Variance Premium, Downside Risk, and Expected Stock Returns

Online Appendix

Bruno Feunou Ricardo Lopez AliouchkinBank of Canada Syracuse University

Roméo Tédongap Lai XuESSEC Business School Syracuse University

June 2018

We thank Bo-Young Chang, Jonathan Witmer, participants to the 2017 OptionMetrics Research Conference,ESSEC Business School’s 4th Empirical Finance Workshop, Frontiers of Factor Investing, 2018 FMA EuropeanConference, 2018 ESSEC–Amundi Annual Workshop on Asset & Risk Management, 11th Workshop on AppliedFinancial Econometrics at MSH de Paris Nord, as well as seminar participants at the Norwegian Business Schooland University of Reading. Feunou gratefully acknowledges financial support from the IFSID. Lopez Aliouchkin andTédongap acknowledge the research grant from the Thule Foundation’s Skandia research programs on “Long-TermSavings.” Send correspondence to Roméo Tédongap, Department of Finance, ESSEC Business School, 3 AvenueBernard Hirsch, CS 50105 CERGY, 95021 Cergy-Pontoise Cedex, France; telephone: +33 1 34 43 97 34. E-mail:tedongap@essec.edu.

Variance Premium, Downside Risk, and Expected Stock Returns

Online Appendix

Abstract

We decompose total variance into its bad and good components and measure the premia

associated with their fluctuations using stock and option data from a large cross-section of

firms. The total variance risk premium (VRP) represents the premium paid to insure against

fluctuations in bad variance (called bad VRP), net of the premium received to compensate for

fluctuations in good variance (called good VRP). Bad VRP provides a direct assessment of the

degree to which asset downside risk may become extreme, while good VRP proxies for the degree

to which asset upside potential may shrink. We find that bad VRP is important economically; in

the cross-section, a one-standard-deviation increase is associated with an increase of up to 24%

in annualized expected excess returns. Simultaneously going long on stocks with high bad VRP

and short on stocks with low bad VRP yields an annualized risk-adjusted expected excess return

of 27%. This result remains significant in double-sort strategies and cross-sectional regressions

controlling for a host of firm characteristics and exposures to regular and downside risk factors.

Keywords: Variance Forecasting, Jump Risk Premium

JEL Classification: G12

A Upside and Downside Moments from OTM Options

Consider the function

F (X) =1

αln (1− δ + δ exp (αX))

with 0 ≤ δ ≤ 1 and α > 0. It can easily be verified that

F (X) =

0 if δ = 0

X if δ = 1

δX if α→ 0, 0 < δ < 1

max (X, 0) if α→∞, 0 < δ < 1

Suppose we are interested in computing the upside risk-neutral moments of the τ -period log

returns defined by rt,t+τ = ln[St+τSt

]. That is, we want to compute

µQ+n (t, τ) ≡ EQt

[rnt,t+τ I (rt,t+τ > 0)

]for n ≥ 2.

Observe that

rnt,t+τ I (rt,t+τ > 0) = (rt,t+τ I (rt,t+τ > 0))n

= (max (rt,t+τ , 0))n

= limα→∞0

Thus we can compute EQt [(F (rt,t+τ ))n] for n ≥ 2 by applying the BKM formula

EQt [exp (−rτ)H (St+τ )] = exp (−rτ)(H (St)− StH ′ (St)

)+ StH

′ (St)

+

St∫0

H ′′ (K)P (t, τ ;K) dK +

∞∫St

H ′′ (K)C (t, τ ;K) dK(A.2)

with the twice differentiable function H (S) =(F(

ln[SSt

]))n. We have

H ′ (S) =nF ′

(ln[SSt

])(F(

ln[SSt

]))n−1S

H ′′ (S) =

n

[(F ′′(

ln[SSt

])− F ′

(ln[SSt

]))F(

ln[SSt

])+ (n− 1)

(F ′(

ln[SSt

]))2](F(

ln[SSt

]))n−2S2

Observe that, since F (0) = 0 and F ′ (0) = δ, for n ≥ 2 we have

H (St) = (F (0))n = 0 and H ′ (St) =

nF ′ (0) (F (0))n−1

St= 0.

This means that

exp (−rτ)(H (St)− StH ′ (St)

)+ StH

′ (St) = 0. (A.3)

Now, we are interested in computing

limα→∞0

and thus

limα→∞0

then it follows that

EQt[exp (−rτ) rnt,t+τ I (rt,t+τ < 0)

]=

St∫0

n(n− 1 + ln

[StK

]) (− ln

[StK

])n−2K2

P (t, τ ;K) dK for n ≥ 2.

(A.8)

Let’s now focus on computing the expression

µQ+1 (t, τ) ≡ EQt [rt,t+τ I (rt,t+τ > 0)] .

Observe that

1

St[exp (−rτ) (St+τ − St) I (St+τ − St > 0)] + exp (−rτ) I (St+τ − St > 0)

= exp (−rτ) St+τSt

I (St+τ − St > 0)

= exp (−rτ) exp (rt,t+τ ) I (rt,t+τ > 0)

= exp (−rτ)

{1 + rt,t+τ +

∞∑n=2

1

n!rnt,t+τ

}I (rt,t+τ > 0)

= exp (−rτ) I (rt,t+τ > 0) + exp (−rτ) rt,t+τ I (rt,t+τ > 0) +∞∑n=2

1

n!exp (−rτ) rnt,t+τ I (rt,t+τ > 0)

Taking risk-neutral expectations of the above equation, we find that

EQt [exp (−rτ) rt,t+τ I (rt,t+τ > 0)] =C (t, τ ;St)

St−∞∑n=2

1

n!EQt[exp (−rτ) rnt,t+τ I (rt,t+τ > 0)

](A.9)

where the summation in the right-hand side can be truncated at n = 4 based on a fourth-order

Taylor approximation of the exponential series (see BKM apendix).

In their appendix, BKM find that

EQt [exp (−rτ) rt,t+τ ] = 1− exp (−rτ)−∞∑n=2

1

n!EQt[exp (−rτ) rnt,t+τ

], (A.10)

where the summation in the right-hand side can be truncated at n = 4 based on a fourth-order

4

Taylor approximation of the exponential series. It follows that

EQt [exp (−rτ) rt,t+τ I (rt,t+τ < 0)] = −P (t, τ ;St)

St−∞∑n=2

1

n!EQt[exp (−rτ) rnt,t+τ I (rt,t+τ < 0)

].

(A.11)

Consistently, we use the notations

µQn (t, τ) ≡ EQt

[rnt,t+τ

]and µQ−n (t, τ) ≡ E

Qt

[rnt,t+τ I (rt,t+τ < 0)

]for n ≥ 1.

B Measuring Bad and Good Variance Risk Premia

For a given stock, we consider its realized return and its realized variance aggregated over a monthly

period based on high-frequency returns, which we define by

rt−1,t =

1/δ∑j=1

rt−1+jδ and RVt−1,t =

1/δ∑j=1

r2t−1+jδ, (A.12)

where 1/δ is the number of high-frequency returns assumed for the monthly period, i.e., δ = 1/21

for daily returns, rt−1+jδ denotes the jth high-frequency return of the monthly period starting at

date t − 1 and ending at date t, rt−1,t is the monthly return, and RVt−1,t is the monthly total

realized variance.

In the literature, a forecasting model of realized variance, such as the HAR-RV model and its

variants, is used to compute an estimate of Et [RVt,t+1] where Et [·] denotes the time-t real-world

conditional expectation operator. Likewise, the BKM formula (A.7) is discretized and used to

obtain an estimate of EQt [RVt,t+1], where EQt [·] denotes the time-t conditional expectation operator

under some risk-neutral measure Q. Notice however that the estimate obtained from a straight

empirical implementation of the discretized BKM formula is truly an estimate of EQt[r2t,t+1

]and

not an estimate of EQt [RVt,t+1]. In that sense, we argue that the empirical estimate of the monthly

variance risk premium from actual returns and options data that has so far been used in the

literature corresponds to an estimate the following quantity:

V RP0t ≡ EQt[r2t,t+1

]− Et [RVt,t+1] . (A.13)

5

There is an inconsistency with this approach of measuring the variance risk premium. In fact, it

is clear from Equation (A.13) that V RP0t is not a premium, since a premium is by definition the

difference between real-world and risk-neutral expectations of the same quantity.

Instead, it would be consistent to define the variance risk premium either as

V RP1t ≡ EQt [RVt,t+1]− Et [RVt,t+1] (A.14)

or

V RP2t ≡ EQt[r2t,t+1

]− Et

[r2t,t+1

]. (A.15)

However, unless we observe variance swap data, measuring EQt [RVt,t+1] is unfeasible from ob-

served options data, since it would require high-frequency-maturing OTM options to implement

the BKM formula. In fact we have

EQt [RVt,t+1] = EQt

1/δ∑j=1

r2t+jδ

= 1/δ∑j=1

EQt[r2t+jδ

]=

1/δ∑j=1

EQt[EQt+jδ−δ

[r2t+jδ

]]

=

1/δ∑j=1

EQt[exp (rδ)µQ2 (t+ jδ − δ, δ)

]= exp (rδ)EQt

1/δ∑j=1

µQ2 (t+ jδ − δ, δ)

(A.16)

where the µQ2 (t+ jδ − δ, δ) , j = 1, . . . , 1/δ can be computed using the BKM formula only if

we observed options with intradaily or daily maturity period. Data on these options obviously

are nonexistent. Even if they were and the µQ2 (t+ jδ − δ, δ) , j = 1, . . . , 1/δ were computed, we

would still have to estimate the risk-neutral conditional expectation of their sum, another layer

of complication. This proves that it is practically unfeasible to estimate V RP1t via options data.

Instead, EQt [RVt,t+1] can be measured as the observed one-month maturity variance swap fixed

rate, but of course the sample would be much smaller compared to that of options data, combined

with the fact that variance swaps data are nonexistent for the vast majority of individual stocks.

Let’s focus on the implications of the actual inconsistency in the definition of V RP0t. Notice

that the squared monthly return may be written

r2t,t+1 =

1/δ∑j=1

rt+jδ

2 = 1/δ∑j=1

r2t+jδ + 2

1/δ−1∑i=1

1/δ−i∑j=1

rt+jδrt+jδ+iδ = RVt,t+1 + 2RAt,t+1 (A.17)

6

where RAt,t+1 denotes the realized autocovariance defined by

RAt,t+1 =

1/δ−1∑i=1

1/δ−i∑j=1

rt+jδrt+jδ+iδ =1

2

(r2t,t+1 −RVt,t+1

). (A.18)

From the decomposition (A.17), it follows that

V RP0t = V RP1t + 2EQt [RAt,t+1] (A.19)

V RP0t = V RP2t + 2Et [RAt,t+1] (A.20)

V RP1t = V RP2t − 2ARPt (A.21)

where ARPt denotes the autocovariance risk premium defined by

ARPt ≡ EQt [RAt,t+1]− Et [RAt,t+1] . (A.22)

Equation (A.17) shows that the squared monthly return is approximately equal to the monthly

realized variance if the realized autocovariance if negligible. A time series analysis of the realized

autocovariance is necessary to verify that assertion. If the realized autocovariance is negligible,

then all three definitions of the variance risk premium are approximately equivalent, and the in-

consistency in the definition of V RP0t would only have a minor effect empirically. More generally,

Equation (A.19) shows that if the risk-neutral conditional expectation of the realized autocovariance

is zero, that is EQt [RAt,t+1] = 0, then V RP0t = V RP1t. Likewise, Equation (A.20) shows that if the

real-world conditional expectation of the realized autocovariance is zero, that is Et [RAt,t+1] = 0,

then V RP0t = V RP2t.

Denoting by I (·) the indicator function that takes the value 1 if the condition is met and 0

otherwise, we split the total realized variance into two components, also referred to as realized

semi-variances, as follows:

RVt−1,t = RVbt−1,t +RV

gt−1,t where

RV bt−1,t =

1/δ∑j=1

r2t−1+jδI (rt−1+jδ < 0) and RVgt−1,t =

1/δ∑j=1

r2t−1+jδI (rt−1+jδ ≥ 0) ,(A.23)

7

where RV bt−1,t captures the part of the variance due to negative returns only, corresponding to

our definition for bad variance, while RV gt−1,t captures the variation due to positive returns only,

corresponding to our definition for good variance. Consistent with the three measures of the total

variance risk premium previous defined, we can equivalently define the bad variance risk premium

V RP b0t ≡ EQt

[r2t,t+1I (rt,t+1 < 0)

]− Et

[RV bt,t+1

]V RP b1t ≡ E

Qt

[RV bt,t+1

]− Et

[RV bt,t+1

]V RP b2t ≡ E

Qt

[r2t,t+1I (rt,t+1 < 0)

]− Et

[r2t,t+1I (rt,t+1 < 0)

],

(A.24)

and the good variance risk premium

V RP g0t ≡ Et[r2t,t+1I (rt,t+1 > 0)

]− EQt

[RV gt,t+1

]V RP g1t ≡ Et

[RV gt,t+1

]− EQt

[RV gt,t+1

]V RP g2t ≡ Et

[r2t,t+1I (rt,t+1 > 0)

]− EQt

[r2t,t+1I (rt,t+1 > 0)

].

(A.25)

Observe that, even if a negligeable magnitude of the realized autocovariance makes the three

measures of the total variance risk premium equivalent, the same those not hold for the three

measures of the bad variance risk premium. Likewise, the three measures of the good variance risk

premium would still be very different. That is, V RP b0t and V RPg0t are inconsistent measures of the

bad and the good variance risk premia, and empirical estimation of V RP b1t and V RPg1t is simply

unfeasible.

To contrary, notice that V RP2t, V RPb2t and V RP

g2t are consistent definitions of premia and

can always be estimated empirically. Using equations (A.7), (A.8) and (A.6), the BKM formula

can be discretized and empirically implemented to compute the risk-neutral expectations of the

squared monthly returns and truncated returns using actual options data. Likewise, a regression

model can be specified with appropriate predictors to estimate the real-world expectations of the

squared monthly returns and truncated returns using actual returns data. To compute these real-

world expectations, we assume that, conditional to time-t information, monthly returns rt,t+1 have

a normal distribution with mean µt = Et [rt,t+1] = Z>t βµ and variance σ2t = Et [RVt,t+1] = Z>t βσ,

8

in which case we have

Et[r2t,t+1

]= µ2t + σ

2t

Et[r2t,t+1I (rt,t+1 < 0)

]=(µ2t + σ

2t

)Φ

(−µtσt

)− µtσtφ

(µtσt

)Et[r2t,t+1I (rt,t+1 > 0)

]=(µ2t + σ

2t

)Φ

(µtσt

)+ µtσtφ

(µtσt

).

(A.26)

An estimate of µt is obtained as the fitted value from a linear regression of monthly returns onto the

vector of predictors, while an estimate of σ2t is obtained as the fitted value from a linear regression

of monthly total realized variance onto the same predictors. Those estimates are further plugged

into the formulas (A.26) to obtain estimates of the real-world expectations of the squared monthly

returns and truncated returns. Predictors in Z include the constant, bad and good realized variances

of the previous month, of the previous five months, and of the previous twenty-four months.

9

References

Amihud, Y. (2002). Illiquidity and Stock Returns: Cross-section and Time-series Effects, Journal

of Financial Markets 5: 31–56.

Ang, A., Hodrick, R. J., Xing, Y. and Zhang, X. (2006). The Cross Section of Volatility and

Expected Returns, Journal of Finance 11(1): 259–299.

Bakshi, G., Kapadia, N. and Madan, D. (2003). Stock Return Characteristics, Skew Laws, and the

Differential Pricing of Individual Equity Options, Review of Financial Studies 16(1): 101 – 143.

Bollerslev, T., Tauchen, G. and Zhou, H. (2009). Expected Stock Returns and Variance Risk

Premia, Review of Financial Studies 22(11): 4463–4492.

Bollerslev, T., Zhengzi, L. S. and Zhao, B. (2017). Good Volatility, Bad Volatility, and the Cross-

Section of Stock Returns, Working Paper .

Chang, B. Y., Christoffersen, P. and Jacobs, K. (2013). Market skewness risk and the cross section

of stock returns, Journal of Financial Economics 107(1): 46 – 68.

Fama, E. F. and French, K. R. (2015). A Five-factor Asset Pricing Model, Journal of Financial

Economics 116: 1–22.

Fama, E. F. and MacBeth, J. D. (1973). Risk, Return, and Equilibrium: Empirical Tests, Journal

of Political Economy 81(3): 607–636.

Farago, A. and Tédongap, R. (forthcoming). Downside risks and the cross-section of asset returns,

Journal of Financial Economics .

Newey, W. and West, K. (1987). A Simple, Positive Semi-definite Heteroskedasticity and Autocor-

relation Consistent Covariance Matrix, Econometrica 55: 703–708.

10

Table 1: Conditional Double Sorts on Exposures to Market Risk Neutral Skewness

Stocks are sorted every month in quintiles based on their exposure to market risk neutral skewness in all panels.Then, in

Panel A stocks within each quintile of exposure to this factor are further sorted in quintiles based on their bad VRP (V RP b).

Similarly, in Panel B, C and D, stocks within each quintile of exposure to these market risk neutral skewness are further sorted

in quintiles based on their good VRP (V RP g), total VRP (V RP ) and jump risk premium (JRP ), respectively. Firm exposures

to market risk-neutral skewness are estimated following the model of Chang et al. (2013) but in this table we use the level of

market risk neutral skewness instead of changes as in the main paper. The table reports average value-weighted excess returns

for the bottom quintile (1), the top quintile (5) and for the second (2), third (3) and fourth (4) quintile. We also report the

difference in average excess returns between the top and the bottom quintile (5-1). T-statistics are computed using Newey and

West (1987) standard errors, and are reported in parentheses. Significant t-statistics at the 95% confidence level are boldfaced.

Data are from January 1996 to December 2015.

Panel A: Firm Bad VRP Panel B: Firm Good VRP

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

1 -1.47 -0.30 -0.01 0.05 -1.04 0.43 (0.78) -1.13 -0.49 0.42 -0.04 -0.54 0.59 (1.12)

2 0.74 0.61 0.52 0.43 0.51 -0.22 (-0.42) 0.55 0.36 0.61 0.57 0.52 -0.03 (-0.07)

3 1.01 1.19 1.08 0.78 0.95 -0.06 (-0.13) 0.97 1.24 0.79 0.46 0.79 -0.18 (-0.37)

4 1.96 1.88 1.42 1.13 1.00 -0.96 (-1.87) 1.35 1.24 1.16 0.78 1.09 -0.25 (-0.41)

5 1.49 1.08 2.08 1.25 1.94 0.45 (0.79) 1.87 1.72 1.28 1.40 1.29 -0.58 (-1.01)

5-1 2.96 1.38 2.10 1.20 2.97 3.00 2.21 0.86 1.43 1.83

(4.34) (3.11) (4.18) (2.75) (5.07) (5.07) (4.35) (2.22) (3.22) (3.85)

Panel C: Firm Total VRP Panel D: Firm JRP

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

1 -0.39 0.54 0.60 0.54 0.01 0.39 (0.63) -1.65 -0.59 -0.26 -0.11 -1.35 0.30 (0.57)

2 0.86 0.66 0.85 0.40 0.47 -0.39 (-0.69) 0.24 0.69 0.65 0.40 0.49 0.26 (0.52)

3 1.20 0.91 0.80 0.53 1.07 -0.12 (-0.25) 1.78 1.09 1.13 0.67 1.07 -0.71 (-1.58)

4 0.87 1.15 1.06 0.74 0.59 -0.28 (-0.60) 1.49 1.59 1.25 1.05 1.43 -0.06 (-0.10)

5 0.85 0.54 0.98 0.36 0.56 -0.29 (-0.48) 2.74 2.21 2.33 2.35 1.98 -0.76 (-1.11)

5-1 1.23 -0.00 0.38 -0.18 0.55 4.39 2.81 2.60 2.46 3.33

(2.36) (-1.5e-3) (0.92) (-0.44) (1.00) (5.49) (6.05) (5.08) (5.64) (6.88)

11

Table 2: Univariate Sorts on Firm Bad VRP Excluding IT- and Financial crises

In Panel A and B, at the end of month t we sort firms into quintiles based on their average bad VRP (V RP b) during month t,

so that Quintile 1 contains the stocks with the lowest V RP b and Quintile 5 the highest. We then form value-weighted portfolios

of these firms, holding the ranking constant for the next month. Subsequently, we compute cumulative returns during month

t + 1 for each quintile portfolio. We report the monthly average cumulative return in percentage of each portfolio. We also

compute the Jensen alpha of each quintile portfolio with respect to the Fama-French five-factor model (Fama and French 2015)

by running a time series regression of the monthly portfolio returns on monthly MKT , SMB, HML, RMW , and CMA. The

t-statistics test the null hypothesis that the average monthly cumulative return of each respective portfolio equals zero, and they

are computed using Newey and West (1987) standard errors to account for autocorrelation, and are reported in parentheses.

Significant t-statistics at the 95% confidence level are boldfaced. V RP and JRP are reported in monthly square percentage

units. The sample period excluding the financial crisis runs from January 1996 until December 2006. The sample period

excluding the IT-crisis runs from January 2003 until December 2015.

Panel A: Excluding Financial Crisis Panel B: Excluding IT-Crisis

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

V RP b -236.94 2.20 33.19 76.69 248.13 -104.16 10.98 29.38 56.02 212.20

E [r] -0.80 0.63 1.22 1.85 2.12 2.91 0.04 0.75 1.04 1.17 1.40 1.37(-1.30) (1.56) (2.92) (2.98) (2.39) (4.54) (0.07) (2.52) (2.75) (2.42) (2.26) (3.37)

alpha -0.58 0.72 1.44 2.18 2.73 3.31 -0.09 0.71 0.95 1.06 1.22 1.31

(-1.02) (1.81) (3.32) (3.31) (2.98) (4.45) (-0.18) (2.33) (2.47) (2.11) (2.03) (3.54)

12

Table 3: Univariate Sorts on Firm Bad VRP: Small, Medium and Large Firms

In Panel A, at the end of month t we sort small firms into quintiles based on their average bad VRP (V RP b) during month

t, so that Quintile 1 contains the stocks with the lowest V RP b and Quintile 5 the highest. Small firms are in the bottom

30% based on market capitalization. We then form value-weighted portfolios of these firms, holding the ranking constant for

the next month. Subsequently, we compute cumulative returns during month t + 1 for each quintile portfolio. We report the

monthly average cumulative return in percentage of each portfolio. Similarly, in Panel B, and C, we sort medium and large

firms into quintiles based on their average bad VRP (V RP g). Medium and large firms are in the middle 40%, and top 30%

based on market capitalization. We also compute the Jensen alpha of each quintile portfolio with respect to the Fama-French

five-factor model (Fama and French 2015) by running a time series regression of the monthly portfolio returns on monthly

MKT , SMB, HML, RMW , and CMA. The t-statistics test the null hypothesis that the average monthly cumulative return

of each respective portfolio equals zero, and they are computed using Newey and West (1987) standard errors to account for

autocorrelation, and are reported in parentheses. Significant t-statistics at the 95% confidence level are boldfaced. V RP and

JRP are reported in monthly square percentage units.

Panel A: Small Firms Panel B: Medium Firms

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

V RP b -323.74 2.80 59.51 122.56 405.44 -156.36 8.37 37.34 71.04 190.14

E [r] -1.75 0.54 1.15 1.66 1.51 3.27 -0.89 0.47 1.09 1.45 1.79 2.68(-2.59) (0.99) (2.28) (3.10) (2.30) (8.08) (-1.49) (1.17) (2.78) (3.36) (3.03) (7.32)

alpha -1.77 0.49 1.13 1.64 1.46 3.23 -0.92 0.42 1.01 1.44 1.76 2.68

(-2.84) (0.89) (2.27) (3.07) (2.33) (7.91) (-1.60) (1.09) (2.59) (3.30) (2.93) (6.96)

Panel C: Large Firms

Quintiles

1 2 3 4 5 5-1

V RP b -61.98 8.90 22.09 39.63 106.95

E [r] 0.04 0.67 0.81 1.18 1.37 1.33(0.11) (2.42) (2.84) (3.27) (2.61) (4.07)

alpha 0.02 0.67 0.82 1.17 1.42 1.40

(0.05) (2.46) (2.86) (3.07) (2.69) (4.00)

13

Table 4: Univariate Sorts on Firm VRP

In Panel A, at the end of month t we sort firms into quintiles based on their average bad VRP (V RP b) during month t, so

that Quintile 1 contains the stocks with the lowest V RP b and Quintile 5 the highest. We then form value-weighted portfolios

of these firms, holding the ranking constant for the next month. Subsequently, we compute cumulative returns during month

t + 1 for each quintile portfolio. We report the monthly average cumulative return in percentage of each portfolio. Similarly,

in Panel B, C and D, we sort firms into quintiles based on their average good VRP (V RP g), total VRP (V RP ) and jump risk

premium (JRP ), respectively. We also compute the Jensen alpha of each quintile portfolio with respect to the Fama-French

five-factor model (Fama and French 2015) by running a time-series regression of the monthly portfolio returns on monthly

MKT , SMB, HML, RMW , and CMA. The t-statistics test the null hypothesis that the average monthly cumulative return

of each respective portfolio equals zero, and they are computed using Newey and West (1987) standard errors to account for

autocorrelation, and are reported in parentheses. Significant t-statistics at the 95% confidence level are boldfaced. V RP and

JRP are reported in monthly square percentage units.

Panel A: Firm Bad VRP Panel B: Firm Good VRP

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

V RP b -180.65 7.17 30.97 68.41 249.24 121.15 39.29 27.99 25.42 -38.71

V RP g 90.82 19.05 13.66 14.62 29.85 -62.18 -7.18 8.29 31.34 197.72

V RP -271.47 -11.88 17.31 53.79 219.39 183.33 46.47 19.70 -5.92 -236.43

JRP -89.82 26.22 44.64 83.04 279.09 58.98 32.11 36.29 56.76 159.01

E [r] 0.06 0.63 1.02 1.28 1.14 1.08 0.17 0.60 1.03 1.00 0.64 0.48(0.15) (2.31) (3.02) (2.76) (1.90) (2.73) (0.44) (2.13) (3.33) (2.42) (1.02) (1.35)

alpha -0.76 -0.16 0.26 0.62 0.71 1.47 -0.38 -0.25 0.24 0.27 0.18 0.56

(-3.92) (-1.90) (2.75) (3.59) (2.71) (3.71) (-2.79) (-2.45) (2.99) (1.87) (0.84) (2.20)

Panel C: Firm Total VRP Panel D: Firm JRP

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

V RP b -159.55 9.99 29.53 62.82 232.36 -151.66 13.95 33.04 65.59 214.23

V RP g 168.33 26.27 9.34 0.17 -36.10 21.18 3.47 8.72 19.57 115.05

V RP -327.88 -16.27 20.19 62.65 268.45 -172.85 10.48 24.32 46.02 99.18

JRP 8.78 36.26 38.86 62.99 196.26 -130.48 17.42 41.77 85.16 329.28

E [r] 0.21 0.68 0.76 1.01 0.94 0.73 0.05 0.66 1.22 1.30 1.43 1.38(0.43) (2.12) (2.61) (2.59) (1.77) (2.46) (0.16) (2.33) (3.15) (2.65) (2.06) (2.63)

alpha -0.42 -0.06 -0.02 0.26 0.42 0.84 -0.74 -0.16 0.44 0.75 1.05 1.78

(-2.25) (-0.52) (-0.19) (1.72) (1.94) (2.51) (-5.27) (-2.29) (3.37) (3.83) (3.80) (5.09)

14

Tab

le5:

Con

dit

ion

alD

ouble

Sor

tson

Exp

osu

res

toG

DA

Fac

tors

Inea

chof

the

five

pan

els

of

the

tab

le,

stock

sare

firs

tso

rted

ever

ym

onth

inqu

inti

les

base

don

thei

rm

ult

ivari

ate

exp

osu

reto

on

eof

the

five

GD

Afa

ctors

.A

sre

ferr

edto

by

Fara

go

an

dT

édon

gap

(fort

hco

min

g),

thes

efa

ctors

are

the

mark

etfa

ctor

(Pan

elA

),th

em

ark

etd

ow

nsi

de

fact

or

(Pan

elB

),th

ed

ow

nst

ate

fact

or

(Pan

elC

),th

evola

tility

fact

or

(Pan

elD

)an

dth

evola

tility

dow

nsi

de

fact

or

(Pan

elE

).N

ext,

stock

sw

ith

inea

chqu

inti

leof

the

giv

enG

DA

fact

or

exp

osu

reare

furt

her

sort

edin

qu

inti

les

base

don

thei

rb

ad

vari

an

ceri

skp

rem

ium

.T

he

tab

lere

port

saver

age

valu

e-w

eighte

dex

cess

retu

rns

for

the

bott

om

qu

inti

le(1

),th

eto

pqu

inti

le(5

)an

dfo

rth

ese

con

d(2

),th

ird

(3)

an

dfo

urt

h

(4)

qu

inti

les.

We

als

ore

port

the

diff

eren

cein

aver

age

exce

ssre

turn

sb

etw

een

the

top

an

dth

eb

ott

om

qu

inti

le(5

-1).

T-s

tati

stic

sb

ase

don

stan

dard

erro

rsco

mp

ute

du

sin

gth

e

New

eyan

dW

est

(1987)

pro

ced

ure

are

rep

ort

edin

pare

nth

eses

.S

ign

ifica

nt

t-st

ati

stic

sat

the

95%

con

fid

ence

level

are

bold

face

d.

Data

are

from

Janu

ary

1996

toD

ecem

ber

2015.

Pan

elA

:M

arke

tF

acto

rP

anel

B:

Mar

ket

Dow

nsi

de

Fac

tor

Pan

elC

:D

ownst

ate

Fact

or

Quin

tile

sQ

uin

tile

sQ

uin

tile

s

12

34

55-

11

23

45

5-1

12

34

55-1

10.

000.

010.

18

-0.0

7-0

.62

-0.6

2(-

0.95

)1

-0.5

70.

120.

36-0

.03

-0.3

20.

25(0

.49)

1-0

.43

-0.2

90.3

2-0

.08

-0.4

5-0

.02

(-0.

04)

20.

490.

530.

69

0.72

0.24

-0.2

5(-

0.47

)2

0.63

0.84

0.60

0.65

0.65

0.01

(0.0

4)2

0.58

0.69

0.7

00.5

40.5

0-0

.08

(-0.2

7)

30.

590.

910.

81

0.61

0.62

0.03

(0.0

6)3

1.33

0.79

1.08

1.05

0.89

-0.4

4(-

1.27

)3

1.16

0.93

0.9

40.9

41.1

4-0

.03

(-0.0

7)

41.

001.

161.

36

1.23

1.44

0.44

(0.7

3)4

1.77

1.05

1.24

0.93

0.70

-1.0

7(-

2.1

1)

41.

03

1.7

60.

980.6

00.6

9-0

.34

(-0.7

8)

51.

040.

881.

32

1.08

1.20

0.16

(0.2

1)5

1.08

1.60

1.20

0.49

0.99

-0.0

9(-

0.20

)5

1.25

1.44

1.0

81.2

00.5

4-0

.71

(-1.2

9)

5-1

1.04

0.87

1.13

1.15

1.82

5-1

1.65

1.48

0.83

0.52

1.31

5-1

1.68

1.7

30.

761.2

80.9

9

(2.4

5)

(2.1

6)

(2.6

5)

(2.8

1)

(3.2

9)

(3.0

8)

(3.1

3)

(1.9

7)

(1.1

2)(2

.35

)(2

.95

)(3

.48

)(1

.90)

(2.6

5)

(1.9

9)

Pan

elD

:V

olat

ilit

yF

acto

rP

anel

E:

Vol

atilit

yD

ownsi

de

Fac

tor

Quin

tile

sQ

uin

tile

s

12

34

55-

11

23

45

5-1

1-0

.31

-0.0

10.

170.2

4-0

.07

0.24

(0.4

7)1

-0.3

60.

300.

07-0

.15

-0.6

4-0

.28

(-0.

57)

20.

340.

760.

63

0.54

0.51

0.16

(0.4

5)2

0.73

0.77

0.83

0.56

0.48

-0.2

6(-

0.73

)

31.

010.

860.

82

0.84

0.97

-0.0

5(-

0.11

)3

1.05

0.87

0.77

0.97

0.84

-0.2

2(-

0.62

)

41.

211.

351.

00

1.25

1.23

0.02

(0.0

4)4

1.51

1.38

0.95

1.07

1.04

-0.4

7(-

1.14

)

51.

031.

110.

95

1.47

0.96

-0.0

7(-

0.12

)5

0.90

1.35

0.95

1.19

0.67

-0.2

3(-

0.44

)

5-1

1.34

1.12

0.78

1.23

1.02

5-1

1.25

1.05

0.88

1.34

1.31

(2.3

1)

(2.3

3)

(1.8

9)(3

.00

)(2

.03

)(2

.67

)(2

.05

)(2

.26

)(2

.92

)(2

.16

)

15

Table 6: Conditional Double Sorts on Exposures to Other Market Factors

Stocks are sorted every month in quintiles based on their exposure to market bad VRP in Panel A, and their exposure to the

market risk neutral skewness in Panel B. Then, stocks within each quintile of exposure to these factors are further sorted in

quintiles based on their bad VRP. Firm exposures to market bad VRP are estimated following the three-factor model implied

by the general equilibrium setting of Bollerslev et al. (2009), i.e, with market excess returns, conditional market variance,

and volatility of volatility, and where we replace volatility of volatility by the bad and good VRP. Firm exposures to market

risk-neutral skewness are estimated following the model of Chang et al. (2013). The table reports average value-weighted excess

returns for the bottom quintile (1), the top quintile (5) and for the second (2), third (3) and fourth (4) quintile. We also

report the difference in average excess returns between the top and the bottom quintile (5-1). T-statistics are computed using

Newey and West (1987) standard errors, and are reported in parentheses. Significant t-statistics at the 95% confidence level

are boldfaced. Data are from January 1996 to December 2015.

Panel A: Market Bad VRP Panel B: Market Risk Neutral Skewness

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

1 -0.64 0.31 0.16 -0.09 -0.34 0.30 (0.73) -0.10 0.27 0.06 0.07 -0.84 -0.74 (-1.30)

2 0.34 0.79 0.63 0.62 0.57 0.23 (0.65) 0.80 0.56 0.50 0.81 0.90 0.10 (0.28)

3 0.94 0.78 0.92 0.91 1.36 0.42 (1.03) 0.96 1.18 0.62 0.87 1.22 0.26 (0.71)

4 1.05 1.28 1.15 1.28 1.12 0.07 (0.14) 0.76 1.03 1.14 1.21 1.32 0.56 (1.31)

5 0.44 1.74 1.44 1.37 0.79 0.34 (0.62) 1.13 1.26 1.24 1.42 0.96 -0.17 (-0.29)

5-1 1.08 1.43 1.28 1.46 1.12 1.23 0.99 1.18 1.35 1.80

(2.35) (3.18) (2.97) (3.57) (2.10) (2.29) (2.31) (2.70) (2.76) (3.32)

16

Table 7: Conditional Double Sorts on Other Firm Characteristics

In each of the four panels of the table, stocks are sorted every month in quintiles based on four different firm characteristics:

illiquidity, idiosyncratic volatility, risk neutral skewness, and relative signed jump variation, respectively. Then, stocks within

each quintile are further sorted in quintiles based on their bad variance risk premium. Illiquidity is measured as in Amihud

(2002). Idiosyncratic volatility is estimated following Ang et al. (2006). Risk neutral skewness is estimated following Bakshi

et al. (2003). Relative signed jump variation is estimated following Bollerslev et al. (2017). T-statistics are computed using

Newey and West (1987) standard errors, and are reported in parentheses. Significant t-statistics at the 95% confidence level

are boldfaced.

Panel A: Illiquidity Panel B: Idiosyncratic Volatility

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

1 0.24 0.08 -0.28 -0.61 -1.10 -1.35 (-3.10) 0.53 0.09 -0.23 -0.61 -1.33 -1.86 (-2.73)

2 0.77 0.73 0.75 0.63 0.40 -0.37 (-0.99) 0.79 0.59 0.41 0.14 0.17 -0.62 (-1.25)

3 0.73 1.02 1.11 0.96 0.95 0.22 (0.67) 0.85 0.91 0.91 1.04 0.60 -0.25 (-0.52)

4 1.06 1.13 1.25 1.09 1.29 0.23 (0.58) 0.97 1.20 1.09 1.41 0.70 -0.27 (-0.44)

5 1.25 1.42 0.79 1.70 0.43 -0.82 (-1.90) 1.22 1.54 1.31 0.91 0.47 -0.75 (-1.16)

5-1 1.01 1.34 1.07 2.31 1.53 0.69 1.45 1.54 1.52 1.80

(2.85) (4.21) (3.43) (6.21) (3.94) (2.46) (4.14) (3.45) (2.97) (2.83)

Panel C: Risk Neutral Skewness Panel D: Relative Signed Jump Variation

Quintiles Quintiles

1 2 3 4 5 5-1 1 2 3 4 5 5-1

1 0.31 0.30 -0.08 -0.57 -0.37 -0.67 (-1.96) -0.87 0.07 -0.07 0.06 0.41 1.28 (2.95)

2 0.62 0.76 0.52 0.72 0.37 -0.25 (-0.86) 0.60 0.66 0.74 0.60 0.58 -0.02 (-0.06)

3 0.75 1.32 1.33 1.25 1.17 0.42 (1.39) 1.27 1.12 1.25 0.94 0.82 -0.45 (-1.31)

4 1.55 1.23 1.43 1.11 1.19 -0.35 (-0.82) 1.49 1.31 1.20 1.08 1.36 -0.12 (-0.24)

5 1.08 1.38 1.30 1.18 1.35 0.28 (0.55) 0.87 0.85 1.08 1.15 1.75 0.87 (1.41)

5-1 0.77 1.08 1.38 1.75 1.72 1.74 0.78 1.14 1.09 1.34

(1.59) (1.98) (2.72) (3.32) (3.33) (3.33) (1.56) (2.21) (2.43) (2.65)

17

Tab

le8:

Fam

a-M

acB

eth

Reg

ress

ion

sC

ontr

olli

ng

for

Syst

emat

icR

isk

This

table

rep

ort

sth

eti

me-

seri

esav

erage

of

the

month

lyes

tim

ate

dco

effici

ents

for

diff

eren

tfa

ctor

model

sin

cludin

gfirm

vari

ance

risk

pre

miu

m(V

RPb,VRPg

andVRP

).In

each

regre

ssio

nfr

om

III

toV

IIw

ein

clude

the

firm

bad

and

good

vari

ance

risk

pre

miu

mw

ith

diff

eren

tfa

ctor

model

s:C

AP

M,

mark

etsk

ewnes

s

fact

or

model

(Chang

etal.;

2013),

mark

etva

riance

risk

pre

miu

mm

odel

(Boller

slev

etal.;

2009),

Carh

art

four-

fact

or

model

,and

the

GD

Afive-

fact

or

model

(Fara

go

and

Téd

ongap;

fort

hco

min

g),

resp

ecti

vel

y.A

llco

effici

ents

are

esti

mate

dusi

ng

the

Fam

aand

MacB

eth

(1973)

two-s

tep

regre

ssio

napplied

on

5150

indiv

idual

firm

s.In

the

firs

tst

ep,

we

regre

sssi

xm

onth

sof

daily

exce

ssre

turn

sof

the

5150

firm

son

the

diff

eren

tfa

ctor

model

sto

obta

inth

eir

resp

ecti

ve

bet

as.

Inth

ese

cond

step

,w

eru

ncr

oss

-sec

tional

regre

ssio

ns

of

month

t+

1firm

exce

ssre

turn

sagain

stth

ees

tim

ate

db

etas

and

firm

vari

ance

risk

pre

miu

m.

T-s

tati

stic

s

are

com

pute

dusi

ng

New

eyand

Wes

t(1

987)

standard

erro

rs,

and

are

rep

ort

edin

pare

nth

eses

.Sig

nifi

cant

t-st

ati

stic

sat

the

95%

confiden

cele

vel

are

bold

face

d.

Adju

sted

R2

isre

port

edin

per

centa

ge.

III

III

IVV

VI

VII

Cst

4.9e

-3C

st3.

3e-3

Cst

6.6e

-3C

st6.

73e-

3C

st7.

1e-3

Cst

7.3e

-3C

st7.

0e-3

(1.1

0)(0

.77)

(2.1

9)

(2.2

4)

(2.4

2)

(2.8

5)

(2.5

0)

VR

P0.

05VRPb

0.17

VRPb

0.17

VRPb

0.17

VRPb

0.18

VRPb

0.17

VRPb

0.18

(1.6

9)

(3.8

4)

(4.2

4)

(4.2

9)

(4.4

9)

(4.3

4)

(4.5

7)

VRPg

0.2

7VRPg

0.29

VRPg

0.29

VRPg

0.31

VRPg

0.32

VRPg

0.31

(3.3

2)

(4.5

4)

(4.5

2)

(4.8

6)

(5.0

4)

(4.8

2)

βm,CAPM

-2.4

e-3

βm,SKEW

-2.5

e-3

βm,BTZ

-2.7

e-3

βm,CH

-2.5

e-3

βW

-2.3

e-3

(-0.

75)

(-0.

79)

(-0.8

8)(-

0.87

)(-

0.74

)βMSKEW

0.04

βMVRPb

-6.0

e-6

βsm

b-1

.5e-

3βX

7.3e

-6(0

.58)

(-0.

94)

(-1.

28)

(1.0

8)βMVRPg

2.9e

-6βhml

-9.5

e-4

βD

0.16

(0.8

1)(-

0.65

)(1

.70)

βVIX

7.8e

-6βmom

1.9e

-3βW

D-3

.2e-

3(1

.06)

(0.8

5)(-

1.46

)βXD

7.6e

-6(1

.32)

Adj.R

20.

701.

755.

205.

566.

469.

096.

63

18

Table 9: Fama-MacBeth Regressions Controlling for Other Firm Characteristics

This table reports the time-series average of the monthly estimated coefficients for different factor models including

firm variance risk premium (V RP b, V RP g and V RP ). In regression VIII we include the firm bad and good variance

risk premium with the relative signed jump variation (RSJ) from Bollerslev et al. (2017). In regression IX we include

the firm bad and good variance risk premium with all the firm characteristics: RSJ , idiosyncratic volatility (IV OL)

computed as in Ang et al. (2006), past 1-month cumulative excess return (P01M), past 12-month cumulative excess

return (P12M), size, book-to-market (B/M), illiquidity (ILLIQ), risk-neutral skewness (SKEW ), the good and bad

realized semi-variances (RV b and RV g), and firm risk neutral skewness. All coefficients are estimated using the

Fama and MacBeth (1973) two-step regression applied on 5150 individual firms. We run cross-sectional regressions of

month t+1 firm excess returns against firm characteristics and firm variance risk premium. T-statistics are computed

using Newey and West (1987) standard errors, and are reported in parentheses. Significant t-statistics at the 95%

confidence level are boldfaced. Adjusted R2 is reported in percentage.

I II VIII IX

Cst 4.9e-3 Cst 3.3e-3 Cst 3.4e-3 Cst 3.2e-3(1.10) (0.77) (0.80) (0.18)

VRP 0.05 V RP b 0.17 V RP b 0.17 V RP b 0.32(1.69) (3.84) (3.81) (4.77)

V RP g 0.27 V RP g 0.27 V RP g 0.61(3.32) (3.34) (5.21)

RSJ -1.5e-3 RSJ 5.2e-3(-0.40) (1.73)

IV OL -0.12(-1.41)

P01M -0.02(-2.48)

P12M 3.9e-3(1.41)

Size 5.0e-5(0.07)

B/M 0.02(3.90)

ILLIQ -0.35(-1.22)

RV b -0.28(-2.38)

RV g -0.06(-0.50)

SKEW 4.5e-3(3.33)

Adj. R2 0.70 Adj. R2 1.75 Adj. R2 2.68 Adj. R2 10.54

19

Table 10: Independent Double Sorts on Good and Bad Firm VRP

Stocks are sorted every month in quintiles independently based on bad (V RP b) and good VRP (V RP g). Then, we form

portfolios by taking the intersection of these quintiles. The table reports average value-weighted excess returns for the bottom

quintile (1), the top quintile (5) and for the second (2), third (3) and fourth (4) quintile. We also report the difference in

average excess returns between the top and the bottom quintile (5-1). T-statistics are computed using Newey and West (1987)

standard errors, and are reported in parentheses. Significant t-statistics at the 95% confidence level are boldfaced. Data are

from January 1996 to December 2015.

Firm Good VRP

Quintiles

Fir

mB

adV

RP

1 2 3 4 5 5-1

1 -3.26 -1.45 -0.93 -0.01 0.08 3.34 (6.65)

2 -1.06 0.34 0.84 0.84 1.45 2.51 (4.28)

3 -0.09 0.87 1.30 1.40 2.94 3.04 (4.48)

4 0.50 1.24 1.02 1.77 2.98 2.48 (5.09)

5 0.55 2.01 2.40 1.30 3.40 2.85 (5.88)

5-1 3.81 3.47 3.33 1.30 3.33

(6.24) (6.49) (5.32) (2.78) (7.12)

20