20 Years of VIX: Fear, Greed and Implications for Alternative Investment Strategies

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In this article, I investigate the statistical properties and relationships of VIX with alternative investment strategies. I find that different VIX states result in very different risk adjusted performance for all strategies and confirm that significant deviations from normality are observed in the states and the full sample, which are not fully captured by traditional risk metrics. I demonstrate that correlations among strategies are unstable and non-linear, leading to highly concentrated diversification benefits at the times of market stress, which a broad set of exposures is likely to negate. I also demonstrate that at certain states, correlations are very high between traditional and alternative investment strategies and performance characteristics are very similar. I establish that the superior, long term performance of such strategies relative to traditional asset classes is not due to higher returns in good times, but rather better preservation of capital in bad times.
Based on empirical data, practical recommendations for investment analysis and risk management are included throughout the article. This is a companion article to ’20 years of VIX: Fear, Greed and Implications for Traditional Asset Classes’ (Munenzon (2010)) available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1583504

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20 Years of VIX: Fear, Greed and Implications for Alternative Investment Strategies

  1. 1. 20 years of VIX: Fear, Greed and Implications for Alternative Investment Strategies The Abstract In this article, I investigate the statistical properties and relationships of VIX with alternative investment strategies. I find that different VIX states result in very different risk adjusted performance for all strategies and confirm that significant deviations from normality are observed in the states and the full sample, which are not fully captured by traditional risk metrics. I demonstrate that correlations among strategies are unstable and non-linear, leading to highly concentrated diversification benefits at the times of market stress, which a broad set of exposures is likely to negate. I also demonstrate that at certain states, correlations are very high between traditional and alternative investment strategies and performance characteristics are very similar. I establish that the superior, long term performance of such strategies relative to traditional asset classes is not due to higher returns in good times, but rather better preservation of capital in bad times. Based on empirical data, practical recommendations for investment analysis and risk management are included throughout the article. This is a companion article to ’20 years of VIX: Fear, Greed and Implications for Traditional Asset Classes’ (Munenzon (2010)) available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1583504 Mikhail Munenzon, CFA, CAIA mikhailmunenzon@gmail.com
  2. 2. Introduction Whaley (1993) introduced the VIX index. In the same year, the Chicago Board Options Exchange (CBOE) introduced the CBOE Volatility Index and it quickly became the benchmark for stock market volatility and, more broadly, investor sentiment. The first VIX was a weighted measure of the implied volatility with 30 days to expiration of eight S&P 100 at-the-money put and call options. Ten years later, it expanded to use options based on a broader index, the SP 500, which allows for a more accurate view of investors' expectations on future market volatility. On March 26, 2004, the first ever trading in futures on the VIX index began on the CBOE. Based on the methodology for SP 500, the index has historical information going back to the start of 1990. Why should an investor care about volatility? Black (1975) suggested that the informed investors would try to take advantage of their views through the options market because of the leverage such instruments provided. Therefore, clues from the options market may have implications for the performance characteristics of a security (for example, see Bali and Hovakimian (2009) and Doran and Krieger (2010)). Moreover, starting with the work of Engle (1982) and Bollerslev(1986), evidence emerged documenting the clustering behavior of volatility1 and its resulting predictability. Consequently, if different volatility states are associated with different performance characteristics of a security, an investor’s investment and risk management policies will need to be flexible enough to incorporate that such information. 1 High volatility is like to be followed by high volatility; low volatility is likely to be followed by low volatility. 2
  3. 3. Munenzon (2010) demonstrates that varying levels of VIX are associated with very different return and risk characteristics of traditional asset classes. In this article, I extend this analysis to 9 common alternative investment strategies as convertible arbitrage (CA), distressed (DS), merger arbitrage (MA)(, commodity trading advisor (CTA), macro, equity long/short (LS), equity market neutral (EMN), emerging markets (EM), event driven (ED) (for more detail on strategies, the reader is referred to Anson (2006)). I also evaluate relationships between traditional assets and alternative investment strategies. This article is structured as follows. After an overview of data, I will present key empirical results; concluding remarks follow. Data and Methods I used data for the following traditional asset classes: equities – SP 500 Total Return Index (SPX); bonds - JPM Morgan Aggregate Bond Total Return Index (JPMAGG); commodities – SP GSCI Commodities Index (GSCI); real estate – FTSE EPRA/NAREIT US Total Return Index (NAREIT)2. Performance data for alternative investment strategies are Center for International Securities and Derivatives Markets (CISDM) indices. The monthly data for the indices was downloaded via Bloomberg. The full historical time horizon for this analysis is 12/31/1991 (the first month available for all CISDM indices via Bloomberg) to 1/29/2010 to allow for all asset classes and strategies to have the same historical time period. Based on the level of VIX, I divided the full historical sample into 6 groups to evaluate any differences in results as compared to the full sample, assuming one remains invested only when VIX is in that particular 2 Some investors consider commodities and real estate alternative asset classes, as compared to stocks and bonds. However, for the purposes of this analysis, I consider all such asset classes to be traditional ingredients in an investment program. 3
  4. 4. state. Such classification is broadly consistent with Figure 1 and practitioners’ views on what constitutes low, medium and high volatility and provides a practical way of judging any changes in performance and other characteristics of asset classes, given a VIX state. Key Empirical Results Figure 1 shows the historical level of VIX and cumulative return graphs for the asset classes and VIX. Though the starting and ending points for VIX are relatively comparable, the range of results is very high; one also finds that there are extended periods of high and low volatility. The figure also suggests that crashes don’t just happen – they are generally preceded by periods of increasing turbulence, which ultimately push markets over the edge. Table 1 presents key statistical information on VIX and alternative investment strategies for the full historical period. For all the asset classes and strategies, cumulative returns are strongly positive, especially real estate3. However, statistical features of strategies have a broad range. As expected and similar to traditional asset classes, strategies’ returns are strongly non-normal. The assumption that returns follow a normal distribution, one of the fundamental assumptions of classical finance, can be strongly rejected for all strategies4. However, not all strategies are equally non – normal and their deviations from normality are not always the ones investors should try to avoid. For example, some strategies such as convertible arbitrage, distressed, emerging markets and event driven have very large kurtosis (fat tails or large extreme events as compared to the normal distribution) and negative skewness (returns below the mean are more likely than 3 Secular decline in long term interest rates and the subsequent real estate bubble, which is still being resolved, also played key roles. 4 For a normal distribution, skewness should be 0 and kurtosis should be 3. 4
  5. 5. above the mean). These empirical features important for portfolio construction and risk management are masked by the deceptively low volatility (see more below). Such features (negative skewness and large kurtosis) are the opposite of what investors typically prefer – positive skewness and small kurtosis, resulting in consistent, positive returns. In contrast, CTA exhibits small positive skewness with relatively small kurtosis while macro exhibits significant positive skewness with larger kurtosis, which is less problematic in this case as extreme positive events are more likely than extreme negative events. Figure 6 provides a graphical representation of the above results. One can observe that most of the probability mass of returns for ED in its histogram is below the typical observation with the left, negative tail being much larger than the right, positive tail. By contrast, CTA’s probability mass is mostly above the typical observation with the negative tail containing a smaller area than the right tail. Not only does one not observe normality, but one also finds serial correlation for most of the time series5, which is inconsistent with a random walk model. In classical finance, correlation6, a linear measure of dependency, plays a key role in portfolio risk measurement and optimization. In Table 4, one can see that in the full sample, correlations within and across asset classes and strategies are relatively low (particularly, between SPX vs GSCI, JPMAGG, CTA, Macro; GSCI vs Macro; EM vs CTA; CTA vs NAREIT, CA and MA). Similar to traditional asset classes, all strategies with the exception of CTA have significant negative correlation with VIX. It is also noteworthy that relative to SPX, most strategies do not offer lower correlations than GSCI or 5 Positive returns are likely to be followed by positive returns and negative returns are likely to be followed by negative returns 6 Throughout the paper, correlation refers to what is more formally known as Pearson product-moment correlation coefficient, which is used extensively by practitioners and academics to model dependence. 5
  6. 6. JPAGG; some do not even improve on NAREIT correlation with SPX. Therefore, depending on an investor’s goals and scenarios, an addition of a broad basket of alternative strategies to a portfolio may not always provide meaningful incremental benefit as compared to other traditional asset classes, which may be available much more cheaply. A more selective addition of alternatives to one’s portfolio may result in greater benefits because of the diversity of behavior of strategies among themselves and with traditional asset classes. Finally, because of fat tails exacerbated by negative skewness, historical VaR significantly understate realistic losses one can experience in adverse scenarios, as measured by historical CVaR7. For instance, CA’s historical VAR at 95% conference level is 1% but its CVaR(95%) is 3 times higher at 3.1%. Moreover, because of high serial correlation resulting in ‘smooth’ returns, volatility masks the true extent of tail losses8. For example, CA’s volatility in the full sample is only a little higher than that of JPMAGG (1.2% vs 1.4%), but its worst loss is over 3 times higher (11.5% vs 3.5%). Similarly, while CA’s volatility is slightly lower than that of Macro in the full sample (1.4% vs 1.6%), its CVaRs and worst losses are much larger. Which states dominated historically? The first state (VIX below 20%) accounted for over 50% of all days in the historical sample due to extended periods of calm in the 90s and, to a lesser extent, in the middle of this decade (Table 6). The first 3 states (VIX at up to 30%) accounted for over 90% of all days. However, as seen in the historical VIX 7 VaR(a) is defined as the probability of a loss less than or equal to quantity Q, with the confidence level of a. Thus, it stops at the start of extreme events and does not analyze the tail. CVaR(a) is defined as the average loss once Q is exceeded, with the confidence level of a. Historical based measures are evaluated based on historical data and thus fully incorporate all features of a distribution of a return series. If one assumes a normal distribution of returns, one can find VaR of a a return series via an analytical formula with just its mean and volatility. However, such a measure will understate the realistic extent of losses even more than the historical VaR. For more detail, the reader is referred to Alexander (2008). 8 Returns can be ‘unsmoothed’ to produce a more realistic picture of volatility and potential losses. For example, see Davies et al (2005). 6
  7. 7. chart, the last decade was far more volatile than the decade of the 90s. In addition, once in states 1-3, VIX is likely to remain in that state for a period of time, as transitions occur gradually. It is not possible to draw strong conclusions with available data for states 4-6 as the number of observations in each state is low. However, such a conclusion is supported with daily data (see Munenzon (2010)). Moreover, even with available data, one can observe that once in a high volatility state, one is likely to remain in one of the high volatility states 4-6. How similar are risk/return properties of strategies in various states and relative to the full historical sample? Very dissimilar (Tables 2 and 3). In fact, evidence of consistent, absolute returns in all market cycles is hard to find for alternative strategies. Only CTA, macro and EMN (and bonds for traditional asset classes) provide cumulatively positive returns across all the states. For all strategies, most of the cumulative returns are made in states 1-3, particularly state 1; returns are mostly flat to negative in higher states. This finding is very similar to that for traditional asset classes, which are also very sensitive to VIX. However, not all strategies are sensitive to VIX in the same way. While the percentage of positive months for strategies drops significantly as VIX rises (Table 7), CTA responds well to a rising VIX and Macro and EMN manage to maintain a high positive percentage even at high VIX levels. Finally, generally superior, long term performance of alternative strategies relative to traditional asset classes in the full sample came not from higher returns in good times but rather in preserving a greater portion of those returns in bad times. For example, LS tracks SPX relatively closely in good times but the downside is much more limited than SPX as managers have full flexibility to adjust their portfolios to a particular environment. Also, 7
  8. 8. in state 1, SPX outperforms virtually all strategies but its losses are very large at stress points of state 6, which significantly affects its cumulative return ranking in the full sample. How consistent are cumulative returns for asset classes in various states (Figures 2-4)? They are very consistent at the extreme states 1 and 6. In state 1, all are positive, especially EM, ED, DS and LS. In state 6, CTA is consistently and meaningfully positive; EMN and Macro are very slightly positive; all other strategies are negative. Strategies exhibit a generally consistent behavior in other states as well. For example, EM, ED and LS generally do not perform well as VIX rises; however, CTA, Macro and EMN are generally positive across all states. Given the prior discussion of returns in different states, it is not surprising to find how unstable correlations are across states (Tables 4 and 5). For example, in state 1 (and state 2 to a lesser extent), all indices (traditional assets and strategies) are highly positively correlated. In state 6 (and state 5 to a lesser extent), most indices are also highly positively correlated with the exception of CTA, Macro, EMN and JPMAGG. In other states, the picture is more complex. However, correlations of strategies among themselves and with traditional asset classes generally remain meaningful even if smaller in magnitude than at extreme times. Evaluation of correlation for the full sample masks such complex behavior. Additionally, such behavior suggests that not only are dependencies among asset classes time varying, but that they are also non-linear. Therefore, correlation may not be an appropriate means of evaluating dependence among asset classes9. Moreover, while at the points of extreme stress, diversification can 9 Correlation will correctly describe dependence structure only in very particular cases, such as multivariate normal distributions. Also, at extremes, correlation should be zero for a multivariate normal distribution, 8
  9. 9. provide downside protection, such benefits are limited only to CTA, Macro and EMN (and bonds for traditional asset classes). Finally, given the non-synchronized relationship of VIX with traditional asset classes and alternative strategies, it should play a useful role in an investment program by helping investors minimize potential losses and thus enhance risk adjusted portfolio performance. Conclusions The level of VIX seems to have important and different implications for return expectations for all alternative investment strategies. This is particularly true for the extreme levels of VIX. Though the historical range for VIX is very broad, it exhibits clustering, which make it useful for forecasting. I further present evidence that during the historical period used in the article, several important assumptions of classical finance – normal distribution, randomness of data (no serial correlation) and the use of correlation to describe dependence – find limited support in empirical data. In the cases of large deviations from normality and in the presence of serial correlation, which are typically larger among alternative investment strategies as compared to traditional asset classes, volatility and VaR metrics fail to capture the risk of losses appropriately. A focus on volatility with alternative strategies may overlook large, potential losses hidden in the tails, which volatility cannot capture. Therefore, a practitioner should generate value by incorporating more realistic assumptions to model markets for investment analysis and risk management, such as non-normal distributions which can incorporate skews and fat tails of returns, adjustments for ‘smooth’ data and copulas which can capture non- linearity of dependencies, particularly in the tails. which is not empirically supported. For a more detailed critique on the use of correlations to model dependence, see Embrechts et al (2002). 9
  10. 10. The diversity of behavior of alternative investment strategies among themselves and with traditional asset classes suggests that such strategies may add significant value in portfolio construction and risk management but each strategy’s role and value may vary significantly depending on an investor’s goals, risk tolerance and investment environment scenarios. Generally superior, long term performance of alternative strategies relative to traditional asset classes is not due to better returns in good times but rather relatively more contained losses. While the return potential of alternative strategies may have eroded due to the significant rise in asset since the early 90s (see Figure 5, particularly CA, Macro, EMN and MA), downside management capabilities should remain intact if managers have flexible investment mandates and risk management discipline. The cost of very low volatility with respectable returns is significant negative skewness and kurtosis resulting in much larger extreme losses than volatility may suggest (e.g., CA, ED). However, some strategies possess positive skewness and well contained tails (e.g., CTA and Macro). Evidence for consistent, absolute returns is limited. Masked within the full sample is the fact that risk/return characteristics of strategies across states are very different – e.g., CTA strongly outperforms in state 6 but EM outperforms in state 1. Most strategies performance deteriorates rapidly as VIX rises; CTA is the only strategy that responds well to a rising level of VIX. Moreover, alternative strategies are much more highly correlated with each other and traditional asset classes than the full sample may suggest, with almost perfect correlation at extremes. At stress points, only CTA, Macro and EMN help preserve and add to capital (particularly, CTA). Thus, as with traditional asset classes, diversification is possible but its benefits are highly concentrated, which a broad set of exposures dilutes. 10
  11. 11. Interestingly, strategies and assets optimal for stressed periods (e.g., bonds, CTA, EMN) are those an investor may want to minimize in a portfolio to optimize returns in a good environment. Also, given the performance characteristics of VIX and its relationship with other assets and strategies, its inclusion in an investment program should provide valuable benefits in risk management. The analytical framework presented in this article can be refined further by adding more factors deemed important, such as inflation or information about the prior VIX state; it can also be extended to sectors within an asset class and alternative investment strategies. Finally, while we do not know which volatility states will dominate in the future or how long they may last, greater awareness of the current investment environment, its implications for risk adjusted performance and flexible investment policies to position portfolios appropriately should help investors produce more consistent results. References Anson, M. 2006. Handbook of Alternative Assets. John Wiley and Sons. Alexander, C. 2008. Value at Risk Models. John Wiley & Sons. Bali, T. and A. Hovakimian. 2009. “Volatility Spreads and Expected stock Returns.” Management Science, vol. 55, no. 11 (November): 1797-1812. Black, F. 1975. “Fact and Fantasy in the Use of Options.” Financial Analysts Journal, vol. 31, no. 4 (July/August): 36-41. Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31: 307-327. 11
  12. 12. Davies, R., Kat, H. and Lu, S. 2005. “Fund of Hedge Funds Portfolio Selection: A Multiple Objective Approach.” Working Paper. Doran J. and K. Krieger. 2010. “Implications for Asset Returns in the Implied Volatility Skew.” Financial Analysts Journal, vol. 66, no, 1 (January/February): 65-76. Engle, R. 1982. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of UK inflation.” Econometrica 50: 987-1007. Embrechts, P, A. McNeil, and D. Straumann. 2002. “Correlation and dependence in risk management: Properties and Pitfalls.” In M. Dempster (e.d), Risk Management: Value at Risk and Beyond. Cambridge University Press. Munenzon, M. 2010. “20 Years of VIX: Fear, Greed and Implications for Traditional Asset Classes.” Working Paper. Whaley, R. 1993. “Derivatives on Market Volatility: Hedging Tools Long Overdue.” Journal of Derivatives, 1 (Fall): 71-84. 12
  13. 13. Cumulative Return VIX 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 0 Figure 1 12/31/1991 12/31/1991 10/26/1992 12/15/1992 8/22/1993 CA 11/30/1993 6/18/1994 11/15/1994 4/14/1995 DS 2/8/1996 10/31/1995 12/4/1996 10/15/1996 MA 9/30/1997 9/30/1997 7/27/1998 9/15/1998 5/23/1999 CTA 8/31/1999 3/18/2000 8/15/2000 1/12/2001 Date Date 11/8/2001 7/31/2001 Macro 9/4/2002 7/16/2002 7/1/2003 7/1/2003 LS 4/26/2004 6/15/2004 Historical VIX (12/31/1991 - 1/29/2010) 2/20/2005 5/31/2005 Cumulative Return - Full Historical Sample EMN 12/17/2005 5/16/2006 10/13/2006 5/1/2007 8/9/2007 EM 6/4/2008 4/15/2008 3/31/2009 3/31/2009 ED 1/25/2010
  14. 14. Cumulative Return Cumulative Return 0 1 2 3 4 5 6 Figure 2 0.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1 '12/31/1991' '2/28/1994' '4/30/1992' '12/31/1996' '9/30/1992' CA CA '3/31/1997' '2/26/1993' '7/30/1993' '7/31/1997' DS DS '12/31/1993' '12/31/1997' '6/30/1994' '4/30/1998' '11/30/1994' MA MA '4/28/1995' '2/26/1999' '9/29/1995' '7/30/1999' CTA CTA '2/29/1996' '11/30/1999' '7/31/1996' '1/30/1998' Date '2/29/2000' Date Macro '2/28/2002' Macro '8/31/2000' '7/31/2003' '4/30/2001' '1/30/2004' LS LS '10/31/2001' '6/30/2004' '11/30/2004' Cumulative Return - State 1 of VIX (<=20) '1/31/2002' '4/29/2005' EMN EMN Cumulative Return - State 2 of VIX (>20 and <=25) '8/29/2003' '9/30/2005' '10/31/2007' '2/28/2006' EM EM '5/30/2008' '7/31/2006' '12/29/2006' '10/30/2009' ED ED '5/31/2007'
  15. 15. Cumulative Return Cumulative Return Figure 3 1 0.9 1.1 1.2 1.3 1.4 1.5 0.95 1.05 1.15 0.9 1.1 1 '8/31/2001' '10/31/1997' '9/30/1998' CA CA '8/31/2001' '12/31/1998' DS DS '3/31/1999' '9/28/2001' '8/31/1999' MA MA '10/31/2000' CTA CTA '6/28/2002' '1/31/2001' Date Date '3/30/2001' Macro Macro '7/31/2002' '10/31/2002' LS LS '1/31/2003' '9/30/2002' '12/31/2007' EMN Cumulative Return - State 4 of VIX (>30 and <=35) Cumulatiave Return - State of VIX (>25 and <=30) EMN '2/29/2008' '12/31/2002' EM '5/29/2009' EM '7/31/2009' ED '9/30/2009' ED
  16. 16. Figure 4 Cumulative Return - State 5 of VIX (>35 and <=40) 1.05 Cumulative Return 1 0.95 0.9 0.85 0.8 '9/30/1997' '9/30/1997' '8/30/2002' '8/29/2008' '11/28/2008' '3/31/2009' Date CA DS MA CTA Macro LS EMN EM ED Cumulative Return - State 6 of VIX (>40) 1.2 1.1 Cumulative Return 1 0.9 0.8 0.7 0.6 0.5 '7/31/1998' '7/31/1998' '8/31/1998' '9/30/2008' '10/31/2008' '12/31/2008' '1/30/2009' '2/27/2009' Date CA DS MA CTA Macro LS EMN EM ED
  17. 17. Figure 5 Cumulative Returns with 4 Year Rolling Window - Selected Alternatives as of 1/29/10 1.2 1 0.8 Cumulative Return 0.6 0.4 0.2 0 12/29/1995 9/24/1998 6/20/2001 3/16/2004 12/11/2006 9/6/2009 -0.2 -0.4 Date CA DS MA CTA EMN Macro EM
  18. 18. Figure 6 Histogram of CISDM Event Driven Index Monthly Returns 12/31/1991 - 1/29/2010 80 70 60 50 40 30 20 10 0 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Histogram of CISDM CTA Index Monthly Returns 12/31/1991 - 1/29/2010 60 50 40 30 20 10 0 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
  19. 19. Table 1 12/31/1991 - 1/31/2010 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return 0.7% 0.7% 1.1% 0.5% 1.5% 0.8% 0.9% 0.8% 0.7% 0.8% 1.0% 0.7% 0.9% 1.0% Compounded avg return 0.6% 0.5% 0.9% 0.5% 0.1% 0.8% 0.9% 0.8% 0.7% 0.8% 0.9% 0.7% 0.8% 0.9% max 9.8% 21.1% 31.7% 4.6% 90.8% 4.7% 5.3% 4.7% 7.9% 8.6% 9.4% 2.8% 12.1% 4.8% min -16.8% -27.8% -32.2% -3.5% -32.7% -11.5% -10.6% -5.6% -5.4% -5.4% -9.4% -2.1% -26.3% -7.3% vol 4.3% 6.1% 6.0% 1.2% 17.9% 1.4% 1.8% 1.1% 2.5% 1.6% 2.2% 0.6% 3.8% 1.7% Normality at 95% confidence level? No No No No No No No No No No No No No No pval 0.1% 0.1% 0.1% 2.3% 0.1% 0.1% 0.1% 0.1% 4.5% 0.1% 0.1% 0.1% 0.1% 0.1% No serial correlation at 95% confidence level? Yes No No Yes No No No No Yes No Yes No No No pval 69% 0% 0% 51% 35% 0% 0% 0% 20% 4% 7% 0% 0% 0% VaR (95%) -7.6% -9.4% -7.8% -1.4% -21.1% -1.0% -1.5% -1.0% -3.3% -1.2% -2.4% -0.1% -4.4% -1.5% VaR (99%) -12.1% -14.3% -22.3% -2.7% -29.9% -4.4% -6.0% -2.4% -4.4% -2.5% -4.5% -1.1% -12.6% -6.9% CVaR(95%) -10.1% -12.9% -15.0% -2.1% -26.4% -3.1% -3.9% -2.0% -4.0% -2.2% -3.9% -0.7% -9.2% -3.7% CVaR(99%) -14.0% -19.2% -25.9% -2.9% -31.1% -7.4% -8.1% -3.5% -4.8% -3.6% -6.3% -1.5% -17.6% -7.1% Skewness -0.8 -0.3 -0.9 -0.2 1.4 -3.9 -1.9 -0.8 0.4 1.2 -0.2 -0.4 -2.1 -1.6 Kurtosis 4.4 5.2 11.6 3.9 6.8 33.6 13.4 8.7 3.0 7.6 5.7 6.3 16.0 9.4 cumulative return for full sample 269.32% 174.79% 604.87% 214.90% 27.50% 435.40% 596.83% 450.51% 306.00% 413.66% 637.00% 322.57% 516.20% 651.63% % of months with positive returns 64.1% 56.2% 65.0% 68.2% 46.5% 85.7% 79.7% 86.2% 55.3% 71.0% 70.0% 92.2% 71.4% 80.2% Notes: Jarque-Bera test was used to evaluate normality of a time series; null hypothesis is stated in the question. Ljung-Box test with 20 lags was used to evaluate serial correlation of a time series; null hypothesis is stated in the question. SPX - SP500 Total Return GSCI - SP GSCI NAREIT - FTSE EPRA/NAREIT US Total Return JPMAGG - JPM Morgan Aggregate Bond Total Return VIX - VIX Index CA - convertible arbitrage DS - distressed LS - equity long/short MA - merger arbitrage EM - emerging markets EMN - equity market neutral ED - event driven
  20. 20. Table 2 State 1 - VIX <= 20 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return 1.5% 1.4% 1.9% 0.5% -1.0% 0.8% 1.3% 1.0% 0.7% 1.0% 1.4% 0.7% 1.5% 1.3% Compounded avg return 1.4% 1.3% 1.8% 0.4% -2.1% 0.8% 1.3% 1.0% 0.6% 1.0% 1.4% 0.7% 1.5% 1.3% max 7.6% 15.7% 8.6% 3.7% 48.0% 3.0% 5.3% 4.7% 6.2% 8.6% 6.3% 2.2% 8.5% 4.4% min -4.4% -9.9% -14.0% -3.5% -32.7% -2.5% -4.2% -1.0% -5.4% -5.4% -3.4% -1.3% -5.8% -1.6% vol 2.4% 5.0% 3.7% 1.1% 14.7% 0.8% 1.5% 1.0% 2.3% 1.8% 1.6% 0.5% 2.4% 1.2% Normality at 95% confidence level? Yes Yes No No No No No No Yes No Yes No Yes Yes pval 50.0% 15.4% 0.1% 2.5% 0.2% 0.1% 0.9% 0.1% 30.5% 0.1% 38.0% 0.1% 5.7% 50.0% No serial correlation at 95% confidence level? Yes Yes No Yes No No No No No Yes Yes Yes No Yes pval 45.5% 5.8% 2.5% 23.6% 3.6% 0.0% 10.3% 1.8% 1.8% 13.3% 59.3% 15.0% 0.4% 25.4% VaR (95%) -2.5% -6.8% -4.0% -1.4% -20.0% -0.7% -1.2% -0.7% -2.7% -1.2% -1.3% 0.1% -2.3% -0.8% VaR (99%) -3.9% -9.5% -10.8% -2.9% -30.3% -1.6% -3.0% -1.0% -4.6% -2.9% -2.7% -0.5% -5.8% -1.5% CVaR(95%) -3.2% -8.4% -7.7% -2.2% -25.4% -1.3% -2.1% -0.9% -3.6% -2.1% -2.1% -0.3% -3.9% -1.2% CVaR(99%) -4.0% -9.7% -11.7% -3.1% -30.9% -1.9% -3.3% -1.0% -4.8% -3.6% -2.9% -0.7% -5.8% -1.5% Skewness -0.1 0.3 -1.0 -0.5 0.9 -1.0 -0.3 0.8 0.3 1.1 -0.1 -0.4 -0.1 0.0 Kurtosis 2.8 3.4 5.1 3.8 4.3 5.1 4.5 5.3 2.8 7.2 3.5 5.7 4.0 3.1 cumulative return 435.1% 342.0% 697.7% 68.9% -91.3% 154.3% 340.7% 229.1% 112.9% 211.6% 398.2% 134.8% 463.0% 340.4% % of months with positive returns 77.1% 62.7% 73.7% 64.4% 40.7% 86.4% 85.6% 91.5% 55.1% 73.7% 79.7% 96.6% 79.7% 86.4% State 2 - VIX > 20 & <= 25 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return 0.7% -0.1% 0.7% 0.5% 2.5% 0.8% 0.9% 0.8% 0.1% 0.6% 1.0% 0.7% 1.1% 1.0% Compounded avg return 0.6% -0.3% 0.6% 0.5% 1.2% 0.8% 0.9% 0.8% 0.0% 0.6% 1.0% 0.7% 1.0% 1.0% max 9.8% 17.7% 10.8% 2.9% 44.9% 2.7% 3.9% 2.8% 6.1% 5.7% 9.4% 2.8% 12.1% 4.4% min -8.4% -11.9% -10.9% -2.0% -29.0% -1.6% -1.5% -1.9% -4.5% -3.0% -2.4% -0.7% -11.3% -1.6% vol 4.4% 5.8% 4.7% 1.1% 16.9% 0.9% 1.3% 1.0% 2.3% 1.4% 2.5% 0.7% 3.7% 1.4% Normality at 95% confidence level? Yes Yes Yes Yes Yes No Yes Yes Yes No No Yes No Yes pval 28.0% 29.4% 50.0% 50.0% 20.7% 2.7% 43.8% 50.0% 50.0% 0.3% 0.5% 6.5% 1.4% 50.0% No serial correlation at 95% confidence level? No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No Yes pval 0.1% 76.8% 20.4% 89.2% 35.2% 6.0% 21.2% 15.5% 80.4% 81.0% 40.8% 0.5% 0.1% 45.7% VaR (95%) -6.0% -8.1% -7.9% -1.4% -19.9% -1.2% -1.3% -1.2% -3.8% -1.2% -2.2% -0.3% -4.3% -1.2% VaR (99%) -8.4% -11.8% -10.9% -2.0% -28.9% -1.6% -1.5% -1.9% -4.5% -3.0% -2.4% -0.7% -11.2% -1.6% CVaR(95%) -6.9% -9.8% -9.4% -1.7% -25.5% -1.5% -1.4% -1.6% -4.2% -2.1% -2.3% -0.5% -7.0% -1.4% CVaR(99%) -8.4% -11.9% -10.9% -2.0% -29.0% -1.6% -1.5% -1.9% -4.5% -3.0% -2.4% -0.7% -11.3% -1.6% Skewness 0.1 0.4 -0.2 -0.1 0.4 -0.8 0.2 -0.2 0.1 0.8 1.2 0.5 -0.4 0.2 Kurtosis 2.1 3.3 2.8 2.9 2.6 3.9 2.4 3.5 3.0 5.6 4.5 3.9 5.1 2.5 cumulative return 40.0% -13.1% 34.9% 32.4% 89.1% 49.3% 59.0% 52.0% 2.3% 34.1% 69.4% 47.0% 68.9% 70.4% % of months with positive returns 50.9% 45.3% 56.6% 71.7% 49.1% 84.9% 79.2% 86.8% 50.9% 66.0% 64.2% 90.6% 67.9% 79.2% State 3 - VIX > 25 & <= 30 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return 0.2% 0.9% 2.5% 0.5% 0.2% 1.3% 0.9% 0.6% 1.4% 0.7% 0.4% 0.6% 0.7% 0.9% Compounded avg return 0.0% 0.6% 2.4% 0.5% -0.8% 1.3% 0.9% 0.6% 1.3% 0.6% 0.4% 0.6% 0.7% 0.8% max 8.1% 21.1% 14.2% 2.1% 28.8% 4.7% 4.9% 2.6% 7.9% 5.6% 4.8% 2.1% 9.5% 4.8% min -9.1% -16.6% -4.7% -2.7% -31.5% -2.4% -2.6% -1.6% -4.3% -2.1% -4.1% -1.0% -6.4% -3.2% vol 5.3% 7.5% 4.7% 1.0% 14.1% 1.4% 1.6% 0.9% 3.3% 1.5% 2.4% 0.7% 3.8% 1.7% Normality at 95% confidence level? Yes Yes Yes No Yes No Yes Yes Yes No Yes Yes Yes Yes pval 20.9% 27.1% 11.7% 1.9% 50.0% 5.0% 50.0% 50.0% 38.0% 0.3% 50.0% 50.0% 50.0% 27.2% No serial correlation at 95% confidence level? Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes pval 33.1% 91.7% 59.0% 68.4% 83.7% 89.1% 84.7% 0.2% 49.0% 80.6% 53.6% 46.7% 95.7% 61.5% VaR (95%) -8.1% -11.1% -3.8% -1.3% -23.3% -0.3% -1.5% -1.3% -3.8% -1.2% -3.3% -0.8% -5.2% -3.1% VaR (99%) -9.1% -16.6% -4.7% -2.7% -31.5% -2.4% -2.6% -1.6% -4.3% -2.1% -4.1% -1.0% -6.4% -3.2% CVaR(95%) -8.5% -13.4% -4.1% -1.9% -26.7% -1.1% -2.0% -1.4% -4.0% -1.6% -3.7% -0.9% -5.7% -3.1% CVaR(99%) -9.1% -16.6% -4.7% -2.7% -31.5% -2.4% -2.6% -1.6% -4.3% -2.1% -4.1% -1.0% -6.4% -3.2% Skewness -0.1 0.3 0.7 -0.9 0.1 0.2 0.3 -0.2 0.4 1.3 0.0 -0.3 0.4 -0.5 Kurtosis 1.8 4.0 2.9 4.8 3.2 4.9 3.3 3.6 2.4 6.1 2.2 3.4 3.0 3.7 cumulative return 0.4% 16.9% 88.5% 14.7% -19.8% 43.0% 26.3% 17.1% 43.4% 18.9% 10.1% 16.7% 19.6% 25.3% % of months with positive returns 51.9% 51.9% 70.4% 74.1% 51.9% 96.3% 74.1% 81.5% 59.3% 74.1% 59.3% 81.5% 63.0% 77.8% Notes: Jarque-Bera test was used to evaluate normality of a time series; null hypothesis is stated in the question. Ljung-Box test with 20 lags was used to evaluate serial correlation of a time series; null hypothesis is stated in the question. SPX - SP500 Total Return GSCI - SP GSCI NAREIT - FTSE EPRA/NAREIT US Total Return JPMAGG - JPM Morgan Aggregate Bond Total Return VIX - VIX Index CA - convertible arbitrage DS - distressed LS - equity long/short MA - merger arbitrage EM - emerging markets EMN - equity market neutral ED - event driven
  21. 21. Table 3 State 4 - VIX > 30 & <= 35 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return -1.3% 0.4% -4.3% 1.0% 9.8% 0.8% 0.4% -0.2% 1.1% 0.1% -0.8% 0.2% 0.2% -0.2% Compounded avg return -1.4% 0.2% -4.3% 1.0% 8.5% 0.8% 0.4% -0.2% 1.0% 0.1% -0.8% 0.2% 0.2% -0.2% max 8.8% 7.8% 0.2% 2.1% 28.1% 2.1% 1.6% 0.8% 3.2% 1.5% 1.1% 0.8% 2.5% 1.3% min -8.1% -10.5% -7.3% -0.3% -21.5% -0.7% -1.3% -2.0% -3.3% -1.8% -3.4% -0.3% -4.0% -2.9% vol 5.9% 6.9% 2.4% 0.9% 17.2% 0.8% 1.2% 1.0% 2.7% 1.1% 1.7% 0.4% 2.3% 1.7% Normality at 95% confidence level? Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes pval 50.0% 50.0% 44.4% 44.6% 48.1% 50.0% 28.4% 20.8% 12.3% 50.0% 21.7% 50.0% 14.6% 11.0% No serial correlation at 95% confidence level? Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes pval 80.1% 98.5% 6.9% 26.3% 80.5% 30.4% 48.2% 12.5% 68.3% 24.2% 16.5% 0.8% 33.0% 31.8% VaR (95%) -8.1% -10.5% -7.3% -0.3% -21.5% -0.7% -1.3% -2.0% -3.3% -1.8% -3.4% -0.3% -4.0% -2.9% VaR (99%) -8.1% -10.5% -7.3% -0.3% -21.5% -0.7% -1.3% -2.0% -3.3% -1.8% -3.4% -0.3% -4.0% -2.9% CVaR(95%) -8.1% -10.5% -7.3% -0.3% -21.5% -0.7% -1.3% -2.0% -3.3% -1.8% -3.4% -0.3% -4.0% -2.9% CVaR(99%) -8.1% -10.5% -7.3% -0.3% -21.5% -0.7% -1.3% -2.0% -3.3% -1.8% -3.4% -0.3% -4.0% -2.9% Skewness 0.4 -0.3 0.7 -0.2 -0.7 -0.4 -0.5 -0.8 -0.8 -0.6 -0.7 0.1 -0.9 -0.8 Kurtosis 2.4 1.8 2.9 1.6 2.6 3.3 1.7 2.3 2.0 2.8 1.9 1.9 2.4 1.9 cumulative return -9.6% 1.4% -26.4% 6.9% 77.2% 5.6% 2.9% -1.6% 7.4% 0.8% -5.6% 1.6% 1.1% -1.4% % of months with positive returns 42.9% 57.1% 14.3% 85.7% 85.7% 85.7% 71.4% 57.1% 71.4% 57.1% 42.9% 85.7% 71.4% 71.4% State 5 - VIX > 35 & <= 40 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return -2.5% -2.4% 8.3% 1.1% 24.1% 0.4% -0.3% 0.3% 0.5% 0.4% -0.9% 0.1% -1.6% -1.1% Compounded avg return -2.8% -2.7% 7.5% 1.1% 16.3% 0.3% -0.3% 0.2% 0.5% 0.4% -0.9% 0.1% -1.8% -1.2% max 9.6% 5.7% 31.7% 3.4% 90.8% 4.6% 3.0% 2.1% 2.5% 1.2% 3.2% 1.2% 8.6% 2.0% min -10.9% -12.1% -4.3% -1.1% -27.6% -8.4% -4.4% -2.6% -1.2% -0.2% -5.4% -2.1% -10.1% -7.3% vol 8.2% 8.4% 15.6% 1.7% 49.2% 5.1% 2.7% 1.8% 1.7% 0.7% 3.2% 1.3% 6.9% 3.6% Normality at 95% confidence level? Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes pval 50.0% 18.9% 24.6% 50.0% 50.0% 5.7% 50.0% 27.6% 44.2% 17.3% 50.0% 7.7% 50.0% 6.4% No serial correlation at 95% confidence level? Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes pval 16.9% 25.5% 19.8% 19.4% 88.5% 54.8% 52.4% 59.3% 4.7% 0.9% 64.8% 22.5% 76.7% 24.8% VaR (95%) -10.9% -12.1% -4.3% -1.1% -27.6% -8.4% -4.4% -2.6% -1.2% -0.2% -5.4% -2.1% -10.0% -7.3% VaR (99%) -10.9% -12.1% -4.3% -1.1% -27.6% -8.4% -4.4% -2.6% -1.2% -0.2% -5.4% -2.1% -10.1% -7.3% CVaR(95%) -10.9% -12.1% -4.3% -1.1% -27.6% -8.4% -4.4% -2.6% -1.2% -0.2% -5.4% -2.1% -10.1% -7.3% CVaR(99%) -10.9% -12.1% -4.3% -1.1% -27.6% -8.4% -4.4% -2.6% -1.2% -0.2% -5.4% -2.1% -10.0% -7.3% Skewness 0.5 -0.3 0.7 0.0 0.3 -1.2 -0.5 -0.9 0.2 0.4 -0.2 -1.1 0.4 -1.2 Kurtosis 1.9 1.2 1.8 1.9 1.6 2.9 2.5 2.5 1.4 1.3 2.1 2.7 2.2 2.9 cumulative return -13.1% -12.6% 43.4% 5.8% 112.6% 1.3% -1.7% 1.2% 2.4% 2.0% -4.4% 0.4% -8.8% -5.8% % of months with positive returns 40.0% 60.0% 40.0% 80.0% 60.0% 80.0% 60.0% 80.0% 60.0% 60.0% 40.0% 80.0% 40.0% 40.0% State 6 - VIX > 40 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED monthly data Arithmetic avg return -6.1% -4.3% -13.7% 1.2% 18.0% -1.1% -3.4% -0.8% 2.5% 0.1% -1.8% 0.4% -6.8% -2.1% Compounded avg return -6.5% -5.2% -14.7% 1.2% 14.4% -1.2% -3.5% -0.8% 2.4% 0.1% -1.8% 0.4% -7.3% -2.1% max 8.8% 12.3% 6.3% 4.6% 78.6% 3.9% 1.1% 1.3% 7.2% 1.1% 1.7% 0.9% 3.5% 1.8% min -16.8% -27.8% -32.2% -2.0% -7.7% -11.5% -10.6% -5.6% -2.0% -2.3% -9.4% 0.0% -26.3% -7.3% vol 9.9% 13.2% 14.3% 2.4% 33.9% 5.0% 4.6% 2.4% 3.4% 1.1% 3.9% 0.3% 10.3% 3.7% Normality at 95% confidence level? Yes Yes Yes Yes Yes No Yes No Yes No Yes Yes Yes Yes pval 27.3% 50.0% 50.0% 50.0% 10.1% 1.8% 14.7% 4.6% 47.3% 1.2% 5.5% 50.0% 8.9% 32.0% No serial correlation at 95% confidence level? Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes pval 46.0% 78.0% 87.0% 27.1% 20.2% 86.1% 82.3% 70.6% 27.0% 89.5% 52.6% 84.1% 69.8% 91.2% VaR (95%) -16.8% -27.8% -32.2% -2.0% -7.7% -11.5% -10.6% -5.6% -2.0% -2.3% -9.4% 0.0% -26.3% -7.3% VaR (99%) -16.8% -27.8% -32.2% -2.0% -7.7% -11.5% -10.6% -5.6% -2.0% -2.3% -9.4% 0.0% -26.3% -7.3% CVaR(95%) -16.8% -27.8% -32.2% -2.0% -7.7% -11.5% -10.6% -5.6% -2.0% -2.3% -9.4% 0.0% -26.3% -7.3% CVaR(99%) -16.8% -27.8% -32.2% -2.0% -7.7% -11.5% -10.6% -5.6% -2.0% -2.3% -9.4% 0.0% -26.3% -7.3% Skewness 0.6 -0.6 0.3 0.0 1.0 -1.5 -0.8 -1.3 0.1 -1.6 -1.2 0.1 -1.1 -0.5 Kurtosis 1.8 2.6 1.7 1.7 2.3 4.1 1.9 3.3 1.6 4.1 3.3 1.7 2.8 1.7 cumulative return -37.4% -31.0% -67.1% 8.6% 155.7% -7.9% -22.2% -5.7% 18.2% 0.7% -12.1% 2.9% -41.2% -13.9% % of months with positive returns 28.6% 42.9% 28.6% 57.1% 57.1% 42.9% 28.6% 42.9% 57.1% 71.4% 42.9% 85.7% 14.3% 28.6% Notes: Jarque-Bera test was used to evaluate normality of a time series; null hypothesis is stated in the question. Ljung-Box test with 20 lags was used to evaluate serial correlation of a time series; null hypothesis is stated in the question. SPX - SP500 Total Return GSCI - SP GSCI NAREIT - FTSE EPRA/NAREIT US Total Return JPMAGG - JPM Morgan Aggregate Bond Total Return VIX - VIX Index CA - convertible arbitrage DS - distressed LS - equity long/short MA - merger arbitrage EM - emerging markets EMN - equity market neutral ED - event driven
  22. 22. Table 4 Correlation Matrices Full Sample SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED SPX 1 GSCI 0.166019 1 NAREIT 0.53491 0.138621 1 JPMAGG 0.050299 0.025469 0.086042 1 VIX -0.62321 -0.15603 -0.34092 -0.04614 1 CA 0.419011 0.291283 0.394769 0.163013 -0.38292 1 DS 0.577209 0.275929 0.46886 0.021617 -0.44175 0.669364 1 MA 0.524669 0.115169 0.355637 0.032789 -0.48047 0.510831 0.673551 1 CTA -0.11897 0.177433 -0.05343 0.252222 0.057434 -0.07251 -0.15321 -0.10711 1 Macro 0.344947 0.092964 0.149527 0.181607 -0.25827 0.232467 0.441577 0.469438 0.259513 1 LS 0.724026 0.249474 0.351369 -0.02816 -0.50077 0.478391 0.753201 0.673489 -0.02296 0.548764 1 EMN 0.384881 0.202311 0.24061 0.129666 -0.25309 0.421363 0.454525 0.541486 0.127153 0.351819 0.614427 1 EM 0.572096 0.29133 0.385921 -0.08274 -0.45825 0.546525 0.725803 0.593944 -0.09523 0.419102 0.725106 0.399661 1 ED 0.666159 0.290115 0.449076 -0.0349 -0.5207 0.678705 0.852948 0.827768 -0.10512 0.444584 0.812982 0.602404 0.754906 1 State 1 - VIX <= 20 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED SPX 1 GSCI 0.941205 1 NAREIT 0.968671 0.962158 1 JPMAGG 0.952314 0.839816 0.903186 1 VIX -0.91204 -0.81929 -0.84961 -0.89769 1 CA 0.957119 0.838031 0.903497 0.988303 -0.885 1 DS 0.991014 0.947677 0.981967 0.956791 -0.8931 0.959535 1 MA 0.979991 0.89966 0.949468 0.985303 -0.89983 0.988699 0.987963 1 CTA 0.981653 0.90269 0.935674 0.97368 -0.90862 0.980252 0.978627 0.987422 1 Macro 0.961245 0.870754 0.920509 0.982752 -0.91242 0.984632 0.972025 0.990924 0.982466 1 LS 0.997082 0.934322 0.964586 0.966029 -0.91526 0.970359 0.994146 0.990164 0.988173 0.976988 1 EMN 0.991518 0.922705 0.960859 0.977024 -0.90646 0.980099 0.993249 0.996267 0.989303 0.983014 0.997565 1 EM 0.979217 0.957366 0.982346 0.928148 -0.87315 0.937548 0.991904 0.973443 0.965357 0.957532 0.981086 0.978201 1 ED 0.993832 0.937688 0.973327 0.966031 -0.90074 0.970734 0.998421 0.993171 0.986048 0.978325 0.997631 0.997408 0.98878 1 State 2 - VIX > 20 & <=25 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED SPX 1 GSCI 0.503598 1 NAREIT 0.281017 -0.38273 1 JPMAGG 0.591286 0.03562 0.655733 1 VIX -0.14834 -0.22626 0.116192 0.421282 1 CA 0.807267 0.264131 0.617629 0.909699 0.171832 1 DS 0.8078 0.248871 0.621926 0.933635 0.210858 0.988517 1 MA 0.860286 0.315384 0.568545 0.873463 0.108972 0.990932 0.978591 1 CTA 0.107005 0.057958 -0.33403 -0.46772 -0.23197 -0.35105 -0.32781 -0.26942 1 Macro 0.817442 0.246765 0.555615 0.937412 0.289504 0.963202 0.983941 0.957752 -0.25147 1 LS 0.887807 0.437052 0.494851 0.827565 0.089189 0.972226 0.965125 0.981538 -0.2491 0.94432 1 EMN 0.833454 0.304335 0.564188 0.911334 0.199824 0.992832 0.990386 0.992594 -0.30432 0.979312 0.978463 1 EM 0.673577 0.238915 0.662337 0.90606 0.237271 0.936357 0.946899 0.899895 -0.47482 0.915813 0.909672 0.923465 1 ED 0.832427 0.290064 0.602019 0.908215 0.154164 0.996239 0.993168 0.994336 -0.31225 0.972878 0.979645 0.996545 0.933914 1
  23. 23. Table 5 Correlation Matrices State 3 - VIX >25 & <= 30 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED SPX 1 GSCI -0.51244 1 NAREIT -0.64832 0.798515 1 JPMAGG -0.75644 0.862869 0.913689 1 VIX -0.88783 0.171959 0.477128 0.492228 1 CA -0.67862 0.807505 0.975273 0.924756 0.49608 1 DS -0.51727 0.78842 0.93875 0.831426 0.33691 0.963878 1 MA -0.63508 0.414459 0.759672 0.590389 0.648542 0.792626 0.801899 1 CTA -0.83769 0.811487 0.899998 0.952449 0.626074 0.931059 0.842053 0.718888 1 Macro -0.80822 0.687251 0.905524 0.879875 0.679876 0.936211 0.877137 0.866432 0.951091 1 LS 0.714872 -0.24221 -0.14454 -0.41182 -0.57735 -0.14418 0.080142 0.052859 -0.40206 -0.23199 1 EMN -0.80927 0.686757 0.918954 0.874346 0.675373 0.947276 0.885721 0.888353 0.935175 0.9852 -0.23079 1 EM 0.211595 0.496111 0.52453 0.311695 -0.33799 0.516199 0.697765 0.381423 0.269815 0.329085 0.667281 0.331143 1 ED -0.50495 0.553775 0.853462 0.670927 0.455153 0.883099 0.932273 0.940233 0.744724 0.869153 0.196033 0.881862 0.626269 1 State 4 - VIX >30 & <= 35 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED SPX 1 GSCI -0.1058 1 NAREIT 0.05963 -0.69457 1 JPMAGG -0.21417 0.504384 -0.89664 1 VIX -0.80139 0.533325 -0.58082 0.650997 1 CA 0.22174 0.840498 -0.87441 0.661421 0.330713 1 DS 0.237762 0.861595 -0.76308 0.522138 0.322799 0.959258 1 MA 0.658235 0.254355 -0.23468 0.205307 -0.28399 0.537104 0.594575 1 CTA -0.53465 0.294564 -0.64778 0.826485 0.719638 0.398464 0.257912 0.056836 1 Macro -0.04478 0.688434 -0.87662 0.824876 0.58073 0.869332 0.844164 0.536615 0.698512 1 LS 0.476773 -0.52424 0.816715 -0.81392 -0.73489 -0.51176 -0.34087 0.320574 -0.68892 -0.5379 1 EMN 0.154582 0.6837 -0.90875 0.857048 0.399624 0.919752 0.836987 0.595298 0.630544 0.944675 -0.56641 1 EM 0.376232 0.695095 -0.82949 0.682518 0.243865 0.927198 0.920912 0.67515 0.318564 0.870815 -0.38087 0.923185 1 ED 0.61897 0.576744 -0.35083 0.102593 -0.18947 0.73999 0.8349 0.844449 -0.11552 0.555226 0.196617 0.591109 0.756747 1 State 5 - VIX > 35 & <=40 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED SPX 1 GSCI 0.481818 1 NAREIT 0.038442 -0.67631 1 JPMAGG -0.38392 -0.74565 0.75955 1 VIX -0.85384 -0.40593 -0.15456 -0.00262 1 CA 0.733765 0.419709 0.292253 0.111476 -0.92906 1 DS 0.845444 0.64987 0.062281 -0.22085 -0.89638 0.943219 1 MA 0.634772 0.066379 0.515918 0.411847 -0.90336 0.91989 0.759959 1 CTA -0.84477 -0.69157 0.374883 0.813796 0.504231 -0.36927 -0.63318 -0.14058 1 Macro -0.57565 -0.77556 0.712102 0.968561 0.225268 -0.06853 -0.38085 0.20252 0.909887 1 LS 0.930158 0.614275 0.02614 -0.27025 -0.94047 0.908423 0.976613 0.763301 -0.71648 -0.45916 1 EMN 0.848494 0.659465 -0.0215 -0.21247 -0.94313 0.937744 0.98204 0.782195 -0.6208 -0.39594 0.98218 1 EM 0.9157 0.621252 0.069516 -0.28128 -0.90024 0.909405 0.987891 0.738582 -0.71923 -0.45032 0.990992 0.968517 1 ED 0.80801 0.881044 -0.38724 -0.5629 -0.77903 0.730836 0.891495 0.47754 -0.79366 -0.70143 0.90516 0.918243 0.890061 1 State 6 - VIX > 40 SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED SPX 1 GSCI 0.947535 1 NAREIT 0.989761 0.955068 1 JPMAGG -0.66685 -0.62371 -0.74051 1 VIX -0.93616 -0.94323 -0.93279 0.503944 1 CA 0.600018 0.785634 0.6409 -0.37895 -0.76855 1 DS 0.88632 0.938534 0.923341 -0.69192 -0.93591 0.867862 1 MA 0.372983 0.555832 0.369027 -0.00897 -0.60503 0.887974 0.632629 1 CTA -0.81826 -0.87646 -0.85576 0.700648 0.874758 -0.86755 -0.96908 -0.69008 1 Macro -0.86627 -0.83157 -0.92302 0.828205 0.837544 -0.59479 -0.8948 -0.24157 0.824027 1 LS 0.950652 0.974157 0.941045 -0.53264 -0.95383 0.766043 0.918142 0.574082 -0.83582 -0.79082 1 EMN -0.80293 -0.72074 -0.8635 0.879599 0.738576 -0.48864 -0.82977 -0.14042 0.780802 0.970528 -0.69509 1 EM 0.91527 0.952187 0.940201 -0.6858 -0.95235 0.841435 0.992965 0.631264 -0.97413 -0.88183 0.928747 -0.81472 1 ED 0.868202 0.952809 0.897106 -0.65446 -0.91142 0.90429 0.984168 0.697127 -0.96479 -0.8197 0.929081 -0.74148 0.981646 1
  24. 24. Table 6 Transition probability matrix for VIX Next day State Current State 1 2 3 4 5 6 1 88.1% 11.0% 0.8% 0.0% 0.0% 0.0% 2 24.5% 52.8% 13.2% 1.9% 3.8% 1.9% 3 0.0% 37.0% 51.9% 11.1% 0.0% 0.0% 4 0.0% 28.6% 28.6% 28.6% 14.3% 0.0% 5 0.0% 0.0% 40.0% 20.0% 0.0% 40.0% 6 0.0% 0.0% 14.3% 0.0% 28.6% 57.1% Average Maximum % of all Months VIX State Duration Duration in State 1 8.4 45 54.6% 2 1.8 6 24.1% 3 2.1 5 12.5% 4 1.4 2 3.2% 5 1.0 1 2.3% 6 2.3 3 3.2% Notes: Based on monthly data. See Munenzon (2010) for the tables above based on daily data.
  25. 25. Table 7 % of months positive SPX GSCI NAREIT JPMAGG VIX CA DS MA CTA Macro LS EMN EM ED VIX state 1 77.1% 62.7% 73.7% 64.4% 40.7% 86.4% 85.6% 91.5% 55.1% 73.7% 79.7% 96.6% 79.7% 86.4% 2 50.9% 45.3% 56.6% 71.7% 49.1% 84.9% 79.2% 86.8% 50.9% 66.0% 64.2% 90.6% 67.9% 79.2% 3 51.9% 51.9% 70.4% 74.1% 51.9% 96.3% 74.1% 81.5% 59.3% 74.1% 59.3% 81.5% 63.0% 77.8% 4 42.9% 57.1% 14.3% 85.7% 85.7% 85.7% 71.4% 57.1% 71.4% 57.1% 42.9% 85.7% 71.4% 71.4% 5 40.0% 60.0% 40.0% 80.0% 60.0% 80.0% 60.0% 80.0% 60.0% 60.0% 40.0% 80.0% 40.0% 40.0% 6 28.6% 42.9% 28.6% 57.1% 57.1% 42.9% 28.6% 42.9% 57.1% 71.4% 42.9% 85.7% 14.3% 28.6%

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