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The Review of Financial Studies
, Volume Advance Article (7) – May 22, 2018

45 pages

/lp/ou_press/risk-everywhere-modeling-and-managing-volatility-fD2lkJDlU9

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press.
- ISSN
- 0893-9454
- eISSN
- 1465-7368
- D.O.I.
- 10.1093/rfs/hhy041
- Publisher site
- See Article on Publisher Site

Abstract Based on high-frequency data for more than fifty commodities, currencies, equity indices, and fixed-income instruments spanning more than two decades, we document strong similarities in realized volatility patterns within and across asset classes. Exploiting these similarities through panel-based estimation of new realized volatility models results in superior out-of-sample risk forecasts, compared to forecasts from existing models and conventional procedures that do not incorporate the similarities in volatilities. We develop a utility-based framework for evaluating risk models that shows significant economic gains from our new risk model. Lastly, we evaluate the effects of transaction costs and trading speed in implementing different risk models. Received March 7, 2016; editorial decision February 3, 2018 by Editor Andrew Karolyi. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. Measuring, forecasting, and controlling risk are at the heart of financial economic theory and practice. Investors and asset managers generally seek to maximize return while limiting risk. Many traders also have specific risk limits or risk targets. Traders in a bank, for example, often face risk limits, while hedge funds often tell their investors that a particular fund is expected to realize a certain risk level. So-called “risk parity strategies” are explicitly designed to equate the risk stemming from the different investments included in a portfolio, rather than the capital allocated to the different components. Such investors therefore continually monitor global risk as it evolves and change their portfolios in response, reducing notional exposures as risk rises and increasing positions as risk fades. This paper presents a framework for measuring, modeling, and forecasting risk across global assets and asset classes. Our main result is that exploiting commonality in risk everywhere has a statistically and economically significant impact. Our results are based on a broad data set of realized volatilities constructed from high-frequency data for more than fifty instruments across four asset classes. We find that our new risk models, new panel-based estimation techniques, and our global volatility factor—all designed to exploit the strong commonalities observed in the volatilities across assets and asset classes—result in statistically significant out-of-sample forecast improvements and nontrivial utility gains compared to more conventional individually estimated asset-specific risk models. We start by constructing a comprehensive database comprising high-frequency intraday data from different global markets spanning more than two decades and covering 20 commodities, 21 equity indices, 8 fixed-income futures, and 9 currencies. We compute the realized volatilities ($$RV$$) for each day and asset in our sample. When qualitatively comparing the estimated $$RV$$s, the differences in risk levels across assets and asset classes immediately stand out. However, when we normalize each asset’s daily realized volatilities by their respective sample averages, striking similarities emerge. Indeed, these “normalized risk measures” have almost identical unconditional distributions and similar highly persistent autocorrelation structures when comparing across assets and asset classes. Hence, the volatilities of different assets—equities, bonds, commodities, or currencies—appear to behave almost the same over time. Going one step further, we document strong volatility spillover effects both within and across different markets and geographical regions. The existence of spillover effects and commonalities in the dynamic dependencies is, of course, well known from the already existing volatility literature and the estimation results obtained with traditional GARCH and stochastic volatility models (see, e.g., Taylor 2005; Andersen et al. 2006; and the references therein). Next, we build risk forecasting models explicitly designed to exploit these strong similarities in the distributions of the volatilities across and within asset classes. The formulation of our models are motivated by the heterogeneous autoregressive (HAR) model of Corsi (2009) and draws on insights from the mixed data sampling (MIDAS) approach of Ghysels et al. (2006) and Ghysels, Sinko, and Valkanov (2007). First, we show how to simultaneously estimate risk models across many assets using panel regressions that add power by exploiting the similarities in the cross-asset risk characteristics.1 An important step needed to allow such a panel-based estimation is to “center” the models and eliminate the asset-specific intercept terms to ensure that all parameters are “scale-free” in the sense that they do not depend on the level of risk. Second, we introduce new “smooth” realized volatility models, in which the forecasted future volatilities depends on the past volatilities in a way that is continuous and decreasing in the lag lengths, thereby eliminating nonmonotonicities arising from estimation noise and predictable jumps in the risk forecast as time passes. Our preferred specification, which we denote the heterogenous exponential realized volatility model (HExp for short), in particular, is based on a simple mixture of exponentially weighted moving average (EWMA) factors. Third, to account for the volatility spillover effects and strong commonalities observed not just across different assets but also across different geographical regions, we augment the asset-specific HExp model with a lagged “global” risk factor. Looking at the in-sample results, we find that all of the $$RV$$ models that we consider perform well compared to models that “only” use daily returns. By construction, when looking in-sample, the models that are tailored to each asset separately have larger predictive power in terms of $$R^2$$ than models that enforce a common risk model across assets. However, when looking at out-of-sample predictability, the models that impose common parameters generally perform better. In particular, enforcing common parameters across models within each asset class produces higher average out-of-sample $$R^2$$s than individually estimated models. Even more surprisingly, enforcing common parameters not just within but across all asset classes, the properly “centered” risk models result in even higher average out-of-sample $$R^2$$s. The basic HExp model and the HExp model with the “global” risk factor (termed “HExpGl”) result in the highest average out-of-sample predictability among all of the models, suggesting that the commonality and “smoothness” embedded in the HExp formulations ensure a robustness beyond that of the standard existing risk models. Last, but not least, we present a simple framework for quantifying the utility benefits of risk modeling. Our approach is linked to the literature that seeks to assess the utility benefits of return predictability in the presence of empirically realistic transaction costs and other practical implementation issues (see, e.g., Balduzzi and Lynch 1999; Sangvinatsos and Wachter 2005; Lynch and Tan 2010). It is also related to the work of Fleming, Kirby, and Ostdiek (2001, 2003), and the idea of using a quadratic utility function to evaluate the benefits of volatility timing.2 In contrast to all of these approaches that explicitly depend on forecasts of—and/or realizations of—both the future returns and volatilities, our method exclusively focuses on volatility forecasting. Specifically, we consider the expected utility of an investor with mean-variance preferences that trades an asset with a constant Sharpe ratio. The investor’s optimal portfolio adjusts the position size to keep a constant volatility (and this “risk target” naturally depends on the investor’s risk aversion). Correspondingly, the investor’s utility is directly related to the volatility: the investor achieves the maximum utility by successfully targeting a constant risk level, while the utility decreases with the volatility-of-volatility. Hence, risk models that help the investor achieve more accurate volatility forecasts are associated with higher levels of utility. We show that, in this situation, under realistic assumptions about the Sharpe ratio and the investor’s risk target, using the HExp risk model augmented with our global risk factor (HExpGl) is worth about 48 basis points (bps) per year relative to using the best possible static risk model. Put differently, the assumed investor would in principle pay 48 bps of her/his wealth each year to have access to the HExpGl risk model developed here rather than using a static risk model. The utility benefit of HExpGl is also significant, but smaller, when compared to other sophisticated risk models. For example, the utility gain of the HExpGl model relative to a simple risk model based on daily data is 19 bps per year and the utility gain over a 21-day rolling average over realized volatilities is 8 bps per year. These utility gains are of the same magnitude as institutional asset management fees. While such fees are often thought as compensation for higher expected returns (“alpha”), our results show that the quality of the risk model can be equally important. Importantly, these benefits remain when we take realistic transaction costs into account since the new “smooth” risk models not only produce more accurate risk forecasts but also more stable forecasts resulting in less spurious trading than risk models based on daily data. In summary, we contribute to the literature by exploring commonality in volatility across a broad set of asset classes, introducing a new class of risk models, showing how risk models can be centered to allow panel-based estimation, developing a utility-based framework to evaluate the economic importance of risk models, and, finally, combining all these to empirically showing that our HExp risk model estimated using our panel method produces statistical and economically significant gains relative to standard risk models. 1. Realized Volatilities: Data Sources and Construction There is a long history in finance of heuristically quantifying the ex post volatility based on the sum of intra-period squared returns.3 This approach may be formally justified by the theory of quadratic variation and the notion of ever-finer sampled returns over fixed time intervals, or so-called “in-fill asymptotic arguments” (see, e.g., the discussion and references in Andersen et al. 2013). 1.1 Realized volatilities and quadratic variation To formally lay out the basic idea underlying the realized volatility concept, we let the unit time interval correspond to a day. The realized variation defined by the summation of high-frequency intraday squared returns, \begin{equation}\label{eq:rv} RV_t ~ \equiv ~ \sum_{i=1}^{1/\Delta} \left[ log(P_{t-1+i\Delta})-log(P_{t-1+(i-1)\Delta}) \right]^2, \end{equation} (1) then consistently estimates the quadratic variation, that is, the true variation, on day $$t$$ as the number of intraday observations increases ($$1/\Delta \to \infty$$), or equivalently the length of the intraday return interval decreases ($$\Delta \to 0$$). This effectively renders the daily variation directly observable on an ex post basis.4 The volatility over longer, say weekly or monthly, horizons may similarly be estimated by summing the intraday squared returns over a week or a month or, equivalently, by summing the daily realized volatilities $$RV_t$$ over the relevant longer multiday horizons. 1.2 Data sources and “cleaning” Our data covers a total of 58 different assets across commodities (20), equities (21), fixed income (8), and foreign exchange (9). The asset universe comprises global equity index futures (both developed and emerging markets), global developed fixed-income futures, commodity futures, and spot market foreign exchange rates. Our specific choice of assets is dictated by liquidity concerns and correspondingly the availability of reliable high-frequency intraday prices. Our primary source of data for equities, fixed income, and commodities is the Thomson Reuters Tick History (TRTH) database. To extend the history for some of the assets, most notably fixed income and commodities, we use data from TickData.com (TDC). For the foreign exchange data, we exclusively rely on Olsen Data (OD). The data for all of the assets run through September 2014. The start of the sample period differs across assets, with some starting as early as October 1992. In general, the data for commodities are available the earliest, followed by equities and fixed income, with foreign exchange having the shortest time span. Table A1 in the appendix summarizes the exact start dates for all of the assets and the relevant data sources.5 Multiple futures contracts for the same underlying asset, but with different expiration, trade at the same time, and new contracts are opened as others expire. We focus on the most liquid contract, “rolling” from one contract to another at a regular schedule as further discussed in the appendix. We organize the resultant single time series of returns for each asset into minute bars based on the last observation prior to the end of each minute. For the TRTH and OD databases, we use the mid-quote price (average between the bid and ask price). For the TDC data we use the observed trade price (TDC does not provide quote-level data prior to 2010). To avoid “polluting” the high-frequency data with quote changes that occur during illiquid periods, we only use the minute bars for which there are at least one valid trade within that minute. Lastly, having organized all of the data into minute bars, we apply a series of “sanity filters” to clean out any obvious data errors. These filters are further discussed in the Appendix. Table A1 Data Sources Asset Class Asset Number of Assets Total Days in Analysis Primary Data Source Used From Secondary Data Source Used From Assumed T-Costs (in bps) COMMODITIES 20 108149 TRTH TDC Brent Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 1.0 Cattle 1 5483 TRTH 12/20/2004 TDC 11/30/1992 3.2 Cocoa 1 5471 TRTH 4/1/2008 TDC 11/11/1992 3.4 Coffee 1 5469 TRTH 4/1/2008 TDC 11/17/1992 8.0 Corn 1 5502 TRTH 8/1/2006 TDC 11/19/1992 5.7 Cotton 1 5453 TRTH 4/1/2008 TDC 11/12/1992 4.6 Crude (WTI) Oil 1 5480 TRTH 9/5/2006 TDC 11/10/1992 1.0 Feeder Cattle 1 5513 TRTH 8/1/2007 TDC 10/29/1992 4.5 Gas Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 2.8 Gold 1 5471 TRTH 12/4/2006 TDC 12/2/1992 0.8 Heating Oil 1 5480 TRTH 9/5/2006 TDC 11/16/1992 1.7 Lean Hogs 1 5486 TRTH 2/15/2005 TDC 11/30/1992 4.5 Natural Gass 1 5442 TRTH 8/23/2006 TDC 1/5/1993 4.0 Silver 1 5412 TRTH 12/4/2006 TDC 1/5/1993 2.6 Soybeans 1 5522 TRTH 8/1/2006 TDC 10/22/1992 2.1 Soymeal 1 5502 TRTH 8/1/2006 TDC 11/19/1992 4.1 Soyoil 1 5501 TRTH 8/1/2006 TDC 11/19/1992 3.0 Sugar 1 5481 TRTH 4/1/2008 TDC 11/3/1992 5.9 Unleaded (RBOB) 1 5475 TRTH 8/22/2006 TDC 11/16/1992 2.0 Wheat 1 5498 TRTH 8/1/2006 TDC 11/19/1992 4.4 EQUITIES 21 80042 TRTH NONE Australia (SPI 200) 1 3472 TRTH 12/18/2000 NA NA 1.9 Germany (DAX 30) 1 4732 TRTH 1/3/1996 NA NA 1.0 Brazil (BOVESPA) 1 4577 TRTH 2/27/1996 NA NA 2.8 China (Hang Seng CEI) 1 2667 TRTH 12/9/2003 NA NA 2.0 Canada (S&P/TSX 60) 1 3773 TRTH 9/14/1999 NA NA 1.3 Spain (IBEX 35) 1 4698 TRTH 1/4/1996 NA NA 2.0 Eurostoxx 1 4130 TRTH 6/23/1998 NA NA 3.2 France (CAC 40) 1 4007 TRTH 1/7/1999 NA NA 1.1 Hong Kong (Hang Seng) 1 4591 TRTH 1/3/1996 NA NA 1.2 India (SGX NIFTY) 1 2213 TRTH 10/11/2005 NA NA 1.7 Italy (FTSE MIB) 1 2617 TRTH 6/15/2004 NA NA 2.4 Japan (TOPIX) 1 4570 TRTH 1/5/1996 NA NA 4.1 South Korea (KOSPI 200) 1 4466 TRTH 5/6/1996 NA NA 1.9 Netherlands (AEX) 1 4499 TRTH 1/9/1997 NA NA 1.3 South Africa (ALSI) 1 2308 TRTH 7/7/2005 NA NA 1.7 Switzerland (SMI) 1 4027 TRTH 9/15/1998 NA NA 1.2 Taiwan (SGX-MSCI Taiwan) 1 4295 TRTH 2/24/1997 NA NA 3.1 UK (FTSE 100) 1 4706 TRTH 1/3/1996 NA NA 0.8 US (S&P 500 E-Mini) 1 4274 TRTH 9/10/1997 NA NA 1.3 US (Russell 2000 E-Mini) 1 2234 TRTH 12/13/2005 NA NA 0.9 US (S&P 400 Mid Cap E-Mini) 1 3186 TRTH 1/29/2002 NA NA 1.5 FIXED INCOME 8 32333 TRTH TDC Australia 10y 1 4734 TRTH 1/3/1996 TDC NA 3.9 Germany 10y 1 4499 TRTH 1/5/1999 TDC 1/3/1997 0.7 Germany 5y 1 4493 TRTH 2/1/1999 TDC 1/3/1997 0.8 Canada 10y 1 2771 TRTH 9/26/2000 TDC NA 0.8 Japan 10y 1 3605 TRTH 1/5/1996 TDC NA 0.7 UK 10y 1 4711 TRTH 1/3/1996 TDC NA 0.9 US 10y 1 3993 TRTH 1/1/2001 TDC 10/20/1998 1.3 US 5y 1 3527 TRTH 7/1/2001 TDC 9/5/2000 0.7 FOREIGN EXCHANGE 9 30161 Olsen Data NONE Australia (AUD-USD) 1 2802 OlsenData 1/1/2004 NA NA 2.2 Eurozone (EUR-USD) 1 4103 OlsenData 1/1/1999 NA NA 0.7 Canada (USD-CAD) 1 3061 OlsenData 1/1/2003 NA NA 2.5 Japan (USD-JPY) 1 3841 OlsenData 1/1/2000 NA NA 1.0 Norway (USD-NOK) 1 2801 OlsenData 1/1/2004 NA NA 8.1 New Zealand (NZD-USD) 1 2803 OlsenData 1/1/2004 NA NA 4.7 Sweden (USD-SEK) 1 3062 OlsenData 1/1/2003 NA NA 7.7 Switzerland (USD-CHF) 1 3844 OlsenData 1/1/2000 NA NA 1.7 UK (GBP-USD) 1 3844 OlsenData 1/1/2000 NA NA 1.5 Asset Class Asset Number of Assets Total Days in Analysis Primary Data Source Used From Secondary Data Source Used From Assumed T-Costs (in bps) COMMODITIES 20 108149 TRTH TDC Brent Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 1.0 Cattle 1 5483 TRTH 12/20/2004 TDC 11/30/1992 3.2 Cocoa 1 5471 TRTH 4/1/2008 TDC 11/11/1992 3.4 Coffee 1 5469 TRTH 4/1/2008 TDC 11/17/1992 8.0 Corn 1 5502 TRTH 8/1/2006 TDC 11/19/1992 5.7 Cotton 1 5453 TRTH 4/1/2008 TDC 11/12/1992 4.6 Crude (WTI) Oil 1 5480 TRTH 9/5/2006 TDC 11/10/1992 1.0 Feeder Cattle 1 5513 TRTH 8/1/2007 TDC 10/29/1992 4.5 Gas Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 2.8 Gold 1 5471 TRTH 12/4/2006 TDC 12/2/1992 0.8 Heating Oil 1 5480 TRTH 9/5/2006 TDC 11/16/1992 1.7 Lean Hogs 1 5486 TRTH 2/15/2005 TDC 11/30/1992 4.5 Natural Gass 1 5442 TRTH 8/23/2006 TDC 1/5/1993 4.0 Silver 1 5412 TRTH 12/4/2006 TDC 1/5/1993 2.6 Soybeans 1 5522 TRTH 8/1/2006 TDC 10/22/1992 2.1 Soymeal 1 5502 TRTH 8/1/2006 TDC 11/19/1992 4.1 Soyoil 1 5501 TRTH 8/1/2006 TDC 11/19/1992 3.0 Sugar 1 5481 TRTH 4/1/2008 TDC 11/3/1992 5.9 Unleaded (RBOB) 1 5475 TRTH 8/22/2006 TDC 11/16/1992 2.0 Wheat 1 5498 TRTH 8/1/2006 TDC 11/19/1992 4.4 EQUITIES 21 80042 TRTH NONE Australia (SPI 200) 1 3472 TRTH 12/18/2000 NA NA 1.9 Germany (DAX 30) 1 4732 TRTH 1/3/1996 NA NA 1.0 Brazil (BOVESPA) 1 4577 TRTH 2/27/1996 NA NA 2.8 China (Hang Seng CEI) 1 2667 TRTH 12/9/2003 NA NA 2.0 Canada (S&P/TSX 60) 1 3773 TRTH 9/14/1999 NA NA 1.3 Spain (IBEX 35) 1 4698 TRTH 1/4/1996 NA NA 2.0 Eurostoxx 1 4130 TRTH 6/23/1998 NA NA 3.2 France (CAC 40) 1 4007 TRTH 1/7/1999 NA NA 1.1 Hong Kong (Hang Seng) 1 4591 TRTH 1/3/1996 NA NA 1.2 India (SGX NIFTY) 1 2213 TRTH 10/11/2005 NA NA 1.7 Italy (FTSE MIB) 1 2617 TRTH 6/15/2004 NA NA 2.4 Japan (TOPIX) 1 4570 TRTH 1/5/1996 NA NA 4.1 South Korea (KOSPI 200) 1 4466 TRTH 5/6/1996 NA NA 1.9 Netherlands (AEX) 1 4499 TRTH 1/9/1997 NA NA 1.3 South Africa (ALSI) 1 2308 TRTH 7/7/2005 NA NA 1.7 Switzerland (SMI) 1 4027 TRTH 9/15/1998 NA NA 1.2 Taiwan (SGX-MSCI Taiwan) 1 4295 TRTH 2/24/1997 NA NA 3.1 UK (FTSE 100) 1 4706 TRTH 1/3/1996 NA NA 0.8 US (S&P 500 E-Mini) 1 4274 TRTH 9/10/1997 NA NA 1.3 US (Russell 2000 E-Mini) 1 2234 TRTH 12/13/2005 NA NA 0.9 US (S&P 400 Mid Cap E-Mini) 1 3186 TRTH 1/29/2002 NA NA 1.5 FIXED INCOME 8 32333 TRTH TDC Australia 10y 1 4734 TRTH 1/3/1996 TDC NA 3.9 Germany 10y 1 4499 TRTH 1/5/1999 TDC 1/3/1997 0.7 Germany 5y 1 4493 TRTH 2/1/1999 TDC 1/3/1997 0.8 Canada 10y 1 2771 TRTH 9/26/2000 TDC NA 0.8 Japan 10y 1 3605 TRTH 1/5/1996 TDC NA 0.7 UK 10y 1 4711 TRTH 1/3/1996 TDC NA 0.9 US 10y 1 3993 TRTH 1/1/2001 TDC 10/20/1998 1.3 US 5y 1 3527 TRTH 7/1/2001 TDC 9/5/2000 0.7 FOREIGN EXCHANGE 9 30161 Olsen Data NONE Australia (AUD-USD) 1 2802 OlsenData 1/1/2004 NA NA 2.2 Eurozone (EUR-USD) 1 4103 OlsenData 1/1/1999 NA NA 0.7 Canada (USD-CAD) 1 3061 OlsenData 1/1/2003 NA NA 2.5 Japan (USD-JPY) 1 3841 OlsenData 1/1/2000 NA NA 1.0 Norway (USD-NOK) 1 2801 OlsenData 1/1/2004 NA NA 8.1 New Zealand (NZD-USD) 1 2803 OlsenData 1/1/2004 NA NA 4.7 Sweden (USD-SEK) 1 3062 OlsenData 1/1/2003 NA NA 7.7 Switzerland (USD-CHF) 1 3844 OlsenData 1/1/2000 NA NA 1.7 UK (GBP-USD) 1 3844 OlsenData 1/1/2000 NA NA 1.5 Table A1 Data Sources Asset Class Asset Number of Assets Total Days in Analysis Primary Data Source Used From Secondary Data Source Used From Assumed T-Costs (in bps) COMMODITIES 20 108149 TRTH TDC Brent Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 1.0 Cattle 1 5483 TRTH 12/20/2004 TDC 11/30/1992 3.2 Cocoa 1 5471 TRTH 4/1/2008 TDC 11/11/1992 3.4 Coffee 1 5469 TRTH 4/1/2008 TDC 11/17/1992 8.0 Corn 1 5502 TRTH 8/1/2006 TDC 11/19/1992 5.7 Cotton 1 5453 TRTH 4/1/2008 TDC 11/12/1992 4.6 Crude (WTI) Oil 1 5480 TRTH 9/5/2006 TDC 11/10/1992 1.0 Feeder Cattle 1 5513 TRTH 8/1/2007 TDC 10/29/1992 4.5 Gas Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 2.8 Gold 1 5471 TRTH 12/4/2006 TDC 12/2/1992 0.8 Heating Oil 1 5480 TRTH 9/5/2006 TDC 11/16/1992 1.7 Lean Hogs 1 5486 TRTH 2/15/2005 TDC 11/30/1992 4.5 Natural Gass 1 5442 TRTH 8/23/2006 TDC 1/5/1993 4.0 Silver 1 5412 TRTH 12/4/2006 TDC 1/5/1993 2.6 Soybeans 1 5522 TRTH 8/1/2006 TDC 10/22/1992 2.1 Soymeal 1 5502 TRTH 8/1/2006 TDC 11/19/1992 4.1 Soyoil 1 5501 TRTH 8/1/2006 TDC 11/19/1992 3.0 Sugar 1 5481 TRTH 4/1/2008 TDC 11/3/1992 5.9 Unleaded (RBOB) 1 5475 TRTH 8/22/2006 TDC 11/16/1992 2.0 Wheat 1 5498 TRTH 8/1/2006 TDC 11/19/1992 4.4 EQUITIES 21 80042 TRTH NONE Australia (SPI 200) 1 3472 TRTH 12/18/2000 NA NA 1.9 Germany (DAX 30) 1 4732 TRTH 1/3/1996 NA NA 1.0 Brazil (BOVESPA) 1 4577 TRTH 2/27/1996 NA NA 2.8 China (Hang Seng CEI) 1 2667 TRTH 12/9/2003 NA NA 2.0 Canada (S&P/TSX 60) 1 3773 TRTH 9/14/1999 NA NA 1.3 Spain (IBEX 35) 1 4698 TRTH 1/4/1996 NA NA 2.0 Eurostoxx 1 4130 TRTH 6/23/1998 NA NA 3.2 France (CAC 40) 1 4007 TRTH 1/7/1999 NA NA 1.1 Hong Kong (Hang Seng) 1 4591 TRTH 1/3/1996 NA NA 1.2 India (SGX NIFTY) 1 2213 TRTH 10/11/2005 NA NA 1.7 Italy (FTSE MIB) 1 2617 TRTH 6/15/2004 NA NA 2.4 Japan (TOPIX) 1 4570 TRTH 1/5/1996 NA NA 4.1 South Korea (KOSPI 200) 1 4466 TRTH 5/6/1996 NA NA 1.9 Netherlands (AEX) 1 4499 TRTH 1/9/1997 NA NA 1.3 South Africa (ALSI) 1 2308 TRTH 7/7/2005 NA NA 1.7 Switzerland (SMI) 1 4027 TRTH 9/15/1998 NA NA 1.2 Taiwan (SGX-MSCI Taiwan) 1 4295 TRTH 2/24/1997 NA NA 3.1 UK (FTSE 100) 1 4706 TRTH 1/3/1996 NA NA 0.8 US (S&P 500 E-Mini) 1 4274 TRTH 9/10/1997 NA NA 1.3 US (Russell 2000 E-Mini) 1 2234 TRTH 12/13/2005 NA NA 0.9 US (S&P 400 Mid Cap E-Mini) 1 3186 TRTH 1/29/2002 NA NA 1.5 FIXED INCOME 8 32333 TRTH TDC Australia 10y 1 4734 TRTH 1/3/1996 TDC NA 3.9 Germany 10y 1 4499 TRTH 1/5/1999 TDC 1/3/1997 0.7 Germany 5y 1 4493 TRTH 2/1/1999 TDC 1/3/1997 0.8 Canada 10y 1 2771 TRTH 9/26/2000 TDC NA 0.8 Japan 10y 1 3605 TRTH 1/5/1996 TDC NA 0.7 UK 10y 1 4711 TRTH 1/3/1996 TDC NA 0.9 US 10y 1 3993 TRTH 1/1/2001 TDC 10/20/1998 1.3 US 5y 1 3527 TRTH 7/1/2001 TDC 9/5/2000 0.7 FOREIGN EXCHANGE 9 30161 Olsen Data NONE Australia (AUD-USD) 1 2802 OlsenData 1/1/2004 NA NA 2.2 Eurozone (EUR-USD) 1 4103 OlsenData 1/1/1999 NA NA 0.7 Canada (USD-CAD) 1 3061 OlsenData 1/1/2003 NA NA 2.5 Japan (USD-JPY) 1 3841 OlsenData 1/1/2000 NA NA 1.0 Norway (USD-NOK) 1 2801 OlsenData 1/1/2004 NA NA 8.1 New Zealand (NZD-USD) 1 2803 OlsenData 1/1/2004 NA NA 4.7 Sweden (USD-SEK) 1 3062 OlsenData 1/1/2003 NA NA 7.7 Switzerland (USD-CHF) 1 3844 OlsenData 1/1/2000 NA NA 1.7 UK (GBP-USD) 1 3844 OlsenData 1/1/2000 NA NA 1.5 Asset Class Asset Number of Assets Total Days in Analysis Primary Data Source Used From Secondary Data Source Used From Assumed T-Costs (in bps) COMMODITIES 20 108149 TRTH TDC Brent Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 1.0 Cattle 1 5483 TRTH 12/20/2004 TDC 11/30/1992 3.2 Cocoa 1 5471 TRTH 4/1/2008 TDC 11/11/1992 3.4 Coffee 1 5469 TRTH 4/1/2008 TDC 11/17/1992 8.0 Corn 1 5502 TRTH 8/1/2006 TDC 11/19/1992 5.7 Cotton 1 5453 TRTH 4/1/2008 TDC 11/12/1992 4.6 Crude (WTI) Oil 1 5480 TRTH 9/5/2006 TDC 11/10/1992 1.0 Feeder Cattle 1 5513 TRTH 8/1/2007 TDC 10/29/1992 4.5 Gas Oil 1 4754 TRTH 1/3/1996 TDC 1/3/1996 2.8 Gold 1 5471 TRTH 12/4/2006 TDC 12/2/1992 0.8 Heating Oil 1 5480 TRTH 9/5/2006 TDC 11/16/1992 1.7 Lean Hogs 1 5486 TRTH 2/15/2005 TDC 11/30/1992 4.5 Natural Gass 1 5442 TRTH 8/23/2006 TDC 1/5/1993 4.0 Silver 1 5412 TRTH 12/4/2006 TDC 1/5/1993 2.6 Soybeans 1 5522 TRTH 8/1/2006 TDC 10/22/1992 2.1 Soymeal 1 5502 TRTH 8/1/2006 TDC 11/19/1992 4.1 Soyoil 1 5501 TRTH 8/1/2006 TDC 11/19/1992 3.0 Sugar 1 5481 TRTH 4/1/2008 TDC 11/3/1992 5.9 Unleaded (RBOB) 1 5475 TRTH 8/22/2006 TDC 11/16/1992 2.0 Wheat 1 5498 TRTH 8/1/2006 TDC 11/19/1992 4.4 EQUITIES 21 80042 TRTH NONE Australia (SPI 200) 1 3472 TRTH 12/18/2000 NA NA 1.9 Germany (DAX 30) 1 4732 TRTH 1/3/1996 NA NA 1.0 Brazil (BOVESPA) 1 4577 TRTH 2/27/1996 NA NA 2.8 China (Hang Seng CEI) 1 2667 TRTH 12/9/2003 NA NA 2.0 Canada (S&P/TSX 60) 1 3773 TRTH 9/14/1999 NA NA 1.3 Spain (IBEX 35) 1 4698 TRTH 1/4/1996 NA NA 2.0 Eurostoxx 1 4130 TRTH 6/23/1998 NA NA 3.2 France (CAC 40) 1 4007 TRTH 1/7/1999 NA NA 1.1 Hong Kong (Hang Seng) 1 4591 TRTH 1/3/1996 NA NA 1.2 India (SGX NIFTY) 1 2213 TRTH 10/11/2005 NA NA 1.7 Italy (FTSE MIB) 1 2617 TRTH 6/15/2004 NA NA 2.4 Japan (TOPIX) 1 4570 TRTH 1/5/1996 NA NA 4.1 South Korea (KOSPI 200) 1 4466 TRTH 5/6/1996 NA NA 1.9 Netherlands (AEX) 1 4499 TRTH 1/9/1997 NA NA 1.3 South Africa (ALSI) 1 2308 TRTH 7/7/2005 NA NA 1.7 Switzerland (SMI) 1 4027 TRTH 9/15/1998 NA NA 1.2 Taiwan (SGX-MSCI Taiwan) 1 4295 TRTH 2/24/1997 NA NA 3.1 UK (FTSE 100) 1 4706 TRTH 1/3/1996 NA NA 0.8 US (S&P 500 E-Mini) 1 4274 TRTH 9/10/1997 NA NA 1.3 US (Russell 2000 E-Mini) 1 2234 TRTH 12/13/2005 NA NA 0.9 US (S&P 400 Mid Cap E-Mini) 1 3186 TRTH 1/29/2002 NA NA 1.5 FIXED INCOME 8 32333 TRTH TDC Australia 10y 1 4734 TRTH 1/3/1996 TDC NA 3.9 Germany 10y 1 4499 TRTH 1/5/1999 TDC 1/3/1997 0.7 Germany 5y 1 4493 TRTH 2/1/1999 TDC 1/3/1997 0.8 Canada 10y 1 2771 TRTH 9/26/2000 TDC NA 0.8 Japan 10y 1 3605 TRTH 1/5/1996 TDC NA 0.7 UK 10y 1 4711 TRTH 1/3/1996 TDC NA 0.9 US 10y 1 3993 TRTH 1/1/2001 TDC 10/20/1998 1.3 US 5y 1 3527 TRTH 7/1/2001 TDC 9/5/2000 0.7 FOREIGN EXCHANGE 9 30161 Olsen Data NONE Australia (AUD-USD) 1 2802 OlsenData 1/1/2004 NA NA 2.2 Eurozone (EUR-USD) 1 4103 OlsenData 1/1/1999 NA NA 0.7 Canada (USD-CAD) 1 3061 OlsenData 1/1/2003 NA NA 2.5 Japan (USD-JPY) 1 3841 OlsenData 1/1/2000 NA NA 1.0 Norway (USD-NOK) 1 2801 OlsenData 1/1/2004 NA NA 8.1 New Zealand (NZD-USD) 1 2803 OlsenData 1/1/2004 NA NA 4.7 Sweden (USD-SEK) 1 3062 OlsenData 1/1/2003 NA NA 7.7 Switzerland (USD-CHF) 1 3844 OlsenData 1/1/2000 NA NA 1.7 UK (GBP-USD) 1 3844 OlsenData 1/1/2000 NA NA 1.5 1.3 Overnight returns and intraday sampling It is well established that volatility tend to be higher during exchange trading hours than during nontrading hours (see, e.g., French and Roll 1986). The theory underlying the consistency of the realized volatility measure portrays prices as evolving continuously through time. In actuality, of course, most markets close on weekends and certain holidays, change their trading hours, and sometimes experience “ghost” hours, where liquidity is very poor despite the markets technically being open. Accordingly, we only retain the trading hours for which the liquidity is sufficiently high to ensure a reasonable quality of the high-frequency data. We use the Financial Calendars (FinCal) database for market open and close times, together with so-called “liquidity plots,” to delineate the periods of actively operating markets; the appendix provides further details. Having determined the period for which reliable high-frequency data are available, we simply add the corresponding “overnight” squared returns to the daily realized volatilities constructed from the “intraday” squared returns to obtain an $$RV$$ measure for the whole day.6 We rely on a common 5-minute sampling frequency for calculating the intraday $$RV$$s for all of the assets. This choice directly mirrors the sampling frequency used in much of the existing realized volatility literature. It may be justified by the volatility signature plots (Andersen et al., 2000) discussed in the appendix. To further enhance the efficiency of the $$RV$$ estimates, we average the five different daily $$RV$$s obtained by starting the day at the first five unique 1-minute marks. A number of other consistent realized volatility estimators, requiring the choice of additional tuning and/or nuisance parameters, have been proposed in the literature.7 However, the theoretical comparisons in Andersen, Bollerslev, and Meddahi (2011) and Ghysels and Sinko (2011) show that from a theoretical forecasting perspective, the simple subsampled 5-minute $$RV$$ estimator that we rely on here performs on par with or better than all of these more complicated estimators. The empirical study by Liu, Patton, and Sheppard (2015), comparing more than 400 different $$RV$$ estimators across multiple assets, similarly concludes that “it is difficult to significantly beat 5-minute $$RV$$.” 2. Risk Characteristics Everywhere To help guide the specification of empirically realistic risk models, we want to understand the distributional characteristics of the risks both within each of the four asset classes, equities, bonds, commodities, and currencies, as well as the similarities and differences across asset classes. 2.1 Unconditional distributions and dynamic dependencies To begin, Figure 1 shows the time series of annualized realized volatilities for four representative assets, one from each asset class: S&P 500, 10-year Treasury bonds, Crude Oil, and Dollar/Euro. Even though the four volatilities obviously exhibit their own distinct behaviors, there is a clear commonality in the dynamic patterns observed across the four assets, with most of the peaks readily associated with specific economic events. Figure 1 View largeDownload slide Monthly realized volatilities This figure shows the time series of 20-day average realized volatilities (annualized) for four representative assets: S&P 500 Futures, U.S. 10-year Bond Futures, Crude Oil (WTI) Futures, and USD/Euro Spot. Figure 1 View largeDownload slide Monthly realized volatilities This figure shows the time series of 20-day average realized volatilities (annualized) for four representative assets: S&P 500 Futures, U.S. 10-year Bond Futures, Crude Oil (WTI) Futures, and USD/Euro Spot. In spite of the similarities in the general patterns, the overall levels of the volatilities clearly differ across the four different assets. This is further evidenced by Figure 2, which plots the unconditional distribution of the daily realized volatilities for the same four representative assets. As the figure shows, Crude Oil is the most volatile on average, followed by the S&P 500, and the Dollar/Euro exchange rate. The volatility of 10-year Treasury bonds is by far the lowest. Figure 2 View largeDownload slide Unconditional daily $$RV$$ distributions This figure shows kernel density estimates of the unconditional daily realized volatility (annualized) for four representative assets: S&P 500 Futures, U.S. 10-year Bond Futures, Crude Oil (WTI) Futures, and USD/Euro Spot. Figure 2 View largeDownload slide Unconditional daily $$RV$$ distributions This figure shows kernel density estimates of the unconditional daily realized volatility (annualized) for four representative assets: S&P 500 Futures, U.S. 10-year Bond Futures, Crude Oil (WTI) Futures, and USD/Euro Spot. This same ranking carries over to the four asset classes more generally. In particular, looking at the summary statistics for the daily realized volatilities averaged across each of the assets within each of the four asset classes reported in Table 1, the average annualized volatility for commodities and equities equal 25.4% and 20.6%, respectively, compared to 10.3% for foreign exchange, and just 3.1% for fixed income. Table 1 $$RV$$ summary statistics Commodities Equities Fixed income Foreign exchange Mean 25.4 20.6 3.1 10.3 SD 12.6 13.7 1.5 5.7 Skewness 2.6 3.4 2.3 3.1 Excess kurtosis 16.9 22.9 11.6 18.5 Maximum 185.6 186.6 19.4 74.1 95th percentile 47.8 44.8 5.8 20.4 50th percentile 22.7 17.0 2.8 9.0 5th percentile 11.6 8.2 1.5 4.6 Minimum 4.9 3.0 0.6 1.2 1-day autocorr. 0.516 0.707 0.481 0.517 20-day autocorr. 0.362 0.480 0.347 0.415 100-day autocorr. 0.195 0.228 0.197 0.221 250-day autocorr. 0.115 0.105 0.073 0.104 Number of assets 20 21 8 9 Avg. number of obs. 5,407 3,812 4,042 3,351 Earliest start date Oct. 22, 1992 Jan. 3, 1996 Jan. 3, 1996 Jan. 1, 1999 Latest start date Jan. 3, 1996 Dec. 13, 2005 Sept. 26, 2000 Jan. 1, 2004 Commodities Equities Fixed income Foreign exchange Mean 25.4 20.6 3.1 10.3 SD 12.6 13.7 1.5 5.7 Skewness 2.6 3.4 2.3 3.1 Excess kurtosis 16.9 22.9 11.6 18.5 Maximum 185.6 186.6 19.4 74.1 95th percentile 47.8 44.8 5.8 20.4 50th percentile 22.7 17.0 2.8 9.0 5th percentile 11.6 8.2 1.5 4.6 Minimum 4.9 3.0 0.6 1.2 1-day autocorr. 0.516 0.707 0.481 0.517 20-day autocorr. 0.362 0.480 0.347 0.415 100-day autocorr. 0.195 0.228 0.197 0.221 250-day autocorr. 0.115 0.105 0.073 0.104 Number of assets 20 21 8 9 Avg. number of obs. 5,407 3,812 4,042 3,351 Earliest start date Oct. 22, 1992 Jan. 3, 1996 Jan. 3, 1996 Jan. 1, 1999 Latest start date Jan. 3, 1996 Dec. 13, 2005 Sept. 26, 2000 Jan. 1, 2004 The table presents summary statistics for daily realized volatilities averaged across all assets within a given asset class. All of the volatility numbers are reported in annualized percentage units. Table 1 $$RV$$ summary statistics Commodities Equities Fixed income Foreign exchange Mean 25.4 20.6 3.1 10.3 SD 12.6 13.7 1.5 5.7 Skewness 2.6 3.4 2.3 3.1 Excess kurtosis 16.9 22.9 11.6 18.5 Maximum 185.6 186.6 19.4 74.1 95th percentile 47.8 44.8 5.8 20.4 50th percentile 22.7 17.0 2.8 9.0 5th percentile 11.6 8.2 1.5 4.6 Minimum 4.9 3.0 0.6 1.2 1-day autocorr. 0.516 0.707 0.481 0.517 20-day autocorr. 0.362 0.480 0.347 0.415 100-day autocorr. 0.195 0.228 0.197 0.221 250-day autocorr. 0.115 0.105 0.073 0.104 Number of assets 20 21 8 9 Avg. number of obs. 5,407 3,812 4,042 3,351 Earliest start date Oct. 22, 1992 Jan. 3, 1996 Jan. 3, 1996 Jan. 1, 1999 Latest start date Jan. 3, 1996 Dec. 13, 2005 Sept. 26, 2000 Jan. 1, 2004 Commodities Equities Fixed income Foreign exchange Mean 25.4 20.6 3.1 10.3 SD 12.6 13.7 1.5 5.7 Skewness 2.6 3.4 2.3 3.1 Excess kurtosis 16.9 22.9 11.6 18.5 Maximum 185.6 186.6 19.4 74.1 95th percentile 47.8 44.8 5.8 20.4 50th percentile 22.7 17.0 2.8 9.0 5th percentile 11.6 8.2 1.5 4.6 Minimum 4.9 3.0 0.6 1.2 1-day autocorr. 0.516 0.707 0.481 0.517 20-day autocorr. 0.362 0.480 0.347 0.415 100-day autocorr. 0.195 0.228 0.197 0.221 250-day autocorr. 0.115 0.105 0.073 0.104 Number of assets 20 21 8 9 Avg. number of obs. 5,407 3,812 4,042 3,351 Earliest start date Oct. 22, 1992 Jan. 3, 1996 Jan. 3, 1996 Jan. 1, 1999 Latest start date Jan. 3, 1996 Dec. 13, 2005 Sept. 26, 2000 Jan. 1, 2004 The table presents summary statistics for daily realized volatilities averaged across all assets within a given asset class. All of the volatility numbers are reported in annualized percentage units. These differences in the mean levels of the volatilities across the different assets and asset classes are, of course, well know. More interesting features arise when we consider the volatilities normalized by their sample mean, $$RV_t/Mean(RV_t)$$, corresponding to the risk of a leveraged (or deleveraged) position. For example, if $$Mean(RV_t)=0.5$$, then the normalized volatility measure corresponds to the risk of a position that is always leveraged 2-to-1. Hence, normalizing each individual contract by its own average volatility is equivalent to measuring risk on a common scale, in the sense that each position is leveraged to the same common average risk level. Interestingly, the unconditional distributions of these daily normalized realized volatilities are remarkably similar, both across assets and asset classes. Indeed, Figure 3 shows that the sampling distributions of the four representative assets are obviously very close.8 The unconditional distributions for the normalized volatility of each of the individual assets within each of the four asset classes are also very similar, as reported in the Online Appendix. Figure 3 View largeDownload slide Normalized unconditional daily $$RV$$ distributions This figure shows kernel density estimates of the normalized daily realized volatility (annualized) of all assets within each of the four asset classes. In particular, we show the density of each asset’s daily RV divided by its average daily RV to account for level differences in individual asset volatilities. Figure 3 View largeDownload slide Normalized unconditional daily $$RV$$ distributions This figure shows kernel density estimates of the normalized daily realized volatility (annualized) of all assets within each of the four asset classes. In particular, we show the density of each asset’s daily RV divided by its average daily RV to account for level differences in individual asset volatilities. It is important to note that the “width” of these distributions are not similar by construction, as would be the case if we normalized by subtracting the mean and dividing by the standard deviation, $$[RV_t-Mean(RV_t)]/std(RV_t)$$, which would match both the mean and standard deviation by construction (or, said differently, such a normalization implies a loss of two degrees of freedom, rather than just one for our normalization). Hence, the fact that the $$RV_t/Mean(RV_t)$$ normalized distributions are so similar is not hard-wired, but rather direct evidence that risks do indeed behave similarly across asset classes. Importantly, these similarities also imply that risk parity strategies designed to match the average volatility of different assets and/or asset classes will not only equate the average volatility levels but also effectively the entire unconditional distributions of the ex post realized volatilities for the leveraged positions. These commonalities in the unconditional distributions carry over to the general dynamic dependencies. The autocorrelations for the daily realized volatilities averaged across the different assets within each of the four asset classes shown in Table 1 and Figure 4 make clear that the general patterns and decay rates are very similar. This is, again, in line with the equities perhaps being slightly different for the short-term lags.9 Similar dynamic dependencies, of course, have been extensively documented in the burgeon volatility literature.10 Figure 4 View largeDownload slide Daily realized volatility autocorrelations This figure shows the average autocorrelation function of the daily realized volatilities averaged across all of the assets within each of the four asset classes. Figure 4 View largeDownload slide Daily realized volatility autocorrelations This figure shows the average autocorrelation function of the daily realized volatilities averaged across all of the assets within each of the four asset classes. 2.2 Cross-asset dependencies, spillovers, and global volatility The averages of the standard sample correlations for the realized volatilities reported in the top panel in Table 2 are all positive. This comovement of risk across assets and asset classes is consistent with the visual impression from the time-series plots for the four representative assets previously discussed in Figure 1. Looking at the actual numbers, commodity volatilities are generally the least correlated, both within the asset class and across other asset classes. That is, the risk of different commodities comove less with each other than do equity risks that comove with each other, and so on, and commodity risks also comove less with the risk of equity, fixed income, and currencies than they comove with each other. In fact, commodity risk comoves about as much across asset classes as within the asset class. Table 2 $$RV$$ contemporaneous and partial correlations Commodities Equities Fixed income Foreign exchange A. Contemporaneous correlations Commodities 0.277 0.298 0.217 0.355 Equities – 0.668 0.407 0.554 Fixed income – – 0.470 0.433 Foreign exchange – – – 0.710 Global 0.410 0.721 0.547 0.633 B. Partial lead-lag correlations and $$\Delta R^2$$ Commodities 0.059 0.063 0.074 0.132 Equities 0.081 0.173 0.122 0.198 Fixed income 0.072 0.078 0.093 0.151 Foreign exchange 0.078 0.108 0.110 0.144 Global 0.171 0.328 0.267 0.456 Commodities 0.005 0.004 0.007 0.017 Equities 0.011 0.021 0.024 0.041 Fixed income 0.006 0.005 0.011 0.016 Foreign exchange 0.007 0.010 0.015 0.014 Global 0.012 0.022 0.033 0.061 Commodities Equities Fixed income Foreign exchange A. Contemporaneous correlations Commodities 0.277 0.298 0.217 0.355 Equities – 0.668 0.407 0.554 Fixed income – – 0.470 0.433 Foreign exchange – – – 0.710 Global 0.410 0.721 0.547 0.633 B. Partial lead-lag correlations and $$\Delta R^2$$ Commodities 0.059 0.063 0.074 0.132 Equities 0.081 0.173 0.122 0.198 Fixed income 0.072 0.078 0.093 0.151 Foreign exchange 0.078 0.108 0.110 0.144 Global 0.171 0.328 0.267 0.456 Commodities 0.005 0.004 0.007 0.017 Equities 0.011 0.021 0.024 0.041 Fixed income 0.006 0.005 0.011 0.016 Foreign exchange 0.007 0.010 0.015 0.014 Global 0.012 0.022 0.033 0.061 The table presents correlations of daily realized volatilities averaged within and across asset classes. Panel A reports the standard contemporaneous correlations. Panel B reports the average partial correlation coefficients obtained from regressions of the daily $$RV$$s for the assets within the asset classes indicated in the columns on their own daily lags and the lagged values of the RVs for the assets in the asset classes indicated in the rows (top panel), together with the average absolute percentage increases in the $$R^2$$s compared to regressions of the assets indicated in the columns on their own daily lags only (bottom panel). The “Global” volatility factor is constructed as a weighted average of the daily $$RV$$s, as further discussed in the main text. Table 2 $$RV$$ contemporaneous and partial correlations Commodities Equities Fixed income Foreign exchange A. Contemporaneous correlations Commodities 0.277 0.298 0.217 0.355 Equities – 0.668 0.407 0.554 Fixed income – – 0.470 0.433 Foreign exchange – – – 0.710 Global 0.410 0.721 0.547 0.633 B. Partial lead-lag correlations and $$\Delta R^2$$ Commodities 0.059 0.063 0.074 0.132 Equities 0.081 0.173 0.122 0.198 Fixed income 0.072 0.078 0.093 0.151 Foreign exchange 0.078 0.108 0.110 0.144 Global 0.171 0.328 0.267 0.456 Commodities 0.005 0.004 0.007 0.017 Equities 0.011 0.021 0.024 0.041 Fixed income 0.006 0.005 0.011 0.016 Foreign exchange 0.007 0.010 0.015 0.014 Global 0.012 0.022 0.033 0.061 Commodities Equities Fixed income Foreign exchange A. Contemporaneous correlations Commodities 0.277 0.298 0.217 0.355 Equities – 0.668 0.407 0.554 Fixed income – – 0.470 0.433 Foreign exchange – – – 0.710 Global 0.410 0.721 0.547 0.633 B. Partial lead-lag correlations and $$\Delta R^2$$ Commodities 0.059 0.063 0.074 0.132 Equities 0.081 0.173 0.122 0.198 Fixed income 0.072 0.078 0.093 0.151 Foreign exchange 0.078 0.108 0.110 0.144 Global 0.171 0.328 0.267 0.456 Commodities 0.005 0.004 0.007 0.017 Equities 0.011 0.021 0.024 0.041 Fixed income 0.006 0.005 0.011 0.016 Foreign exchange 0.007 0.010 0.015 0.014 Global 0.012 0.022 0.033 0.061 The table presents correlations of daily realized volatilities averaged within and across asset classes. Panel A reports the standard contemporaneous correlations. Panel B reports the average partial correlation coefficients obtained from regressions of the daily $$RV$$s for the assets within the asset classes indicated in the columns on their own daily lags and the lagged values of the RVs for the assets in the asset classes indicated in the rows (top panel), together with the average absolute percentage increases in the $$R^2$$s compared to regressions of the assets indicated in the columns on their own daily lags only (bottom panel). The “Global” volatility factor is constructed as a weighted average of the daily $$RV$$s, as further discussed in the main text. In addition to the within and across asses class correlations, the last row reports the average correlations with a “global risk factor” that we denote $$GlRV$$. The global risk factor is defined as the average normalized $$RV$$s across all assets. Since we will also use $$GlRV$$ for forecasting, we construct this factor on an asset-specific basis to prevent any look-ahead biases due to time-zone effects. In particular, for any specific asset, we construct the corresponding $$GlRV$$ so that today’s $$GlRV$$ does not use any data that overlap with tomorrow’s trading hours for the specific asset, lagging by 1 day any asset that would otherwise create such an overlap.11 Based on this lagging convention, on each day and for each asset $$i$$, we compute the asset-specific $$GlRV$$ as the average normalized $$RV$$ scaled back to the asset’s own level of volatility; that is, $$\left(\frac{1}{J}\sum_{j=1,...,J}\frac{RV_{t,j}}{ \overline{RV_j}}\right)\overline{RV_i}$$. This global volatility factor captures well the overall volatility dynamic across asset classes as seen in the last row of the top panel in Table 2. Indeed, as a sign of the strong commonalities in the realized volatilities, the average correlations with this new global risk factor systematically exceed the across asset class correlations. With the exception of foreign exchange, the global risk factor correlations are also all higher than the within-asset class correlations. The literature is rife with studies seeking to model these strong dependencies and possible volatility spillover effects using multivariate GARCH and related procedures (see, e.g., Engle et al. 1990; Karolyi 1995 for some of the earliest evidence). In contrast to these earlier studies, which rely on parametric volatility models for inferring the dependencies, our use of realized volatilities allow us to directly quantify the strengths of any spillover effects. To this end, the middle panel in Table 2 reports the average partial lead-lag correlation coefficients obtained from regressions of the daily $$RV$$s for each of the assets within the asset classes indicated in the columns on their own daily lagged value and the lagged values of the $$RV$$s for the assets in the asset classes indicated in the rows. To allow for a scale-invariant interpretation, we further normalize the realized volatility $$RV_j$$ of any asset $$j$$ by its sample mean $$\overline{RV_j}$$ and regress the normalized $$RV$$ for asset $$j$$ on a constant, its own lag, and the normalized lagged $$RV$$ for asset $$i$$, where the latter regression coefficient is the partial correlation of interest. Specifically, the first-order partial autocorrelation between asset $$j$$ and asset $$i$$ is the estimated $$b_{2,ij}$$ coefficient from the regressions $$RV_{t,j}/ \overline{RV_j} = b_{0,ij} + b_{1,ij} RV_{t-1,j}/ \overline{RV_j} + b_{2,ij} RV_{t-1,i}/ \overline{RV_i} + u_{t,ij}$$, with Table 2 reporting the average of these within-asset classes. The bottom panel shows the resultant average increases in the $$R^2$$s compared to simple first order autoregressions that only control for the own lagged dynamic dependencies; that is, $$RV_{t,j}/ \overline{RV_j} = b_{0,j} + b_{1,j} RV_{t-1,j}/ \overline{RV_j} + u_{t,j}$$. Consistent with the presence of strong cross-market linkages and spillover effects, all of the average partial correlations are positive.12 Comparing the results across the different asset classes, equity volatilities as a whole tend to exert the largest impact on the other asset class volatilities. Meanwhile, the magnitude of the equity partial correlations and the resultant increases in the $$R^2$$s are all dominated by those of the global risk factor. This naturally raises a few questions: What is behind these linkages? And, in particular, what might explain the dynamic variation in the global risk factor? The economic forces behind volatility clustering per se remain poorly understood, and a full-fledged analysis of that question is beyond the scope of the present paper. However, in an effort to shed some light on the mechanisms at work, Table 3 reports the monthly (end-of-month) correlations between an exponentially smoothed version of the global volatility factor $$ExpGlRV$$, as used in our preferred HExpGl model formally defined in Section 3.6, and four other variables naturally related to volatility. Figure 5 plots the global volatility factor, together with three of the four variables. Figure 5 View largeDownload slide Global volatility, sentiment, VRP, and news This figure shows the global volatility factor (ExpGlRV), unusual monthly sentiment, the volatility risk premium (VRP), and the absolute news surprise variable, as formally defined in the main text. All of the variables are plotted at a monthly frequency. Figure 5 View largeDownload slide Global volatility, sentiment, VRP, and news This figure shows the global volatility factor (ExpGlRV), unusual monthly sentiment, the volatility risk premium (VRP), and the absolute news surprise variable, as formally defined in the main text. All of the variables are plotted at a monthly frequency. Table 3 Global volatility correlations Sentiment Unusual sentiment VRP News Global Volatility $$-$$0.123$$^*$$ 0.197$$^{**}$$ $$-$$0.454$$^{**}$$ 0.183$$^{**}$$ Sentiment – 0.568$$^{**}$$ $$-$$0.097 0.171$$^{**}$$ Unusual sentiment – – 0.051 0.136$$^*$$ VRP – – – $$-$$0.087 Sentiment Unusual sentiment VRP News Global Volatility $$-$$0.123$$^*$$ 0.197$$^{**}$$ $$-$$0.454$$^{**}$$ 0.183$$^{**}$$ Sentiment – 0.568$$^{**}$$ $$-$$0.097 0.171$$^{**}$$ Unusual sentiment – – 0.051 0.136$$^*$$ VRP – – – $$-$$0.087 The table reports the monthly (end-of-month) contemporaneous correlations between the global volatility factor $$ExpGlRV$$ formally defined in Section 3.6 and four other variables. Sentiment refers to the “orthogonalized” investor sentiment measure of Baker and Wurgler (2006). Unusual sentiment is defined as the absolute value of the Sentiment measure minus its sample mean. The variance risk premium (VRP) is equal to the difference between the VIX and the realized U.S. equity volatility over the past month. The News surprise variable is constructed as the average value of the standardized absolute news announcement surprises observed over the month. Statistical significance at the 5% and 1% levels are indicated by $$*$$ and $${**}$$, respectively. Table 3 Global volatility correlations Sentiment Unusual sentiment VRP News Global Volatility $$-$$0.123$$^*$$ 0.197$$^{**}$$ $$-$$0.454$$^{**}$$ 0.183$$^{**}$$ Sentiment – 0.568$$^{**}$$ $$-$$0.097 0.171$$^{**}$$ Unusual sentiment – – 0.051 0.136$$^*$$ VRP – – – $$-$$0.087 Sentiment Unusual sentiment VRP News Global Volatility $$-$$0.123$$^*$$ 0.197$$^{**}$$ $$-$$0.454$$^{**}$$ 0.183$$^{**}$$ Sentiment – 0.568$$^{**}$$ $$-$$0.097 0.171$$^{**}$$ Unusual sentiment – – 0.051 0.136$$^*$$ VRP – – – $$-$$0.087 The table reports the monthly (end-of-month) contemporaneous correlations between the global volatility factor $$ExpGlRV$$ formally defined in Section 3.6 and four other variables. Sentiment refers to the “orthogonalized” investor sentiment measure of Baker and Wurgler (2006). Unusual sentiment is defined as the absolute value of the Sentiment measure minus its sample mean. The variance risk premium (VRP) is equal to the difference between the VIX and the realized U.S. equity volatility over the past month. The News surprise variable is constructed as the average value of the standardized absolute news announcement surprises observed over the month. Statistical significance at the 5% and 1% levels are indicated by $$*$$ and $${**}$$, respectively. The first entry in Table 3, in particular, shows that the global volatility factor is negatively correlated with the U.S. investor sentiment index of Baker and Wurgler (2006).13Baker, Wurgler, and Yuan (2012) have previously argued that U.S. investor sentiment is contagious, and that international capital flows may in part account for this contagion. The well-established strong link between trading volume and return volatility (see, e.g., for a survey of some of the earliest empirical evidence Karpoff 1987) may help explain the connection between investor sentiment and global volatility. Further along these lines, Karolyi, Lee, and van Dijk (2012) have previously documented strong commonalities in equity trading volumes across countries, arguably driven by correlated trading activity among institutional investors and common investor sentiment. The previous correlation portrays a monotonic relationship between volatility and sentiment, possibly driven by correlated trading. However, if “noise” traders acting on sentiment affect prices, unusually high or low levels of sentiment should both be associated with high levels of volatility (see, e.g., Brown 1999). Consistent with this idea, the monthly correlation between the global volatility factor and “unusual” sentiment, defined as the absolute value of the same sentiment measure minus its sample mean, equals 0.197.14 Thus, whereas the general level of U.S. investor sentiment is negatively correlated with global market volatility, unusual U.S. investor sentiment is positively correlated with global market volatility. The variance risk premium, formally defined as the difference between the actual and risk-neutral expectation of the future return variation, is naturally interpreted as a measure of aggregate risk aversion (see, e.g., Bakshi and Madan 2006). Supporting the idea that risk aversion, and in turn risk bearing capacity, influence volatility, the global volatility factor is strongly negatively correlated with the U.S. equity variance risk premium.15 Numerous studies have sought to relate low-frequency variation in return volatility to directly observable macroeconomic variables or indicators (see, e.g., Schwert 1989; Engle and Rangel 2008). The reported relations, however, are weak at best. At the other end of the spectrum, a number of studies have documented a sharp, but short lived, increase in intraday volatility following macroeconomic and other public news announcements (see, e.g., Ederington and Lee 1993; Andersen et al. 2003). The last entry in Table 3 reports the correlation between the global volatility factor and a news surprise variable, constructed as the average of the standardized absolute surprises for five of the most important U.S. macroeconomic news announcements released over the month.16 Corroborating the idea that the volatility in global financial markets may in part be attributed to “news” about the U.S. economy, the correlation between this news surprise variable and the global volatility factor equals 0.183. Taken as a whole, the results in Table 3 support the idea that commonalities in trading behavior possibly induced by changes in investor sentiment and/or notions of risk aversion, together with unanticipated news, all serve important roles in accounting for the strong commonalities in the dynamic dependencies observed in the volatilities within and across asset classes. We will not pursue this any further here. Instead, we turn to a discussion of the practical risk models that we use for modeling and forecasting these dynamic dependencies. 3. Risk Modeling Our new risk models are explicitly designed to incorporate the strong similarities observed in risk characteristics across assets and asset classes. 3.1 Omnibus RV-based risk models: AR($$\infty$$) To facilitate the discussion, it is instructive to consider the omnibus AR($$\infty$$) risk model, \begin{equation}\label{eq:ARinf} RV_{t+1} ~= ~b_0 ~+ ~b_1 RV_t ~+~ b_2 RV_{t-1} ~+~ ...~ +~ \epsilon_t ~= ~ b_0~ + ~b(L)RV_t ~+ ~\epsilon_t, \end{equation} (2) in which the realized volatility on day $$t+1$$ is determined by a distributed lag of past realized volatilities.17 The estimation of an infinite number of $$b_i$$ coefficients implicit in this representation is, of course, not practically feasible, and the different risk models in effect represent alternative ways of restricting the $$b(L)$$ lag polynomial to allow for its meaningful estimation, as exemplified by the $$RV$$-based ARIMA model originally proposed by Andersen et al. (2003), and the MIDAS model advocated by Ghysels, Santa-Clara, and Valkanov (2006) in which $$b(L)$$ is parameterized in terms of beta functions. The notion of multiple volatility components, or factors, is commonly used for parsimony, representing the $$b(L)$$ lag polynomial. The HAR model of Corsi (2009), for example, is based on a weighted sum of a daily, weekly and monthly volatility factors.18 In this situation, what ultimately matters from a practical forecasting perspective is the factors’ ability to capture the influence of the lagged $$RV$$’s. Ideally, we want a set of factors that “span” the $$b(L)$$ lag space well, while still enforcing some commonality and “smoothness,” thereby enabling a unified set of factors to be used for all assets and asset classes by simply altering the weights of the different factors. Regardless of the way in which the $$b(L)$$ lag polynomial is parameterized, the $$b_i$$ coefficients are usually estimated on an asset-by-asset basis. This ignores the cross-asset similarities in the dynamic dependencies discussed in the previous section. We therefore also explore the use of panel regression techniques that force the coefficients to be the same within and across different asset classes. As demonstrated below, doing so imbues the resultant risk models with a built-in robustness and statistically significant superior out-of-sample forecast performance. 3.2 “Centering”: Eliminating the level parameter in a robust fashion Even though one might naturally restrict the dynamic $$b(L)$$ lag coefficients to be the same across assets to exploit the commonalities in the dynamic dependencies and distribution of the standardized volatilities, $$RV_t/Mean(RV_t)$$, the very different volatility levels for different asset classes means that it is unreasonable to force the $$b_0$$ intercepts to be the same. To circumvent this and allow for meaningful cross-asset estimation, we “center” the risk models by replacing the intercept with a long-run volatility factor $$RV_t^{LR}$$, equal to the expanding sample mean of the daily $$RV$$’s from the start of the sample up until day $$t$$. Forcing all of the $$b_i$$ coefficients to sum to one, including the coefficient for the $$RV_t^{LR}$$ factor, ensures that the iterated long-run forecasts from the model constructed on day $$t$$ converges to this day $$t$$ estimate of the “unconditional” volatility.19 Although seemingly complicated to implement, this “centering” of the risk models is easily enforced by subtracting the long-run volatility factor from all of the $$RV$$s, including the left-hand-side $$RV$$ forecast target: \begin{equation}\label{eq:center} RV_{t+1} - RV_t^{LR} ~= ~b_1 (RV_t - RV_t^{LR})~ + ~b_2 (RV_{t-1} - RV_t^{LR}) ~+~ ...~ + ~ \epsilon_t. \end{equation} (3) When the regression is run in this way, the coefficients are free (i.e., need not sum to one), but, if we collect terms for $$RV_t^{LR}$$ on the right-hand side, then $$RV_t^{LR}$$ has an implied coefficient of $$1-b_1-b_2-\ldots = 1 - b(1)$$ such that all the implied coefficients do sum to one. By eliminating the level of the volatility, this alternate representation allows for the meaningful estimation of common dynamic $$b_i$$ coefficients by panel regression techniques. More complicated Bayesian shrinkage-type procedures, in which the dynamic coefficients are allowed to differ across assets, of course, could be applied. However, we purposely restrict the coefficients to be the same within-asset classes or across all assets, to allow for a direct comparison with the individually estimated risk models.20 3.3 Multiperiod and other volatility forecasts The AR($$\infty$$) model in (2) and the centered version thereof in (3) are directly geared to forecasting the one-day-ahead variance. Longer-run forecasts, say over weekly or monthly horizons, may be obtained by recursively substituting the forecasts for the future daily $$RV$$’s into the right-hand side of the model, subsequently adding up the resultant one, two, three, etc., days-ahead forecasts to achieve the multiperiod $$RV$$ forecast over the requisite horizon. Instead, a much simpler approach for constructing multi-day-ahead forecasts is to estimate the risk model as such from the start. That is, by replacing the daily variance $$RV_{t+1}$$ on the left-hand side of the risk model, with the realized variance over the forecast horizon $$h$$ of interest, say $$RV_t^h \equiv \frac{1}{h}\sum_{i=1}^h RV_{t-h+i}$$. In the forecasting literature, this approach is commonly referred to as direct as opposed to iterated forecasts.21 In particular, for the “monthly,” or 20-day, forecast that we focus on below, and the generic risk model in (2), we have \begin{equation}\label{eq:mperiod} RV_{t+h}^{h} ~= ~ b_0^h ~+ ~b^h(L) RV_t ~ + ~ \epsilon_t^h, \end{equation} (4) with $$h=20$$. The $$b_i^h$$ coefficients, which dictate the “speed” of the model, will obviously depend on the forecast horizon, as indicated by the superscripts $$h$$. For notational simplicity, however, we will drop this superscript in the sequel. Also, even though $$\epsilon_t^{h}$$ generally will be serially correlated up to the order of $$h-1$$, we will simply denote the residuals in all of the models discussed below as $$\epsilon_t$$ for short.22 In theory, if the model for the 1-day-ahead $$RV_{t+1}$$ in (2) is correctly specified, the iterated forecasts from that model would be the most efficient. However, ample empirical evidence shows that even minor model misspecifications tend to be amplified in iterated volatility forecasts, and, as a result, the direct forecasts produced from a model, such as (4), are often superior in practice (see, e.g., Andersen et al. 2003; Ghysels et al. 2009; Sizova 2011).23 This same basic idea also may be used for forecasting other functions of the variance, by simply replacing $$RV_{t+h}^{h}$$ on the left-hand side in Equation (4) with the volatility object of interest. For instance, the future volatility as opposed to the variance, or the inverse of the variance, are often of primary import. Unless the volatility is constant or perfectly predictable, simply transforming the forecast for the variance will result in a systematically biased forecast.24 We turn next to a brief discussion of the specific risk models, old and new, that we actually rely on. 3.4 HAR models The original HAR model of Corsi (2009) has proven very successful. It has emerged as somewhat of a benchmark in the financial econometrics literature for judging other $$RV$$-based forecasting procedures. The model may be succinctly expressed in variance form as \begin{equation}\label{eq:HARd} RV_{t+h}^{h} = \beta_0 + \beta_D RV_{t} + \beta_W RV_t^{W} +\beta_M RV_t^{M} + \epsilon_t, \end{equation} (5) where $$RV_t^{W}$$ and $$RV_t^{M}$$ denote the 5-day (weekly) and 20-day (monthly) realized volatilities, respectively, thus implying a step function for the $$b_i$$ coefficients in the omnibus AR($$\infty$$) representation in (2).25 In addition to the results from individual asset-by-asset estimation of this “uncentered” HAR model, we report the results from a fixed effects panel-based estimation of a “centered” version of the model, in which we restrict the $$\beta_D$$, $$\beta_W$$ and $$\beta_M$$ coefficients to be the same, but allow the $$\beta_0$$ coefficients to differ across different assets. The stepwise nature of the volatility factors employed in the HAR models, imply that the forecasts from the models are subject to potentially abrupt changes as an unusually large/small daily lagged $$RV$$ drops out of the sums for the longer-horizon lagged volatility factors.26 Our remaining risk models rely on alternative $$b(L)$$ polynomials for “smoothing” out these problems. 3.5 MIDAS models The original HAR model may be interpreted as a special case of MIDAS regressions with step functions (see, e.g., the discussion in Andersen et al. 2007, Ghysels et al. 2007, Corsi 2009), while the HAR-free model discussed in the footnote above is closely related to the so-called “U-MIDAS model” (Foroni et al. 2015). In contrast to HAR models, however, one of the main objectives of the MIDAS approach is the specification of “smooth” distributed lag polynomials for representing the dynamic dependencies. Another main theme of the MIDAS literature, of course, relates to the use of data sampled at different frequencies, and the choice of sampling frequency for the regressor variables. Addressing both of those issues, Ghysels, Santa-Clara, and Valkanov (2006) conclude that the direct modeling of high-frequency data does not result in systematically better volatility forecasts compared to the forecasts from a model of the form in (2) based on the daily $$RV$$s only.27 They also propose a specific parameterization for the $$b(L)$$ lag polynomial based on beta functions. This representation has now emerged as somewhat of a standard in the MIDAS literature. Thus, directly following Ghysels, Santa-Clara, and Valkanov (2006), we implement a MIDAS model of the form, \begin{equation}\label{eq:midas1} RV_{t+h}^{h} ~ = ~\beta_0 ~ + ~ \beta_1 [a(1)^{-1} a(L) ] RV_{t} + \epsilon_t, \end{equation} (6) in which the nonzero coefficients in the $$a(L)$$ lag polynomial are defined by scaled beta functions: \begin{equation}\label{eq:midas2} a_i ~ = ~ \left( \frac{i}{k} \right)^{\theta_1 - 1} \left( 1 - \frac{i}{k} \right)^{\theta_2 - 1} \Gamma(\theta_1+\theta_2) \Gamma(\theta_1)^{-1} \Gamma(\theta_2)^{-1},~~~~~~i=1,...,k, \end{equation} (7) where $$\Gamma( \cdot )$$ denotes the Gamma function. The normalization by $$a(1) \equiv a_1 + ... + a_k$$ in Equation (6) ensures that the coefficients in the $$[ a(1)^{-1} a(L)]$$ lag polynomial sum to unity, so that the $$\beta_1$$ coefficient is uniquely identified. Implementation of the model still requires a choice of the cutoff $$k$$, and the two tuning parameters $$\theta_1$$ and $$\theta_2$$. Again, directly mirroring Ghysels, Santa-Clara, and Valkanov (2006), we fix the cutoff at $$k =50$$, and set the tuning parameter $$\theta_1 =1$$. This choice of $$\theta_1$$ is now also commonly employed in the MIDAS literature more generally. Lastly, following the approach of Ghysels and Qian (2016), we determine the remaining $$\theta_2$$ tuning parameter by a grid search, in which we profile the predictive $$R^2$$ from the model as a function of $$\theta_2$$, together with the freely estimated $$\beta_0$$ and $$\beta_1$$ parameters, and choose the value of $$\theta_2$$ that maximizes the predictability over the full sample.28 As a result, our subsequent predictive MIDAS analysis is not truly out-of-sample. However, the computational burden does not allow us to perform a rolling grid search for the $$\theta_2$$ parameter. 3.6 HExp models The “smooth” beta polynomial employed in the MIDAS model avoids the stepwise changes inherent in the forecast from the HAR component-type structure. Further extending this idea, our last set of risk models rely on a mixture of “smooth” exponentially weighted moving averages (EWMA) of the past realized volatilities. Simple EWMA filters with a pre-specified center of mass are often used in practice. Instead, we explicitly estimate the relative importance of different EWMA factors constructed from the past daily $$RV$$’s: \begin{equation}\label{eq:ewma} ExpRV_{t}^{CoM(\lambda)} \equiv \sum_{i=1}^{500}\frac{e^{- i \lambda}}{e^{-\lambda} + e^{-2 \lambda} + ...+ e^{-500 \lambda}}RV_{t+1-i}, \end{equation} (8) where $$\lambda$$ defines the decay rate of the weights, and $$CoM(\lambda)$$ denotes the corresponding center-of-mass, $$CoM(\lambda)=e^{-\lambda}/(1-e^{-\lambda})$$.29 The center-of-mass of each exponential $$RV$$ measure effectively summarizes the “average” horizon of the lagged realized volatilities that it uses. Conversely, for each center-of-mass, we can compute the corresponding rate of decay as $$\lambda=\log(1+1/CoM)$$. Hence, we can think of $$\lambda=\log(1+1/125)=0.008$$ as an annual $$ExpRV$$ risk measure because the corresponding center of mass is 125 trading days, that is, about half a year, just like an annual equal-weighted average realized volatility, $$RV_t^{A}$$. Focussing on similar horizons to the ones used in the HAR model augmented with an annual volatility factor, we rely on the exponential $$RV$$’s to “span” the universe of past $$RV$$’s in a way that is both parsimonious and “smooth,” mixing four $$ExpRV_{t}^{CoM(\lambda)}$$ factors with $$\lambda$$ chosen to equate the center-of-mass to 1, 5, 25, and 125 days, respectively. Further “centering” the model around the expanding long-run volatility factor, we obtain the following new risk model: \begin{equation}\label{eq:expHAR} RV_{t+h}^{h} - RV_t^{LR} ~ = \sum_{j=1,5,25,125} \beta_j (ExpRV_{t}^{j} - RV_t^{LR}) ~+ ~\epsilon_t. \end{equation} (9) We will refer to this specification as the Heterogeneous Exponential, or HExp, model. This model, of course, is still nested in the omnibus distributed-lag model in (3). In contrast to the MIDAS specification above, this HExp model uses pre-specified volatility factors for characterizing the volatility dynamics that do not depend on any unknown tuning parameters, which implies that the model is straightforward to estimate by standard OLS on an asset-by-asset basis or panel regression procedures that restrict the beta coefficients to be the same across groups of assets.30 Motivated by the cross-asset and cross-market volatility spillover effects discussed in Section 2, our final risk model augments the asset-specific HExp model in (9) with a global risk factor. Specifically, $$ExpGlRV_t^{5}$$ is defined as the 5-day center-of-mass EWMA of the realizations of the $$GlRV_t$$ global risk factor defined in Section 2.2, yielding the following model: \begin{align} \label{eq:GlHExp} RV_{t+h}^{h} - RV_t^{LR} &= \sum_{j=1,5,25,125} \beta_j (ExpRV_{t}^{j} - RV_t^{LR})\nonumber\\ &\quad +\,\beta_{5}^{Gl} (ExpGlRV_t^{5} - RV_t^{LR}) +\epsilon_t. \end{align} (10) As discussed in Section 2.2, the global risk factor is purposely defined on an asset-specific basis to avoid any overlap between $$RV_t$$ and $$GlRV_{t-1}$$, and further normalized to have the same time $$t$$ expanding sample mean as the specific asset. The inclusion of the $$ExpGlRV_t^{5}$$ risk factor thus naturally enforces a degree of commonality over-and-above that afforded by restricting the beta coefficients to be the same. We will refer to this specification as the HExpGl risk model. 4. Risk Inference: Model Estimation and Forecasting We will focus our discussion on a 1-month (i.e., 20-day) forecast horizon. We report both in-sample results, in which we rely on all the available data, as well as out-of-sample forecasts, in which we estimate the models based on an expanding window of the data up to that point in time.31 We rely on the explained sum of squares divided by the total sum of squares within and across asset classes as way to succinctly summarize the performance of the different models. To allow for meaningful comparisons between the in- and out-of-sample results, we use the expanding long-run sample mean, or the $$RV_{t}^{LR}$$ factor, in the calculations of the out-of-sample $$R^2$$’s.32 All of the models are estimated by OLS on an individual asset-by-asset basis, by panel regressions that restrict the coefficients to be the same for all of the assets within an asset class, and by panel regressions that restrict the coefficients to be the same across all assets.33 We refer to these alternative estimation schemes as “Individual Asset,” “Panel,” and “Mega” panel, respectively. 4.1 Basic estimation and in-sample forecasting results We begin our discussion by considering the in-sample estimation results, and the different risk models’ ability to forecast the future variance, as measured by the in-sample predictive $$R^2$$’s. It is instructive to think about the results in terms of different types, or “generations,” of risk models: static (based on the idea that risk is constant, as exemplified by the expanding sample average of the realized $$RV$$s); first-generation dynamic (based on daily data, as exemplified by the 21-day rolling sums of the daily squared returns); second-generation dynamic (based on the use of high-frequency intraday data, as exemplified by the 21-day rolling $$RV$$s, corresponding to a simple random-walk-type forecast); and state-of-the-art dynamic (based on directly modeling the $$RV$$s, as exemplified by the HAR, MIDAS, and HExp risk models). Table 4 shows that, not surprisingly, having a static risk model performs the worst. The use of the 21-day rolling sample variance constructed from the daily squared returns is obviously better, and for foreign exchange, in particular, by quite a wide margin. Still, the simple 21-day $$RV$$ performs substantially better, while the more sophisticated dynamic $$RV$$-based models perform the best. At the same time, the differences across the sophisticated risk models are modest, with the HExp models generally performing the best overall. Table 4 In-sample predictions Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. $$-$$2.3% 2.5% 27.1% 45.3% 46.1% 45.9% 47.1% Commodities Panel – – – 43.4% 43.6% 43.8% 44.8% Mega – – – 43.2% 43.5% 43.7% 44.4% Indiv. $$-$$3.2% 2.3% 20.9% 43.6% 43.4% 43.8% 46.9% Equities Panel – – – 42.1% 41.6% 42.2% 45.2% Mega – – – 41.9% 41.5% 42.0% 44.1% Indiv. –4.8% 2.5% 27.6% 43.1% 44.1% 46.7% 47.4% Fixed income Panel – – – 42.5% 43.4% 45.9% 46.1% Mega – – – 41.6% 42.2% 43.9% 42.3% Indiv. $$-$$1.8% 28.9% 39.9% 53.7% 54.3% 54.4% 61.1% Foreign exchange Panel – – – 53.0% 53.5% 53.5% 58.8% Mega – – – 52.4% 52.7% 52.6% 56.2% Indiv. –2.8% 2.6% 24.2% 44.5% 44.8% 44.9% 47.1% All assets Panel – – – 42.8% 42.7% 43.1% 45.1% Mega – – – 42.6% 42.6% 42.9% 44.3% DM $$t$$-tests Indiv. $$-$$3.86 $$-$$7.09 $$-$$4.60 $$-$$2.22 $$-$$0.46 NA 1.68 Commodities Panel – – – $$-$$2.45 $$-$$2.22 $$-$$2.00 $$-$$1.27 Mega – – – $$-$$2.55 $$-$$2.32 $$-$$2.11 $$-$$1.64 Indiv. $$-$$2.86 $$-$$2.78 $$-$$2.40 $$-$$0.15 $$-$$0.65 NA 1.29 Equities Panel – – – $$-$$1.30 $$-$$1.32 $$-$$1.41 0.28 Mega – – – $$-$$1.31 $$-$$1.36 $$-$$1.44 $$-$$0.36 Indiv. $$-$$5.02 $$-$$3.63 $$-$$3.46 $$-$$2.10 $$-$$1.42 NA 1.44 Fixed income Panel – – – $$-$$2.30 $$-$$1.75 $$-$$2.65 $$-$$1.92 Mega – – – $$-$$3.10 $$-$$2.84 $$-$$3.66 $$-$$2.07 Indiv. $$-$$1.96 $$-$$2.31 $$-$$1.28 $$-$$0.60 $$-$$0.15 NA 1.08 Foreign exchange Panel – – – $$-$$1.20 $$-$$0.97 $$-$$1.79 0.66 Mega – – – $$-$$2.01 –1.79 –1.54 0.98 Indiv. $$-$$3.42 $$-$$4.64 $$-$$3.64 $$-$$0.86 $$-$$0.23 NA 1.39 All assets Panel – – – $$-$$2.20 $$-$$2.02 $$-$$2.20 0.17 Mega – – – $$-$$2.24 $$-$$2.10 $$-$$2.21 $$-$$0.65 Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. $$-$$2.3% 2.5% 27.1% 45.3% 46.1% 45.9% 47.1% Commodities Panel – – – 43.4% 43.6% 43.8% 44.8% Mega – – – 43.2% 43.5% 43.7% 44.4% Indiv. $$-$$3.2% 2.3% 20.9% 43.6% 43.4% 43.8% 46.9% Equities Panel – – – 42.1% 41.6% 42.2% 45.2% Mega – – – 41.9% 41.5% 42.0% 44.1% Indiv. –4.8% 2.5% 27.6% 43.1% 44.1% 46.7% 47.4% Fixed income Panel – – – 42.5% 43.4% 45.9% 46.1% Mega – – – 41.6% 42.2% 43.9% 42.3% Indiv. $$-$$1.8% 28.9% 39.9% 53.7% 54.3% 54.4% 61.1% Foreign exchange Panel – – – 53.0% 53.5% 53.5% 58.8% Mega – – – 52.4% 52.7% 52.6% 56.2% Indiv. –2.8% 2.6% 24.2% 44.5% 44.8% 44.9% 47.1% All assets Panel – – – 42.8% 42.7% 43.1% 45.1% Mega – – – 42.6% 42.6% 42.9% 44.3% DM $$t$$-tests Indiv. $$-$$3.86 $$-$$7.09 $$-$$4.60 $$-$$2.22 $$-$$0.46 NA 1.68 Commodities Panel – – – $$-$$2.45 $$-$$2.22 $$-$$2.00 $$-$$1.27 Mega – – – $$-$$2.55 $$-$$2.32 $$-$$2.11 $$-$$1.64 Indiv. $$-$$2.86 $$-$$2.78 $$-$$2.40 $$-$$0.15 $$-$$0.65 NA 1.29 Equities Panel – – – $$-$$1.30 $$-$$1.32 $$-$$1.41 0.28 Mega – – – $$-$$1.31 $$-$$1.36 $$-$$1.44 $$-$$0.36 Indiv. $$-$$5.02 $$-$$3.63 $$-$$3.46 $$-$$2.10 $$-$$1.42 NA 1.44 Fixed income Panel – – – $$-$$2.30 $$-$$1.75 $$-$$2.65 $$-$$1.92 Mega – – – $$-$$3.10 $$-$$2.84 $$-$$3.66 $$-$$2.07 Indiv. $$-$$1.96 $$-$$2.31 $$-$$1.28 $$-$$0.60 $$-$$0.15 NA 1.08 Foreign exchange Panel – – – $$-$$1.20 $$-$$0.97 $$-$$1.79 0.66 Mega – – – $$-$$2.01 –1.79 –1.54 0.98 Indiv. $$-$$3.42 $$-$$4.64 $$-$$3.64 $$-$$0.86 $$-$$0.23 NA 1.39 All assets Panel – – – $$-$$2.20 $$-$$2.02 $$-$$2.20 0.17 Mega – – – $$-$$2.24 $$-$$2.10 $$-$$2.21 $$-$$0.65 This table reports the in-sample results for predicting the 20-day future realized volatility using the different predictor variables and risk models. The top panel reports the average in-sample regression $$R^2$$s by asset class and across all assets. The bottom panel reports Diebold-Mariano (DM) $$t$$-statistics for testing the significance of the average standardized loss differentials relative to the individually estimated HExp model. A positive (negative) DM-test statistic indicates that the model and the estimation procedure outperform (underperform) the individually estimated HExp model in-sample. Table 4 In-sample predictions Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. $$-$$2.3% 2.5% 27.1% 45.3% 46.1% 45.9% 47.1% Commodities Panel – – – 43.4% 43.6% 43.8% 44.8% Mega – – – 43.2% 43.5% 43.7% 44.4% Indiv. $$-$$3.2% 2.3% 20.9% 43.6% 43.4% 43.8% 46.9% Equities Panel – – – 42.1% 41.6% 42.2% 45.2% Mega – – – 41.9% 41.5% 42.0% 44.1% Indiv. –4.8% 2.5% 27.6% 43.1% 44.1% 46.7% 47.4% Fixed income Panel – – – 42.5% 43.4% 45.9% 46.1% Mega – – – 41.6% 42.2% 43.9% 42.3% Indiv. $$-$$1.8% 28.9% 39.9% 53.7% 54.3% 54.4% 61.1% Foreign exchange Panel – – – 53.0% 53.5% 53.5% 58.8% Mega – – – 52.4% 52.7% 52.6% 56.2% Indiv. –2.8% 2.6% 24.2% 44.5% 44.8% 44.9% 47.1% All assets Panel – – – 42.8% 42.7% 43.1% 45.1% Mega – – – 42.6% 42.6% 42.9% 44.3% DM $$t$$-tests Indiv. $$-$$3.86 $$-$$7.09 $$-$$4.60 $$-$$2.22 $$-$$0.46 NA 1.68 Commodities Panel – – – $$-$$2.45 $$-$$2.22 $$-$$2.00 $$-$$1.27 Mega – – – $$-$$2.55 $$-$$2.32 $$-$$2.11 $$-$$1.64 Indiv. $$-$$2.86 $$-$$2.78 $$-$$2.40 $$-$$0.15 $$-$$0.65 NA 1.29 Equities Panel – – – $$-$$1.30 $$-$$1.32 $$-$$1.41 0.28 Mega – – – $$-$$1.31 $$-$$1.36 $$-$$1.44 $$-$$0.36 Indiv. $$-$$5.02 $$-$$3.63 $$-$$3.46 $$-$$2.10 $$-$$1.42 NA 1.44 Fixed income Panel – – – $$-$$2.30 $$-$$1.75 $$-$$2.65 $$-$$1.92 Mega – – – $$-$$3.10 $$-$$2.84 $$-$$3.66 $$-$$2.07 Indiv. $$-$$1.96 $$-$$2.31 $$-$$1.28 $$-$$0.60 $$-$$0.15 NA 1.08 Foreign exchange Panel – – – $$-$$1.20 $$-$$0.97 $$-$$1.79 0.66 Mega – – – $$-$$2.01 –1.79 –1.54 0.98 Indiv. $$-$$3.42 $$-$$4.64 $$-$$3.64 $$-$$0.86 $$-$$0.23 NA 1.39 All assets Panel – – – $$-$$2.20 $$-$$2.02 $$-$$2.20 0.17 Mega – – – $$-$$2.24 $$-$$2.10 $$-$$2.21 $$-$$0.65 Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. $$-$$2.3% 2.5% 27.1% 45.3% 46.1% 45.9% 47.1% Commodities Panel – – – 43.4% 43.6% 43.8% 44.8% Mega – – – 43.2% 43.5% 43.7% 44.4% Indiv. $$-$$3.2% 2.3% 20.9% 43.6% 43.4% 43.8% 46.9% Equities Panel – – – 42.1% 41.6% 42.2% 45.2% Mega – – – 41.9% 41.5% 42.0% 44.1% Indiv. –4.8% 2.5% 27.6% 43.1% 44.1% 46.7% 47.4% Fixed income Panel – – – 42.5% 43.4% 45.9% 46.1% Mega – – – 41.6% 42.2% 43.9% 42.3% Indiv. $$-$$1.8% 28.9% 39.9% 53.7% 54.3% 54.4% 61.1% Foreign exchange Panel – – – 53.0% 53.5% 53.5% 58.8% Mega – – – 52.4% 52.7% 52.6% 56.2% Indiv. –2.8% 2.6% 24.2% 44.5% 44.8% 44.9% 47.1% All assets Panel – – – 42.8% 42.7% 43.1% 45.1% Mega – – – 42.6% 42.6% 42.9% 44.3% DM $$t$$-tests Indiv. $$-$$3.86 $$-$$7.09 $$-$$4.60 $$-$$2.22 $$-$$0.46 NA 1.68 Commodities Panel – – – $$-$$2.45 $$-$$2.22 $$-$$2.00 $$-$$1.27 Mega – – – $$-$$2.55 $$-$$2.32 $$-$$2.11 $$-$$1.64 Indiv. $$-$$2.86 $$-$$2.78 $$-$$2.40 $$-$$0.15 $$-$$0.65 NA 1.29 Equities Panel – – – $$-$$1.30 $$-$$1.32 $$-$$1.41 0.28 Mega – – – $$-$$1.31 $$-$$1.36 $$-$$1.44 $$-$$0.36 Indiv. $$-$$5.02 $$-$$3.63 $$-$$3.46 $$-$$2.10 $$-$$1.42 NA 1.44 Fixed income Panel – – – $$-$$2.30 $$-$$1.75 $$-$$2.65 $$-$$1.92 Mega – – – $$-$$3.10 $$-$$2.84 $$-$$3.66 $$-$$2.07 Indiv. $$-$$1.96 $$-$$2.31 $$-$$1.28 $$-$$0.60 $$-$$0.15 NA 1.08 Foreign exchange Panel – – – $$-$$1.20 $$-$$0.97 $$-$$1.79 0.66 Mega – – – $$-$$2.01 –1.79 –1.54 0.98 Indiv. $$-$$3.42 $$-$$4.64 $$-$$3.64 $$-$$0.86 $$-$$0.23 NA 1.39 All assets Panel – – – $$-$$2.20 $$-$$2.02 $$-$$2.20 0.17 Mega – – – $$-$$2.24 $$-$$2.10 $$-$$2.21 $$-$$0.65 This table reports the in-sample results for predicting the 20-day future realized volatility using the different predictor variables and risk models. The top panel reports the average in-sample regression $$R^2$$s by asset class and across all assets. The bottom panel reports Diebold-Mariano (DM) $$t$$-statistics for testing the significance of the average standardized loss differentials relative to the individually estimated HExp model. A positive (negative) DM-test statistic indicates that the model and the estimation procedure outperform (underperform) the individually estimated HExp model in-sample. In addition to the results for the individually estimated risk models, the table also shows the results from our panel-based estimation techniques that restricts the parameters in the dynamic risk models to be the same for all assets within a given asset class (panel) and across all assets (mega). By construction, of course, the individually estimated risk models always result in larger in-sample $$R^2$$’s compared to any of the panel-based versions of the identical models. Interestingly, however, the differences in $$R^2$$’s across the individual versus panel-based models are fairly small. Given the robustness afforded by the panel estimation, this therefore also suggests that the results may look different out-of-sample. To more formally assess the statistical significance of the differences in in-sample predictability, the bottom panel of Table 4 reports the results of Diebold and Mariano (1995) (DM) tests. Specifically, taking the individually estimated HExp model as the benchmark, we test the null hypotheses of equal predictive ability by calculating robust $$t$$-statistics for the sample means of the time series comprising the average standardized (by the mean of the realized variation) squared error losses for each of the different models, estimation procedures, and asset classes.34 With the exception of the predictions pertaining to the foreign exchange market and the predictions from the HExpGl model, the majority of the $$t$$-statistics are significant at the usual 5% level. Looking across the different estimation procedures, the by-construction higher in-sample $$R^2$$s for the individually estimated models also generally translate into statistically significant lower losses compared to the panel and mega models that restrict the coefficients to be the same across groups of assets. As a case in point, the $$t$$-statistic for the individually estimated HExp model versus the mega HExp model for all assets equals $$-2.21$$. Among the mega-based models, only the mega HExpGl model does not result in statistically inferior in-sample predictions compared to the individually estimated HExp model. The different specifications of the models complicate any direct comparisons of the estimated regressions coefficients. However, all of the risk models, except for the HExpGl model, are nested in the univariate AR($$\infty$$) representation in (2). Hence, whereas the estimated $$\beta$$ coefficients are not directly comparable, the dynamics of the different risk models may be meaningfully compared in terms of the implied $$b_i$$ coefficients in that representation. To this end, Figure 6 depicts the impled $$b(L)$$ polynomials out to a lag length of 25 days for the mega-panel-based estimated models.35 For comparison purposes, we also include the weights for the 21-day $$RV$$, or equivalently a HAR model with $$\beta_M=1$$ and $$\beta_0=\beta_D=\beta_W=0$$, together with a HAR-free model in which we freely estimate the impact of the first six daily lagged $$RV$$s. As the figure shows, with the exception of the flat weights for the 21-day $$RV$$, the estimates are generally fairly close. Nonetheless, the HAR-free and HExp models both appear slightly “faster” than the MIDAS model, with a more rapid initial decay and less weight assigned to the intermediate lags ranging between five days and two weeks. Figure 6 View largeDownload slide Implied lag coefficients for different risk models This figure shows the lag coefficients implied by the regression coefficients of full-period models pooled across all assets for each of five different RV-based models: 21-day-RV, HAR, HAR-free, MIDAS, and HExp. Figure 6 View largeDownload slide Implied lag coefficients for different risk models This figure shows the lag coefficients implied by the regression coefficients of full-period models pooled across all assets for each of five different RV-based models: 21-day-RV, HAR, HAR-free, MIDAS, and HExp. The figure also visualizes the stepwise nature of the implied $$b_i$$ coefficients for both of the HAR models. As a result, forecasts constructed from these models are more susceptible to abrupt changes, and therefore potentially also more costly to implement, than the forecasts from the “smoother” risk models.36 We will return to this issue and the “speed” of the models in our discussion of the utility-based comparisons in Section 5 below. 4.2 Out-of-sample forecasting results Before this discussion, however, Table 5 reports on the out-of-sample performance of the identical set of risk models using the same $$R^2$$ metric as above.37 Looking across the columns, we see a similar ranking as for the in-sample results: the static risk model (the expanding average $$RV$$) naturally performs the worst, the first-generation risk model based on daily data is second, followed by the simple 21-day $$RV$$, while the more sophisticated $$RV$$ models perform the best, with the HExp models again preforming the very best overall. Table 5 Out-of-sample predictions Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. 1.2% 5.9% 29.7% 44.5% 45.0% 46.4% 47.6% Commodities Panel – – – 45.5% 45.7% 46.9% 47.9% Mega – – – 45.6% 45.8% 47.5% 48.1% Indiv. 2.9% 8.1% 25.6% 41.2% 41.9% 48.3% 51.6% Equities Panel – – – 45.8% 46.4% 50.7% 53.8% Mega – – – 47.0% 47.8% 51.0% 53.3% Indiv. $$-$$1.5% 5.5% 29.9% 42.6% 44.3% 47.2% 45.7% Fixed income Panel – – – 43.3% 44.7% 47.5% 46.1% Mega – – – 43.2% 43.1% 47.0% 44.0% Indiv. 1.4% 31.1% 41.8% 28.6% 30.4% 46.8% 54.6% Foreign exchange Panel – – – 27.8% 30.4% 46.5% 55.1% Mega – – – 47.7% 49.2% 52.6% 56.8% Indiv. 2.1% 7.1% 27.7% 42.8% 43.4% 47.3% 49.5% All assets Panel – – – 45.5% 45.9% 48.7% 50.7% Mega – – – 46.2% 46.7% 49.2% 50.6% DM $$t$$-tests Indiv. $$-$$3.93 $$-$$7.01 $$-$$3.68 $$-$$3.18 $$-$$1.82 $$-$$0.76 0.29 Commodities Panel – – – $$-$$3.34 $$-$$3.68 $$-$$2.31 0.85 Mega – – – $$-$$4.01 $$-$$4.09 NA 1.31 Indiv. $$-$$2.42 $$-$$2.46 $$-$$1.85 $$-$$1.69 $$-$$1.72 $$-$$1.40 $$-$$0.07 Equities Panel – – – $$-$$2.91 $$-$$3.71 $$-$$0.77 1.03 Mega – – – $$-$$2.95 $$-$$3.62 NA 1.16 Indiv. $$-$$4.70 $$-$$3.07 $$-$$2.50 $$-$$1.32 $$-$$0.96 $$-$$0.23 $$-$$0.78 Fixed income Panel – – – $$-$$1.78 $$-$$1.29 $$-$$0.52 $$-$$0.77 Mega – – – $$-$$1.71 $$-$$1.78 NA $$-$$0.86 Indiv. $$-$$1.96 $$-$$2.01 $$-$$0.93 $$-$$1.21 $$-$$1.15 $$-$$0.93 0.67 Foreign exchange Panel – – – $$-$$1.20 $$-$$1.15 $$-$$0.90 0.68 Mega – – – $$-$$1.30 $$-$$1.32 NA 1.30 Indiv. $$-$$3.42 $$-$$3.70 $$-$$2.56 $$-$$2.01 $$-$$2.02 $$-$$1.93 $$-$$0.23 All assets Panel – – – $$-$$2.83 $$-$$3.43 $$-$$2.05 0.91 Mega – – – $$-$$3.64 $$-$$4.80 NA 1.21 Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. 1.2% 5.9% 29.7% 44.5% 45.0% 46.4% 47.6% Commodities Panel – – – 45.5% 45.7% 46.9% 47.9% Mega – – – 45.6% 45.8% 47.5% 48.1% Indiv. 2.9% 8.1% 25.6% 41.2% 41.9% 48.3% 51.6% Equities Panel – – – 45.8% 46.4% 50.7% 53.8% Mega – – – 47.0% 47.8% 51.0% 53.3% Indiv. $$-$$1.5% 5.5% 29.9% 42.6% 44.3% 47.2% 45.7% Fixed income Panel – – – 43.3% 44.7% 47.5% 46.1% Mega – – – 43.2% 43.1% 47.0% 44.0% Indiv. 1.4% 31.1% 41.8% 28.6% 30.4% 46.8% 54.6% Foreign exchange Panel – – – 27.8% 30.4% 46.5% 55.1% Mega – – – 47.7% 49.2% 52.6% 56.8% Indiv. 2.1% 7.1% 27.7% 42.8% 43.4% 47.3% 49.5% All assets Panel – – – 45.5% 45.9% 48.7% 50.7% Mega – – – 46.2% 46.7% 49.2% 50.6% DM $$t$$-tests Indiv. $$-$$3.93 $$-$$7.01 $$-$$3.68 $$-$$3.18 $$-$$1.82 $$-$$0.76 0.29 Commodities Panel – – – $$-$$3.34 $$-$$3.68 $$-$$2.31 0.85 Mega – – – $$-$$4.01 $$-$$4.09 NA 1.31 Indiv. $$-$$2.42 $$-$$2.46 $$-$$1.85 $$-$$1.69 $$-$$1.72 $$-$$1.40 $$-$$0.07 Equities Panel – – – $$-$$2.91 $$-$$3.71 $$-$$0.77 1.03 Mega – – – $$-$$2.95 $$-$$3.62 NA 1.16 Indiv. $$-$$4.70 $$-$$3.07 $$-$$2.50 $$-$$1.32 $$-$$0.96 $$-$$0.23 $$-$$0.78 Fixed income Panel – – – $$-$$1.78 $$-$$1.29 $$-$$0.52 $$-$$0.77 Mega – – – $$-$$1.71 $$-$$1.78 NA $$-$$0.86 Indiv. $$-$$1.96 $$-$$2.01 $$-$$0.93 $$-$$1.21 $$-$$1.15 $$-$$0.93 0.67 Foreign exchange Panel – – – $$-$$1.20 $$-$$1.15 $$-$$0.90 0.68 Mega – – – $$-$$1.30 $$-$$1.32 NA 1.30 Indiv. $$-$$3.42 $$-$$3.70 $$-$$2.56 $$-$$2.01 $$-$$2.02 $$-$$1.93 $$-$$0.23 All assets Panel – – – $$-$$2.83 $$-$$3.43 $$-$$2.05 0.91 Mega – – – $$-$$3.64 $$-$$4.80 NA 1.21 This table reports the out-of-sample results for predicting the 20-day future realized volatility using the different predictor variables and risk models. The top panel reports the average out-of-sample predictive $$R^2$$s by asset class and across all assets. The bottom panel reports Diebold-Mariano (DM) $$t$$-statistics for testing the significance of the average standardized loss differentials relative to the mega HExp model that restricts the coefficients to be the same across all assets. A positive (negative) DM-test statistic indicates that the model and the estimation procedure outperform (underperform) the mega HExp model out-of-sample. Table 5 Out-of-sample predictions Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. 1.2% 5.9% 29.7% 44.5% 45.0% 46.4% 47.6% Commodities Panel – – – 45.5% 45.7% 46.9% 47.9% Mega – – – 45.6% 45.8% 47.5% 48.1% Indiv. 2.9% 8.1% 25.6% 41.2% 41.9% 48.3% 51.6% Equities Panel – – – 45.8% 46.4% 50.7% 53.8% Mega – – – 47.0% 47.8% 51.0% 53.3% Indiv. $$-$$1.5% 5.5% 29.9% 42.6% 44.3% 47.2% 45.7% Fixed income Panel – – – 43.3% 44.7% 47.5% 46.1% Mega – – – 43.2% 43.1% 47.0% 44.0% Indiv. 1.4% 31.1% 41.8% 28.6% 30.4% 46.8% 54.6% Foreign exchange Panel – – – 27.8% 30.4% 46.5% 55.1% Mega – – – 47.7% 49.2% 52.6% 56.8% Indiv. 2.1% 7.1% 27.7% 42.8% 43.4% 47.3% 49.5% All assets Panel – – – 45.5% 45.9% 48.7% 50.7% Mega – – – 46.2% 46.7% 49.2% 50.6% DM $$t$$-tests Indiv. $$-$$3.93 $$-$$7.01 $$-$$3.68 $$-$$3.18 $$-$$1.82 $$-$$0.76 0.29 Commodities Panel – – – $$-$$3.34 $$-$$3.68 $$-$$2.31 0.85 Mega – – – $$-$$4.01 $$-$$4.09 NA 1.31 Indiv. $$-$$2.42 $$-$$2.46 $$-$$1.85 $$-$$1.69 $$-$$1.72 $$-$$1.40 $$-$$0.07 Equities Panel – – – $$-$$2.91 $$-$$3.71 $$-$$0.77 1.03 Mega – – – $$-$$2.95 $$-$$3.62 NA 1.16 Indiv. $$-$$4.70 $$-$$3.07 $$-$$2.50 $$-$$1.32 $$-$$0.96 $$-$$0.23 $$-$$0.78 Fixed income Panel – – – $$-$$1.78 $$-$$1.29 $$-$$0.52 $$-$$0.77 Mega – – – $$-$$1.71 $$-$$1.78 NA $$-$$0.86 Indiv. $$-$$1.96 $$-$$2.01 $$-$$0.93 $$-$$1.21 $$-$$1.15 $$-$$0.93 0.67 Foreign exchange Panel – – – $$-$$1.20 $$-$$1.15 $$-$$0.90 0.68 Mega – – – $$-$$1.30 $$-$$1.32 NA 1.30 Indiv. $$-$$3.42 $$-$$3.70 $$-$$2.56 $$-$$2.01 $$-$$2.02 $$-$$1.93 $$-$$0.23 All assets Panel – – – $$-$$2.83 $$-$$3.43 $$-$$2.05 0.91 Mega – – – $$-$$3.64 $$-$$4.80 NA 1.21 Static 21-daily 21-day $$RV$$ HAR MIDAS HExp HExpGl $$R^2$$ Indiv. 1.2% 5.9% 29.7% 44.5% 45.0% 46.4% 47.6% Commodities Panel – – – 45.5% 45.7% 46.9% 47.9% Mega – – – 45.6% 45.8% 47.5% 48.1% Indiv. 2.9% 8.1% 25.6% 41.2% 41.9% 48.3% 51.6% Equities Panel – – – 45.8% 46.4% 50.7% 53.8% Mega – – – 47.0% 47.8% 51.0% 53.3% Indiv. $$-$$1.5% 5.5% 29.9% 42.6% 44.3% 47.2% 45.7% Fixed income Panel – – – 43.3% 44.7% 47.5% 46.1% Mega – – – 43.2% 43.1% 47.0% 44.0% Indiv. 1.4% 31.1% 41.8% 28.6% 30.4% 46.8% 54.6% Foreign exchange Panel – – – 27.8% 30.4% 46.5% 55.1% Mega – – – 47.7% 49.2% 52.6% 56.8% Indiv. 2.1% 7.1% 27.7% 42.8% 43.4% 47.3% 49.5% All assets Panel – – – 45.5% 45.9% 48.7% 50.7% Mega – – – 46.2% 46.7% 49.2% 50.6% DM $$t$$-tests Indiv. $$-$$3.93 $$-$$7.01 $$-$$3.68 $$-$$3.18 $$-$$1.82 $$-$$0.76 0.29 Commodities Panel – – – $$-$$3.34 $$-$$3.68 $$-$$2.31 0.85 Mega – – – $$-$$4.01 $$-$$4.09 NA 1.31 Indiv. $$-$$2.42 $$-$$2.46 $$-$$1.85 $$-$$1.69 $$-$$1.72 $$-$$1.40 $$-$$0.07 Equities Panel – – – $$-$$2.91 $$-$$3.71 $$-$$0.77 1.03 Mega – – – $$-$$2.95 $$-$$3.62 NA 1.16 Indiv. $$-$$4.70 $$-$$3.07 $$-$$2.50 $$-$$1.32 $$-$$0.96 $$-$$0.23 $$-$$0.78 Fixed income Panel – – – $$-$$1.78 $$-$$1.29 $$-$$0.52 $$-$$0.77 Mega – – – $$-$$1.71 $$-$$1.78 NA $$-$$0.86 Indiv. $$-$$1.96 $$-$$2.01 $$-$$0.93 $$-$$1.21 $$-$$1.15 $$-$$0.93 0.67 Foreign exchange Panel – – – $$-$$1.20 $$-$$1.15 $$-$$0.90 0.68 Mega – – – $$-$$1.30 $$-$$1.32 NA 1.30 Indiv. $$-$$3.42 $$-$$3.70 $$-$$2.56 $$-$$2.01 $$-$$2.02 $$-$$1.93 $$-$$0.23 All assets Panel – – – $$-$$2.83 $$-$$3.43 $$-$$2.05 0.91 Mega – – – $$-$$3.64 $$-$$4.80 NA 1.21 This table reports the out-of-sample results for predicting the 20-day future realized volatility using the different predictor variables and risk models. The top panel reports the average out-of-sample predictive $$R^2$$s by asset class and across all assets. The bottom panel reports Diebold-Mariano (DM) $$t$$-statistics for testing the significance of the average standardized loss differentials relative to the mega HExp model that restricts the coefficients to be the same across all assets. A positive (negative) DM-test statistic indicates that the model and the estimation procedure outperform (underperform) the mega HExp model out-of-sample. Meanwhile, looking across the rows, we see that the systematic ordering of the individual versus panel-based estimated models is completely reversed relative to the in-sample results in Table 4. The mega-panel estimation that restricts the coefficients to be the same across all assets now typically results in the highest $$R^2$$’s. This is especially true for foreign exchange, where the individually estimated dynamic risk models perform rather poorly, clearly underscoring the importance of more robust forecasting procedures. The DM-tests based on the average out-of-sample standardized squared error losses, reported in the bottom part of the table, again corroborate these conclusions.38 Now taking the mega HExp model as the benchmark, the pairwise tests show that the forecasts from that model result in significantly lower losses than the forecasts from all of the other models, except for the different HExpGl models. Importantly, the forecasts from the mega HExp model also result in significantly lower losses than the forecasts from the individually estimated HExp models and panel HExp models that allow for different coefficients across asset classes. Intuitively, the mega estimation approach provides a built-in robustness against influential outliers, and in turn superior out-of-sample forecast. To further appreciate this point, the expected squared forecast error loss may be expressed as the sum of the squared forecast bias, the variance of the forecasts, plus the variance of the “irreducible error” associated with the true (unknown) conditional expectation $$RV_{t+h}^h - E_t(RV_{t+h}^h)$$ (see, e.g., the discussion in Hastie et al. 2009). Looking at the bias-variance trade-off implicit in the squared forecast error losses thus help explain why the pooling and panel-based estimation that exploit the commonalities and reduce the parameter estimation error uncertainty generally works the best from an out-of-sample forecasting perspective and result in the highest predictive $$R^2$$s. In particular, while the squared out-of-sample forecast biases are very small for all of the different models and estimation methods and effectively immaterial, the individually estimated risk models systematically result in the most variable forecasts by quite a wide margin. For the HExp model, for example, the average variance of the forecasts is reduced by almost 30% for the mega model that restrict the coefficients to be the same for all assets compared to the average forecast variance for the individually estimated HExp models; the Online Appendix provides more detailed results along these lines for all of the different models. Importantly, by the same reasoning, this does not necessarily imply that the best performing forecasting model is somehow closest to the “true” model, only that more parsimonious risk models tend to produce better out-of-sample forecasts. The practical uses of risk models, of course, face a host of other issues and tradeoffs related to the actual costs and benefits of implementing the forecasts from the models. To illustrate these issues, we turn next to a utility-based framework for evaluating the benefits of an investment strategy involving the notion of equal risk shares. To keep the results simple and directly comparable to the ones discussed in the previous sections, we purposely focus on the risk models estimated to forecast the variance. 5. Risk Models in Action: Quantifying the Utility Benefits We consider a simple utility-based framework: an investor with mean-variance preferences investing in an asset with time-varying volatility and a constant Sharpe ratio. In contrast to the related approach of Fleming, Kirby, and Ostdiek (2001, 2003), which depends on forecasts for both returns and volatilities, our framework relies exclusively on volatility forecasts. 5.1 Expected utility and risk targeting By standard arguments, the time-$$t$$ expected utility may, up to a factor of proportionality, conveniently be approximated as (dropping constant terms that only depend on time-$$t$$ variables): \begin{equation}\label{eq:utility0} E_{t}(u(W_{t+1})) = E_{t}(W_{t+1})-\frac{1}{2}\gamma^A \,Var_t(W_{t+1}), \end{equation} (11) where $$\gamma^A \equiv -u''/u'$$ denotes the absolute risk aversion of the investor. We will assume that the investor allocates a fraction $$x_t$$ of his current wealth to a risky asset with return $$r_{t+1}$$ and the rest to a risk-free money market account earning $$r^f_t$$. Correspondingly, his future wealth becomes $$W_{t+1}=W_t(1+x_t r_{t+1}+(1-x_{t})r^f_t)=W_t(1+r^f_t)+W_tx_{t}r^e_{t+1}$$, where $$r^e_{t+1} \equiv r_{t+1}-r^f_t$$ denotes the excess return, resulting in an expected utility of (again dropping constant terms): \begin{equation}\label{eq:utility1b} \begin{split} U(x_{t})~=~&W_t \left( x_t E_t(r^e_{t+1})-\frac{\gamma}{2}x_t^2 Var_{t}(r^e_{t+1})\right)\\ ~=~&W_t \left( x_t E_t(r^e_{t+1})-\frac{\gamma}{2}x_t^2 E_t(RV_{t+1})\right), \end{split} \end{equation} (12) where $$\gamma \equiv \gamma^AW_t$$ refers to the investor’s relative risk aversion. To focus on risk modeling, we assume that the conditional Sharpe ratio, defined as $$SR \equiv E_t(r^{e}_{t+1}) / \sqrt{E_t(RV_{t+1})}$$, is constant.39 Under this assumption, the expected utility simply depends on the position $$x_t$$, together with the expected realized volatility $$E_t(RV_{t+1})$$: \begin{equation}\label{eq:utility1c} U(x_{t})=W_t \left( x_t SR \sqrt{E_t(RV_{t+1})}-\frac{\gamma}{2}x_t^2 E_t(RV_{t+1})\right). \end{equation} (13) The optimal portfolio that maximizes this utility is obtained by investing the fraction of wealth $$x_t^* = E_t(r^{e}_{t+1}) /( \gamma E_t(RV_{t+1}))$$ in the risky asset, or, alternatively, \begin{equation}\label{eq:opt:x:a} x_t^*=\frac{SR/\gamma}{\sqrt{ E_t(RV_{t+1})}}. \end{equation} (14) In other words, the investor optimally targets a volatility of $$SR/\gamma$$, since the conditional standard deviation of the $$x_t^*$$ portfolio equals $$\sqrt{Var_t(x_t^* r^e_{t+1})} = SR/\gamma.$$ When the predicted volatility $$\sqrt{ E_t(RV_{t+1})}$$ is above the “risk target” $$SR/\gamma$$, the agent only invests part of his wealth in the risky asset ($$x_t^*<1$$). Conversely, when the predicted risk is below the target, the investor applies leverage ($$x_t^*>1$$) to reach his risk target. This “volatility-timing" behavior mimics in a simple way the actual trading behavior of many hedge funds with explicit volatility targets and so-called “risk parity investors.” This in turn results in an expected utility of \begin{equation}\label{eq:utility2} U(x_t^*) =\frac{SR^2}{2\gamma}\,W_t= \underbrace{\frac{1}{2}}_{ \scriptsize\begin{array}{c} \text{fraction of expected} \text{return not lost to} \text{disutility of risk} \end{array}}\times \underbrace{\underbrace{SR}_{\text{reward-to-risk}}\times \underbrace{\frac{SR}{\gamma}}_{\text{risk target}}}_{\text{expected excess return}}\,W_t. \end{equation} (15) We see that the expected utility (as a fraction of wealth) is half of the expected return of the optimal position size; the other half of the expected return is “lost” to disutility of risk. For concreteness and in parallel to the forecasting results discussed in the previous section, we focus on a monthly forecast horizon. Guided by the typical results reported in the extant investment literature, we take the corresponding annualized Sharpe ratio and coefficient of risk aversion to be $$SR=0.4$$ and $$\gamma=2$$, respectively (see, e.g., Pedersen 2015 for empirical evidence pertaining to the same broad set of assets analyzed here).40 Correspondingly, the investor optimally targets an annualized volatility of 20% (similar to the average volatility across all of the assets considered here): \begin{equation}\label{eq:opt:x:b} x_t^*=\frac{ 20\%}{\sqrt{E_t(RV_{t+1})}}. \end{equation} (16) By substitution into (13) or (15), the associated utility for this optimally targeted portfolio equals \begin{equation}\label{eq:utility3} U(x_t^*) = 4\%\,W_t, \end{equation} (17) meaning that the investor would be willing to give up 4% of his wealth to have access to the $$x_t^*$$ portfolio rather than simply investing in the risk-free asset. Put differently, since the utility from (13) of a risk-free position equals $$U(0)=0$$, the investor would receive the same utility by either (1) trading the risky asset optimally while paying a fee of 4% times his wealth or (2) putting all of his money in the risk-free asset. To further appreciate this number, consider the expected return of the investor’s strategy. Given a Sharpe ratio of 0.4 and a 20% risk target, the investor expects to make an excess return of 8% per year. However, by the decomposition in (15), at the optimally targeted position half of this return is “lost” due to the disutility of risk, so the investor is left with a benefit of only 4%.41 To explicitly quantify the utility gains from different risk models, let $$E_t^\theta( \cdot )$$ denote the expectations from model $$\theta$$. Also, let $$E_t( \cdot)$$ denote the expectations from the true (unknown) risk model. Assuming that the investor uses model $$\theta$$, to choose the position $$x_t^{\theta}=20\%/\sqrt{E^\theta_t(RV_{t+1})}$$, the expected utility per unit of wealth, $$UoW_t^{\theta} \equiv U_t(x_t^{\theta})/W_t$$, may be expressed as \begin{equation}\label{eq:utility4} UoW_t^{\theta} ~=~ 8\% \frac{\sqrt{E_t(RV_{t+1})}}{\sqrt{E^\theta_t(RV_{t+1})}} -4\%\frac{E_t(RV_{t+1})}{E^\theta_t(RV_{t+1})}. \end{equation} (18) We evaluate this expected utility empirically by averaging the corresponding realized expressions over the same rolling out-of-sample risk mo