
alternative of being paid the return of the replicating optioned portfolio.Consequently, this replicating return has to be exceeded by other investmentsexhibiting similar risk characteristics.The efficiency gain/loss measure (the excess replicating return) takes theformERRp = E(rp) − RR(rp)
(16.1)
where E(rp) = expected return on portfolio pRR(rp) = expected replicating returnThe excess replicating return can be directly interpreted as an efficiencygain, if it is positive, or an efficiency loss, if it is negative. If the replicatingoptioned return is higher than the expected return for an arbitrary CTA portfolio, this underlying asset offers an inferior performance (compared to thebenchmark asset). That is, the comparable investment in form of an optionedportfolio offers a higher expected return for the same risk characteristicsof returns. The fund’s shareholder would do better with a different fund (ofcourse, as stand-alone investment only). The excess replicating return provides a simple measure in assessing whether a portfolio outperformed otherson an ex-post basis.This measure is in a close relationship with the excess return measureproposed by Ang and Chau (1979), which is an alpha-like composite performance ratio. An important distinction is that, in the replicating case,individual portfolios do not have to possess the same systematic risk characteristics as the benchmark asset. It is sufficient if both share the samereturn distribution shapes.
Construction of Replicating PortfoliosAs defined earlier, replicating portfolios are portfolios that have the samerisk structure in terms of some statistical moments (of order three andhigher) as the portfolio being assessed. The foundation for including replicating portfolios in the performance assessment is the assumption that portfolios can be created to “mimic” the risk structure of the underlying assetas benchmarks.The present replicating framework will be termed partial, because onlya reduced set of the return characteristics (the moments) is used for describing any return distributions, thereby reducing the return distribution’sdimensionality.
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Choosing the Right CTA
It is very important to note that the term “replication” as we use it doesnot intend to create the same payoff profile in terms of identical probability distributions, nor does it intend to create portfolios having the same payoff in every possible state of nature.For the construction of individual replicating portfolios, we used theStandard and Poor’s (S&P) 500 index as underlying. Based on the assumption that returns from the index are independent and follow a lognormaldistribution—a simplification that greatly facilitates the use of contingentclaim–based performance evaluation but is not essential—we calculatedprices for a specified number of Black-Scholes call options. Consideringonly call options ensures that asset returns are not linearly dependent. Forthe sake of simplicity, a holding period of one year is assumed.In the next step, we used nonlinear programming for generating returnson replicating portfolios, with variance and (a predefined number of) higherstatistical moments being set to that of the CTA under investigation.This approach provides a relatively simple and robust means for calculating individual benchmark returns. This idea parallels the work of Aminand Kat (2001). They propose a point-by-point optimization algorithmwith a 500-pins-setting, that is, they match 500 separate points of thereturn distribution, to calculate hedge fund efficiency gains/losses.The optimization algorithm that produces replicating portfolio weightsxi, can be formulated:Max Z = ∑ xi E(ri )xi
(16.2)
i
subject to
σ 2 = σ 02 target variances 3 = s03 target skewness
(16.3)
and the constraints on the portfolio weights
∑ xi = 1i
where E(ri ) = expected return on asset is 02 s03 = target values for variance and skewness, respectively.The constraints in equation 16.3 can be expanded to include momentsof order higher than three.
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PROGRAM EVALUATION, SELECTION, AND RETURNS
MARKET DATA USEDFor the testing, we used CTA data publicly available from TradeView(www.tradeview.com). The chosen data set contains 110 CTAs with amonthly return history of five years, from January 1998 to December 2002.From the monthly returns, “semi ex-ante” annual discrete returns weregenerated with a bootstrap-like methodology i.e., drawing 12 samples withreplacement from the set of monthly data, using 1,000 repetitions for eachfund. This bootstrapping methodology is in the vein of the technique appliedby Ederington (1995).We used the Standard and Poor’s monthly return series as underlying.As proxy for the risk-free rate, we took the one-month U.S. Dollar (USD)London Interbank Offered Rate (LIBOR).
RESULTSNonnormality of ReturnsWe test for nonnormality of returns with the Jarque and Bera (1987) test(see Greene 2000). Analyzing the samples, we find that the null hypothesisof normally distributed returns cannot be rejected at the 1 percent level foronly 14 cases (11 percent of the observations) and at the 5 percent levelfor only 11 cases (8.7 percent). Clearly, the sample of CTA funds is highlynonnormal. This should underline the need for a performance measure thataccounts for nonnormality of returns.
Portfolio Efficiency Rankingsby the Excess Replicating ReturnThe excess replicating return (ERR) is, in much the same way as the Sharperatio, a composite—risk-adjusted—performance measure. It is risk adjustedbecause the ERR is calculated always to a given level of risk. Thus riskadjustment takes place indirectly by applying an additive, not multiplicative, rule.The ERR, again, denotes the return differential between the expectedreturn of a particular asset and its replicating counterpart. It is designed toassess the value added by the portfolio manager—that is, the efficiency gainor loss. Negative values would mean that the investor is better off buyingthe same risk structure through options instead of investing in the givenasset/CTA and vice versa.
301
Choosing the Right CTA
We investigated the efficiency of the CTA sample. Figures 16.1 and16.2 summarize the main results of the analysis. The first diagram displaysthe excess replicating returns for the second moment case (variance only),sorted by magnitude. It is evident that, for the sample being investigated,CTAs provided a risk-adjusted performance that is—to a large extent—notaccessible on the stock markets. That is, about 80 percent of the CTAs perform better than the replicating optioned portfolios based on the S&P 500.Nonetheless, two factors have to be considered.1. In the time period investigated, the S&P delivered an annual return ofabout 5.6 percent, which is barely higher that the estimated risk-freerate (4.2 percent).2. For technical reasons, we have not accounted for possible survivorshipbias, which may be expected to have a substantial impact on the overall performance.Figure 16.2 displays the excess replicating returns for the nine-momentcase, but in the same ranking order, as in Figure 16.1. It is noticeable thatthe basic performance characteristics of the CTAs are mirrored fairly wellwith the two-moment method; this suggests that a large part of the replicating return is attributable to the variance itself. Yet we certainly can alsoascertain some significant discrepancies between the rankings of the twocases that should be subject to a closer look.2nd Moment Case60%50%Efficiency Gain/Loss
40%30%20%10%0%−10%−20%−30%
FIGURE 16.1Sorted
Ranked CTA
Efficiency Gain/Loss Measure (Variance Only) for the CTA Samples,
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PROGRAM EVALUATION, SELECTION, AND RETURNS
9th Moment Case60%50%
Efficiency Gain/Loss
40%30%20%10%0%−10%−20%−30%
Ranked CTA
FIGURE 16.2 Efficiency Gain/Loss Measure for the CTA Samples, Sorted byRankings of the Variance Only Case
Rank Correlation StatisticsLet us assume that the more complete description of the CTA return distributions (the use of more moments) enables a more robust and exact performance measurement. We would like to investigate the overall propertiesof the moment-based replicating measures and determine how the rankingresult is affected by calculating replicating returns with a lower number ofmoments. Using the optimization algorithm, we obtained eight portfolios,M2 through M9, by specifying 2 through 9 moments in the constraints,respectively. We then replicated returns in each portfolio and obtained theSharpe ratio and ERR measures of the replicated returns. Next, we calculated rank correlations (Spearman correlations) between the particularERR measures and Sharpe ratio, which evaluates the closeness of the rankings produced by the different methods of performance evaluation. Table16.1 sums up the results of the calculation. As can be seen, the momentbased measures lie within a limited range, that is, the rankings provided bythem are very close to each other: The rank correlations are always higherthan 0.99. It is also noticeable that the rank correlation between the Sharperatio and the moment-based measures is high but lower than the rank correlation between the moment-based measures themselves.Then we repeated the analysis with a slightly different frame, drawingsmall samples repeatedly and comparing the percentage of identical decisions regarding the best possible CTA. Not surprisingly, when drawing sam-
303
Sharpe RatioERR M2ERR M3ERR M4ERR M5ERR M6ERR M7ERR M8ERR M9
TABLE 16.1
1.00000.76750.76690.76960.75950.75750.74970.74390.7288
Sharpe Ratio
1.00000.99990.99970.99810.99740.99520.99580.9935
ERR M2
1.00000.99980.99840.99780.99560.99610.9939
ERR M3
1.00000.99850.99790.99560.99580.9935
ERR M4
1.00000.99950.99640.99590.9930
ERR M5
Simple Rank Correlation Between Particular Performance Measures
1.00000.99660.99590.9928
ERR M6
1.00000.99560.9927
ERR M7
1.00000.9958
ERR M8
1.0000
ERR M9
304
Sharpe RatioERR M2ERR M3ERR M4ERR M5ERR M6ERR M7ERR M8ERR M9
TABLE 16.2
10.3530.3940.4090.3540.3620.3740.3360.326
Sharpe Ratio
11.0000.9980.9220.9410.8610.920.871
ERR M2
11.0000.9630.9480.9030.9170.918
ERR M3
10.9430.9490.9020.9370.928
ERR M4
10.9940.8960.9030.918
ERR M5
10.9190.9490.931
ERR M6
10.9370.899
ERR M7
10.945
ERR M8
Ratio of Identical Decisions (Sample Size 10, 1,000 Times) with Particular Performance Measures
1
ERR M9
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Choosing the Right CTA
TABLE 16.3
Contribution of Higher Moments to the Replicating Returns
AbsolutecontributionRelativecontribution
Moment 3
Moment 4
Moment 5
Moment 6
−0.062%
−0.032%
−0.261%
−0.069%
−1.540%
−0.730%
−5.974%
−1.348%
AbsolutecontributionRelativecontribution
Moment 7
Moment 8
Moment 9
−0.602%
−0.477%
−0.422%
−15.530%
−11.985%
−13.001%
ples of 10 CTAs 1,000 times, it turns out that the moment-based replicatingmeasures produce also very similar results, as it can be seen in Table 16.2.
How Many Moments?Thus the different moment-based replicating measures produce very similar results. The most important question is: How many moments aresufficient to reproduce the results of the quasi-benchmark (i.e., the ninemoment case)?To cast some light on this question, we calculated the absolute as wellas the relative contributions of every higher moment to the replicatingreturns. That is, we tried to determine whether the absolute as well as therelative contribution (i.e., the absolute difference between two corresponding moment replicating returns divided by the level of the replicatingreturn) diminishes as the number of moments increases.As Table 16.3 shows, this is, unfortunately, not the case. Somewhat surprisingly, the moments 7, 8, and 9 have all an average relative contributionto the replicating return of over 10 percent. This fact indicates that considering the first few moments only may provide a good correlation in theranking properties but not necessarily a good approximation in terms ofabsolute value of the replicating return.Put differently, for such nonnormal cases, we found only weak evidence for supporting the use of only the variance in the calculation of replicating returns.
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PROGRAM EVALUATION, SELECTION, AND RETURNS
CONCLUSIONWe investigated the ranking properties of the moment-based replicatingefficiency measures on a sample of CTA managed funds.Summing up, we found that (neglecting any possible survivorship bias)using these measures, the majority of the funds investigated had a performance superior to the S&P 500. We also found that the moment-based replicating measures of efficiency gain/loss produce results very close to eachother in terms of rank correlation. In addition, the Sharpe ratio and themoment-based efficiency measures produce a higher rank correlation but alower rate of identical decisions.However, we also found that the higher moments have a high absoluteand relative contribution to the replicating return. This fact implies thatalthough the replicating measures may be very useful for ranking assetswith nonnormally distributed returns, the magnitude of the replicatingreturns is not robust to the number of moments used in the replication. Inparticular, important contributions to replicating returns will be omitted ifmoments of low order only (like the variance) are used. Thus, furtherresearch is needed to evaluate the usefulness of approximations of the truereplicating return based on two (or three) moments.
CHAPTER
17
CTAs andPortfolio Diversification:A Study through TimeNicolas Laporte
he standard mean/variance framework and the concept of efficient frontiers are one way of assessing the portfolio added value of a hedge fundstrategy such as CTAs. However, even if it provides interesting results, thisframework is a two-dimensional one and it gives a static vision of theCTAs’ industry. Changes in correlation or volatility over time are ignored.To provide a more dynamic approach, this chapter presents a three-dimensional framework with time as the third variable. It assesses the evolutionof the CTAs’ diversification abilities in a portfolio environment over thelast decade.
T
INTRODUCTIONCommodity trading advisors (CTAs) are professional money managers.They manage the assets of their clients using derivative instruments(futures, forwards, and options) on commodities and money marketsaround the world. As an asset category in the alternative investment industry, they are classified as “managed futures.” CTAs’ strategies range fromsystematic models to discretionary approaches, the first one being the mostcommon. CTAs are, most of the time, considered trend followers.Even though CTAs have existed for a while, only a few studies havebeen published about them. The term “CTAs” appears regularly in publications but, most of the time, is far from being the main topic. UsuallyCTAs are mentioned because of their affiliation to the hedge fund industry.Looking at the practitioner side, the same conclusion can be made.
307
308
PROGRAM EVALUATION, SELECTION, AND RETURNS
Although it is true that most of the financial players are familiar with CTAs(CTAs, in fact, have the reputation of being low/negatively correlated toany asset family, including hedge funds), most of the time, this interestingfeature is all they know about them.Based on these findings, it is interesting to propose a study focusinguniquely on the CTA industry with, as main objective, the definition of theiradded value in portfolio allocation. Different statistics and portfolio frameworks (with two or three dimensions) are then considered. Each brings newinformation and helps in understanding the managed futures universe. Notethat the three-dimensional framework used with portfolio allocations isdefinitively the “pioneering” part of this study.The chapter is organized in three parts. The first part compares CTAswith other assets. Two types of values are computed: plain statistics (staticview of the CTA industry) and rolling statistics (dynamic approach, whichtakes into account time evolutions). The second part focuses on portfoliooptimization and efficient frontiers. Its objective is to assess the CTAs’diversification capacity. As in the first part, CTAs are considered under astatic and a dynamic perspective. The dynamic perspective considers timeevolutions using a three-dimensional representation.
CTAsCTAs’ Quantitative DescriptionAs for any financial asset, the CTAs’ universe is assessable through indicescompiled by several providers. In theory, these indices should match eachother in terms of volatility and performance since they are constructed on thesame original universe (they are supposed to proxy the same industry). Inpractice, it is rarely the case. Indices are constructed using different methodologies (each methodology defines rebalancing dates, index componentselections, survivorship bias correction, etc.) and, even more important, different data sources. It generates, most of the time, significant patterns dissimilarity between them.In the case of CTAs, there are two major index providers: CSFB/Tremontand Barclay Group. It is interesting to consider these two indices1 (see Figure 17.1). Because they are traceable from December 1993, our historical
1Of course, the purpose of this comparison is not to run an index quality test. As
mentioned, it is logical to find differences between indices since their methodologiesand universe selection process are different.
309
The CSFB/Tremont Hedge Fund Indices are assetweighted. CSFB/Tremont uses the TASS database.The CSFB/Tremont universe consists only of fundswith a minimum of US $10 million under managementand a current audited financial statement. Funds areseparated into primary subcategories based on theirinvestment style. Managed Futures proxies the CTAs’universe. Funds are not removed from the index until theyare liquidated or fail to meet the financial reportingrequirements. The index is calculated on a monthly basis.Funds are reselected quarterly.
FIGURE 17.1
Financial/Metals
Currency Traders
Diversified Traders
Agricultural Traders
System Traders
Discretionary Traders
Long/Short
Convertible
Event Driven
Barclay CTAs
Fixed Income Arbitrage
CSFB/Tremont HedgeFunds
Emerging Markets
BarclayIndices
Dedicated Short Bias
CSFB/TremontHedge FundsIndices
Equity Market Neutral
Arbitrage
Managed Futures
CTAs and Portfolio Diversification
The Barclay CTA Index is unweighted and rebalanced at thebeginning of each year. To qualify for inclusion in the CTAIndex, an advisor must have four years of prior performancehistory. The restrictions offset high turnover rates of tradingadvisors as well as artificially high short-term performancerecords.The Barclay CTA Index also includes six separate subindicesof managed futures programs, based on portfolio compositionand trading style.For a managed program to be included in any of these subindices, they must have at least 12 full months of priorperformance history, with no extracted performance.
CSFB/Tremont CTA Index versus Barclay
index database goes from this date to December 2002, which correspondsto 109 monthly index levels. From this database, it is easy to extract somestatistics. They are displayed in Figure 17.2 and provide a first step in theCTAs’ performance assessment.Clearly, the CTA index and the managed futures index present similarannualized returns (respectively 6.44 percent and 6.26 percent). However,their volatilities differ significantly: The annualized standard deviations are,respectively, 8.39 percent and 11.94 percent (40 percent superior to the Barclay volatility). CSFB/Tremont provides a riskier (or more volatile) view ofthe industry than Barclay Group.In a risk/return framework, CTAs do not have an exceptional profilecompared to other hedge fund investment strategies (e.g., global macro) oreven some traditional equity groups (e.g., real estate investment trust[REIT] equities).Actually, only two hedge funds families have lower returns than CTAs:the dedicated short bias and the emerging markets. Such a finding is notthat surprising, and the origin of their poor results is related to their invest-
310
PROGRAM EVALUATION, SELECTION, AND RETURNS
0.150CSFB Global Macro0.125
CSFB Equity Mkt. Ntrl.CSFB Event Driven
0.100
CSFB Long/ShortCSFB Tremont Hedge Fund
Annualized Return
CSFB Convertible Arbitrage
NAREITNorth Americas eq.
0.075CSFB Fixed Inc. Arb.
North Americas bds.World bds.
Europe bds.CTA Barclay
CSFB Managed FuturesCSFB Emerging Markets
0.050Europe eq.World eq.
0.025
CSFB Ded. Short Bias0.000
0.00
0.05
0.10
0.15
0.20
–0.025
–0.050Far East eq.–0.075
Annualized Standard Deviation
World bonds
Annualizedreturns
Annualizedstd. dev.
Min
Max
0.0581
0.0628
–0.0354
0.05850.0384
North Americas bonds
0.0703
0.0441
–0.0249
Europe bonds
0.0682
0.0887
–0.0500
0.0849
World equities
0.0314
0.1503
–0.1445
0.0853
North Americas equities
0.0715
0.1652
–0.1548
0.0938
Europe equities
0.0384
0.1586
0.0037
0.0033
Far East equities
–0.0649
0.1980
–0.1295
0.1676
NAREIT
0.0926
0.1197
0.0075
0.0076
CSFB Tremont Hedge Fund
0.1054
0.0882
0.0096
0.0097
Convertible Arbitrage
0.1013
0.0488
0.0090
0.0090
Ded. Short Bias
0.0079
0.1805
–0.0010
–0.0009
Emerging Markets
0.0489
0.1885
0.0047
0.0042
Equity Mkt. Ntrl.
0.1095
0.0316
0.0092
0.0094
Event Driven
0.1038
0.0643
0.0087
0.0087
Fixed Inc. Arb.
0.0661
0.0416
0.0059
0.0060
Global Macro
0.1396
0.1260
0.0125
0.0127
Long/ShortManaged Futures
0.1149
0.1139
0.0105
0.0105
0.0626
0.1194
0.0045
0.0046
CTA Barclay
0.0644
0.0839
0.0055
0.0057
FIGURE 17.2 Statistics for CTA Performance AssessmentAll calculations are based on monthly data from December 1993 to December2002. Data sources are Morgan Stanley Capital International (equity and bondindices), NAREIT (REIT index), CSFB/Tremont (hedge fund indices, including themanaged futures index), and Barclay Group (CTA index).
CTAs and Portfolio Diversification
311
ment styles and their ensuing relation with the markets. Concerning thededicated short bias, this strategy lost most of its interest during the strongtelecom/information technology bull period. The emerging markets fundsfocus on hazardous businesses; because they invest in debt, equity, andtrade claims of companies located in emerging countries, they deal with animportant lack of transparency and have many uncertainties linked to economical, political, and social factors. (The Russian bond default is oneextreme example.)As with the two previous hedge funds strategies, the relative underperformance of CTAs is, in large part, explainable by the specificities of theirbusiness. Future managers focus on a few highly volatile and speculativemarkets, which reduces their physical investment opportunities. CTAs didnot really take advantage of the increasing markets globalization (comparedto some other hedge funds families). Moreover, most of them are trend followers, meaning that they go long or short with a lag compared to the markets. In the best cases, this lag reduces their benefits; in the worst cases, itgenerates heavy losses. It is true that managers significantly leverage theirpositions to increase their returns, but the use of leverage does not compensate for the lack of diversification and the important risk bearing.Besides these negative issues, investors see in CTAs an interestinginvestment vehicle because they have been historically low/negatively correlated to the other financial assets. This characteristic is the logic consequence of their business (CTAs do not invest in standard assets but insteaddeal with futures, a product not frequently used by the other hedge fundmanagers), and it is clearly verified Figure 17.3. CTAs do provide a lowcorrelation level with standard assets (stocks and bonds) and hedge fundstrategies. Of course, because of the index methodologies differences,results differ from one index to the other. The values range from −0.207 to0.376 for the CTA index and from −0.283 to 0.339 for the managedfutures index. The difference in methodologies and data sources betweenthe two indices is assessed by the managed futures/CTAs index correlation:The value is 0.805 (a relatively low result for two products proxying thesame industry).
CTAs through TimeEven if findings are interesting and help in defining the CTAs’ behavior relative to other assets, they give a static view of this investment strategy, sothey are unable to detect any temporal change in return, volatility, or correlation. Time variations are simply ignored.A dynamic approach that uses rolling windows is therefore warranted.This technique uses moving subsamples as inputs for the statistics’ compu-
312EuropeEquity
10.7810.5140.286–0.207–0.197
10.073–0.2830.295–0.1640.158–0.269–0.1200.239–0.093
NorthAmericasBonds
1–0.117–0.037–0.1090.0050.3660.339
CTA Barclay1Managed Futures 0.805CSFB Tremont Hedge Fund 0.219Convertible Arbitrage –0.100Ded. Short Bias 0.213Emerging Markets –0.120Equity Mkt. Ntrl. 0.218Event Driven –0.156Fixed Inc. Arb. 0.041Global Macro 0.376Long/Short –0.018
10.5210.292–0.193–0.269
NorthAmericasEquity
1–0.2240.3490.3240.5980.5460.3000.267
Conv.Arb.
10.2360.6990.2990.4050.591
10.089–0.094–0.0351–0.047–0.093
10.805
CTABarclay
10.3910.0910.2070.353
Emerging EquityMarkets Mkt. Ntrl.
Far East NAREITEquityEquity
1–0.568–0.391–0.609–0.063–0.113–0.738
Ded.ShortBias
1
Futures
10.3810.3650.662
EventDriven
10.4630.205
FixedInc.Arb.
10.422
GlobalMacro
1
LongShort
FIGURE 17.3 Correlation of CTAs with Standard Assets and Hedge Fund StrategiesAll calculations are based on monthly data from December 1993 to December 2002. Data sources are MorganStanley Capital International (equity and bond indices), NAREIT (REIT index), CSFB/Tremont (hedge fundindices, including the Managed Futures index) and Barclay Group (CTA index).
EuropeBondsEurope Bonds1North Americas Bonds 0.394Europe Equity 0.054North Americas Equity –0.189Far East Equity 0.014NAREIT 0.035CTA Barclay 0.170Managed Futures 0.310
10.408–0.4710.6470.3350.6570.4530.8610.781
Futures
CSFBHedgeFund
CTABarclay
313
CTAs and Portfolio Diversification
tation. From the practical perspective, the choice of the subsample length(or rolling window) is the sensitive step. A large window limits the numberof statistics and smooths results while increasing the econometric significance. A small one does exactly the opposite. With a database going fromDecember 1993 to December 2002, a time period of 36 months is a goodcompromise. It allows the generation of 73 sets of statistics (starting inDecember 1996).This time approach provides interesting results on the CTAs’ standarddeviation for two reasons (Figure 17