Are higher-order factors useful in pricing the cross-section of hedge fund returns?
| Autor | Fang, Elaine |
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Introduction
Hedge fund returns are usually associated with a variety of risk factors, each with a risk premium. Most of the literature that specializes in performance analysis of such funds searches for additional risk factors that could linearly explain their cross-section of expected returns. Standard practice is to choose a set of excess returns representing risk factors and to linearly regress hedge fund excess returns on these factors. Improving on initial models that used the market plus a few linear factors as a benchmark, Fung & Hsieh (2001) argue that hedge funds typically generate option-like returns and propose a seven-factor model that has greater explanatory power than models with only standard asset indices. Fung and Hsieh's seven-factor model is regarded as one of the main benchmarks for modeling hedge fund returns. (1)
Despite a large effort to find new factors, one drawback of these models and their variations is that they constrain the relationship between the risk factors and returns to be linear--and linear models cannot price assets whose payoffs (returns) are nonlinear functions of the risk factors. Several studies address this issue with a nonlinear asset-pricing framework. The pioneering work of Bansal, Hsieh & Viswanathan (1993) extends the arbitrage pricing theory (APT) by allowing returns to be nonlinearly related to risk factors. Harvey & Siddique (2000a, 2000b) propose a model in which returns are related to the quadratic market return. Unfortunately, this strand of literature focuses mainly on standard asset indices and mutual funds, which employ less dynamic trading strategies and have different risk exposures, compared to hedge funds.
In this study, we contribute to the hedge fund literature by extending the analysis from an augmented linear model based on Fung and Hsieh's seven-factor model (hereinafter referred to as the linear model) to models that include all second-order terms (second-order models). We investigate four models--the Fama-French three-factor model (as a basic sanity check), the linear model, and two second-order models (which include linear, quadratic and interaction terms)--and use a data-driven method to select the final sets of risk factors for each. We then investigate hedge funds' exposures to the factors and examine the performance of each model in explaining and predicting the cross-sectional variation in returns. Including second-order factors increases the flexibility of a model's functional form and decreases the danger of model misspecification. In addition, second-order factors (quadratic and cross terms) may be associated to an attempt to do factor timing.
The aim of this study is to address two major sets of research questions. The first set relates to the performance of the second-order models: (1) Do second-order models provide more explanatory and/or predictive power than linear models based on commonly-used factors, including those proposed by Fama & French (1993) and Fung & Hsieh (2001)? (2) Are the comparison results consistent across all funds? The second set of questions relates to the variation between different hedge fund strategies: (3) Is there a single set of linear and/or second-order factors that can capture the risk dynamics of all hedge fund strategies? (4) Are the hedge fund returns on some investment strategies more difficult to model than others? (5) Do our second-order models still exhibit missing factors for some strategies?
Note that the main goal of this paper is not to search for new factors, but rather to investigate whether second-order factors provide more explanatory and/or predictive power than their linear counterparts, and if they are sufficient for modeling cross-sectional hedge fund returns. Moreover, we are more interested in hedge fund exposure to various risk factors than in the returns performance of the individual funds.
Our linear and second-order models all start with the following set of factors: the three Fama & French (1993) factors representing basic dynamics of the stock market, the momentum factor (Carhart, 1997), five option-like factors (Fung & Hsieh, 2001), the 10-year Treasury bill, and the MSCI World and Emerging Markets indices. Our empirical analysis is based on 10 years of monthly hedge fund returns data, spanning the time period between January 2006 and December 2015. The hedge fund data was obtained from the Lipper Hedge Fund Database (TASS), accessible via the Wharton Research Data Services (WRDS). Figure 1 presents the distribution of hedge funds with complete returns data in TASS by investment strategy (see Table 2 for a more detailed breakdown). This categorization allows us to answer our second set of proposed research questions. As we can see, TASS contains little data for a number of strategies. Thus, we only include the six most represented strategies in our empirical analysis: fund of funds, long/short equity hedge, multi-strategy, managed futures, emerging markets and event-driven (See Appendix A for a description of each strategy).
There are some important highlights of this study. First, contrary to most existing literature in which the risk factors are fixed for all funds, we allow the risk factors to be strategy-specific. Different strategies vary in investment style and are hence exposed to distinct sets of risk factors. Following Bali, Brown & Caglayan (2011), we classify emerging markets, event-driven and managed futures as directional strategies, and fund of funds, long/short equity hedge and multi-strategy as semi-directional strategies. (2) Directional funds willingly take direct market exposure and risk, while semi-directional funds take both long and short positions to try to diversify the market risk. Due to the diverse set of strategies used in this study, we include only Fama and French's three factors in all of the final models as a set of factors representative of the stock market. (3) We select all other factors separately for each strategy based on a data-driven method that makes use of a combination of LASSO to select factors on a first stage, with Fama-Macbeth regressions to identify factors' risk-premium on a second stage. (4) Using strategy-specific sets of factors allows us to address the second set of proposed research questions.
We first examine the extent to which aggregate risk factors can explain the time series of hedge fund returns across investment strategies. Similar to Bali, Brown & Caglayan (2012), we quantify the risk hedge funds face by dividing the total risk into its systematic and fund-specific components. This decomposition provides micro-level explanations for different hedge fund strategies. We then investigate how well hedge funds' risk exposures can predict the cross-sectional variation in returns using Fama & MacBeth's (1973) cross-sectional regressions of one-month-ahead excess returns on the factor betas. If the slope coefficient for a certain risk factor is significant, we can conclude that the risk factor's beta has significant predictive power over future hedge fund returns. (5)
Our analysis provides the following findings regarding our proposed research questions. First, although several quadratic and interaction terms are statistically significant (for some strategies), there is no statistical evidence that the second-order models have more overall explanatory or predictive power than the linear model. Second, evidence strongly indicates that the set of risk factors should be strategy-specific. Our analysis suggests that both the linear and second-order models can effectively explain the predictive power for directional funds. On the other hand, they fail to explain the predictability of semi-directional funds, implying that missing factors may still remain. Ultimately, our results imply that searching for additional factors may be more effective than adding higher-order or interaction terms.
The rest of this paper is organized as follows. Section 2 reviews the relevant hedge fund and asset pricing literature. Section 3 describes the hedge fund returns and factor data used in our analysis and Section 4 presents our multistep estimation procedure and model performance measures. Finally, we present and discuss our empirical analysis results in Section 5 and conclude in Section 6.
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Literature Review
Asset pricing theory is a central topic of discussion in financial literature. While many methods of pricing different securities have emerged, a universally-accepted model describing cross-sectional hedge fund returns does not currently exist.
2.1 CAPM and Fama-French Factor Models
The Capital Asset Pricing Model (CAPM), introduced by William Sharpe (1964) and John Lintner (1965), is the foundation of most modern asset pricing theories. When applied to funds, CAPM describes the following relationship between the systematic risk and expected return of fund i:
[R.sub.i]t - [r.sub.ft] = [[alpha].sub.i] + [[beta].sub.i]([R.sub.mt] - [[r.sub.ft]) + [[epsilon].sub.it], t = 1,..., T, (1)
where [R.sub.it] is the return of fund i at time t, [r.sub.ft] is the risk-free rate at time t, the intercept [[alpha].sub.i] is the fund's excess return after accounting for market risk, the slope [[beta].sub.i] is the fund's "beta," and [R.sub.mt] is the expected market return (commonly defined as the S&P 500 return) at time t.
In particular, the expected return of a portfolio depends on the risk-free return and the fund's risk premium, which is derived from its beta. The risk-free rate compensates for the time-value of money.
As we can see, CAPM prices only the market (or systematic) risk. Eugene Fama and Kenneth French (1993) made great strides in the asset pricing literature by introducing additional factors that capture other risks priced in the cross-section of U.S. stock returns. In their three-factor model (equation 2), Fama and French build upon the CAPM by adding two factors based on size (SMB--small minus big) and value...
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