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## Abstract

This article analyzes the portfolio management implications of using drawdown-based measures in allocation decisions. The authors introduce *modified conditional expected drawdown* (MCED), a new risk measure that is derived from portfolio drawdowns, or peak-to-trough losses, of demeaned constituents. They show that MCED exhibits the attractive properties of coherent risk measures that are present in conditional expected drawdown (CED) but are lacking in the historical maximum drawdown (MDD) commonly used in the industry. This article introduces a robust block bootstrap approach to calculating CED, MCED, and marginal contributions from portfolio constituents. First, the authors show that MCED is less sensitive to sample error than CED and MDD. Second, they evaluate several drawdown-based minimum risk and equal-risk allocation approaches within the large-scale simulation framework of Molyboga and L’Ahelec via a subset of hedge funds in the managed futures space that contains 613 live and 1,384 defunct funds over the 1993–2015 period. The authors find that the MCED-based equal-risk approach dominates the other drawdown-based techniques but does not consistently outperform the simple equal volatility–adjusted approach. This finding highlights the importance of carefully accounting for sample error, as reported by DeMiquel et al., and cautions against overreliance on drawdown-based measures in portfolio management.

Institutional investors make investment decisions based on a variety of measures of risk and risk-adjusted performance, with *maximum historical drawdown*, defined as the largest peak-to-valley loss, among the most popular measures. In fact, Greenwich Roundtable’s *Best Practices in Alternative Investments: Due Diligence* [2010] require drawdown analysis as part of quantitative due diligence.

The purpose of this article is to evaluate portfolio implications of using drawdowns in allocation decisions by examining several established and new drawdown-based approaches. We introduce a new drawdown-based measure, *modified conditional expected drawdown* (MCED), and demonstrate its attractive characteristics, suggest a robust block bootstrap approach for its calculation, and investigate the portfolio implications of utilizing this measure. We evaluate MCED using the large-scale simulation framework and the realistic constraints Molyboga and L’Ahelec [2016] imposed on a subset of hedge funds in the managed futures space that contains 613 live and 1,384 defunct funds over the 1993–2015 period. We find that the MCED-based equal-risk approach dominates the other drawdown-based techniques while underperforming the equal volatility–adjusted approach^{1} (EVA) highlighted by Molyboga and L’Ahelec [2016].

Goldberg and Mahmoud [2014] formalized drawdown risk as *conditional expected drawdown* (CED), the tail mean of maximum drawdown (MDD) distributions; showed that CED is a coherent measure as defined by Artzner et al. [1999]; and investigated two portfolio construction approaches that either minimize CED or equalize constituent contributions to portfolio CED. The former approach represents a minimum risk approach while the latter is a variation of risk-parity introduced by Qian [2006]. The historical performance of portfolio constituents is factored into the CED calculation through the constituents’ total cumulative performance, or the slope of the cumulative return function, and their contribution to portfolio performance during bad periods. Although contribution to portfolio performance is likely to detect diversifying portfolio constituents, incorporating total cumulative performance reflects performance chasing and is likely to have negative implications. We modify CED to capture its diversifying characteristics while eliminating performance chasing aspects and refer to the new risk measure MCED. This modification eliminates the slopes of individual portfolio constituents by demeaning returns while preserving all the attractive properties of CED documented by Goldberg and Mahmoud [2014]. We demonstrate that MCED is less sensitive to sample error than CED and maximum drawdown (MDD), using 1,000 simulations with five hypothetical portfolio constituents, each with a 36-month track record. The standard deviation of errors of the MCED approach is 12.2%, lower than the 17.39% of the CED approach and the 17.07% of the MDD approach.

Chekhlov, Uryasev, and Zabarankin [2005] introduced a family of risk measures called *conditional drawdown* (CDD), the tail mean of drawdown distributions; investigated its mathematical characteristics; and discussed applications of the new measure to asset allocation decisions. They also suggested a block bootstrap procedure for the calculation of CDD and showed that the procedure is robust after approximately 100 simulations. They used the bootstrap approach because analytic solutions are not feasible, and, as shown by Douady, Shiryaev, and Yor [2000], even the calculation of expected drawdown for standard Brownian motion can be very complex. Block bootstrap is particularly attractive because it preserves the serial and cross-correlational characteristics of the original dataset, likely making it superior to the rolling historical MDD algorithm of Goldberg and Mahmoud [2014] that was used for illustrative purposes. We apply a similar block bootstrap procedure to CED and MCED calculations, using 200 simulations for robustness.

Drawdown-based measures can be applied to portfolios of traditional and alternative investments such as stocks, bonds, real estate, and hedge funds. In this article, we evaluate MCED within the large-scale simulation framework, which contains 613 live and 1,384 defunct funds over the 1993–2015 period, that Molyboga and L’Ahelec [2016] imposed on a subset of hedge funds in the managed futures space. The framework incorporates the standard requirements of institutional investors regarding track record length, the amount of assets under management (AUM), and the number of funds in the portfolio. The methodology closely mimics the portfolio management decisions of institutional investors and incorporates investment constraints and investor preferences to produce results that are relevant for investors.^{2}

Following Molyboga and L’Ahelec [2016], we evaluate out-of-sample performance with several commonly used measures of stand-alone performance and marginal portfolio contribution.^{3} We use annualized return, Sharpe and Calmar^{4} ratios, and MDD to measure stand-alone performance. We evaluate marginal portfolio contribution by measuring the improvement in Sharpe and Calmar ratios that results from replacing a modest 10% allocation to the investor’s original portfolio with a 10% allocation to a simulated commodity trading advisor (CTA) portfolio. In this article, we use the standard 60/40 portfolio of stocks and bonds with monthly returns from January 1999 through June 2015 as the investor’s original portfolio. Although this portfolio is often used in the literature, the framework is flexible to the choice of investor benchmark.

We find that a modest 10% allocation to CTA portfolios improves the performance of the original 60/40 portfolio of stocks and bonds for all portfolio construction methodologies considered in the study. For the out-of-sample period of January 1999–June 2015, a 10% allocation to managed futures improves the Sharpe ratio of the original portfolio from 0.365 to 0.390–0.404, on average, depending on the portfolio construction methodology. The MCED-based equal-risk approach results in a Sharpe ratio of 0.400. Similarly, the Calmar ratio improves from 0.088 to 0.097–0.104, on average, with the MCED-based equal-risk approach delivering an average Calmar ratio of 0.101. Blended portfolios have higher Sharpe ratios in at least 89.9% of simulations and higher Calmar ratios in at least 91.9% of simulations. We find that on a stand-alone basis the MCED-based equal-risk approach dominates the other drawdown-based techniques but does not outperform the EVA highlighted by Molyboga and L’Ahelec [2016]. For the out-of-sample period of January 1999–June 2015, the MCED-based equal-risk portfolio delivered an average Sharpe ratio of 0.339, which is higher than the 0.286–0.296 achieved by the other drawdown-based approaches, the 0.308 of random portfolios, and the 0.326 of a naïve 1/*N* approach, but slightly lower than the 0.347 of the EVA method. Calculating the Calmar ratio produces relative results similar to those of the MCED-based risk-parity equal-risk approach, delivering an average Calmar ratio of 0.163—higher than the ratios of the other approaches, with the exception of EVA, which delivers an average Calmar ratio of 0.166.

The remainder of the article is organized as follows: The first section discusses several drawdown-based measures and introduces MCED; the second section presents the block bootstrap methodology for calculating MCED and CED; the third section describes the CTA data and accounts for biases within the data; the fourth section describes the simulation framework; the fifth section presents empirical out-of-sample results; and the final section concludes.

**DRAWDOWN-BASED MEASURES**

In this section, we describe several measures of drawdown that are either commonly used in the industry or documented in the academic literature. We then introduce the MCED and discuss its characteristics.

Institutional investors use a variety of measures of risk, with MDD, defined as the largest peak-to-valley loss, among the most popular measures. Greenwich Roundtable [2010] requires drawdown analysis as an important part of quantitative due diligence. There may be very significant value in constructing portfolios with low drawdowns; however, minimizing maximum historical drawdowns has several known issues. First, the issue of *overfitting*—excessive optimization for noise in past returns without accounting for potential alternative outcomes—typically results in poor out-of-sample performance, as documented by DeMiquel, Garlappi, and Uppal [2009] for mean-variance optimization approaches. Second, the numerical calculation of MDD can be quite involved. Finally, focusing on maximum historical drawdown, the worst case scenario from within the drawdown distribution, can potentially result in suboptimal behavior.

Chekhlov, Uryasev, and Zabarankin [2005] and Goldberg and Mahmoud [2014] suggested looking beyond a single point of historical MDD and introduced drawdown-based measures that are based on the left tail of the drawdown distribution. In this article, we consider CED, which was introduced by Goldberg and Mahmoud [2014] and defined as the tail mean of MDD distributions

1where µ(?) represents the MDD distribution of random variable ?^{5} and *DT*
_{a}(?) is the quantile of the MDD distribution that corresponds to probability a.

This measure resembles expected shortfall, or conditional value at risk (CVaR), but uses the distribution of MDD instead of the distribution of returns. Similar to expected shortfall, CED is a coherent risk measure that possesses a number of attractive characteristics, as described by Artzner et al. [1999]. It is also a homogeneous function of order one, which allows simple calculation of the contribution of constituents to portfolio CED via the Euler equation.

Let us denote the demeaned portfolio component
, where
is the sample mean of portfolio component *i.* The demeaned portfolio is, therefore,
.

We define the MCED as follows:

2By construction, MCED has the same properties of coherence and positive homogeneity as CED, but as we show in the next section, MCED is less sensitive to sample error than is CED or MDD. Moreover, we present a methodology for generating the MDD distribution µ using the historical performance of the portfolio constituents and their weights.

**SIMULATION-BASED CALCULATION OF CED AND MCED**

In this section, we describe previously documented approaches for calculating CED and introduce a block bootstrap methodology for estimating the MDD distribution that is used in the calculation of CED and MCED. Bootstrap methodologies, introduced by Efron [1979] and Efron and Gong [1983], are commonly used in statistics to estimate distributions when analytic solutions are unsuitable or difficult to obtain. An analytic solution for the drawdown distribution is not feasible for realistic time series with serial correlation and fat tails. Even the relatively simple case of standard Brownian motion involves a very complex derivation of expected drawdown, as documented by Douady, Shiryaev, and Yor [2000].

The block bootstrap technique is particularly attractive because it preserves the serial and cross-correlational characteristics of the original dataset.^{6} The bootstrap procedure is likely superior to the rolling historical MDD algorithm of Goldberg and Mahmoud [2014] that was used for illustrative purposes because rolling historical MDD relies heavily on a single path and uses overlapping observations that are likely to include the same MDD multiple times. We apply a similar block bootstrap procedure to the CED and MCED calculations and show that, unlike CED and MDD, MCED is robust to sample error. Finally, we demonstrate an efficient approach to calculating marginal contribution to CED and MCED based on the algorithm of Goldberg and Mahmoud [2014].

**Block Bootstrapping Methodology**

Chekhlov, Uryasev, and Zabarankin [2005] suggested a block bootstrap procedure for the calculation of drawdown-based measures and demonstrated that the procedure is robust after approximately 100 simulations. Block bootstrap is a technique that applies multiple samplings with replacement to blocks of cross sections of an original dataset to produce realistic “what if” scenarios. Each scenario has the same number of constituents and the same length of time series as the original dataset.

We generate 200 simulations to calculate drawdown-based measures to ensure that calculations are robust. Simulations are used to calculate potential distributions of MDD, CED, and MCED for any non-negative portfolio weights such that *w*
_{1} + … + *w*
_{n} = 1 using simulated portfolio returns ? = *w*
_{1}?_{1} + … + *w*
_{n}?_{n} Drawdown contributions from individual components are calculated as described in the next subsection.

**Algorithm for Calculating Contribution to Portfolio MCED and CED**

Following Goldberg and Mahmoud [2014], we denote marginal contribution to portfolio risk measure ? as the derivative of the risk measure with respect to its component *i*:

For any homogeneous function of order one, such as MCED and CED, the portfolio risk can be decomposed using Euler’s formula:

4where
is the total risk contribution from component *i*. Equal-risk approaches, such as classic risk parity, use weights that result in an equal total return contribution from each component.

Goldberg and Mahmoud [2014] showed that

5where the MDD of the portfolio with a cumulative return path of {?^{t}, *t* = 1, …, *T*} is achieved between time *t _{K}
* and

*t*with

_{J}This formulation simplifies the calculation of marginal contribution to CED and MCED for datasets generated using the block bootstrap methodology.

**Sensitivity to Sampling Error**

We investigate the sensitivity of MDD, CED, and MCED to sampling error by generating 1,000 random scenarios, calculating weights that minimize MDD, CED, and MCED for each scenario, and evaluating them against the true optimal weights. Each scenario uses five uncorrelated assets with 36 monthly returns that are independent and identically distributed and follow a standard normal *N*(0, 1) distribution. By construction, true optimal weights are equal to 20%, but the introduction of sampling error results in differing sets of weights. We calculate the average distance between the calculated optimal weights and the true optimal weights to evaluate the sensitivity of the measures to sampling error.

Exhibit 1 presents the density functions of the sample errors of the three methodologies, and Exhibit 2 reports the results of the sensitivity testing. The table presents the ranges of sample errors, the percentages of errors between -20% and +20%, and the standard deviations of errors.

The range of MCED errors is 64.71%, whereas both CED and MDD have ranges of 100%. The percentage of relatively small errors between -20% and +20% is 94.14% for MCED, which is higher than the 86.90% for CED and the 86.56% for MDD. The standard deviation of its errors is 12.20%, which is lower than the 17.39% for CED and the 17.07% for MDD. Each metric shows that MCED is the most robust to sample error among the three drawdown-based measures.

**DATA**

In this study, we use the BarclayHedge database, the largest publicly available database of CTAs with 989 active and 3,784 defunct funds from December 1993 through June 2015.^{7}
Appendix A outlines the standard data-processing procedures used to address biases in the data and to limit the scope of the study to those funds that are relevant for institutional investors who make direct investments. We include the graveyard database, which contains defunct funds, to account for survivorship bias. We also explicitly account for backfill and incubation biases that arise due to the voluntary nature of self-reporting in CTA and hedge fund databases.^{8} We combine two standard methodologies to mitigate these biases. The first methodology, introduced by Fama and French [2010], limits the tests to those funds that managed at least USD 10 million normalized to December 2014 values. Because a significant portion of CTAs reported only net returns for an extended period of time prior to their initial inclusion of AUM data, using the Fama and French [2010] technique exclusively would eliminate a significant portion of the dataset. To include these data, we apply the methodology of Kosowski, Naik, and Teo [2007], which eliminates only the first 24 months of data for such funds. We further incorporate a liquidation bias of 1% as suggested by Ackermann, McEnally, and Ravenscraft [1999]. After accounting for the biases, our dataset includes 613 live and 1,384 defunct funds for the period December 1995–June 2015.

This study uses the Fung–Hsieh five-factor model of primitive trend-following systems, documented by Fung and Hsieh [2001], as benchmarks for measuring the performance of CTA portfolios. The factors include PTFSBD (bonds), PTFSFX (foreign exchange), PTFSCOM (commodities), PTFSIR (interest rates), and PTFSSTK (stocks). The three-month secondary market rate Treasury bill series with ID TB3MS from the Board of Governors of the Federal Reserve System serves as a proxy for the risk-free rate. Exhibit 3 reports summary statistics and tests of normality, heteroskedasticity, and serial correlation in CTA returns by strategy and status.

The 60/40 portfolio of stocks and bonds is used as the benchmark portfolio, following Anson [2011], who suggested that the portfolio represents a typical starting point for a U.S. institutional investor. This blend is constructed using the S&P 500 Total Return Index and the JPM Global Government Bond Index. Exhibit 4 presents the annualized excess return, standard deviation, MDD, Sharpe ratio, and Calmar ratio of the 60/40 portfolio for the period January 1999 and June 2015. Over this period, the portfolio delivered a Sharpe ratio of 0.365 and a Calmar ratio of 0.088.

Exhibit 5 shows the performance of the portfolio from January 1999 to June 2015.

**METHODOLOGY**

In this section, we describe the portfolio construction approaches evaluated in this study and the large-scale simulation framework employed.

**Review of Portfolio Construction Approaches Considered in the Study**

In this article, we evaluate five drawdown-based approaches. Three are minimum-risk portfolios: minimum MDD (MinMDD), minimum CED (MinCED), and minimum MCED (minMCED). Two additional drawdown-based approaches are the equal-risk approaches, RP_CED and RP_MCED, which use portfolio weights that result in equal total contribution to risk from each component. We use three benchmark portfolio construction approaches, including an equal notional (EN) approach, which is a naïve diversification 1/*N* method highlighted by DeMiquel, Garlappi, and Uppal [2009]; an EVA approach documented by Hallerbach [2012]; and a random portfolio selection approach (RANDOM). The approaches are evaluated using a large-scale simulation framework with realistic constraints.^{9}

**Large-Scale Simulation Framework**

This study uses the large scale simulation framework introduced by Molyboga and L’Ahelec [2016] with 1,000 simulations for the out-of-sample period of January 1999–June 2015. The methodology mimics the portfolio management decisions of institutional investors who rebalance portfolios at the end of each month. The first allocation decision is made in December 1998. Due to the delay in CTA reporting, documented by Molyboga, Baek, and Bilson [2015], the investor has return information available only through November 1998 at the time of decision making. Therefore, the investor considers all funds that have a complete set of monthly returns from December 1995 through November 1998. Following Molyboga and L’Ahelec [2016], the investor excludes all funds in the bottom quintile of AUM among the funds considered.^{10} The investor then randomly chooses five funds from the remaining pool of CTAs and allocates to them using the five drawdown-based approaches and the three benchmark methods. Monthly returns are recorded for each portfolio construction approach for January 1999 using the liquidation bias adjustment for funds that liquidate during the month. At the end of January 1999, the pool of CTAs is updated and constituents of the original portfolio not included in the new CTA pool, either due to liquidation or decrease in relative AUM level, are randomly replaced with funds from the new pool. Each portfolio is then rebalanced again using the original portfolio construction methodologies. The process is repeated until the end of the out-of-sample period in June 2015. A single simulation results in eight out-of-sample return streams for the period January 1999–December 2015—one for each of the portfolio construction approaches. Finally, performance results are evaluated based on out-of-sample results across 1,000 simulations.

Exhibit 6 reports the average AUM threshold level for each year and the average number of funds that meet that threshold. The AUM threshold represents the 20th percentile of AUM among all active fund managers with a track record of at least 36 months. Variation in the AUM threshold and the number of active funds over time reflects industry dynamics, which are driven primarily by recent performance and industry maturity.

**Evaluation of Out-of-Sample Results**

We evaluate out-of-sample performance using Sharpe and Calmar ratios as stand-alone performance measures^{11} as well as portfolio contribution metrics. Performance contribution is calculated as the difference in Sharpe ratio and Calmar ratio that results from replacing 10% of the original benchmark portfolio with portfolios of CTA funds constructed within the simulation framework. In this study, we use the standard 60/40 portfolio of stocks and bonds with monthly returns from January 1999 through June 2015.

**EMPIRICAL RESULTS**

In this section, we present empirical out-of-sample results obtained within the large-scale simulation framework with realistic constraints. We demonstrate that the MCED-based equal-risk portfolio is superior to the other drawdown-based methods considered in the study but fails to outperform a volatility-based equal-risk approach. We also show that the marginal benefit of CTA portfolios to the traditional 60/40 portfolio of stocks and bonds is positive and robust across all performance measures and portfolio methodologies considered in the study.

**Analysis of Out-of-Sample Performance of CTA Portfolios as Stand-Alone Investments**

We evaluate out-of-sample performance using means and medians of the distributions generated using the large-scale simulation framework. A bootstrapping methodology^{12} is used to draw statistical inference because simulations are not independent and, thus, classic statistics are not appropriate.

Exhibit 7 reports means and medians of the distributions of returns, volatilities, Sharpe and Calmar ratios, and maximum historical drawdowns for each portfolio construction approach. Most drawdown-based approaches except RP_MCED, the MCED-based equal-risk approach, produce Sharpe ratios that are inferior to the 0.308 of the RANDOM methodology. Bootstrapping suggests that the lower Sharpe ratios are statistically different at the 99% level for MinMDD, MinCED, and RP_CED and at the 98% level for the MinMCED. By contrast, EN, EVA, and RP_MCED outperform random portfolios at the 99% confidence level, with the equal volatility–adjusted approach leading with an average Sharpe ratio of 0.347, followed by the MCED-based equal-risk approach with an average Sharpe ratio of 0.339, and the naïve 1/*N* approach with an average Sharpe ratio of 0.326. This finding is consistent with those of DeMiquel, Garlappi, and Uppal [2009], who documented the superior out-of-sample performance of the naïve 1/*N* (EN) approach,^{13} and those of Molyboga and L’Ahelec [2016], who documented the superior performance of equal-risk approaches relative to random and minimum risk methodologies. Median values reported in Panel B demonstrate consistent results.

The large-scale simulation framework produces distributions of out-of-sample performance that can be visualized using standard box and whisker plots to provide additional insights.^{14}
Exhibit 8 displays the distributions of Sharpe ratios for each portfolio construction approach.

The breadth of each distribution represents the role of chance and highlights the importance of using a large-scale simulation framework to evaluate portfolio techniques, as discussed in detail by Molyboga and L’Ahelec [2016]. For example, minimum risk portfolios tend to have wider distributions of outcomes with large negative outliers—something that risk-averse investors may want to consider when they make their investment decisions.^{15}

Exhibit 9 displays the distributions of Calmar ratios for each portfolio approach. Although the choice of a portfolio construction methodology based on the distribution of outcomes ultimately depends on the preferences of a specific investor, Exhibit 8 indicates that the role of chance is significant and the EVA and RP_MCED methodologies look more attractive than any of the traditional minimum drawdown methodologies.

**Evaluation of Portfolio Contribution**

We further analyze the marginal contribution of the portfolio construction approaches to a traditional 60/40 portfolio of stocks and bonds by comparing the performance of blended portfolios that replace a modest 10% allocation to the benchmark portfolio with 10% to the hypothetical CTA portfolios.^{16}
Exhibit 10 reports the Sharpe and Calmar ratios of the blended portfolios and the percentage of scenarios that result in blended portfolios having higher Sharpe and Calmar ratios than the original 60/40 portfolio. Panel A reports mean results for the Sharpe and Calmar ratios. Panel B presents median values.

The marginal benefit of managed futures is very robust across portfolio construction approaches and simulations. A modest allocation to hypothetical CTA portfolios improves the performance of the original 60/40 portfolio of stocks and bonds for all portfolio construction methodologies considered in the study. For the out-of-sample period of January 1999–June 2015, the 0.365 Sharpe ratio of the original portfolio increases to between 0.390 and 0.404, on average, depending on the portfolio construction methodology. The MCED-based equal-risk approach yields a Sharpe ratio of 0.40. Similarly, the Calmar ratio improves from 0.088 to 0.0970–0.104, on average, with the MCED-based equal-risk approach delivering an average Calmar ratio of 0.101. The improvement in the Sharpe ratio is observed across at least 89.9% of simulations, and the improvement in the Calmar ratio is even more robust with at least 91.9% of simulations resulting in superior Calmar ratios for the blended portfolios versus the original portfolio.

The results are striking but consistent with the academic literature on managed futures.^{17} Our findings are consistent with those of Kat [2004]; Lintner [1996]; Abrams, Bhaduri, and Flores [2009]; and Chen, O’Neill, and Zhu [2005], who reported a positive contribution of managed futures to traditional portfolios, on average. Molyboga and L’Ahelec [2016] demonstrated that the positive contribution of managed futures to a traditional 60/40 portfolio of stocks and bonds is robust across all risk-based approaches considered in their study.

**CONCLUDING REMARKS**

This article evaluated the portfolio management implications of using drawdown-based measures in allocation decisions. We introduced MCED and showed that MCED exhibits the attractive properties of coherent risk measures present in CED but lacking in the historical MDD commonly used in the industry. We introduced a robust block bootstrap approach that uses historical performance to calculate MCED and the marginal contributions to MCED from portfolio constituents. MCED is less sensitive to sample error than are either CED or MDD, which makes it more attractive as an implementable portfolio management methodology.

We examined several drawdown-based minimum-risk and equal-risk approaches within the large-scale simulation framework of Molyboga and L’Ahelec [2016] using a subset of hedge funds in the managed futures space that contains 613 live and 1,384 defunct funds over the 1993–2015 period. We found that CTA investments make a significant portfolio contribution to a 60/40 portfolio of stocks and bonds over the 1999–2015 out-of-sample period. This result is robust across portfolio methodologies and performance measures.

We found that the MCED-based equal-risk approach dominates the other drawdown-based techniques but does not consistently outperform the simple equal volatility–adjusted approach. This finding highlights the importance of carefully accounting for sample error, as reported by DeMiquel, Garlappi, and Uppal [2009], and cautions against overreliance on drawdown-based measures in portfolio management.

**APPENDIX A**

**DATA CLEANING**

Because the article focuses on the evaluation of direct fund investments, we excluded all funds from the BarclayHedge database that are multi-advisors or report returns gross-of-fees. These exclusions reduced the fund universe to 4,773 funds with 989 active and 3,784 defunct funds for the period December 1993–June 2015. We then performed a few additional data filtering procedures to account for biases and potential errors in the data and produce results that are relevant for institutional investors. First, we eliminated all null returns at the end of the track records of defunct funds, which is a typical reporting issue inherent within hedge fund databases. We then excluded managers with less than 24 months of data, which limited the dataset to 3,321 funds. Additionally, we eliminated all funds with maximum AUM of less than USD10 million, which further limited the dataset to 2,009 funds. Finally, we excluded funds with one or more monthly returns in excess of 100%, which resulted in the final pool of 1,997 funds, 613 of which were active and 1,384 of which were defunct.

**APPENDIX B**

**ALLOCATION APPROACHES**

In this study, we consider three minimum drawdown and several equal-risk approaches. They include EN, EVA, MDD, minimum CED, minimum MCED, CED-based equal-risk portfolio (RP_CED), and MCED-based equal-risk portfolio (RP_MCED).

1. EN allocation is a simple equal-weighting (or naïve diversification) approach: B1

where

*N*is the number of funds in the portfolio and*w*is the weight of fund_{i}*i*.2. Equal EVA allocation is similar to the equal notional approach with exposure to each fund adjusted for the fund’s volatility, which is estimated using the standard deviation of its in-sample excess returns: B2

3. The MinMDD approach produces an allocation with the minimum historical drawdown.

4. The MinCED approach produces an allocation with the minimum CED for a confidence level of 90%.

5. The MinMCED approach produces an allocation with the minimum MCED for a confidence level of 90%.

6. The RP_CED is the solution to the following optimization problem: B3

such that and

*w*_{i}= 0, and ? = CED.7. RP_MCED is the solution to the following optimization problem: B4

such that , and

*w*_{i}= 0, and ? = MCED.8. A random portfolio (RANDOM) is used as a benchmark approach for portfolio allocation. First, a random number

*x*between 0 and 1 is generated. Then random portfolio weights are normalized by setting ._{i}

**APPENDIX C**

**BOOTSTRAPPING PROCEDURE**

The bootstrapping procedure follows each step of the simulation framework but limits the set of portfolio construction approaches to the RANDOM portfolio methodology against which we compare all other approaches.^{18} Each simulation set consists of 1,000 simulations. The bootstrapping procedure includes 400 sets of simulations, a sufficient number to estimate P-values with high precision. A comparison of the performance metrics of the original simulation to the set of bootstrapped simulations gives the P-values reported in the empirical results section.

## ENDNOTES

↵

^{1}The equal volatility–adjusted approach is similar to the naïve 1/*N*allocation methodology, which gives equal dollar allocation to each portfolio constituent; EVA, however, gives equal risk allocations with sample volatility used as the measure of risk. Appendix B provides detailed technical definitions of all allocation approaches, including EVA.↵

^{2}The framework of Molyboga and L’Ahelec [2016] is customizable to the preferences and constraints of individual investors, such as investment objectives, number of funds in the portfolio, and rebalancing frequency.↵

^{3}The framework is flexible and can incorporate customized performance measures selected by the investor.↵

^{4}*Calmar ratio*is defined as the ratio of the annualized excess return to the maximum historical drawdown.↵

^{5}Portfolio return is a weighted average of its components’ returns ? =*w*_{1}?_{1}+ … +*w*?_{n}_{n}, such that*w*_{1}+ … +*w*_{n}= 1. In this study, portfolio weights are non-negative because manager allocations cannot be negative, but the MCED can be applied to long–short portfolios that have both positive and negative weights.↵

^{6}Fama and French [2010] applied a bootstrap methodology that uses single-month cross sections of data. Blocher, Cooper, and Molyboga [2015] used a block bootstrap technique for CTAs that specialize in trading commodities.↵

^{7}Joenväärä, Kosowski, and Tolonen [2012] reported that the BarclayHedge database provides the highest-quality data from among CISDM (Morningstar), Lipper (formerly TASS), Eurekahedge, and BarclayHedge.↵

^{8}Fund managers start reporting to a CTA database to raise capital from outside investors only if the track record generated using proprietary capital during the incubation period is attractive. They then typically backfill the returns over the incubation period. Because funds with poor performance are unlikely to report returns to the database, incubation/backfill bias arises.↵

^{9}See Appendix B for technical definitions of the allocation approaches.↵

^{10}Molyboga, Baek, and Bilson [2015] argued that this relative AUM threshold is more appropriate than the fixed AUM approach commonly used in the literature because the average level of AUM has increased substantially over the last 20 years.↵

^{11}Molyboga and L’Ahelec [2016] used several additional measures of performance to demonstrate the flexibility of the framework. This study uses only two of the measures for brevity.↵

^{12}Appendix C describes the bootstrapping procedure that is used to draw statistical inference about the relative performance of the portfolio construction approaches and estimation of*P*-values.↵

^{13}DeMiquel, Garlappi, and Uppal [2009] also pointed out that a naïve 1/*N*approach should dominate a random portfolio in terms of Sharpe ratio due to the concavity of the Sharpe ratio. Jensen’s inequality states that*Eg*(*X*) =*g*(*Ex*) for any concave function*g*such as the Sharpe ratio. See Rudin [1986] for a detailed explanation of Jensen’s inequality.↵

^{14}The box contains the middle two quartiles, the thick line inside the box represents the median of the distribution, and the whiskers are displayed at the top and bottom 5% of the distribution.↵

^{15}Molyboga, Baek, and Bilson [2015] used stochastic dominance and utility functions to compare distributions. Both approaches capture the risk-aversion characteristics in investors’ preferences and can account for the differences in distributions shown in Exhibit 8.↵

^{16}Molyboga and L’Ahelec [2016] argued that marginal contribution analysis is particularly important for investors with exposure to a large number of systematic sources of return.↵

^{17}It is important to note that relative performance results depend on the evaluation window as described by Anderson, Bianchi, and Goldberg [2012]. Therefore, subperiod analysis can provide additional insights into key factors, such as market environment, that can potentially affect the relative performance of strategies.↵

^{18}The framework is flexible in comparing any two approaches to each other but requires performing additional bootstrapping simulations based on an investor’s particular areas of interest.

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