.9 18.5 375 18.9 18.6 1.02 11.4 1.3 0.1 0.01OLS A 528 22.6 14.3 527 22.6 14.3 1.58 33.2 3.7 0.3 0.04Nonparametric 242 14.6 10.6 242 14.6 10.6 1.38 9.6 1.1 0.1 0.01OLS R 403 19.7 9.8 401 19.6 10.2 1.92 154.8 17.2 1.5 0.17Nonparametric 319 17.3 7.8 319 17.2 7.9 2.17 47.0 5.2 0.5 0.05OLS T Kelly 1342 34.5 18.4 1341 34.5 18.7 1.84 154.8 17.2 1.5 0.17Nonparametric 313 17.1 20.0 312 17.0 20.1 0.85 19.4 2.2 0.2 0.02OLS A 689 25.8 17.1 689 25.8 17.5 1.48 44.5 4.9 0.4 0.05Nonparametric 237 14.5 12.7 237 14.5 12.7 1.14 17.5 1.9 0.2 0.02OLS R 369 18.7 9.2 367 18.7 9.7 1.93 149.7 16.6 1.5 0.17Nonparametric 335 17.7 6.3 334 17.7 6.4 2.75 56.0 6.2 0.6 0.06

Lag8 Simplex Optimizer 931 29.6 28.1 929 29.6 28.0 1.06 190.3 21.1 1.9 0.21Equal 765 27.1 22.1 764 27.1 22.1 1.22 126.4 14.0 1.3 0.14Kelly 1748 38.3 22.2 1747 38.3 22.1 1.74 153.2 17.0 1.5 0.17

Notes* “T” denotes fund selection based on ‘Total’ Return ranking. “A” denotes fund selection based on ‘Alpha’ Return ranking. “R” denotes fund selection based on “Risk-adjusted total return” ranking.Table 7 reports performance of simulationwith the following settings: portfolio holdings= 10, implementation lag= 0months, rebalancing frequency=1month. Results from other simulationswith combinations of portfolio holdings between 10and 50 in increments of 5, rebalancing frequencies of 1, 2, 3, 6, 12 and 24 months, implementation lags of 0,1,2, and 3 months are available upon request.

196M.Subbiah,F.J.Fabozzi/InternationalReview

ofFinancialAnalysis

45(2016)

189–201

Page 9: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Table 8Kelly based simplex forecasts, altering the portfolio size.

Portfoliosize, n

Top 1:nth fund: Portfolio simulation Alphacaptured

Average return, %,per annum I.R.

Net annualizedreturn, % I.R.

10 64.5 1.71 38.26 1.74 59%15 56.1 4.91 26.88 1.34 48%20 50.8 2.6 22.75 1.16 45%25 47.2 1.35 21.59 1.17 46%30 44.3 2.18 20.69 1.13 47%35 41.8 1.85 20.83 1.17 50%40 39.9 1.68 20.81 1.20 52%45 38.3 2.87 20.23 1.20 53%50 37 1.71 19.36 1.17 52%

197M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

January 2000, using the 199820–1999 forecasts to determine the firstKelly weighted portfolio in January 2000.

We also compare the effect of different rebalancing frequencies. Wepresent results for rebalancing at 1, 2, 3, 6, 12, and 24months. Given thenecessity to create 24-month forecasts for the Kelly weighted portfolio,we then have an overlapping window to compare all the portfolioscreated between January 2002 and December 2012. At rebalancing fre-quencies of greater than onemonth, we initiate the rebalancing at eachof the valid startingmonths, rebalance accordingly, and then report sta-tistics based on the average of themultiple simulations. For example, fora quarterly rebalance simulation, we initiate three simulations with astarting date in January, February and March, and then report theaverage results of the three simulations.21 We apply a 1% transactioncost22 for turnover, and report the results accordingly. We also reportthe results of a lagged implementation. Having created forecasts, wethen lag the implementation of the forecasts in the portfolio construc-tion process by 1, 2 and 3 months and report these results. We alsoshow the results where the number of funds held in the portfolio isset at 10 to 50 funds (in increments of five funds).

7. Results

In this section, we report our empirical findings.

7.1. Index level

Using the CISDM index classifications, Table 2 provides index-levelstatistics over the period January 2000–December 2013. The indicesare Convertible Arbitrage Index, Distressed Debt Index, Long ShortIndex, Equity Market Neutral Index, Fixed Income Arbitrage Index,Global Macro Index, and CTA Index.

In Table 3 we report for our three hedge fund return forecastingmethodologies, a “hit rate” statistic which indicates the percentage ofdirectionally accurate forecasts and the sum of squared errors (SSE)for each method. Our independent variables for the regressions arethe FH8 variables. The simplex projection uses eight lags of the depen-dent series as predictors. Table 3 provides the statistics for each of theeight CISDM hedge fund indices and the average statistic for each ofthe eight indices.

From the results reported in Table 3 we can observe a higher hit rateusing the nonparametric methodology compared to OLS (FH8: 74% vs72%) is achieved, with the simplex approach producing a similar 74% hitrate.

Using FH8 as independent variables,we can see fromTable 3 that theOLS on average produces a lower SSE than NPR, which holds for all theindices except CTA. The simplex approach generally produces lowerSSEs than both OLS and NPR formost indices. The table does not includethe results for hit rates and SSEs for the alpha forecast or risk-adjustedforecast compared to the observed “total” return. However, the fore-casts using the “alpha” and “risk-adjusted” return using both OLS andNPRwith FH8 as predictor variables are included as a basis for selectionin the portfolio construction process, thereby providing an additionaleight sets of forecasts.

We use the following seven sets of forecasts as our fund selectioncriteria:

1. OLS regression of total return with FH8 predictors.2. Nonparametric regression of total return with FH8 predictors.3. OLS regression of “alpha” return with FH8 predictors.4. Nonparametric regression of “alpha” return with FH8 predictors.

20 This is consistent with Fixed Income Arbitrage Index start date in 1998.21 We found that there is very little difference between simulations with alternative ini-tial start dates.22 Some hedge funds charge a subscription/redemption fee payable to the fund to pro-tect existing investors from the cost of trading underlying positions induced by subscrip-tions/redemptions. We apply a 1% transaction cost to allow for such charges.

5. OLS regression of risk-adjusted total return with FH8 predictors.6. Nonparametric regression of risk-adjusted total return with FH8

predictors.7. Simplex projection.

Using each forecast, we construct a backtest to assess the perfor-mance of each strategy.Weuse three portfolioweighting schemesmen-tioned earlier, rebalancing on a monthly basis. The results are reportedin Table 4.

The analysis produces similar net annualized performance perannum of around 12%23 from all three methods using the total returnas the fund selection criteria with I.R's of around 3 in many cases. Weobserve that the alpha return based portfolios generally produce inferi-or returns compared to their equivalent total return based portfolio, al-though on a risk-adjusted basis, the I.R's of the alpha return basedportfolios using the Kelly criteriaweighting generally produce better re-sults than their analogous total return based portfolios. The risk-adjusted total return portfolios produce slightly better annualized netperformance compared to the total return portfolios, with slightly infe-rior I.R's for the OLS regressions and slightly better I.R's for the nonpara-metric regressions. Given the very similar performance and risk levelsfrom the “risk” indices shown in Table 2,we analyze all themethods fur-ther at the individual fund level.

7.2. Fund level

To analyze the data at the individual hedge fund level we use the1243 individual hedge fund returns from the three databases as the de-pendent variable. Table 5 shows the OLS and nonparametric regressionsresults using total return as the dependent variables with the FH8 vari-ables as the independent variables and the simplex projection usingeight lags.

We find that the results for the nonparametric methodology pro-duce more accurate (lower SSE) forecasts than the OLS (0.156 versus0.188). However, directional accuracy produces a lower hit rate underthe nonparametric than the OLS method. The SSE results favor the useof nonparametric methods over OLS but the lower hit rates do not sug-gest the methods to be conclusively superior.

Because a monthly rebalance frequency may be unrealistic in prac-tice, we also report regression statistics using forecasts over differenthorizons for the three methodologies. Reported in Table 6 are statisticsfor j month forecasts, where j = 2, 3, 6, 12 and 24 months with corre-sponding jmonth rebalancing frequency. Not surprisingly, as the rebal-ance frequency extends, we find that the hit rate for all threemethodologies increases given the positive bias in the return series.We observe that as the rebalance frequency increases from 1 monthto 12months, the percentage of positive returns in the underlying seriesincreases from 60% to 69%, with the median monthly return increasing

23 Therewere 345 individual hedge funds that have greater than 12%net annualized per-formance over their lifetime.

Page 10: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Fig. 7. Information ratio as holdings increase.

Fig. 6. Performance as holdings increase.

198 M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

from 0.64% to a median annual return of 7.67%. As with the one-monthforecasts in Table 5 extending the rebalancing frequency, we continueto observe the hit rate from the nonparametric regression and simplexprojection compared toOLS is inferior, and even though the SSE natural-ly becomes larger, the SSE's continue to consistently favor the nonpara-metric regression over the OLS (except at a forecast horizon of24 months). This finding suggests that more accurate forecasts are pro-duced by applying the nonparametric regression up to horizons of12 months. Similarly, the simplex projection produces a SSE similar tothe nonparametric regression and better than the OLS regression.

We then created more than 5000 simulated portfolios using combi-nations of forecastmethodologies and portfolio constructionmethodol-ogies. For the portfolio methodologies we also vary the followingparameters – number of portfolio holdings, and portfolio horizon/rebal-ance frequency. We created portfolios where the forecasts used in theportfolio construction are lagged by 1, 2 or 3 months.

The following lists the combinations of parameters that wereinvestigated:

• Three sets of dependent variables: total return, “alpha” return andrisk-adjusted total return.

• Three forecast methodologies: OLS regression (OLS, nonparametricregression (NPR), and simplex projection (SP)

• Three portfolio construction/weighting methodologies: optimizer(Opt), Equally weighted (Eql), and Kelly criterion (Kel)

• Number of portfolio holdings: from 10 to 50 funds in increments of 5funds.

• Portfolio horizon/rebalance frequency: 1, 2, 3, 6, 12 and 24 months.• Lagged implementation - 0, 1, 2 and 3 months.

Fig. 5 graphically shows the parameter selections for dependent var-iables, forecast method, portfolio constructionmethod, number of port-folio holdings, portfolio horizon, and implementation lag.

Combinations of these six parameters in the simulations lead to over5000+ simulations. (The results from the 5000+ simulations are avail-able upon request.) In this paper, for conciseness, we report in Table 7only the results of the first simulation where the portfolio holdings areset to 10 (ourmost focused set), with a monthly rebalancing frequency,and without any lag.

The results show superior I.R.'s using NPR to predict the “alpha” re-turn versus the respective predictions of the total return, while the op-posite is true for the OLS regression. We can infer that the source of thealpha return is more time (regime) dependent and the NPR is a bettermethod for predicting this component of the return. As with the indexlevel observations, we also note the annualized net performance ofthe “alpha” return portfolios are generally inferior to their equivalent“total” return portfolios.

We suggest the following possible interpretation of this observation:expanding on our earlier proposition: if we consider the FH8 variablesas proxies for the market, then a manager may demonstrate skill(alpha) in factor selection in excess of themarket return but notmarkettiming. The superior result using NPR methodology on the “alpha” re-turn, may assist in exploiting the regime switches captured in theman-agers' nonsystematic returns. As the manager is unable to capturemarket timing, the NPR methodology using the total return fails to ex-ploit the systematic portion of the fund's total return. For example, inthe case of a long-short equity hedge fund: a manager's skill inswitching styles (e.g. value to growth) or sector rotation through time(or regimes) is best forecasted using the NPR methodology. Converselythe market timing portion of the manager's return is best forecastedusing the betas from the OLS regression. As the “alpha” based portfolioreturns are inferior to the total return based portfolios, we infer that amanager's ability to accurately forecast style is overshadowed by themanager's poormarket timing ability and the returns from poormarkettiming dominate the style return.

The I.R using risk-adjusted total return with the NPR methodologyand FH8 predictors is superior using all three portfolio constructionmethods than the corresponding OLS methodology. The I.R. using risk-adjusted total return with the NPR methodology and FH8 predictors isalso superior to the alternative dependent variables of “alpha” and“total” return. When analyzing mutual fund performance Zhao (2006)finds that managers with a quantitative process exhibit higher Sharperatios than traditional managers. Extrapolating this conclusion ontoour hedge fund sample, we might conclude that a fund manager withhigher risk-adjusted total returnsmay be following a systematic invest-ment process, and hence the NPRmethodology is able to exploit this re-peatable performance and deliver superior I.R.'s relative to theequivalent OLS simulation.

The Kelly portfolio construction method with FH8 predictors, “risk-adjusted total return” dependent variables and NPR methodology pro-duces the highest I.R. (2.75) of any of the portfolios with 10 holdings.The 2.75 I.R. compares favorably to the 1.34 and 1.71 I.R.'s of the firstand tenth best performing funds in the universe observed inTable 1(a). The net annualized performance is only superior using therisk-adjusted total return compared to the alternative two dependentreturns in the case of a nonparametric Kelly combination or a nonpara-metric optimizer combination. The remaining combinations of forecastmethod (OLS or NPR) and portfolio construction method (equal, Kellyor optimized) all produce inferior net annualized performance usingthe risk-adjusted total return compared to total return, but theselower returns are more than offset by the reduction in risk to producesuperior I.R.'s.

The choice of portfolio construction methodology favors the use ofthe Kelly criterion method with superior net annualized performanceand generally better I.R.'s from this method compared to equallyweighted and optimized portfolios.

The nonparametric simplex projection also produces superior netannualized performance and I.R.'s using the Kelly criterion comparedto the equal and optimized weighting methods. We have used eight

Page 11: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Fig. 8. Lagged implementation effect across weighting methods based on simplexprojection forecasts.

199M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

lags of the dependent variables as predictors in the simplex projection.The good performance from the use of the lagged variables as predictorsmay be explained by Getmansky et al. (2004) that serial correlation inthe returns reflects illiquidity exposure. This conclusion is also support-ed by Anson (2011) who provides a framework for measuring liquidityrisk and calculating a premium for that risk across a variety of assets. In astudy of private equity funds, Anson (2013) notes that because beta co-efficients are linearly additive, the inclusion of lagged returns shouldlead to a better estimate of alpha, and concludes that the alpha interceptthat measures the manager skill declines significantly when laggedmarket returns are included in the regression.

In Table 8we expand on the last rowof Table 7. Table 8 reports resultsfrom the simplex forecasts with a Kelly criterion portfolio constructionmethod, varying the number of holdings in the portfolio from 10 to 50by increments of 5. As a basis of comparison we also report fromTable 1(a), the top n funds average performance and I.R., and observethat the simulations capture 45%–60% of the available performance.

The results in Table 8 show the portfolios with less than 20 fundsproduce performance in excess of 25% with a I.R. greater than 1.3; theperformance of portfolios with 20 to 50 funds plateaus around net an-nualized performance of 20% with a I.R. of approximately 1.17, suggest-ing that diversification benefits are subdued for portfolios with morethan 20 funds.

Fig. 6 shows the performance of all three forecasting methodologiesas the number of holdings increase. We report the Kelly criterionweighting methodology and FH8 as predictor variables in the regres-sions with the total return as the dependent variable. Consistent withthe results reported in Table 8, we observe in Fig. 6 similar plateaus inperformance using the OLS forecasts and the simplex projection fore-casts, and a generally similar performance for NPR forecasts regardlessof portfolio size.

In Fig. 7 we report the I.R. for all three forecasting methodologies asthe number of holdings increases. We use the Kelly criterion weightingmethodology in this figure. The variation of the I.R with holding sizesupports the findings in Fig. 6 using the net annualized performance.

Using the simplex forecasts with portfolio holdings set to 10, andmonthly rebalance frequency,

Fig. 8 shows the effect of lagging the implementation of the forecastin the portfolio construction process by 1, 2 and 3 months respectivelyacross all three weighting methodologies.

Fig. 8 shows the natural decay of lagged implementation. One possi-ble explanation for the kink observed in the month 2 lag together withthe 1 month delay we applied for data collection in the analysis is thatthere is a quarterly mean-reversion effect in the data.

Using the Kelly criterion methodology as a portfolio constructiontechnique, with portfolio holdings set to 10, andmonthly rebalance fre-quency and total return as the dependent variable, Fig. 9 shows the ef-fect of lagging the implementation of the forecast in the portfolioconstruction process by 1, 2 and 3 months respectively across all threeforecasting methodologies using both FH8, PC7 and simplex projectionas independent predictors for the regression methods.

As with Fig. 8, Fig. 9 shows a natural decay with increasing laggedimplementation using all three forecast methodologies, with a kink ob-served for the month 2 lag, which we again attribute to a quarterlymean-reversion effect.

Again, using the Kelly criterionmethodology as a portfolio construc-tion technique with portfolio holdings set to 10, andmonthly rebalancefrequency with FH8 variables as the predictors, Table 9 reports the im-pact of extending the initial training window from two years to fiveyears in annual increments,24 using the three sets of returns (total,alpha and risk-adjusted total) under consideration as the dependentvariable/fund selection criteria.

24 Extending our initial training window reduces the universe by approximately 200funds for each incremental year.

We observe from Table 9 that as the initial trainingwindow for gen-erating forecasts expands, using OLS regression, annualized perfor-mance (gross and net) and I.R.'s of portfolios decrease using forecastsgenerated from both “total” and “alpha” returns. Conversely, usingNPR, as the initial training window expands, both performance andI.R.'s generally increase, supporting the explanation that the NPRmeth-odology in this analysis is hindered by the “curse of dimensionality”problem. Using risk-adjusted total returns, both OLS and NPR simula-tions at each training window from two years to five years produceI.R.'s that are superior to the equivalent simulationsusing either total re-turn or “alpha” return.

8. Conclusions

In this paper, we present evidence that nonparametric methodolo-gies are capable of exploiting the time varying “alpha” component inAsian hedge funds returns. The beta components of fund returns canbe captured through an OLS regression using the Fung–Hsieh eight-factor variables as predictors. The simulated performance of the“alpha” portfolios versus the “total” return portfolios suggests that thebeta components of a hedge fund's performance dominate the alphacomponents. Using the Fung–Hsieh eight-factor variables as predictors,we generally find that higher I.R.'s can be achieved using “risk-adjustedtotal returns” compared to either “total” or “alpha” returns for eachanalogous set of portfolio construction methods (Kelly, equally weight-ed or optimized) and forecast methodology (NPR or OLS) combination.We also find that for each portfolio constructionmethod, predicting therisk-adjusted returns using theNPRmethodology generates higher I.R.'sthan the equivalent OLS simulation.

We find evidence to support the following view — a nonparametricregression as a methodology to construct an Asian fund of hedgefunds may suffer from the curse of dimensionality problem as the

Fig. 9. Lagged implementation effect across forecast methods based on Kelly Criterionportfolio.

Page 12: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

Table 9Extending the training window.

ForecastMethodology

Fund SelectionCriteria*

Training WindowLength, years

GrossPerformance,%

Annualized GrossPerformance, %

AnnualizedGross Risk

NetPerformance,%

Annualized NetPerformance, %

AnnualizedNet Risk

I.R Turnover,%

Turnover PerAnnum, %

TransactionCosts, %

Transaction CostsPer Annum,%

OLS regression

T 2 1342 34.52 18.45 1341 34.50 18.73 1.84 154.80 17.20 1.55 0.17T 3 1131 32.17 16.46 1129 32.15 16.46 1.95 150.86 16.76 1.51 0.17T 4 787 27.45 15.61 786 27.42 15.64 1.75 148.77 16.53 1.49 0.17T 5 713 26.22 15.35 712 26.20 15.56 1.68 143.07 15.90 1.43 0.16A 2 689 25.80 17.09 689 25.79 17.46 1.48 44.52 4.95 0.45 0.05A 3 360 18.48 14.15 360 18.47 14.20 1.30 35.41 3.93 0.35 0.04A 4 351 18.21 14.36 350 18.20 14.62 1.25 35.19 3.91 0.35 0.04A 5 286 16.20 16.93 286 16.18 17.02 0.95 36.63 4.07 0.37 0.04R 2 369 18.73 9.24 367 18.69 9.66 1.93 149.73 16.64 1.50 0.17R 3 343 17.99 6.79 342 17.95 7.32 2.45 132.90 14.77 1.33 0.15R 4 271 15.69 6.47 270 15.64 7.05 2.22 125.38 13.93 1.25 0.14R 5 293 16.42 8.56 292 16.38 8.58 1.91 129.05 14.34 1.29 0.14

Nonparametric Regression

T 2 313 17.05 19.98 312 17.05 20.11 0.85 19.38 2.15 0.19 0.02T 3 344 18.01 19.98 344 18.01 20.09 0.90 20.10 2.23 0.20 0.02T 4 372 18.81 19.63 371 18.80 19.87 0.95 22.41 2.49 0.22 0.02T 5 355 18.34 19.57 355 18.33 19.80 0.93 25.04 2.78 0.25 0.03A 2 237 14.46 12.72 237 14.46 12.72 1.14 17.55 1.95 0.18 0.02A 3 219 13.77 10.96 219 13.76 10.99 1.25 13.23 1.47 0.13 0.01A 4 287 16.24 10.37 287 16.24 10.32 1.57 15.17 1.69 0.15 0.02A 5 451 20.89 16.18 451 20.88 16.23 1.29 20.74 2.30 0.21 0.02R 2 335 17.74 6.30 334 17.73 6.44 2.75 55.96 6.22 0.56 0.06R 3 233 14.32 4.82 233 14.30 4.94 2.90 43.39 4.82 0.43 0.05R 4 190 12.57 6.45 190 12.55 6.60 1.90 43.95 4.88 0.44 0.05R 5 229 14.14 8.29 228 14.13 8.43 1.67 44.07 4.90 0.44 0.05

Note: * “T” denotes fund selection based on ‘Total’ Return ranking. “A” denotes fund selection based on ‘Alpha’ Return ranking. “R” denotes fund selection based on “Risk-adjusted total return” rankings.

200M.Subbiah,F.J.Fabozzi/InternationalReview

ofFinancialAnalysis

45(2016)

189–201

Page 13: Hedge fund allocation: Evaluating parametric and ... · forecasts using alternative portfolio construction techniques ... trillion asset under management managed by 11,000 funds by

201M. Subbiah, F.J. Fabozzi / International Review of Financial Analysis 45 (2016) 189–201

majority of hedge funds in Asia do not have sufficiently long history tovalidate the methodology.

We find that using the Kelly criterion portfolio construction methodgenerally produces a slightly improved performance compared to theequally weighted portfolio construction method. Conversely a basicoptimization portfolio construction process, using the CISDM data as abasis for risk model, is unable to produce superior performancecompared to an equally weighted portfolio. Of course, there are also amyriad of optimization enhancements that could be evaluated.

The backtests reported in this paper generally produce good results,with fund level backtests producing net annualized returns in the rangeof 20% to 40%, and index level back tests of 12% compared to the individ-ual hedge fund index returns of below 10% per annum. This finding sup-ports the conclusion of Anand et al. (2011) on the importance of tacticalasset allocation at the fund level to enhance returns.

Using the Fung–Hsieh eight-factor variables as predictors of individ-ual hedge fund returns, we find that the OLS forecasts produce higherdirectional accuracy than nonparametric methods but nonparametricmethods produce more accurate (better sum of squared errors) fore-casts compared to the forecasts generated by OLS. In backtests, thehighest I.R. to predict hedge fund returns is achieved using a combina-tion of the OLS regression with the Fung–Hsieh eight-factor variablesas predictors through the Kelly criterion portfolio construction method.We also find that the combination of nonparametric regression usingthe Fung–Hsieh eight-factor model variables as predictors of risk-adjusted returns with the Kelly criterion portfolio construction methodproduces the best I.R. The benefits of diversification are observed to pla-teauwith portfolios containingmore than 20 hedge funds. These resultsgenerally hold with portfolio implementation lags up to 12 months.

Acknowledgments

The authors are grateful to Bing Liang of the University of Massachu-setts and René Garcia of EDHEC Business School for their valuablefeedback on earlier versions of this paper.

References

Agarwal, V., & Naik, N. (2004). Risks and portfolio decisions involving hedge funds. Reviewof Financial Studies, 9, 63–98.

Agarwal, V., Bakshi, G. S., & Huij, J. (2009). Do higher-moment equity risks explain hedgefund returns? 1st Annual Conference on Econometrics of Hedge Funds, Research PaperNo. RHS 06–066. Robert H. Smith School, University of Maryland.

Agnon, Y., Golan, A., & Shearer, M. (1999). Nonparametric, nonlinear, short-term forecast-ing: Theory and evidence for nonlinearities in the commodity markets. EconomicLetters, 65, 293–299.

Ammann, M., Schmid, M., & Huber, O. (2011). Has hedge fund alpha disappeared? Journalof Investment Management, 9(1), 50–71.

Anand, G., Kutsarov, I., Maier, T., & Storr, M. (2011). Importance of tactical strategy allocationon fund-of-hedge-funds allocations. Journal of Wealth Management, 14(2), 49–58.

Anand, G., Kutsarov, I., Maier, T., & Storr, M. (2013). The influence of macroeconomic andbehavioral factors on tactical strategy allocation (TSA) for funds of hedge funds.Journal of Wealth Management, 16(2), 63–76.

Anson, M. (2011). Measuring a premium for liquidity risk. Journal of Private Equity, 13(2),6–16.

Anson, M. (2013). Performance measurement in private equity: another look at thelagged beta effect. Journal of Private Equity, 17(1), 1–16.

Asness, C., Krail, R., & Liew, J. (2001). Do hedge funds hedge? Journal of PortfolioManagement, 28(1), 6–19.

Bali, T. G., Brown, S. J., & Caglayan, M. O. (2012). Systematic risk and the cross section ofhedge fund returns. Journal of Financial Economics, 106(1), 114–131.

Beckers, S., & Blair, B. (2002). Nonparametric forecasting for conditional asset allocation.Journal of Asset Management, 3(3), 213–228.

Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing ofkernel density estimates. Biometrika, 71(2), 353–360.

Breiman, L. (1961). Optimal gambling systems for favorable games. Fourth Berkeley Sympo-sium on Mathematical Statistics and Probability. Vol. 1., . Berkeley, CA.: Univ. Calif. Press.

Crego, J. A., & Galvez, J. (2014). Hedge funds and asset markets: Tail or two state depen-dence? CEMFI Working Paper Series.

DeFusco, R., McLeavey, D., Pinto, & Runkle, D. E. (2001). Quantitative methods for invest-ment analysis. Baltimore, MD: United Book Press, Inc.

Delevingne, L. (2014). Hedge fund industry snapshot: $2.6 trillion in 11,000 funds. CNBC.com Available at http://www.cnbc.com/2014/08/29/industry-snapshot-26-trillion-in-11000-funds.html.

DeMiguel, V., Garlappi, L., & Uppal, R. (2009). Optimal versus naive diversification: Howinefficient is the 1/N portfolio strategy? Review of Financial Studies, 22(5), 1915–1953.

Edwards, F., & Caglayan, M. (2001). Hedge fund performance and manager skill. Journal ofFutures Markets, 21(11), 1003–1028.

Epanechnikov, V. A. (1969). Nonparametric estimation of a multidimensional probabilitydensity. Theory of Probability and its Applications, 14(1), 153–158.

Ethier, S. N. (2004). The Kelly system maximizes median fortune. Journal of AppliedProbability, 41(5), 1230–1236.

Faraway, J., & Jhun, M. (1990). Bootstrap choice of bandwidth for density estimation.Journal of the American Statistical Association, 85(412), 1110–1122.

Fung, W., Hsieh, D., Naik, N., & Ramadorai, T. (2008). Hedge funds: Performance, risk andcapital formation. Journal of Finance, 63(4), 1777–1803.

Fung, W., & Hsieh, D. (1997). Empirical characteristics of dynamic trading strategies:Thecase of hedge funds. Review of Financial Studies, 10, 275–302.

Fung, W., & Hsieh, D. (2000). Performance characteristics of hedge funds and commodityfunds: Natural vs. spurious biases. Journal of Financial and Quantitative Analysis, 35(3),291–307.

Fung,W., & Hsieh, D. (2001). The risk in hedge fund strategies: Theory and evidence fromtrend followers. Review of Financial Studies, 14, 313–341.

Fung, W., & Hsieh, D. (2002). The risk in fixed-income hedge fund styles. Journal of FixedIncome, 12(2), 6–27.

Fung, W., & Hsieh, D. (2004). Hedge fund benchmarks: A risk based approach. FinancialAnalysts Journal, 60(5), 65–80.

Fung, W., & Hsieh, D. (2004b). The risk in hedge fund strategies: Theory and evidencefrom long/short equity hedge funds. Working Paper, London Business School.

Fung, W., & Hsieh, D. (2007). Will hedge funds regress towards index-like products?Journal of Investment Management, 5(2), 56–80.

Getmansky, M., Lo, A., & Makarov, I. (2004). An econometric model of serial correlationand illiquidity in hedge fund returns. Journal of Financial Economics, 74, 529–609.

Hsieh, C. H., Glaser, S. M., Lucas, A. J., & Sugihara, G. (2005). Distinguishing random envi-ronmental fluctuations from ecological catastrophes for the North Pacific Ocean.Nature, 435, 336–340.

Kelly, B. T., & Jiang, H. (2012). Tail risk and hedge fund returns. Chicago Booth ResearchPaper No. 12-44. Fama-Miller Working Paper. University of Chicago.

Kelly, J. L. (1956). A new interpretation of information rate. Bell System Technical Journal,35(4), 917–926.

Lack, S. (2012). The hedge fund miracle. Hoboken, NJ: John Wiley & Sons.Laureti, P., Medo, M., & Zhang, Y. C. (2010). Analysis of Kelly-optimal portfolios.

Quantitative Finance, 10(7), 689–697.Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.Maslov, S., & Zhang, Y. C. (1998). Optimal investment strategy for risky assets.

International Journal of Theoretical and Applied Finance, 1(3), 377–387.Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and its Applications,

9(1), 141–142.Parzen, E. (1962). On estimation of a probability density function and mode. Annals of

Mathematical Statistics, 33(3), 1065–1076.Rodríguez, F., Sosvilla-Rivero, S., & Andrada-Félix, J. (2000). Technical analysis in foreign

exchange markets: Linear versus nonlinear trading rules. Working Paper on Interna-tional Economics and Finance 00-02. FEDEA.

Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function.Annals of Mathematical Statistics, 27(3), 832–837.

Sadka, R. (2010). Liquidity risk and the cross-section of hedge-fund returns. Journal ofFinancial Economics, 98(1), 54–71.

Sain, S. R. (2002). Multivariate locally adaptive density estimation. ComputationalStatistics & Data Analysis, 39(2), 165–186.

Sharpe,W. F. (1992). Asset allocation:Management style and performancemeasurement.Journal of Portfolio Management, 18(1), 7–19.

Sheather, S. J., & Jones, M. C. (1991). A reliable data-based bandwidth selection methodfor kernel density estimation. Journal of the Royal Statistical Society, Series B, 53(3),683–690.

Sheather, S. J. (2004). Density estimation — statistical science. Institute of MathematicalSciences, 19(4), 588–597.

Silverman, B. W. (1986). Density estimation. London: Chapman and Hall.Sugihara, G., & May, R. M. (1990). Nonlinear forecasting as a way of distinguishing chaos

from measurement error in time series. Nature, 344, 734–741.Teo, M. (2009). The geography of hedge funds. Review of Financial Studies, 22(9),

3531–3561.Titman, S., & Tiu, C. (2011). Do the best hedge funds hedge? Review of Financial Studies,

24(1), 123–168.Turlach, B. (1993). Bandwidth selection in kernel density estimation: A review. Dis-

cussion paper 9307. Institut für Statistik und Ökonometrie, Humboldt-Universitätzu Berlin.

Watson, G. S. (1964). Smooth regression analysis. Sankhyā Ser. A, 26(15), 175–184.Zhao, J. (2006). Quant jocks and tire kickers: Does the stock selection process matter? (PhD

dissertation) University of Arizona.


Recommended