Rejoinder on: nonparametric tail risk, stock returns, and the macroeconomy

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Rejoinder on: Nonparametric Tail Risk,

Stock Returns, and the Macroeconomy

Caio Almeida

1

, Kym Ardison

2

, Rene´ Garcia

3

and Jose Vicente

4 1_{EPGE/FGV, Rio de Janeiro, E-mail: calmeida@fgv.br,} 2_{EPGE/FGV, Rio de Janeiro, E-mail:}

kym.ardison@fgvmail.br, 3_{Universite´ de Montre´al and Toulouse School of Economics, E-mail:}

rene.garcia@umontreal.ca and 4_{IBMEC Business School and Banco Central do Brasil, E-mail:}

jose.valentim@bcb.gov.br

Address correspondence to Caio Almeida, FGV-EPGE, Praia de Botafogo, 190, Rio de Janeiro, Brazil, e-mail: caio.almeida@fgv.br.

Abstract

The discussions focus on different aspects of the paper and are quite complemen-tary. Dobrev and Schaumburg look closely at our implementation choices and ana-lyse the sensitivity of the measure to these choices. Camponovo, Scaillet, and Trojani propose to use robust predictive regression methods to analyze our results. From a theoretical point of view, Kris Jacobs addresses the applicability of our risk neutralization procedure from a risk management perspective. Finally, Turan Bali proposes a handful of future research topics. This rejoinder provides additional material to the main paper and addresses the points raised by the discussants. Key words: economic predictability, prediction of market returns, risk factor, risk-neutral prob-ability, tail risk

JEL classification: G12, G13, G17

Let us first thank the editors, Federico Bandi and Andrew Patton, for selecting such a distinguished panel of discussants. Of course we are grateful to the discussants for hav-ing put such an effort to read the paper in detail and come up with challenghav-ing and use-ful comments. The discussions point to different aspects of the paper and are quite complementary. We could not have hoped for a better set of points of view. All discuss-ants agree with the main contribution of the paper. We propose a new measure of tail risk under the risk-neutral distribution of stock returns computed with an option-free methodology. Of course the methodology involves some choices to arrive at a comput-able index of tail risk. The discussion by Dobrev and Schaumburg looks closely at these implementation choices and analyses the sensitivity of the measure to these choices. We will summarize their main conclusions in Section 2 and provide more robustness VC_{The Author, 2017. Published by Oxford University Press. All rights reserved.}

For Permissions, please email: journals.permissions@oup.com

doi: 10.1093/jjfinec/nbx006 Advance Access Publication Date: 16 March 2017 Commentary

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evidence in response. This will be complementing the robustness results already included in the paper. Another important issue raised by Dobrev and Schaumburg has to do with the precision of the tail risk estimator we propose. Camponovo, Scaillet, and Trojani propose to use robust predictive regression methods to analyze our results in a wide roll-ing window framework over the period 1980–2010 that complements nicely our full-sample regressions over various full-sample periods. We discuss their conclusions in Section 3. Another issue raised by the same discussants and by Kris Jacobs addresses the risk-neutral nature of the tail-risk measure and questions its usefulness and relevance with re-spect to an objective risk measure easier to compute. We will discuss this issue in Section 4. The last section comments on the suggestions for future work by Tarun Bali and dis-cusses briefly the issue of understanding better the economic story that may lie behind our empirical findings.

1 Robustness of Measure to Implementation Choices

Dobrev and Schaumburg recompute the tail-risk index with our methodology for different choices of assets (the all 25 Fama–French portfolios instead of the first five principal com-ponents on these 25 portfolios), a window of 60 days instead of 30 days, and a threshold of 5% instead of 10%. They exhibit in Table A.1 the correlations of the resulting measures with our benchmark tail-risk measure, respectively, 0.7406, 0.8076, and 0.7849. They con-clude that there is a lack of sufficient invariance to the implementation choices and that this should be kept in mind when applying the measure in different contexts. In our paper, we have focused more directly on the robustness of results to different implementation choices. We have presented in Section 5 several robustness checks regarding mainly the equity pre-mia resulting for several variations including the number of principal components, window, and threshold. For space considerations we left unreported many other combinations and their impact on risk premia and predictability regressions. We take the opportunity of this rejoinder to report our results for the particular choices made by our discussants. We report our results in two tables.Table 1includes the risk premia results for the different tail-risk measures, whileTable 2shows the t-statistics in the macroeconomic regressions for hori-zons 1–12 months.1Results inTable 1show that the patterns, magnitudes, and statistical significance of risk premia for the different variants of the tail-risk measure are pretty ro-bust. Including all 25 portfolios improves slightly the significance as it should be expected since we rely on more information.2_{The window length is not changing results significantly}

for 45 and 60 days. It does however for 90 days. As the measure becomes more persistent we lose an important ingredient of our empirical strategy. The main goal of extracting risk-neutral probabilities is to capture the recent change in economic conditions and in in-vestors’ risk aversion. In times of turmoil averaging over a longer window makes our

1 We leave aside the predictability regressions of the market returns for space considerations. These regressions will be considered in the next section with the contribution of Camponovo, Scaillet, and Trojani.

2 The decision to extract the first five principal components was based on a theoretical concern. The extraction of the risk neutral probabilities with a large number of basis assets introduces pric-ing errors that we wanted to avoid since the recursive nature of our measure did not allow for a systematic check of the pricing errors.

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measure less responsive and not as good a measure of tail risk. Finally, including less obser-vations (5%) is less damaging than including too many for the tail nature of our measure.

Table 2on macroeconomic regressions exhibits interesting findings. Predictability results

are overall stronger with the baseline measure (five principal components) than with the measure based on the 25 portfolios (all assets). This is especially true for the financial and macroeconomic uncertainty measures. The strength of the signal is much stronger with the principal components and it matters for capturing future uncertainty. For the window length the message is similar, but there are a few cases worth mentioning. Increasing the

Table 1. Sorted portfolios: robustness

Panel A: One month holding period

Baseline 45 days 60 days 90 days 20% 5% All assets

Average return 1.52 1.68 1.42 0.65 1.03 1.52 1.70 (3.41) (3.68) (3.15) (1.82) (3.62) (3.34) (3.80) FF3 1.15 1.19 1.01 0.37 0.65 0.99 1.25 (3.09) (3.02) (2.46) (1.26) (2.59) (2.61) (3.33) FF3þMOM 1.20 1.19 0.99 0.36 0.64 1.03 1.29 (3.10) (2.96) (2.38) (1.22) (2.53) (2.63) (3.33) FF3þMOMþLIQ 1.28 1.29 1.04 0.34 0.62 1.13 1.39 (2.79) (2.72) (2.13) (1.11) (2.41) (2.44) (3.01) FF3þMOMþLIQþVOL 0.83 0.53 0.26 0.52 0.14 0.68 0.88 (1.80) (1.23) (0.63) (1.47) (0.47) (1.63) (2.08) Panel B: One year holding period

Baseline 45 days 60 days 90 days 20% 5% All assets

Average return 13.84 15.57 13.30 8.06 11.67 14.71 13.04 (3.10) (3.54) (3.48) (2.02) (3.50) (3.63) (3.07) FF3 3.17 3.63 3.05 0.50 2.82 4.18 3.26 (1.16) (1.33) (1.32) (0.18) (1.11) (1.55) (1.21) FF3þMOM 4.99 4.54 4.02 1.25 2.87 4.88 5.11 (2.01) (1.87) (1.94) (0.47) (1.30) (1.94) (2.15) FF3þMOMþLIQ 4.98 4.82 4.03 0.40 3.26 6.76 7.03 (2.05) (1.88) (1.98) (0.15) (1.48) (2.79) (2.80) FF3þMOMþLIQþVOL 11.56 8.63 16.85 23.34 8.19 8.13 0.59 (1.03) (0.67) (1.43) (1.78) (0.78) (0.84) (0.06)

Notes: This table presents the results for the high minus low portfolio returns formed by sorting CRSP stocks with code 10-11 into 10 decile portfolios according to their tail-risk hedging capacity for alternative tail-risk measures. For each month in our sample from February 1967 to December 2013, we sort the stocks and track their returns one-month post formation (Panel A) or one-year post formation (Panel B). In the first line we re-port the average re-portfolio returns. In the following lines we rere-port the a0_{s of regressions where we control for}

the factors indicated in the first column. The first column reproduces the results for the baseline measure re-ported in the paper. Subsequent columns present the results for alternative choices in the tail-risk parameters: number of days over which tail risk is computed; percentage of tail observations; all assets refer to the 25 Fama–French portfolios. Newey–West t-statistics reported between parentheses are computed with one lag for monthly results and 12 lags for yearly results.

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Table 2. Macroeconomic regressions robustness

Forecasting ADS KCFED NBER CFNAI Fin.

Uncert. Macro. Uncert. St. Louis. EPU Panel A: Baseline 1 (2.70) (0.27) (3.17) (2.80) (2.44) (2.35) (1.81) (4.18) 2 (3.09) (0.55) (2.50) (3.37) (4.17) (2.59) (2.21) (3.80) 3 (2.40) (1.05) (3.27) (2.97) (4.47) (2.17) (2.51) (3.15) 4 (1.57) (1.39) (3.45) (1.89) (3.40) (2.16) (2.47) (0.82) 5 (1.27) (1.40) (3.24) (1.68) (2.66) (1.66) (1.63) (1.05) 6 (0.76) (1.46) (3.27) (1.89) (2.63) (1.83) (1.79) (0.90) 7 (0.45) (1.36) (3.65) (0.87) (2.66) (2.16) (1.80) (2.24) 8 (0.65) (1.23) (3.56) (0.70) (3.38) (1.90) (0.55) (1.05) 9 (0.57) (1.15) (3.35) (1.66) (2.83) (2.31) (1.07) (0.86) 10 (0.39) (1.28) (3.57) (1.26) (3.05) (1.90) (2.23) (1.05) 11 (0.79) (1.03) (3.80) (1.37) (3.06) (2.18) (0.35) (1.75) 12 (0.96) (0.70) (3.74) (1.69) (1.81) (1.52) (0.59) (2.43) Panel B: 45 days 1 (2.49) (0.86) (2.30) (3.60) (2.75) (3.23) (2.71) (1.78) 2 (2.83) (0.41) (2.34) (3.04) (4.73) (2.26) (1.90) (1.91) 3 (2.44) (1.41) (2.88) (3.67) (4.45) (2.12) (2.45) (2.08) 4 (1.61) (1.49) (3.26) (1.50) (3.11) (2.24) (2.13) (0.87) 5 (1.16) (1.61) (3.46) (2.06) (3.18) (2.32) (2.08) (0.41) 6 (0.28) (1.49) (3.46) (1.35) (2.39) (2.45) (1.51) (1.08) 7 (0.27) (1.30) (3.48) (1.14) (3.48) (2.26) (1.06) (1.96) 8 (0.45) (1.15) (3.48) (0.92) (3.25) (1.71) (0.86) (1.25) 9 (0.30) (1.22) (3.53) (1.38) (2.07) (1.79) (2.81) (1.14) 10 (0.34) (1.22) (3.63) (1.28) (3.11) (1.80) (1.92) (1.41) 11 (0.91) (0.97) (3.74) (1.49) (3.27) (1.46) (0.89) (1.89) 12 (1.63) (0.95) (3.63) (1.74) (1.88) (0.60) (1.50) (2.29) Panel C: 60 days 1 (2.49) (1.02) (2.03) (3.05) (3.54) (2.85) (2.91) (1.42) 2 (2.22) (1.21) (2.00) (2.54) (5.74) (2.27) (2.02) (1.06) 3 (1.64) (1.57) (2.39) (2.10) (4.41) (2.12) (2.25) (0.54) 4 (1.24) (1.66) (2.74) (1.45) (3.71) (2.21) (2.08) (0.04) 5 (0.91) (1.56) (3.03) (1.54) (3.40) (2.19) (1.40) (0.21) 6 (0.66) (1.43) (3.10) (1.40) (3.34) (2.04) (1.56) (0.12) 7 (0.63) (1.28) (3.10) (1.43) (3.16) (1.86) (1.14) (0.53) 8 (0.53) (1.21) (3.28) (1.07) (2.83) (1.64) (1.59) (0.55) 9 (0.35) (1.15) (3.26) (1.17) (3.04) (1.23) (1.90) (0.70) 10 (0.50) (1.07) (3.45) (1.14) (3.08) (0.91) (1.03) (0.87) 11 (1.10) (0.88) (3.46) (1.35) (2.53) (0.91) (0.85) (1.19) 12 (1.56) (0.80) (3.45) (1.44) (1.44) (0.53) (0.11) (1.29) (continued)

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Table 2. Continued

Uncert. Macro. Uncert. St. Louis. EPU Panel D: 90 days 1 (1.66) (0.89) (1.32) (2.64) (3.52) (3.22) (3.62) (0.90) 2 (1.60) (1.18) (1.94) (1.89) (4.94) (2.56) (2.42) (0.19) 3 (1.34) (1.56) (2.54) (2.08) (4.20) (2.23) (2.20) (0.33) 4 (0.61) (1.58) (2.98) (1.35) (3.46) (2.30) (2.36) (0.15) 5 (0.47) (1.48) (3.35) (1.25) (2.91) (2.01) (1.66) (0.39) 6 (0.73) (1.37) (3.31) (1.47) (3.01) (1.87) (1.64) (0.44) 7 (0.52) (1.30) (3.11) (1.41) (2.82) (1.68) (1.71) (0.56) 8 (0.26) (1.15) (3.17) (0.94) (3.23) (1.48) (1.66) (0.34) 9 (0.32) (1.01) (3.16) (0.87) (3.23) (1.25) (1.36) (0.67) 10 (0.67) (0.93) (3.26) (1.12) (3.40) (0.82) (1.22) (1.05) 11 (1.13) (0.81) (3.45) (1.34) (2.45) (0.72) (1.21) (1.11) 12 (1.46) (0.81) (3.07) (1.53) (1.43) (0.52) (0.34) (1.21) Panel E: 20% 1 (2.43) (0.17) (2.81) (2.67) (2.35) (4.43) (1.78) (3.49) 2 (2.98) (0.32) (3.39) (3.22) (4.44) (3.04) (1.99) (2.30) 3 (2.57) (0.90) (4.33) (2.80) (5.14) (2.37) (2.17) (1.95) 4 (1.57) (1.34) (3.99) (1.87) (5.48) (1.94) (3.02) (0.34) 5 (1.39) (1.27) (4.36) (1.54) (5.05) (1.78) (2.11) (0.85) 6 (1.40) (1.43) (4.10) (2.43) (4.66) (2.11) (2.11) (1.11) 7 (0.74) (1.39) (4.12) (1.26) (3.44) (2.25) (2.16) (1.35) 8 (0.92) (1.23) (3.90) (1.06) (2.45) (1.67) (0.30) (1.21) 9 (0.89) (1.24) (2.90) (1.73) (3.58) (2.25) (1.72) (0.84) 10 (0.50) (1.35) (3.05) (1.43) (1.58) (2.06) (2.00) (1.44) 11 (0.87) (1.19) (3.30) (1.24) (2.63) (2.09) (1.71) (1.46) 12 (1.41) (1.14) (3.30) (1.74) (1.49) (1.72) (1.23) (1.32) Panel F: 5% 1 (2.20) (1.88) (2.24) (1.61) (3.50) (2.86) (0.63) (2.21) 2 (2.17) (1.46) (0.95) (1.91) (6.96) (2.01) (1.54) (2.11) 3 (1.70) (1.23) (1.54) (1.87) (4.49) (1.83) (1.23) (1.69) 4 (1.19) (0.30) (2.13) (1.18) (5.91) (2.64) (2.11) (0.37) 5 (1.30) (0.58) (2.01) (0.85) (4.30) (1.93) (0.98) (0.56) 6 (1.57) (0.23) (1.83) (1.44) (3.31) (1.73) (1.25) (0.50) 7 (1.24) (1.40) (2.03) (1.35) (2.43) (1.67) (1.51) (0.62) 8 (1.00) (1.17) (2.22) (0.68) (1.81) (1.33) (0.66) (1.02) 9 (0.65) (0.95) (1.79) (1.54) (0.84) (0.39) (1.27) (1.22) 10 (0.10) (0.44) (2.03) (0.29) (0.92) (0.02) (0.22) (0.72) 11 (0.08) (0.31) (2.17) (0.76) (2.20) (0.61) (0.93) (0.26) 12 (0.11) (1.55) (1.89) (0.49) (1.11) (0.64) (0.71) (0.58) (continued)

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window to 90 or 60 days is in general worse than our baseline case, except for financial un-certainty. Again increasing the information in the cross-section of stock returns appears useful for capturing future financial uncertainty, but it is not for predicting economic activ-ity and macroeconomic uncertainty. For the threshold in macroeconomic predictabilactiv-ity re-gressions, including less observations in the tail (5%) is more damaging than including more observations. While the significance patterns are similar between our baseline and 20%, the significance is often higher with the 20% threshold.

These complementary results refine the point raised by Dobrev and Schaumburg about im-plementation choices. We agree that some choices may be better for a particular application, but we hope to have shown that our particular choices for the particular applications we chose (equity premia and predictability regressions for market returns and macroeconomic in-dicators) seem to be overall quite robust to the variations suggested by the discussants.

2 Robust Regression Methods and Estimation Uncertainty

Camponovo, Scaillet, and Trojani decided to revisit our predictive regressions by using ro-bust resampling tests developed in Camponovo, Scaillet, and Trojani (2015). This is

Table 2. Continued

Uncert.

Macro. Uncert.

St. Louis. EPU

Panel G: All assets

1 (2.14) (1.40) (2.89) (1.82) (1.95) (1.50) (1.19) (4.06) 2 (3.00) (1.26) (3.21) (1.89) (3.02) (2.51) (0.92) (2.96) 3 (2.52) (0.66) (3.70) (2.88) (2.43) (1.40) (2.13) (1.98) 4 (1.50) (1.16) (3.87) (1.25) (2.53) (1.45) (1.46) (0.11) 5 (1.44) (0.98) (3.10) (1.76) (3.07) (1.22) (1.16) (0.46) 6 (1.08) (1.21) (2.71) (1.87) (2.08) (1.16) (1.16) (0.75) 7 (1.06) (1.14) (2.95) (1.04) (0.21) (0.98) (0.39) (1.24) 8 (1.55) (0.66) (2.47) (1.21) (1.31) (0.54) (1.06) (0.88) 9 (1.42) (0.34) (1.84) (1.79) (1.25) (0.96) (0.71) (0.48) 10 (1.06) (0.45) (2.02) (1.30) (0.93) (0.48) (1.18) (1.01) 11 (1.11) (0.70) (2.45) (1.37) (1.72) (0.88) (0.88) (1.40) 12 (0.99) (1.01) (2.40) (1.36) (1.60) (0.06) (1.51) (1.85)

Notes: This table presents the results of predictive regressions for a variety of macroeconomic indicators over different samples for each indicator. ADS denotes the Aruoba, Diebold, and Scotti macroeconomic activity in-dicator (02/1960–04/2014), KCFED the Kansas City FED macroeconomic inin-dicator index (01/1990–04/2014), NBER a recession period dummy (07/1926–04/2014), CFNAI the Chicago FED National Activity Index (02/ 1967–04/2014), Fin and Macro Uncert. the Jurando, Ludvigson, and Ng (2015) financial and macroeconomic uncertainty indices, respectively (07/1960–04/2014), St. Louis the St. Louis FED Financial Stress Index (01/ 1994–04/2014), and EPU the Economic Policy Uncertainty Index of Backer, Bloom, and Davis (2015) (01/ 1985–04/2014). All regressions control for 12 lags of the endogenous variable. All t-statistics are calculated using Newey and West matrix with 24 lags. For the NBER variable we include the Probit regression results. Panel A reproduces the baseline results in the paper. Subsequent panels present the results for alternative choices in the tail-risk parameters: number of days over which tail risk is computed; percentage of tail observa-tions; all assets refer to the 25 Fama–French portfolios.

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certainly an important addition to our paper in several respects. First, it is a particularly important methodology to apply in a context where data can include rare influential obser-vations. Second, it is also an important element to take into consideration when construct-ing confidence bands around the estimated tail-risk measure. Dobrev and Schaumburg raise the issue of estimation uncertainty and compute standard bootstrap confidence bands around our tail-risk measure. They stress the high degree of uncertainty around our meas-ure, especially around more pronounced spikes. They acknowledge though that their confi-dence bands may be inflated by outliers and cite the work of Camponovo, Scaillet, and Trojani. The later discussants do not compute confidence bands with their outlier-robust resampling methods but devote their discussion on checking the predictability of market returns by our tail-risk measure, with and without dividend yield as an additional pre-dictor. They use rolling windows of 180 monthly observations over the period 1980–2010. Both their robust subsampling and bootstrap procedures conclude that their confidence bands point to the rejection of no predictability, which is not the case with standard resam-pling procedures. They also add value to our paper by investigating out-of-sample predict-ability (that we did not include in our paper to leave space for more applications). They conclude that our tail-risk measure provides more accurate out-of-sample predictions than the sample mean of market returns, irrespective of the robust or non-robust methods. This certainly reinforces the evidence about the reliability of the predictive power of our measure.

3 Objective Versus Risk-Neutral Tail-Risk Measure

The discussions by Dobrev and Schaumburg and by Jacobs point to the complexity of our risk-neutralization procedure relative to a simpler approach based on the physical distribu-tion of returns. Jacobs makes a distincdistribu-tion between two general applicadistribu-tions of the tail-risk measure, namely risk management and financial or economic forecasting with a cross-sectional approach. He goes on by saying that for risk management what matters is the per-formance of the measure in the tails only, while forecasting perper-formance is assessed on average at every point in the sample. While he admits that a risk-neutral measure based on a single day of option data may be preferable to a traditional risk measure based on the physical measure, he is convinced by our procedure that risk-neutralizes returns because of the short window (since extreme events are scarce in historical samples). We agree that we did not produce any convincing evidence of the reliability of our measure for risk manage-ment and on its ability to measure the frequency of extreme events. However, it is worth mentioning that using the cross-section of returns may help in capturing an information that is present in options because of some long-short strategies concentrated on some secur-ities or sectors. But the point is well taken and more work is needed to verify this point. What we showed clearly is that the risk neutralization was very important for the assess-ment of risk premia attached to tail risk.

Dobrev and Schaumburg go further and propose a conditionally normal approxi-mation to our tail-risk measure. Indeed under conditional normality the physical and the risk-neutral measures coincide. This approximate measure is a linear combination of the conditional variances. As such, it is more persistent and stable through time than our measure. It is also naturally more correlated with the volatility Bloom index (0.69 instead of 0.46 for our baseline measure) and with the VIX (0.84 instead of

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0.56 for our baseline measure). However, in our view it misses important tail informa-tion. Section 4.2.1 shows the additional economic effects of tail information in a vec-tor auvec-toregressive framework with respect to stock market variance and Bloom big-event volatility.

4 Future Work

Our paper has left several stones unturned. The discussion by Bali points to useful exten-sions and robustness checks of our work. Estimating the price of tail risk by Fama– MacBeth cross-sectional regressions is certainly a worthwhile exercise although it will ne-cessitate choices of factors and test portfolios. Our strategy based on a long-short portfolio risk premium avoids these choices but has the usual shortcomings attached to the approach. Another extension will be to check if our tail-risk measure predicts returns for other catego-ries of assets such as bonds, hedge funds, or mutual funds. Yet another is the persistence of the individual tail-risk betas. But the main one is probably to better understand the source of return predictability. What is the economic mechanism that will rationalize why secur-ities with low tail-risk betas have such a capacity to predict future returns? The question is also present in the discussion by Jacobs and Dobrev and Schaumburg. There are at least two ways to address the problem. One is to better identify the securities included in the low-tail-risk beta portfolios through their other characteristics such as size, liquidity, value, and volatility. Another one is to come up with economic models that can produce empirical findings similar to the ones exhibited in the paper. This is a much more challenging task but certainly worth the effort.

Dobrev and Schaumburg go mid-way by proposing a statistical toy model and checking correlation and predictability properties of a tail-risk measure built according to a closed-form risk neutralization specific to the model used and one based on our methodology. Their parametric market factor dynamics is a two-component model with separate volatil-ity and jump dynamics. The returns of the K basis assets are linearly related to the market factor and their idiosyncratic shocks are normally distributed. Empirically they regress the excess returns of the 25 Fama–French size and book-to-market portfolios on the market ex-cess returns to estimate the portfolio betas and covariance matrix. The discussants provide several comparisons between variants of our measure (length of window and number of assets as discussed before) in two different contexts, one with observed past returns simu-lated from the model and the other based on the infeasible conditionally drawn returns. Their main results are: i) the poor correlation of our 30-day baseline model with the true tail-risk index; ii) the absence of predictability of market returns with our baseline model; and iii) the consistency of our procedure since both correlation and predictability increase with the length of the window. These results illustrate that our parameter choices may not universally apply and that the returns generating process ultimately determines what will work best. We have however shown in Section 2 that with the data set at hand our results were in general robust to the parameter choices. We believe that the toy model establishes a tight link between volatility and jump risk. Therefore, the true tail-risk measure exhibits persistence and longer windows are necessary to recover the true process. The challenge therefore is to find a model that will be more in line with our empirical results. Certainly jumps in basis assets will need to be modeled differently. Jacobs gives an important hint,

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saying that for the index the third moment is more important while it is the fourth moment for individual stocks. We leave this challenging task for future work.

Reference

Camponovo, L., Scaillet, O., and F. Trojani, 2015. Predictability hidden by anomalous observa-tions, working paper, University of Geneva & Swiss Finance Institute.