EDHEC RISK AND ASSET MANAGEMENT RESEARCH CENTRE 393-400 promenade des Anglais 06202 Nice Cedex 3 Tel.: +33 (0)4 93 18 32 53 E-mail: [email protected] Web: www.edhec-risk.com Assessing and Valuing the Non-Linear Structure of Hedge Fund Returns November 2007 Antonio Diez de los Rios Bank of Canada René Garcia EDHEC Business School ABSTRACT Several studies have put forward that hedge fund returns exhibit a non-linear relationship with equity market returns, captured either through constructed portfolios of traded options or piece-wise linear regressions. This paper provides a statistical methodology to unveil such non-linear features with the returns on any selected benchmark index. We estimate a portfolio of options that best approximates the returns of a given hedge fund, account for this search in the statistical testing of the contingent claim features, and test whether the identifed non-linear features have a positive value. We find that not all indexes for categories of funds exhibit significant non-linearities, and that only a few strategies as a group provide significant value to investors. Our methodology helps identify individual funds that provide value in an otherwise poorly performing category. Keywords: Hedge Funds, Non-linear Return Structure, Valuation of Contingent Claims, Performance of Hedge Funds JEL Classification: C1,C5,G1 Address for correspondence: René Garcia, EDHEC Business School, Finance, Law and Accounting Department, 393, Promenade des Anglais, BP 1116, 06200 Nice France, E-mail : diez@bankofcanada. ca and [email protected]. This paper started while the first author was a post-doctoral fellow at CIREQ and CIRANO; he is thankful for their hospitality. The second author gratefully acknowledges financial support from the Fonds Québécois de la recherche sur la société et la culture (FQRSC), the Social Sciences and Humanities Research Council of Canada (SSHRC), the Network of Centres of Excellence (MITACS), Hydro-Québec, and the Bank of Canada. Both authors also thank Greg Bauer, Georges Hübner, Michael King, Andrew Patton, Enrique Sentana, Marno Verbeek, Jun Yang, and seminar participants at the Bank of Canada, CEMFI, the 2005 CIREQ Conference on Time Series Models, and the Second Annual Empirical Asset Pricing Retreat at the University of Amsterdam for their comments. We also thank Vikas Agarwal and Narayan Y. Naik for kindly providing us with an updated time series of the returns on their option- based factors. The views in this paper are those of the authors and do not necessarily reflect those of the Bank of Canada. EDHEC is one of the top five business schools in France owing to the high quality of its academic staff (110 permanent lecturers from France and abroad) and its privileged relationship with professionals that the school has been developing since its establishment in 1906. EDHEC Business School has decided to draw on its extensive knowledge of the professional environment and has therefore concentrated its research on themes that satisfy the needs of professionals. EDHEC pursues an active research policy in the field of finance. Its “Risk and Asset Management Research Centre” carries out numerous research programs in the areas of asset allocation and risk management in both the traditional and alternative investment universes. Copyright © 2008 EDHEC 2 1. Introduction Since the burst of the Internet stock bubble in 2000, many pension funds have decided to invest in hedge funds with the hope of improving their performance on a path to full funding of their commitments. Assessing the performance of hedge funds has therefore become a topic of major social relevance. Will hedge funds with a typical fee structure of 2% of asset value and a 20% performance fee be able to fulfill institutional investors' expectations? While a cursory look at historical performance suggests that even modest allocations to hedge funds may improve significantly the efficiency of pension fund portfolios, episodes like the near-bankruptcy of Long-Term Capital Management (LTCM) in 1998 have raised questions about the true nature of their risks. In this paper we pursue two main objectives. First, we want to better characterize and understand the risks associated with the different hedge fund strategies. Second, we want to determine whether, given these risks, the value of the cash flows generated by a fund, net of management fees, is greater than the amount entrusted to the fund manager. Fulfilling these two objectives is not an easy task. Hedge funds often engage in short-selling and derivatives trading, while leveraging their positions. More importantly, hedge funds are not transparent to the investor, since they have no obligation to disclose their positions. Therefore, any assessment of hedge fund performance can rely only on analyzing their ex-post returns. However, the return databases suffer from several biases and they do not go back very far in time. Recent literature suggests that hedge fund returns exhibit non-linear structures and option-like features, or, in other words, risks that are typically ignored by mean-variance approaches. Fung and Hsieh (2001) analyze trend-following strategies and show that their payoffs are related to those of an investment in a lookback straddle. Mitchell and Pulvino (2001) show that returns to risk arbitrage are similar to those obtained from selling un-covered index put options. Agarwal and Naik (2004) extend these results and show that, in fact, a wide range of equity-oriented hedge fund strategies exhibit this non-linear payoff structure. In particular, they use a stepwise regression procedure to identify the significant risk factors. To account for non-linearities, they include option-based risk factors that consist of returns obtained by buying, and selling one month later, liquid put and call options on the Standard & Poor's (S&P) 500 index. Finally, Hasanhodzic and Lo (2007) introduce as a risk factor the first difference in the end-of-month value of the CBOE volatility index (VIX), which can be interpreted as a proxy for the return on the portfolio of options used to compute the VIX. In all these studies hedge fund returns are regressed on a complex set of risk factors whose determination involves an implicit or explicit search over a large set of potential candidate variables to increase the R2. In the stepwise regression approach used by Agarwal and Naik (2004), variables are added or deleted in a sequential way based on the value of the F-statistic. However, such a search makes it impossible to rely on standard statistical inference to determine if the hedge-fund alpha is positive or not. To capture the non-linear risk exposure, these studies have added to the regression the returns on a set of well-chosen traded option indices. For example, Agarwal and Naik (2004) choose at-the-money and out-of-the-money puts and calls on the S&P 500 index. However, using the same portfolio of options for different funds may not capture well the particular strategy associated with a category of funds and it can lead to a biased assessment of the value provided to investors. This prompts the question of how many options and which strike prices should be used for each fund (see Amin and Kat, 2003). Also, managers can use strategies to replicate synthetically the payoffs of options on a benchmark portfolio for which no liquid options exist. Finally, it can be the case that hedge funds present non-linearities with respect to risk factors for which no liquid options exist. We propose a new method that makes it possible to overcome the difficulties mentioned above. In particular, our method allows us to (i) use options on any benchmark portfolio deemed to best characterize the strategies of the fund (and not simply traded options on the S&P 500 or other liquid options), (ii) estimate whether the options that best characterize the returns of a particular fund are 3 puts or calls, or both, as well as their corresponding moneyness, (iii) assess whether the presence of the estimated non-linearities is statistically significant, over and above the linear factors, (iv) value the performance of a fund by valuing the portfolio of options that have been found to be significant in characterizing the hedge fund returns, and (v) provide a reliable test for a positive valuation of the fund. The starting point of the methodology is based on Glosten and Jagannathan (1994). We estimate a flexible piece-wise linear function to capture the potentially non-linear relationship between the returns of a hedge fund and those of benchmark portfolios. These portfolios can be chosen among the risk factors that enter linearly in the characterization of hedge fund returns. Following Hasanhodzic and Lo (2007), we choose the following set of factors as our benchmark model: stocks, bonds, currencies, commodities, and credit.1 For example, the stock market factor will be an important driving factor for equity-oriented hedge funds, while the bond and the credit spread factors are chosen to explain fixed- income-oriented funds (see Fung and Hsieh 2002). In addition and motivated by the work of Fung and Hsieh (2004) on long-short hedge funds, we also report results based on these five factors plus the spread between returns on large-capitalization stocks and returns on small-capitalization stocks. We believe that, given the small sample of hedge fund observations, this set of factors provides a reasonable trade-off between the right number of risk exposures for a typical hedge fund and a potential over-fitting of the model. To start our analysis, we consider one option on a well-diversified equity index. The coefficients of such a non-linear regression are interpretable by practitioners, since they correspond to a position on a risk-free asset, a position on the selected index, one or more positions on options on this equity portfolio, and the effective strikes of such options.