A multitude of investment fund performance models and metrics exist in part because some measures are more appropriate for certain purposes than others.
For example, the Sharpe ratio is arguably more appropriate when analyzing an entire portfolio, while the Treynor ratio is appropriate when evaluating a security or investment that is part of a larger portfolio.3The multitude of per- formance measures and approaches also suggests that more than one meas- ure of risk may be needed to accurately assess performance. Conversely, some measures can be redundant. For example, Daglioglu and Gupta (2003b) find that returns of hedge fund portfolios constructed on the basis of some risk measures are often highly correlated, and sometimes perfectly correlated, with returns of portfolios constructed on the basis of others. Burghart, Dun- can, and Liu (2003) illustrate that the theoretical distribution of drawdowns can be replicated with a high degree of accuracy given only a manager’s aver- age return, standard deviation of returns, and length of track record.
In this section we begin by briefly reviewing some of the traditional portfolio performance measures and analysis techniques. We review single parameter risk measures based on modern portfolio theory, we discuss expanded performance models that account for time-varying risk, discuss concerns over assuming mean-variance sufficiency, and consider multifactor models of style and performance attribution. This short review exposes a plethora of performance measures. The question of appropriateness and redundancy is revisited in the section that describes the data used in this study. The current section also discusses the seemingly paradoxical issue of using benchmarks to evaluate absolute return strategies4 and concludes with a discussion of potential determinants of performance.
Alpha and Benchmarks
Traditional asset managers seek to outperform a benchmark, and their per- formance is measured relative to that benchmark in terms of an alpha.
3The Sharpe measure is appropriate when analyzing an entire portfolio, because the standard deviation, or total risk, is in the denominator whereas beta is the denomi- nator of the Treynor measure, and beta measures the systematic risk that will con- tribute to the risk of a well-diversified portfolio.
4Absolute return strategies seek to make positive returns in all market conditions.
In contrast, relative return strategies seek only to outperform a benchmark.
While CTAs follow absolute return strategies that seek to make positive returns in all market conditions, benchmarks now exist for CTAs and other hedge fund strategies. Before considering benchmarks for absolute return strategies, we first review the concepts in the context of traditional asset management. Jensen’s (1968) alpha is generally a capital asset pricing model (CAPM)-based performance measure of an asset’s average return in excess of that predicted by the CAPM, given its systematic risk (beta)5 and the market (benchmark) return. Alphas also may be measured relative to addi- tional sources of risk in multi-index models.
Whereas various single-index models are based on the CAPM and assume that security returns are a function of their co-movements6with the market portfolio, multi-index (or multifactor) models assume that returns are also a function of additional influences.7For example, Chen, Roll, and Ross (1986) develop a model where returns are a function of factors related to cash flows and discount rates such a gross national product and infla- tion. The purposes of multi-index models are varied and, in addition to performance attribution, include forming expectations about returns and identifying sources of returns.
Sharpe (1992) decomposes stock portfolio returns into several “style”
factors (more narrowly defined asset classes such as growth and income stocks, value stocks, high-yield bonds) and shows that the portfolio’s mix accounts for up to 98 percent of portfolio returns. Similarly, Brinson, Singer, and Beebower (1991) show that rather than selectivity or market timing abilities, it is the portfolio mix (allocation to stocks, bonds, and cash) that determines over 90 percent of portfolio returns. However, Brown and Goetzmann (1995) identify a tendency for fund returns to be correlated across managers, suggesting performance is due to common strategies that are not captured in style analysis.
Schneeweis and Spurgin (1998) use various published indexes (Gold- man Sachs Commodity Index, the Standard & Poor’s 500 stock index, the CTA Performance Evaluation with Data Envelopment Analysis 83
5Within the Markowitz (1952) framework, total risk is quantified by the standard deviation of returns. Tobin (1958) extended the Markowitz efficient frontier by adding the risk-free asset, resulting in the capital market line (CML) and paving the way for the development of the capital asset pricing model, developed by Sharpe (1964), Lintner (1965), and Mossin (1966). The CAPM defines systematic risk, measured by beta (b), as the relevant portion of total risk since investors can diver- sify away the remaining portion.
6Usually CAPM-based performance models describe covariance with the market portfolio, however, as noted earlier, they can attempt to describe coskewness and cokurtosis as well.
7Arbitrage pricing theory (APT) establishes the conditions under which a multi- index model can be an equilibrium description (Ross, 1976).
Salomon Brothers government bond index, and U.S. dollar trade-weighted currency index, the MLM Index8) with absolute S&P 500 returns and intramonth S&P return volatility in a multifactor regression analysis to describe the sources of return to hedge funds, managed futures, and mutual funds. The index returns employed in the regression analysis are intended to be risk factors that explain the source of natural returns. The explana- tory variable, absolute equity returns, captures the source of return that derives from the ability to go short or long. Returns from the use of options or intramonth timing strategies are proxies for the intramonth standard deviation. The MLM Index, an active index designed to mimic trend- following strategies, is used to capture returns from market inefficiencies in the form of temporary trends.
Seigel (2003) provides a comprehensive review of benchmarking and investment management. Despite the fact that CTAs and many hedge fund managers follow absolute return strategies, various CTA benchmarks now exist, as described by Seigel (2003).
Addressing Time-Varying Risk
Single-parameter risk measures are problematic if managers are changing fund betas over time, as they would if they were attempting to time the mar- ket. For example, when equity prices are rising, the manager might increase the fund’s beta and vice versa. Although market risk can be measured if the portfolio weights are known, this information is generally not publicly available and other techniques must be employed.9
8Mount Lucas Management IndexTMis based on a concept conceived in 1988 of an index methodology that involves changing (commodity) market sides long and short to measure economic return.
9Treynor and Mazuy (1966) added a quadratic term to the basic linear regression model to capture nonlinearities in beta resulting from market timing activities. Kon and Jen (1978, 1979) use a switching regression technique. Merton (1981) and Hen- riksson and Merton (1981) develop nonparametric and parametric option-based methods to test for directional market timing ability. The nonparametric approach requires knowledge of the managers’ forecasts. The more commonly employed parametric approach involves adding an extra term to the usual linear regression model and is CAPM based. Ferson and Schadt (1996) note that fund betas may change in response to changes in betas of the underlying assets as well as from changing portfolio weights. They modify the classic CAPM performance evaluation techniques to account for time variation in risk premiums by using a conditional CAPM framework. This method removes the perverse negative performance often found in earlier tests and suggests that including information variables in perform- ance analysis is important.
Mitev (1998) uses a maximum likelihood factor analysis technique to classify CTAs according to unobservable factors. Similarly, Fung and Hsieh (1997b) also use a factor-analytic approach to classify hedge funds. In both cases, the results identify general investment approaches or trading strate- gies (e.g., trend-following, spread strategies, or systems approaches) as sources of returns to these alternative investment classes. Factor analysis and multifactor regression analysis differ in their approach to identifying the factors (benchmarks) that serve as proxies for risk. In multifactor regression analysis, the factors are specified in advance. Factor analysis will identify funds that have common yet unobservable factors, although the factors can be inferred from the qualitative descriptions of the funds. While this may seem redundant, the clustering of funds is done independently of the qualitative descriptions in a formal data-driven process.
The data envelopment analysis methodology used in this chapter, and described in more detail in Wilkens and Zhu (2001, 2004), incorporates multiple criteria and “benchmarks” funds or other securities according to these criteria. This is distinctly different from multifactor analysis. Here benchmarks are not risk factors but rather are efficient securities as defined inn dimensions where each dimension represents risk and return criteria.
Recently Gregoriou (2003) used the DEA method in the context of bench- marking hedge funds.
Skewness and Kurtosis:
Questioning Mean-Variance Sufficiency
The standard CAPM framework assumes that investors are concerned with only the mean and variance of returns. Ang and Chau (1979) argue that skewness in returns distributions should be incorporated into the perform- ance measurement process. Even if the returns of the risky assets within a portfolio are normally distributed, dynamic trading strategies may produce nonnormal distributions in portfolio returns. Both Prakash and Bear (1986) and Stephens and Proffitt (1991) also develop higher-moment performance measurements.
Fishburn (1977), Sortino and van der Meer (1991), Marmer and Ng (1993), Merriken (1994), Sortino and Price (1994), and others also have developed measures that take into account downside risk (or semivariance) rather than the standard deviation of returns. Although some differences exist among these measures, the Sortino ratio captures their essence.
Whereas the Sharpe ratio is defined as excess return10divided by standard CTA Performance Evaluation with Data Envelopment Analysis 85
10Return minus the risk-free rate.
deviation, the Sortino ratio is defined as return divided by downside devia- tion. Downside deviation (DD) measures the deviations below some mini- mal accepted return (MAR). Of course, when the MAR is the average return and returns are normally distributed, the Sharpe and Sortino ratios will measure the same thing. Martin and Spurgin (1998) illustrate that even if individual asset or fund returns are skewed, the skewness tends to be diversified away at the portfolio level. However, they also illustrate that managers may choose to follow strategies that produce skewed returns as a form of signaling their skill. Note that coskewness remains irrelevant if it can be diversified away, but skewness may have some signaling value. Addi- tionally, the popularity of the related value at risk (VaR) measure11and the common practice of reporting drawdown12 information for various alter- native investments suggest that skewness may be important, whether in terms of investor utility or skill signaling.
Beta-Squared Coefficient The classic paper by Fama and MacBeth (1973), and several other early papers (e.g., Carroll and Wei 1988; Shanken 1992) empirically test a two-pass regression methodology for stock returns.
Assuming a nonlinear relationship between stock returns, the tests include beta-squared in the second-pass regression. These tests find that the coeffi- cient for beta-squared is negative and statistically significant, providing evi- dence of a nonlinearity in stock returns.
Schneeweis and Georgiev (2002, p. 7) provide evidence that CTAs have nonlinear returns with respect to the equity market: “When S&P 500 returns were ranked from low to high and divided into four thirty-three month sub-periods, managed futures offered the opportunity of obtaining positive returns in months in which the S&P 500 provided negative returns as well as in months in which the S&P 500 reported positive returns.”
We include equity beta-squared in our Tobit regressions where the dependent variable is not the expected return of the CTA, but is rather the efficiency score obtained in the DEA models. Although the dependent variable is not the same as in the earlier stock studies, we might hypothe- size that CTA efficiency scores are also negatively related to beta-squared.
11See Chung (1999) for a concise review of VaR methodologies.
12Drawdown information is generally reported as the maximum drawdown over a period and is defined as the return from a fund’s net asset value peak to trough. The Calmar ratio is a similar measure that CTA investors are often interested in and is defined as the average annual return over the past three years divided by the absolute value of the maximum drawdown during that period.
We infer a direct correspondence between the efficiency score and expected return. The CTA returns observed by Schneeweis and Georgiev (2002), therefore, imply a positive coefficient. Finally, we note that the efficiency scores used in this study minimize variability. This leads to the hypothesis that the beta-squared coefficient is negatively correlated with the efficiency score, unless the enhanced return from high (absolute) betas is an offset- ting factor.
Fund Size In his chapter “The Lure of the Small,” Jaeger (2003) describes how small firms and small portfolios are desirable features of hedge funds.
Small firms satisfy hedge fund managers’ entrepreneurial spirit, and small portfolios are often necessary to enable hedge funds to implement their strategies, especially if they trade in markets that are sometimes illiquid.
Gregoriou and Rouah (2002) find, however, that fund size does not matter to hedge fund performance. Being a subset class of hedge funds, CTAs are examined in this chapter to see if fund size or length of manager track record is related to the DEA efficiency scores.
Determinants of Performance Based on the discussion above, we choose as bases for performance evaluation in a DEA model monthly returns, kurto- sis, minimum return, skewness, standard deviation of returns, and percent- age of negative monthly returns. We then investigate the potential of fund size, length of track record, strategy, and style to impact performance scores of funds created by the DEA model.