variance premium, downside risk, and expected stock returns … · 2018. 6. 16. · variance...
Post on 05-Feb-2021
4 Views
Preview:
TRANSCRIPT
-
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
top related