Forecasting hedge fund volatility: a Markov regime-switching approach

Forecasting hedge fund volatility: a Markov regime-switching approach The article addresses forecasting volatility of idiosyncratic Hedge Fund (HF) returns using a non-linear Markov Switching GARCH (MS-GARCH) framework in which the conditional mean and volatility may exhibit dynamic MS behavior. The out-of-sample multi-step ahead volatility forecasting performance of MS-GARCH(1,1) and GARCH(1,1) models is compared when applied to twelve global HF indices over the period of January 1990 to October 2010. The results show significant MS dynamics for all HF strategies with exception of Macro, Equity Market Neutral and Global HF Index and clearly identify different regimes with periods of high and low volatility. In-sample estimation reveals superiority of MS-GARCH framework for all HF strategies and shows that regime switching is related to structural changes in the market factor for most of the strategies. The forecast accuracy and forecast encompassing tests provide evidence of superior performance of MS-GARCH models in out-ofsample volatility forecasting for most forecast horizons and HF strategies. Similar analysis performed during crisis and non-crisis periods shows that both models perform better during crisis periods and that inclusion of MS dynamics in GARCH specification highly improves volatility forecasts for strategies particularly sensitive to general macroeconomic conditions such as Distressed Restructuring (during both crisis and non-crisis periods) and Merger Arbitrage (during non-crisis periods). Keywords: Hedge Fund (HF) risk, idiosyncratic returns, Markov Switching (MS) model, volatility forecasting, financial crisis JEL Classification: G11, G17, G23

1. Introduction In the last two decades, Hedge Funds (HFs) have established their position in the global financial markets. The industry has grown from a small number of funds to over 10,000 HFs and funds of HFs during the last decades. Currently, these funds manage assets of more than USD1.9 trillion (see Figure 1). HFs operate in loose regulatory environments and can employ a broad range of investment strategies involving unrestricted positions in derivatives, short sales and leverage. HFs are reserved for high net worth individuals and institutional investors with large financial resources, require high minimum investment and charge high management and performance fees. Many sophisticated investors have incorporated these investment vehicles in their portfolios in order to increase returns, to obtain exposure to broader markets, to different asset classes or investment strategies and to diversify risk. (See Fung and Hsieh (1999) for a review of HF organizations.) Historically, HFs outperformed major equity markets showing low correlation with traditional asset classes and providing safe shelter during moments of financial distress. This situation changed substantially in 2008 when the HF industry experienced unprecedented losses and withdraws of capital. ‘Hedge Fund Research, Inc.’ (HFR) reported that assets under management in the entire HF industry declined by more than thirty percent in 2008 and underperformed the equity markets in the following years (see Figures 1, 2 and 3). 1

This new situation has cast a doubt on the ability of HF managers to produce superior returns, which has justified high performance fees. Their traditionally ‘hedged’ status has been questioned and, as a consequence, they are subject to closer scrutiny by regulators, investors and academics. Regulators have recognized the impact of HFs’ activities on the global financial markets as it has become clear that through their excessive use of leverage, large and concentrated positions in derivatives and short sales of the stocks of distressed financial firms, they might have contributed to deepen the financial crisis in 2007 and 2008. Investors and academics have started to pay more attention to the drivers of HFs’ performance, the HF managers’ ability to produce ‘alpha’ and the characteristics of HFs’ risk exposure. HF risk exposure may vary in different market conditions as HFs’ managers are free to change allocation of resources to different asset classes, to choose alternative trading strategies or levels of leverage in response to new macroeconomic conditions and investment opportunities (e.g. different phases of the business cycle, Mergers & Acquisitions (M&A) activity or the surge of companies in distress). This phenomenon has been observed after the recent financial crisis when many HF managers reduced their leverage substantially, reallocated portfolios to safer assets or focused on new investment opportunities, such as Asian opportunities and distressed debt. For example, according to HFR and Eurekahedge, assets under management of Distressed Debt were the best performing strategy, gaining 33.0% in 2009 and 21.3% in 2010, when the average HF gained only 19.7% and 10.9%, respectively. HF risk profiles will probably change in the future due to different portfolio composition, amount of leverage or scrutiny level of regulatory institutions. After the recent financial crisis, new regulatory framework for HFs is in preparation both in the US and Europe and is likely to be approved at the end of 2011. Therefore, we believe that it is interesting to study HF risk dynamics and their relationship with the economic conditions and different market risk factors. This article has two main objectives. First, to identify an efficient econometric technique for measuring changes in HF volatility, which is useful especially in periods of economic distress. Second, to determine the economic variables that might drive HF volatility changes. In order to achieve these objectives, regime switching dynamics are incorporated in the traditional GARCH model to study if such a specification can be helpful in determining the shifts in HF risk. More precisely, the article focuses on forecasting volatility of the idiosyncratic HF returns using the non-linear Markov Switching GARCH (MS-GARCH) framework in which the conditional mean and volatility of systematic and idiosyncratic HF returns components may exhibit regime switching behavior. The article compares the out-of-sample multi-step ahead (3, 6, 9 and 12 months ahead) forecasting performance of two competing conditional volatility specifications: the traditional GARCH model and the regime switching MS-GARCH specification. The data used in this study consists of return series of twelve HF indices obtained from HFR and comprises the period between January 1990 and October 2010 to produce volatility 2

forecasts for the period January 1999 to October 2010. There are several contributions of this article. First, it evaluates the impact of market risk factors on the returns of different HF indices in order to evidence common observable return drivers in the period of 1990 to 2010. Second, it employs unit root test, cointegration and latent common trends modeling techniques to estimate several common factors and their effects on the returns of different HF indices over the period 1990 to 2010. Third, the article associates the latent factors with the observable HF return drivers. Fourth, the article provides likelihood based in-sample comparisons of the competing GARCH and MS-GARCH specifications estimated for the period from 1990 to 2010. Fifth, it provides some evidence about the in-sample determinants of the switching HF regimes by evaluating the influence of switching equity market index returns on HF regimes. Sixth, the article compares the out-of-sample and multi-step ahead idiosyncratic volatility forecasting performance of the GARCH and MS-GARCH formulations over the period 1999 to 2010 and, supported by several significance tests as proposed by the econometric literature, evidences superiority of the MS-GARCH model for several HF strategies. Seventh, the work compares the out-of-sample and multi-step ahead idiosyncratic volatility forecasting performance of the GARCH and MS-GARCH formulations over the crisis and non-crisis periods over the 1999 to 2010 period and shows improvement of volatility forecasts for both specifications. For example, results reveal significant superiority of GARCH models with regime MS dynamics in predicting volatility of Distressed Restructuring strategy both during crisis and non-crisis periods. The remaining part of this article is organized as follows. Section 2 reviews the corresponding HF literature. Section 3 summarizes the data set employed. Section 4 gives a brief overview of the two competing econometric models employed, presents the statistical inference procedure and reports the volatility forecasting formulas applied. Section 5 summarizes the empirical findings. Finally, Section 6 concludes. [APPROXIMATE LOCATION OF FIGURES 1, 2 AND 3] 2. Literature review In the HF literature, several previous works have investigated how market-specific, i.e. systematic risk factors influence HF returns. These papers evidence that a significant part of the volatility in HF returns can be explained by market-related factors (e.g. Fung and Hsieh (1997, 2001, 2002, 2004), Agarwal and Naik (2004, 2005) and Hasanhodzic and Lo (2007)). The first HF return models have been founded on the traditional linear factor models such as the capital asset pricing model (Sharpe (1964), Lintner (1965) and Mossin (1966)), the Fama and French (1993) model, the Carhart (1997) model and the arbitrage pricing theory (Ross, 1976). These models can identify the exposure of HF returns to systematic risk factors but they assume that the relation between the risk factors and returns is linear. However, linear factor 3

models cannot price securities whose payoffs are non-linear, as is the case with HFs, which are free to implement dynamic trading strategies, make use of derivatives or leverage. Several papers inform about this issue and propose alternative non-linear econometric approaches, where higher moments of HF returns are considered as well. See Fung and Hsieh (2001, 2004), Agarwal and Naik (2004), Adcock (2005), Diez de los Rios and Garcia (2006), Co¨en and H¨ ubner (2009), Meligkotsidou et al. (2009), Haglund (2009) and Kang et al. (2010). In order to capture the non-linear nature of HF returns, numerous articles consider time varying systematic factor loadings in the HF return equation. Fung and Hsieh (2004) and Fung et al. (2008) identify structural breaks in factor loadings in the HF return model using time dependent dummy variables. Dupleich et al. (2009) analyze the systematic exposures of equity HF managers’ portfolios and evidence time varying factor loadings for equity HF returns. Bollen and Whaley (2009) model HF returns with time dependent factor loadings, which may capture regime switches of HF returns. They apply two non-linear models: First, the changepoint regression model of Andrews et al. (1996) in which discrete structural changes are identified. Second, the stochastic beta model where the latent factor loadings follow an AR(1) process. An alternative approach to capture non-linearity considers that the effects of both systematic and idiosyncratic components are time dependent. Chan et al. (2005) consider MS dynamics of HF returns but do not separate the impact of the systematic and idiosyncratic components. Billio et al. (2006, 2009) extend Chan et al. (2005) to explicitly consider MS structure in both the systematic and idiosyncratic components of HF returns and evidence regime switching dynamics in both components. In a recent study, Ben-David et al. (2010) show that HFs allocate on average more capital to high idiosyncratic risk stocks than to low idiosyncratic risk stocks. Thus, they are significantly exposed to idiosyncratic risk sources as well as to market specific risk factors. Ben-David et al. (2010) results are in line with Billio et al. (2009) who evidence that the 2008 financial crisis had significant impact on all HF strategies and affected both the systematic and idiosyncratic components of HF returns. According to Billio et al. (2009), assuming that only systematic risk factor exposures are important during crisis periods greatly underestimates the impact of financial crises on HF risk. Since HF returns process may exhibit a regime switching behavior, it may be interesting for practitioners to forecast its volatility. Volatility forecasts are broadly used in risk management, derivative pricing and hedging, market making, market timing, portfolio selection and many other financial market activities (Engle and Patton, 2001). In the context of HFs, more precise volatility forecasts may help risk or portfolio managers to evaluate the future risk of HF portfolios. Many risk measures commonly used by practitioners to evaluate portfolio risk, such as Value-at-Risk and Conditional-Value-at-Risk, require precise estimates of volatility. Forecasts of future volatilities can also allow portfolio managers to control 4

the risk temporally, for example, by selling an asset or portfolio before a dramatic increase in volatility takes place. Volatility forecasts can also be useful for individual HFs in their market timing activities. Furthermore, precise volatility prediction can support portfolio decisions on including HFs in diversified portfolios. Interestingly, there are very few papers on forecasting HF risk or performance in the finance literature. Some papers focus on forecasting the conditional mean of HF returns (e.g. Amenc et al. (2003) and Dash and Kajiji (2003)). These works find evidence of predictability of expected HF returns. Furthermore, Liang and Park (2010) forecast HF failure by implementing a duration model to examine its determinants. Finally, F¨ uss et al. (2007) forecast the volatility of HF returns in order to estimate value at risk of HF portfolios. The authors model HF return volatility by GARCH and Exponential GARCH (EGARCH) specifications and provide evidence of the predictability of volatility. Nevertheless, F¨ uss et al. (2007) do not include systematic components in the HF return mean equation and do not consider changing regimes. 3. Data This section describes the database used in the article, provides data description of the variables employed in the empirical section, analyzes the HF return drivers proposed by the literature and summarizes the unit root tests, cointegration relationships and common factor representation of HF index levels. 3.1 HF database Several previous papers in the HF literature report possible biases in HF data that arise due to the lack of regulatory requirement to provide transparent, uniform and audited reports on a regular basis. Due to the voluntary disclosure of HF managers, it is a standard practice in the HF industry to produce partially transparent reports with respect to the underlying portfolio. For example, Hedges (2005) notes: ‘As private entities, HFs are not allowed to advertise, but they are exempt from disclosure requirements facing traditional investment funds. As a result, there are no incentives for managers to be forthcoming with their performance data and fund’s holdings.’ See also Fung and Hsieh (1999), Goodworth and Jones (2007), Fung et al. (2008) and Bollen and Pool (2009). Moreover, the paper of Straumann (2009) gives an overview on HF database quality issues. As a consequence, two important features of HF data are backfill bias and survivorship bias. Backfill bias arises when HFs bring their history with them when they join a database. Because only funds with relatively superior historical performance enter a database, ignoring possible backfilling of data results in a bias toward mistakenly assigning superior ability to managers of funds in their earlier years. See Ackermann et al. (1999) and Brown et al. (1999) for the discussion of the backfill bias. Survivorship bias exists because ‘unlike mutual funds, HFs need 5

not register with the Securities and Exchange Commission, nor does a HF industry association exist that can document the entry and exit of funds’ (Fung and Hsieh, 1997). Therefore, it is not known exactly how many funds presented in the database exist at a given point in time. Fung and Hsieh (1997) provide a discussion on survivorship bias. Fung and Hsieh (2000) suggest a technique to mitigate the biases in HF data. They recommend using data on funds-of-funds (i.e., funds that invest in HFs), arguing that fund-of-fund returns are a more accurate representation of the returns earned by HF investors. In this article, similarly to Agarwal and Naik (2000) and Jagannathan et al. (2010), we use HF indices data obtained from HFR. The HFR Monthly Indices (HFRI) are equally weighted performance indices used as a benchmark for the HF industry. Because the HFR data contain information on when funds actually joined the database, it is corrected for backfill bias. Moreover, HFR database is constructed so as to account for survivorship bias, i.e. if a fund liquidates/closes, that fund’s performance will be included in the HFRI as of that fund’s last reported performance update. 3.2 Data description The HF index data, used in this article, consist of 250 observations of monthly returns of twelve HF indices comprising the period from January 1990 to October 2010. The article studies HF indices, which represent the performance of the HF industry at various levels. First, the global HF index (Fund Weighted Composite) is considered, which represents the overall performance of the HF industry. Second, the five subgroups of HF indices are analyzed, which include HFs with similar investment strategies. Third, specific HF strategies are studied within some subgroups. The HF indices are classified as follows: Fund Weighted Composite (HF1) (a) Equity Hedge (HF2) Equity Market Neutral (HF3) Short Bias (HF4) (b) Macro (HF5) (c) Emerging Markets (HF6) (d) Event-Driven (HF7) Distressed Restructuring (HF8) Merger Arbitrage (HF9) (e) Relative Value (HF10) Fixed Income - Asset Backed (HF11) Fixed Income - Convertible Arbitrage (HF12) The description of the five subgroups and specific HF strategies, as provided by HFR, is 6

presented in Appendix I. (See also Agarwal and Naik (2000) and Chan et al. (2005) who provide descriptions of specific HF strategies.) In the HF literature, there exist alternative classifications of HF strategies (e.g. Purcell and Crowley (1999), Agarwal and Naik (2000), Kat and Lu (2002), Ineichen (2003), Capocci et al. (2005) and F¨ uss et al. (2007)). In this article, we follow the classification of HFR because the data has been obtained from this company. This classification is similar to the HF groups defined in several previous studies (e.g. Agarwal and Naik (2000) and Ineichen (2003)). In addition, one-month US Treasury bill rate data obtained from Ibbotson Associates is used to compute the monthly excess HF returns over the risk-free rate. Monthly excess HF returns are denoted by yt in the remaining part of this article. In order to approximate the true volatility of HF returns we follow the standard approach suggested by Pagan and Schwert (1990) and Day and Lewis (1992): A proxy for the true volatility is given by σt∗ = |yt − y|, where y is the average excess return over the sample period. In the empirical analysis of HF returns, excess HF returns are controlled by monthly returns on the Morgan Stanley Capital International (MSCI) global equity market index. The MSCI data is collected for the same time span as the HF return data. In the econometric models, the MSCI monthly returns are adjusted for Treasury bill returns. In the following part of this article, MSCIt denotes the excess MSCI return. The MSCI global equity market factor is used for two reasons. First, several papers of the HF literature evidence that the equity market index is a common risk factor of HF indices (e.g. Agarwal and Naik (1999, 2004), Liang (1999), Chan et al. (2005) and Billio et al. (2006)). Second, the MSCI is a global index. Thus, it seems to be an appropriate choice for the global HF indices studied in this article compared to a countryspecific equity index, for example, the S&P500. The evolution of MSCI return and HFR global HF index return, i.e. Fund Weighted Composite return, over 1990-2010 is presented in Figure 3. The descriptive statistics, presented in Table 1, indicate that the best performing HF strategy in terms of returns is Emerging Markets with 0.9% average monthly return for the sample period, while the poorest performing category is Short Bias with −0.1% mean monthly return. A comparison of the HFs returns with the MSCI index reveals that all strategies, except for Short Bias, outperformed the broader market index for that time period. Equity Market Neutral achieved a higher average monthly return than the market index (0.3% HF versus 0.1% MSCI) with lower volatility (0.9% HF versus 4.5% MSCI). It is also the least risky strategy as compared with other indices. The most risky strategy, as measured by standard deviation, is Short Bias (5.6%) followed by Emerging Markets (4.2%). In all but two categories (Short Bias and Macro) the skewness statistics are negative, indicating that the tail on the left side of the density function is longer than the right side and mass of the distribution is concentrated on the right of the mean. The range of skewness varies from 0.410 for Macro to −2.924 for Fixed Income - Convertible 7

Arbitrage. For a non-symmetric distribution, the skewness statistic is negative when extreme positive returns are more likely than sharp negative returns. The figures on the magnitude of skewness indicate that there is a higher probability of extreme positive price changes (skewness lower than −2.0) in the arbitrage strategies such as Merger Arbitrage, Relative Value and Fixed Income strategies and lower in other categories. A more direct measure of tail-risk is kurtosis. The range of kurtosis varies from 3.860 for Macro to 30.986 for Fixed Income - Convertible Arbitrage. High kurtosis values strongly indicate the presence of fat tails in the HFs return distribution. The results presented in Table 1 show that the kurtosis is higher than 3 for all HF strategies, i.e. the tail thickness of HF return distributions is more important than that of the normal distribution. The Bera and Jarque (BJ, 1982) test for normal distribution rejects normality for all return series. These findings are consistent with Chan et al. (2005), Ranaldo and Favre (2005) and Billio et al. (2009). The presence of significant negative skewness and high kurtosis is caused by the nature of investment strategies employed by HFs. Many of them make use of derivatives and employ high leverage (e.g. Relative Value) or invest in illiquid assets (e.g. Event-Driven, Distressed Debt, Emerging Markets or Assets-Backed), which means that in the situation of adverse price movements it is difficult to close the positions and, in this way, limit the losses. In the case of Merger Arbitrage, high kurtosis and large negative skewness reflect large exposure to event risk. In line with other studies (e.g. Brooks and Kat (2001), Lo (2001), Kat and Lu (2002), Getmansky et al. (2004), Loudon et al. (2006) and Jagannathan et al. (2010)), high serial correlation is observed in the HF indices return series. The Ljung and Box (LB, 1978) serial correlation test statistic, computed up to the 20th-order autocorrelation, evidences significant serial correlation of excess returns for several HF indices. One possible explanation of this fact is that the nature of HF strategies leads their returns to be inherently related to those of the preceding months. An alternative explanation lies in the difficulty for HF managers to obtain up-to-date valuations of their positions in illiquid and complex over-the-counter securities. In this case, HFs either use the last reported transaction price or an estimate of the current market price, which may easily create lags in the evolution of their net asset value. This would explain why the Fixed Income - Convertible Arbitrage, Fixed Income - Assets Backed, Emerging Markets and Distressed Restructuring HF indices exhibit the most significant autocorrelation. Finally, as expected, we observe a high positive correlation with the stock market in the case of Fund Weighted Composite (0.74), Equity Hedge (0.71), Emerging Markets (0.69), EventDriven (0.68), Distressed Restructuring (0.51), Merger Arbitrage (0.51) and Relative Value (0.52). Furthermore, Short Bias exhibits a high negative correlation with the equity markets (−0.67). Correlation with the bond market is relatively low, though significant, for most of the strategies. 8

[APPROXIMATE LOCATION OF TABLE 1] 3.3 HF return drivers This section describes the HF return drivers reported in the literature (e.g. Agarwal and Naik (1999, 2004), Liang (1999), Chan et al. (2005) and Billio et al. (2006)) and evaluates their impact on HF returns. HF return drivers suggested by previous works can be classified into the following groups of factors: (a) Equity factors (b) Volatility factors (c) Interest rate and credit risk factors (d) Fama and French (1993)-type factors (e) Currency factors (f) Commodity factors (g) Emerging markets factors The specific HF return factors of each group analyzed by this article are presented in Appendix II. Table 2 presents correlation coefficients between HF index returns and HF return drivers and their statistical significance. The following risk factors have significant correlation coefficients at the 10% level for all HF indices: MSCI, S&P500, Russel2000 (R2000) returns, return on Large minus Small (LS) factor, return on Commodity Research Bureau Continuous Commodity Index (CCI), return on Emerging Market Stock Index (EMS) and return on Emerging Market Bond Index (EMB). From all the factors considered, only performance of the US dollar index is not significant for any of the HFs strategies. Table 2 evidences the following interactions among observable risk factors and HF strategies. Consistently with previous studies, returns of HFs taking positions in equity markets (e.g. Equity Hedge, Short Bias, Event-Driven, Merger Arbitrage, Distressed Restructuring, Relative Value) are mainly driven by performance of equity risk factors (e.g. MSCI, S&P500, R2000 and LS factor). These strategies perform worse in the periods of high market volatility except for the Short Bias strategy, which is positively affected by volatility due to the short position held in equity markets. The Equity Market Neutral index has relatively low correlation with the market factors (e.g. equity and Fama-French-type factors) because this strategy usually matches long and short positions in equity markets in order to have a low level of market risk. Similarly, Macro funds that aim to profit from major economic trends and events in the global economy by investing in very different investment instruments, for example futures and options, do not show significant exposure to any specific market, except for a moderate correlation with equity factors. 9

Furthermore, increased volatility affects negatively the Emerging Markets strategy because in the moments of uncertainty, investors tend to abandon emerging markets to search for safer investments in global financial markets. Emerging Markets are mainly driven by the performance of emerging equity and fixed income markets as these are the main investment areas of HF managers following this strategy. Finally, rising commodities prices have positive influence on this strategy because many emerging countries are suppliers of commodities. [APPROXIMATE LOCATION OF TABLE 2] 3.4 Unit root tests, cointegration and common factors In this section, a further analysis of the statistical properties of HFs index data is reported for the period between January 1990 and October 2010. First, results of the Augmented Dickey and Fuller (ADF, 1979) unit root test are summarized. Second, the main implications of the Johansen (1988, 1991) cointegration test are presented. Third, the common trends representation of the cointegrating system of Stock and Watson (1988) is considered and the estimation results of the common factors model are studied. The unit root test (Table A1), cointegration test (Table A2), common factor model parameter estimates (Table A3), analysis of common factors versus HF regime drivers (Table A4) and the evolution of common factors over the period 1990 to 2010 (Figure A1) are presented in Appendix III. Table A1 presents the ADF test results for each HF index. The order of integration of HF prices is verified because the cointegration test is applied for unit root processes. Three ADF tests are performed: ADF1 (without constant, without trend), ADF2 (with constant, without trend) and ADF3 (with constant, with trend). In the ADF tests, the significance of the parameter estimates of the constant and trend parameters is also verified and a general to specific approach is followed to determine the appropriate ADF test statistic to be used. First, the ADF3 test is evaluated and the significance of the trend parameter is checked. If it is not significant then the ADF2 test is considered. The ADF2 test is evaluated in a similar manner by checking the significance of the constant parameter. In Table A1, the ADF test statistic applied is indicated by box. The unit root null hypothesis is rejected at the 10% level for two HF indices: Equity Market Neutral and Relative Value, while for the other ten HF indices the null hypothesis cannot be rejected. Table A2 shows the Johansen (1988, 1991) cointegration test results for ten HF index series where the unit root null hypothesis is not rejected. The table presents two alternative test statistics: the trace and the maximum eigenvalue statistics. The results obtained are robust for both tests because four cointegrating equations are evidenced at the 10% level for both test statistics, i.e. there is cointegrating relationship among ten HF indices. The existence of cointegrating relationships suggest that the movements of the ten HF indices are influenced by 10

some common factors. The Johansen (1988, 1991) test suggests the existence of 10 − 4 = 6 common factors (see Hamilton, 1994, Chapter 19), which drive these ten HF indices. Therefore, the cointegrating system of the ten HF indices can be represented by the common trends representation of Stock and Watson (1988). The common factor model and the time series of the latent common trends are estimated by the Kalman filter approach (Hamilton, 1994, Chapter 13). The parameter estimates are presented in Table A3. The evolution of the common latent factors over the period 1990 to 2010 is presented in Figure A1. In order to relate the latent factors with some observable HF return drivers, Table A4 shows the correlation coefficients between the HF return drivers and the first differences of the common factors and identifies which HF return determinants are significantly correlated with the latent factors. The significantly correlated HF return drivers are listed for each latent factor as follows: (a) Factor 1 : Interest rate and credit risk factors (US6L, CS, TS); Commodity factors (OIL, CCI); (b) Factor 2 : Equity factors (MSCI, R2000); Interest rate and credit risk factors (CS, TS); Fama and French (1993)-type factor (LS); Emerging markets factor (EMS); (c) Factor 3 : Interest rate and credit risk factors (CS, TS); Fama and French (1993)-type factor (LS); (d) Factor 4 : Volatility factor (VIX); Interest rate factor (US6L); Fama and French (1993)-type factors (LS, VG, MU); Emerging markets factor (EMS); (e) Factor 5 : Equity factor (R2000); Volatility factor (VIX); Interest rate and credit risk factors (US6L, CS, TS); Fama and French (1993)-type factors (LS, VG, MU); Emerging markets factor (EMS); (f) Factor 6 : Credit risk rate factor (CS); Fama and French (1993)-type factor (LS). (See Appendix II for the abbreviations of HF regime drivers.) Notice that the S&P500, USD Index (USD), Gold spot price index (GOLD) and Emerging Market Bond Index (EMB) HF return drivers are not correlated significantly with any of the common factors. 4. Models, estimation and volatility forecast formulas This section summarizes the models of excess HF returns used in this article, briefly presents the statistical inference procedures of these models and the formulae for computing n-step ahead forecasts are also summarized. As most of the methodologies employed are well documented in the literature, each model is presented only very briefly, starting with our benchmark model: the GARCH(1,1) model. 11

4.1 GARCH(1,1) model of idiosyncratic HF returns The excess returns of a HF index yt is modeled over t = 1, . . . , T periods. It is assumed that excess HF returns are normally distributed with time dependent conditional mean µt and conditional variance σt2 : yt = µt + ²t = µt + σt ut , (1) where ut in the last equation is a N(0,1) distributed i.i.d. error term. In this article, the volatility of the idiosyncratic component of returns is forecasted. Therefore, the MSCI global equity market factor is included into µt as follows: µt = c + ζyt−1 + φMSCIt ,

(2)

where the AR(1) term is included to account for the serial correlation of HF returns reported previously. See F¨ uss et al. (2007) and Kang et al. (2010) who consider various ARMA structures of the conditional mean for similar reasons. The covariance stationarity of the AR(1) model requires that |ζ| < 1. Furthermore, φ measures the impact of the MSCI global equity market factor on µt . The benchmark model of σt2 employed in this article is the GARCH (1,1) model of Bollerslev (1986) and Taylor (1986). In the standard notation used in the literature, σt2 can be formalized as: 2 σt2 = ω + α²2t−1 + βσt−1 , (3) where the α + β < 1 condition is required for stationarity. This specification is considered for forecasting conditional volatility at least for two reasons. First, the GARCH(1,1) model is widely applied in the financial econometrics literature. Dunis et al. (2003) include the GARCH(1,1) specification to compare alternative models’ volatility forecasts and in several cases find superior forecast performance to other models. Preminger et al. (2006) use the GARCH(1,1) model as a benchmark to evaluate the forecasting performance of their extended switching regression models. Furthermore, Muzzioli (2010) employs the GARCH(1,1) model to check for robustness of option-based volatility forecasts. Hansen and Lunde (2005) compare the out-of-sample performances of more than 300 different volatility models to the GARCH(1,1) model, concluding: ‘Interestingly, the best models do not provide a significantly better forecast than the GARCH(1,1) model.’ See also Donaldson and Kamstra (1997) for similar results regarding the GARCH(1,1) model. Second, the first-order specification of GARCH is used because the forecasting performance of the GARCH(1,1) specification is found to be superior to alternative GARCH models with more complicated lag structure. The parameters of the benchmark model are estimated by the maximum likelihood method. The vector Yt = (y1 , . . . , yt ) for t = 1, . . . , T , denotes excess returns observed until period t. The 12

likelihood function of Yk with 1 ≤ k ≤ T is given by the product of conditional density functions f: · ¸ k k Y Y 1 (yt − µt )2 p L(Yk ; θ) = f (yt |Yt−1 ) = exp − , (4) 2 2 2σ 2πσ t t t=1 t=1 where θ = (c, ζ, φ, ω, α, β) denotes the vector of parameters. The parameter estimates θˆ are obtained by maximizing ln L numerically in θ. Engle and Bollerslev (1986) and Baillie and Bollerslev (1992) give the n-step ahead variance 2 forecast, σ ˆt+n , for a GARCH (1,1) process: 2 2 ˆ n (ht − sˆ2 ), σ ˆt+n = E(σt+n |Yt−1 ) = sˆ2 + (ˆ α + β)

(5)

ˆ where sˆ2 = ω ˆ /(1 − α ˆ − β). 4.2 MS-GARCH(1,1) model of idiosyncratic HF returns MS models have been developed as a way of allowing data to arise from a combination of two or more distinct data generating processes (Hamilton, 1989). At each time t, the actual process generating the data is determined by the realization of a latent random discrete variable denoted by st , which is called a state variable or regime. In the MS models, the st over t = 1, . . . , T is assumed to form a Markov process. In this article, a combination of the original MS model developed by Hamilton (1989) and the GARCH(1,1) framework is applied. This makes it possible to include MS dynamics into the conditional volatility equation of the idiosyncratic return of HF indices. Consideration of the MS parameters in the returns model is motivated for example by Diebold (1986) who notes that the GARCH specification can be improved by including regime dummy variables for the conditional variance intercept. Moreover, Freidman and Laibson (1989) note that the GARCH model does not differentiate between the persistence of large and small shocks. In the past two decades, various MS-GARCH specifications have been proposed in the financial econometrics literature. There have been two streams of the MS-GARCH literature: (1) path dependent MS-GARCH models and (2) non-path dependent MS-GARCH models. In the path dependent MS-GARCH models, the conditional density of yt depends on all previous values of st , which complicates the model estimation. Dueker (1997) estimates an MSGARCH model using the collapsing procedure introduced by Kim (1994). Recently, Bauwens et al. (2010) and Henneke et al. (2011) suggest Bayesian Markov Chain Monte Carlo (MCMC) methods for the inference of the path dependent MS-GARCH. In the non-path dependent MS-GARCH models, the conditional density of yt depends only on the current regime st . For example, Gray (1996), Klaassen (2002) and Haas et al. (2004) use a recombining structure for the regime path tree, where the past values of st are integrated 13

out from the likelihood function. The estimation of these specifications can be done using the standard MS inference method (see Kim and Nelson, 1999). In this article, the non-path dependent MS volatility model of Klaassen (2002) is estimated for two reasons. First, the MS-GARCH model of Klaassen (2002) can be estimated more rapidly than the path dependent MS models. Therefore, it is more appropriate for the repeated out-ofsample forecasting purposes of this article. Second, we have found the forecasting performance of Klaassen’s (2002) model has been superior to alternative MS-ARCH (e.g. Hamilton and Susmel (1994) and Cai (1994)) and other non-path dependent MS-GARCH models (e.g. Gray (1996) and Haas et al. (2004)). In the remaining part of this section, the most important details of the Klaassen (2002) model are presented. It is assumed that excess HF returns are normally distributed with time and regime dependent conditional mean µt (st ) and conditional variance σt2 (st ): yt = µt (st ) + ²t (st ) = µt (st ) + σt (st )ut ,

(6)

where ut in the second equation is a N(0,1) distributed i.i.d. error term and st ∈ {1, 2} indicates the regime at time t, which forms a Markov chain with the 2 × 2 transition probability matrix P = {ηij }. The transition probability matrix of st is given by the next four parameters: Pr[st = 1|st−1 = 1] = η11 Pr[st = 1|st−1 = 2] = η12 , Pr[st = 2|st−1 = 1] = η21 Pr[st = 2|st−1 = 2] = η22 ,

(7)

where η11 + η21 = 1 and η22 + η12 = 1. The MSCI global equity market factor is included into µt (st ) as follows: µt (st ) = c(st ) + ζ(st )yt−1 + φ(st )MSCIt . (8) Francq and Zakoian (2001) give the stationarity condition for the MS-AR model, which can be applied for this specification. The conditional variance of excess returns, σt2 (st ) is formulated as follows: 2 σt2 (st ) = ω(st ) + α(st )²2t−1 (st ) + β(st )σt−1 (st ), ²t−1 (st ) = E[²t−1 (st−1 )|st , Yt−1 ], 2 2 σt−1 (st ) = E[σt−1 (st−1 )|st , Yt−1 ].

(9)

The stationarity condition for Klaassen’s (2002) model has been derived by Abramson and Cohen (2007). The expressions for the MS-AR and MS-GARCH covariance stationarity conditions and the computation of the expectations in the last two equations of the MS-GARCH formulation are summarized in Appendix IV. The parameters of the model are estimated by the maximum likelihood method. The likelihood function of Yk with 1 ≤ k ≤ T is given by: L(Yk ; θ) =

k X Y

f (yt |st = i, Yt−1 ) Pr[st = i|Yt−1 ] =

t=1 i=1,2

14

k X Y

½

[yt − µt (st = i)]2 p = exp − 2σt2 (st = i) 2πσt2 (st = i) t=1 i=1,2 1

¾ Pr[st = i|Yt−1 ],

(10)

where θ = (c1 , c2 , ζ1 , ζ2 , φ1 , φ2 , ω1 , ω2 , α1 , α2 , β1 , β2 , η11 , η22 ) denotes the vector of parameters. The probability Pr[st = i|Yt−1 ] can be computed by standard MS inference method (Kim and Nelson, 1999). Appendix IV summarizes some details of the computation of the likelihood function. The parameter estimates θˆ are obtained by maximizing ln L numerically in θ. 2 Klaassen (2002) gives a recursive formula for the n-step ahead variance forecast, σ ˆt+n (st+n ), for a GARCH (1,1) process: X 2 2 σ ˆt+n (st+n = i) Pr[st+n = i|Yt−1 ], (11) σ ˆt+n = i=1,2

where 2 2 σ ˆt+n (st+n ) = E[σt+n (st+n )|Yt−1 ] = 2 = ω(st+n ) + [α(st+n ) + β(st+n )]E[σt+n−1 (st+n−1 )|st+n , Yt−1 ].

(12)

The details of the volatility forecast computation are presented in Klaassen (2002). 5. Results This section summarizes the in-sample estimation results, the in-sample HF regime determinants, the out-of-sample forecasting procedure used, the model diagnostics procedures, the forecasting precision measure employed, the forecast accuracy and forecast encompassing test results, and compares the forecast performance of competing econometric models for crisis and non-crisis subperiods. 5.1 In-sample model comparisons This subsection reports the in-sample estimation results for the GARCH and MS-GARCH specifications for the period 1990 to 2010. Tables A5-A7 of Appendix V present the in-sample estimation results of the GARCH and MS-GARCH specifications. For three HF indices MS dynamics are not significant: Fund Weighted Composite, Equity Market Neutral and Macro. This may be explained by the fact that these strategies are not driven by any specific market factor (see Section 3.3). In the remaining part of this article, only the GARCH estimates are reported for these three series. Moreover, for some HF indices the non-significant α and β parameters of the GARCH and MS-GARCH models are restricted to zeros and are not reported in Appendix V. Several previous works employ likelihood based measures for model selection purposes in the MS models literature (e.g. Gray, 1996; Smith, 2002; Kalimipalli and Susmel, 2004). Motivated by these works, Tables A5-A7 include the next model selection metrics: Log Likelihood (LL), 15

Akaike Information Criterion (AIC), AIC corrected for finite sample bias (see Burnham and Anderson, 2002) (AICc) and Bayesian Information Criterion (BIS). The AIC, AICc and BIC measures are computed as follows: AIC = 2K − 2LL, 2K(K + 1) , T −K −1 BIC = K ln(T ) − 2LL,

AICc = AIC +

(13) (14) (15)

where K is the number of parameters of the model and T is the sample size. The LL measure is higher for the MS-GARCH model than for the GARCH specification for all HF indices. The AIC, AICc and BIC measures are lower, i.e. indicating a better model, for all HF indices besides the Emerging Markets index. Nevertheless, the differences between the previous in-sample model quality measures for the MS-GARCH and GARCH formulations may not be statistically significant. Therefore, this article applies the Likelihood Ratio (LR) test statistic to evidence significant differences in explanatory power. Because the GARCH and MS-GARCH specifications of this article are nonnested models (see for example Smith, 2002), the non-nested LR test approach of Vuong (1989) is used. Tables A5-A7 show the significance of the LR statistic according to Vuong (1989). Besides the Emerging Markets index, the LR statistic is significant showing better forecast performance of the MS-GARCH model for all HF indices. Although the AIC, AICc and BIC indicate worse performance of the MS-GARCH model than the GARCH for the Emerging Markets index, the quality difference tested by the LR statistic is not significant. Finally, Figures A2-A10 of Appendix V present the evolution of the filtered probability of the first regime over the period 1990 to 2010, i.e. Pr[st = 1|Yt−1 ] over t = 1, . . . , T with the corresponding HF index excess return series. Notice on the figures that the first regime represents periods with higher volatility for all HF indices. 5.2 In-sample HF regime determinants Practitioners may be interested in the following question: Are regime switches of HF returns caused by systematic factors or by idiosyncratic determinants? As an initial attempt to answer this question, this section studies whether regime switches of the HF returns are influenced by regime switches of the systematic component, i.e. MSCI excess returns. First, the following MS-GARCH(1,1) specification is estimated for MSCIt : MSCIt = c(st ) + ²t (st ) = c(st ) + σt (st )ut ,

(16)

2 σt2 (st ) = ω(st ) + α(st )²2t−1 (st ) + β(st )σt−1 (st ).

(17)

16

Figure 4 reports the evolution of filtered probability of regime 1 during this period. The figure shows that significant MS behavior of the market factor is evidenced over the period 1990 to 2010. Second, the filtered probabilities of regime 1 are transformed applying the {f : (0, 1) → R} with f (x) = ln[x/(1 − x)] function for all assets in order to get unrestricted values for the original filtered probabilities. Notice that regime 1 corresponds to the high volatility regime for all HFs and the MSCI index. Third, the correlation coefficient between the transformed filtered probability of each HF index and the transformed filtered probability of MSCI is computed. Fourth, the following regression is estimated for each HF index: π ˜t = c + ψ˜ πMSCIt + et ,

(18)

where π ˜t and π ˜MSCIt denote the transformed filtered probabilities of the HF and MSCI indices, respectively, and et is the error term. Table 3 shows that HF regime switching is related to structural changes in the market factor for the following HF indices: Equity Hedge, Short Bias, Event-Driven, Distressed Restructuring, Relative Value and Fixed Income - Asset Backed. [APPROXIMATE LOCATION OF FIGURE 4 AND TABLE 3] 5.3 Out-of-sample forecasting procedure and model diagnostics The out-of-sample forecasts are derived by dividing the 20-year study period into two subsamples. The first subsample contains 108 observations from January 1990 to December 1998, which are used to estimate the parameters of the two competing models in order to produce out-of-sample volatility forecasts. Next, the data is updated by adding the first month of 1999 to the previous subsample and the model parameters are re-estimated in order to produce the out-of-sample forecasts for the subsequent months of 1999. This procedure is repeated until volatility forecasts are obtained for each month for the period January 1999 to October 2010. Multi-step-ahead volatility forecasts of volatility are estimated for one, two, three and four 2 quarters. More formally, n-step ahead forecasts of σt+n with n = 3, 6, 9 and 12 months are computed for the historical HF data. We choose these time horizons as they coincide with typical HF redemption periods for the investors. In the forecasting procedure described, after the estimation of each competing model, model diagnostics are checked in five steps. First, the significance of the φ parameter measuring the impact of MSCI on the HF index return is checked. The φ parameter is significant for all indices, which justifies the consideration of this market factor in the conditional mean equation. Second, the significance of the parameters of the dynamic terms in the volatility equation, i.e. α and β, is analyzed. In some subsamples, the Merger Arbitrage HF index return series do not evidence significant single-regime GARCH dynamics. In these cases, the GARCH model 17

is re-estimated with α = β = 0 for this HF index. Furthermore, in the case of Emerging Markets, Event-Driven, Distressed Restructuring and Fixed Income - Asset Backed either one or both regimes do not have significant GARCH dynamics for some subsamples. In these cases, depending on the general MS-GARCH model estimation result, the non-GARCH regime is restricted for constant variance, i.e. α(i) = β(i) = 0, and the restricted model is re-estimated. Third, the significance of the parameters of the transition probability matrix, i.e. η11 and η22 , of the MS-GARCH specification is studied in order to see whether two different regimes of the return process exist. For nine HF indices, MS dynamics of idiosyncratic excess HF returns are found. However, two separate regimes of idiosyncratic excess returns are not identified for the Fund Weighted Composite, Equity Market Neutral and Macro indices. In these cases, only the single-regime GARCH model is estimated. Fourth, covariance stationarity of the dynamic models is verified. The AR(1) and MS-AR(1) conditional mean formulations are found to be stationary. Regarding the volatility equations, the estimated GARCH and MS-GARCH models are stationary. Nevertheless, for some HF indices the single-regime GARCH(1,1) model is close to non-stationarity probably due to high volatility during the financial market crisis periods included in the 20-year sample. The MS-GARCH specification is weak stationary for all HF indices for the period studied. Fifth, the residuals corresponding to the error term ut are computed, the desired properties of the first and second moments are checked and residuals are tested for the absence of serial correlation using the LB(20) statistic. The results confirm the assumptions of the GARCH and MS-GARCH models. 5.4 Out-of-sample forecasting precision The n-step ahead forecast performance of the models over the period 1999 to 2010 is verified using the root mean squared error (RMSE) loss function. The RMSE is given as follows: v u t+n u1 X RMSEn = t (σ ∗ − σ ˆτ )2 . (19) n τ =t+1 τ The RMSE loss function is used because Hansen and Lunde (2006) and Patton (2011) show that the MSE is a robust loss function (Patton, 2011) of volatility forecasts evaluation. Table 4 reports the RMSE of volatility forecasts for all HF indices. The table suggests that the volatility forecasting precision of the MS-GARCH formulation is superior to the singleregime GARCH model for most HF indices for most forecast time horizons. The exceptions are Equity Hedge and Emerging Markets, which show higher RMSE for the MS-GARCH model for all forecast horizons. Furthermore, the Fixed Income - Asset Backed index shows higher RMSE for the MS-GARCH framework for the 9-month forecast horizon. [APPROXIMATE LOCATION OF TABLE 4] 18

5.5 Out-of-sample forecast accuracy and forecast encompassing tests The forecasting precision measure, presented in the previous section, cannot determine if a given forecasting framework is in fact significantly better than another. In order to evaluate the statistical significance of alternative models, two alternative forecast comparison tests are conducted. First, the forecast accuracy test of Diebold and Mariano (1995) is performed. Second, the forecast encompassing test of Chong and Hendry (1986) is applied, which is used by several past works to compare the forecasting performance of competing forecasting models (e.g. Donaldson and Kamstra, 1997; Darrat and Zhong, 2000; and Preminger et al., 2006). First, the Diebold and Mariano (1995) test statistics are presented in Table 5. The null hypothesis of this test is that the forecast accuracy of the competing models is equal. The Diebold and Mariano (1995) statistic is computed by the difference of the squared forecast errors of the GARCH and the MS-GARCH specifications. Positive values of the test statistic in Table 5 indicate superior performance of the MS-GARCH model, while negative values reflect better forecasting performance of the GARCH specification. The table shows that the MSGARCH produces significantly more accurate volatility forecasts than the GARCH model for the Event-Driven (for 6, 9, 12 months), Distressed Restructuring (for 3, 6, 9, 12 months), Merger Arbitrage (for 3, 6, 9, 12 months), Relative Value (for 3, 6, 9 months) and Fixed Income - Convertible Arbitrage (for 3 months). For other HF indices and forecast horizons, the MSGARCH specification does not yield significantly more accurate idiosyncratic volatility forecasts than the GARCH model. Although the RMSE of the MS-GARCH model forecast is higher in the case of Equity Hedge and Emerging Markets indices, the Diebold and Mariano (1995) test does not find significant the superior performance of the GARCH model. [APPROXIMATE LOCATION OF TABLE 5] Second, the forecast encompassing tests of this article are based on a set of two linear regressions of the forecast error from one model on the forecast obtained from the other model. The regressions can be formulated as follows: ∗ (σt+n −σ ˆ1,t+n ) = c1 + δ1n σ ˆ2,t+n + e1t ,

(20)

∗ (σt+n −σ ˆ2,t+n ) = c2 + δ2n σ ˆ1,t+n + e2t ,

(21)

where e1t and e2t are the error terms. The subindices 1 and 2 in the volatility forecast terms refer to the GARCH and MS-GARCH volatility specifications, respectively. The null hypothesis of the forecast encompassing test is that neither model encompasses the other, which is accepted when both δ1n and δ2n are not significant or neither δ1n nor δ2n are significant. When δ1n is significant at some predetermined level but δ2n is not significant, then the null hypothesis will be rejected in favor of the alternative hypothesis. In this case, the 19

MS-GARCH model encompasses the GARCH model. Conversely, when δ2n is significant but δ1n is not significant, then the null hypothesis will be rejected in favor of the alternative hypothesis. In this case, the GARCH model encompasses the MS-GARCH model. The forecast encompassing tests’ results are presented in Table 6 for nine HF indices for which significant MS dynamics have been found during the forecast period. The name of the model forecasting error is presented on the left side of the table, while the name of n-step ahead forecast model is listed on the top of the table. The table contains p-values associated with the Newey and West (1987) heteroscedasticity and autocorrelation robust covariance matrix. A p-value less than 0.10 indicates that the forecast from the model listed along the top of the table explains, with 10% significance, the forecast error from the model listed down the left side of the table. Therefore, the model listed on the left side cannot encompass the model listed on the top. The forecast encompassing results stated in Table 6 evidence that the MS-GARCH volatility forecast encompasses the GARCH forecast at the 10% level of significance (for different forecast horizons) in the following cases: Short bias (for 12 months), Event-Driven (for 3, 6, 9, 12 months), Distressed Restructuring (for 6, 9 months), Merger Arbitrage (for 3, 6, 9, 12 months), Relative Value (for 6 months) and Fixed Income - Asset Backed (for 3 months). [APPROXIMATE LOCATION OF TABLE 6] 5.6 Forecast accuracy in ‘crisis’ and ‘non-crisis’ periods This section compares the predictive performance of GARCH and MS-GARCH specifications for out-of-sample volatility forecasts over the period 1999 to 2010 for crisis and non-crisis subperiods. The full sample period is divided into crisis and non-crisis periods by using the filtered probability estimated in the MS-GARCH framework. If Pr[st = 1|Yt−1 ] ≥ 0.5 then period t will be a ‘crisis’ period. On the other hand, when Pr[st = 1|Yt−1 ] < 0.5 then period t will be defined as a ‘non-crisis’ period. The first regime, i.e. st = 1 corresponds to the high volatility periods for all HF indices (see Figures A2-A10). Therefore, this definition associates the term crisis with the period of high volatility and non-crisis with a period of low volatility in financial markets. The previous definition is in line with the general concept of financial crisis that can apply to a variety of situations in which some financial institutions or assets suddenly lose a significant part of their value. In the past, many financial crises were associated with banking panics and in some cases (but not all) coincided with recession periods. Other situations that are often called financial crises include stock market crashes, bursting of financial bubbles, currency crises or sovereign defaults. Financial crises imply large losses of asset values but may only influence the real economy when they are followed by a recession or depression. (See for instance Kindleberger and Aliber (2005) and Kolb (2010).) Independent of the causes of financial crises, 20

such periods are characterized by high volatility compared to the historical average. Figure 5 shows the evolution of the market volatility VIX index and its unprecedented high levels during the financial crises in the last 20 years. [APPROXIMATE LOCATION OF FIGURE 5] The forecast performance results of GARCH and MS-GARCH models are presented in Tables A8 and A9 of Appendix VI. Table A8 shows the RMSE of nine HF indices computed for crisis and non-crisis periods for both models. The table exhibits that the RMSE for volatility forecasts of the crisis period is higher than the RMSE computed for the non-crisis period for all HF indices and for all forecast horizons. In order to check the significance of the differences between the RMSEs of the GARCH and MS-GARCH models, the forecast precision test of Diebold and Mariano (1995) is performed. Table A9 presents the Diebold and Mariano (1995) forecast accuracy test statistics for nine HF indices for crisis and non-crisis periods. This test compares the predictive accuracy of GARCH and MS-GARCH models in a similar way, as is described in Section 5.5. Table A9 evidences some interesting findings. First, including regime MS dynamics in the traditional GARCH model significantly improves volatility forecasts during both crisis and noncrisis periods in the case of the Distressed Restructuring strategy for all time horizons. This is an important result because this strategy is particularly sensitive to the economic conditions and the phase of the business cycle, e.g. moments of market distress or recession periods are characterized by an increasing number of companies in distress, low general level of investment, low credit availability and low or non-existent M&A activity. Similarly, the MS-GARCH model’s forecasting performance is superior to the GARCH formulation for other strategies, which depend strongly on the macroeconomic conditions, such as Event-Driven (for 6 months and for crisis periods) and Merger Arbitrage (for all time horizons and the non-crisis periods). Results in Table A9 show that MS-GARCH better predicts volatility during non-crisis periods in the case of the Relative Value (for 3, 6, 9 months), Fixed Income - Asset Backed (for 9 and 12 months) and Fixed Income - Convertible Arbitrage (for all time horizons) strategies. Finally, Table A9 exhibits for the Short Bias index that the MS-GARCH model forecasts significantly better the volatility for most time horizons during the crisis periods. 6. Summary and conclusions In this article, the out-of-sample and multi-step ahead idiosyncratic HF return volatility forecasting performance of GARCH and MS-GARCH models has been compared. The conditional mean equation of both models has considered AR dynamics to control for the serial correlation of HF returns. Moreover, the systematic, equity market factor has been

21

captured by the MSCI index. The conditional variance equations have considered GARCH (Bollerslev, 1986) and non-path dependent MS-GARCH (Klaassen, 2002) specifications. The two competing models have been estimated for twelve global HF indices: Fund Weighted Composite, Equity Hedge, Equity Market Neutral, Short Bias, Macro, Emerging Markets, Event-Driven, Distressed Restructuring, Merger Arbitrage, Relative Value, Fixed Income - Asset Backed and Fixed Income - Convertible Arbitrage. The article has employed data collected over the period January 1990 to October 2010. The MS dynamics have not been significant for three HF indices: Fund Weighted Composite, Equity Market Neutral and Macro. Therefore, 3, 6, 9 and 12 month ahead out-of-sample volatility forecasts have been produced over the period January 1999 to October 2010 for nine HF indices. The forecasting precision has been evaluated by the robust RMSE loss function, which has evidenced superior forecasting performance of the MS-GARCH formulation for most HF indices and for most forecast horizons. In order to see the significance of the superior forecast accuracy, a robustness check has been done by the Diebold and Mariano (1995) forecast accuracy test and the forecast encompassing test of Chong and Hendry (1986). These tests suggest that the volatility of the Short Bias, Event-Driven, Distressed Restructuring, Merger Arbitrage, Relative Value, Fixed Income - Asset Backed and Fixed Income - Convertible Arbitrage indices have been forecasted significantly better out-of-sample by the MS-GARCH model than the volatility of the Equity Hedge and Emerging Markets indices over the period 1999 to 2010. Finally, the article has also compared the out-of-sample forecasts of the GARCH and MSGARCH frameworks for crisis and non-crisis periods between 1999 and 2010. Results have revealed the superiority of the MS-GARCH model in predicting volatility of Distressed Restructuring strategy both during crisis and non-crisis periods. The Diebold and Mariano (1995) test has also suggested that the volatility of the Short Bias and Event-Driven indices can be forecasted more accurately than the rest of the strategies studied in the article during crisis periods of 1999 to 2010. Furthermore, the same test statistics have evidenced superior performance in volatility forecasting during the non-crisis periods of 1999 to 2010 in the case of Merger Arbitrage, Relative Value, Fixed Income - Asset Backed and Fixed Income - Convertible Arbitrage. The present article can be extended in several ways. First, additional systematic factors besides the global equity market index (MSCI) may be considered. For example, future works could include the Fama and French (1993) factors, the Carhart (1997) factors or other observable or latent HF regime drivers. Second, further empirical works should investigate the determinants of HF regimes. Future research may evaluate the importance of HF return drivers proposed by past articles as determinants of HF regimes. Third, future papers may consider extended MS models with endogenous switching regimes, for example extending Kim et al. (2008), to study 22

the importance of alternative HF regime drivers. Fourth, subsequent HF volatility papers may also consider multivariate econometric frameworks of the HF return series motivated by the existence of common HF regime drivers and cointegrating relationships among HF indices. Acknowledgements The authors thank the Editor, the anonymous referees, Luc Bauwens, Helmuth Ch´avez, Christine Choriat, Georges Gallais-Hamonno, Germ´an L´opez, Yuliya Lovcha, Bertrand Maillet, Carlos M´endez, Anna Nasz´odi, Alejandro P´erez-Laborda, H´el`ene Raymond, and participants of the following conferences: Forecasting Financial Markets 2009 Conference, Luxembourg, May 2009; Monetary Economics, Banking and Finance Conference, Orl´eans, June 2009; European Economics and Finance Society Conference, Warsaw, June 2009; 29th Annual International Symposium on Forecasting, Hong Kong, June 2009; XVII Spanish Finance Association Meeting, Madrid, November 2009. Additional thanks go to seminar participants at Universidad de Navarra, Spain and Universidad Francisco Marroqu´ın, Guatemala for helpful comments and discussions on previous versions of this article. The first author acknowledges the research financing of the PIUNA project from Universidad de Navarra. The second author acknowledges financial support: S2009/ESP1685, ECO2009-14457-C04-03 and ECO2010-17625 of the Spanish Ministry of Science and Innovation.

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F¨ uss, R., D. G. Kaiser, and Z. Adams. 2007. Value at risk, GARCH modelling and the forecasting of hedge fund return volatility. Journal of Derivatives & Hedge Funds 13, no. 1: 2-25. Getmansky, M., A. W. Lo, and I. Makarov. 2004. An econometric model of serial correlation and illiquidity in hedge fund returns. Journal of Financial Economics 74: 529-609. Goodworth, T. R. J., and C. M. Jones. 2007. Factor-based, non-parametric risk measurement framework for hedge funds and fund-of-funds. The European Journal of Finance 13, no. 7: 645-655. Gray, S. 1996. Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42: 27-62. Haas, M., S. Mittink, and M. S. Paolella. 2004. A new approach to Markov-switching GARCH models. Journal of Financial Econometrics 2, no. 4, 493-530. Haglund, M. 2009. Higher moment diversification benefits of hedge fund strategy allocation. Journal of Derivatives & Hedge Funds 16, no. 1: 53-69. Hamilton, J. D. 1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, no. 2: 357-384. Hamilton, J. D. 1994. Time Series Analysis. Princeton University Press, Princeton, New Jersey. Hamilton, J. D., and R. Susmel. 1994. Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics 64: 307-333. Hansen, P. R., and A. Lunde. 2005. A forecast comparison of volatility models: does anything beat a GARCH(1,1)? Journal of Applied Econometrics 20, no. 7: 873-889. Hansen, P. R., and A. Lunde. 2006. Consistent ranking of volatility models. Journal of Econometrics 131, no. 12: 97-121. Hasanhodzic J., and A. W. Lo. 2007. Can hedge-fund returns be replicated? The linear case. Journal of Investment Management 5, no. 2: 5-45. Hedges, I. V. J. R. 2005. Hedge fund transparency. The European Journal of Finance 11, no. 5: 411-417. Henneke, J. S., S. T. Rachev, F. J. Fabozzi, and M. Nikolov. 2011. MCMC-based estimation of Markov switching ARMA-GARCH models. Applied Economics 43, no. 3: 259-271. Ineichen, A. M. 2003. Absolute Returns - The Risk and Opportunities of Hedge Fund Investing. John Wiley & Sons, London. Jagannathan, R., A. Malakhov, and D. Novikov. 2010. Do hot hands exist among fedge fund managers? An empirical evaluation. The Journal of Finance 65, no. 1: 217-255. Johansen, S. 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12: 231-254. Johansen, S. 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59: 1551-1580. Kalimipalli, M, and R. Susmel. 2004. Regime-switching volatility and short-term interest rates. Journal of Empirical Finance 11: 309-329. Kang, B. U., F. In, G. Kim, and T. S. Kim. 2010. A longer look at the asymmetric dependence between hedge funds and the equity market. Journal of Financial and Quantitative Analysis 45, no. 3: 763-789.

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Kat, H. M., and S. Lu. 2002. An excursion into the statistical properties of hedge funds. ICMA Centre, Discussion Papers in Finance, icma-dp2002-12, Henley Business School, Reading University. Kim, C. J. 1994. Dynamic linear models with Markov-switching. Journal of Econometrics 60: 1-22. Kim, C. J., and C. R. Nelson. 1999. State-Space Models with Regime Switching. The MIT Press. Kim, C. J., J. Piger, and R. Startz. 2008. Estimation of Markov regime-switching regression models with endogenous switching. Journal of Econometrics 143: 263-273. Kindleberger Ch. P., and R. Aliber. 2005. Manias, Panics, and Crashes: A History of Financial Crises. 5th ed., John Wiley and Sons, Hoboken, NJ. Klaassen, F. 2002. Improving GARCH volatility forecasts with regime-switching GARCH. Empirical Economics 27: 363-394. Kolb, R. 2010. Lessons from the Financial Crisis: Causes, Consequences, and Our Economic Future. John Wiley and Sons, UK. Liang, B. 1999. On performance of HFs. Financial Analysts Journal 55: 72-85. Liang, B., and H. Park. 2010. Predicting hedge fund failure: a comparison of risk measures. Journal of Financial and Quantitative Analysis 45, no. 1, 199-222. Lintner, J. 1969. The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47, no. 1: 13-25. Ljung, G., and G. Box. 1978. On a measure of lack of fit in time-series models. Biometrika 65: 297-303. Lo, A. 2001. Risk management for hedge funds: introduction and overview. Financial Analysts Journal 57: 16-33. Loudon, G., J. Okunev, and D. White. 2006. Hedge fund risk factors and the value at risk of fixed income trading strategies. Journal of Fixed Income 16: 46-61. MacKinnon, J. G., A. Haug, and L. Michelis. 1999. Numerical distribution functions of likelihood ratio tests for cointegration. Journal of Applied Econometrics 14: 563-577. Meligkotsidou, L., I. D. Vrontos, and S. D. Vrontos. Quantile regression analysis of hedge fund strategies. Journal of Empirical Finance 16: 264-279. Meyn, S., and R. Tweedie. 1993. Markov Chains and Stochastic Stability. London, Springer Verlag. Mossin, J. 1966. Equilibrium in a capital asset market. Economerica 34, no. 4: 768-783. Muzzioli S. 2010. Option-based forecasts of volatility: an empirical study in the DAX-index options market. The European Journal of Finance 16, no. 6: 561-586. Newey, K., and K. D. West. 1987. A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, no. 3: 703-708. Pagan, A. R., and G. W. Schwert. 1990. Alternative models for conditional stock volatility. Journal of Econometrics 45: 267-290. Patton, A. J. 2011. Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics 160: 246-256.

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Preminger, A., U. Ben-Zion, and D. Wettstein. 2006. Extended switching regression models with time-varying probabilities for combining forecasts. The European Journal of Finance 12, no. 6-7: 455-472. Purcell, D., and P. Crowley. 1999. The reality of hedge funds. Journal of Investing 8, no. 3: 26-44. Ranaldo, A., and L. Favre. 2005. HF performance and higher-moment market models. The Journal of Alternative Investments 8, no. 3: 37-51. Ross, S. A. 1976. Risk, return and arbitrage. In: I. Friend, and J. Bicksler (Eds.) Risk and Return in Finance. Cambridge MA, Ballinger. Smith, D. R. 2002. Markov-switching and stochastic volatility diffusion models of short-term interest rates. Journal of Business & Economic Statistics 20, no. 2: 183-197. Sharpe, W. 1964. Capital asset prices: a theory of market equilibrium under conditions of risk. The Journal of Finance 19, no. 3: 425-442. Stock, J. H., and M. W. Watson. 1988. Testing for common trends. Journal of the American Statistical Association 83: 1097-1107. Straumann, D. 2009. Measuring the Quality of Hedge Fund Data. The Journal of Alternative Investments 12, no. 2: 26-40. Taylor, S. J. 1986. Modelling Financial Time Series. Chichester: John Wiley & Sons. Vuong, Q. H. 1989. Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57: 303-333.

28

29

Mean 0.001 0.007 0.008 0.003 −0.001 0.008 0.009 0.007 0.007 0.004 0.006 0.004 0.005

Std. Dev. 0.045 0.020 0.026 0.009 0.056 0.022 0.042 0.020 0.019 0.012 0.013 0.012 0.020

Skewness −0.649 −0.773 −0.250 −0.213 0.134 0.410 −0.908 −1.359 −0.980 −2.491 −2.099 −2.899 −2.924

Kurtosis 4.248 5.612 4.779 4.336 4.920 3.860 6.656 7.202 7.728 13.818 16.337 22.981 30.986

BJ 33.76∗∗∗ 95.95∗∗∗ 35.56∗∗∗ 20.48∗∗∗ 39.16∗∗∗ 14.73∗∗∗ 173.60∗∗∗ 260.86∗∗∗ 272.81∗∗∗ 1477.65∗∗∗ 2036.56∗∗∗ 4508.80∗∗∗ 8514.92∗∗∗

LB(20) 16.85 38.43∗∗∗ 27.43 40.85∗∗∗ 28.35 34.44∗∗ 71.52∗∗∗ 53.34∗∗∗ 119.26∗∗∗ 19.62 75.53∗∗∗ 157.5∗∗∗ 145.66∗∗∗

MSCI corr NA 0.74∗∗∗ 0.71∗∗∗ 0.20∗∗∗ −0.67∗∗∗ 0.37∗∗∗ 0.69∗∗∗ 0.68∗∗∗ 0.51∗∗∗ 0.51∗∗∗ 0.52∗∗∗ 0.17∗∗∗ 0.48∗∗∗

Bond corr NA 0.14∗∗ 0.14∗∗ 0.00 −0.11∗ −0.17∗∗∗ 0.14∗∗ 0.19∗∗∗ 0.24∗∗∗ 0.05 0.19∗∗∗ 0.14∗∗ 0.12∗

statistic significant at the 10, 5 and 1 percent levels, respectively.

Not Available (NA). Critical Values (CV) of the correlation coefficients: CV10% = ±0.105, CV5% = ±0.125, CV1% = ±0.164. *, ** and *** denote

on US Treasury securities at 10-year constant maturity. Bera and Jarque (1982) (BJ). Ljung and Box (1978) (LB) for 20th-order serial correlation.

same statistic between a representative US bond yield and the excess return of the HF indices. The representative bond yield used is the market yield

The MSCI corr column presents the correlation coefficients between the excess returns of MSCI and HF indices. The Bond corr column presents the

Index MSCI Fund Weighted Composite Equity Hedge Equity Market Neutral Short Bias Macro Emerging Markets Event-Driven Distressed Restructuring Merger Arbitrage Relative Value F. I. - Asset Backed F. I. - Convertible Arbitrage

Table 1. Summary statistics of monthly HF index returns over the period 1990 to 2010.

30

−0.48∗∗∗ −0.07 −0.09 0.24∗∗∗

−0.01 0.11 0.11∗ 0.34∗∗∗

−0.20∗∗∗ −0.21∗∗∗ −0.05 0.15∗∗ 0.23∗∗∗ −0.18∗∗∗ −0.18∗∗∗ 0.04 0.17∗∗∗ 0.03 0.25∗∗∗ 0.44∗∗∗ 0.32∗∗∗

0.50∗∗∗ 0.05 0.03 −0.07 −0.54∗∗∗ 0.51∗∗∗ 0.51∗∗∗ −0.01 −0.00 −0.04 −0.16∗∗ −0.60∗∗∗ −0.42∗∗∗

0.89∗∗∗ 0.59∗∗∗

0.31∗∗∗ −0.20∗∗∗ −0.20∗∗∗

0.69∗∗∗ 0.63∗∗∗ 0.66∗∗∗

0.37∗∗∗ 0.33∗∗∗ 0.39∗∗∗

−0.67∗∗∗ −0.71∗∗∗ −0.82∗∗∗

HF6

HF5

HF4

0.71∗∗∗ 0.51∗∗∗

0.07 0.15∗∗ 0.33∗∗∗

−0.01

0.44∗∗∗ −0.15∗∗ −0.15∗∗

−0.06 −0.23∗∗∗ 0.22∗∗∗

−0.54∗∗∗

0.68∗∗∗ 0.70∗∗∗ 0.80∗∗∗

HF7

0.61∗∗∗ 0.41∗∗∗

0.07 0.17∗∗∗ 0.31∗∗∗

−0.01

0.39∗∗∗ −0.01 −0.01

−0.01 −0.24∗∗∗ 0.36∗∗∗

−0.40∗∗∗

0.51∗∗∗ 0.51∗∗∗ 0.62∗∗∗

HF8

0.52∗∗∗ 0.41∗∗∗

0.02 0.03 0.23∗∗∗

0.01

0.27∗∗∗ −0.06 −0.06

−0.05 −0.12∗ 0.03

−0.48∗∗∗

0.51∗∗∗ 0.53∗∗∗ 0.59∗∗∗

HF9

0.57∗∗∗ 0.45∗∗∗

0.15∗∗ 0.23∗∗∗ 0.44∗∗∗

0.00

0.27∗∗∗ −0.00 −0.00

−0.07 −0.17∗∗∗ 0.25∗∗∗

−0.48∗∗∗

0.52∗∗∗ 0.50∗∗∗ 0.55∗∗∗

HF10

0.18∗∗∗ 0.13∗∗

0.06 0.16∗∗ 0.12∗

−0.06

0.16∗∗ 0.06 0.06

−0.01 −0.05 0.31∗∗∗

−0.04

0.17∗∗∗ 0.14∗∗ 0.18∗∗∗

HF11

0.52∗∗∗ 0.43∗∗∗

0.19∗∗∗ 0.26∗∗∗ 0.46∗∗∗

−0.00

0.17∗∗∗ −0.02 −0.02

−0.19∗∗∗ −0.05 0.20∗∗∗

−0.44∗∗∗

0.48∗∗∗ 0.45∗∗∗ 0.44∗∗∗

HF12

HF indices: Fund Weighted Composite (HF1), Equity Hedge (HF2), Equity Market Neutral (HF3), Short Bias (Hf4), Macro (HF5), Emerging Markets (HF6), Event-Driven (HF7), Distressed Restructuring (HF8), Merger Arbitrage (HF9), Relative Value (HF10), F. I. - Asset Backed (HF11), F. I. - Convertible Arbitrage (HF12). See the abbreviations of HF return drivers in Appendix II. Critical Values (CV) of correlation coefficients: CV10% = ±0.105, CV5% = ±0.125, CV1% = ±0.164. *, ** and *** denote statistic significant at the 10, 5 and 1 percent levels, respectively.

Return driver HF1 HF2 HF3 Equity factors MSCI 0.74∗∗∗ 0.71∗∗∗ 0.21∗∗∗ ∗∗∗ ∗∗∗ S&P500 0.74 0.72 0.21∗∗∗ R2000 0.83∗∗∗ 0.82∗∗∗ 0.26∗∗∗ Volatility factors VIX −0.56∗∗∗ −0.53∗∗∗ −0.12∗ Interest rate and credit risk factors US6L −0.09 −0.08 −0.08 ∗∗ ∗∗∗ CS −0.16 −0.18 −0.25∗∗∗ TS 0.18∗∗∗ 0.09 0.02 Fama-French (1993)-type factors LS 0.48∗∗∗ 0.48∗∗∗ 0.12∗ ∗∗∗ ∗∗∗ VG −0.33 −0.35 0.02 ∗∗∗ ∗∗∗ MU −0.33 −0.35 0.02 Exchange rate factors USD −0.01 −0.00 0.05 Commodity factors GOLD 0.11∗ 0.07 0.03 ∗∗∗ ∗∗∗ OIL 0.18 0.24 0.17∗∗∗ ∗∗∗ ∗∗∗ CCI 0.38 0.36 0.20∗∗∗ Emerging markets factors EMS 0.80∗∗∗ 0.70∗∗∗ 0.15∗∗ ∗∗∗ ∗∗∗ EMB 0.55 0.45 0.12∗

Table 2. Correlation coefficients between the returns of HF indices and the HF return drivers.

Table 3. In-sample HF regime determinants over the period 1990 to 2010. Index Equity Hedge Short Bias Emerging Markets Event-Driven Distressed Restructuring Merger Arbitrage Relative Value F. I. - Asset Backed F. I. - Convertible Arbitrage

MSCI corr 0.132∗∗ 0.180∗∗∗ 0.075 0.453∗∗∗ 0.465∗∗∗ 0.045 −0.282∗∗∗ 0.333∗∗∗ 0.095

ψ 0.117 0.690 0.051 0.253∗∗∗ 0.256∗∗∗ 0.015 −0.116∗∗∗ 0.123∗∗∗ 0.047

The MSCI corr column reports the correlation coefficient between the transformed filtered probability of each HF index and the transformed filtered probability of the MSCI index. Critical Values (CV) of the correlation coefficient: CV10% = ±0.105, CV5% = ±0.125, CV1% = ±0.164. The ψ column reports the estimates of the ψ parameter of the following regression: π ˜t = c + ψ˜ πMSCIt + et , where π ˜t and π ˜MSCIt are the transformed filtered probabilities of the HF and MSCI indices, respectively. *, ** and *** denote significance at the 10%, 5% and 1% levels, respectively, evaluated by the framework of Newey and West (1987).

31

32

Not Available (NA).

Index Fund Weighted Composite Equity Hedge Equity Market Neutral Short Bias Macro Emerging Markets Event-Driven Distressed Restructuring Merger Arbitrage Relative Value F. I. - Asset Backed F. I. - Convertible Arbitrage

3-month forecast GARCH MS-GARCH 1.319% NA 1.711% 1.755% 0.627% NA 3.563% 3.544% 1.476% NA 2.879% 2.880% 1.336% 1.313% 1.686% 1.489% 0.826% 0.767% 1.132% 0.955% 1.142% 1.062% 1.822% 1.537%

6-month forecast GARCH MS- GARCH 1.330% NA 1.706% 1.762% 0.645% NA 3.718% 3.597% 1.525% NA 2.860% 2.890% 1.337% 1.309% 1.768% 1.485% 0.828% 0.770% 1.079% 0.975% 1.112% 1.055% 1.568% 1.513%

9-month forecast GARCH MS-GARCH 1.315% NA 1.707% 1.773% 0.646% NA 3.772% 3.607% 1.524% NA 2.767% 2.796% 1.302% 1.275% 1.837% 1.486% 0.777% 0.714% 1.133% 1.031% 1.072% 1.122% 1.729% 1.646%

Table 4. RMSE of competing out-of-sample volatility forecasts for the period 1999 to 2010. 12-month forecast GARCH MS- GARCH 1.320% NA 1.714% 1.784% 0.648% NA 3.817% 3.620% 1.527% NA 2.779% 2.811% 1.302% 1.274% 1.911% 1.480% 0.778% 0.716% 1.155% 1.154% 1.036% 1.002% 1.826% 1.710%

Table 5. Diebold and Mariano (1995) test of predictive accuracy of out-of-sample volatility forecasts for the period 1999 to 2010. Index Equity Hedge Short Bias Emerging Markets Event-Driven Distressed Restructuring Merger Arbitrage Relative Value F. I. - Asset Backed F. I. - Convertible Arbitrage

3-month forecast −0.928 0.217 −0.030 1.625 4.937∗∗∗ 4.178∗∗∗ 2.014∗∗ 0.554 2.070∗∗

6-month forecast −0.977 1.200 −0.677 1.998∗ 5.501∗∗∗ 4.084∗∗∗ 3.248∗∗∗ 0.553 0.457

9-month forecast −1.097 1.310 −0.600 1.804∗ 6.199∗∗∗ 3.989∗∗∗ 2.106∗∗ −1.144 0.836

12-month forecast −1.100 1.452 −0.605 1.957∗∗ 6.598∗∗∗ 3.988∗∗∗ 0.004 0.578 1.078

The Diebold and Mariano (1995) statistic is computed by the difference of the squared forecast errors of the GARCH and the MS-GARCH specifications. Positive values of the test statistic indicate superior performance of the MS-GARCH model, while negative values reflect better forecasting performance of the GARCH specification. Critical Values (CV) of the test statistic: CV10% = ±1.645, CV5% = ±1.960, CV1% = ±2.576. *, ** and *** denote rejection of equal predictive accuracy of the competing forecasting models at the 10%, 5% and 1% levels, respectively.

33

34 0.099 NA 0.000 NA 0.050 NA 0.000 NA

NA 0.066 NA 0.129 NA 0.000

0.027 NA

NA 0.761

NA 0.236

0.247 NA

NA 0.000

0.000 NA

0.000 NA

NA 0.004

NA 0.000

0.026 NA

NA 0.002

3-month forecast from GARCH MS-GARCH

NA 0.000

NA 0.156

NA 0.173

NA 0.186

NA 0.175

NA 0.793

NA 0.000

NA 0.083

NA 0.006

0.000 NA

0.261 NA

0.000 NA

0.098 NA

0.004 NA

0.027 NA

0.247 NA

0.000 NA

0.031 NA

6-month forecast from GARCH MS-GARCH

NA 0.000

NA 0.000

NA 0.001

NA 0.191

NA 0.115

NA 0.833

NA 0.000

NA 0.064

NA 0.010

0.003 NA

0.000 NA

0.002 NA

0.098 NA

0.027 NA

0.027 NA

0.247 NA

0.000 NA

0.188 NA

9-month forecast from GARCH MS-GARCH

NA 0.001

NA 0.002

NA 0.054

NA 0.188

NA 0.587

NA 0.785

NA 0.000

NA 0.125

NA 0.008

0.035 NA

0.044 NA

0.372 NA

0.098 NA

0.118 NA

0.027 NA

0.247 NA

0.000 NA

0.164 NA

12-month forecast from GARCH MS-GARCH

The table presents the p-values associated to the δin parameter estimates of the forecast encompassing test. The p-values are calculated using the Newey and West (1987) standard errors. The p-values in bold font indicate that the MS-GARCH encompasses the forecast of the GARCH at the 10% level of significance. Not Available (NA).

p-values Forecast error from Equity Hedge GARCH MS-GARCH Short Bias GARCH MS-GARCH Emerging Markets GARCH MS-GARCH Event-Driven GARCH MS-GARCH Distressed Restructuring GARCH MS-GARCH Merger Arbitrage GARCH MS-GARCH Relative Value GARCH MS-GARCH F. I. - Asset Backed GARCH MS-GARCH F. I. - Convertible Arbitrage GARCH MS-GARCH

Table 6. Forecast encompassing test results of out-of-sample volatility forecasts for the period 1999 to 2010.

Figure 1. Total value of HF assets over the period 1990 to 2010. (Source: HFR)

Figure 2. Performance of several HF indices in 2008. (Source: HFR) Notes: HFR HF index: 1 - Short Bias, 2 - Macro, 3 - Fixed Income - Asset Backed, 4 - Merger Arbitrage, 5 - Equity Market Neutral, 6 - Relative Value, 7 - Fund Weighted Composite, 8 - Event-Driven, 9 - Distressed Restructuring, 10 - Equity Hedge, 11 - Fixed Income - Convertible Arbitrage, 12 - Emerging Markets

35

Figure 3. Performance of global HFs and Morgan Stanley Capital International (MSCI) global equity market index over the period 1990 to 2010. (Sources: HFR and Bloomberg)

1

0

−0.2 1990

0.5

1992

1994

1996

1998

2000

2002

2004

2006

2008

Filtered prob of regime 1

Return

0.2

0 2010

Figure 4. Monthly MSCI return over the period 1990 to 2010. (Source: Bloomberg) Notes: The figure presents the monthly systematic excess return, MSCIt (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS-GARCH model.

36

Figure 5. Evolution of VIX over the period 1990 to 2010. (Source: Bloomberg)

37

Appendix I. Description of HF indices In this appendix, the description of the HF subgroups and indices studied in the article is summarized. The descriptions are provided by HFR. Equity Hedge: Equity Hedge strategies maintain positions both long and short in primarily equity and equity derivative securities. A wide variety of investment processes can be employed to arrive at an investment decision, including both quantitative and fundamental techniques; strategies can be broadly diversified or narrowly focused on specific sectors and can range broadly in terms of levels of net exposure, leverage employed, holding period, concentrations of market capitalizations and valuation ranges of typical portfolios. Equity Market Neutral : These strategies employ sophisticated quantitative techniques of analyzing price data to ascertain information about future price movement and relationships between securities, select securities for purchase and sale. These can include both factor-based and statistical arbitrage/trading strategies. Factor-based investment strategies include strategies in which the investment thesis is predicated on the systematic analysis of common relationships between securities. Statistical arbitrage/trading strategies consist of strategies in which the investment thesis is predicated on exploiting pricing anomalies which may occur as a function of expected mean reversion inherent in security prices; high frequency techniques may be employed and trading strategies may also be employed on the basis on technical analysis or opportunistically to exploit new information the investment manager believes has not been fully, completely or accurately discounted into current security prices. Short Bias: These strategies employ analytical techniques in which the investment thesis is predicated on assessment of the valuation characteristics on the underlying companies with the goal of identifying overvalued companies. Short Bias strategies may vary the investment level or the level of short exposure over market cycles, but the primary distinguishing characteristic is that the manager maintains consistent short exposure and expects to outperform traditional equity managers in declining equity markets. Investment theses may be fundamental or technical and nature and manager has a particular focus, above that of a market generalist, on identification of overvalued companies and would expect to maintain a net short equity position over various market cycles. Macro: The investment process in this strategy is predicated on movements in underlying economic variables and the impact these have on equity, fixed income, hard currency and commodity markets. Managers employ a variety of techniques, both discretionary and systematic analysis, combinations of top down and bottom up theses, quantitative and fundamental approaches and long and short term holding periods. Emerging Markets: These strategies have a regional investment focus in one of the following geographic areas: Asia ex-Japan, Eastern Europe, Latin America, Africa or the Middle East. These funds invest in equities or fixed income securities in emerging markets around the world and they tend to be long only because in many emerging markets short selling is not allowed and financial derivatives are not available. Event Driven: These strategies focus on positions in companies currently or prospectively involved in corporate transactions of a wide variety including but not limited to mergers, restructurings, financial distress, tender offers, shareholder buybacks, debt exchanges, security issuance or other capital structure adjustments. Security types can range from most senior in the capital structure to most junior or subordinated, and frequently involve additional derivative securities. Event Driven exposure includes a combination of sensitivities to equity markets, credit markets and idiosyncratic, company specific developments.

38

Distressed Restructuring: These HF strategies employ an investment process focused on corporate fixed income instruments, primarily on corporate credit instruments of companies trading at significant discounts to their value at issuance or obliged (par value) at maturity as a result of either formal bankruptcy proceeding or financial market perception of near term proceedings. Managers are typically actively involved with the management of these companies, frequently involved on creditors’ committees in negotiating the exchange of securities for alternative obligations, either swaps of debt, equity or hybrid securities. Merger Arbitrage: These strategies employ an investment process primarily focused on opportunities in equity and equity related instruments of companies which are currently engaged in a corporate transaction. Merger Arbitrage involves primarily announced transactions, typically with limited or no exposure to situations which pre-, post-date or situations in which no formal announcement is expected to occur. Opportunities are frequently presented in cross border, collared and international transactions which incorporate multiple geographic regulatory institutions and typically involve minimal exposure to corporate credits. Relative Value: Investment managers who maintain positions in which the investment thesis is predicated on realization of a valuation discrepancy in the relationship between multiple securities. Managers employ a variety of fundamental and quantitative techniques to establish investment theses, and security types range broadly across equity, fixed income, derivative or other security types. Fixed income strategies are typically quantitatively driven to measure the existing relationship between instruments and, in some cases, identify attractive positions in which the risk adjusted spread between these instruments represents an attractive opportunity for the investment manager. Fixed Income - Asset Backed : This index includes strategies in which the investment thesis is predicated on realization of a spread between related instruments in which one or multiple components of the spread is a fixed income instrument backed physical collateral or other financial obligations (loans, credit cards) other than those of a specific corporation. Strategies employ an investment process designed to isolate attractive opportunities between a variety of fixed income instruments specifically securitized by collateral commitments which frequently include loans, pools and portfolios of loans, receivables, real estate, machinery or other tangible financial commitments. Fixed Income - Convertible Arbitrage: In these HF strategies, the investment thesis is predicated on realization of a spread between related instruments in which one or multiple components of the spread is a convertible fixed income instrument. Strategies employ an investment process designed to isolate attractive opportunities between the price of a convertible security and the price of a non-convertible security, typically of the same issuer. Convertible arbitrage positions maintain characteristic sensitivities to credit quality of the issuer, implied and realized volatility of the underlying instruments, levels of interest rates and the valuation of the issuer’s equity, among other more general market and idiosyncratic sensitivities.

39

Appendix II. HF return drivers The HF return drivers considered in this article are as follows: (a) Equity factors: (1) Monthly return on MSCI equity index (MSCI) (2) Monthly return on S&P500 (S&P500) (3) Monthly return on Russel 2000 (R2000) (b) Volatility factors: (4) Monthly first difference in the CBOE VIX implied volatility index (VIX) (c) Interest rate and credit risk factors: (5) Monthly change in US 6-month LIBOR (US6L) (6) Credit Spread (CS) monthly yield difference between seasoned BAA and AAA corporate provided by Moodys (7) Term Spread (TS): difference between 10 year US Treasury bond redemption yield and the 6 month LIBOR rate (d) Fama and French (1993)-type factors: (8) Large minus Small (LS): monthly return difference between Russell 1000 and Russell 2000 indexes (9) Value minus Growth (VG): difference in monthly return between Russell 1000 Value index and Russell Growth index (10) Momentum (MU): the momentum factor is based on six value-weighted portfolios formed using independent sort on size and prior returns of NYSE, AMEX and NASDAQ stocks (e) Exchange rate factors: (11) Monthly return on Bank of England USD Trade Weighted Index (USD) (f) Commodity factors: (12) Monthly return on gold spot price index (GOLD) (13) Monthly change in WTI oil price index (OIL) (14) Monthly return on Commodity Research Bureau Continuous Commodity Index (CCI) (g) Emerging markets factors: (15) Emerging Market Stock Index (EMS): monthly return on MSCI Emerging Market Stock Index (16) Emerging Market Bond Index (EMB): monthly return on JPMorgan EMBI Global Index All HF return drivers were collected from the following sources: Bloomberg, Reuters and DataStream.

40

Appendix III. Unit root tests, cointegration and common factors Table A1. ADF unit root test results. Index Fund Weighted Composite Equity Hedge Equity Market Neutral Short Bias Macro Emerging Markets Event-Driven Distressed Restructuring Merger Arbitrage Relative Value F. I. - Asset Backed F. I. - Convertible Arbitrage Critical values: 1% 5% 10%

ADF1 2.323 1.669 3.834 -1.044 4.308 1.358 2.240 1.867 4.177 2.712 2.603 1.372

ADF2 -0.643 -0.929 −2.622∗ -1.793 -0.361 -0.387 -0.371 -0.478 -0.46 -0.228 1.245 -0.714

ADF3 -2.946 -2.204 -0.016 -3.075 -2.800 -2.499 -2.823 -2.472 -2.110 −3.442∗∗ -1.787 -3.027

-2.574 -1.942 -1.616

-3.457 -2.873 -2.573

-3.995 -3.428 -3.137

ADF tests: ADF1 (without constant, without trend), ADF2 (with constant, without trend) and ADF3 (with constant, with trend). *, ** and *** denote the rejection of the null hypothesis of unit root at the 10%, 5% and 1% levels, respectively. Test statistics in bold font indicate that in the unit root test equation at least one of the parameters, i.e. constant or trend parameter, is not significant at the 10% level. Test statistics indicated by boxes are used to evaluate the null hypothesis of unit root.

41

Table A2. Johansen (1988, 1991) cointegration test: trace and maximum eigenvalue test statistics. Hypothesized no. of CE(s) None At most 1 At most 2 At most 3 At most 4 At most 5 At most 6 At most 7 At most 8 At most 9

Trace statistic 392.361∗∗∗ 261.480∗∗∗ 174.995∗∗∗ 122.220∗ 76.305 49.893 30.784 13.775 6.090 0.887

Critical value 239.235 197.371 159.530 125.615 95.754 69.819 47.856 29.797 15.495 3.841

p-value 0.000 0.000 0.005 0.079 0.495 0.642 0.678 0.853 0.685 0.346

Max-eig. statistic 130.881∗∗∗ 86.485∗∗∗ 52.775∗∗ 45.916∗ 26.412 19.109 17.009 7.685 5.203 0.887

Critical value 64.505 58.434 52.363 46.231 40.078 33.877 27.584 21.132 14.265 3.841

p-value 0.000 0.000 0.045 0.054 0.674 0.815 0.579 0.922 0.716 0.346

Cointegrating Equation (CE). *, ** and *** denote rejection of the hypothesis at the 10%, 5% and 1% levels, respectively. MacKinnon, Haug and Michelis (1999) p-values are applied.

42

43

Factor 1 0.054∗∗∗ 0.073∗∗∗ 0.011∗∗ 0.068∗∗∗ 0.086∗∗∗ 0.069∗∗∗ 0.083∗∗∗ 0.030∗∗∗ 0.037∗∗∗ 0.045∗∗∗ 0.782∗∗∗

Factor 3 0.046∗∗∗ 0.075∗∗∗ 0.010 0.059∗∗ 0.031∗∗∗ 0.036∗∗∗ 0.029∗∗ 0.033∗∗∗ 0.009 0.019∗∗∗ 0.696∗∗∗

Factor model:

Factor 2 0.029∗∗∗ 0.011 0.014∗∗∗ 0.055∗∗∗ 0.098∗∗∗ 0.031∗∗ 0.045∗∗∗ 0.018∗∗∗ 0.021∗∗∗ 0.000 0.853∗∗∗

Factor 5 0.043∗∗∗ 0.072∗∗∗ −0.017∗∗∗ 0.059∗∗∗ 0.083∗∗∗ 0.057∗∗∗ 0.062∗∗∗ 0.016∗∗∗ 0.015∗∗∗ 0.016∗∗ 0.760∗∗∗

Factor 6 0.044∗∗∗ 0.074∗∗∗ 0.020∗∗∗ 0.045∗∗∗ 0.001 0.037∗∗∗ 0.032∗∗∗ 0.034∗∗∗ 0.018∗∗∗ 0.040∗∗∗ 0.868∗∗∗

HFt = H 0 Ft + vt , vt ∼ N (0, Σv ), Ft = Ft−1 + ²t , ²t ∼ N (0, Σ² ),

Factor 4 0.012 0.050∗∗∗ −0.032∗∗∗ 0.010 −0.007 0.013 0.000 −0.002 −0.016∗∗∗ −0.005 0.838∗∗∗

Variance of HF error 1.510∗∗∗ 4.324∗∗∗ 0.035∗∗∗ 2.624∗∗∗ 3.476∗∗∗ 1.931∗∗∗ 2.094∗∗∗ 0.385∗∗∗ 0.162∗∗∗ 0.337∗∗∗ NA

where HFt denotes the 10 × 1 vector of HF index levels, Ft denotes the 6 × 1 vector of common factors, vt is a 10 × 1 vector of errors of the HF ˆ 0 for each HF index equation and ²t is a 6 × 1 vector of errors of the factor equation. The table shows the estimated 10 × 6 factor loadings matrix H and the variance of the errors for each HF index and factor. *, ** and *** denote parameter significance at the 10%, 5% and 1% levels, respectively. Not Available (NA).

Index Fund Weighted Composite Equity Hedge Short Bias Macro Emerging Markets Event-Driven Distressed Restructuring Merger Arbitrage F. I. - Asset Backed F. I. - Convertible Arbitrage Variance of factor error

Table A3. Parameter estimates of the common factor model for the period 1990 to 2010.

Table A4. Correlation coefficients between the HF return drivers and the first differences of common factors. Return driver ∆F1 ∆F2 ∆F3 Equity factors MSCI 0.043 −0.119∗ −0.065 S&P500 0.082 −0.098 −0.008 R2000 0.066 −0.144∗∗ −0.064 Volatility factors VIX 0.018 −0.041 −0.032 Interest rate and credit risk factors US6L 0.172∗∗∗ −0.011 −0.035 CS −0.301∗∗∗ 0.268∗∗∗ −0.167∗∗∗ TS 0.351∗∗∗ −0.280∗∗∗ −0.397∗∗∗ Fama-French (1993)-type factors LS −0.003 −0.126∗∗ −0.120∗ VG 0.099 −0.092 0.033 MU 0.099 −0.092 0.033 Exchange rate factors USD −0.008 −0.016 −0.019 Commodity factors GOLD −0.005 0.017 −0.072 OIL 0.136∗∗ −0.024 0.048 ∗∗∗ CCI 0.191 0.016 −0.003 Emerging markets factors EMS 0.034 −0.134∗∗ −0.102 EMB 0.012 −0.036 −0.064

∆F4

∆F5

∆F6

0.076 0.045 0.102

0.076 0.055 0.108∗

−0.019 0.045 −0.017

0.109∗

0.126∗∗

−0.080

0.094 −0.164∗∗∗ 0.072

0.181∗∗∗ −0.209∗∗∗ 0.186∗∗∗

−0.030 −0.235∗∗∗ −0.016

0.123∗ 0.120∗ 0.120∗

0.117∗ 0.148∗∗ 0.148∗∗

−0.105∗ 0.011 0.011

0.012

0.007

−0.015

−0.008 0.016 −0.011

0.003 0.061 0.072

−0.050 0.080 0.068

0.120∗ 0.002

0.124∗∗ 0.003

−0.073 −0.016

∆Fit = Fit − Fit−1 for i = 1, . . . , 6. See the abbreviations of HF return drivers in Appendix II. Critical Values (CV) of the correlation coefficient: CV10% = ±0.105, CV5% = ±0.125, CV1% = ±0.164. *, ** and *** denote parameter significant at the 10%, 5% and 1% levels, respectively.

44

Figure A1. Evolution of common factors over the period 1990 to 2010. Notes: The factors are normalized to start from the value one in January 1990.

45

Appendix IV. MS model inference In this Appendix, some details of the MS model’s statistical inference are summarized. MS-AR(1) stationarity conditions Francq and Zakoian (2001) give the covariance stationarity condition for the MS-ARMA model, which can be used for the MS-AR specification of Section 4. The condition is given by: X πi ln(|ζ(i)|) < 0, (A.1) i=1,2

where πi is the invariant probability (Meyn and Tweedie, 1993) of regime i, which is given by π1 =

1 − η22 , 2 − η11 − η22

π2 =

1 − η11 . 2 − η11 − η22

(A.2)

Computation of expectations in the MS-GARCH(1,1) model The recombining MS-GARCH(1,1) model of Klaassen (2002) is formulated as follows: 2 (st ), σt2 (st ) = ω(st ) + α(st )²2t−1 (st ) + β(st )σt−1 ²t−1 (st ) = E[²t−1 (st−1 )|st , Yt−1 ], 2 2 (st−1 )|st , Yt−1 ]. (st ) = E[σt−1 σt−1

(A.3)

The expectations in the last two equations are evaluated using the Pr[st−1 = i|st = j, Yt−1 ] probabilities computed as follows: Pr[st = j|st−1 = i] Pr[st−1 = i|Yt−1 ] Pr[st−1 = i|st = j, Yt−1 ] = P , (A.4) i=1,2 Pr[st = j|st−1 = i] Pr[st−1 = i|Yt−1 ] where Pr[st−1 = i|Yt−1 ] is computed by f (yt−1 |st−1 = i, Yt−2 ) Pr[st−1 = i|Yt−2 ] . i=1,2 f (yt−1 |st−1 = i, Yt−2 ) Pr[st−1 = i|Yt−2 ]

Pr[st−1 = i|Yt−1 ] = P

(A.5)

The initial value of Pr[s0 = i|Y0 ] is approximated by the invariant probabilities of st : Pr[s0 = 1|Y0 ] =

1 − η22 , 2 − η11 − η22

Pr[s0 = 2|Y0 ] =

1 − η11 . 2 − η11 − η22

(A.6)

MS-GARCH(1,1) stationarity conditions The conditions for covariance stationarity for the model of Klaassen (2002) have been established by Abramson and Cohen (2007). In particular, for the MS-GARCH(1,1) with two regimes applied in this article, the condition of covariance stationarity is as follows. Define the 2 × 2 matrix V = {Vij } as Vij = (αi + βi )

πj ηij , πi

(A.7)

where πi is the invariant probability of regime i, which is given by π1 =

1 − η22 , 2 − η11 − η22

π2 =

46

1 − η11 . 2 − η11 − η22

(A.8)

Then, the MS-GARCH equation is stationary when all eigenvalues of V are inside the unit circle. Computation of the likelihood function of the MS model The likelihood function of Yk with 1 ≤ k ≤ T for the HF return model with MS-GARCH volatility dynamics is given by: k X Y L(Yk ; θ) = f (yt |st = i, Yt−1 ) Pr[st = i|Yt−1 ]. (A.9) t=1 i=1,2

The weighting terms Pr[st = j|Yt−1 ] are calculated as follows: X Pr[st = j|Yt−1 ] = Pr[st = j|st−1 = i] Pr[st−1 = i|Yt−1 ],

(A.10)

i=1,2

where

f (yt−1 |st−1 = i, Yt−2 ) Pr[st−1 = i|Yt−2 ] . i=1,2 f (yt−1 |st−1 = i, Yt−2 ) Pr[st−1 = i|Yt−2 ]

Pr[st−1 = i|Yt−1 ] = P

(A.11)

The initial value of Pr[s0 = i|Y0 ] is approximated by the invariant probabilities of st : Pr[s0 = 1|Y0 ] =

1 − η22 , 2 − η11 − η22

Pr[s0 = 2|Y0 ] =

47

1 − η11 . 2 − η11 − η22

(A.12)

48 660.000 688.000 −1308.000 −1348.000 −1307.654 −1346.213 −1286.871 −1298.700 56.000∗∗∗

0.001 0.000 0.167∗∗∗ 0.132∗∗∗ 0.584∗∗∗ 0.706∗∗∗ 0.973∗∗∗ 0.980∗∗∗ 479.250 505.750 −946.500 −983.500 −946.154 −981.713 −925.371 −934.200 53.000∗∗∗

0.000 NA 0.353∗∗∗ NA 0.603∗∗∗ NA NA NA

0.002 0.000 0.127∗∗∗ 0.232∗∗∗ −0.945∗∗∗ −0.658∗∗∗

*, ** and *** is significance at the 10%, 5% and 1% levels, respectively. Not Available (NA). Log Likelihood (LL). Akaike Information Criterion (AIC). AIC correction for finite sample bias (AICc). Bayesian Information Criterion (BIC). Likelihood Ratio (LR).

MS-GARCH(1,1) specification:

NA NA NA NA

NA NA NA NA NA NA NA NA

0.001 NA 0.152∗∗∗ NA −0.743∗∗∗ NA

Short Bias GARCH MS- GARCH

yt = c + ζyt−1 + φM SCIt + σt ut , 2 σt2 = ω + α²2t−1 + βσt−1 . yt = c(st ) + ζ(st )yt−1 + φ(st )M SCIt + σt (st )ut , 2 σt2 (st ) = ω(st ) + α(st )²2t−1 (st ) + β(st )σt−1 (st ).

847.500 −1683.000 −1682.654 −1661.871 NA

0.000 0.000 0.242∗∗∗ 0.179∗∗∗ 0.437∗∗∗ 0.661∗∗∗ 0.986∗∗∗ 0.943∗∗∗

0.000 NA 0.319∗∗∗ NA 0.435∗∗∗ NA NA NA

0.000 NA 0.214∗∗∗ NA 0.671∗∗∗ NA NA NA

0.003 NA 0.143∗∗∗ NA 0.035∗ NA

0.016 0.003 0.280∗∗∗ 0.212∗∗∗ 0.279∗∗∗ 0.441∗∗∗

0.006 NA 0.264∗∗∗ NA 0.421∗∗∗ NA

NA NA NA NA NA NA

Equity Market Neutral GARCH MS-GARCH

Equity Hedge GARCH MS- GARCH

GARCH(1,1) specification:

Fund Weighted Composite GARCH MS-GARCH Mean eqation c(1) 0.006 NA c(2) NA NA ζ(1) 0.318∗∗∗ NA ζ(2) NA NA φ(1) 0.329∗∗∗ NA φ(2) NA NA Volatility equation ω(1) 0.000 NA ω(2) NA NA α(1) 0.219∗∗∗ NA α(2) NA NA ∗∗∗ β(1) 0.413 NA β(2) NA NA η11 NA NA η22 NA NA Model diagnostics LL 724.750 NA AIC −1437.500 NA AICc −1437.154 NA BIC −1416.371 NA LR NA

Table A5. In-sample estimation results for the period 1990 to 2010.

Appendix V. In-sample estimation results

49 546.500 −1073.000 −1072.079 −1037.785 6.500

0.001 0.001 NA NA NA NA 0.884∗∗∗ 0.989∗∗∗

0.000 NA 0.322∗∗∗ NA 0.598∗∗∗ NA NA NA 543.250 −1074.500 −1074.154 −1053.371

−0.042∗ 0.012∗ 0.131∗∗∗ 0.170∗∗∗ 1.031∗∗∗ 0.624∗∗∗

0.011 NA 0.153∗∗∗ NA 0.589∗∗∗ NA

Emerging Markets GARCH MS- GARCH

0.000 0.000 NA NA NA NA 0.849∗∗∗ 0.974∗∗∗

−0.004 0.009 0.840∗∗∗ 0.261∗∗∗ 0.327∗∗∗ 0.247∗∗∗ 0.000 0.000 0.929∗∗∗ NA 0.011∗ NA 0.891∗∗∗ 0.917∗∗∗

0.015 0.004 1.136∗∗∗ 0.468∗∗∗ 0.162∗∗∗ 0.219∗∗∗

702.000 742.750 −1392.000 −1461.500 −1391.654 −1460.184 −1370.871 −1419.242 81.500∗∗∗

0.000 NA 0.822∗∗∗ NA 0.057∗∗ NA NA NA

0.008 NA 0.583∗∗∗ NA 0.173∗∗∗ NA

Distressed Restructuring GARCH MS- GARCH

yt = c + ζyt−1 + φM SCIt + σt ut , 2 σt2 = ω + α²2t−1 + βσt−1 . yt = c(st ) + ζ(st )yt−1 + φ(st )M SCIt + σt (st )ut , 2 σt2 (st ) = ω(st ) + α(st )²2t−1 (st ) + β(st )σt−1 (st ).

703.250 722.500 −1394.500 −1425.000 −1394.154 −1424.095 −1373.371 −1389.785 38.500∗∗∗

0.000 NA 0.125∗∗∗ NA 0.223∗∗∗ NA NA NA

0.007 NA 0.391∗∗∗ NA 0.296∗∗∗ NA

Event-Driven GARCH MS-GARCH

*, ** and *** is significance at the 10%, 5% and 1% levels, respectively. Not Available (NA). Log Likelihood (LL). Akaike Information Criterion (AIC). AIC correction for finite sample bias (AICc). Bayesian Information Criterion (BIC). Likelihood Ratio (LR).

MS-GARCH(1,1) specification:

GARCH(1,1) specification:

Macro GARCH MS-GARCH Mean eqation c(1) 0.006 NA c(2) NA NA ζ(1) 0.171∗∗∗ NA ζ(2) NA NA ∗∗∗ φ(1) 0.167 NA φ(2) NA NA Volatility equation ω(1) 0.000 NA ω(2) NA NA ∗∗∗ α(1) 0.172 NA α(2) NA NA β(1) 0.697∗∗∗ NA β(2) NA NA η11 NA NA η22 NA NA Model diagnostics LL 624.000 NA AIC −1236.000 NA AICc −1235.654 NA BIC −1214.871 NA LR NA

Table A6. In-sample estimation results for the period 1990 to 2010.

50 815.750 846.500 −1619.500 −1665.000 −1619.154 −1663.213 −1598.371 −1615.700 61.500∗∗∗

0.000 0.000 0.254∗∗∗ 0.247∗∗∗ 0.822∗∗∗ 0.652∗∗∗ 0.750∗∗∗ 0.926∗∗∗

0.000 NA 0.536∗∗∗ NA 0.374∗∗∗ NA NA NA

0.000 0.000 0.382∗∗∗ 0.244∗∗∗ 0.769∗∗∗ 0.538∗∗∗ 0.704∗∗∗ 0.942∗∗∗

−0.010 0.008 0.467∗∗∗ 0.556∗∗∗ 0.156∗∗∗ 0.047∗

yt = c + ζyt−1 + φM SCIt + σt ut , 2 σt2 = ω + α²2t−1 + βσt−1 . yt = c(st ) + ζ(st )yt−1 + φ(st )M SCIt + σt (st )ut , 2 σt2 (st ) = ω(st ) + α(st )²2t−1 (st ) + β(st )σt−1 (st ).

761.750 806.000 −1511.500 −1584.000 −1511.154 −1582.213 −1490.371 −1534.700 88.500∗∗∗

0.000 NA 0.486∗∗∗ NA 0.181∗∗∗ NA NA NA

0.007 NA 0.613∗∗∗ NA 0.071∗∗ NA

F. I. - Convertible Arbitrage GARCH MS- GARCH

*, ** and *** is significance at the 10%, 5% and 1% levels, respectively. Not Available (NA). Log Likelihood (LL). Akaike Information Criterion (AIC). AIC correction for finite sample bias (AICc). Bayesian Information Criterion (BIC). Likelihood Ratio (LR).

MS-GARCH(1,1) specification:

0.000 0.000 0.135∗∗∗ NA 0.658∗∗∗ NA 0.946∗∗∗ 0.967∗∗∗

812.750 847.000 −1613.500 −1670.000 −1613.154 −1668.684 −1592.371 −1627.742 68.500∗∗∗

0.000 NA 0.314∗∗∗ NA 0.502∗∗∗ NA NA NA

0.004 NA 0.452∗∗∗ NA 0.058∗∗ NA

0.000 0.006 0.432∗∗∗ 0.394∗∗∗ 0.347∗∗∗ 0.054∗

0.006 NA 0.435∗∗∗ NA 0.089∗∗ NA

−0.005 0.007 0.353∗∗∗ 0.462∗∗∗ 0.003 0.034∗

F. I. - Asset Backed GARCH MS-GARCH

Relative Value GARCH MS- GARCH

GARCH(1,1) specification:

Merger Arbitrage GARCH MS-GARCH Mean eqation c(1) 0.004 −0.001 c(2) NA 0.007 ζ(1) 0.242∗∗∗ 0.232∗∗∗ ζ(2) NA 0.051∗ φ(1) 0.135∗∗∗ 0.239∗∗∗ φ(2) NA 0.049∗∗∗ Volatility equation ω(1) 0.000 0.000 ω(2) NA 0.000 α(1) NA 0.177∗∗∗ α(2) NA 0.044∗∗ β(1) NA 0.178∗∗∗ β(2) NA 0.259∗∗∗ η11 NA 0.678∗∗∗ η22 NA 0.892∗∗∗ Model diagnostics LL 791.500 837.000 AIC −1575.000 −1646.000 AICc −1574.837 −1644.213 BIC −1560.914 −1596.700 LR 91.000∗∗∗

Table A7. In-sample estimation results for the period 1990 to 2010.

1

0.1

0.8

0.05

0.6

0

0.4

−0.05

0.2

−0.1 1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Filtered prob of regime 1

Return

0.15

0 2010

Figure A2. Equity Hedge. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS-GARCH model.

1

0.2

0.9

0.15

0.8

0.1

0.7

0.05

0.6

0

0.5

−0.05

0.4

−0.1

0.3

−0.15

0.2

−0.2

0.1

−0.25 1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Filtered prob of regime 1

Return

0.25

0 2010

Figure A3. Short Bias. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

51

1

0.1

0.8

0

0.6

−0.1

0.4

−0.2

0.2

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

Filtered prob of regime 1

Return

0.2

0 2010

Figure A4. Emerging Markets. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

1

0

−0.1 1990

0.5

1992

1994

1996

1998

2000

2002

2004

2006

2008

Filtered prob of regime 1

Return

0.1

0 2010

Figure A5. Event-Driven. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

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Figure A6. Distressed Restructuring. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

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Figure A7. Merger Arbitrage. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

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Figure A8. Relative Value. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

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Figure A9. Fixed Income - Asset Backed. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

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Figure A10. Fixed Income - Convertible Arbitrage. (Source: HFR) Notes: The figure presents the monthly excess return yt on the HF index (left axis) and the corresponding probability Pr[st = 1|Yt−1 ] (right axis) over the period January 1990 to October 2010 for the MS model.

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Period crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis

6-month forecast GARCH MS-GARCH 1.868% 1.815% 1.642% 1.742% 4.423% 4.226% 2.389% 2.444% 5.495% 5.248% 2.576% 2.647% 2.372% 2.244% 1.225% 1.211% 1.781% 1.603% 1.748% 1.405% 1.595% 1.588% 0.681% 0.602% 1.534% 1.442% 1.002% 0.893% 1.523% 1.307% 0.985% 0.657% 3.237% 2.772% 1.355% 1.146%

9-month forecast GARCH MS-GARCH 1.838% 1.811% 1.658% 1.760% 4.475% 4.255% 2.488% 2.439% 5.396% 5.248% 2.476% 2.533% 2.357% 2.250% 1.191% 1.175% 1.847% 1.619% 1.820% 1.398% 1.341% 1.341% 0.682% 0.601% 1.679% 1.562% 1.037% 0.935% 1.666% 1.474% 0.769% 0.657% 3.407% 3.141% 1.469% 1.278%

12-month forecast GARCH MS-GARCH 1.861% 1.824% 1.662% 1.771% 4.479% 4.282% 2.669% 2.450% 5.325% 5.248% 2.498% 2.547% 2.547% 2.409% 1.191% 1.175% 1.938% 1.602% 1.864% 1.403% 1.363% 1.360% 0.682% 0.602% 1.548% 1.806% 1.089% 1.032% 1.435% 1.284% 0.877% 0.669% 3.439% 3.156% 1.588% 1.354%

The crisis and non-crisis periods of the sample are determined by the value of the filtered probability of the MS-GARCH model. When Pr[st = 1|Yt−1 ] ≥ 0.5 then period t is a ‘crisis’ period. Conversely, when Pr[st = 1|Yt−1 ] < 0.5 then period t is a ‘non-crisis’ period.

F. I. - Convertible Arbitrage

F. I. - Asset Backed

Relative Value

Merger Arbitrage

Distressed Restructuring

Event-Driven

Emerging Markets

Short Bias

Index Equity Hedge

3-month forecast GARCH MS-GARCH 1.873% 1.791% 1.643% 1.741% 4.241% 4.142% 2.249% 2.432% 5.629% 5.248% 2.582% 2.639% 2.237% 2.158% 1.225% 1.211% 1.841% 1.652% 1.577% 1.373% 1.595% 1.588% 0.681% 0.600% 1.638% 1.507% 1.046% 0.855% 1.501% 1.285% 1.050% 0.687% 3.120% 3.003% 1.627% 1.222%

Table A8. RMSE of competing out-of-sample volatility forecasts for the period 1999 to 2010 for ‘crisis’ and ‘non-crisis’ periods.

Appendix VI. Forecasting performance of models in ‘crisis’ and ‘non-crisis’ periods

Table A9. Diebold and Mariano (1995) test of predictive accuracy of out-of-sample volatility forecasts for the period 1999 to 2010 for ‘crisis’ and ‘non-crisis’ periods.

Index Equity Hedge Short Bias Emerging Markets Event-Driven Distressed Restructuring Merger Arbitrage Relative Value F. I. - Asset Backed F. I. - Convertible Arbitrage

Period crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis crisis non-crisis

3-month forecast 1.473 −1.609 0.968 −0.917 0.961 −0.810 1.173 1.206 2.291∗∗ 6.152∗∗∗ 0.666 4.338∗∗∗ 1.309 2.383∗∗ 0.474 1.285 0.451 2.076∗∗

6-month forecast 1.210 −1.316 2.195∗∗ −0.214 0.769 −1.015 1.982∗∗ 1.205 2.583∗∗∗ 6.477∗∗∗ 0.673 4.229∗∗∗ 0.974 3.336∗∗∗ 0.631 1.520 0.551 2.539∗∗

9-month forecast 0.561 −1.253 2.577∗∗∗ 0.158 0.550 −0.765 1.395 1.347 3.996∗∗∗ 5.857∗∗∗ 0.037 4.319∗∗∗ 0.985 2.243∗∗ 0.128 2.087∗∗ 0.880 1.890∗

12-month forecast 0.752 −1.284 1.852∗ 0.648 0.325 −0.652 1.074 1.347 4.486∗∗∗ 7.441∗∗∗ 0.224 4.248∗∗∗ −0.990 0.738 0.209 2.094∗∗ 0.830 2.151∗∗

The crisis and non-crisis periods of the sample are determined by the value of the filtered probability of the MS-GARCH model. When Pr[st = 1|Yt−1 ] ≥ 0.5 then period t is a ‘crisis’ period. Conversely, when Pr[st = 1|Yt−1 ] < 0.5 then period t is a ‘non-crisis’ period. The Diebold and Mariano (1995) statistic is computed by the difference of the squared forecast errors of the GARCH and the MS-GARCH specifications. Positive values of the test statistic indicate superior performance of the MS-GARCH model, while negative values reflect better forecasting performance of the GARCH specification. Critical Values (CV) of the test statistic: CV10% = ±1.645, CV5% = ±1.960, CV1% = ±2.576. *, ** and *** denote rejection of equal predictive accuracy of the competing forecasting models at the 10%, 5% and 1% levels, respectively.

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