
FuturesFundIndex
−0.30−1.373.679.99-0.34
0.52%
0.43%0.29%
27
MSCIGlobal
40–50%
355
(continued)
Sample Size
MeanStd. Dev.DD byMAR
SkewnessKurtosisSharpeSortinoCorrelation
TABLE 19.4
Futures
FundIndexvs.MSCI
Global
−0.52−1.2517.2312.560.21
1.57%
4.32%
0.630.820.873.24
1.51%0.30%
28
MSCIGlobal
1.18%4.15%
28
FuturesFundIndex
50–60%
0.642.070.191.05
3.90%
0.41%3.88%
28
FuturesFundIndex
−0.30−1.3228.8013.37−0.01
2.49%
2.42%0.32%
28
MSCIGlobal
60–70%
−0.480.18−0.49−1.23
3.33%
0.12−1.4534.4714.290.31
3.66%
283.57%0.43%
28
MSCIGlobal
−0.29%3.31%
FuturesFundIndex
70–80%
1.614.081.986.28
5.88%
2.77%5.15%
28
FuturesFundIndex
0.47−1.1650.8215.510.38
5.04%
4.94%0.44%
28
MSCIGlobal
80–90%
0.930.440.652.47
5.71%
1.25%5.56%
28
FuturesFundIndex
0.82−0.3424.4117.870.26
8.02%
7.71%1.68%
28
MSCIGlobal
90–100%
356
0/10010/9020/8030/7040/6050/5060/4070/3080/2090/10100/0
Combination
Returns
−3.36%−2.86%−2.36%−1.85%−1.35%−0.85%−1.02%0.16%0.66%1.16%1.67%
−7.57%−6.57%−5.57%−4.57%−3.57%−2.57%−3.08%−0.57%0.43%1.44%2.44%
10–20%
Returns
0–10%
−1.86%−1.71%−1.56%−1.42%−1.27%−1.12%−1.34%−0.83%−0.68%−0.53%−0.38%
Returns
20–30%
−0.58%−0.49%−0.40%−0.31%−0.22%−0.14%−0.16%0.04%0.13%0.22%0.31%
Returns
30–40%
0.43%0.43%0.44%0.45%0.45%0.46%0.55%0.47%0.48%0.49%0.50%
Returns
40–50%
1.51%1.48%1.44%1.41%1.38%1.34%1.61%1.28%1.25%1.21%1.18%
Returns
50–60%
TABLE 19.5 Combining Futures and Equity Indices in Different Proportions
2.42%2.22%2.02%1.82%1.62%1.42%1.70%1.01%0.81%0.61%0.41%
Returns
60–70%
3.57%3.18%2.79%2.41%2.02%1.64%1.96%0.87%0.48%0.09%−0.29%
Returns
70–80%
90–100%
4.94%4.72%4.50%4.29%4.07%3.85%4.62%3.42%3.20%2.99%2.77%
7.71%7.06%6.41%5.77%5.12%4.48%5.37%3.18%2.54%1.89%1.25%
Returns Returns
80–90%
7.19%7.45%7.72%7.98%8.25%8.51%10.22%9.04%9.31%9.58%9.84%
PortfolioReturns
CTA Strategies for Returns-Enhancing Diversification
357
were to look for the least number of negative periods, then the combinationof 90/10 would almost ensure that there would only be a 1 in 10 chance ofnegative returns.The results illustrate a useful idea: If we are concerned about event risk,we may wish to define our objective function as one that has the least number of negative returns during the investment horizon, with the constraintthat the correlation at first decile should be the lowest. This could be a useful framework to carry out constrained optimization of portfolio returns.
CONCLUSIONOur results indicate that the risk-adjusted returns as measured by Sharpeand Sortino ratios are always higher in CTA strategies than in most traditional asset classes for the entire sample period under study. Unlike hedgefunds, the correlation coefficients of the CTAs with the equity markets arenegative during bad times (worst performance period of the equity markets). Yet the volatility (measured by downside deviation) of CTA strategiesis lower compared to equity indices. For the up-market months, CTA strategies are associated with high Sortino ratios.The negative correlations of CTAs with equity indices during periods ofmarked downturns of equity markets indicate that CTAs can provide aneffective hedge against catastrophic event risks. While hedge funds also provide diversification, they have positive correlation with equity indices indown markets, especially when extreme events occur. Hence our findingssuggest that adding more tightly regulated CTA strategies to an equity portfolio can improve its overall risk-return profile. Such strategies not onlyprovide the usual portfolio diversification effects, but, given the negativecorrelation in down markets, CTAs are returns-enhancing diversifiers.Although our findings present strong reasons to use CTAs, their use maynot be without a cost. Liang (2003) found that attrition rates are higher forCTAs when compared with hedge fund and hedge fund of funds. However,the reasons why CTAs are return-enhancing diversifiers deserve furtherinvestigation. The level of liquidity risk borne may be an important difference between hedge funds and CTAs.
CHAPTER
20
Incorporating CTAs intothe Asset Allocation Process:A Mean-Modified Valueat Risk FrameworkMaher Kooli
alue at risk has become a heavily used risk management tool, and animportant approach for setting capital requirements for banks. In thisstudy, we examine the effect of including a CTA in a traditional portfolio.Using a mean-modified value at risk framework, we examine the case of aCanadian pension fund and compute the optimal portfolio by minimizing themodified value at risk at a given confidence level.
V
INTRODUCTIONFor the individual or the institutional investor who is simultaneouslyperformance-oriented and risk-conscious, the key question is how best toachieve a higher overall rate of return with acceptable risk. The answer maybe a diversified investment portfolio with some portion of the total assetsinvested in alternative investments. According to a survey by Nakakubo(2002), the alternative investment market reached $550 to $600 billion at theend of 2001. Pension funds also are increasing the proportion of alternativeinvestments in their asset allocation.For many institutional investors, alternative investments are viewedlargely as private, illiquid, alternative investments that include venture capital, leveraged buyout, distressed securities, private equity, private debt, oil andgas programs, and timber or farmland. However, other alternative investmentvehicles, such as hedge funds and commodity trading advisors (CTAs), alsohave observed a dramatic increase in investment and often provide access to
358
Incorporating CTAs into the Asset Allocation Process
359
investment not easily available from traditional stock and bond investment.For instance, the Managed Accounts Reports (MAR) cites an increase in managed futures1 from less than $1 billion in 1980 to almost $35 billion in 1999;hedge fund investment is now estimated to be over $300 billion. Further, Lintner (1983) uses the composite performance of 15 trading advisors and showthat the return/risk ratio of a portfolio of trading advisors (or futures funds)is higher than a well-diversified stock/bond portfolio. Furthermore, he finds alow correlation between the returns of trading advisors and those of stocks,bonds, or a combined stock/bond portfolio. Lintner examines the 1979 to1982 period. Schneeweis and Spurgin (1997) show that various CTA andhedge fund, energy-based investment provide risk and return opportunitiesnot available from a wide range of traditional commodity investments or realestate investments. The Chicago Mercantile Exchange (1999) showed that forthe 1980 to 1998 period, managed futures investments (as measured by theBarclay CTA Index) had a compound annual return of 15.8 percent. Thatcompares very favorably with the 17.7 percent return that common stockshad during the same period, one of the strongest stock markets in U.S. history. Further, it exceeded the 11.8 percent return on bonds. Moreover, duringa similar period (1980 to 1997), analysis shows that a portfolio that comprised some managed futures had similar profitability with far less risk. Liang(2003) finds that CTAs are good hedging instruments for hedge funds, fundof funds, and equity markets when the others are not well hedged. This isespecially true in down markets. Schneeweis and Georgiev (2002), in examining the benefits of managed funds, show that CTAs reduce portfolio volatility risk, enhance portfolio returns in economic environments in whichtraditional stock and bond investment media offer limited opportunities, andparticipate in a wide variety of new financial products and markets not available in traditional investor products. However, they note that for managedfutures to grow as an investment alternative, individuals need to increase theirknowledge and comfort level regarding the use of managed futures in theirinvestment portfolios. For instance, there is still some confusion about theperformance of CTAs as supply has expanded. In this study we first analyzethe risk and return benefits of CTAs, as an alternative investment, using amore precise measure of risk. Then, we show how CTAs can be integratedinto existing investment strategies and how to determine the optimal proportion of assets to invest in such products.
1
The term “managed futures” describes an industry made up of professional moneymanagers known as commodity trading advisors. These trading advisors manageclient assets on a discretionary basis using global futures markets as an investmentmedium.
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PROGRAM EVALUATION, SELECTION, AND RETURNS
MEAN-MODIFIED VALUE AT RISK FRAMEWORKInvestment decisions are made to achieve an optimal risk/return trade-offfrom the available opportunities. To meet this objective, the portfolio manager has to identify the set of assets that are the most efficient, in the senseof providing the lowest level of risk for a desired level of expected return,and then to select one combination that is consistent with the risk aversionof the investor. Mean-variance analysis has been increasingly applied toasset allocation and is now the standard formulation of the investment decision problem. Although the principle of identifying portfolios with therequired risk and return characteristics is clear, the proper definition of riskis vague. Risk may be defined differently according to the sensibility and theobjectives of the portfolio manager. One manager might define risk asthe probability of underperformance relative to some benchmark level ofreturn, while another may be more sensitive to the overall magnitude of aloss. In a mean-variance framework, risk is defined in terms of the possiblevariation of expected portfolio returns. The focus on standard deviation asthe appropriate measure for risk implies that investors weigh the probability of negative returns equally against positive returns. However, it is highlyunlikely that the perception of investors to downside risk faced on investments is the same as the perception to the upward potential. Thus, investorsneeded a more precise measure of downside risk.With the value at risk (VaR) approach, it is possible to measure theamount of portfolio wealth that can be lost over a given period of time witha certain probability. VaR has become a widely used risk management tool.The Basel Accord of 1988, for example, requires commercial banks to compute VaR in setting their minimum capital requirements (see Jorion 2001).One of the main advantages of VaR is that it works across different assetclasses such as stocks and bonds. Further, VaR often is used as an ex-postmeasure to evaluate the current exposure to market risk and determinewhether this exposure should be reduced.Our objective consists in drawing the efficient frontiers based on theVaR framework. We also use the Cornish-Fisher (1937) expansion toadjust the traditional VaR with the skewness and kurtosis of the return distribution, which often deviates from normality.2 We call the VaR with theCornish-Fisher expansion modified VaR. Favre and Galeano (2002b) showthat risk measured only with volatility will be lower than risk measured
2Mina and Ulmer (1999) provide four methods—Johnson transformations, Cornish-
Fisher expansion, Fourier method, partial Monte-Carlo—to compute the VaR fornonnormally distributed assets.
Incorporating CTAs into the Asset Allocation Process
361
with volatility, skewness, and kurtosis. Thus, results with modified VaR willbe less biased. For details on obtaining the normal VaR, the Cornish-Fisherexpansion to VaR, and other VaR methods, see Christoffersen (2003).
CHARACTERISTICS OF CTABefore we engage in a detailed analysis of the risk-return properties of theCTA, a word of caution is necessary: Unlike traditional asset classes (bondsand equity), where performance data and benchmarks are readily and reliably available, the infrastructure and reliability of performance data foralternative investments, in general, and CTAs, in particular, are still ratherunderdeveloped. In this chapter, the CTA Qualified Universe index3 (CTAQU) is used to give an overall picture of CTA, as it is more representativeof the performance of trading advisors as a whole and cannot be criticizedas having selection bias.The sample portfolio is made up of CTA, Canadian, U.S., and international equities as well as domestic bonds. Canadian equities are representedby the Standard & Poor’s (S&P)/Toronto Stock Exchange index, the CTAby the CTA QU Index (from CISDM database), the U.S. equities asset bythe S&P 500 Index, the international equities asset by the Morgan StanleyCapital Index for Europe, Asia, and the Far East (MSCI EAFE), and thebonds by the Scotia McLeod universe bond index. We use monthly datafrom January 1990 to February 2003.Within the assets considered (see Table 20.1), the CTA index is lessrisky than the S&P 500, the S&P/TSX, and the MSCI EAFE indices. Inaddition, CTA QU index possesses a higher Sharpe ratio than equity indices,indicating that CTAs offer superior risk-adjusted returns. These estimatesmay understate true risk, so monthly modified Sharpe ratios (using VaRinstead of standard deviation) is also presented and confirms the advantageof the CTA QU index. Using VaR and modified VaR to measure risk, theCTAs are still less risky than equity indices. For instance, a one percent VaRof −5.3 percent for CTA QU index means that there is a 1 percent chancethat the loss will be greater that 5.3 percent next month (or a 99 percentchance that it will be less than 5.3 percent).Besides very attractive risk adjusted return characteristics, one of themost important features of CTAs is their favorable correlation structure totraditional assets classes (see Table 20.2). By including CTAs in their portfolios, traditional asset managers are given the opportunity to producemore consistent returns with lower levels of risk in their global portfolio by
3See www.cisdm.org for data and description of CTA Qualified Universe Index.
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PROGRAM EVALUATION, SELECTION, AND RETURNS
TABLE 20.1 Characteristics of CTA and Traditional Asset Classes, January1990 to February 2003Assets
AnnualMean
AnnualVolatility
Skewness
ExcessKurtosis
CTA QU IndexSCM Bond IndexS&P/TSX IndexS&P 500 IndexMSCI EAFE Index
11.8%9.8%7.3%11.1%1.9%
10.2%5.5%15.7%15.2%16.4%
0.7−0.2−0.7−0.5−0.5
2.20.72.30.60.8
Assets
MonthlyNormalVaR
MonthlyModifiedVaR
MonthlySharpeRatio
ModifiedSharp Ratio
CTA QU IndexSCM Bond IndexS&P/TSX IndexS&P 500 IndexMSCI EAFE Index
−5.9%−2.9%−9.9%−9.3%−10.8%
−5.3%−3.4%−13.9%−11.2%−13.1%
0.180.230.040.11−0.05
0.100.110.010.04−0.02
means of diversification. CTA QU index has negative correlation to equitymarkets (−0.19 correlation to MSCI EAFE, −0.13 correlation to the S&P500, and −0.12 correlation to the TSX/S&P). Furthermore, CTAs demonstrate remarkably low correlation with the bond market (0.20). Thus,including CTAs in a diversified asset portfolio may provide additionaldiversification benefits.
TABLE 20.2 Correlations Across CTA and Traditional Asset Classes,January 1990 to February 2003
CTA QU IndexSCM Bond IndexS&P/TSXS&P 500MSCI EAFE
CTA QUIndex
SCM
S&P/TSX
S&P500
MSCIEAFE
10.20−0.12−0.13−0.19
10.320.260.20
10.750.66
10.70
1
363
Incorporating CTAs into the Asset Allocation Process
INCORPORATING CTA TO THE ASSETALLOCATION PROCESSIn this section, we show the results obtained by applying the mean-VaRframework explained previously. We compute the efficient frontier and theoptimal portfolio allocation for a Canadian pension fund assuming that theportfolio manager has a VaR limit, that is, the manager does not want tolose more than a specified amount each month, with a specified probability(typically 1 or 5 percent).The individual asset classes can vary within specific limits. As a result,a relatively conservative asset allocation was chosen to match the allocations of conservative investors, pension funds, and institutions. Theweightings of individual asset classes are then changed within the permitted margins to minimize the normal VaR (see Table 20.3). This first steppermits us to examine the effect of including a CTA in a traditional portfolio. In the second step, modified VaR values are used to measure riskmore precisely.Table 20.4 shows that CTAs take the place of U.S equities. Once theweights of the tangent portfolios are obtained, we compute the monthlyreturns that each portfolio would have yielded from January 1990 to February 2003. Based on these monthly returns, we compute the average returnover the period and the modified VaR. We obtain the results shown in Table20.5, which shows that while the average return of the portfolio with 10percent CTA is less than the one with 0 percent CTA, the level of risk, measured with the modified VaR, is decreased by adding CTA. The modifiedSharpe ratio is also improved by adding CTA investments in the traditionalportfolio.
TABLE 20.3 Upper and Lower Limits forIndividual Asset ClassesAsset ClassCommoditytrading advisorsCanadian equitiesU.S. equitiesInternationalequitiesCanadian bonds
Minimum
Maximum
0%10%0%
10%40%30%
0%25%
30%50%
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PROGRAM EVALUATION, SELECTION, AND RETURNS
TABLE 20.4
Portfolio Weights from Mean-VaR
OptimizationNo CTAAvailable
CTA InvestmentLimit of 10%
CTA QU IndexMSCI EAFES&P 500SCMBond IndexS&P/TSX
0%6%29%
10%8%17%
50%15%
50%15%
Total
100%
100%
Asset Class
Further, Figure 20.1 shows the degree to which the sample portfoliowith a CTA portion of maximum 10 percent is represented too positively ifwe do not take into account the skewness and kurtosis of the return distributions—in other words, if we do not use modified VaR. It is assumed thatthe investor is seeking an annual return of 7.2 percent with this sampleportfolio. Our calculation using the Cornish-Fisher expansion shows thatthe investor will underestimate the risk by 14.28 percent if he or she is looking to achieve this return with normal VaR.The crucial question for an investor is whether including CTAs as analternative investment makes sense for his or her portfolio. To assess this,we use both normal and modified VaR with traditional and nontraditionalportfolios (with CTA).The arrows in Figure 20.2 show the shift in efficiency lines or, rather,the positive effect on including CTA QU index in a traditional portfolio.Figure 20.3 shows the added value of CTAs if skewness and kurtosis aretaken into account (by using modified VaR as a risk measurement). The two
TABLE 20.5 Average Return, Modified VaR, and Modified Sharpe Ratio
Portfolio with 0% CTAPortfolio with 10% CTA
AverageReturn
ModifiedVaR
ModifiedSharpe Ratio
0.593%0.581%
5.93%4.56%
0.1000.128
365
Incorporating CTAs into the Asset Allocation Process
0.85%
b
0.80%
Monthly Returns
0.75%
a
Annual return of 7.20%
0.70%0.65%0.60%0.55%0.50%0.45%0.00%
a
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
9.00%
10.00%
Normal and Modified VaRbEfficient frontier with modified VaR.Efficient frontier with normal VaR.
FIGURE 20.1
Pension Fund Portfolio with 10% CTA
figures show the classic picture, as can be seen in a mean-variance diagram. It is obvious that including CTAs with high negative skewness andkurtosis values in a portfolio does bring a benefit in the sense of betterrisk-adjusted returns.
0.65%
with CTA
without CTA
Monthly Returns
0.60%
0.55%
0.50%
0.45%
0.40%0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
Normal VaR
FIGURE 20.2
Pension Fund Portfolio with and without 10% CTA
9.00%
366
PROGRAM EVALUATION, SELECTION, AND RETURNS
0.75%0.70%
with CTA
Monthly Returns
0.65%
without CTA
0.60%0.55%0.50%0.45%0.40%0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
Modified VaR
FIGURE 20.3
Pension Fund Portfolio with and without 10% CTA
CONCLUSIONNowadays it is clear that a traditional strategy that divides investments intoasset classes is no longer sufficient. The results of this study provide important information to the investment community about the benefits of CTAs.We show that an efficiently allocated portfolio consisting of CTAs and traditional assets should provide a better reward/risk ratio than an investmentin traditional assets. We showed, as did Favre and Galeano (2002), that itis possible to use modified VaR risk measure to build a portfolio composedof traditional and alternative assets and that has the lowest probability oflosing more than the modified VaR at a defined confidence level. However,investors must be very cautious in CTA selection. There are various CTAswith different characteristics and strategies. These differences need to be amajor consideration, perhaps even more important than the decision ofwhether to invest in the asset class itself. Finally, analysis of alternativemethods of measuring risk for alternative investments, in general, and CTAand hedge funds, in particular, is, of course, required.
CHAPTER
21
ARMA Modelingof CTA ReturnsVassilios N. Karavas and L. Joe Moffitt
n this chapter, we extend previous attempts to model hedge fund returnsusing ARMA models to the case of CTAs. We show that for the period1996 to 2003, the return series of the largest CTAs are stationary and thatARMA models in certain cases provide adequate representation of thereturn series. Comparing to the hedge fund case, we see that a higher orderof ARMA model usually is required. We also test for structural changes inthe return processes, and we fit similar models for the period 2000 to 2003.Results appear to be no drastically different from those reported in previous studies for hedge funds.
I
INTRODUCTIONThe period 1996 to 2003 offered a number of surprises to investors, withthe excellent performance of the equity market during the first four years ofthe period and the subsequent drawdown for three consecutive years until2003, when the long-expected economic recovery finally appeared. Commodity trading advisors (CTAs) did not suffer many years of losses, anddefinitely not at the magnitude of the equity markets’ losses. The CTAindices showed that all years (included in this study) were profitable for theCTAs with the exception of 1999, when small losses were reported. CTAsoffered investors a safe harbor for the years during which control was lostin the equity markets. In the next section we show pieces of historical evidence that CTAs were more stable over time, from a performance point ofview, not only when compared to equity markets but also when comparedto hedge funds.Over the past few years, a large number of hedge fund managers weredragged toward an increased equity exposure, which in several cases
367
368
PROGRAM EVALUATION, SELECTION, AND RETURNS
100%80%60%40%20%0%–20%
1996
1997
1998
1999
2000
2001
2002
2003
–40%–60%–80%–100%S&P 500 Annual ReturnCSFB Short BiasCSFB DSCSFB GM
FIGURE 21.1
CSFB MF Annual ReturnCSFB Em.MCSFB ED MultiCSFB L/S
CSFB CompositeCSFB EMNCSFB MACSFB MF
CSFB CACSFB EDCSFB FIA
CSFB Multi
CTA and Hedge Fund Strategies Correlations with S&P 500
appeared to be rather catalytic for their existence, as the expected economicrecovery, after the tech boom, did not arrive until 2003. In Figures 21.1 to21.3, it is obvious that CTAs (as proxied by Credit Suisse First Boston Managed Futures Index [CSFB MF]) have strongly resisted the downward trendin equity markets. At the same time they have offered positive returnsexcept in 1999, when they suffered mild losses. Figure 21.1 shows theannual correlation of each of the hedge fund strategies and CTAs relative toS&P 500. It also shows how the changes in the correlation with the S&P
100%80%60%40%20%0%–20%
1996
1997
1998
1999
2000
2001
2002
2003
–40%–60%–80%–100%Nasdaq Annual ReturnCSFB Short BiasCSFB DSCSFB GM
FIGURE 21.2
CSFB MF Annual ReturnCSFB Em.MCSFB ED MultiCSFB L/S
CSFB CompositeCSFB EMNCSFB MACSFB MF
CSFB CACSFB EDCSFB FIACSFB Multi
CTA and Hedge Fund Strategies Correlations with Nasdaq
369
ARMA Modeling of CTA Returns
100%80%60%40%20%0%–20%
1996
1997
1998
1999
2000
2001
2002
2003
–40%–60%–80%–100%Lehman Agg Annual ReturnCSFB Short BiasCSFB DSCSFB GM
FIGURE 21.3
CSFB MF Annual ReturnCSFB Em.MCSFB ED MultiCSFB L/S
CSFB CompositeCSFB EMNCSFB MACSFB MF
CSFB CACSFB EDCSFB FIACSFB Multi
CTA and Hedge Fund Strategies Correlations with Lehman
Aggregate
have affected the annual returns of the CSFB MF. Figures 21.