2 and 21.3show the corresponding results for Nasdaq and Lehman Aggregate BondIndex respectively.These historical performance comparative results indicate that CTAsare an investment vehicle worth exploring and can offer unique risk/returncharacteristics in a stock/bond portfolio as well as in a stock/bond hedgefunds portfolio. A number of studies have explored the benefits of managedfutures (CISDM 2002), so we limit the analysis of managed futures toshowing the importance of modeling their return series.In the next section we examine whether CTAs generate stationaryreturn time series, and we attempt to fit auto-regressive moving average(ARMA) models.
METHODOLOGYWe test for second-order (weak) stationarity in our return time series {yt }t = 1.In other words, we test whether its first and second moments and its autocorrelations are invariant in time. For comparison purposes, we carry out allthe tests that appeared in Gregoriou and Rouah (2003a) for hedge funds,among others. However, we examine a more complete set of CTAs that satisfy certain track record and assets under management requirements, as wehave included all the CTAs that report their performance in the database fromthe Center for International Securities and Derivatives Markets (CISDM). Wealso extend the analysis to the manager’s excess returns as a proxy for determining stationarity of manager’s alpha. We use the Augmented Dickey-Fuller∞
370
PROGRAM EVALUATION, SELECTION, AND RETURNS
(ADF) test to test for the presence of a unit root in the series. In our examples, intercept and time trend have been taken into account.N
∆yt = µ + λt + (γ − 1)yt − 1 + ∑ ϕ i ∆yt − i + ε t
(21.1)
i =1
After we test for stationarity, we model the return series usingARMA(p,q) processes of different orders using correlograms for each seriesas a guide. Finally we perform stability tests using the Chow test to investigate possible structural changes in the parameters of the specified ARMAprocesses.
DATAFor this study we have chosen the 10 largest CTAs from the CISDM database that have complete data series (monthly) for the period from January1996 to December 2003. Their average assets under management were over$100 million during the fourth quarter of 2003. For comparison purposes,we required that the return series are complete, and we wanted to examineCTAs with relatively long historical track records and that are of significantsize (based on the most recent information available). The effects of lengthof track record as well as fund size have been extensively examined bySchneeweis, Kazemi, and Karavas (2003a, b) for hedge funds. Althoughsimilar analysis for CTAs, to the best of our knowledge, is not available, weanticipate that the benefits of larger hedge funds with long track recordsapply to CTAs, too. Briefly, a long track record provides evidence of manager performance under different market conditions, while high assetsunder management indicate that the strategy followed can be replicable atlarger scale. The latter is important especially for CTAs because of theimpact on prices due to trade of high volumes of specific futures; managerswith low assets under management impact the prices to a lesser extent.For the calculation of the excess returns used in the tests, we calculatedthe excess CTA monthly return from the CISDM Equally Weighted TradingAdvisor Qualified Universe Index (CISDM CTA). The CISDM CTA Indexis the median return of all CTAs and commodity pool operators (CPOs)reporting to the CISDM CTA database. At the end of 2003, there existedapproximately 600 CTAs and CPOs each having approximately an equalshare in the database.The CTA returns, as well the returns of the CSFB/Tremont and CISDMindices, used in this analysis have not been adjusted to eliminate biases inher-
ARMA Modeling of CTA Returns
371
ent in them. A description of potential biases in the indices, some of whichapply to the databases as well, can be found in Karavas and Siokos (2004).The following tables provide descriptive statistics for the data set usedin the simulations as well as for the corresponding excess returns. As we seein Table 21.1, CTAs offer a wide range of performance characteristics. It isnoteworthy to see that the risk-adjusted return as proxied by the information ratio varies significantly relative to the information ratio of the CISDMCTA index. This means that across the 10 largest CTAs in existence for atleast eight years, the majority of them offer returns that are not justified forthe amount of risk they undertake (see Table 21.2). Information ratios inbold denote values below the information ratio of the CISDM CTA index.
RESULTSThe ADF tests showed that for all CTAs included in this study, the error termswere white noise; thus all series were stationary. With the exception of oneCTA (#3), we could reject the null hypothesis of unit root for all CTAs at 99percent confidence level (#3: at 90 percent). All the ADF tests were run forfour lags; the results are shown in Table 21.3. Similar tests were performedon CTAs’ excess returns and are shown in Table 21.4. The results using CTAreturns were consistent with those in Gregoriou and Rouah (2003a) for hedgefunds. Those authors did not examine excess returns, however, this studyshows that the added alpha relative to the strategy (as proxied by the CISDMCTA index) for the 10 largest funds is indeed stationary.Using the correlograms, we determined that in several cases the autocorrelations did not fade after the first lag, so more lags needed to beincluded in the models. As we see in Table 21.3, the CTA returns studiedcarry the effect of previous months return levels. The table shows the different orders of ARMA models that have been utilized to better representthe corresponding return series. In certain cases (CTA: 2, 5, 10) the representation is adequate, evidenced by relatively high R2 values and significantcoefficients. For CTA #9, although there is a relatively high R2, the MAprocess is noninvertible. For CTA #3, although we have not rejected theexistence of unit root at 95 percent, we have used an ARMA (2,2) modelwith a low R2. We note that CTAs #2 and #3 are the only ones that are lownegatively correlated with the CTA Index.Table 21.4 presents similar results to Table 21.3 using excess returns.The benefit of studying CTAs’ excess returns is it allows us to see whetherand how individual CTAs outperform the strategy to which they belong. Itis rather useful when managers of specific strategies are evaluated for inclu-
372
9.26%
14.47%5.98%14.77%6.00%10.32%12.58%13.84%14.42%15.53%15.11%1.04
1.180.991.071.431.241.321.110.781.340.74
InformationStandardRatioa
7.80%
59%
41%
b
1.00
0.81−0.12−0.110.050.540.780.830.790.750.52
−4.45%
35%32%38%31%42%42%40%44%39%45%
−9.62%−5.50%−11.97%−2.29%−4.49%−7.95%−10.33%−9.68%−6.99%−8.03%65%68%63%69%58%58%60%56%61%55%
12.22%9.65%9.44%10.39%9.48%16.50%9.54%11.01%14.36%12.23%
MonthlyMinReturn
CTA information ratios that are lower than the CISDM CTA index information ratio are in bold.Assets under management.
9.60%
CISDM
a
17.04%5.92%15.73%8.60%12.79%16.58%15.35%11.20%20.87%11.22%
AnnualizedAnnualized Rate ofReturnDeviation
CorrelationMonthly% of% ofwithMax Profitable Non-profitableCISDMReturn MonthsMonthsCTA
Statistics for the 10 Largest CTAs, January 1996 to December 2003
CTA1CTA2CTA3CTA4CTA5CTA6CTA7CTA8CTA9CTA10
TABLE 21.1
$4,523$2,103$1,463$1,385$1.192$1,134$ 950$ 890$ 839$ 821
AUMb inmillions(USD)Q3-03
373
ARMA Modeling of CTA Returns
TABLE 21.2 Statistics for the Excess Returns of the 10 Largest CTAs, January1996 to December 2003
CTA1CTA2CTA3CTA4CTA5CTA6CTA7CTA8CTA9CTA10
AnnualizedExcessReturn
AnnualizedStandardDeviation
MonthlyMinReturn
MonthlyMaxReturn
% ofMonthswithPositiveExcessReturn
7.07%−4.26%4.58%−1.73%2.59%6.46%5.51%1.66%10.59%1.34%
8.90%11.60%18.30%10.79%9.43%7.83%8.01%9.10%10.54%13.01%
−5.42%−8.71%−12.73%−9.26%−5.59%−4.04%−5.96%−6.96%−4.59%−12.43%
8.25%11.28%11.08%7.81%6.86%9.29%5.96%6.43%10.98%12.21%
60%49%59%50%60%57%55%52%59%48%
% ofMonthswithNegativeExcessReturn40%51%41%50%40%43%45%48%41%52%
sion in portfolios of CTAs (CPOs) or in portfolio of mixed strategies andthe objective is to maximize alpha.Table 21.2 shows that CTA #2 has underperformed the CISDM CTAIndex, but Table 21.4 shows its series (excess returns) appears to be stationary. Excess returns of CTA #9 and #10 are adequately represented bythe ARMA models shown in Table 21.4, as evidenced by high R2 and significant coefficients. Both CTAs have outperformed the CISDM CTA Index,but they were the most volatile of the 10 CTAs and the index.We then performed a stability test on the ARMA model parameters toinvestigate possible structural changes. For this purpose we utilized theChow test before and after January 2000. The justification for this breakpoint is that 1999 was a very profitable year for the equity indices; CTAsdid not perform as well afterward.Chow test statistics appear in Table 21.3. The F-statistics for threeCTAs are relatively high, indicating structural changes. For CTA #9, we didnot test for structural changes as the MA process was noninvertible, and themodel did not fit better even for the period 2000 to 2003.For the three CTAs with relatively high F-statistics, we fitted the corresponding ARMA models for the period 2000 to 2003. As shown in Table 21.5,
374
0.01440.00060.00150.22880.01410.00550.00650.00000.00960.00340.01380.00050.01110.00000.00970.00000.01600.00030.00980.0006
M−0.87780.0000−0.51060.0000−0.79100.0000−0.14930.1632−0.9215−0.0000
−0.44470.0000−0.56180.00001.12310.0000−0.83220.0000−0.4473−0.0000−0.92490.0000−0.85660.00000.94020.0000−0.89770.0000−1.55090.0000−1.32940.0000
−0.14790.1546
AR(2)A2
AR(1)A1
−0.58110.0022
0.74820.0000
AR(3)A3
−0.27690.0039
AR(4)A40.45110.00000.66380.0000−1.16840.0000−0.83780.00000.55980.00000.98010.00000.97400.0000−0.98140.0000−1.12740.00001.35080.0000
MA(1)B1
0.93440.0000
1.04300.0000
0.97990.0000−0.62490.00000.97990.0000
MA(2)B2
All ADF tests are at 99 percent confidence level. CTA3 rejects hypothesis of unit root at 90 percent.
−4.5596
−5.1140
−5.4682
−4.7019
−4.9529
−4.9350
−5.7926
−3.4275
−5.6161
−5.6629
ADFTests
CTA Returns: ARMA Models
CTA1CTA1: p-valueCTA2CTA2: p-valueCTA3CTA3: p-valueCTA4CTA4: p-valueCTA5CTA5: p-valueCTA6CTA6: p-valueCTA7CTA7: p-valueCTA8CTA8: p-valueCTA9CTA9: p-valueCTA10CTA10: p-value
TABLE 21.3
0.15290.0000
−0.95810.0000
MA(3)B3
0.15
0.20
0.06
0.03
0.01
0.15
0.03
0.06
0.31
0.04
R2
0.340.93
1.490.201.400.222.250.063.950.010.230.971.950.130.530.662.210.07
ChowF-Statp-value
375
ADFTests
M
0.92930.0000−0.52020.0004
−0.78770.0000−0.89450.0000−0.75390.0000−0.47240.0000−0.56440.0376
AR(2)A2
AR(4)A4
MA(1)B1
MA(2)B2
MA(3)B3
0.71090.0262−0.85920.00000.59470.98000.00000.0000−0.41870.96170.00000.0000−1.22200.96380.00000.0000−0.70670.57220.5721 0.96610.00000.00000.0000 0.0000−0.82710.69830.00090.00530.57680.07060.1356 −0.6643 −0.4929 − 1.0160 −0.41150.2974 0.0000 0.00000.0000 0.0000−1.10910.38890.00000.0300
AR(3)A3
All ADF tests are at 99 percent confidence level. CTA9 rejects hypothesis of unit root at 95 percent.
−0.72030.0191−0.71320.0000−0.52930.0000−0.36770.00001.07160.0000−0.59970.0000−0.78900.0039−0.45600.17710.54980.00000.77680.0000
AR(1)A1
CTA Excess Returns: ARMA Models
CTA Exc10.0059−4.7461CTA Exc1: p-value0.0259CTA Exc2−0.0043−5.3094CTA Exc2: p-value0.0178CTA Exc30.0051−4.3574CTA Exc3: p-value0.3588CTA Exc4−0.0012−5.3569CTA Exc4: p-value0.7033CTA Exc50.0025−4.1900CTA Exc5: p-value0.3855CTA Exc60.0046−4.2616CTA Exc6: p-value0.0473CTA Exc70.0051−4.8709CTA Exc7: p-value0.0572CTA Exc80.0014−5.1742CTA Exc8: p-value0.6276CTA Exc90.0100−3.9901CTA Exc9: p-value0.0000CTA Exc100.0016−6.7692CTA Exc10: p-value0.2435
TABLE 21.4
0.00
0.01
0.11
0.07
0.03
0.12
0.07
0.01
R2
0.20
0.92480.160.0000
MA(4)B4
376TABLE 21.5
PROGRAM EVALUATION, SELECTION, AND RETURNS
CTA Returns, 2000 to 2003: ARMA Models
CTA3CTA3: p-valueCTA4CTA4: p-valueCTA8CTA8: p-value
M
AR(1)A1
AR(2)A2
MA(1)A1
MA(2)A2
0.01230.08950.00500.02880.01200.0831
−0.80420.0000−0.57340.0126−0.70180.0000
−0.65460.00000.09560.5748−0.14820.3521
0.99940.00000.87310.00000.95290.0000
0.98000.0000
R20.160.040.09
there is a significant improvement for CTA #3 and #8 (evidenced by theincreased R2). For CTA #4, ARMA (1,1) (results not shown) appear to better model the return series during 2000 to 2003 than the ARMA (2,1)model utilized for 1996 to 2003 and 2000 to 2003.
CONCLUSIONIn this study, we investigated the return series behavior of the 10 largestCTAs in the CISDM database and utilized a number of ARMA models.Results showed that the series are in general stationary (using ADF tests),as are the excess returns of the same CTAs relative to the CISDM CTAIndex. ARMA models for the largest CTAs tended to be of higher ordersthan those in the case of hedge funds (Gregoriou and Rouah 2003b). Inspite of the significant parameters in most cases, very few of these CTAmodels were accompanied by substantial R2. Unfortunately, this impliesthat the models have little forecasting power. A few indicated possible structural changes, evidenced by Chow tests. For two CTAs the same modelsoffered a better representation for the period after the breakpoint (January2000), while for the third CTA a different ARMA model appears to offerbetter results.
CHAPTER
22
Risk-Adjusted Returns of CTAs:Using the Modified Sharpe RatioRobert Christopherson and Greg N. Gregoriou
any institutional investors use the traditional Sharpe ratio to examinethe risk-adjusted performance of CTAs. However, this could pose problems due to the nonnormal returns of this alternative asset class. A modified VaR and modified Sharpe ratio solves the problem and can provide asuperior tool for correctly measuring risk-adjusted performance. Here werank 30 CTAs according to the Sharpe and modified Sharpe ratio and findthat larger CTAs possess high modified Sharpe ratios.
M
INTRODUCTIONThe assessment of portfolio performance is fundamental for both investors and funds managers, as well as commodity trading advisors(CTAs). Traditional portfolio measures are of limited value when appliedto CTAs. For instance, applying the traditional Sharpe ratio will overstatethe excess reward per unit of risk as measure of performance, with riskrepresented by the variance (standard deviation) because of the nonnormal returns of CTAs.The mean-variance approach to the portfolio selection problem developed by Markowitz (1952) has been criticized often due to its utilizationof variance as a measure of risk exposure when examining the nonnormalreturns of CTAs. The value at risk (VaR) measure for financial risk hasbecome accepted as a better measure for investment firms, large banks,and pension funds. As a result of the recurring frequency of down markets since the collapse of Long-Term Capital Management (LTCM) inAugust 1998, VaR has played a paramount role as a risk managementtool and is considered a mainstream technique to estimate a CTA’s exposure to market risk.
377
378
PROGRAM EVALUATION, SELECTION, AND RETURNS
With the large acceptance of VaR and, specifically, the modified VaR asa relevant risk management tool, a more suitable portfolio performancemeasure for CTAs can be formulated in term of the modified Sharpe ratio.1Using the traditional Sharpe ratio to rank CTAs will underestimate the tail risk and overestimate performance. Distributions that arehighly skewed will experience greater-than-average risk underestimation.The greater the distribution is from normal, the greater is the risk underestimation.In this chapter we rank 30 CTAs according to the Sharpe ratio andmodified Sharpe ratio. Our results indicate that the modified Sharpe ratiois more accurate when examining nonnormal returns. Nonnormality ofreturns is present in the majority of CTA subtype classifications.
LITERATURE REVIEWMany CTAs produce statistical reports that include the traditional Sharperatio, which can be misleading because funds will look better in terms ofrisk-adjusted returns. The drawback of using a traditional Sharpe ratio isthat it does not distinguish between upside and downside risk.VaR has emerged in the finance literature as a ubiquitous measure ofrisk. However, its simple version presents some limitations. Methods tomeasure VaR such as, the Delta-Normal method described in Jorion (2000),are simple and easy to apply. However, the formula has a drawback sincethe assumption of normality of the distributions is violated due to the useof short-selling and derivatives strategies such as futures contracts frequently used by CTAs.Several methods have been proposed recently to correctly assess theVaR for nonnormal returns (Rockafellar and Uryasev 2001). Using a conditional VaR for general loss distributions, Agarwal and Naik (2004) con-
1The standard VaR, which assumes normality and uses the traditional standard
deviation measure, looks only at the tails of the distribution of the extreme events.This is common when examining mutual funds, but when applying this techniqueto funds of hedge funds, difficulty arises because of the nonnormality of returns(Favre and Galeano 2002a, b). The modified VaR takes into consideration themean, standard deviation, skewness, and kurtosis to correctly evaluate the riskadjusted returns of funds of hedge funds. Computing the risk of a traditional investment portfolio consisting of 50 percent stocks and 50 percent bonds with thetraditional standard deviation measure could underestimate the risk in excess of 35percent (Favre and Singer 2002).
379
Risk-Adjusted Returns of CTAs
struct a mean conditional VaR demonstrating that mean-variance analysisunderestimates tail risk. Favre and Galeano (2002b) also have developed atechnique to properly assess funds with nonnormal distributions. Theydemonstrate that the modified VaR (MVaR) does considerably improve theaccuracy of the traditional VaR. The difference between the modified VaRand the traditional VaR is that the latter only considers the mean and standard deviation, while the former takes into account higher moments suchas skewness and kurtosis.The modified VaR allows one to calculate a modified Sharpe ratio,which is more suitable for CTAs. For example, when two portfolios havethe same mean and standard deviation, they still may be quite different dueto their extreme loss potential. If a traditional portfolio of stocks and bondswas equally split, using the standard deviation as opposed to modified VaRto calculate risk-adjusted performance could underestimate the risk bymore than 35 percent (Favre and Galeano 2002b).
DATA AND METHODOLOGYThe data set consists of 164 CTAs who reported monthly performance figures, net of all fees, to the Barclay Trading Group database. The data spansthe period January 1997 to November 31, 2003, for a total of 83 months.We selected this period because of the extreme market event of August 1998(Long-Term Captial Management collapse) as well as the September 11,2001, attacks. From this we extracted and ranked the top 10, middle 10,and bottom 10 funds according to ending assets under management. We usethis comparison to see if there exist any differences between groups in termsof the Sharpe and modified Sharpe ratio. We use the Extreme metrics software available on the www.alternativesoft.com web site to compute theresults using a 99 percent VaR probability, and we assume that we are ableto borrow at a risk-free rate of 0 percent.The difference between the traditional and modified Sharpe ratio isthat, in the latter, the standard deviation is replaced by the modified VaR inthe denominator. The traditional Sharpe ratio, generally defined as theexcess return per unit of standard deviation, is represented by this equation:Sharpe Ratio =
Rp − RF
σ
where RP = return of the portfolioRF = risk-free rate ands = standard deviation of the portfolio
(22.1)
380
PROGRAM EVALUATION, SELECTION, AND RETURNS
A modified Sharpe ratio can be defined in terms of modified VaR:Modified Sharpe Ratio =
Rp − RFMVaR
(22.2)
The derivation of the formula for the modified VaR is beyond thescope of this chapter. Readers are guided to Favre and Galeano (2002b)and Christoffersen (2003) for a more detailed explanation.
EMPIRICAL RESULTSDescriptive StatisticsTable 22.1 displays monthly statistics on CTAs during the examinationperiod, including mean return, standard deviation, skewness, excess kurtosis,and compounded returns.The average of the compounded returns and mean monthly returns isgreatest in the top group (Panel A) and the lowest in the bottom group, asexpected. In addition, we find that negative skewness is more pronouncedin the bottom group, yielding more negative extreme returns, whereasthe middle group (Panel B) has the greatest positive skewness. A likelyexplanation is that the middle-size CTA may better control skewness during down markets and will have on average fewer negative monthlyreturns. Large CTAs may have a harder time getting in and out of investment positions.The bottom group (Panel C) has the highest volatility (standard deviation 32.56 percent) and lowest compounded returns (18.29 percent), likelyattributable to CTAs taking on more risk to achieve greater returns.
Performance DiscussionTable 22.2 presents market risk and performance results. First, observe thatthe top group (Panel A) has, in absolute value, the lowest normal and modified VaR (i.e., is less exposed to extreme market losses). Furthermore, thebottom group (Panel C) has in absolute value the highest normal and modified VaR, implying that CTAs with small assets under management aremore susceptible to extreme losses. This is not surprising, because they havethe lowest monthly average returns, as seen in Table 22.1.Concerning performance, the bottom group has the lowest traditionalmodified and modified Sharpe ratios. It appears that large CTAs do a better job of controlling risk-adjusted performance than can small CTAs. Comparing the results of the traditional and the modified Sharpe ratios, we findthat the traditional Sharpe ratio is higher, confirming that tail risk is underestimated when using the traditional Sharpe ratio.
381
Risk-Adjusted Returns of CTAs
TABLE 22.1 Descriptive Statistics
FundName
AverageAverageAssets Annualized AnnualizedCompounded(EndingReturnStd. Dev.ExcessReturnMillions $)(%)(%)Skewness Kurtosis(%)
Panel A: Subsample 1: Top 