Commodity Trading Advisors: Risk, Performance Analysis, and Selection Chapter 16 pdf

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 highe

Choosing the Right CTA:

A Contingent Claim Approach

Zsolt Berenyi

Managed 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

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 comparicompari-son 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,

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

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 replirepli-cating 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.

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

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

s02s03= target values for variance and skewness, respectively The constraints in equation 16.3 can be expanded to include moments

of order higher than three

i

0

=

σ2 σ

0

= Max

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

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

FIGURE 16.1 Efficiency Gain/Loss Measure (Variance Only) for the CTA Samples, Sorted

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

9th Moment Case

FIGURE 16.2 Efficiency Gain/Loss Measure for the CTA Samples, Sorted by Rankings of the Variance Only Case

TABLE 16.1

Simple Rank Correlation Between Particular Performance Measures Sharpe Ratio

303