1. No category

Online Appendix to Accompany “Downside Variance Risk
Premium”
Bruno Feunou
Bank of Canada
Mohammad R. Jahan-Parvar
Federal Reserve Board
March 2015
1
C´edric Okou
UQAM
1
1.1
The Theoretical Results
Preferences
We consider a endowment economy in discrete time. The representative agent’s preferences over
the future consumption stream are characterized by Kreps and Porteus (1978) intertemporal preferences, as formulated by Epstein and Zin (1989) and Weil (1989):
θ
1 1−γ
1−γ
θ
1−γ
Ut = (1 − δ)Ct θ + δ Et Ut+1
,
(1)
where Ct is the consumption bundle at time t, δ is the subjective discount factor, γ is the coefficient
of risk aversion, and ψ is the elasticity of intertemporal substitution (IES). Parameter θ is defined
1−γ
as θ ≡ 1−
1 . If θ = 1, then γ = 1/ψ and EZ preferences collapse to expected power utility, which
ψ
implies an agent who is indifferent to the timing of resolution of uncertainty of the consumption
path. With γ > 1/ψ, the agent prefers early resolution of uncertainty. For γ < 1/ψ, the agent
prefers late resolution of uncertainty. Epstein and Zin (1989) show that the logarithm of stochastic
discount factor (SDF) implied by these preferences is given by:
ln Mt+1 = mt+1 = θ ln δ −
θ
∆ct+1 + (θ − 1)rc,t+1 ,
ψ
(2)
is the log growth rate of aggregate consumption, and rc,t is the log return
where ∆ct+1 = ln CCt+1
t
of the asset that delivers aggregate consumption as dividends. This asset represents the returns on
a wealth portfolio. The Euler equation states that
Et [exp (mt+1 + ri,t+1 )] = 1,
(3)
where ri,t i = c, d represents the log returns for the consumption generating asset (rc,t ) or dividend
generating asset (rd,t ), respectively. The risk-free rate, which represents the returns of an asset
that delivers a unit of consumption in the next period with certainty, is defined as:
1
f
rt = ln
.
(4)
Et (Mt+1 )
1.2
Consumption Dynamics
Our specification of consumption dynamics incorporates elements from Bansal and Yaron (2004),
Eraker and Shaliastovich (2008), Bekaert, Engstrom, and Ermolov (2014), and especially Bollerslev,
Tauchen, and Zhou (2009) and Segal, Shaliastovich, and Yaron (2015).
Fundamentally, we follow Bansal and Yaron (2004) in assuming that consumption growth has
a predictable component. We differ from Bansal and Yaron in assuming that the predictable
component is proportional to consumption growth’s upside and downside volatility components:1
∆ct+1 = µ0 + µ1 Vu,t + µ2 Vd,t + σc (εu,t+1 − εd,t+1 ) ,
(5)
1
Segal, Shaliastovich, and Yaron (2015) maintain this assumption in their definition of their long-run risk component.
2
where εu,t+1 and εd,t+1 are two mean-zero shocks that affect both the realized and expected consumption growth.2 εu,t+1 represents upside shocks to consumption growth, and εd,t+1 stands for
downside shocks. Following Bekaert, Engstrom, and Ermolov (2014) and Segal, Shaliastovich, and
Yaron (2015), we assume that these shocks follow a demeaned Gamma distribution and model them
as
εi,t+1 = ε˜i,t+1 − Vi,t i = {u, d},
(6)
where ε˜i,t+1 ∼ Γ(Vi,t , 1). These distributional assumptions imply that volatilities of upside and
downside shocks are time-varying and driven by shape parameters Vu,t and Vd,t . In particular, we
posit that
V art εi,t+1 ≡ Vi,t , i = {u, d}.
(7)
Naturally, total variance of consumption growth when εu,t+1 and εd,t+1 are linearly independent is
simply σc2 (Vu,t + Vd,t ).
As a result, sign and size of µ1 and µ2 matter in this context. With µ1 = µ2 , we have a
stochastic volatility component in the conditional mean of consumption growth process, similar to
the classic GARCH-in-Mean structure for modeling risk-return trade off in equity returns. With
both slope parameters equal to zero, the model yields the BTZ unpredictable consumption growth.3
If |µ1 | = |µ2 |, with µ1 > 0 and µ2 < 0, we have Skewness-in-Mean, similar in spirit to Feunou,
Jahan-Parvar, and T´edongap (2013) formulation for equity returns. With µ1 6= µ2 , we have free
parameters that have an impact on loadings of risk factors on risky asset returns and the stochastic
discount factor, which we demonstrate in this section. Both µ1 and µ2 are real-valued. Intuitively,
we expect µ1 > 0, meaning that a rise in upside volatility at time t implies higher consumption
growth at time t + 1, all else equal. By the same logic, we intuitively expect a negative-valued
µ2 , implying an expected fall in consumption growth following an up-tick in downside volatility –
following bad economic outcomes, households curb their consumption. In what follows, we buttress
our intuition with theory and derive the analytical bounds on these parameters that ensure closedform solutions for our results.
We observe that
ln Et exp (νεi,t+1 ) = f (ν)Vi,t ,
(8)
where f (ν) = −(ln(1 − ν) + ν). Both Bekaert, Engstrom, and Ermolov (2014) and Segal, Shaliastovich, and Yaron (2015) use this compact functional form for the Gamma-distribution cumulant.
It simply follows that f (ν) > 0, f 00 (ν) > 0, and f (ν) > f (−ν) for all ν > 0.
We assume that Vi,t follow a time-varying, square root process with time-varying volatilityof-volatility, similar to the specification of the volatility process in Bollerslev, Tauchen, and Zhou
(2009):
2
This assumption is for the sake of brevity. Violating this assumption adds to algebraic complexity, but does not
affect our analytical findings.
3
It can be shown that assuming an unpredictable consumption growth process does not support the existence
of distinct upside and downside variance risk premia that are supported either by theory or empirical evidence,
if we assume agents endowed with Epstein and Zin (1989) preferences. Using asymmetric preferences, such as
smooth ambiguity aversion preferences of Klibanoff, Marinacci, and Mukerji (2009) or disappointment aversion of
Gul (1991), it may be possible to derive upside and downside variance risk premia for an economy with unpredictable
consumption growth. The cost we pay is the loss of closed form analytical results. Miao, Wei, and Zhou (2012) use
smooth ambiguity aversion preferences to motivate their study of variance risk premium, but assume time-variation
in the conditional mean of the consumption growth.
3
qu,t+1
Vd,t+1
qd,t+1
√
u
qu,t zt+1
,
√
1
= γu,0 + γu,1 qu,t + ϕu qu,t zt+1
,
√
d
= αd + βd Vd,t + qd,t zt+1 ,
√
2
= γd,0 + γd,1 qd,t + ϕd qd,t zt+1
,
Vu,t+1 = αu + βu Vu,t +
(9)
(10)
(11)
(12)
where zti and i = {u, d, 1, 2} are standard normal innovations. The parameters must satisfy the
following: αu > 0, αd > 0, γu,0 > 0,
γd,0
>
0,|βu | <1, |β
d | < 1, |γu,1 | < 1, |γd,1 | < 1, ϕu > 0, ϕd > 0,
in addition we assume that {ztu } , ztd , zt1 and zt2 are independent i.i.d. N (0, 1) and they are
jointly independent from {εu,t } and {εd,t }.
The assumptions above yield time-varying uncertainty and asymmetry in consumption growth.
Through volatility-of-volatility processes qu,t and qd,t , the set up induces additional temporal variation in consumption growth. Temporal variation in volatility-of-volatility process is necessary for
generating sizable variance risk premium. Asymmetry is needed to generate upside and downside
variance risk premia, as we show in what follows.
We solve the model following the same methodology proposed by Bansal and Yaron (2004),
Bollerslev, Tauchen, and Zhou (2009), Segal, Shaliastovich, and Yaron (2015), and many others
and assume that the logarithm of the price-dividend ratio, wt , is affine with respect to state variables
Vi,t and qi,t . Equivalently, wt can represent wealth-consumption ratio or price-consumption ratio
(pct = ln
Pt
Ct
) for the asset that pays the consumption endowment {Ct+i }∞
i=1 .
We then posit that the consumption-generating returns are approximately linear with respect
to the log price-consumption ratio. Thus, we represent them as:
rc,t+1 = κ0 + κ1 wt+1 − wt + ∆ct+1 ,
(13)
wt = A0 + A1 Vu,t + A2 Vd,t + A3 qu,t + A4 qd,t ,
(14)
where κ0 and κ1 are log-linearization coefficients and A0 , A1 , A2 , A3 and A4 are factor loading
coefficients to be determined. We solve for the consumption-generating asset returns, rc,t , using
the Euler equation (3):
θ
Et exp θ ln δ − ∆ct+1 + (θ − 1)rc,t+1 + rc,t+1 = 1
ψ
and
ln Et
θ
exp θ ln δ − ∆ct+1 + θrc,t+1
ψ
4
= 0.
Substituting for ∆ct+1 , rc,t+1 , Vi,t+1 and qi,t+1 , we get:
"
h
ln Et exp θ ln δ + (1 − γ) [µ0 + µ1 Vu,t + µ2 Vd,t ] + θ[κ0 + (κ1 − 1)A0 ] + θ(αu A1 + αd A2 + γu,0 A3 + γd,0 A4 )
+ (1 − γ)σc (εu,t+1 − εd,t+1 )
h
i
+ θ A1 (κ1 βu − 1)Vu,t + A2 (κ1 βd − 1)Vd,t + A3 (κ1 γu,1 − 1)qu,d + A4 (κ1 γd,1 − 1)qd,t
#
√
√
√
√
u
d
1
d
+ θκ1 A1 qu,t zt+1 + A2 qd,t zt+1 + A3 ϕu qu,t zt+1 + A4 ϕd qd,t zt+1
= 0,
(15)
and then proceed to compute the expectations and coefficients, as follows:
"
#
1. ln Et exp σc (1 − γ)εu,t+1
"
#
2. ln Et exp − σc (1 − γ)εd,t+1
"
= f σc (1 − γ) Vu,t ,
√
= f − σc (1 − γ) Vd,t
√
√
√
u +A
d
1
d
3. ln Et exp θκ1 A1 qu,t zt+1
2 qd,t zt+1 + A3 ϕu qu,t zt+1 + A4 ϕd qd,t zt+1
1 2 2
2 θ κ1
"
#
=
A21 + ϕ2 A23 qu,t + A22 + ϕ2 A24 qd,t ,
h
4. ln Et exp (1 − γ) [µ1 Vu,t + µ2 Vd,t ] + θ A1 (κ1 βu − 1)Vu,t + A2 (κ1 βd − 1)Vd,t + A3 (κ1 γu,1 −
#
i
=
1)qu,d + A4 (κ1 γd,1 − 1)qd,t
h
(1 − γ) [µ1 Vu,t + µ2 Vd,t ] + θ A1 (κ1 βu − 1)Vu,t + A2 (κ1 βd − 1)Vd,t + A3 (κ1 γu,1 − 1)qu,d +
i
A4 (κ1 γd,1 − 1)qd,t ,
"
#
5. ln Et exp θ ln δ + (1 − γ)µ0 + θ[κ0 + (κ1 − 1)A0 ] + θ(αu A1 + αd A2 + γu,0 A3 + γd,0 A4 )
h
θ ln δ +
1−γ
θ µ0
i
+ κ0 + (κ1 − 1)A0 + κ1 αu A1 + αd A2 + γu,0 A3 + γd,0 A4 .
We gather the terms for Vu,t , Vd,t , qu,t and qd,t , and solve for A0 to A4 :
5
=
f σc (1 − γ) + (1 − γ)µ1
A1 = −
,
θ(κ1 βu − 1)
f − σc (1 − γ) + (1 − γ)µ2
(16)
A2 = −
(17)
A3 =
(18)
A4 =
A0 =
,
θ(κ1 βd − 1)
p
(1 − κ1 γu,1 ) − (1 − κ1 γu,1 )2 − θ2 ϕ2u κ41 A21
,
θκ21 ϕ2u
q
(1 − κ1 γd,1 ) − (1 − κ1 γd,1 )2 − θ2 ϕ2d κ41 A22
,
θκ21 ϕ2d
ln δ + 1 − ψ1 µ0 + κ0 + κ1 αu A1 + αd A2 + γu,0 A3 + γd,0 A4
1 − κ1
(19)
.
(20)
Following sone simple algebraic manipulation, we obtain the following representations for conditional equity premium and innovations of conditional equity premium:
µ1 f [σc (1 − γ)]
µ2 f [−σc (1 − γ)]
µ0
+
−
Vu,t +
−
Vd,t
rc,t+1 = ln δ +
ψ
ψ
θ
ψ
θ
+σc (εu,t+1 − εd,t+1 ) + (κ1 γu,1 − 1)A3 qu,t + (κ1 γd,1 − 1)A4 qd,t
(21)
i
h
√
√
d
2
u
1
qu,t + A2 zt+1
+ ϕd A4 zt+1
qd,t ,
+κ1 A1 zt+1
+ ϕu A3 zt+1
rc,t+1 − Et (rc,t+1 ) = σc (εu,t+1 − εd,t+1 )
h
√ i
√
u
1
d
2
+κ1 A1 zt+1
+ ϕu A3 zt+1
qu,t + A2 zt+1
+ ϕd A4 zt+1
qd,t . (22)
It is immediately obvious that there is significant correspondence between our characterization
of risky returns and equation (10) of BTZ. The differences mainly rest with different distributional assumptions regarding consumption growth shocks and the fact that we model upside and
downside
uncertainty
explicitly
h
i h
i rather than targeting aggregate uncertainty as in BTZ. Notice that
f [σc (1−γ)]
f [−σc (1−γ)]
−
< −
and both terms are positive-valued. Thus, the impact of Vu,t and
θ
θ
Vd,t on expected returns depend on µ1 and µ2 , and as a result, they are ambiguous at this point.
In what follows, we offer a crisp determination of equity premium to complement the analysis so
far.
Similarly, we obtain the following characterizations for conditional log stochastic discount factor
and innovations in conditional log SDF:
1−θ
µ1
1−θ
µ2
µ0
+
[σc (1 − γ)] −
Vu,t +
[−σc (1 − γ)] −
Vd,t
mt+1 = ln δ −
ψ
θ
ψ
θ
ψ
+(κ1 γu,1 − 1)A3 qu,t + (κ1 γd,1 − 1)A4 qd,t − γσc (εu,t+1 − εd,t+1 )
(23)
h
i
√
√
u
1
d
2
+κ1 (θ − 1) A1 zt+1
+ ϕu A3 zt+1
qu,t + A2 zt+1
+ ϕd A4 zt+1
qd,t ,
mt+1 − Et (mt+1 ) = −γσc (εu,t+1 − εd,t+1 )
(24)
h
√ i
√
u
1
d
2
+κ1 (θ − 1) A1 zt+1
+ ϕu A3 zt+1
qu,t + A2 zt+1
+ ϕd A4 zt+1
qd,t .
6
Due to differences in distributional assumptions, we do not follow BTZ or Bansal and Yaron methods
for deriving equity premium and various variance risk premia. The dynamics specified so far are
all under the physical measure (P). We need to compute the dynamics under the risk-neutral
measure (Q) to derive the formulae for upside and downside variance risk premia and skewness risk
premium. We thus begin by deriving the risk-neutral distribution of all the shocks, εu,t+1 , εd,t+1 ,
u , zd , z1
2
zt+1
t+1
t+1 and zt+1 . In this computation, we construct the characteristic function for each
shock and exploit the properties of characteristic functions to derive the expectations under the
risk-neutral measure. Thus, our derivations yield exact equity and risk premia measures, in contrast
to approximate values reported by, for example, in equation (15) of Bollerslev, Tauchen, and Zhou
(2009) or in Drechsler and Yaron (2011).
We start from εu,t+1 . The SDF is the Radon-Nikodym change of measure and ln Et (Mt+1 ) is
the risk-neutral drift term. We have:
=
=
=
=
=
=
=
≡
EQ
t (exp (νεu,t+1 ))
Mt+1
Et
exp (νεu,t+1 )
Et (Mt+1 )
Et [exp (νεu,t+1 + mt+1 − ln(Et (Mt+1 )))]
θ
Et exp νεu,t+1 + θ ln δ − ∆ct+1 + (θ − 1)rc,t+1 − ln(Et (Mt+1 ))
ψ
νεu,t+1 + θ ln δ − ψθ ∆ct+1
Et exp
+(θ − 1) (κ0 + κ1 wt+1 − wt + ∆ct+1 ) − ln(Et (Mt+1 ))
νεu,t+1 + θ ln δ + (θ − 1 − ψθ )∆ct+1
Et exp
+(θ − 1) (κ0 + κ1 wt+1 − wt ) − ln(Et (Mt+1 ))
νεu,t+1 + θ ln δ − γ [µ0 + µ1 Vu,t + µ2 Vd,t + σc (εu,t+1 − εd,t+1 )]
Et exp
+(θ − 1) (κ0 + κ1 wt+1 − wt ) − ln(Et (Mt+1 ))
(ν − γσc ) εu,t+1 + θ ln δ − γ (µ0 + µ1 Vu,t + µ2 Vd,t − σc εd,t+1 )
Et exp
+(θ − 1) (κ0 + κ1 wt+1 − wt ) − ln(Et (Mt+1 ))
i
h
Et exp (ν − γσc ) εu,t+1 + Bt∗,1
where
B∗,1
t
θ ln δ − γ (µ0 + µ1 Vu,t + µ2 Vd,t − σc εd,t+1 )
= ln Et exp
.
+(θ − 1) (κ0 + κ1 wt+1 − wt ) − ln(Et (Mt+1 ))
Hence, it follows that:
∗,1
EQ
(exp
(νε
))
=
exp
B
+
f
(ν
−
γσ
)V
.
u,t+1
c
u,t
t
t
With ν = 0, we have
1 = exp B1t + f (−γσc )Vu,t ,
hence, we obtain:
B1t = −f (−γσc )Vu,t .
In conclusion
EQ
t (exp (νεu,t+1 )) = exp ((f (ν − γσc ) − f (−γσc )) Vu,t )
7
Similarly, for εd,t+1 we deduce that:
EQ
t (exp (νεd,t+1 )) = exp ((f (ν + γσc ) − f (γσc )) Vd,t )
It follows that:
γσc
Vu,t
1 + γσc
γσc
=
Vd,t
1 − γσc
0
EQ
t [εu,t+1 ] = f (−γσc )Vu,t = −
0
EQ
t [εd,t+1 ] = f (γσc )Vd,t
u , zd , z1
2
We now derive the expectation of Gaussian shocks, zt+1
t+1 t+1 and zt+1 , under the risk-neutral
u :
measure, Q. We start by deriving the characteristic function for zt+1
=
=
=
=
=
=
=
=
=
=
=
u
EQ
t (exp νzt+1 )
Mt+1
u
Et
exp νzt+1
Et (Mt+1 )
u
Et exp νzt+1
+ mt+1 − ln(Et (Mt+1 ))
θ
u
Et exp νzt+1 + θ ln δ − ∆ct+1 + (θ − 1)rc,t+1 − ln(Et (Mt+1 ))
ψ
u + θ ln δ − θ ∆c
νzt+1
t+1
ψ
Et exp
+(θ − 1) (κ0 + κ1 wt+1 − wt + ∆ct+1 ) − ln(Et (Mt+1 ))
u + θ ln δ + (θ − 1 − θ )∆c
νzt+1
t+1
ψ
Et exp
+(θ − 1) (κ0 + κ1 wt+1 − wt ) − ln(Et (Mt+1 ))
u + θ ln δ − γ∆c
νzt+1
t+1
Et exp
+(θ − 1) (κ0 + κ1 wt+1 − wt ) − ln(Et (Mt+1 ))
u + θ ln δ − γ∆c
νzt+1
t+1
Et exp
+(θ − 1) (κ0 + κ1 (wt+1 − A1 Vu,t+1 + A1 Vu,t+1 ) − wt ) − ln(Et (Mt+1 ))
u + (θ − 1)κ A V
νzt+1
1 1 u,t+1 + θ ln δ − γ∆ct+1
Et exp
+(θ − 1) (κ0 + κ1 (wt+1 − A1 Vu,t+1 ) − wt ) − ln(Et (Mt+1 ))
√
u
u + (θ − 1)κ A α + β V
νzt+1
1 1
u
u u,t + qu,t zt+1 + θ ln δ − γ∆ct+1
Et exp
+(θ − 1) (κ0 + κ1 (wt+1 − A1 Vu,t+1 ) − wt ) − ln(Et (Mt+1 ))
√ u
ν + (θ − 1)κ1 A1 qu,t zt+1
+ (θ − 1)κ1 A1 (αu + βu Vu,t ) + θ ln δ − γ∆ct+1
Et exp
+(θ − 1) (κ0 + κ1 (wt+1 − A1 Vu,t+1 ) − wt ) − ln(Et (Mt+1 ))
h
i
√ u
Et exp ν + (θ − 1)κ1 A1 qu,t zt+1
+ B∗,g1
t
where
Bt∗,g1
= ln Et exp
(θ − 1)κ1 A1 (αu + βu Vu,t ) + θ ln δ − γ∆ct+1
+(θ − 1) (κ0 + κ1 (wt+1 − A1 Vu,t+1 ) − wt ) − ln(Et (Mt+1 ))
It follows that:
u
EQ
t (exp νzt+1
√ 2 !
ν
+
(θ
−
1)κ
A
qu,t
1
1
) = exp Bt∗,g1 +
.
2
8
.
Setting ν = 0, we have
√ 2 !
qu,t
(θ
−
1)κ
A
1
1
,
1 = exp Bg1
t +
2
and hence:
Bg1
t =−
(θ − 1)2 κ21 A21 qu,t
.
2
Thus:
u
EQ
t (exp νzt+1
√ 2
√ 2 !
ν + (θ − 1)κ1 A1 qu,t − (θ − 1)κ1 A1 qu,t
) = exp
2
2
ν
√
= exp
+ ν(θ − 1)κ1 A1 qu,t .
2
u
Based on the last result, we characterize the distribution of zt+1
under the risk-neutral measure as:
√
u
zt+1
∼Q N (θ − 1)κ1 A1 qu,t , 1
Similarly, we characterize the distributions for the remaining shocks:
√
d
zt+1
∼ Q N (θ − 1)κ1 A2 qd,t , 1
√
1
zt+1
∼ Q N (θ − 1)κ1 A3 ϕu qu,t , 1
√
2
zt+1
∼ Q N (θ − 1)κ1 A4 ϕd qd,t , 1
Thus far, we have derived the distribution of shock processes under the risk-neutral measure,
Q. Since any premium – whether equity, variance risk, or skewness risk premia – can be defined as
the difference between the physical and risk-neutral expectations of processes, we are now ready to
compute all the premia of interest. We start with the equity risk premium:
ERPt ≡ Et [rc,t+1 ] − EQ
t [rc,t+1 ]
= Et [κ0 + κ1 wt+1 − wt + ∆ct+1 ] − EQ
t [κ0 + κ1 wt+1 − wt + ∆ct+1 ]
Q
= κ1 Et [wt+1 ] − EQ
t [wt+1 ] + Et [∆ct+1 ] − Et [∆ct+1 ]
As it is clear from the expressions above, we need to compute both Et [∆ct+1 ] − EQ
t [∆ct+1 ] and
Q
Et [wt+1 ] − EQ
[w
].
Starting
with
E
[∆c
]
−
E
[∆c
],
we
have:
t+1
t
t+1
t+1
t
t
Q
Et [∆ct+1 ] − EQ
t [∆ct+1 ] = −σc Et [εu,t+1 − εd,t+1 ]
γσc
γσc
= σc
Vu,t +
Vd,t
1 + γσc
1 − γσc
1
1
= γσc2
Vu,t +
Vd,t
1 + γσc
1 − γσc
Similarly, for Et [wt+1 ] − EQ
t [wt+1 ] we get:
Q
Q
E
[V
]
−
E
[V
]
+
A
E
[V
]
−
E
[V
]
Et [wt+1 ] − EQ
[w
]
=
A
t+1
1
t u,t
u,t
2
t d,t
d,t
t
t
t
Q
+A3 Et [qu,t ] − EQ
[q
]
+
A
E
[q
]
−
E
[q
]
u,t
4
t d,t
d,t
t
t
9
At this stage, we need the premia for each risk factor (Vu,t , Vd,t , qu,t and qd,t ) to compute Et [wt+1 ] −
EQ
t [wt+1 ]. We start with Vu,t :
√
√
√
Q u Et [Vu,t ] − EQ
t [Vu,t ] = − qu,t Et zt+1 = − qu,t (θ − 1)κ1 A1 qu,t = −(θ − 1)κ1 A1 qu,t .
Thus, we derive the premia accrued to each risk factor as:
Et [Vu,t ] − EQ
t [Vu,t ] = (1 − θ)κ1 A1 qu,t ,
Et [Vd,t ] − EQ
t [Vd,t ] = (1 − θ)κ1 A2 qd,t ,
2
Et [qu,t ] − EQ
t [qu,t ] = (1 − θ)κ1 A3 ϕu qu,t ,
2
Et [qd,t ] − EQ
t [qd,t ] = (1 − θ)κ1 A4 ϕd qd,t .
Collecting these terms and substituting in the expression for Et [wt+1 ] − EQ
t [wt+1 ], we get:
2
2 2
2
2 2
Et [wt+1 ] − EQ
t [wt+1 ] = (1 − θ)κ1 A1 + A3 ϕu qu,t + A2 + A4 ϕd qd,t .
It easily follows that the equity premium in our model is:
ERPt ≡
γσc2
γσc2
Vu,t +
Vd,t + (1 − θ)κ1 A21 + A23 ϕ2u qu,t + (1 − θ)κ1 A22 + A24 ϕ2d qd,t . (25)
1 + γσc
1 − γσc
From this expression for equity premium, it is immediately clear that our model implies unequal
loadings for upside and downside volatility factors for equity premium. The slope coefficients for
volatility-of-volatility factors are also – in general – unequal.
Given our findings so far, we derive upside and downside variance risk premia in closed form.
In the first step, we characterize variance of the consumption-generating asset. From, equation
(22) we know that
2
σr,t
≡ V art [rc,t+1 ]
h
h
√
√ ii
u
1
d
2
= V art σc (εu,t+1 − εd,t+1 ) + κ1 (A1 zt+1
+ ϕu A3 zt+1
) qu,t + (A2 zt+1
+ ϕd A4 zt+1
) qd,t
2
2
= σc,t
+ κ21 σw,t
= σc2 Vu,t + σc2 Vd,t + κ21 A21 + A23 ϕ2u qu,t + κ21 A22 + A24 ϕ2d qd,t
2
u 2
d
≡ σr,t
+ σr,t
where upside and downside variances are defined as:
u 2
σr,t
= σc2 Vu,t + κ21 A21 + A23 ϕ2u qu,t ,
2
d
= σc2 Vd,t + κ21 A22 + A24 ϕ2d qd,t .
σr,t
Using the definition of variance risk premium, we define upside variance risk premium as:
h
h
2 i
2 i
u
u
− Et σr,t+1
,
V RPtu ≡ EQ
σr,t+1
t
= σc2 (θ − 1)κ1 A1 qu,t + κ21 A21 + A23 ϕ2u (θ − 1)κ1 A3 ϕ2u qu,t ,
= (θ − 1) σc2 κ1 A1 + κ31 A21 + A23 ϕ2u A3 ϕ2u qu,t .
10
(26)
(27)
(28)
Similarly, we define downside variance risk premium as:
2 2 Q
d
d
d
V RPt ≡ Et
σr,t+1
− Et σr,t+1
= (θ − 1) σc2 κ1 A2 + κ31 A22 + A24 ϕ2d A4 ϕ2d qd,t .
(29)
Empirical evidence requires that V RPtu < 0 and V RPtd > 0, hence it follows that
σc2 κ1 A1 + κ31 A21 + A23 ϕ2u A3 ϕ2u > 0
σc2 κ1 A2 + κ31 A22 + A24 ϕ2d A4 ϕ2d < 0
Since A4 < 0, A2 < 0 is a sufficient condition for σc2 κ1 A2 + κ31 A22 + A24 ϕ2d A4 ϕ2d < 0.
f − σc (1 − γ)
A2 < 0 ⇔ µ 2 <
γ−1
In particular we have
µ2 ≤ 0 ⇒ A2 < 0 ⇒ V RPtd > 0
Since A3 < 0, A1 > 0 is a necessary condition for σc2 κ1 A1 + κ31 A21 + A23 ϕ2u A3 ϕ2u > 0. It is easily
shown that
U
σc2 κ1 A1 + κ31 A21 + A23 ϕ2u A3 ϕ2u > 0 ⇔ AL
1 < A1 < A1
with
AL
1 =
−σc2 κ1 +
q
2
σc4 κ21 − 4 κ31 A3 ϕ2u A23 ϕ2u
2κ31 A3 ϕ2u
, AU
1 =
−σc2 κ1 −
q
2
σc4 κ21 − 4 κ31 A3 ϕ2u A23 ϕ2u
2κ31 A3 ϕ2u
L
It is worth nothing that both AU
1 and A1 are positive. In addition, it is easy to see that
f σc (1 − γ) + (1 − γ)µ1
L
< AU
AL
< A1 < AU
1,
1
1 ⇔ A1 < −
θ(κ1 βu − 1)
L
U
< A1 < AU
AL
1
1 ⇔ µ1 < µ1 < µ1 ,
with
f σc (1 − γ) + θ(κ1 βu − 1)AL
1
µL
=
1
γ−1
f σc (1 − γ) + θ(κ1 βu − 1)AU
1
> 0,
µU
1
=
γ−1
> 0,
which implies
µ1 > 0.
Consequently, for upside variance risk-premium to be negative, it must be the case that expected
consumption growth increases with the upside variance. Similarly, a non-positive relation between
11
expected consumption growth and downside variance is sufficient to have a positive downside
variance risk-premium.
In the next step, we derive the skewness risk premium. Following Feunou, Jahan-Parvar, and
T´edongap (2013) and Feunou, Jahan-Parvar, and T´edongap (2014), define the skewness as
u
skr,t = σr,t
2
2
d
.
− σr,t
As mentioned earlier, we define all premia as the difference between risk-neutral and physical
expectations of the process. Similarly, skewness risk premia is defined as
SRPt ≡ V RPtu − V RPtd
h h
h
2 2 2 i
2 ii
Q
Q
u
u
d
d
= Et σr,t+1
− Et σr,t+1
− Et
σr,t+1
− Et σr,t+1
,
= (σ1) σc2 κ1 A1 + κ31 A21 + A23 ϕ2u A3 ϕ2u qu,t − (θ − 1) σc2 κ1 A2 + κ31 A22 + A24 ϕ2d A4 ϕ2d qd,t ,
= (θ − 1) σc2 κ1 A1 + κ31 A21 + A23 ϕ2u A3 ϕ2u qu,t − σc2 κ1 A2 + κ31 A22 + A24 ϕ2d A4 ϕ2d qd,t
(30)
We rewrite the equity risk-premium as
ERPt
γσc2
γσc2
Vu,t +
Vd,t ,
=
1 + γσc
1 − γσc
−(θ − 1)κ1 A21 + A23 ϕ2u qu,t − (θ − 1)κ1 A22 + A24 ϕ2d qd,t ,
γ
u 2
=
σr,t
− κ21 A21 + A23 ϕ2u qu,t ,
1 + γσc
2
γ
d
2
2
2 2
σr,t − κ1 A2 + A4 ϕd qd,t ,
+
1 − γσc
−(θ − 1)κ1 A21 + A23 ϕ2u qu,t − (θ − 1)κ1 A22 + A24 ϕ2d qd,t ,
2 γκ
γ
γ
1
u 2
d
=
σr,t +
σr,t −
+ θ − 1 κ1 A21 + A23 ϕ2u qu,t ,
1 + γσc
1 − γσc
1 + γσc
γκ1
+ θ − 1 κ1 A22 + A24 ϕ2d qd,t ,
−
1 − γσc
and
12
we have
ERPt
2 + sk
2 − sk
σr,t
σr,t
γ
γ
r,t
r,t
=
+
1 + γσc
2
1 − γσc
2
γκ1
+ θ − 1 κ1 A21 + A23 ϕ2u qu,t
−
1 + γσc
γκ1
−
+ θ − 1 κ1 A22 + A24 ϕ2d qd,t
1 − γσc
γ 2 σc
γ
2
σ
−
skr,t
=
r,t
1 − γ 2 σc2
1 − γ 2 σc2
γκ1
−
+ θ − 1 κ1 A21 + A23 ϕ2u qu,t
1 + γσc
γκ1
−
+ θ − 1 κ1 A22 + A24 ϕ2d qd,t
1 − γσc
It is important to notice that this set-up is well defined if and only if
γσc < 1
Indeed, f (ν), is defined if only if ν < 1. Since,
EQ
t (exp (νεd,t+1 )) = exp ((f (ν + γσc ) − f (γσc )) Vd,t )
It must be the case that
γσc < 1
Hence
σc <
1
γ
Under such condition, as expected, the expected excess return increase with the conditional variance
γ 2 σc
(since the slope 1−γγ2 σ2 > 0) and decrease with the conditional skewness (since the slope − 1−γ
2 σ 2 ).
c
c
Upside and downside variance of variance impact the expected excess returns positively if
γκ1
+θ−1 < 0
1 + γσc
γκ1
+θ−1 < 0
1 − γσc
Given that
γκ1
γκ1
+θ−1<
+θ−1
1 + γσc
1 − γσc
it is sufficient to find condition under which
γκ1
+θ−1<0
1 − γσc
γκ1
(1 − θ) (1 − γσc )
+ θ − 1 < 0 ⇔ κ1 <
1 − γσc
γ
13
Hence, if
σc <
1
(1 − θ) (1 − γσc )
U
, κ1 <
, µ2 ≤ 0 and µL
1 < µ1 < µ1
γ
γ
then the upside variance risk-premium is negative and the downside variance risk-premium is positive. The equity risk-premium responds positively to the conditional variance and the downside
variance risk-premium, and negatively to the conditional skewness and the upside variance riskpremium, all in accordance with the empirical findings.
14
Table 1: Macroeconomic and Financial Time Series
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
Series Name
’Industrial Production Index, Total Index’
’IPI, Final Products and Nonindustrial Supplies (Total Products)’
’IPI, Consumer Goods’
’IPI, Durable Consumer Goods’
’IPI, Nondurable Consumer Goods’
’IPI, Automotive Products’
’IPI, Business Equipment’
’IPI, Defense & Space Equipment’
’IPI, Final Products’
’IPI, Materials’
’IPI, Durable Goods Materials’
’IPI, Nondurable Goods Materials’
’IPI, Energy (special aggregate)’
’IPI, Energy Materials (market group)’
’IPI, Primary Energy (market group)’
’IPI, Converted Fuel (market group)’
’IPI, Fuels (market group, within consumer energy products)’
’IPI, Manufacturing (SIC)’
’ISM Production Index ’
’ISM Purchasing Mangers” Index (Manufacturing)’
’Capacity Utilization, Manufacturing (SIC)’
’Personal Income’
’Disposable Personal Income’
’Personal Income Less Transfer Payments’
’Personal Saving’
’Personal Saving as % of Disposable Income’
’Total Civilian Employment’
’Total Civilian Nonagricultural Employment’
’Total Civilian Unemployment’
’Unemployment Rate’
’Unemployment Rate, 16-19 yrs’
’Avg. Weeks Unemployed’
No Unemployed ¡ 5 weeks
No Unemployed 5-14 weeks
No Unemployed 15-26 weeks
No Unemployed 15+ weeks
No Unemployed 27+ weeks
’ISM Employment Index (Manufacturing)’
’Nonfarm Payrolls, Total Private’
’Nonfarm Payrolls, Goods-Producing’
’Nonfarm Payrolls, Mining and Logging (Natural Resources)’
’Nonfarm Payrolls, Construction’
’Nonfarm Payrolls, Manufacturing’
’Nonfarm Payrolls, Durable Goods’
’Nonfarm Payrolls, Nondurable Goods’
’Nonfarm Payrolls, Services’
’Nonfarm Payrolls, Transportation, Trade & Utilities’
’Nonfarm Payrolls, Wholesale Trade’
’Nonfarm Payrolls, Retail Trade’
’Nonfarm Payrolls, Financial Sector’
’Nonfarm Payrolls, Government’
’Avg. Week Hrs Production/Non-Supervisory Empl, Goods-Producing’
Avg. Week Hrs Production/Non-Supervisory Empl, Const
Avg. Week Hrs Production/Non-Supervisory Empl, Manuf
Avg. Week OT Hrs Production/Non-Supervisory Empl, Manuf’
’Avg. Week Hrs Production/Non-Supervisory Empl, Durable Goods’
’Avg. Hourly Earnings Production/Non-Supervisory Empl, Goods-Producing’
’Avg. Hour Earnings, Production/Non-Supervisory Empl, Construction’
Avg. Hour Earnings Production/Non-Supervisory Empl, Manuf
’Personal Consumption Expenditures’
’Durable Goods’
’Nondurable Goods’
’Services’
’Manufacturers” New Orders- Consumer Goods & Materials ’
’New Orders- Nondefense Capital Goods’
’ISM Supplier Deliveries Index (Manufacturing)’
’ISM New Orders Index (Manufacturing)’
’ISM Inventories Index (Manufacturing)’
’Total New Private Housing Starts’
’Midwest Housing Starts’
’Northeast Housing Starts’
’South Housing Starts’
’West Housing Starts’
’USDCAD’
’GBPUSD’
No
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
Series Name
’USD Effective Exchange Rate Index
’ISM Prices Index (Manufacturing)’
’PPI, Crude Petroleum (Domestic Production)’
’PPI, Crude Materials’
’PPI, Finished Consumer Goods’
’PPI, Finished Goods’
’PPI, Intermediate Materials, Supplies & Components’
’CPI, All Items’
’CPI, Services’
’CPI, All Items Less Food’
’CPI, All Items Less Shelter’
’CPI, All Items Less Medical Care’
’CPI, All Items Less Food & Energy’
’CPI, Commodities’
’CPI, Durables’
’CPI, Apparel’
’CPI, Transportaiton’
’CPI, Medical Care’
’Dow Jones Industrial Average’
’S&P 500 Dividend Yield’
’S&P 500 Index’
’S&P 500 Industrials Index ’
’S&P 500 PE’
’M1 Index’
’M2 Index’
’Consumer Credit Outstanding (nonrevolving)’
’Conference Board Consumer Confidence Index’
’University of Michigan Consumer Sentiment Index’
’3-Month T-Bill’
’6-Month T-Bill’
’1-Year Treasury’
5-Year Treasury
3-Year Treasury
10-Year Treasury
20-Year Treasury
’Prime Rate’
’Moody”s AAA Corp. Index’
’Moody”s AAA Muni. 10y Index’
’Moody”s A Corporate Index’
’Moody”s Average Corporate Index’
’Moody”s BAA Corp. Index’
’3M-Fed Funds Spread’
’6M-Fed Funds Spread’
’1y-Fed Funds Spread’
’5y-Fed Funds Spread’
’10y-Fed Funds Spread’
’AAA Corp. - Fed Funds Spread’
’BAA Corp. - Fed Funds Spread’
’Effective Federal Funds Rate’
’BAA Corp. - 10y Treasury Spread’
This table displays the names and the number associated with 125 macroeconomic and financial time series used in the analysis
of relationship between variance and skewness risk premia and macroeconomic factors.
15
Figure 1: Counter-Cyclical Risk Factors
Downside Semi−Variance Premium
0.5
0.5
0.4
0.4
Adjusted R2
Adjusted R2
Variance Risk Premium
0.3
0.3
0.2
0.2
0.1
0.1
0
0
20
40
60
80
100
0
120
0
20
0.5
0.5
0.4
0.4
0.3
0.2
0.1
0.1
0
20
40
60
80
60
80
100
120
100
120
0.3
0.2
0
40
Downside Relative Semi−Variance Premium
Adjusted R2
Adjusted R2
Upside Semi−Variance Premium
100
0
120
16
0
20
40
60
80
References
Bansal, R., and A. Yaron. 2004. Risks for the long run: A potential resolution of asset pricing
puzzles. Journal of Finance 59:1481–1509.
Bekaert, G., E. Engstrom, and A. Ermolov. 2014. Bad environments, good environments: A nongaussian asymmetric volatility model. Journal of Econometrics, forthcoming .
Bollerslev, T., G. Tauchen, and H. Zhou. 2009. Expected stock returns and variance risk premia.
Review of Financial Studies 22:4463–92.
Drechsler, I., and A. Yaron. 2011. Whats Vol got to do with it?
24:1–45.
Review of Financial Studies
Epstein, L. G., and S. E. Zin. 1989. Substitution, risk aversion, and the temporal behavior of
consumption and asset returns: A theoretical framework. Econometrica 57:937–69.
Eraker, B., and I. Shaliastovich. 2008. An equilibrium guide to designing affine pricing models.
Mathematical Finance 18:519–543.
Feunou, B., M. R. Jahan-Parvar, and R. T´edongap. 2013. Modeling Market Downside Volatility.
Review of Finance 17:443–81.
———. 2014. Which parametric model for conditional skewness? European Journal of Finance,
forthcoming .
Gul, F. 1991. A Teory of Disappointment Aversion. Econometrica 59:667–86.
Klibanoff, P., M. Marinacci, and S. Mukerji. 2009. Recursive smooth ambiguity preferences. Journal
of Economic Theory 144:930–76.
Kreps, D. M., and E. L. Porteus. 1978. Temporal Resolution of Uncertainty and Dynamic Choice
Theory. Econometrica 46:185–200.
Miao, J., B. Wei, and H. Zhou. 2012. Ambiguity aversion and variance premium. Working Paper,
Federal Reserve Board and Boston University .
Segal, G., I. Shaliastovich, and A. Yaron. 2015. Good and bad uncertainty: Macroeconomic and
financial market implications. Journal of Financial Economics, forthcoming .
Weil, P. 1989. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary
Economics 24:401–21.
17