294 CHAPTER 16 Choosing the Right CTA: A Contingent Claim Approach Zsolt Berenyi M anaged futures have enjoyed a significant increase as investments dur- ing the last one and a half decades, both on a stand-alone basis and as part of a well-diversified portfolio. Managed futures accounts, indeed, seem to offer investors significant advantages not accessible elsewhere. Yet rank- ing such investment opportunities either on an ex-ante or an ex-post basis is still difficult because the risk and return structure of managed futures accounts often differs from that of (more or less) common benchmarks, and the risk structure of such investments may be unstable since CTAs may change the risk exposure of the funds individually. In this chapter we investigate the ex-post performance ranking of CTAs based on a contingent claim performance approach. In this approach, the performance of each managed futures fund is compared to individually cre- ated benchmark assets having the same risk profile in terms of particular higher moments. Benchmark assets are constructed (“replicated”) using the S&P 500, options, and the risk-free asset. Using benchmark assets, we esti- mate the efficiency gain or loss each CTA produces and analyze the robust- ness of this kind of efficiency measurement with respect to the number of moments used. INTRODUCTION Commodity funds, which are managed by commodity trading advisors (CTAs), belong to the modern alternative investment class. Managed com- modity funds (managed futures) are publicly offered investment vehicles that invest in futures and options of a wide range of financial assets as well as commodities and may employ a variety of leverage-creating techniques. c16_gregoriou.qxd 7/27/04 11:36 AM Page 294 Choosing the Right CTA 295 Managed futures accounts offer investors significant advantages not accessible elsewhere, due to their unconventional investment strategies. These forms of investment offer, in much the same way other modern alter- native investment forms do, both diversification advantages and return pro- files different from traditional investments. From this background, research on alternative investments, predomi- nantly on CTAs and hedge funds, has mushroomed. A particularly interest- ing field continues to be the performance evaluation of those alternative investments. Because they may offer highly nonnormal and optionlike return profiles, traditional performance measures used elsewhere suffer from seri- ous disadvantages (i.e., they produce controversial results and, in particular, may be subject to gambling behavior). The performance of CTAs and hedge funds remains, however, a particularly important issue because, in spite of the somewhat controversial theoretical results on persistence in CTA perfor- mance (see, e.g., Schneeweiss 1996), investors evaluate investments, at least partially, based on past performance. This chapter reviews the performance of a series of managed futures funds with a contingent claim–based efficiency measure, which is based on a moment-based performance evaluation methodology. First, we investigate the efficiency of CTAs as stand-alone investments based on the compari- son to option-based strategies. The basis for the comparison is the risk pro- file of the given CTA asset, where risk is defined as some set of statistical moments. Then we compare the moment-based efficiency measures to find out whether using a more complete replication pays off in terms of mone- tary advantages and accuracy. MOMENT-BASED EFFICIENCY MEASURE Distributional Performance Evaluation Assessing performance in case of opaque or continuously changing portfo- lios such as managed funds remains difficult because finding or creating a proper benchmark is still not an easy task. Here we propose a methodology in which the performance of CTA funds will be measured using syntheti- cally created benchmarks. The main idea is to compare any investment portfolio (especially those with nonnormal return distributions like man- aged funds) to artificial, so-called replicating benchmarks possessing risk characteristics similar to the primary investment. The idea that investors compare portfolios based on some statistical (or other) risk profile should not be very surprising. In the most funda- mental consideration about investments, investors buy risky time, that is, c16_gregoriou.qxd 7/27/04 11:36 AM Page 295 296 PROGRAM EVALUATION, SELECTION, AND RETURNS a particular portfolio return profile provided by holding risky assets for a predefined period. Throughout this chapter, the distributional features of return streams will be called risk characteristics (valid for the particular holding period). Return distributions certainly can be arbitrary, not just normal (as would be the usual assumption in case of equity investments). Going one step further, the particular portfolio payoff and return distribution are lim- ited in their shape by the available investment opportunities. Performance measurement, consequently, denotes the evaluation of the particular risk characteristics of the individual payoff profiles. The payoff distribution pricing model of Dybvig (1988a), provides a related perspective. Dybvig develops a pricing framework for assets with arbitrary return distributions. The basic idea of his work is that agents min- imize the cost of any one-period return distribution, regardless of the fac- tors that drive state probabilities. He calls the price of the minimum cost portfolio for any return distribution the distributional price (to distinguish it from the normal asset price). That is, economic agents compare return distributions resulting from any kind of investment opportunity directly. This approach neglects the underlying structure of portfolios, consider- ing it as irrelevant for performance comparison. However, because investors usually use cash returns from the noncash investments for consumption, we argue it is legitimate to do so. That is, it is of no relevance whether a port- folio contains common stocks or hedge funds, because only the distribution of the investment returns for the holding period is important for the per- formance assessment. This approach also may be justified by acknowledg- ing factors like investment barriers and relative illiquidity. Contingent Claim–Based Performance Evaluation The possibility to create and transform arbitrary return distributions is an important property of options that has been known and used by practi- tioners for a long time (cf. Reback 1975). Reback (1975) states that deriv- ative assets are able to alter the pattern of any portfolio return to create any desired shape of return distribution. Thus it is possible to create optioned portfolios mimicking other portfolios in risk characteristics by using options. Because the return distribution of optioned portfolios can be shaped arbitrarily, they can be used as a common benchmark asset. Thus, the use of optioned markets as the reference point suggests extending the performance evaluation framework to multiple asset classes as well. Doing this facilitates the broadening of the classical one asset view to more asset classes compet- ing with each other. In addition, also multimanager funds theoretically could c16_gregoriou.qxd 7/27/04 11:36 AM Page 296 Choosing the Right CTA 297 be analyzed with optioned markets, if the underlying structure of the invest- ments remains immaterial. Indeed, the use of optioned benchmark portfolios for performance measurement purposes itself is not a novel idea. The work of Dybvig (1988a, b) also signifies implicitly that optioned portfolios can be compared to portfolios of other asset classes, regardless of the underlying asset. Glosten and Jagannathan (1994) propose the use of options to re-create con- tingent claims for mutual fund performance evaluation. Dynamic strategies, in much the same way as options, also can be used to create any particular payoff profile. Recently Amin and Kat (2001) have proposed a similar meth- odology to evaluate hedge fund returns in using path-independent dynamic strategies that have positive correlation with the underlying index. The novelty of this chapter in proposing optioned portfolios for benchmark purposes lies, however, in reducing the risk characteristics of the replicating portfolio to a handful of higher statistical moments. EFFICIENCY GAIN/LOSS MEASURE This section proposes using so-called replicating portfolios for benchmark purposes. Replicating portfolios are optioned portfolios designed to repro- duce the risk characteristics of a given asset by combining a benchmark asset with options and the risk-free asset. The expected return on a replicating portfolio for a given risk shape (of a particular asset) will be called the replicating return. The replicating return can be interpreted as the alternative return an investor may achieve if, hold- ing the risk exposure (defined here in terms of return variance, skewness, and a number of higher moments) constant, she chooses to invest in the optioned market instead of investing in a given portfolio. The efficiency gain/loss measure or excess replicating return is simply the difference between the expected return of the asset under investigation and that of its synthetic benchmark asset. The expected return of this repli- cating benchmark asset will be termed as the replicating return. 1 This asset-specific replicating return embodies, at the same time, the minimum acceptable return on investments having the same risk structure, and serves thus as a natural benchmark. That is, investors always have the 1 Certainly the replicating benchmark asset will have to be computed to achieve maximum expected return within the set of possible replicating assets with the same (moment-based) risk characteristics. c16_gregoriou.qxd 7/27/04 11:36 AM Page 297 alternative of being paid the return of the replicating optioned portfolio. Consequently, this replicating return has to be exceeded by other investments exhibiting similar risk characteristics. The efficiency gain/loss measure (the excess replicating return) takes the form ERR p = E(r p ) − RR(r p ) (16.1) where E(r p ) = expected return on portfolio p RR(r p ) = expected replicating return The excess replicating return can be directly interpreted as an efficiency gain, if it is positive, or an efficiency loss, if it is negative. If the replicating optioned return is higher than the expected return for an arbitrary CTA port- folio, this underlying asset offers an inferior performance (compared to the benchmark asset). That is, the comparable investment in form of an optioned portfolio offers a higher expected return for the same risk characteristics of returns. The fund’s shareholder would do better with a different fund (of course, as stand-alone investment only). The excess replicating return pro- vides a simple measure in assessing whether a portfolio outperformed others on an ex-post basis. This measure is in a close relationship with the excess return measure proposed by Ang and Chau (1979), which is an alpha-like composite per- formance ratio. An important distinction is that, in the replicating case, individual portfolios do not have to possess the same systematic risk char- acteristics as the benchmark asset. It is sufficient if both share the same return distribution shapes. Construction of Replicating Portfolios As defined earlier, replicating portfolios are portfolios that have the same risk structure in terms of some statistical moments (of order three and higher) as the portfolio being assessed. The foundation for including repli- cating portfolios in the performance assessment is the assumption that port- folios can be created to “mimic” the risk structure of the underlying asset as benchmarks. The present replicating framework will be termed partial, because only a reduced set of the return characteristics (the moments) is used for de- scribing any return distributions, thereby reducing the return distribution’s dimensionality. 298 PROGRAM EVALUATION, SELECTION, AND RETURNS c16_gregoriou.qxd 7/27/04 11:36 AM Page 298 It is very important to note that the term “replication” as we use it does not intend to create the same payoff profile in terms of identical probabil- ity distributions, nor does it intend to create portfolios having the same pay- off in every possible state of nature. For the construction of individual replicating portfolios, we used the Standard and Poor’s (S&P) 500 index as underlying. Based on the assump- tion that returns from the index are independent and follow a lognormal distribution—a simplification that greatly facilitates the use of contingent claim–based performance evaluation but is not essential—we calculated prices for a specified number of Black-Scholes call options. Considering only call options ensures that asset returns are not linearly dependent. For the sake of simplicity, a holding period of one year is assumed. In the next step, we used nonlinear programming for generating returns on replicating portfolios, with variance and (a predefined number of) higher statistical moments being set to that of the CTA under investigation. This approach provides a relatively simple and robust means for calcu- lating individual benchmark returns. This idea parallels the work of Amin and Kat (2001). They propose a point-by-point optimization algorithm with a 500-pins-setting, that is, they match 500 separate points of the return distribution, to calculate hedge fund efficiency gains/losses. The optimization algorithm that produces replicating portfolio weights x i , can be formulated: (16.2) subject to target variance target skewness (16.3) and the constraints on the portfolio weights where E(r i ) = expected return on asset i s 0 2 s 0 3 = target values for variance and skewness, respectively. The constraints in equation 16.3 can be expanded to include moments of order higher than three. x i i ∑ = 1 ss 3 0 3 = σσ 2 0 2 = Max xi i i i ZxEr= ∑ () Choosing the Right CTA 299 c16_gregoriou.qxd 7/27/04 11:36 AM Page 299 MARKET DATA USED For the testing, we used CTA data publicly available from TradeView (www.tradeview.com). The chosen data set contains 110 CTAs with a monthly return history of five years, from January 1998 to December 2002. From the monthly returns, “semi ex-ante” annual discrete returns were generated with a bootstrap-like methodology i.e., drawing 12 samples with replacement from the set of monthly data, using 1,000 repetitions for each fund. This bootstrapping methodology is in the vein of the technique applied by 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). RESULTS Nonnormality of Returns We test for nonnormality of returns with the Jarque and Bera (1987) test (see Greene 2000). Analyzing the samples, we find that the null hypothesis of normally distributed returns cannot be rejected at the 1 percent level for only 14 cases (11 percent of the observations) and at the 5 percent level for only 11 cases (8.7 percent). Clearly, the sample of CTA funds is highly nonnormal. This should underline the need for a performance measure that accounts for nonnormality of returns. Portfolio Efficiency Rankings by the Excess Replicating Return The excess replicating return (ERR) is, in much the same way as the Sharpe ratio, a composite—risk-adjusted—performance measure. It is risk adjusted because the ERR is calculated always to a given level of risk. Thus risk adjustment takes place indirectly by applying an additive, not multiplica- tive, rule. The ERR, again, denotes the return differential between the expected return of a particular asset and its replicating counterpart. It is designed to assess the value added by the portfolio manager—that is, the efficiency gain or loss. Negative values would mean that the investor is better off buying the same risk structure through options instead of investing in the given asset/CTA and vice versa. 300 PROGRAM EVALUATION, SELECTION, AND RETURNS c16_gregoriou.qxd 7/27/04 11:36 AM Page 300 Choosing the Right CTA 301 We investigated the efficiency of the CTA sample. Figures 16.1 and 16.2 summarize the main results of the analysis. The first diagram displays the 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—not accessible on the stock markets. That is, about 80 percent of the CTAs per- form 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 of about 5.6 percent, which is barely higher that the estimated risk-free rate (4.2 percent). 2. For technical reasons, we have not accounted for possible survivorship bias, which may be expected to have a substantial impact on the over- all performance. Figure 16.2 displays the excess replicating returns for the nine-moment case, but in the same ranking order, as in Figure 16.1. It is noticeable that the basic performance characteristics of the CTAs are mirrored fairly well with the two-moment method; this suggests that a large part of the repli- cating return is attributable to the variance itself. Yet we certainly can also ascertain some significant discrepancies between the rankings of the two cases that should be subject to a closer look. −30% −20% −10% 0% 10% 20% 30% 40% 50% 60% Ranked CTA 2nd Moment Case Efficiency Gain/Loss FIGURE 16.1 Efficiency Gain/Loss Measure (Variance Only) for the CTA Samples, Sorted c16_gregoriou.qxd 7/27/04 11:36 AM Page 301 302 PROGRAM EVALUATION, SELECTION, AND RETURNS Rank Correlation Statistics Let us assume that the more complete description of the CTA return distri- butions (the use of more moments) enables a more robust and exact per- formance measurement. We would like to investigate the overall properties of the moment-based replicating measures and determine how the ranking result is affected by calculating replicating returns with a lower number of moments. 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 the Sharpe ratio and ERR measures of the replicated returns. Next, we calcu- lated rank correlations (Spearman correlations) between the particular ERR measures and Sharpe ratio, which evaluates the closeness of the rank- ings produced by the different methods of performance evaluation. Table 16.1 sums up the results of the calculation. As can be seen, the moment- based measures lie within a limited range, that is, the rankings provided by them are very close to each other: The rank correlations are always higher than 0.99. It is also noticeable that the rank correlation between the Sharpe ratio and the moment-based measures is high but lower than the rank cor- relation between the moment-based measures themselves. Then we repeated the analysis with a slightly different frame, drawing small samples repeatedly and comparing the percentage of identical deci- sions regarding the best possible CTA. Not surprisingly, when drawing sam- −30% −20% −10% 0% 10% 20% 30% 40% 50% 60% Ranked CTA Efficiency Gain/Loss 9th Moment Case FIGURE 16.2 Efficiency Gain/Loss Measure for the CTA Samples, Sorted by Rankings of the Variance Only Case c16_gregoriou.qxd 7/27/04 11:36 AM Page 302 TABLE 16.1 Simple Rank Correlation Between Particular Performance Measures Sharpe Ratio ERR M2 ERR M3 ERR M4 ERR M5 ERR M6 ERR M7 ERR M8 ERR M9 Sharpe Ratio 1.0000 ERR M2 0.7675 1.0000 ERR M3 0.7669 0.9999 1.0000 ERR M4 0.7696 0.9997 0.9998 1.0000 ERR M5 0.7595 0.9981 0.9984 0.9985 1.0000 ERR M6 0.7575 0.9974 0.9978 0.9979 0.9995 1.0000 ERR M7 0.7497 0.9952 0.9956 0.9956 0.9964 0.9966 1.0000 ERR M8 0.7439 0.9958 0.9961 0.9958 0.9959 0.9959 0.9956 1.0000 ERR M9 0.7288 0.9935 0.9939 0.9935 0.9930 0.9928 0.9927 0.9958 1.0000 303 c16_gregoriou.qxd 7/27/04 11:36 AM Page 303 [...]... calculation of replicating returns 306 PROGRAM EVALUATION, SELECTION, AND RETURNS CONCLUSION We investigated the ranking properties of the moment-based replicating efficiency 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... between two corresponding moment replicating returns divided by the level of the replicating return) 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 contribution to the replicating return of over 10 percent This fact indicates that considering the first few moments only... M8 ERR M9 TABLE 16. 2 1 0.353 0.394 0.409 0.354 0.362 0.374 0.336 0.326 Sharpe Ratio 1 1.000 0.998 0.922 0.941 0.861 0.92 0.871 ERR M2 1 1.000 0.963 0.948 0.903 0.917 0.918 ERR M3 1 0.943 0.949 0.902 0.937 0.928 ERR M4 1 0.994 0.896 0.903 0.918 ERR M5 1 0.919 0.949 0.931 ERR M6 1 0.937 0.899 ERR M7 1 0.945 ERR M8 Ratio of Identical Decisions (Sample Size 10, 1,000 Times) with Particular Performance Measures... gain/loss produce results very close to each other in terms of rank correlation In addition, the Sharpe ratio and the moment-based efficiency measures produce a higher rank correlation but a lower rate of identical decisions However, we also found that the higher moments have a high absolute and relative contribution to the replicating return This fact implies that although the replicating measures... 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 replicating measures 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 are sufficient to reproduce the results of the... 1 0.919 0.949 0.931 ERR M6 1 0.937 0.899 ERR M7 1 0.945 ERR M8 Ratio of Identical Decisions (Sample Size 10, 1,000 Times) with Particular Performance Measures 1 ERR M9 305 Choosing the Right CTA TABLE 16. 3 Contribution of Higher Moments to the Replicating Returns Moment 3 Absolute contribution Relative contribution Moment 4 Moment 5 Moment 6 −0.062% −0.032% −0.261% −0.069% −1.540% −0.730% −5.974% −1.348% . funds individually. In this chapter we investigate the ex-post performance ranking of CTAs based on a contingent claim performance approach. In this approach, the performance of each managed. based on past performance. This chapter reviews the performance of a series of managed futures funds with a contingent claim–based efficiency measure, which is based on a moment-based performance. formulated: (16. 2) subject to target variance target skewness (16. 3) and the constraints on the portfolio weights where E(r i ) = expected return on asset i s 0 2 s 0 3 = target values for variance and