.4). First, it highlights the strong instability of volatility through time, which was not assessable with the previous statistics (see Figure 17.2). Second, even if the range of values differsfrom one index to the other (from 0.067 to 0.092 for the CTA index and0.099 to 0.131 for the managed futures index), the trend is similar, whichis comforting (the two indices are a proxy of the same universe). Note thatthe managed futures standard deviation is more volatile than the CTAstandard deviation. It confirms observations obtained with the previousstatistics.Another interesting application for the rolling statistics is on correlations.Based on a 36-month window, the correlation is estimated exactly as for thestandard deviation. The main results are shown in Figures 17.5 and 17.6.Figure 17.5 illustrates the evolution of the CTAs’ correlation with severalequity indices. Similar to the standard deviation, there is clearly instabilitythrough time, and the two indices have a similar trend. The correlationhad a strong move-down in late summer 1998 and decreased since thatperiod. In December 2002, CTAs are negatively related to the equity industry. For the CTAs’ index, the values range from −0.59 (North Americas/
0.14
CTAs' Barclay
CSFB/Tremont Managed Futures
0.130.120.110.100.090.080.07Dec-02
Jun-02
Dec-01
Jun-01
Dec-00
Jun-00
Dec-99
Jun-99
Dec-98
Jun-98
Dec-97
Jun-97
Dec-96
0.06
FIGURE 17.4 Evolution of CTA Standard DeviationThe standard deviation is annualized and estimated on a 36-month rolling basis.The database covers the period December 1993 to December 2002 and the firststandard deviation is estimated in December 1996.
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0.60Eur_eq/M_Fut
N_Am_eq/M_Fut
F_East_eq/M_Fut
NAREIT/M_Fut
0.400.200.00–0.20–0.40–0.600.600.400.200.00–0.20–0.40Dec-02
Jun-02
Dec-01
Jun-01
Dec-00
Jun-00
Dec-99
Jun-99
Dec-98
Jun-98
Dec-97
Jun-97
Dec-96
–0.60
FIGURE 17.5 Evolution of CTA Correlation with Equity IndicesThe correlation is estimated on a 36-month rolling basis. The database covers theperiod December 1993 to December 2002, and the first correlation is estimated inDecember 1996.
CTAs) to −0.21 (REIT/CTAs). This negative correlation implies that CTAsprovide positive returns when equities do not, which is a nice feature.More generally, when looking at the overall period covered by Figure17.5, CTAs tend to be positively or neutrally correlated to markets in bullish periods while being negatively correlated in bearish markets.
1.000.750.50CSFB / CTA
CSFB / M.Fut
M.Fut / CTA
0.25
FIGURE 17.6
Dec-02
Jun-02
Dec-01
Jun-01
Dec-00
Jun-00
Dec-99
Jun-99
Dec-98
Jun-98
Dec-97
Jun-97
Dec-96
0.00–0.25
Correlation at the Two CTA Indices with Each Other and with theCTA/Tremont Hedge Fund IndexThe correlation is estimated on a 36-month rolling basis. The database covers theperiod December 1993 to December 2002, and the first correlation is estimated inDecember 1996.
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Figure 17.6 shows the correlation of the two CTAs indices with eachother and the correlation with the Credit Suisse First Boston FBCS/TremontHedge Fund index. The two CTAs versus hedge funds profiles are identical, but the correlation through time (as for the standard deviation) fluctuates. With the progression of the years, the managed futures indices areless and less related to the hedge fund industry. In December 2002 (basedon the last 36-month values), the correlation is around zero. With suchresults, CTAs also can be expected to be a source of diversification forhedge fund portfolios.Note that the correlation between the two CTAs indices ranges from0.6 at the beginning of 1997 to almost 1 in December 2002. This convergence is consistent with the previously observed common trends on standard deviation and correlation for the two indices. It reflects increasingsimilarities on the different index provider’s universes. (The current dataavailable for the index computations are definitively more transparent andaccessible for any index provider than they were six or eight years ago.)
CTAs AND PORTFOLIO OPTIMIZATIONOur findings lend support to the claim that CTAs are without a doubt anextra source of diversification in portfolios. This claim is far from beingnew and is actually the main market players’ belief about CTAs. However,because something everyone believes is not necessarily true, we now focuson verifying this assumption through a simple portfolio optimization framework. This framework is based on three steps:1. Creation of different pools of assets, including pools without CTAs.2. Construction of efficient frontiers with each of the pools.3. Comparison of the efficient frontiers built with CTAs to those constructed without CTAs and determination if this hedge fund strategyadds value at the portfolio level or not in terms of risk/returns.Recall that, in a risk/return framework, the efficient frontier represents allthe risk/return combinations where the risk is minimized for a specificreturn (or the return is maximized for a specific risk). Each minima (ormaxima) is reached thanks to an optimal asset allocation. The process ofconstructing efficient frontiers through an asset weight optimization is summarized in this definition:For all possible target portfolio returns, find portfolio weights (i.e.,asset allocation) such as the portfolio volatility is minimized and thefollowing constraints are respected: no short sale, full investment, andweight limits if any.
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Clearly, the resulting efficient frontier depends on the returns, volatility, and correlations of the considered assets, but it also depends on the constraints (maximum and minimum weight limit, no short selling, and fullinvestment) fixed by the portfolio manager.Assets used in this chapter are indices only. There are advantages inconsidering indices for a portfolio optimization, because they cover marketareas large enough to avoid an excessive number of elements in the pooland cover the most relevant asset classes. Of course, they must be selectedin such a way they do not overlap each other. Practically, the chosen assetsare either standard indices (equities and bonds) or alternative investmentindices (hedge funds):Bonds indices: MSCI North Americas, MSCI Europe.Stock indices: MSCI North Americas, MSCI Europe, MSCI Far East,NAREIT index.■ Hedge funds indices: CSFB/Tremont and its nine subindices (Convertible Arbitrage, Dedicated Short Bias, Emerging Markets, EquityMarket Neutral, Event Driven, Fixed Income Arbitrage, Global Macro,Long/Short, Managed Futures).■ Two CTAs indices are available. To avoid the multiplication of figures,only the managed futures index from CSFB/Tremont is considered forthe portfolio optimizations.■■
Based on a database of 108 monthly returns (December 1993 toDecember 2002), four pools of indices are created (see Figure 17.7). Thefirst one contains only traditional assets (stocks and bonds). The secondcorresponds to the first one plus CTAs. The third one has traditional assetsand all the hedge funds strategies except CTAs. Finally, the last one is madeof all traditional assets and hedge funds strategies including CTAs. Whetherto consider or not consider CTAs in the pools should affect the generatedefficient frontiers and highlight any diversification capacity of CTAs.Concerning the portfolio optimizations, two frameworks are used: aclassical two-dimensional risk-return framework and a three-dimensionalone (a risk/return/time framework; the time being introduced with rollingstatistics). The three-dimensional framework should capture time changes,which are rarely presented in portfolio allocation studies.
Portfolio Optimization and ConstraintsBefore being specific about CTAs, it is important to have a brief reminderof portfolio optimization and constraints. As mentioned, the efficient frontier’s shape strongly depends on the weight threshold applied during the
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x
x
x
x
x
x
x
x
x
x
x
x
Pool of assets IV
x
x
x
x
x
x
Long/Short
Pool of assets IIPool of assets III
Fixed Income Arbitrage
x
Event Driven
x
Equity Market Neutral
x
Emerging Markets
REIT equity
x
Dedicated Short Bias
Far East equity
x
Arbitrage
North American equity
x
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North American bonds
Pool of assets I
Managed Futures
Europe bonds
Europe equity
Hedge Funds Indices
x
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x
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x
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x
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x
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FIGURE 17.7 Four Pools of IndicesEach pool is made of several assets: equity indices (MSCI), bond indices (MSCI),a REIT index (NAREIT), different hedge funds strategies (CSFB/Tremont), and aCTA index (Barclay Group).
portfolio optimization. The “best” efficient frontiers always are built whenthere are no weight limits (see Figures 17.8a and b.) However, unconstrainedefficient frontiers do not represent real investment conditions. Most of thetime, a “free” optimization allocates unrealistic weights to the assets. Theyalso do not fit, most of the time, either the investor’s legal requirementsand/or the risk profile (see Figure 17.9).
CTAs’ Portfolio Optimization, Full DataThis section determines whether the common belief about CTAs, that CTAsare an attractive investment vehicle and they bring diversification to portfolios, is true. We have seen that, from the statistical point of view, there isa high probability of CTAs adding value to portfolios. But, do they reallyadd value for all types of portfolios, or only under a particular asset allocation environment? To answer to these questions, several efficient frontiersare generated with various baskets of assets and constraints.Varying the asset to be included in portfolios and the constraints highlights several interesting features about CTAs. In a traditional asset universe (no hedge funds), CTAs do in fact add value to conservative
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B0.110
0.15
0.105
0.14
Return (annualized)
Return (annualized)
A
0.1000.0950.0900.0850.080
0.130.120.110.10
0.075
0.09
0.070
0.08
0.0650.03
0.04
0.05
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0.07
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Risk (annualized standard deviation)
0.09
0
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Risk (annualized standard deviation)
FIGURE 17.8 Efficient Frontiers and Asset Allocation ConstraintsTwo efficient frontiers are built on the pool of assets II (Figure A). The two optimizations assume a full investment and no short sale. The first efficient frontier (solidline) is generated without weight constraints and the second one (dashed line) withweight constraints (a maximum 50 percent allocation per asset). If the weightincreases (the constraint is less strict), the efficient frontier tends to be similar tothe unconstrained efficient frontier (Figure B, dashed line). Four efficient frontiers (solid lines) are built on the pool of assets IV (portfolio fully invested and noshort sale).
portfolios (they significantly increase low-risk portfolio returns). But thisreturn enhancement rapidly decreases and becomes null when consideringhigher risk portfolios (see Figures 17.10 and 17.11a). The return enhancement is verified in constrained and unconstrained environments. With aconstant asset universe, the lower the weight threshold is, the more important is the CTAs’ added value, which was expected (assets such as hedgefunds are rapidly capped).Finally, when considering a portfolio mixing traditional and alternativeassets, CTAs also add value but only if the optimization process is constrained (see Figures 17.10 and 17.11b). The added value itself is muchsmaller than when constructing efficient frontiers with standard indices.CTAs apparently cannot compete with the other hedge funds strategies ona free asset allocation construction. Once again, this conclusion wasexpected. In fact, even if CTAs are low correlated with the hedge fundsindustry, their returns are not exceptionally impressive. And correlation isonly one of the factors to be considered in a portfolio optimization. Weightconstraints, returns, and volatility (standard deviation) definitely influencethe definition of the optimal weight of an optimal portfolio.
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Return
0.110.100.090.080.070.061.0
Constraint (percentweight allocation)
0.750.500.25
0.080.060.04Risk0 0.02
0.10
0.12
FIGURE 17.9 Consequence of Weight Constraints on the Efficient SurfaceThe efficient surface is built on the pool of assets II. Leverage is not allowed andthe portfolio is fully invested. For each efficient frontier, the same asset isconstrained (the North Americas index) with an increasing fixed weight in theportfolio. The range of weight goes from 0 percent (unconstrained portfolio) to100 percent (portfolio made of a single asset). As a constraint increases, the efficient surface is reduced and tends to a single risk/return combination (100 percentallocation in a single asset).
As an example, let us focus on one portfolio optimization (Figures17.11a and b). These assumptions are applied on the pool of assets IV (15members): no constraints, full investment, and no short sell. The optimization includes assets having the best risk/return profiles. Hedge funds strategies like global macro or the REIT equities are immediately selected, whichis not the case for CTAs. CTAs are not included in any efficient portfolioconstruction. The asset allocation is totally different when weight restrictions are applied: The best risk/return assets are rapidly capped, and theoptimization process considers other assets such as CTAs.Note that in the real world, portfolio allocations are weight-capped.No investor takes the risk to be fully invested in a single asset family(absence of diversification). Moreover, most of the time, investors have todeal with regulations that forbid excessive weights. As it has been demonstrated that, in a constrained universe, CTAs add diversification to portfolios (especially when the original portfolio is made of standard assets) thisstrategy is worth being considered. It confirms the results discussed earlierand also investors’ belief about CTAs as a diversification vehicle.
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A
B0.095a
0.110
0.0850.080
c
b
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0.075
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Risk (annualized standard deviation)
FIGURE 17.10 Comparative CTA Portfolio OptimizationIn the first figure (A), four efficient frontiers are built with different weightthresholds (absence of constraints (line a), maximum weight per asset of 0.4 (linesb), 0.45 (lines c) and 0.5 (lines d)). The assets considered below to the pools I(standard assets, without CTAs (dashed lines)) and II (standard assets, with CTAs(solid lines)). The optimizations assume a full investment of the portfolio with noshort sell. With a low cap level, the inclusion of CTAs significantly improve theperformance of risk averse investors.Similar results are obtained with the pool of assets III/IV (standard and alternativeassets, without/with CTAs) for two of the three efficient frontiers (Figure B). Theperformance improvement is less significant than the one observed with the firstfigure. CTAs are not included in the unconstrained portfolio (single solid line).
B0.100
0.15
0.095
0.14
Return (annualized)
Return (annualized)
A
0.0900.0850.0800.075
0.130.120.110.10
0.070
0.09
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0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13
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0.02
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Risk (annualized standard deviation)
FIGURE 17.11 Comparative Unconstrained CTA Portfolio OptimizationIn figure A, two unconstrained efficient frontiers are generated on the pools ofassets I (without CTAs, dashed line) and II (with CTAs, solid line). Managedfutures add diversification to low-risk portfolios. The inclusion of CTAs on a poolof assets including standard (equities and bonds) and alternative (hedge funds)vehicles is useless in an unconstrained environment (Figure B).
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CTAs’ Portfolio Optimization, Rolling WindowBecause the previous efficient frontiers use the full database, the results givean interesting but static view of the CTAs’ diversification ability. Furthermore, time is ignored. Over the period December 1993 to December 2002,many economical, financial, and even political events impacted the marketsand influenced the CTAs’ industry. Consequently, the inclusion of time as aparameter in an efficient frontier study should provide interesting results.In practice, this time perspective is included in the efficient frontier construction simply by combining the portfolio optimization with the rollingwindows technique. For each of the rolling windows (subsample of the historical database), an efficient frontier is computed. Each efficient frontierreflects the subsample allocation structure.Because this approach considers three different variables, the clearestway to represent results is to construct a three-dimensional framework, theaxes being the risk, the return, and the time. The resulting surface is generated based on a sequence of efficient frontiers, and it can be consideredas a three-dimensional efficient surface. Practically, the surface is built inthis way:Select a 36-month data range, starting in December 1993.Compute portfolio statistics for this range of data.■ Construct the efficient frontier.■ Move to the next month and start the same process again.■ Repeat the same procedure for each month until December 2002.■■
The efficient surface presented in Figure 17.12 is the result of the constrained portfolio optimization through time on pool of assets II. The surface is unstable; significant jumps in values and some brutal lengthreductions are observable. Nevertheless, this instability is logical. Becausethe input needed for each of the efficient frontiers (the rolling statistics) varied significantly through time, so does the resulting efficient surface. Thisinstability implies that, every month, the available efficient portfolios are different. They evolve from one month to the next. Some risk/return combinations are not reachable anymore (combinations not accessible by any weightallocation2) or no longer efficient. New combinations also may emerge.
2The same portfolio construction rules are kept through time. The introduction of
leverage, for example, would substantially modify the results.
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0.30
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0.250.200.150.100.05Dec 02Dec 01Dec 00Dec 99TimeDec 98Dec 97Dec 96 0
0.250.200.150.100.05
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Dec 02 Dec 01 Dec 00
Dec 99 Dec 98 Dec 97 Dec 96
Dec 02Dec 01Dec 00Dec 99Dec 98TimeDec 970.4 Dec 96
Time
FIGURE 17.12 Efficient SurfaceThe efficient surface is built on the pool of assets II with a 30 percent weightconstraints. The efficient surface is an interpolation derived from a sequence ofefficient frontiers (generated on a monthly basis, starting December 1996). Thesurfaces highlight the instability of efficient portfolios through time.
For example, there were no high risk/return portfolios “available” inthe years 1997 and 1998. High risk portfolios were attainable during theperiod 1999 to late 2001, but in December 2002, it was not possible toinvest in such portfolios anymore (in other terms, in December 2002, therewere no weight allocations that enabled the creation of a portfolio with ahigh risk/return profile).Aside from the instability of the efficient frontier through time, it isinteresting to note that the lowest risk level for a portfolio based on thepool of assets II remains relatively stable (but the returns fluctuate).The concept of efficient surface and its visualization through a threedimensional framework is meaningful. It illustrates the importance of time
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variations in portfolio construction. And it explicitly highlights the importance, for portfolio managers, of considering dynamic allocations.But there is another way to look at the efficient surface and to extractinformation; it consists of comparing two efficient surfaces: one built on apool of assets including CTAs and one built on a similar pool withoutCTAs. Such an approach determines periods where CTAs did add value tothe portfolios, and at the same time it quantifies the extra returns. Thiscomparison can be performed easily because the efficient frontiers are builton the same framework. Note that interpolation is used during the creationof the efficient surfaces. It is therefore important to choose a sufficientlyfine partition (or grid) for the range of standard deviations.Compared to Figure 17.12, the interpretation of the resulting surface isstraightforward and much more explicit in Figure 17.13. When there is noadded value in including CTAs in a portfolio, the surface (a “diversificationsurface”) is flat. When it is penalizing to include CTAs (in terms ofrisk/return), the surface goes below zero.3 When CTAs add value to the portfolio, the surface has a positive shape. The shape itself depends on how muchthe asset adds in terms of returns. With this representation, “abnormal”reliefs can be seen. They appear only if the efficient frontier including CTAshas a wider range of risk/return combinations than the one without CTAs orif it is the contrary (the efficient frontier without CTAs has a wider range ofcombinations than the one with CTAs). In these cases, peaks (respectivelypositive and negative) emerge on the diversification surface; the peaks’height are equal to the return provided by the considered portfolio.The diversification surface presented in Figure 17.13 reflects the differences between the efficient portfolios generated with the pool of assets II(pool that includes CTAs) and the ones generated with the pool of assets I(no CTAs in this pool). The constraints are no short selling, full investment,and a weight threshold (30 percent maximum per asset).From the diversification surface, one clearly sees that, except for theperiod October 1996 to March 1997, managed futures generated new lowrisk portfolios (peaks on the surface with a height equal to the new portfolio’s return).CTAs also occasionally bring diversification to medium-risk portfolios.It is significant during 1999 and then decreases a lot the first quarter of2000. Therefore, Figure 17.13 confirms the previous observations on CTAs.They add value to portfolios, but this diversification ability is not constantover time and not verified for all efficient portfolios.
3This case was not observable because no minimum weight was imposed.
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0.10
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0.080.060.040.020Dec 02Dec 01
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FIGURE 17.13 Diversification SurfaceDiversification surface generated using two efficient surfaces; the first one has someexposure to CTAs while the second one does not invest in any CTAs. The shape ofthe surface highlights the CTAs’ diversification capacity through time and fordifferent risk levels; the higher the “relief,” the more important the CTAs’ addedvalue for a portfolio.
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CONCLUSIONThis chapter confirms that CTAs add value to portfolios but only under certain conditions. Results demonstrated that CTAs bring diversification aslong as the asset allocation environment is constrained. This diversificationability clearly increases the lower the weight threshold is (the stricter theconstraints are) and if the included assets are only standard assets.Aside from the demonstration of the CTAs’ added value, the rollingwindow analyses illustrate the time variability of the efficient frontiers. Thisfinding was expected because the input factors are themselves evolvingthrough time, proving the necessity of using dynamic asset allocation.Moreover, the analyses also reveal that CTAs did not systematicallyimprove portfolio returns over the period 1996 to 2002.More generally, the three-dimensional graphs presented are one of thefirst attempts at using surfaces as a visualization and assessment tool forasset allocation. The frameworks prove interesting for decision making,understanding efficient portfolio constructions, and temporal dynamics. Itis exciting to represent simultaneously the evolution of three variables. Andwith the growing information technology resources, this graphical representation should be used more frequently in the future.
CHAPTER
18
Random Walk Behaviorof CTA ReturnsGreg N. Gregoriou and Fabrice Rouah
his chapter examines whether CTA percent changes in NAVs follow random walks with drift. Monthly data from January 1994 to December2000 are tested for nonstationarity and random walk with drift, using theAugmented Dickey-Fuller test. All classifications (except the diversifiedsubindex) are found to behave like random walks, but many of the seriesshow evidence of a positive drift parameter, an indication that trends couldbe present in the series. The effectiveness of CTAs in enhancing risk-returncharacteristics of portfolios could be compromised when pure random walkbehavior is identified.
T
INTRODUCTIONThis chapter investigates whether monthly percent changes in net asset values (NAVs) of commodity trading advisor (CTA) classifications follow random walks. Previous econometric studies of financial time series haveemployed unit root tests, such as the Augmented Dickey-Fuller test (ADF),to identify random walk behavior in stock prices and market indices, forexample. The characteristics of CTAs are such that investment into thisalternative investment class can enhance portfolio returns, but these characteristics are likely to be mitigated if pure random walk behavior is present because that would imply a lack of evidence of value added to theportfolio (differential manager skill).Research into the performance persistence of CTAs is sparse, so thereis little information on the long-term diligence of these managers (EdwardsThis article previously appeared in Journal of Alternative Investments, No. 2, 2003.Reprinted by permission of the publisher.
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and Ma 1988; Irwin, Krukemeyer, and Zulauf 1992; Irwin, Zulauf, andWard 1994; Kazemi 1996). However, it is generally agreed that during bearmarkets, CTAs provide greater downside protection than hedge funds, andhave higher returns along with an inverse correlation with stock returns inbear markets (Edwards and Caglayan 2001). The benefits of CTAs are similar to those of hedge funds, in that they improve and can offer a superiorrisk-adjusted return trade-off to stock and bond indices while acting asdiversifiers in investment portfolios (Schneeweis, Savayana, and McCarthy1991; Schneeweis 1996).During the 1990 to 1998 period the correlation of managed futures tothe Standard & Poor’s (S&P) 500 during its best 30 months was 0.33 and−0.25 during its worst 30 months (Worthington 2001). The benefit of lowcorrelation cannot be easily replicated through other investment alternatives. According to Schneeweis, Spurgin, and Georgiev (2001), CTAs areknown to short stock markets regularly. One drawback, however, is thatduring bull markets their performance is generally inferior to those of hedgefunds (Georgiev 2001).Investors who choose to include CTAs in their portfolios usually allocate only a small portion of their assets (Georgiev 2001). Others are unaware that during increased periods of stock market volatility, carefulinclusion of CTA managers into investment portfolios can enhance theirreturn characteristics, especially during severe bear markets (Schneeweisand Georgiev 2002). Moreover, international financial markets in times ofextreme volatility this past decade, such as the Asian currency crisis of 1997and the Russian ruble crisis of August 1998, did not really affect CTAs. Infact, CTAs make their money and produce superior returns during suchperiods of high volatility. One must consider the possibility that CTA percent changes in NAVs follow random walks, for that would indicate thattemporary shocks in the NAVs are persistent and not reverting to the meanlevel. Such behavior would likely affect the timing of CTA allocation in andout of investment portfolios.Time series that are “mean reverting” are usually defined as secondorder stationary, weak stationary, or simply stationary in the statistical literature. These are characterized by a constant mean and variance and byautocorrelations that depend only on the time lag. Nonstationarity (the violation at least one