Equity Tail Risk in the Treasury Bond Market
aa r X i v : . [ q -f i n . P R ] J u l Equity Tail Risk in the Treasury Bond Market ∗ Mirco Rubin † , Dario Ruzzi ‡ July 14, 2020
Abstract
This paper quantifies the effects of equity tail risk on the US government bond market.We estimate equity tail risk with option-implied stock market volatility that stems fromlarge negative price jumps, and we assess its value in reduced-form predictive regressions forTreasury returns and a term structure model for interest rates. We find that the left tailvolatility of the stock market significantly predicts one-month excess returns on Treasuriesboth in- and out-of-sample. The incremental value of employing equity tail risk as a re-turn forecasting factor can be of economic importance for a mean-variance investor tradingbonds. The estimated term structure model shows that equity tail risk is priced in the USgovernment bond market and, consistent with the theory of flight-to-safety, Treasury pricesincrease when the perception of tail risk is higher. Our results concerning the predictivepower and pricing of equity tail risk extend to major government bond markets in Europe.
JEL classification:
C52, C58, G12, E43.
Keywords:
Bond return predictability, equity tail risk, bond risk premium, flight-to-safety,affine term structure model. ∗ The authors are grateful to Torben Andersen, Nicola Fusari and Viktor Todorov for valuable comments andhelpful discussions of preliminary results. This paper has been written when Ruzzi was a Research Fellow at theBank of Italy. The views expressed in this paper are those of the authors and do not necessarily reflect those ofthe Bank of Italy. † EDHEC Business School ( [email protected] ) ‡ Bank of Italy ( [email protected] ) Introduction
In times of financial distress, the disengagement from risky assets, such as stocks, and thesimultaneous demand for a safe haven, such as top-tier government bonds, generate a flight-to-safety (FTS) event in the capital markets. A large body of literature examines the linkagesbetween the stock and bond markets during crisis periods and their implications for asset pricing,see Hartmann et al. (2004), Vayanos (2004), Chordia et al. (2005), Connolly et al. (2005) andAdrian et al. (2019), among others. We add to this literature by studying how Treasury bondprices and returns respond to changes in the perceived tail risk in the stock market. If top-tiergovernment bonds are a major beneficiary of the FTS flows occurring when the stock market ishit by heavy losses, then we expect the downside tail risk of equity to affect bond risk premia anddetermine both stock and bond prices during distress periods. We investigate this conjectureby considering a Gaussian affine term structure model (ATSM) for US interest rates where thepricing factors are the principal components of the yield curve combined with the risk-neutralvolatility of the US stock market that stems from large negative price jumps. Further, we addto the existing empirical literature on bond return predictability by assessing the improvementsin forecasting accuracy obtained with equity tail risk and examining whether they translate intohigher risk-adjusted portfolio returns. Although evidence of bond return predictability basedon measures of stock market uncertainty and skew has previously been found (Feunou et al.,2014; Adrian et al., 2019; Crump and Gospodinov, 2019), this is, to the best of our knowledge,the first study to assess the economic gains of employing equity tail risk for predicting bondreturns and examine in detail its implications for pricing Treasuries in a term structure model.Understanding the dynamics of bond yields is particularly useful for forecasting financial andmacro variables, for making debt and monetary policy decisions and for derivative pricing. Mostof these applications require the decomposition of yields into expectations of future short rates(averaged over the lifetime of the bond) and term premia, i.e. the additional returns required1y investors for bearing the risk of long-term commitment. Gaussian affine term structuremodels have long been used for this purpose, see, e.g., Duffee (2002), Kim and Wright (2005)and Abrahams et al. (2016). In the setup of a Gaussian ATSM, a number of pricing factors thataffect bond yields are selected and assumed to evolve according to a vector autoregressive (VAR)process of order one. The yields of different maturities are all expressed as linear functions ofthe factors with restrictions on the coefficients that prevent arbitrage opportunities, implyingthat long-term yields are merely risk-adjusted expectations of future short rates.The selection of pricing factors typically starts by extracting from the cross-section of bondyields a given number of principal components (PCs), which are linear combinations of therates themselves. Since the seminal work of Litterman and Scheinkman (1991), the first threePCs have been prime candidates in this regard as they generally explain over 99% of the vari-ability in the term structure of bond yields and, due to their loadings, may be interpreted asthe level, slope and curvature factor. As for the second principal component, Fama and Bliss(1987) and Campbell and Shiller (1991) showed that variables related to the slope of the yieldcurve are highly informative about future bond returns. Despite the important role of thelevel, slope and curvature, it is well established in the literature that additional factors areneeded to explain the cross-section of bond returns. For this reason, the first five principalcomponents of the US Treasury yield curve are used as pricing factors in Adrian et al. (2013),while Malik and Meldrum (2016) adopt a four-factor specification for UK government bondyields. In a recent study focused on the US bond market, Feunou and Fontaine (2018) showthat a term structure model that includes the first three principal components and their ownlags delivers better forecasts of excess returns than a specification using the first five principalcomponents of yields as risk factors. Furthermore, several studies suggest that a great deal ofinformation about expected excess returns – the bond risk premium – can be found in factorsthat are not principal components of the yield curve. Cochrane and Piazzesi (2005) discover anew linear combination of forward rates which is a strong predictor of future excess bond re-2urns and, based on this evidence, Cochrane and Piazzesi (2008) use it in an ATSM along withthe classical level, slope and curvature factors. More recently, Cooper and Priestley (2008),Ludvigson and Ng (2009), Duffee (2011), Joslin et al. (2014), Cieslak and Povala (2015) andHuang et al. (2019) show that valuable information about bond risk premia is located outsideof the yield curve and contained, for example, in macro variables that have little or no impacton current yields but strong predictive power for future bond returns.This paper explores the use of factors, other than combinations of yields, to drive the curveof US Treasury rates and explain bond returns. In contrast to the vast majority of previousstudies, however, we draw on the literature that deals with comovement in the equity and bondmarkets and we consider the possibility that pricing factors of Treasury bonds originate also inthe stock market. The findings of Connolly et al. (2005) and Baele et al. (2010) indicate thatmeasures linked to stock market uncertainty explain time variation in the stock-bond returnrelation and have important cross-market pricing effects. Therefore, we select a risk measurewhich is known to predict the equity risk premium and we examine its role in the Treasury bondmarket. The existing literature suggests that the variance risk premium (VRP) forecasts thestock market returns at shorter horizons than do other predictors like dividend yields or price-to-earning ratios, see Bollerslev et al. (2009), Bollerslev et al. (2014) and Bekaert and Hoerova(2014), among others. In view of recent studies showing that the predictive power of the VRPfor the equity risk premium stems from a jump tail risk component that capture the investors’fear of a market crash (see, e.g., Andersen et al. (2015, 2019a), Bollerslev et al. (2015) andLi and Zinna (2018)), we opt for the left jump volatility measure of Bollerslev et al. (2015)to assess the impact of equity tail events on US Treasury bonds. Building on the findings of Connolly et al. (2005) find that when the implied volatility from equity index options, measured by the VIX,increases to a considerable extent, bond returns tend to be higher than stock returns (flight-to-quality) and thecorrelation between the two assets over the next month is lower. Baele et al. (2010) show that the time-varyingand sometimes negative stock-bond return correlations cannot be explained by macro variables but instead byliquidity factors and the variance risk premium, which represents the compensation demanded by investors forbearing variance risk and is defined as the difference between the risk-neutral and statistical expectations of thefuture return variation. Although the variance risk premium is a major contributor to the stock-bond returncorrelation dynamics, Baele et al. (2010) find significant exposures to it only for stock but not for bond returns. As opposed to Crump and Gospodinov (2019), we do not rely on risk-neutral skewness tomeasure equity tail risk as the computation of moments higher than the second is prone to nu-merical errors and instability. Instead, we rely on the procedures put forth by Bollerslev and Todorov(2011) to proxy investor fears for jump tail events. Specifically, we estimate equity tail risk withthe model-free measure of left tail volatility developed by Bollerslev et al. (2015) and calculatedfrom short-dated deep out-of-the-money put options on the S&P 500 market index. By doingso, we gauge the market’s perception of jump tail risk over the following month based on therisk-neutral expectation of future return volatility associated with large negative price jumps. The equity tail risk factor so obtained is by construction a measure of downside tail risk and inthis it also differs from the CBOE VIX Index which is a symmetric risk measure that reflectscompensation for both diffusive and jump risk. With the Bollerslev et al. (2015) measure inhand, we test whether equity tail risk is priced in the US term structure and examine whether We stress that our pricing methodology differs from that of Farago and T´edongap (2018), who price Treasurybonds (and many other types of assets) using a consumption-based general equilibrium model that includes anon-risk-neutralized measure of downside risk. Liu and van der Heijden (2016) discuss the difficulties associated with the computation of risk-neutral skew-ness using the method by Bakshi et al. (2003), on which the CBOE Skew Index is also based. They note howdifferent approaches to the implementation of the Bakshi et al. (2003) method have led to mixed results in theliterature of stock return predictability. With regard to this, the negative relationship between the Bakshi et al.(2003) measure of skewness and future returns found by Bali and Murray (2013) and Conrad et al. (2013) con-trasts sharply with the positive relationship found by Rehman and Vilkov (2012) and Stilger et al. (2016). As a robustness check, we also used a simple alternative measure of downside risk perceptions, the S&P 500implied volatility skew (or smirk), defined as the difference between the out-of-the-money put implied volatility(with delta of 0.20) and the average of the at-the-money call and put implied volatilities (with deltas of 0.50),both calculated from options with an expiration of 30 days (An et al., 2014; Xing et al., 2010). The results,which are available upon request, are very similar to those described here with the left jump volatility measureof Bollerslev et al. (2015). The econometric framework consists of reduced-form predictiveregressions that use the measure of equity tail risk to forecast monthly excess Treasury returns,and a Gaussian ATSM that uses equity tail risk to drive the curve of US interest rates. Moreover,the novel three-pass method of Giglio and Xiu (2019), which delivers an estimate of a factor’srisk premium that is robust to the omitted variable and measurement error problems, allows usto corroborate our conclusions regarding the pricing of equity tail risk in the bond market.Our results can be summarized as follows. First, there exist significant interactions betweenthe future one-month returns of the US government bond market and the option-implied lefttail volatility of the stock market. The frequency at which we uncover the predictive power ofequity tail risk for bond risk premia is considerably higher than that of the business cycle, whichis normally used to interpret return predictability over forecast horizons of one quarter or longer.By contrast, the short-term predictability documented in this paper may be associated with theinstantaneous reactions of market participants that, fearing a stock market crash, flock to theperceived safety of Treasuries. Second, the predictability afforded by the equity tail factorcontinues to hold out-of-sample and can sometimes yield substantial economic value to a mean- The option-implied left tail volatilities are computed daily and then the month-end value is recorded. Tominimize the impact of outliers and help smooth out the estimation error, we also considered monthly estimatesof equity tail risk obtained by averaging over the last five days of the month with the results being very similarto the ones reported below for their end-of-month counterparts. Adrian et al. (2019) find that a nonlinear function of the VIX can predict both stock and bond returns atforecast horizons of about five months or longer. We show that the predictive power of the VIX for the futureone-month returns on bonds is completely subsumed by the equity tail factor. Our study is also related to thework of Kaminska and Roberts-Sklar (2015), who document the importance of global market sentiment for theterm structure of UK government bonds using a VRP-based proxy of risk aversion. The short-term predictability of the US term structure that we find is also in agreement with the fact thatthe investors’ fear of a market crash decreases with the time horizon (Li and Zinna, 2018).
Data
In this section we present the data sources and methods used to construct the monthly timeseries of excess Treasury returns and equity tail risk measure. All time series are generated overthe period January 1996 to December 2018 with data recorded at the end of each month.
We compute Treasury bond returns using the G¨urkaynak et al. (2007) zero-coupon bondyield curve derived from observed US government bond prices. We consider maturities up toten years, for which we construct non-overlapping one-month holding period returns. Followingthe studies of Adrian et al. (2013), Abrahams et al. (2016) and Gargano et al. (2019), we definethe monthly return of the bond with maturity n (in months) as the return from buying an n -maturity bond and selling it as an ( n − n = 1 month yield, the monthly excess log-return at date t + 1 (i.e., from theend of month t to the end of month t + 1) for the generic bond with maturity n at time t getscomputed as rx ( n − t +1 = − ( n − y ( n − t +1 + n y ( n ) t − y (1) t , (1)where y ( i ) t is the annualized (but not in percentage) continuously compounded yield on thezero-coupon bond with maturity i at time t , provided by G¨urkaynak et al. (2007).Table 1 provides descriptive statistics for one-month excess returns on US Treasury bondswith maturity n = 12 , , , , , ,
120 months. A quick inspection of Panel A revealsthat longer-term bonds are characterized by higher mean excess returns and higher volatility. The G¨urkaynak et al. (2007) yield data are available at a daily frequency for an-nually spaced maturities ranging from 1 to 30 years from the Federal Reserve website . The parameters of theNelson-Siegel-Svensson model used by G¨urkaynak et al. (2007) are also published, thus allowing to retrieveyields for any desired maturity, including the longer ones. The advantages of using non-overlapping one-month returns instead of the more conventional overlappingone-year returns are explained in Gargano et al. (2019). Throughout the rest of the paper, the terms “returns” and “excess returns” are used interchangeably toindicate excess returns unless otherwise indicated by the particular context.
The equity tail risk factor of this paper corresponds to the Bollerslev et al. (2015) measureof left jump tail volatility implied by short-dated deep out-of-the-money (OTM) put optionson the US stock market index. This measure is essentially model-free and exploits extremevalue theory to characterize the density of the risk-neutral return tails. The intuition behindit is that short-maturity OTM options remain worthless unless the investors believe that a bigjump in the underlying price will occur before the option expires. Since diffusive risk doesnot affect their price, these contracts are fundamentally suitable to estimate jump tail risk(Bollerslev and Todorov, 2011, 2014). The calculation of the Bollerslev et al. (2015) measure isbased on two parameters that must be estimated period-per-period and represent two separatesources of independent variation in the jump intensity process. The first parameter is α − t whichcontrols the time-varying rate of decay, or shape, of the left tail. Lower values of α − t areassociated with a slower rate at which the put option prices decay for successively deeper OTMcontracts, implying a fatter left tail of the risk neutral density. Bollerslev and Todorov (2014)and Bollerslev et al. (2015) show that α − t can be estimated as follow,ˆ α − t = arg min α − N − t N − t X i =2 (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) O t,τ ( k t,i ) O t,τ ( k t,i − ) (cid:19) ( k t,i − k t,i − ) − − (1 + α − ) (cid:12)(cid:12)(cid:12)(cid:12) , (2) The interested reader is directed to Bollerslev et al. (2015) for an in-depth description of the theoreticalframework since here we limit ourselves to highlighting the distinctive features and to discussing the estimationand implementation procedures. O t,τ ( k ) is the time t price of the OTM put option with time to expiration τ and (negative)log-forward moneyness k , and N − t is the total number of OTM puts used in the estimation withmoneyness 0 < − k t, < ... < − k t,N − t . The second source of variation in the jump tails comesfrom parameter φ − t which shifts the level of the jump intensity process through time. Given anestimate for α − t , the estimate of φ − t can be calculated as follows,ˆ φ − t = arg min φ − N − t N − t X i =1 (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) e r t,τ O t,τ ( k t,i ) τ F t,τ (cid:19) − (1 + ˆ α − t ) k t,i + log( ˆ α − t + 1) + log( ˆ α − t ) − log( φ − ) (cid:12)(cid:12)(cid:12)(cid:12) , (3)where r t,τ is the risk-free interest rate over the [ t, t + τ ] time interval, F t,τ is the forward priceof the underlying asset at time t and with maturity date t + τ , and the rest of the notation isas before. Following Andersen et al. (2019b), we estimate α − t at a weekly frequency, while weallow φ − t , which is less sensitive to outliers, to vary each trading day. Furthermore, we pooldata across multiple maturities for more robust estimation of both parameters.When defining left jump tail variation, Bollerslev et al. (2015) focus on asset price movesthat are unusually large relative to the current level of risk in the economy. To this end, theyuse a time-varying cutoff k t for the log-jump size that identifies, for each trading day, the startof the left tail based on the market volatility level. In our study we let k t be the threshold for anegative tail jump at the one-month horizon and we fix it at three times the maturity-normalized30-day at-the-money Black-Scholes implied volatility at time t . By substituting ˆ α − t , ˆ φ − t and k t in the expression proposed by Bollerslev et al. (2015) for the predictable risk-neutral left jumptail variation, we construct the equity tail risk measure of this paper as,TR ( eq ) t = q ˆ φ − t e − ˆ α − t | k t | ( ˆ α − t k t ( ˆ α − t k t + 2) + 2) / ( ˆ α − t ) . (4) The threshold that we use for the log-jump size, although smaller than that of Bollerslev et al. (2015)and Andersen et al. (2019b), is still able to define as jumps asset price moves of greater magnitude than thosecorresponding to the levels of moneyness used in Bollerslev and Todorov (2011) and considered sufficiently “deep”in the tails to guarantee that the effect of the diffusive price components is minimal, and that the extreme valuedistribution provides a good approximation to the jump tail probabilities . Nevertheless, we also considered largervalues for the tail cutoff, resulting in similar, but less significant, interactions between the left tail volatility ofthe stock market and future bond returns. These results are available upon request.
9o compute the equity tail risk measure in (4), which represents the (annualized) volatilitythat stems from negative return jumps greater than a threshold k t , we rely on daily data reportedby OptionMetrics IvyDB US for the European style S&P 500 equity-index options. We applythe following standard filters to our dataset. We discard options with a tenor of less than eightdays or more than forty-five days. We discard options with missing prices, options with non-positive bid prices and options with non-positive bid-ask spread. The price of the survivingcontracts is obtained as the average of bid and ask quotes. For each day in the sample, weretain only option tenors for which we have at least five pairs of call and put contracts with thesame strike price. We exploit these cross sections to derive, via put-call parity, the underlyingasset price adjusted for the dividend yield that apply to a given option tenor on a given day. We discard all in-the-money options and we retain only out-of-the-money put options withvolatility-adjusted log-forward moneyness less than or equal to − .
5. Finally, we omit anyout-of-the-money options for which the price does not decrease with the strike price. Using thedata obtained from the filtering process, we compute the end-of-month values of the S&P 500option-implied left tail volatility TR ( eq ) , which we plot in Figure 1 against the 3-month movingaverage of the Chicago National Activity Index (CFNAI) and the National Bureau of EconomicResearch (NBER) based recession periods.[ Insert Figure 1 here ]From Figure 1 it is clear that our equity tail risk measure is higher during periods of economiccontraction. However, we note that TR ( eq ) spikes also in periods when the CFNAI is above itsmean level, for instance during the Russian financial crisis in 1998 and the intensification ofthe European sovereign debt crisis in 2010 and 2011. Now turning to the descriptive statisticsreported in Table 1, we find that the annualized left tail volatility of the stock market is on The risk-free rates used in the estimation of TR ( eq ) t come from the G¨urkaynak et al. (2007) dataset describedin Section 2.1. Data for the 30-day at-the-money implied volatility used to calculate k t is from the volatilitysurface file of IvyDB OptionMetrics. ( eq ) to gauge the market’s perception of jump tail risk andexamine the response of US Treasury bonds to the downside tail risk of the stock market. In this section we describe the techniques and evaluation criteria used to investigate thepredictive content of equity tail risk for future bond returns and we outline the procedures usedin the assessment of equity tail risk pricing in the US government bond market.
The econometric framework that we adopt to evaluate bond return predictability is based onreduced-form predictive regressions that include the equity tail risk measure in (4) and, possibly,a certain number of PCs of bond yields that control for the forecasting information contained inthe yield curve. With respect to the yield predictors, we consider both the traditional level,slope and curvature factors, which are standard in the literature on bond return predictability,and the two higher-order principal components used by Adrian et al. (2013) to explain Treasuryreturn variation. Therefore, our bond return prediction models take the following form, rx ( n − t +1 = β + β TR ( eq ) t + ǫ t +1 , (5a) rx ( n − t +1 = β + β TR ( eq ) t + β PC1 t + β PC2 t + β PC3 t + ǫ t +1 , (5b) rx ( n − t +1 = β + β TR ( eq ) t + β PC1 t + β PC2 t + β PC3 t + β PC4 t + β PC5 t + ǫ t +1 , (5c) In Section 4 we assess the in-sample explanatory power of equity tail risk for bond risk premia by con-trolling for other successful return predictive factors found in the literature. Specifically, we consider theCochrane and Piazzesi (2005) bond return predictor obtained as a linear combination of forward rates, theCieslak and Povala (2015) risk-premium factor obtained from a decomposition of Treasury yields into inflationexpectations and maturity-specific interest-rate cycles, and the orthogonal component of the CBOE VIX withrespect to TR ( eq ) . ( eq ) represents the perceived tail risk in the US stock market, and PC1–PC5 are thefirst five principal components estimated from an eigenvalue decomposition of the variance-covariance matrix of zero-coupon bond yields. We include in the analysis the univariate modelof equation (5a) not only because it is a quick and inexpensive method to gauge the strengthand sign of the relation between bond returns and equity tail risk, but also because simplermodels might generate more accurate out-of-sample forecasts. In the following, we will assessthe forecasting performance of model (5a) relative to that of the Expectation Hypothesis (EH)model. The EH assumes no predictability of bond risk premia, implying that the out-of-samplemodel forecasts of bond returns are equal to a recursively updated constant based on thehistorical return mean. The performance of models (5b) and (5c) will be compared to that ofa model that includes, respectively, the first three and five PCs of bond yields alone.The relationship between equity tail risk and bond risk premia is firstly assessed by testingthe statistical significance of the coefficient of TR ( eq ) over the full sample period. The test of β = 0 is carried out not only by means of conventional inference, for which we compute theNewey-West p -values with a 12-lag standard error correction, but even with the more robustinference method developed by Bauer and Hamilton (2018). The latter addresses the small-sample distortions in bond return predictive regressions that are induced, among others, by thehigh persistence of the predictive variables. Bauer and Hamilton (2018) propose a parametricbootstrap that generates yield curve data assuming that a given factor structure underlies thebond yields and that the relevant predictive information for bond returns is entirely contained inthe yield curve. We compute Bauer and Hamilton (2018) p -values with 5,000 artificial samplesand two separate 1-month VAR processes for TR ( eq ) and the principal components of yields. As a robustness check, we have also evaluated the strength of the relationship between equity tail risk andfuture Treasury bond returns using the inference method recently proposed by Crump and Gospodinov (2019).This is a non-parametric bootstrap that accounts for the time-series and cross-sectional dependence in bondyields and generates data while remaining agnostic about the exact factor structure in the data. Based onthe Crump and Gospodinov (2019) p -values computed with resampled data from 999 boostrap replications, wecontinue to observe statistically significant relationships at the 0.10 level or lower across all maturities considered.Because of space considerations, these results are not reported in the paper, but are available upon request fromthe authors.
12o check whether the in-sample interactions between one-month-ahead bond risk premiaand equity tail risk translate into positive real-time predictive ability, we consider an out-of-sample exercise in which forecasts are recursively generated at a monthly frequency based oninformation available only at the forecast time. We estimate the models in (5a), (5b) and (5c)– and corresponding benchmarks that do not include TR ( eq ) – recursively over expanding androlling samples, where the first half of observations (1996:01-2007:06) constitutes the initialestimation period and the second half (2007:07-2018:12) constitutes the forecast evaluationperiod. Within this out-of-sample setting, we follow the approach used by Eriksen (2017) andGargano et al. (2019), among others, and we assess both the statistical and the economic valueof bond return predictability with equity tail risk. We evaluate statistical significance withthe Campbell and Thompson (2008) R OS statistic that measures the percentage reduction inmean squared prediction error (MSPE) for the out-of-sample forecasts generated by a givenmodel relative to a benchmark. For each one of the preferred models in (5a), (5b) and (5c), wecompute the Campbell and Thompson (2008) statistic as, R OS = 1 − T X t =1 (cid:16) rx ( n − t +1 − c rx ( n − t +1 (cid:17) T X t =1 (cid:16) rx ( n − t +1 − f rx ( n − t +1 (cid:17) , (6)where c rx ( n − t +1 and f rx ( n − t +1 denote, respectively, the forecasts from one of the preferred modelsthat include TR ( eq ) and the forecasts from its benchmark (either the PCs-only or EH model),and T is the number of out-of-sample forecasts. Positive values of R OS indicate higher predictiveaccuracy for the bond return prediction model that includes equity tail risk. We formally testfor predictive superiority of the preferred models using the Clark and West (2007) test. This isa statistical test of the null hypothesis of R OS ≤ R OS > t -statistic13f regressing CW t +1 = (cid:16) rx ( n − t +1 − f rx ( n − t +1 (cid:17) − h(cid:16) rx ( n − t +1 − c rx ( n − t +1 (cid:17) − (cid:16)f rx ( n − t +1 − c rx ( n − t +1 (cid:17) i , (7)on a constant term, and then computing its p -value according to the Newey-West and Bauer and Hamilton(2018) inference procedures described above. The statistic in (7) is the difference in the pre-ferred and benchmark model’s squared prediction errors adjusted for the upward bias inducedby having to estimate the additional parameter β that is 0 under the null hypothesis.Finally, we examine the economic value of the predicting capability of the models in (5a),(5b) and (5c) by looking for sizeable risk-adjusted returns in asset allocation. To this end, weconduct a portfolio exercise with a mean-variance investor that every month allocates his or herwealth between a 1-month Treasury (risk-free) bond and an n -month Treasury (risky) bond.By solving the same expected utility maximization problem as in Eriksen (2017), at time t , theinvestor optimally allocates a proportion of: w ( n ) t = 1 γ E t h rx ( n − t +1 i Var t h rx ( n − t +1 i , (8)of his or her wealth to the n -month bond, and (1 − w ( n ) t ) to the 1-month bond. E t h rx ( n − t +1 i denotes the conditional expectation of the n -month bond return, for which the investor canuse the out-of-sample forecasts generated either by one of the models that include TR ( eq ) or byits benchmark that does not use the equity tail factor as predictor. Var t h rx ( n − t +1 i denotes theconditional variance of the n -month bond return, which we estimate with the sample varianceof the returns observed over the past 10 years. γ represents the investor’s level of risk aversion.Following Thornton and Valente (2012) and Gargano et al. (2019), we assume a risk aversioncoefficient of γ = 5 but we also consider a less risk-averse investor characterized by γ = 3.Furthermore, as in the study of Huang et al. (2019), we prevent extreme positions by restricting We use the Bauer and Hamilton (2018) procedure to also bootstrap the p -values of the R OS statistic. w ( n ) t on the risky bond to lie in the interval [ − , t + 1 is given by r ( n ) P ,t +1 = y (1) t + w ( n ) t rx ( n − t +1 . (9)where y (1) t is the yield of the zero-coupon bond with 1-month maturity. The certainty equivalentreturn (CER) of the portfolio, which is defined as the average utility realized by the investorfrom using the optimal weights w ( n ) t , is given byCER ( n ) P = µ ( n ) P − γ σ n ) P , (10)where µ P = T − P Tt =1 r ( n ) P ,t +1 and σ n ) P = T − P Tt =1 (cid:16) r ( n ) P ,t +1 − µ P (cid:17) . In order to establishwhether an investor that relies on the investment signals generated by TR ( eq ) is able to improveupon the economic utility realized by an investor whose portfolio allocations do not rely onequity tail risk, we compute the difference between the CER for the investor that uses one ofthe preferred models in (5a), (5b) and (5c) and the CER for the investor that uses the corre-sponding benchmark. This difference, which we denote by ∆ ( n ) and we express in terms of anannualized percentage CER gain, can be interpreted as the portfolio management fee that aninvestor is willing to pay for the bond return forecasts produced with equity tail risk. FollowingThornton and Valente (2012), Eriksen (2017) and Huang et al. (2019), we assess portfolio per-formance using also the manipulation-proof performance (MPP) measure of Goetzmann et al.(2007). For each of the preferred models, we compute the MPP improvement relative to itsbenchmark asΘ ( n ) = 11 − γ " ln T − T X t =1 " r ( n ) P ,t +1 , y (1) t +1 − γ ! − ln T − T X t =1 " r ( n ) P ,t +1 , y (1) t +1 − γ ! , (11)where r ( n ) P ,t +1 , and r ( n ) P ,t +1 , are the realized portfolio returns associated with the preferred and15enchmark models. As with the CER gain, we report annualized percentage values for Θ ( n ) . We now introduce the term structure framework adopted in this paper and we present itsestimation procedure. To set up the model, we rely on the approach suggested by Adrian et al.(2013), which has the advantage that the pricing factors of bonds are not restricted to linearcombinations of yields. Factors can indeed also be of different origin, such as the internationalequity tail risk measure TR ( eq ) defined in Section 2.2. After deriving the data generating processof log excess bond returns from a dynamic asset pricing model with an exponentially affinepricing kernel, Adrian et al. (2013) propose a new regression-based estimation technique forthe model parameters. The linear regressions of this simple estimator avoid the computationalburden of maximum likelihood methods, which have previously been the standard approach tothe pricing of interest rates.The formulation and estimation of the Gaussian ATSM in Adrian et al. (2013) can be sum-marized as follows. A K × X t , is assumed to evolve according to aVAR process of order one: X t +1 = µ + φ X t + v t +1 , (12)where the shocks v t +1 ∼ N ( , Σ ) are conditionally Gaussian with zero mean and variance-covariance matrix Σ . Letting P ( n ) t denote the price of a zero-coupon bond with maturity n attime t , the assumption of no-arbitrage implies the existence of a pricing kernel M t +1 such that, P ( n ) t = E t h M t +1 P ( n − t +1 i . (13)The pricing kernel M t +1 is assumed to have the following exponential form: M t +1 = exp (cid:16) − r t − λ ′ t λ t − λ ′ t Σ − / v t +1 (cid:17) , (14)16here r t = − ln P (1) t is the continuously compounded one-period risk-free rate and λ t is the K × λ t = Σ − / ( λ + λ X t ) . (15)The log excess one-period return of a bond maturing in n periods is defined as follows, rx ( n − t +1 = ln P ( n − t +1 − ln P ( n ) t − r t . (16)After assuming the joint normality of { rx ( n − t +1 , v t +1 } , Adrian et al. (2013) derive the returngenerating process for log excess returns, which takes the form , rx ( n − t +1 = β ( n − ′ ( λ + λ X t ) −
12 ( β ( n − ′ Σ β ( n − + σ ) + β ( n − ′ v t +1 + e ( n − t +1 , (17)where the return pricing errors e ( n − t +1 ∼ i.i.d. (0 , σ ) are conditionally independently and identi-cally distributed with zero mean and variance σ . Letting N be the number of bond maturitiesavailable and T be the number of time periods at which bond returns are observed, Adrian et al.(2013) rewrite equation (17) in the stacked form, rx = β ′ ( λ ι ′ T + λ X ) −
12 ( B ∗ vec( Σ ) + σ ι N ) ι ′ T + β ′ V + E , (18)where rx is an N × T matrix of excess bond returns, β = h β (1) β (2) ... β ( N ) i is a K × N matrixof factor loadings, ι T and ι N are a T × N × X = [ X X ... X T − ] is a K × T matrix of lagged pricing factors, B ∗ = h vec( β (1) β (1) ′ ) ... vec( β ( N ) β ( N ) ′ ) i ′ is an N × K matrix, V is a K × T matrix and E is an N × T matrix.The main novelty of the approach taken by Adrian et al. (2013) to model the term structure of For the full derivation of the data generating process see Section 2.1 in Adrian et al. (2013). Stack the estimates of the innovations ˆ v t +1 into matrix ˆ V and use this to construct anestimator of the variance-covariance matrix ˆ Σ = ˆ V ˆ V ′ /T .2. From the excess return regression equation rx = a ι ′ T + β ′ ˆ V + cX + E , obtain estimatesof ˆ a , ˆ β and ˆ c . Use ˆ β to construct ˆ B ∗ . Stack the residuals of the regression into matrix ˆ E and use this to construct an estimator of the variance ˆ σ = tr( ˆ E ˆ E ′ ) /N T .3. Noting from equation (18) that a = β ′ λ − ( B ∗ vec( Σ ) + σ ι N ) and c = β ′ λ , estimatethe price of risk parameters λ and λ via cross-sectional regressions,ˆ λ = ( ˆ β ˆ β ′ ) − ˆ β (cid:16) ˆ a + 12 ( ˆ B ∗ vec( ˆ Σ ) + ˆ σ ι N ) (cid:17) , (19)ˆ λ = ( ˆ β ˆ β ′ ) − ˆ β ˆ c . (20)The analytical expressions of the asymptotic variance and covariance of ˆ β and ˆ Λ = [ˆ λ ˆ λ ],which we do not report here to save space, are provided in Appendix A.1 of Adrian et al.(2013). From the estimated model parameters, Adrian et al. (2013) show how to generate ayield curve. Indeed, within the proposed framework, bond prices are exponentially affine in thepricing factors. Consequently, the yield of a zero-coupon bond with maturity n at time t , y ( n ) t ,can be expressed as follows, y ( n ) t = − n [ a n + b ′ n X t ] + u ( n ) t , (21) For estimation purposes, Adrian et al. (2013) advise to set µ = 0 in case of zero-mean pricing factors. a n and b n are obtained from the following no-arbitrage recursions, a n = a n − + b ′ n − ( µ − λ ) + 12 ( b ′ n − Σb n − + σ ) − δ , (22) b ′ n = b ′ n − ( φ − λ ) − δ ′ , (23)subject to the initial conditions a = 0, b n = , a = − δ and b = − δ . The parameters δ and δ are estimated by regressing the short rate, r t = − ln P (1) t , on a constant and contemporaneouspricing factors according to, r t = δ + δ X t + ǫ t , ǫ t ∼ i.i.d. (0 , σ ǫ ) . (24)By setting the price of risk parameters λ and λ to zero in equation (22) and (23), Adrian et al.(2013) obtain a RN n and b RN n , which they use to generate the risk-neutral yields, y ( n ) RN t . Theseyields reflect the average expected short rate over the current and the subsequent ( n −
1) periodsand are computed as follows, y ( n ) RN t = 1 n n − X i =0 E t [ r t + i ] = − n [ a RN n + b RN ′ n X t ] . (25)Given equation (21) and (25), the term premium T P ( n ) t , which is the additional compensationrequired for investing in long-term bonds relative to rolling over a series of short-term bonds,can be calculated as follows, T P ( n ) t = y ( n ) t − y ( n ) RN t . (26)In the next sections we specify and estimate a term structure model for US interest rates fol-lowing the procedure outlined above. The difference between the Gaussian ATSM in Adrian et al.(2013) and ours is that we use a different set of pricing factors. Indeed, we include in X t notonly the PCs of bond yields but also the equity tail factor TR ( eq ) described in Section 2.2.19 .3 Consistent Risk Premium Estimation In this section we briefly review the method of Giglio and Xiu (2019), GX hereafter, toestimate the risk premium of an observable factor (TR ( eq ) in our case), which is valid evenwhen the observed factor is measured with noise and the model does not fully account for allpriced sources of risk in the economy. The new GX three-pass methodology combines principalcomponent analysis (PCA) with two-pass regressions (Fama and MacBeth, 1973) to consistentlyestimate the risk premium of any observed factor. The estimator relies on a large cross sectionof test assets and is valid as long as PCA can recover the entire factor space of test asset returns.In our paper we apply the GX three-pass method to the whole term structure of Treasury bondreturns to estimate and test the significance of the risk premium of the equity tail factor TR ( eq ) .Unlike the term structure model described above where the pricing kernel is an exponentialfunction of the state variables, Giglio and Xiu (2019) assume a linear stochastic discount factor.Working with a linear asset pricing model they can exploit the so-called “rotation invariance”property that allows them to estimate the risk premium γ g of an observable factor g t withoutnecessarily observing or knowing all the true factors v t entering the pricing kernel. Written inmatrix form, the GX model consists of the following two equations:¯ R = β ¯ V + ¯ U , (27)¯ G = η ¯ V + ¯ Z , (28)where ¯ R is the n × T matrix of demeaned excess returns of the test assets, ¯ V is the p × T matrixof demeaned true factors, β is the n × p matrix of factor risk exposures, ¯ U is the n × T matrixof idiosyncratic errors, ¯ G is the d × T matrix of demeaned observed factors, the risk premium ofwhich has to be estimated, η is the d × p matrix of the loadings of the observed factors on theunobserved true factors, and ¯ Z is the d × T matrix of measurement errors. The GX estimatorproceeds in three steps which can be summarized as follows:20. PCA step . The first pass consists of estimating the true factors and factor risk exposuresby extracting the first p principal components and their respective loadings from the crosssection of test asset returns. The estimators can therefore be written as: b V = T / ( ξ : ξ : ... : ξ p ) ⊺ and b β = T − ¯ R b V ⊺ , (29)where ξ , ..., ξ p are the eigenvectors corresponding to the largest p eigenvalues of n − T − ¯ R ⊺ ¯ R .2. Cross-sectional regression step . The second pass consists of estimating the risk premia ofthe latent factors by running a cross-sectional ordinary least square regression of averagerealized excess returns, ¯ r , onto the previously estimated factor loadings, b β : b γ = ( b β ⊺ b β ) − b β ⊺ ¯ r . (30)3. Time-series regression step . The third pass consists of estimating the risk premia of thefactors of interest by first running a time series regression of the demeaned candidatefactors onto the space of the latent factors and then combining these estimates with thoseof the second step. The estimator b η of the loadings on the latent factors and the estimator b γ g of the risk premia of the observed factors of interest can therefore be written as: b η = ¯ G b V ⊺ ( b V b V ⊺ ) − , (31) b γ g = b η b γ . (32)Due to space considerations, we do not provide analytical expressions for the asymptotic varianceof the risk-premium estimates and we refer the reader to Section 4 in Giglio and Xiu (2019). Giglio and Xiu (2019) propose a consistent estimator of p in their Online Appendix I.1. They also demon-strate that as long as the number of principal components used is greater than or equal to the true number offactors, the estimator of the risk premium is consistent. In our empirical analysis we report results with respectnot only to the number of principal components selected with the Giglio and Xiu (2019) criterion but also tohigher numbers of factors to ensure robustness of the estimates. η is small) orwhether it is dominated by noise ( z t is large), Giglio and Xiu (2019) define the R of the time-series regressions in the third-pass, R g = b η b V b V ⊺ b η ⊺ ¯ G ¯ G ⊺ . Furthermore, they provide a Wald test for thenull that the observed factor g is weak by formulating the hypotheses H : η = 0 vs H : η = 0.In our empirical analysis we report the R g and Wald p -value for the strength of the observedfactor g = TR ( eq ) with respect to the cross section of Treasury returns, alongside the estimateand significance of the factor’s risk premium. In this section we present our empirical results. We first consider in Section 4.1 the full-sample least-squares estimates for the bond return prediction models with equity tail risk. Weempirically show that the equity tail factor TR ( eq ) significantly predicts monthly bond returnsin- and out-of-sample and the more accurate forecasts can be of economic importance for aninvestor facing portfolio decisions. In Section 4.2 we discuss the estimates of the Gaussian ATSMwhich allow to explore in detail the effects of equity tail risk on bond prices and determinewhether TR ( eq ) is a priced source of risk in the term structure of US interest rates. Section 4.3corroborates the existence of a significant market price of equity tail risk in the US governmentbond market using the GX three pass method. Finally, Section 4.4 investigates to what extentequity tail risk affects the government bond market of countries other than the United States. We start by examining the interactions between the one-month returns of US Treasury bondsand the S&P 500 option-implied volatility that stems from large negative price jumps, TR ( eq ) .22sing the full sample (1996:01-2018:12) of monthly data, we run the predictive regressions in(5a), (5b) and (5c), for which we report in, respectively, Panels A, B, and C of Table 2 theleast-squares estimates of the slope coefficients and their corresponding p -values. Numbers atthe bottom of each panel correspond to the adjusted R -squared of the predictive regressions thatinclude and exclude TR ( eq ) as predictor, and to the p -value of an F -test of the null hypothesisthat the regression that includes TR ( eq ) does not give a significantly better fit to the datathan does a regression without it. In order to ease interpretation of the results, all predictors,including those discussed later, have been normalized to have a zero mean and a standarddeviation of one. Here and in the rest of this section, evidence is presented for returns on theone-, two-, three-, four-, five-, seven- and ten-year Treasury bonds ( n = 12 , , , , , , ( eq ) is statistically significant at well below the 0.05 level across the whole yield curve. Looking atthe size of the coefficient, we observe that the impact of equity tail risk on bond risk premia ismonotonically increasing with the bond maturity. Our estimates suggest that a one standarddeviation increase in the equity tail factor raises the expected annualized return on the 1-yearand 10-year Treasury bonds by about 0.5% and 6.2%, respectively. Furthermore, we note thatfor all maturities considered, the sign of the coefficient is positive. This result is in sharp contrastwith that obtained by Crump and Gospodinov (2019) with a conceptually very different measureof equity tail risk. It can however be explained in light of the opposite movements in equity andbond prices observed in times of stress and the considerations raised by previous studies thatfound a negative relation between future stock returns and measures of option-implied volatility,23ee, among others, Xing et al. (2010) and An et al. (2014). That is, if we believe that informedtraders with negative news choose the option market to trade first, then an increase in tail riskis later accompanied by lower and higher prices on, respectively, the equity and bond markets,which are slow in incorporating the information embedded in the option volatility surface.Since the literature on bond return predictability is more often interested in the forecastingpower of a variable beyond that of the information contained in the yield curve, we now discussthe results reported in Panels B and C of Table 2. When controlling for yield curve factorswith the first 3 and 5 PCs, the coefficient associated with TR ( eq ) remains positive and highlysignificant for all bond maturities. We find strong significance not only with the standardNewey-West p -values but also with the more robust p -values computed with the bootstrapprocedure of Bauer and Hamilton (2018). Furthermore, we note that the inclusion of equitytail risk in the predictive regressions determines sizeable changes in the adjusted R s, whichnearly double in Panel B and increase by about 50% in Panel C. Finally, the F -test resultsconfirm the importance of TR ( eq ) for explaining the one-month-ahead variation in bond riskpremia.In addition to our baseline regressions in (5a), (5b) and (5c), we examine whether equity tailrisk remains a strong predictor of future bond returns even when controlling for other suc-cessful return forecasting factors found in the literature. Specifically, we report in PanelsD and E of Table 2 the results of regressions that use the equity tail factor in combinationwith, respectively, the Cochrane and Piazzesi (2005) and Cieslak and Povala (2015) factors.The Cochrane and Piazzesi (2005) bond return predictor is obtained as a linear combinationof forward rates, while the Cieslak and Povala (2015) risk-premium factor is obtained from adecomposition of Treasury yields into inflation expectations and maturity-specific interest-ratecycles. Due to the low correlation that exists between the covariates, Treasury risk premia In results available upon request, we also considered specifications of the regression equations (5b) and (5c)that make use of the orthogonal component of TR ( eq ) with respect to the principal components. The coefficientin front of the equity tail factor continues to be statistically significant at the 0.05 level or lower for all maturities. ( eq ) and the CBOE VIX unspanned by TR ( eq ) aspredictors. The immediate point that stands out here is that the VIX components that are notrelated to our equity tail factor, i.e. continuous return variation and right jump variation, arehighly insignificant for almost all bond maturities. Based on this result, we can conclude thatthe VIX does not have predictive power over-and-above TR ( eq ) for future bond returns.We now discuss the out-of-sample performance of the models in (5a), (5b) and (5c), whichpredict bond returns with the S&P 500 option-implied tail risk measure TR ( eq ) . The accuracyof the bond return forecasts of model (5a) is measured relative to the recursively updatedforecasts from the EH model that projects returns on a constant, while the accuracy of theforecasts of models (5b) and (5c) is measured relative to the forecasts of the models that onlyinclude the principal components as predictors. Table 3 reports the Campbell and Thompson(2008) out-of-sample R OS values for each model, alongside the p -value of the Clark and West(2007) MSPE-adjusted statistic for testing H : R OS ≤ H : R OS >
0. We reportresults for both increasing and rolling windows of past data used in the estimation method. Theout-of-sample period is 2007:07–2018:12.[ Insert Table 3 here ]Overall, the results in Table 3 suggest that the good in-sample fit provided by TR ( eq ) anddiscussed above translates into positive out-of-sample performance. For instance, when thebenchmark is the EH model, we find that equity tail risk improves the out-of-sample bond returnpredictions across all maturities. The gains are in the range of 1.6% to 4.3% for both windowestimations, with the largest improvements observed for medium-maturity bonds. We note thatwith the robust inference method developed by Bauer and Hamilton (2018) the increases in the R OS s are significant in a statistical sense for bond maturities greater than 2 years, while the p -25alues of the Clark and West (2007) MSPE-adjusted statistic are lower than 10% for maturitiesof 5 years or longer. Similarly, we observe positive values of R OS in Panels B and C indicatinghigher predictive accuracy for the bond return prediction models that include TR ( eq ) comparedto their PCs-only benchmark specifications. Except for the 10-year bond, the bootstrap p -valuesof both R OS and Clark and West (2007) MSPE-adjusted statistic are below 0.1, thus provingthe statistical significance of the results.Next, we examine the economic value of using equity tail risk to make one-month-aheadpredictions of Treasury bond returns. Table 4 reports values for the CER gain (∆) andGoetzmann et al. (2007) MPP improvement (Θ) that an investor can achieve by switchingfrom a benchmark to a model that uses the equity tail factor TR ( eq ) to predict bond returns.Results are based on the out-of-sample model forecasts produced for the period 2007:07–2018:12with predictive models that are recursively estimated with a rolling window approach.[ Insert Table 4 here ]From an investment perspective, the results in Table 4 indicate that predicting bond returnswith equity tail risk can generate substantial risk-adjusted returns. This is particularly thecase for an investor that can use TR ( eq ) alongside the first 5 PCs of bond yields to predictthe one-month-ahead returns of Treasuries with maturities in the range of two to seven years.Specifically, we find that the investor is willing to pay from 80 up to 360 basis points per yearto switch from the 5 PCs-only benchmark to the model that forecasts bond returns also withequity tail risk. Even when the benchmark is the EH model, we find that an investor tradingsome specific medium-term bonds is better off following the return forecasts based on equitytail risk. On the other hand, when the benchmark is the 3 PCs-only model, the investor cannotachieve any asset allocation gains by switching to the predictive model with equity tail risk.Finally, we briefly discuss how the forecast performance of the models in (5a), (5b) and (5c)is related to the real economy. Panels A and B of Table 5 report contemporaneous correlations26etween the out-of-sample forecasts of one-month-ahead Treasury bond returns and the CFNAIand the macroeconomic uncertainty index ( U MACRO ) constructed by Jurado et al. (2015). Wenote that the bond risk premia implied by any of the three models are countercyclical as theyare negatively correlated with macroeconomic condition. This is a common result found inthe literature on bond return predictability and is consistent with economic theories in whichinvestors require compensation for bearing business cycle risk, see, e.g., Eriksen (2017) andreferences therein. In order to understand whether the models that include TR ( eq ) as predictorperform well in recessions or expansion periods, Panels C and D of Table 5 report contempo-raneous correlations between the models’ relative forecast and portfolio performance and theCFNAI. The relative forecast performance is defined as the difference in cumulative squaredprediction error (DCSPE), while the relative portfolio performance is defined as the differencein cumulative realized utilities (DCRU). As we can see, the forecasting performance of the threemodels tends to be positively correlated with the CFNAI, indicating superior model perfor-mance in good times when the CFNAI is high. Looking at the relative portfolio performancegives less clear-cut results since the correlations vary substantially across maturities. In fact,asset allocation gains seem to be achievable during expansion periods for short-term bonds andduring recessions for long-term bonds.[ Insert Table 5 here ] On the basis of the significant interactions observed between future Treasury returns andthe equity tail factor TR ( eq ) , it is of interest to examine to what extent the left tail volatilityof the stock market also affects the current level of bond prices. Figure 2 shows the time trendof Treasury bond yields against periods of elevated equity tail risk, corresponding to whenTR ( eq ) is above its historical 85-th percentile. As it can be seen in the graph, many of the most27emarkable declines in Treasury rates occurred at times of elevated equity tail risk. In fact, theaverage contemporaneous correlation between bond yields and TR ( eq ) is about -0.15.[ Insert Figure 2 here ]To investigate the role of equity jump tail risk in pricing US government bonds, we now estimatethe Gaussian ATSM of Section 3.2 with the inclusion of our equity tail factor in the vector ofstate variables. In addition to TR ( eq ) , however, we also need pricing factors that summarizethe information contained in the yield curve. To this end, we extract the first five principalcomponents of the US yield curve, which have proven to be remarkably effective in fitting thecross-section of bond yields and returns in Adrian et al. (2013). Based on this evidence, we letthese PCs drive the interest rates of our model as well, but with a slight modification of themethodology. Indeed, in order to have pricing factors that are uncorrelated with each other,we follow Cochrane and Piazzesi (2008) and extract the principal components not from theconventional yields, but instead from the yields orthogonalized to the extra factor, which in ourstudy is TR ( eq ) . By doing so, we obtain yield curve factors that are unrelated to the pricing oftail risk in the stock market, which is entirely ascribed to the TR ( eq ) factor. In view of theseconsiderations, we employ the following set of pricing factors in our Gaussian ATSM, X t = h TR ( eq ) t , PC1 t , PC2 t , PC3 t , PC4 t , PC5 t i ′ , (33)where TR ( eq ) is the S&P 500 option-implied measure of left tail volatility, and PC1–PC5 are thefirst five principal components estimated from an eigenvalue decomposition of the covariancematrix of zero-coupon bond yields of maturities n = 3 , , ...,
120 months, orthogonal to TR ( eq ) .All factors have mean zero and unit variance, and they are plotted in Figure 3. The panels ofPC1–PC5 also present the principal components of the conventional non-orthogonalized bondyields. We find that estimates of the factors extracted using the two yield curves track each28ther quite closely, with the largest differences occurring for PC2 and PC3 at the onset of thefinancial crisis. Therefore, the orthogonalization of the rates with respect to TR ( eq ) does notappear to significantly alter the interpretation and role of the principal components in describingthe characteristics of the US Treasury yield curve.[ Insert Figure 3 here ]Given the vector of state variables in (33), we estimate our Gaussian ATSM using themethod put forward by Adrian et al. (2013) and discussed in Section 3.2. In particular, we useone-month excess returns for Treasury bonds with maturities n = 6 , , ...,
120 months to fitthe cross-section of yields. The summary statistics of the pricing errors implied by our termstructure model, which accounts for equity tail risk, and a benchmark model based on only thefirst five PCs of the yield curve are provided in Table 6. Overall the results indicate a good fitbetween the data and the proposed model with equity tail risk. Indeed, both the mean and thestandard deviation of our yield pricing errors remain well below a basis point for all maturitiesand they never exceed, in absolute value, those of the benchmark. As for the return pricingerrors, we notice that explicitly including the equity tail risk factor TR ( eq ) in a Gaussian ATSMcan improve the fit especially to the short end of the US yield curve. Moreover, consistentwith the way Adrian et al. (2013) construct their framework for the term structure of interestrates, we observe a strong autocorrelation in the yield pricing errors and a negligible one in thereturn pricing errors, except for the 3-year bond. The success of our model in fitting the yieldcurve is shown graphically in the left panels of Figure 4. In these plots, the solid black linesof observed yields are visually indistinguishable from the dashed gray lines of model-impliedyields. Similarly, the right panels of Figure 4 display the tight fit between actual and fittedexcess Treasury returns. The dashed red lines plot the model-implied dynamics of bond term In results available upon request, we have found significant relationships only between TR ( eq ) and PC2 andPC3 of the conventional non-orthogonalized bond yields. Both correlation coefficients were around − . β i be the i -th column of β ′ , theWald statistic, under the null H : β i = N × , is defined as follows, W β i = ˆ β ′ i ˆ V − β i ˆ β i α ∼ χ ( N ) , (34)where ˆ V β i is an N × N diagonal matrix that contains the estimated variances of the ˆ β i coefficientestimates. The results of the Wald test on the pricing factors of both the proposed ATSMwith equity tail risk and the benchmark PC-only specification are shown in Table 7. As we cansee, we strongly reject the hypothesis of unspanned factor for each of our state variables. Thismeans that the data support the use of the equity tail factor TR ( eq ) , together with the yieldcurve factors indicated by Adrian et al. (2013), for pricing government bonds in the US marketover the period 1996 – 2018. [ Insert Table 7 here ]We now examine whether the risk factors that we use in our Gaussian ATSM are priced inthe cross-section of Treasury returns. To this end, we follow Adrian et al. (2013) and performa Wald test of the null hypothesis that the market price of risk parameters associated with a See Appendix A.1 in Adrian et al. (2013) for the analytical expressions of the asymptotic variance of theestimators. λ ′ i be the i -th row of Λ = [ λ λ ], theWald statistic, under the null H : λ ′ i = × ( K +1) , is defined as follows, W Λ i = ˆ λ ′ i ˆ V − λ i ˆ λ i α ∼ χ ( K + 1) , (35)where ˆ V λ i is a square matrix of order ( K + 1) that contains the estimated variances of the ˆ λ i coefficient estimates. In addition, in order to test whether the market prices of risk are time-varying, Adrian et al. (2013) propose the following Wald test which focuses on λ and excludesthe contribution of λ . Letting λ ′ i be the i -th row of λ , the Wald statistic of this second test,under the null H : λ ′ i = × ( K ) , is defined as follows, W λ i = ˆ λ ′ i ˆ V − λ i ˆ λ i α ∼ χ ( K ) . (36)In Table 8, we report the estimates and t -statistics for the market price of risk parameters inthe proposed Gaussian ATSM, together with the Wald statistics and p -values for the two testsjust described. Examining the first row of the table, we note that equity tail risk, as measuredby exposure to TR ( eq ) , is strongly priced in our term structure model with a p -value of 8.5%.We detect statistically significant time variations in the market price of equity tail risk, whichare mostly explained by the level and curvature components of bond yields. Furthermore, whenlooking at the t -statistics in the second column of the table, we note that TR ( eq ) is an importantdriver of the market price of level risk. Finally, we observe that PC2 carries a significant priceof risk in our term structure model. This result, together with the fact that Adrian et al. (2013)find a significant market price of slope risk only after adding an unspanned real activity factorto their framework, corroborates the hypothesis that valuable information about bond premiais located outside of the yield curve. See Appendix A.1 in Adrian et al. (2013) for the analytical expressions of the asymptotic variance of theestimators.
31 Insert Table 8 here ]We now discuss the impact of the state variables of our Gaussian ATSM on the pricingof Treasury bonds. The loadings of the yields on all model factors are reported in Figure 5,whereas the loadings of the expected one-month excess returns are displayed in Figure 6. Froman examination of the state variables that are in common with the work of Adrian et al. (2013),we can see that our results are broadly consistent with the well-established role of these factors.Indeed, given the sign of the yield loadings on PC1, PC2 and PC3, we can argue that the firstthree principal components of yields preserve in our study the interpretation of, respectively,level, slope and curvature of the term structure. Moreover, the yield loadings on PC4 and PC5are both quite small, reflecting the modest variability of bond rates explained by these factors.As can be seen from Figure 6, however, all the principal components, including the higherorder ones, are important to explain variation in Treasury returns. Specifically, in line withprevious findings concerning the predictability of bond returns with yield spreads, our evidencesuggests that an increase in the slope factor forecasts higher expected excess returns on bondsof all maturities. Now turning to the new pricing factor that we propose in this paper, weobserve from the top left panel of Figure 5 that the yield loadings on TR ( eq ) are negative acrossall maturities. These results imply that bond prices, which move inversely to yields, rise inresponse to a contemporaneous shock to the equity left tail factor. And since, by construction,TR ( eq ) is associated with a downturn in the stock market, we confirm the hypothesis that USTreasury bonds benefit from flight-to-safety flows during periods of turmoil. Judging by themagnitude of the coefficients, the immediate flight-to-safety effect is stronger on shorter-termbonds. We find that a one standard deviation increase in the TR ( eq ) factor is associated witha reduction of about 40 basis point in the yields of Treasuries with maturities ranging from six In results available upon request, we found that the contemporaneous correlation between TR ( eq ) and theFama and French (1993) market factor is -0.35. Also, there is a negative but insignificant relation between TR ( eq ) and the one-month-ahead stock market returns, as measured by the Fama and French (1993) market factor. ( eq ) displayed in the top left panel of Figure 6 confirm the previously established positiverelation between the left tail volatility of the stock market and the one-month-ahead risk premiaof the US government bond market. Due to the convenient orthogonalization of pricing factorsdescribed at the start of this section, we are able to quantify the effects of a shock to the equitytail factor on the bond risk premia. In particular, we find that a one standard deviation increasein the TR ( eq ) factor raises the annualized expected excess return by approximately 1% for the2-year bond and 6% for the 10-year bond. The effect is linearly related to the bond maturity.[ Insert Figure 5 here ][ Insert Figure 6 here ]We conclude this section by discussing how equity tail risk has affected the trend of yields,risk-neutral rates and term premia over the course of time. To this end, we calculate thecomponent of fitted yields in equation (21) and the component of their risk-neutral counterpartsin equation (25) that the model attributes to the equity left tail factor TR ( eq ) . Similarly, wedetermine the contribution of equity tail risk to the bond term premia in equation (26) asthe difference between the component of fitted yields and the component of their risk-neutralcounterparts that the model ascribes to TR ( eq ) . The left panels of Figure 7 illustrate the effectof the equity left tail factor TR ( eq ) on the dynamics of the 1-, 5- and 10-year Treasury yields,whereas the right panels display the effects on the expected future short rate and term premiumembedded in those rates. The following remarks can be made by observing Figure 7. The effectof equity tail risk is much smaller (in absolute value) for the bond term premium than for theexpectation of future short rates. Therefore, when the equity left tail factor TR ( eq ) increases, the33eduction in the expected future short rate more than offsets the increase in the term premium.As a result, bond yields fall in periods of elevated equity tail risk. However, it is interesting to seethat, although the same pattern is observed for all yields in Figure 2, the equity left tail factorTR ( eq ) has influenced the downward trend of rates differently depending on the bond maturity.Indeed, from the left panels of Figure 7, it appears that the dynamics of short-maturity bondyields was strongly affected by equity tail risk, whereas the response of longer-maturity rateswas consistently negligible. This further corroborates our previous conclusion that short-termbonds provide a more effective shelter against equity market losses than long-term bonds do.[ Insert Figure 7 here ]To better visualize how the impact of equity tail risk varies across maturities and in time,Figure 8 shows the effect of the TR ( eq ) factor for the whole term structure calculated on selecteddates: August 1998, October 2008, September 2011, and May 2013. Interest rates fell on alldates except for May 2013, when yields markedly rose with the announcement of the FederalReserve’s “taper tantrum”. On that occasion, as it can be seen from the figure, TR ( eq ) did notplay any role in the yield changes. On the other hand, at the peak of the 2008-09 financial crisis,we measure the impact of equity tail risk on bond yields to be larger than -200 basis points forTreasuries with maturities up to four years, while it is reduced to only -66 basis points for the10-year Treasury. The rates showed strong downward oscillations also in the summer of 1998and the second half of 2011, when the equity left tail factor increased in response to, respectively,the collapse of Long Term Capital Management fund and the intensification of the Europeansovereign debt crisis. In both these instances, the extent of the reduction in short-term bondrates that can be credited to equity tail risk is approximately 100 basis points.[ Insert Figure 8 here ]34n conclusion, we can state that equity jump tail risk has been a dominant factor for theevolution of the short end of the US Treasury yield curve. In particular, while the unconventionalmonetary policies introduced by central banks to mitigate the severity of the financial crisis havebeen a major force in lowering longer-term yields (Kaminska and Zinna, 2018), the reduction inshorter-term yields can be associated with the investors’ increased fear of a stock market crash. To address the concern that the rows in the price of risk parameters λ and λ in (15)corresponding to TR ( eq ) can only be weakly identified because our equity tail factor is weaklyspanned by bond yields, we now provide further evidence for a significant price of equity tailrisk in the US government bond market. This is done by estimating the risk premium of TR ( eq ) with the novel three-pass procedure of Giglio and Xiu (2019). The results of the GX three-passmethod applied to the whole term structure of Treasury bond returns are reported in Table 9.[ Insert Table 9 here ]We start by examining the results reported in column p = 5, which corresponds to the numberof principal components of bond returns selected with the criterion of Giglio and Xiu (2019). For this number of latent factors, we find that the estimated risk premium of TR ( eq ) in the USTreasury bond market is statistically significant at the 10% level. Although obtained with adifferent asset pricing model, this results is well in line with the estimates of the ATSM presentedin the previous section. Furthermore, we provide evidence against the hypothesis that TR ( eq ) is measured with noise or weakly reflected in the cross section of government bond returns. Infact, the R of the time-series regression in the third-pass of the GX procedure amounts to 0.09and we reject, at the 5% significance level, the null of TR ( eq ) being a weak factor. If we now lookat the estimates obtained with a higher number of latent factors, we observe robustness of our See Online Appendix I.1 in Giglio and Xiu (2019) for a consistent estimator of p . p . Even when using eight principal components,the market price of equity tail risk is still significant at the 0.1 level. However, including theprincipal components beyond the fifth one does not result in further noticeable improvement inthe regression R . On the other hand, we find that much of the information about equity tailrisk is contained in the slope factor, with the R that jumps from 0.04 to 0.08 when the secondprincipal component is included in the model. In this subsection, we extend our empirical analysis of bond pricing and return predictabilityto the Treasury market of United Kingdom, Germany, Switzerland, France, Italy and Spain.First, we explore to what extent the S&P 500 option-implied tail risk measure TR ( eq ) affectsthe Treasury market of countries other than the United States. Then, we estimate country-specific measures of equity tail risk and investigate the relation between these measures and thegovernment bond market in the corresponding European country. To compute the one-monthholding period returns on Treasuries in Europe, we construct a data set of end-of-month zero-coupon interest rates that extends from January 1996 to December 2018. We collect data forthe United Kingdom (UK) from the Bank of England, for Germany (DE) from the Bundesbankand BIS database, for Switzerland (CH) from the Swiss National Bank and BIS database, forItaly (IT) and Spain (ES) from the BIS database, while for France (FR) we fit a Nelson-Siegel-Svensson model to the constant maturity yields from Datastream. As for the country-specificmeasures of equity tail risk, we follow the methodology outlined in Section 2.2 and estimateoption-implied volatility that stems from large negative price jumps using daily data reportedby OptionMetrics IvyDB Europe for the European style FTSE 100 (UK), DAX 30 (DE), SMI(CH), CAC 40 (FR), FTSE MIB (IT), and IBEX 35 (ES) equity-index options. Data is availablefrom January 2002 to December 2018 for UK, DE and CH, from January 2007 to December 2018for IT and FR, and from May 2007 to December 2018 for ES. We use option-implied left tail36olatilities recorded at the end of the month for UK, DE and CH, while we use the average valueover the last five days of the month for FR, IT, and ES since their less liquid option marketsyield a much noisier measure of equity tail risk. Figure 9 displays the time series of theseinternational equity tail risk measures along with the S&P 500 option-implied measure TR ( eq ) .Comparing the left tail volatility of the US stock market to that of the UK, German, Swiss andFrench stock markets, we note a strong coherence between the series with all the correlationcoefficients above 0.70. At the same time, however, there are also some important differences.In particular, we note that in 2002-03 the UK, DE and CH tail risk measures attained highervalues and remained elevated for a much longer period of time than TR ( eq ) , which howeverexhibits more pronounced peaks in the aftermath of the recent financial crisis. With regard tothe equity tail risk measures of Italy and Spain, their series diverge quite substantially fromthat of the US measure with correlation coefficients of only 0.50 and 0.20, respectively.[ Insert Figure 9 here ]We begin by assessing the predictive power of the left tail volatility of the US stock marketfor future one-month returns on the government bond market of the European countries. To thisend, we estimate the predictive regressions in (5a), (5b) and (5c) using international bond re-turns on the left hand side of the equations and TR ( eq ) , combined with the principal componentsof the country-specific yield curves, on the right hand side. For each Treasury market, Table10 reports the full-sample estimates of the coefficient of TR ( eq ) and the corresponding p -valuescomputed with both Newey-West and Bauer and Hamilton (2018) inference procedures.[ Insert Table 10 here ]Overall, the results in Table 10 indicate that the perceived tail risk in the US stock markethas significant explanatory power for future returns on Treasury bonds in the UK, Germany,37witzerland and France. When we do not control for yield curve factors in the return predictiveregressions, the coefficient of TR ( eq ) , assessed with the robust inference method developed byBauer and Hamilton (2018), is statistically significant at the 0.05 level or lower for all maturitiesof UK, DE and CH bonds, and at the 0.10 level or lower for all maturities of FR bonds.Controlling with the first three or five principal components of bond yields does not change theresults for the UK and DE Treasuries, while it reduces the significance for the longer maturitiesof CH and FR bonds. Consistent with the results in Table 2, the sign of the coefficient is positive,implying that higher equity tail risk is associated with an increase in the one-month-ahead bondrisk premia. In contrast to the results obtained with the UK, DE, CH and FR bonds, the equitytail risk factor TR ( eq ) does not seem to help explain time variations in the bond risk premiaof Italy and Spain. In fact, the explanatory power of TR ( eq ) is never statistically significant atthe 10% level for IT bonds with maturity greater than one year, and is at most significant atthat level for the short-term ES bonds. These results point to the possible role that countryrisk may play in the identification of a safe asset when the equity market tumbles. It is indeedpossible that, in periods of stress, international investors shift their holdings into instrumentslike the “safe” German Bund rather than debt issued by fiscally weak sovereigns, such as Italyand Spain. Due to the mostly insignificant interactions observed in-sample between TR ( eq ) andTreasury bonds of Italy and Spain, we do not consider the out-of-sample forecast improvementsafforded by equity tail risk for bond returns in these two countries. For all other countries,Table 11 reports the out-of-sample relative forecast and portfolio performance of the models in(5a), (5b) and (5c), which predict international bond returns with the S&P 500 option-impliedtail risk measure TR ( eq ) . Results are based on the out-of-sample setting described in Section3.1, with predictive regressions that are recursively estimated with a rolling window approachand the assumption that the investor’s level of risk aversion is γ = 5.[ Insert Table 11 here ]38rom an examination of the Campbell and Thompson (2008) out-of-sample R OS s in Table 11,we note that the models that include equity tail risk systematically outperform the benchmarksin predicting returns of the UK and Germany Treasury markets. The same holds true forshort- and medium-maturity bonds in Switzerland and France. The reductions in the MSPEfor the forecasts generated by the model that includes TR ( eq ) are in the range of 4% to 24% forUK bond returns and in the range of 0.5% to 11% for DE bond returns. On the basis of theClark and West (2007) test results, however, the gains of predictability in international bondreturns are only marginally statistically significant. When assessing the portfolio performanceafforded by equity tail risk, we observe that TR ( eq ) can generate substantial risk-adjusted returnsfor investors trading bonds in all four countries, but especially in the UK and Germany. Forinstance, when the benchmark is the 3 PCs-only model, we find that an investor trading the5-year UK (DE) Treasury bond is willing to pay approximately 165 (213) basis points per yearto switch from the benchmark to the model that predicts bond returns with equity tail risk.Having identified significant associations between the left tail volatility of the US stockmarket and the future returns on some of the major international government bond markets,the natural question that arises is whether equity tail risk is also a key determinant of thecurrent level of prices of those bonds. To answer this question, we estimate the risk premiumof TR ( eq ) by applying the GX three-pass method to the term structures of Treasury bonds inthe UK, Germany, Switzerland, France, Italy and Spain. The results are reported in Table 12.[ Insert Table 12 here ]As we did for the US term structure, we assess robustness of the estimates by reporting resultsalso for a higher number of latent factors than those selected with the Giglio and Xiu (2019)criterion, which points to 5 principal components for all Treasury markets except for the UKwhere 4 factors are selected. Examining the significance of the risk premium estimates γ g , wecan see that TR ( eq ) carries a significant price of risk in the Treasury bond market of not only39ermany, Switzerland and France, for which we found strong return predictability, but alsoSpain, where the evidence on predictability was much weaker. However, we do not reject thenull of TR ( eq ) being a weak factor for the ES term structure. Surprisingly, equity tail risk is notpriced in the UK Treasury market, where TR ( eq ) has strong predictive power for future returns.As for the Italian government bond market, we confirm the lack of a connection with equity tailrisk. Furthermore, it can be seen from the time-series regression R s that the equity left tailfactor is mostly spanned by the second and third principal components of the Treasury returns.We end this section by relating the returns of the international government bond marketsto the perceived tail risk in the stock market of the home country. We do this by running thepredictive regressions in (5a), (5b) and (5c) with the country-specific equity tail risk measuresdisplayed in Figure 9 and estimating their risk premium with the GX three-pass procedure.Due to the limited availability of option data on the European stock market indices, we onlyconsider the in-sample performance of the predictive models in (5a), (5b) and (5c). The full-sample estimates of the coefficients of the country-specific equity tail risk measures are reportedin Table 13 while the results of the GX three-pass regression procedure are shown in Table 14.[ Insert Table 13 here ][ Insert Table 14 here ]A quick inspection of Table 13 reveals that the future one-month returns of UK, DE and CHTreasury bonds are strongly associated not only with the S&P 500 option-implied left tail factorTR ( eq ) but also with the corresponding country-specific measure of equity tail risk. On the otherhand, we do not find any statistically significant relationship between the FR, IT and ES bondreturns and the perceived tail risk in the stock market of the home country. Finally, the resultsin Table 14 support our previous observations on the existence of a significant market price ofequity tail risk in the Treasury bond market of Germany, Switzerland and France.40n conclusion, our findings concerning the predictive power and pricing of equity tail riskare robust to alternative data sets. In fact, there is clear evidence that equity tail risk carriessignificant information about the dynamics of Treasury bond yields and returns not only in theUS but also in major government bond markets in Europe. In this paper, we study how US Treasury bonds respond to changes in the perceived tail riskin the stock market. We estimate equity tail risk with the risk-neutral expectation of futurevolatility that stems from large negative price jumps and we examine how it relates to the futureone-month returns on bonds in reduced-form predictive regressions. Also, we propose an affineterm structure model in which the main drivers of interest rates are the principal components ofthe zero-coupon yield curve and the equity tail risk factor. While earlier approaches to pricingbonds with factors other than combinations of yields have proven useful when macro variablesare considered, we focus here on the observed comovement in stock and bond markets duringcrisis periods and use a state variable that originates in the equity option market.The results of our main application to the US government bond and S&P 500 index optionmarkets are summarized as follows. First, there exist significant interactions between the one-month-ahead risk premia in Treasury bonds and the left tail volatility of the stock market.Second, the strong predictive power of equity tail risk for future bond returns is confirmed ina real-time out-of-sample exercise, where this predictability can be exploited to improve theeconomic utility of a mean-variance investor. Third, the left tail volatility of the stock market isa priced state variable in the US term structure. We find evidence of a significant market priceof equity tail risk not only with the ATSM but also with the novel three-pass method proposedby Giglio and Xiu (2019). Fourth, consistent with the theory of flight-to-safety, bond pricesrise in response to a contemporaneous shock to the equity left tail factor. Fifth, large dropsin short-term bond yields and expected future short rates are attributable to equity tail risk.41inally, our results concerning the predictive power and pricing of equity tail risk are robustto alternative data sets. When extending the analysis to major government bond marketsin Europe, we find that equity tail risk carries significant information about the dynamics ofTreasury bond yields and returns in United Kingdom, Germany, Switzerland and France, whilethe evidence is considerably weaker in Spain and non-existent in Italy.Given our findings with a measure of downside tail risk of the stock market, a naturaldirection for future research would be to assess the impact on the yield curve of a tail factorimplied by Treasury options. For instance, it would be interesting to see whether the downside,or even the upside, tail risk of the bond market receives compensation in a term structuremodel and how its pricing differs from that of equity tail risk. This would contribute to therecent literature on the auxiliary role of Treasury variance and jump risk in explaining bondrisk premia, see (Wright and Zhou, 2009; Mueller et al., 2016). 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Journal of Financial and Quantitative Anal-ysis , 45, 641–662. 48 able 1 – Descriptive statistics: bond risk premia and equity tail risk RX (12) t +1 RX (24) t +1 RX (36) t +1 RX (48) t +1 RX (60) t +1 RX (84) t +1 RX (120) t +1 TR ( eq ) t Panel A: Descriptive StatisticsMean 0 .
385 0 .
953 1 .
511 2 .
030 2 .
498 3 .
274 4 .
097 0 . .
597 1 .
535 2 .
556 3 .
561 4 .
530 6 .
382 8 .
998 0 . .
386 0 .
489 0 .
140 0 . − .
029 0 .
020 0 .
071 2 . .
542 4 .
604 3 .
894 3 .
693 3 .
752 4 .
187 4 .
981 10 . ρ (1) 0 .
214 0 .
156 0 .
113 0 .
085 0 .
067 0 .
048 0 .
028 0 . ρ (6) 0 . − . − . − . − . − . − .
083 0 . ρ (12) 0 .
082 0 .
133 0 .
137 0 .
127 0 .
110 0 .
065 0 .
010 0 . .
645 0 .
621 0 .
591 0 .
570 0 .
551 0 .
513 0 . (12) t +1 . (24) t +1 .
926 1 . (36) t +1 .
849 0 .
981 1 . (48) t +1 .
790 0 .
946 0 .
990 1 . (60) t +1 .
739 0 .
905 0 .
966 0 .
993 1 . (84) t +1 .
652 0 .
821 0 .
899 0 .
949 0 .
980 1 . (120) t +1 .
549 0 .
711 0 .
799 0 .
865 0 .
915 0 .
975 1 . ( eq ) t .
223 0 .
190 0 .
185 0 .
189 0 .
194 0 .
201 0 .
199 1 . ( n ) t +1 ,with maturity n = 12 , , , , , ,
120 months, and for the S&P 500 option-implied equity tail risk measureTR ( eq ) t used as predictor in the empirical analyses. Panel A reports the sample mean, standard deviation,skewness, kurtosis and autocorrelation coefficients of order one, six and twelve for each of the variables. Returnmeans and standard deviations are expressed in annualized percentage terms. The annualized Sharpe ratio (SR)is also reported for the Treasury bonds. Panel B reports the correlation coefficients calculated with the futurebond returns and contemporaneous TR ( eq ) factor. The sample uses end-of-month data for 1996:01–2018:12. able 2 – In-sample forecasts of Treasury returns with equity tail risk n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel A: No control for bond return forecasting factors TR ( eq ) t β .
460 1 .
009 1 .
638 2 .
322 3 .
032 4 .
429 6 . p -value 0 .
000 0 .
000 0 .
001 0 .
001 0 .
001 0 .
002 0 . p -value (b) .
008 0 .
037 0 .
012 0 .
010 0 .
006 0 .
001 0 . R .
622 3 .
258 3 .
081 3 .
204 3 .
394 3 .
676 3 . R ( eq ) .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . F -test 0 .
000 0 .
002 0 .
002 0 .
002 0 .
001 0 .
001 0 . Panel B: Control for yield curve factors with 3 PCs TR ( eq ) t β .
419 0 .
910 1 .
490 2 .
115 2 .
751 3 .
970 5 . p -value 0 .
001 0 .
008 0 .
011 0 .
011 0 .
010 0 .
010 0 . p -value (b) .
001 0 .
010 0 .
007 0 .
008 0 .
007 0 .
007 0 . t β .
355 0 .
542 0 .
692 0 .
834 0 .
968 1 .
208 1 . p -value 0 .
015 0 .
126 0 .
218 0 .
277 0 .
316 0 .
370 0 . t β .
240 0 .
746 1 .
246 1 .
743 2 .
234 3 .
182 4 . p -value 0 .
032 0 .
007 0 .
004 0 .
002 0 .
001 0 .
000 0 . t β .
158 0 . − . − . − . − . − . p -value 0 .
471 0 .
929 0 .
864 0 .
791 0 .
784 0 .
835 0 . R .
477 5 .
211 4 .
629 4 .
638 4 .
779 4 .
994 4 . R ( eq ) .
198 2 .
986 2 .
491 2 .
404 2 .
429 2 .
510 2 . F -test 0 .
001 0 .
007 0 .
008 0 .
007 0 .
006 0 .
005 0 . Panel C: Control for yield curve factors with 5 PCs TR ( eq ) t β .
411 0 .
895 1 .
453 2 .
042 2 .
635 3 .
770 5 . p -value 0 .
001 0 .
006 0 .
010 0 .
010 0 .
010 0 .
010 0 . p -value (b) .
002 0 .
009 0 .
007 0 .
009 0 .
010 0 .
009 0 . t β .
353 0 .
540 0 .
686 0 .
822 0 .
950 1 .
177 1 . p -value 0 .
014 0 .
122 0 .
232 0 .
310 0 .
363 0 .
426 0 . t β .
241 0 .
747 1 .
250 1 .
753 2 .
253 3 .
219 4 . p -value 0 .
045 0 .
009 0 .
006 0 .
004 0 .
003 0 .
002 0 . t β .
160 0 . − . − . − . − . − . p -value 0 .
450 0 .
921 0 .
870 0 .
802 0 .
802 0 .
861 0 . t β .
194 0 .
569 0 .
958 1 .
282 1 .
515 1 .
698 1 . p -value 0 .
239 0 .
092 0 .
075 0 .
095 0 .
134 0 .
257 0 . t β − . − . − . − . − . − . − . p -value 0 .
178 0 .
213 0 .
121 0 .
059 0 .
032 0 .
015 0 . R .
399 6 .
221 5 .
984 6 .
366 6 .
783 7 .
112 6 . R ( eq ) .
250 4 .
075 3 .
962 4 .
300 4 .
645 4 .
895 4 . F -test 0 .
001 0 .
008 0 .
010 0 .
009 0 .
008 0 .
007 0 . able 2 – In-sample forecasts of Treasury returns with equity tail risk (continued) n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel D: Control for Cochrane-Piazzesi (CP) factor TR ( eq ) t β .
365 0 .
798 1 .
303 1 .
853 2 .
425 3 .
563 5 . p -value 0 .
002 0 .
008 0 .
009 0 .
008 0 .
007 0 .
007 0 . t β .
473 1 .
053 1 .
668 2 .
339 3 .
026 4 .
312 5 . p -value 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . R .
357 6 .
704 6 .
170 6 .
337 6 .
647 7 .
013 6 . R ( eq ) .
673 4 .
871 4 .
419 4 .
499 4 .
680 4 .
842 4 . F -test 0 .
003 0 .
012 0 .
014 0 .
012 0 .
010 0 .
007 0 . Panel E: Control for Cieslak-Povala (CiP) factor TR ( eq ) t β .
438 0 .
934 1 .
507 2 .
136 2 .
792 4 .
086 5 . p -value 0 .
001 0 .
006 0 .
007 0 .
007 0 .
006 0 .
005 0 . t β .
307 1 .
029 1 .
794 2 .
554 3 .
292 4 .
695 6 . p -value 0 .
005 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . R .
488 6 .
669 6 .
852 7 .
158 7 .
465 7 .
857 7 . R ( eq ) .
340 3 .
923 4 .
292 4 .
495 4 .
633 4 .
772 4 . F -test 0 .
000 0 .
003 0 .
004 0 .
003 0 .
002 0 .
002 0 . Panel F: Control for VIX ⊥ TR ( eq ) t β .
460 1 .
009 1 .
638 2 .
322 3 .
032 4 .
429 6 . p -value 0 .
000 0 .
001 0 .
001 0 .
001 0 .
001 0 .
002 0 . ⊥ t β .
427 0 .
701 0 .
815 0 .
812 0 .
711 0 . − . p -value 0 .
018 0 .
241 0 .
380 0 .
392 0 .
374 0 .
371 0 . R .
578 4 .
658 3 .
580 3 .
286 3 .
247 3 .
338 3 . R ( eq ) .
622 3 .
258 3 .
081 3 .
204 3 .
394 3 .
676 3 . F -test 0 .
000 0 .
026 0 .
121 0 .
268 0 .
445 0 .
830 0 . p -values from predictive regressions of one-month US Treasurybond returns on the S&P 500 option-implied equity tail risk measure TR ( eq ) . n denotes the bond maturity inmonths. Panel A reports the results of a regression that only uses TR ( eq ) as predictor. Panels B to E reportthe results of regressions that control for bond return predictors identified in the literature: PC1 – PC5 are thefirst five principal components extracted from the Treasury bond yields, CP is the Cochrane and Piazzesi (2005)bond return predictor obtained as a linear combination of forward rates, CiP is the Cieslak and Povala (2015)risk-premium factor obtained from a decomposition of Treasury yields into inflation expectations and maturity-specific interest-rate cycles. Panel F reports the results of a regression that uses TR ( eq ) and the orthogonalcomponent of the CBOE VIX with respect to TR ( eq ) . All predictors have been normalized to have mean zeroand unit variance. For all predictors we report the Newey-West p -values computed with a 12-lag standard errorcorrection. In addition, for the TR ( eq ) factor used alone or alongside the principal components in the predictiveregressions, we report the p -value (b) computed with the bootstrap procedure of Bauer and Hamilton (2018). Foreach regression we report the adjusted R -squared in percentage. This measure is also reported for a regressionthat excludes the TR ( eq ) factor as predictor in Panels A to E, and for a regression that only uses TR ( eq ) aspredictor in Panel F. We also report the p -value of an F -test of the null hypothesis that the regression thatincludes the TR ( eq ) as predictor does not give a significantly better fit to the data than does a regression withoutit in Panels A to E, and the p -value of an F -test of the null hypothesis that the regression that includes VIX ⊥ does not give a significantly better fit to the data than does a regression that only uses TR ( eq ) in Panel F. Thein-sample period is 1996:01–2018:12. able 3 – Out-of-sample forecasts of Treasury returns with equity tail risk n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel A: Benchmark predictor is EH model (no predictability)
Panel A1: Increasing windows R OS (%) 1 .
642 2 .
112 3 .
074 3 .
876 4 .
295 4 .
081 2 . p -value ( CW ) .
118 0 .
107 0 .
093 0 .
079 0 .
066 0 .
043 0 . p -value (b) ( CW ) .
161 0 .
213 0 .
134 0 .
111 0 .
090 0 .
073 0 . p -value (b) ( R OS ) .
102 0 .
097 0 .
024 0 .
014 0 .
006 0 .
003 0 . R OS (%) 0 .
867 1 .
571 2 .
671 3 .
581 4 .
096 4 .
039 2 . p -value ( CW ) .
129 0 .
114 0 .
094 0 .
074 0 .
056 0 .
028 0 . p -value (b) ( CW ) .
175 0 .
224 0 .
139 0 .
107 0 .
085 0 .
063 0 . p -value (b) ( R OS ) .
166 0 .
135 0 .
036 0 .
018 0 .
009 0 .
004 0 . Panel B: Benchmark predictor is 3 PCs-only model
Panel B1: Increasing windows R OS (%) 1 .
837 1 .
272 1 .
984 2 .
681 3 .
062 2 .
924 1 . p -value ( CW ) .
039 0 .
061 0 .
060 0 .
051 0 .
041 0 .
022 0 . p -value (b) ( CW ) .
040 0 .
067 0 .
059 0 .
067 0 .
055 0 .
058 0 . p -value (b) ( R OS ) .
022 0 .
045 0 .
017 0 .
013 0 .
006 0 .
008 0 . R OS (%) 1 .
967 1 .
370 1 .
767 2 .
233 2 .
472 2 .
191 0 . p -value ( CW ) .
056 0 .
069 0 .
053 0 .
036 0 .
024 0 .
016 0 . p -value (b) ( CW ) .
049 0 .
066 0 .
052 0 .
052 0 .
046 0 .
060 0 . p -value (b) ( R OS ) .
024 0 .
045 0 .
027 0 .
021 0 .
017 0 .
022 0 . Panel C: Benchmark predictor is 5 PCs-only model
Panel C1: Increasing windows R OS (%) 1 .
067 1 .
460 1 .
975 2 .
338 2 .
481 2 .
155 1 . p -value ( CW ) .
081 0 .
123 0 .
130 0 .
124 0 .
112 0 .
080 0 . p -value (b) ( CW ) .
051 0 .
080 0 .
078 0 .
087 0 .
084 0 .
104 0 . p -value (b) ( R OS ) .
060 0 .
042 0 .
019 0 .
017 0 .
013 0 .
020 0 . R OS (%) 2 .
759 3 .
155 3 .
448 3 .
663 3 .
654 3 .
002 1 . p -value ( CW ) .
055 0 .
058 0 .
053 0 .
043 0 .
030 0 .
011 0 . p -value (b) ( CW ) .
039 0 .
043 0 .
038 0 .
047 0 .
045 0 .
057 0 . p -value (b) ( R OS ) .
016 0 .
013 0 .
006 0 .
006 0 .
005 0 .
012 0 . R OS s of predicting one-monthreturns on the n -month US Treasury bond with the S&P 500 option-implied equity tail risk measure TR ( eq ) .These R OS statistics represent the percentage reduction in the MSPE for the forecasts generated by a preferredmodel that includes TR ( eq ) relative to a benchmark that does not use it as predictor. Panel A: the preferredmodel uses the TR ( eq ) factor alone, while the benchmark model complies with the expectation hypothesis thatassumes no predictability of bond returns. Panel B: the preferred model includes TR ( eq ) and the first 3 principalcomponents of bond yields, while the benchmark model only includes the 3 principal components. Panel C: thepreferred model includes TR ( eq ) and the first 5 principal components of bond yields, while the benchmark modelonly includes the 5 principal components. Predictive regressions are recursively estimated with both expandingand rolling window approach. The out-of-sample period is 2007:07–2018:12. Statistical significance for R OS isbased on the p -value of the Clark and West (2007) MSPE-adjusted statistic ( CW ) for testing H : R OS ≤ H : R OS >
0. For the CW statistics we report both the Newey-West p -value computed with a 12-lagstandard error correction and the p -value (b) computed with the bootstrap procedure of Bauer and Hamilton(2018). For the out-of-sample R OS we only report the bootstrap p -value (b) . able 4 – Asset allocation gains of equity tail risk n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel A: Benchmark predictor is EH model (no predictability)
Panel A1: Risk aversion γ = 3∆ (%) . − . − . − . − .
054 0 . − . (%) . − . − . − . − .
002 1 . − . γ = 5∆ (%) . − . − .
410 0 .
787 0 . − . − . (%) . − . − .
421 0 .
877 0 . − . − . Panel B: Benchmark predictor is 3 PCs-only model
Panel B1: Risk aversion γ = 3∆ (%) − . − . − . − . − . − . − . (%) − . − . − . − . − . − . − . γ = 5∆ (%) − . − . − . − . − . − . − . (%) − . − . − . − . − .
359 0 . − . Panel C: Benchmark predictor is 5 PCs-only model
Panel C1: Risk aversion γ = 3∆ (%) − . − .
079 0 .
627 1 .
973 3 .
559 3 . − . (%) − . − .
080 0 .
630 2 .
029 3 .
719 3 . − . γ = 5∆ (%) − .
708 0 .
048 0 .
883 1 .
862 2 .
492 1 . − . (%) − .
719 0 .
048 0 .
895 2 .
027 2 .
909 0 . − . ( eq ) . n denotes the maturity of the bond in months. Weassume a mean-variance investor with risk aversion γ = 3 or γ = 5 that every month allocates his or her wealthbetween a 1-month Treasury (risk-free) bond and an n -month Treasury bond. Investment decisions are basedon the expected return forecasts of the n -month bond which are generated by a preferred model that includesTR ( eq ) or by a benchmark model that does not use TR ( eq ) as predictor. Panel A: the preferred model usesthe TR ( eq ) factor alone, while the benchmark model complies with the expectation hypothesis that assumes nopredictability of bond returns, implying that model forecasts are based on historical return means. Panel B: thepreferred model includes TR ( eq ) and the first three principal components of bond yields, while the benchmarkmodel only includes the three principal components. Panel C: the preferred model includes TR ( eq ) and the firstfive principal components of bond yields, while the benchmark model only includes the five principal components.Predictive models are recursively estimated with a rolling window approach. The (out-of-sample) investmentperiod is 2007:07–2018:12. We report two measures for the performance of the preferred model relative to thatof the benchmark model: certainty equivalent return gain (∆) and Goetzmann et al. (2007) manipulation-proofperformance improvement (Θ). Both measures are expressed in annualized percentage terms. able 5 – Expected returns, forecasting performance and macroeconomic condition n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel A: ρ ( E t [RX ( n ) t +1 ] , CFNAI t )TR ( eq ) − . − . − . − . − . − . − . ( eq ) + 3PCs − . − . − . − . − . − . − . ( eq ) + 5PCs − . − . − . − . − . − . − . Panel B: ρ ( E t [RX ( n ) t +1 ] , U MACRO t )TR ( eq ) .
697 0 .
657 0 .
632 0 .
615 0 .
599 0 .
566 0 . ( eq ) + 3PCs 0 .
590 0 .
547 0 .
527 0 .
522 0 .
525 0 .
526 0 . ( eq ) + 5PCs 0 .
522 0 .
364 0 .
291 0 .
277 0 .
291 0 .
347 0 . Panel C: ρ (DCSPE t , CFNAI t )TR ( eq ) − . − . − .
079 0 .
056 0 .
140 0 .
235 0 . ( eq ) + 3PCs 0 .
444 0 .
390 0 .
357 0 .
343 0 .
335 0 .
303 0 . ( eq ) + 5PCs 0 .
412 0 .
407 0 .
372 0 .
351 0 .
341 0 .
313 0 . Panel D: ρ (DCRU t , CFNAI t )TR ( eq ) .
182 0 .
372 0 . − . − . − . − . ( eq ) + 3PCs 0 . − . − . − . − . − . − . ( eq ) + 5PCs 0 .
078 0 .
159 0 . − .
004 0 . − . − . ( eq ) . n denotes the bond maturity in months. Panels A and B report contemporaneous correlations between the out-of-sample forecasts of the one-month-ahead Treasury bond returns obtained by one of the three models that useTR ( eq ) as predictor and the Chicago Fed National Activity Index (CFNAI) and the macroeconomic uncertaintyindex ( U MACRO ) constructed by Jurado et al. (2015). Panels C and D report contemporaneous correlationsbetween relative forecast and portfolio performance obtained by one of the three models that use TR ( eq ) aspredictor (relative to its benchmark that does not use TR ( eq ) to predict bond returns) and the CFNAI. Relativeforecast performance is defined as the difference in cumulative squared prediction error (DCSPE) and portfolioperformance is defined as the difference in cumulative realized utilities (DCRU). The out-of-sample evaluationperiod is 2007:07–2018:12. The predictive models are recursively estimated with a rolling window approach.The investor’s risk aversion coefficient is γ = 5. able 6 – Fit diagnostics of the ATSM with equity tail risk Panel A: Equity Tail Risk ATSM n = 12 n = 24 n = 36 n = 60 n = 84 n = 120Panel A1: Yield Pricing ErrorsMean − .
001 0 .
000 0 . − . − . − . .
004 0 .
005 0 .
003 0 .
004 0 .
003 0 . − .
390 0 .
843 0 . − .
236 0 . − . .
399 4 .
182 1 .
994 3 .
292 3 .
166 3 . ρ (1) 0 .
867 0 .
807 0 .
909 0 .
897 0 .
839 0 . ρ (6) 0 .
530 0 .
370 0 .
767 0 .
587 0 .
451 0 . .
000 0 . − . − .
004 0 . − . .
047 0 .
074 0 .
069 0 .
113 0 .
117 0 . − . − . − . − . − . − . .
650 6 .
903 13 .
366 5 .
479 5 .
999 4 . ρ (1) 0 .
020 0 .
050 0 . − . − . − . ρ (6) 0 .
153 0 .
214 0 .
274 0 .
031 0 .
132 0 . Panel B: PC-only ATSM n = 12 n = 24 n = 36 n = 60 n = 84 n = 120Panel B1: Yield Pricing ErrorsMean − . − . − . − . − . − . .
006 0 .
006 0 .
003 0 .
005 0 .
003 0 . − .
080 0 . − . − .
089 0 . − . .
875 3 .
994 1 .
838 3 .
181 2 .
301 3 . ρ (1) 0 .
902 0 .
812 0 .
952 0 .
920 0 .
896 0 . ρ (6) 0 .
606 0 .
398 0 .
875 0 .
649 0 .
690 0 . − .
001 0 . − . − . − . − . .
052 0 .
076 0 .
067 0 .
128 0 .
114 0 . − . − . − . − .
078 0 . − . .
474 7 .
669 13 .
211 5 .
848 6 .
281 5 . ρ (1) 0 .
118 0 .
009 0 .
348 0 . − . − . ρ (6) 0 .
104 0 .
218 0 .
334 0 .
003 0 .
139 0 . ( eq ) (Panel A) and by the benchmark modelthat only uses the first five PCs of the yield curve (Panel B). Models are estimated over the period 1996 to 2018.Reported are the sample mean, standard deviation, skewness, kurtosis and the autocorrelation coefficients oforder one and six. Panels A1 and B1: properties of the yield pricing errors ˆ u . Panels A2 and B2: properties ofthe return pricing errors ˆ e . n denotes the maturity of the bonds in months. able 7 – Factor risk exposures in the ATSM with equity tail risk Equity Tail Risk ATSM PC-only ATSM
Factor W β i p -value W β i p -valueTR ( eq ) .
154 0.000 - -PC1 29773988 .
504 0.000 31625802 .
379 0.000PC2 5640992 .
750 0.000 6114464 .
179 0.000PC3 933067 .
335 0.000 942985 .
226 0.000PC4 174656 .
368 0.000 176667 .
454 0.000PC5 33311 .
223 0.000 33261 .
513 0.000Notes: This table provides the Wald statistics and corresponding p -values for the Wald test of whether theexposures of bond returns to a given model factor are jointly zero. Under the null H : β i = N × the i -thpricing factor is unspanned, i.e. Treasury returns are not exposed to it. The test is conducted on the pricingfactors of both the proposed ATSM specified with the S&P 500 option-implied equity tail risk measure TR ( eq ) ,and a benchmark PC-only model specification. able 8 – Market prices of risk in the ATSM with equity tail risk Factor λ λ , λ , λ , λ , λ , λ , W Λ i W λ i TR ( eq ) . − .
164 0 .
413 0 . − . − .
103 0 .
227 12 .
493 12 . . − . . . − . − . . . . . − .
054 0 .
051 0 . − . − .
032 0 .
018 12 .
201 12 . . − . . . − . − . . . . − .
047 0 . − . − .
069 0 .
122 0 . − .
097 15 .
532 14 . − . . − . − . . . − . . . .
004 0 . − .
150 0 .
009 0 .
038 0 . − .
068 12 .
150 12 . . . − . . . . − . . . . − . − .
064 0 . − . − . − .
074 18 .
343 17 . . − . − . . − . − . − . . . − . − . − . − .
127 0 . − . − .
141 16 .
187 14 . − . − . − . − . . − . − . . . λ and λ in equation (15)for the Gaussian ATSM specified with the S&P 500 option-implied equity tail risk measure TR ( eq ) . Estimated t -statistics are reported in parentheses. Wald statistics for tests of the rows of Λ and of λ being different fromzero are reported along each row, with the corresponding p -values in parentheses below. The null hypothesisunderlying W Λ i is that the risk related to a given factor is not priced in the term structure model. The nullhypothesis underlying W λ i is that the price of risk associated with a given factor does not vary over time. able 9 – Market price of equity tail risk with GX procedure p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = 8 γ g . . . . . . . . . . . . . . . . R g .
038 0 .
079 0 .
085 0 .
087 0 .
093 0 .
094 0 .
097 0 . p -value 0 .
000 0 .
006 0 .
011 0 .
023 0 .
019 0 .
025 0 .
041 0 . g weakNotes: This table reports the results of the three-pass regression procedure of Giglio and Xiu (2019) to estimatethe risk premium of the S&P 500 option-implied equity tail risk measure TR ( eq ) in the US Treasury bond market. p denotes the number of latent factors used in the three-pass estimator. For each number of latent factors, wereport the estimate of the market price of risk γ g of the observable factor g = TR ( eq ) with standard errors inparentheses, the R -squared of the time series regression of the observable factor g onto the p latent factors, andthe p -value of the Wald test of testing the null hypothesis that the observable factor is weak. * (resp. **, and***) denote statistical significance at the 10% (resp. 5%, and 1%) level. able 10 – In-sample forecasts of international bond returns with equity tail risk n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel A: No control for bond return forecasting factors UK β .
975 1 .
806 2 .
450 3 .
002 3 .
549 4 .
520 5 . p -value 0 .
001 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . p -value (b) .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . β .
542 1 .
030 1 .
448 1 .
827 2 .
172 2 .
758 3 . p -value 0 .
002 0 .
001 0 .
001 0 .
001 0 .
002 0 .
004 0 . p -value (b) .
000 0 .
001 0 .
001 0 .
003 0 .
002 0 .
010 0 . β .
547 0 .
654 0 .
799 1 .
033 1 .
295 1 .
776 2 . p -value 0 .
036 0 .
002 0 .
000 0 .
001 0 .
001 0 .
002 0 . p -value (b) .
000 0 .
011 0 .
035 0 .
037 0 .
032 0 .
034 0 . β .
468 0 .
893 1 .
231 1 .
511 1 .
754 2 .
170 2 . p -value 0 .
003 0 .
003 0 .
004 0 .
007 0 .
012 0 .
035 0 . p -value (b) .
003 0 .
008 0 .
010 0 .
014 0 .
024 0 .
042 0 . β .
402 0 .
454 0 .
679 1 .
008 1 .
336 1 .
811 2 . p -value 0 .
103 0 .
343 0 .
350 0 .
285 0 .
237 0 .
205 0 . p -value (b) .
116 0 .
417 0 .
402 0 .
317 0 .
269 0 .
245 0 . β .
619 1 .
179 1 .
662 2 .
079 2 .
445 3 .
067 3 . p -value 0 .
000 0 .
000 0 .
000 0 .
001 0 .
002 0 .
005 0 . p -value (b) .
017 0 .
045 0 .
054 0 .
072 0 .
085 0 .
112 0 . Panel B: Control for yield curve factors with 3 PCs (country-specific) UK β .
966 1 .
758 2 .
351 2 .
862 3 .
376 4 .
289 5 . p -value 0 .
001 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . p -value (b) .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . β .
548 1 .
009 1 .
393 1 .
732 2 .
034 2 .
527 3 . p -value 0 .
002 0 .
001 0 .
001 0 .
002 0 .
003 0 .
007 0 . p -value (b) .
000 0 .
000 0 .
001 0 .
003 0 .
004 0 .
010 0 . β .
559 0 .
693 0 .
828 1 .
008 1 .
194 1 .
523 1 . p -value 0 .
034 0 .
003 0 .
001 0 .
002 0 .
004 0 .
011 0 . p -value (b) .
000 0 .
007 0 .
030 0 .
041 0 .
041 0 .
055 0 . β .
452 0 .
830 1 .
099 1 .
295 1 .
447 1 .
678 1 . p -value 0 .
007 0 .
010 0 .
015 0 .
028 0 .
050 0 .
123 0 . p -value (b) .
000 0 .
002 0 .
007 0 .
019 0 .
035 0 .
094 0 . β .
446 0 .
597 0 .
841 1 .
117 1 .
344 1 .
549 1 . p -value 0 .
111 0 .
283 0 .
312 0 .
288 0 .
272 0 .
290 0 . p -value (b) .
068 0 .
269 0 .
299 0 .
279 0 .
260 0 .
318 0 . β .
500 1 .
015 1 .
416 1 .
746 2 .
031 2 .
512 3 . p -value 0 .
003 0 .
003 0 .
008 0 .
014 0 .
020 0 .
031 0 . p -value (b) .
051 0 .
082 0 .
111 0 .
143 0 .
170 0 .
199 0 . able 10 – In-sample forecasts of international bond returns with equity tail risk (continued) n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel C: Control for yield curve factors with 5 PCs (country-specific) UK β .
941 1 .
752 2 .
343 2 .
877 3 .
431 4 .
481 5 . p -value 0 .
002 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . p -value (b) .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . β .
479 0 .
908 1 .
279 1 .
615 1 .
915 2 .
399 2 . p -value 0 .
000 0 .
000 0 .
000 0 .
001 0 .
002 0 .
006 0 . p -value (b) .
000 0 .
001 0 .
003 0 .
006 0 .
008 0 .
017 0 . β .
464 0 .
605 0 .
767 0 .
939 1 .
081 1 .
281 1 . p -value 0 .
014 0 .
001 0 .
001 0 .
005 0 .
010 0 .
027 0 . p -value (b) .
000 0 .
018 0 .
047 0 .
057 0 .
068 0 .
110 0 . β .
441 0 .
817 1 .
093 1 .
298 1 .
454 1 .
673 1 . p -value 0 .
016 0 .
015 0 .
017 0 .
026 0 .
047 0 .
127 0 . p -value (b) .
000 0 .
003 0 .
007 0 .
017 0 .
034 0 .
096 0 . β .
586 0 .
837 1 .
134 1 .
426 1 .
657 1 .
900 1 . p -value 0 .
027 0 .
099 0 .
123 0 .
119 0 .
118 0 .
140 0 . p -value (b) .
017 0 .
115 0 .
155 0 .
170 0 .
170 0 .
214 0 . β .
678 1 .
189 1 .
605 2 .
009 2 .
395 3 .
085 3 . p -value 0 .
008 0 .
009 0 .
012 0 .
015 0 .
017 0 .
027 0 . p -value (b) .
006 0 .
039 0 .
071 0 .
094 0 .
100 0 .
118 0 . p -values associated with the S&P 500 option-implied equity tailrisk measure TR ( eq ) used in return predictive regressions of Treasury bonds of United Kingdom (UK), Germany(DE), Switzerland (CH), France (FR), Italy (IT), and Spain (ES). n denotes the bond maturity in months.Panel A reports the results of a regression that only uses TR ( eq ) t as predictor. Panel B (resp. C) reports theresults of a predictive regression that controls for country-specific yield curve factors represented by the firstthree (resp. five) principal components of Treasury bond yields. Predictors have been normalized to have meanzero and unit variance. We report the Newey-West p -values computed with a 12-lag standard error correction,and the p -value (b) computed with the bootstrap procedure of Bauer and Hamilton (2018). The in-sample periodis 1996:01–2018:12. able 11 – Out-of-sample forecasts of international bond returns with equity tail risk n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel A: No control for bond return forecasting factors UK R OS (%) 24 .
245 18 .
207 14 .
287 12 .
136 10 .
904 8 .
411 5 . p -value 0 .
103 0 .
078 0 .
064 0 .
056 0 .
051 0 .
044 0 . (%) .
009 0 .
243 1 .
012 2 .
093 3 .
433 1 . − . (%) .
007 0 .
264 1 .
110 2 .
284 3 .
774 2 . − . R OS (%) 9 .
637 7 .
644 6 .
923 6 .
498 6 .
038 4 .
941 3 . p -value 0 .
046 0 .
046 0 .
041 0 .
034 0 .
028 0 .
020 0 . (%) − .
011 0 .
070 0 .
188 0 .
420 0 .
785 1 .
359 0 . (%) − .
011 0 .
072 0 .
220 0 .
505 0 .
986 1 .
858 0 . R OS (%) 9 .
422 5 .
735 2 .
465 2 .
594 3 .
182 3 .
362 2 . p -value 0 .
100 0 .
056 0 .
033 0 .
032 0 .
030 0 .
016 0 . (%) .
105 0 .
332 0 . − .
192 0 .
070 0 .
112 1 . (%) .
104 0 .
330 0 . − .
251 0 .
028 0 .
183 1 . R OS (%) 7 .
269 5 .
474 4 .
397 3 .
480 2 .
757 1 .
763 0 . p -value 0 .
102 0 .
116 0 .
118 0 .
117 0 .
120 0 .
139 0 . (%) − .
001 0 .
005 0 .
080 0 .
140 0 .
140 0 . − . (%) − .
001 0 .
004 0 .
093 0 .
178 0 .
250 0 . − . Panel B: Control for yield curve factors with 3 PCs (country-specific) UK R OS (%) 20 .
834 14 .
966 11 .
143 9 .
289 8 .
254 6 .
173 4 . p -value 0 .
096 0 .
097 0 .
097 0 .
090 0 .
081 0 .
063 0 . (%) .
422 0 .
449 0 .
526 1 .
553 1 .
649 0 . − . (%) .
435 0 .
503 0 .
621 1 .
680 1 .
794 1 . − . R OS (%) 7 .
717 5 .
934 5 .
218 4 .
500 3 .
714 2 .
205 0 . p -value 0 .
084 0 .
103 0 .
105 0 .
101 0 .
096 0 .
089 0 . (%) .
285 1 .
671 2 .
281 2 .
433 2 .
134 1 .
769 0 . (%) .
284 1 .
677 2 .
301 2 .
497 2 .
243 2 .
053 0 . R OS (%) 8 .
870 5 .
372 2 .
211 1 .
918 1 .
832 0 . − . p -value 0 .
102 0 .
071 0 .
081 0 .
102 0 .
111 0 .
106 0 . (%) .
839 0 .
566 1 .
104 1 .
175 1 .
184 0 .
941 0 . (%) .
839 0 .
572 1 .
120 1 .
203 1 .
232 1 .
056 0 . R OS (%) 6 .
493 4 .
489 3 .
136 1 .
950 1 . − . − . p -value 0 .
115 0 .
151 0 .
173 0 .
197 0 .
231 0 .
339 0 . (%) . − . − . − . − . − . − . (%) . − . − . − . − . − . − . able 11 – Out-of-sample forecasts of international bond returns with equity tail risk (continued) n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel C: Control for yield curve factors with 5 PCs (country-specific) UK R OS (%) 20 .
461 15 .
758 12 .
521 11 .
005 10 .
215 8 .
208 5 . p -value 0 .
096 0 .
086 0 .
080 0 .
072 0 .
063 0 .
049 0 . (%) .
481 0 .
701 0 .
497 1 .
682 2 .
288 0 .
996 1 . (%) .
494 0 .
761 0 .
586 1 .
773 2 .
266 0 .
793 0 . R OS (%) 10 .
968 8 .
402 6 .
647 5 .
373 4 .
320 2 .
580 0 . p -value 0 .
030 0 .
048 0 .
051 0 .
051 0 .
051 0 .
054 0 . (%) . − . − .
311 0 .
138 0 .
428 0 . − . (%) . − . − .
294 0 .
190 0 .
473 0 . − . R OS (%) 8 .
631 3 .
860 1 .
082 0 .
770 0 .
775 0 . − . p -value 0 .
059 0 .
027 0 .
060 0 .
097 0 .
116 0 .
136 0 . (%) − .
114 0 .
398 0 .
593 0 .
543 0 .
611 0 .
639 0 . (%) − .
114 0 .
397 0 .
605 0 .
562 0 .
617 0 . − . R OS (%) 7 .
846 5 .
277 3 .
554 2 .
268 1 . − . − . p -value 0 .
103 0 .
149 0 .
179 0 .
204 0 .
233 0 .
334 0 . (%) − . − .
101 0 .
766 1 .
339 0 . − . − . (%) − . − .
091 0 .
801 1 .
483 1 . − . − . R OS s of predicting one-monthTreasury bond returns in United Kingdom (UK), Germany (DE), Switzerland (CH), and France (FR) with theS&P 500 option-implied equity tail risk measure TR ( eq ) . These R OS statistics represent the percentage reduc-tion in the MSPE for the forecasts generated by a preferred model that includes TR ( eq ) relative to a benchmarkthat does not use it as predictor. The preferred model uses the TR ( eq ) factor alone in Panel A, and alongsidethe country-specific first three (resp. five) principal components of bond yields in Panel B (resp. C). Statisti-cal significance for R OS is based on the Clark and West (2007) MSPE-adjusted statistic, for which we reportNewey-West p -values computed with a 12-lag standard error correction. To assess the portfolio performanceafforded by TR ( eq ) relative to the benchmark models, we report the certainty equivalent return gain (∆) andGoetzmann et al. (2007) manipulation-proof performance improvement (Θ) in annualized percentage terms. Theout-of-sample period is 2007:07–2018:12. Predictive regressions are recursively estimated with a rolling windowapproach. The investor’s risk aversion coefficient γ is set equal to 5. able 12 – Market price of equity tail risk in international bond markets p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = 8UK γ g . .
068 0 . .
080 0 .
082 0 .
085 0 .
085 0 . . . . . . . . . R g .
066 0 .
152 0 .
218 0 .
237 0 .
254 0 .
259 0 .
260 0 . p -value 0 .
000 0 .
001 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . g weakDE γ g . . . . . . . . . . . . . . . . R g .
074 0 .
156 0 .
212 0 .
238 0 .
241 0 .
255 0 .
257 0 . p -value 0 .
002 0 .
004 0 .
001 0 .
000 0 .
000 0 .
000 0 .
000 0 . g weakCH γ g . . . . . . . . . . . . . . . . R g .
047 0 .
065 0 .
080 0 .
129 0 .
158 0 .
159 0 .
164 0 . p -value 0 .
001 0 .
000 0 .
000 0 .
001 0 .
001 0 .
001 0 .
000 0 . g weakFR γ g . . . . . . . . . . . . . . . . R g .
035 0 .
138 0 .
142 0 .
163 0 .
194 0 .
215 0 .
218 0 . p -value 0 .
006 0 .
004 0 .
005 0 .
014 0 .
007 0 .
007 0 .
013 0 . g weakIT γ g − . − .
001 0 .
001 0 .
004 0 .
007 0 .
007 0 .
007 0 . . . . . . . . . R g .
000 0 .
006 0 .
008 0 .
018 0 .
022 0 .
023 0 .
023 0 . p -value 0 .
803 0 .
487 0 .
592 0 .
717 0 .
748 0 .
293 0 .
245 0 . g weakES γ g .
008 0 .
024 0 .
026 0 .
040 0 .
049 0 . . . . . . . . . . . R g .
002 0 .
027 0 .
037 0 .
042 0 .
044 0 .
074 0 .
074 0 . p -value 0 .
328 0 .
227 0 .
370 0 .
550 0 .
695 0 .
441 0 .
582 0 . g weakNotes: This table reports the results of the three-pass regression procedure of Giglio and Xiu (2019) to estimatethe risk premium of the S&P 500 option-implied equity tail risk measure TR ( eq ) in the Treasury bond market ofUnited Kingdom (UK), Germany (DE), Switzerland (CH), France (FR), Italy (IT), and Spain (ES). p denotesthe number of latent factors used in the three-pass estimator. For each number of latent factors, we report theestimate of the market price of risk γ g of the observable factor g = TR ( eq ) with standard errors in parentheses,the R -squared of the time series regression of the observable factor g onto the p latent factors, and the p -valueof the Wald test of testing the null hypothesis that the observable factor is weak. * (resp. **, and ***) denotestatistical significance at the 10% (resp. 5%, and 1%) level. able 13 – In-sample forecasts of international bond returns with country-specific equity tail risk n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel A: No control for bond return forecasting factors UK β .
864 1 .
612 2 .
290 2 .
955 3 .
623 4 .
810 6 . p -value 0 .
004 0 .
000 0 .
000 0 .
000 0 .
000 0 .
002 0 . p -value (b) .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
001 0 . β .
581 1 .
128 1 .
563 1 .
927 2 .
231 2 .
667 2 . p -value 0 .
000 0 .
000 0 .
001 0 .
002 0 .
006 0 .
018 0 . p -value (b) .
001 0 .
002 0 .
004 0 .
006 0 .
010 0 .
022 0 . β .
600 0 .
741 0 .
826 1 .
008 1 .
270 1 .
921 3 . p -value 0 .
117 0 .
012 0 .
009 0 .
016 0 .
015 0 .
010 0 . p -value (b) .
000 0 .
003 0 .
035 0 .
059 0 .
049 0 .
023 0 . β .
432 0 .
561 0 .
558 0 .
557 0 .
614 0 .
966 2 . p -value 0 .
017 0 .
107 0 .
221 0 .
296 0 .
308 0 .
217 0 . p -value (b) .
008 0 .
115 0 .
310 0 .
458 0 .
517 0 .
479 0 . β .
205 1 .
837 2 .
076 2 .
105 2 .
048 1 .
920 1 . p -value 0 .
000 0 .
007 0 .
046 0 .
123 0 .
218 0 .
379 0 . p -value (b) .
000 0 .
028 0 .
123 0 .
219 0 .
306 0 .
436 0 . β .
726 0 .
752 0 .
560 0 .
515 0 .
531 0 .
487 0 . p -value 0 .
295 0 .
439 0 .
672 0 .
765 0 .
803 0 .
868 0 . p -value (b) .
127 0 .
492 0 .
749 0 .
804 0 .
831 0 .
892 0 . Panel B: Control for yield curve factors with 3 PCs (country-specific) UK β .
758 1 .
348 1 .
862 2 .
386 2 .
924 3 .
851 4 . p -value 0 .
015 0 .
001 0 .
001 0 .
002 0 .
006 0 .
034 0 . p -value (b) .
000 0 .
000 0 .
000 0 .
001 0 .
001 0 .
004 0 . β .
634 1 .
160 1 .
512 1 .
754 1 .
915 2 .
036 1 . p -value 0 .
000 0 .
001 0 .
007 0 .
025 0 .
055 0 .
148 0 . p -value (b) .
000 0 .
001 0 .
005 0 .
012 0 .
030 0 .
097 0 . β .
588 0 .
783 0 .
860 0 .
948 1 .
076 1 .
469 2 . p -value 0 .
139 0 .
019 0 .
004 0 .
009 0 .
022 0 .
045 0 . p -value (b) .
000 0 .
002 0 .
034 0 .
074 0 .
094 0 .
096 0 . β .
320 0 .
280 0 . − . − . − .
346 0 . p -value 0 .
105 0 .
483 0 .
885 0 .
826 0 .
688 0 .
735 0 . p -value (b) .
062 0 .
468 0 .
892 0 .
861 0 .
767 0 .
816 0 . β .
603 0 .
458 0 . − . − . − . − . p -value 0 .
061 0 .
621 0 .
941 0 .
852 0 .
717 0 .
607 0 . p -value (b) .
049 0 .
605 0 .
944 0 .
868 0 .
768 0 .
689 0 . β .
419 0 . − . − . − . − . − . p -value 0 .
555 0 .
845 0 .
892 0 .
841 0 .
846 0 .
843 0 . p -value (b) .
371 0 .
854 0 .
911 0 .
858 0 .
869 0 .
858 0 . able 13 – In-sample forecasts of international bond returns with country-specific equity tail risk (cont.) n = 12 n = 24 n = 36 n = 48 n = 60 n = 84 n = 120 Panel C: Control for yield curve factors with 5 PCs (country-specific) UK β .
747 1 .
346 1 .
847 2 .
360 2 .
894 3 .
849 4 . p -value 0 .
018 0 .
002 0 .
001 0 .
001 0 .
005 0 .
032 0 . p -value (b) .
000 0 .
000 0 .
001 0 .
001 0 .
001 0 .
003 0 . β .
604 1 .
151 1 .
514 1 .
751 1 .
903 2 .
018 1 . p -value 0 .
000 0 .
001 0 .
009 0 .
028 0 .
059 0 .
156 0 . p -value (b) .
000 0 .
001 0 .
007 0 .
013 0 .
036 0 .
107 0 . β .
515 0 .
729 0 .
815 0 .
886 0 .
987 1 .
342 2 . p -value 0 .
015 0 .
000 0 .
001 0 .
014 0 .
036 0 .
046 0 . p -value (b) .
000 0 .
002 0 .
042 0 .
100 0 .
122 0 .
132 0 . β .
254 0 . − . − . − . − . − . p -value 0 .
174 0 .
810 0 .
511 0 .
201 0 .
132 0 .
209 0 . p -value (b) .
138 0 .
825 0 .
636 0 .
413 0 .
355 0 .
411 0 . β .
639 0 .
669 0 .
545 0 .
353 0 . − . − . p -value 0 .
066 0 .
467 0 .
677 0 .
821 0 .
932 0 .
924 0 . p -value (b) .
037 0 .
443 0 .
691 0 .
841 0 .
945 0 .
948 0 . β .
430 0 . − . − . − .
149 0 .
004 0 . p -value 0 .
509 0 .
787 0 .
970 0 .
924 0 .
947 0 .
999 0 . p -value (b) .
354 0 .
793 0 .
973 0 .
933 0 .
955 0 .
999 0 . p -values associated with country-specific equity tail riskmeasures used in return predictive regressions of Treasury bonds in United Kingdom (UK), Germany (DE),Switzerland (CH), France (FR), Italy (IT), and Spain (ES). n denotes the maturity of the bonds in months.The country-specific equity tail risk measures are calculated using options on the FTSE 100 (UK), DAX (DE),SMI (CH), CAC 40 (FR), FTSE MIB (IT) and IBEX 35 (ES) equity index. Panel A reports the results of aregression that only uses the country-specific equity tail risk measure as predictor. Panel B (resp. C) reportsthe results of a predictive regression that controls for country-specific yield curve factors represented by thefirst three (resp. five) principal components of Treasury bond yields. All predictors have been normalized tohave mean zero and unit variance. We report the Newey-West p -values computed with a 12-lag standard errorcorrection, and the p -value (b) computed with the bootstrap procedure of Bauer and Hamilton (2018). Thein-sample period is 2002:01–2018:12 in UK, DE and CH, 2007:01–2018:12 in IT and FR, 2007:05–2018:12 in ES. able 14 – Market price of country-specific equity tail risk in international bond markets p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = 8UK γ g . . .
086 0 .
085 0 .
083 0 .
084 0 .
087 0 . . . . . . . . . R g .
087 0 .
180 0 .
231 0 .
239 0 .
269 0 .
275 0 .
278 0 . p -value 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . g weakDE γ g . .
093 0 .
099 0 . . . . . . . . . . . . . R g .
081 0 .
189 0 .
246 0 .
263 0 .
279 0 .
324 0 .
327 0 . p -value 0 .
003 0 .
001 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . g weakCH γ g . . . . . . . . . . . . . . . . R g .
083 0 .
099 0 .
149 0 .
204 0 .
216 0 .
216 0 .
259 0 . p -value 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . g weakFR γ g . . . . . . . . . . . . . . . . R g .
042 0 .
213 0 .
214 0 .
283 0 .
283 0 .
299 0 .
322 0 . p -value 0 .
027 0 .
008 0 .
016 0 .
007 0 .
014 0 .
000 0 .
013 0 . g weakIT γ g − . − .
006 0 .
010 0 .
044 0 .
046 0 .
047 0 .
048 0 . . . . . . . . . R g .
005 0 .
011 0 .
062 0 .
146 0 .
149 0 .
155 0 .
156 0 . p -value 0 .
369 0 .
506 0 .
047 0 .
017 0 .
013 0 .
004 0 .
000 0 . g weakES γ g .
009 0 .
020 0 .
016 0 .
004 0 .
020 0 .
065 0 .
073 0 . . . . . . . . . R g .
004 0 .
009 0 .
018 0 .
020 0 .
022 0 .
029 0 .
035 0 . p -value 0 .
286 0 .
564 0 .
660 0 .
762 0 .
788 0 .
851 0 .
840 0 . g weakNotes: This table reports the results of the three-pass regression procedure of Giglio and Xiu (2019) to estimatethe risk premium of country-specific equity tail risk measures in the Treasury bond market of United Kingdom(UK), Germany (DE), Switzerland (CH), France (FR), Italy (IT), and Spain (ES). The country-specific equitytail risk measures are calculated using options on the FTSE 100 (UK), DAX (DE), SMI (CH), CAC 40 (FR),FTSE MIB (IT) and IBEX 35 (ES) equity index. p denotes the number of latent factors used in the three-passestimator. For each number of latent factors, we report the estimate of the market price of risk γ g of theobservable factor g = TR ( eq ) with standard errors in parentheses, the R -squared of the time series regression ofthe observable factor g onto the p latent factors, and the p -value of the Wald test of testing the null hypothesisthat the observable factor is weak. * (resp. **, and ***) denote statistical significance at the 10% (resp. 5%,and 1%) level. igure 1 – Time series of the S&P 500 option-implied equity tail risk measure
96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19
Year −6−5−4−3−2−10123456 TR (eq)t CFNAI t The figure displays the end-of-month values of the S&P 500 option-implied equity tail risk mea-sure (TR ( eq ) t ) and 3-month moving average of the Chicago National Activity Index (CFNAI t )from January 1996 to December 2018. For convenience, both series have been normalized tohave mean zero and unit variance. Contemporaneous correlation between TR ( eq ) and CFNAIis − .
49. Vertical gray bars denote the National Bureau of Economic Research (NBER) basedrecession periods. 67 igure 2 – Time series of US Treasury bond yields
96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 1901234567 Y i e l d ( % ) The figure displays the end-of-month values of 1- to 10-year Treasury bond yields from January1996 to December 2018. Vertical gray bars indicate periods of elevated ( > = 85%-ile) equitytail risk implied by S&P 500 index options. 68 igure 3 – Time series of the pricing factors of US Treasuries
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123456 TR (eq)
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123
PC1
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123
PC2
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123
PC3
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123
PC4
96 98 00 02 04 06 08 10 12 14 16 18
Year −4−3−2−10123456
PC5
The figure displays the monthly time series of the pricing factors of the proposed GaussianATSM with equity tail risk. The top-left panel shows the S&P 500 option-implied equity tailrisk factor TR ( eq ) . The remaining panels show the first five principal components extractedfrom the US Treasury yields orthogonal to the TR ( eq ) factor. The light-colored dashed linesshow the principal components extracted from non-orthogonalized yields, which however arenot used as pricing factors in our model. All factors have been normalized to have mean zeroand unit variance. 69 igure 4 – Observed and model-implied US Treasury bond yields and returns
96 98 00 02 04 06 08 10 12 14 16 18
Year −101234567 Y i e l d ( % )
96 98 00 02 04 06 08 10 12 14 16 18
Year −6−4−2024681012 R e t u r n ( % )
96 98 00 02 04 06 08 10 12 14 16 18
Year −101234567 Y i e l d ( % )
96 98 00 02 04 06 08 10 12 14 16 18
Year −60−40−20020406080 R e t u r n ( % )
96 98 00 02 04 06 08 10 12 14 16 18
Year −101234567 Y i e l d ( % )
96 98 00 02 04 06 08 10 12 14 16 18
Year −150−100−50050100150 R e t u r n ( % ) The figure displays the observed and model-implied time series of yields and one-month excessreturns on US Treasury bonds with 1-, 5- and 10-year maturities. In the left panels, the solidblack lines show the observed yields, the dashed gray lines plot the model-implied yields, whilethe dashed red lines indicate the model-implied term premia. In the right panels, the solidblack lines show the observed excess returns, the dashed gray lines plot the model-impliedexcess returns, while the dashed red lines indicate the model-implied expected excess returns.70 igure 5 – Model-implied yield loadings on the pricing factors of US Treasuries
Maturity (months) −0.45−0.40−0.35−0.30−0.25−0.20−0.15−0.10 TR (eq) Maturity (months)
PC1
Maturity (months) −0.6−0.4−0.20.00.20.4
PC2
Maturity (months) −0.10−0.050.000.050.100.150.20
PC3
Maturity (months) −0.04−0.03−0.02−0.010.000.010.020.03
PC4
Maturity (months) −0.015−0.010−0.0050.0000.0050.0100.0150.0200.025
PC5
The figure displays the model-implied yield loadings on the pricing factors of the proposedATSM with equity tail risk. These coefficients are calculated as − (1 /n ) b n and can be inter-preted as the response of the n -month yield (expressed in annualized percentage terms) to acontemporaneous shock to the respective factor. TR ( eq ) represents the S&P 500 option-impliedequity tail risk factor, normalized to have mean zero and unit variance. PC1 – PC5 denote thefirst five standardized principal components extracted from the US Treasury yields orthogonalwith respect to the TR ( eq ) factor. 71 igure 6 – Model-implied return loadings on the pricing factors of US Treasuries Maturity (months) TR (eq) Maturity (months)
PC1
Maturity (months)
PC2
Maturity (months) −0.07−0.06−0.05−0.04−0.03−0.02−0.010.000.01
PC3
Maturity (months)
PC4
Maturity (months)
PC5
The figure displays the model-implied excess return loadings on the pricing factors of theproposed ATSM with equity tail risk. These coefficients are calculated as b ′ n λ and can beinterpreted as the response of the expected one-month excess return (expressed in percentagenot annualized terms) on the n -month bond to a contemporaneous shock to the respectivefactor. TR ( eq ) represents the S&P 500 option-implied equity tail risk factor, normalized tohave mean zero and unit variance. PC1 – PC5 denote the first five standardized principalcomponents extracted from the US Treasury yields orthogonal with respect to the TR ( eq ) factor. 72 igure 7 – Impact over time of equity tail risk on US Treasury bond yields and components
96 98 00 02 04 06 08 10 12 14 16 18
Year −300−200−1000100200300 B a s i s p o i n t s
96 98 00 02 04 06 08 10 12 14 16 18
Year −300−200−1000100200300 B a s i s p o i n t s
96 98 00 02 04 06 08 10 12 14 16 18
Year −300−200−1000100200300 B a s i s p o i n t s
96 98 00 02 04 06 08 10 12 14 16 18
Year −300−200−1000100200300 B a s i s p o i n t s
96 98 00 02 04 06 08 10 12 14 16 18
Year −300−200−1000100200300 B a s i s p o i n t s
96 98 00 02 04 06 08 10 12 14 16 18
Year −300−200−1000100200300 B a s i s p o i n t s The figure displays the impact over time of the S&P 500 option-implied equity tail risk factorTR ( eq ) t on the 1-, 5- and 10-year US Treasury bond yields (black lines) and on their twocomponents, i.e average expected future short rate (red lines) and term premium (blue lines).73 igure 8 – Impact of equity tail risk on US Treasury bond yields Maturity (months) −300−200−1000100200300 B a s i s p o i n t s LTCM co apse (A(g-98)Financial crisis (Oct-08)Eurozone debt crisis (Sep-11)Fed's Taper Tantrum (May-13)
The figure displays the impact (in basis points) of the S&P 500 option-implied equity tail riskfactor TR ( eq ) t on the term structure of US interest rates for selected dates: Russian financialcrisis and collapse of Long Term Capital Management fund (Aug-98), onset of 2008-09 financialcrisis with bankruptcy of Lehman Brothers (Oct-08), intensification of European sovereign debtcrisis (Sep-11), announcement of the Federal Reserve’s “taper tantrum” (May-13). Interestrates fell on all dates except for May-13, when yields markedly rose.74 igure 9 – Time series of international equity tail risk measures
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123456 UK
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123456 DE
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123456 CH
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123456 FR
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123456 IT
96 98 00 02 04 06 08 10 12 14 16 18
Year −3−2−10123456 ES The figure displays the international equity tail risk measures calculated using options on theFTSE 100 (UK), DAX (DE), SMI (CH), CAC 40 (FR), FTSE MIB (IT) and IBEX 35 (ES)equity index. All series have been normalized to have mean zero and unit variance. The solidblack lines show the equity tail risk measure of the country of interest, while the dashed graylines show, for comparison, the S&P 500 option-implied equity tail risk measure TR ( eq ))