Pricing Kernels with Stochastic Skewness and Volatility Risk
Online Appendix
In this online appendix, I first provide a brief comparison of the small noise expansion series and the
Taylor expansion series. I then use separably Taylor expansion series and the small noise expansion series
to derive the pricing kernel in a one-market model with two dates, and the pricing kernel in a two-market
model with three dates. I thereafter, investigate whether one obtains a pricing kernel that depends on the
market co-skewness and the market volatility in a long run risk model and also in a model where the
representative investor chooses his optimal allocation in the presence of the market equity and the variance
swap contracts. The remaining part of the appendix contains additional tables and figures used in the paper.
Appendix A: A Brief Comparison of the Small Noise Expansion and the Taylor Expansion Series
To approximate the investor’s utility u [W ] with Taylor expansion series, the usual route is to assume that
W − EW is small
W − E (W ) < δ for δ > 0,
(A1)
and expand u [.] around the expected value E (W ). Similarly to the Taylor expansion series, the small noise
expansion framework would assume
W − E (W ) = εY
(A2)
and drive ε toward zero without making any restrictions on the distribution of Y . A simple comparison of
both approaches in (A1) and (A2) shows that under the restriction
|Y | < δ/ε,
(A3)
the small noise expansion series and the Taylor expansion series are equivalent. The small noise expansion
approach drives ε toward 0. In order words, ε → 0 is equivalent to
ε=
1
,
2n
(A4)
where n is very large. Replacing (A4) in (A2) produces
Y = 2n (W − E [W ]) .
(A5)
The random variable Y represents the innovation in the aggregate wealth scaled by a large number. The
probability density function of Y and W − E [W ] are proportional.
In this paper, I consider the small noise approximation of innovations in returns
Rkt+1 − Et Rkt+1 = εYkt+1 and Rkt+2 − Et+1 Rkt+2 = εYkt+2
(A6)
and derive the pricing kernel in a one-period model with two dates (see Appendix C), and the pricing kernel
in a two-period model with three dates (see Appendix D).
2
Appendix B: Pricing Kernel Obtained with the Standard Taylor Series in
a Two-Period Model with Three Dates
I assume that the representative agent maximizes his expected utility in a two-period model with three dates
(
Vt = max Et
{ωt }
)
max Et+1 (u [Wt+2 ]) ,
(B1)
{ωt+1 }
where the investor’s wealth is
(
)(
e
e
Wt+2 = R f + ωt Rt+1
R f + ωt+1 Rt+2
)
(B2)
and Reτ+1 = (Rτ+1 − R f ), τ = t, t + 1. The initial wealth Wt = 1. I use the first-order condition of (B1) with
respect to ωt
( ′
)
Et u [Wt+2 ] (Rkt+1 − R f ) = 0,
(B3)
and use the decomposition Covt (X,Y ) = Et (XY ) − Et (X) Et (Y ) to expand (B3) to derive the risk premium
(
at time t
Et (Rkt+1 − R f ) = −Covt
)
′
u [Wt+2 ]
, Rkt+1 .
Et u′ [Wt+2 ]
(B4)
I show in subsection B.1 and B.2 that the static Taylor expansion series and the recursive Taylor expansion
series of the investor’s marginal utility produce pricing kernels where the co-skewness risk and the volatility
risk factor have identical prices.
B.1. Static Taylor Expansion Series
′
The standard Taylor expansion series of u [Wt+2 ] around (ωt Rt+1 , ωt+1 Rt+2 ) = (R f , R f ) produces
]
]
]
′
′[
′′ [
′′ [
u [Wt+2 ] = u R2f + u R2f (ωt Rt+1 − R f ) + u R2f (ωt+1 Rt+2 − R f )
1 ′′′ [ ]
1 ′′′ [ ]
+ u R2f (ωt Rt+1 − R f )2 + u R2f (ωt+1 Rt+2 − R f )2 .
2
2
3
(B5)
I replace (B5) in the risk premium (B4) and show
′′
Et (Rkt+1 − R f ) = −
′′
u [R2f ]
′
Et u [Wt+2 ]
Covt (ωt Rt+1 , Rkt+1 ) −
′′′
−
u [R2f ]
′
2Et u [Wt+2 ]
′′′
−
u [R2f ]
′
u [R2f ]
2Et u [Wt+2 ]
′
Et u [Wt+2 ]
(
)
Covt (ωt Rt+1 − R f )2 , Rkt+1
Covt (ωt+1 Rt+2 , Rkt+1 ) (B6)
(
)
Covt (ωt+1 Rt+2 − R f )2 , Rkt+1 .
I notice that
(
)
(
) ( (
))
Covt (RMt+2 − R f )2 , Rkt+1
= Et (RMt+2 − R f )2 Rkt+1 − Et (RMt+2 − R f )2 (Et (Rkt+1 ))
(B7)
(
(
)) ( (
))
= Et Rkt+1 Et+1 (RMt+2 − R f )2 − Et Et+1 (RMt+2 − R f )2 (Et (Rkt+1 ))
(
)
= Covt Et+1 (RMt+2 − R f )2 , Rkt+1 ,
Since ωt+1 Rt+2 = RMt+2 , Covt (RMt+2 , Rkt+1 ) = Covt (Et+1 RMt+2 , Rkt+1 ), and σ2Mt+1 ≈ Et+1 (ωt+1 Rt+2 − R f )2 ,
the pricing kernel consistent with (B6) is
Mt+1 =
1
+ G1 (RMt+1 − Et+1 RMt+1 ) + G1 (Et+1 RMt+2 − Et RMt+2 )
Rf
(
)
(
)
+G2 (RMt+1 − R f )2 − Et (RMt+1 − R f )2 + G2 σ2Mt+1 − Et σ2Mt+1
with
′′
G1 =
u [R2f ]
R f Et u′ [Wt+2 ]
(B8)
′′′
and G2 =
u [R2f ]
2R f Et u′ [Wt+2 ]
.
The pricing kernel in (B8) resembles the pricing kernel with co-skewness and volatility risk derived in
Proposition 1, except that the prices of co-skewness and volatility risk in (B8) are identical.
B.2. Recursive Taylor Expansion Series
I notice that Wt+2 can be written as Wt+2 = Wt+1 (ωt+1 Rt+2 ) and use Taylor expansion series recursively
to derive the pricing kernel.
First, at time t + 1, I notice that Wt+1 = ωt Rt+1 and use Taylor series to expand the investor’s marginal
4
′
utility u [Wt+2 ] around ωt+1 Rt+2 = R f as follows
′
′
′′
1 ′′′
2
u [Wt+2 ] = u [R f Wt+1 ] + u [R f Wt+1 ] Wt+1 (ωt+1 Rt+2 − R f ) + u [R f Wt+1 ] (Wt+1 ) (ωt+1 Rt+2 − R f )2 .
2
(B9)
I denote by h (Wt+1 ) the right hand side of (B9), and hence
′
u [Wt+2 ] = h (Wt+1 ) .
(B10)
Second, at time t, I expand (B9) around Wt+1 = R f
h (Wt+1 ) = h (R f ) +
1 ′
1 ′′
2
h (R f ) (Wt+1 − R f ) + h (R f ) (Wt+1 − R f ) ,
1!
2!
(B11)
where
∂ (h (Wt+1 ))
,
∂Wt+1
∂2 (h (Wt+1 ))
∂2 Wt+1
′
h (Wt+1 ) =
′′
h (Wt+1 ) =
When Wt+1 = R f , I use (B10) and (B9) to obtain
]
]
′[
′′ [
1 ′′′ [ ]
h (R f ) = u R2f + u R2f R f (ωt+1 Rt+2 − R f ) + u R2f R2f (ωt+1 Rt+2 − R f )2
2
(B12)
and
])
] ( ′′ [ ]
′′′ [
′
′′ [
h (R f ) = R f u R2f + u R2f + R2f u R2f (ωt+1 Rt+2 − R f )
(
)
′′′ [ 2 ]
1 3 ′′′′ [ 2 ]
+ R f u R f + R f u R f (ωt+1 Rt+2 − R f )2
2
(B13)
and
′′
′′′
h (R f ) = R2f u
[ 2 ] ( 3 ′′′′ [ 2 ]
])
′′′ [
R f + R f u R f + 2R f u R2f (ωt+1 Rt+2 − R f )
]
′′′′′ [
1
+ R2f u R2f R2f (ωt+1 Rt+2 − R f )2
2
]
]
′′′ [
′′′′ [
+u R2f (ωt+1 Rt+2 − R f )2 + u R2f R2f (ωt+1 Rt+2 − R f )2 .
5
(B14)
I replace (B12), (B13), and (B14) in (B11) and obtain
(
)
′ [ 2]
′′ [ 2 ]
1 ′′′ [ 2 ] 2
2
u [Wt+2 ] =
u R f + u R f R f (ωt+1 Rt+2 − R f ) + u R f R f (ωt+1 Rt+2 − R f )
(B15)
2

[ ] ( ′′ [ ]
[ ])

′′
2
2
2 ′′′
2
1  R f u R f + u R f + R f u R f (ωt+1 Rt+2 − R f ) 
(
[ ]
[ ])
+
(ωt Rt+1 − R f )
′′′
′′′′
1!
R f u R2f + 12 R3f u R2f (ωt+1 Rt+2 − R f )2


[ ] (
[ ]
[ ])
′′′
′′′′
′′′
2
2
3
2
2
 R f u R f + R f u [R f ]+ 2R f u R f (ωt+1 Rt+2 − R f ) 

1
2
1 2 ′′′′′
2 R2 (ω
 (ωt Rt+1 − R f )2
+ 
R
u
R
R
−
R
)
+
t+1
t+2
f


f
f
f
2
2! 
[ ]
[ ]

′′′
′′′′
2
2
2
2
2
+u R f (ωt+1 Rt+2 − R f ) + u R f R f (ωt+1 Rt+2 − R f ) .
′
Replacing (B15) in the risk premium (B4) allows me to show that the contributions of the market, the
co-skewness, and the volatility factors to the risk premium on asset k are respectively

−Covt 

−Covt 
and

−Covt 
′′
1
1! R f u
[
]
R2f (ωt Rt+1 − R f )
′
Et u [Wt+2 ]
1 2 ′′′
2! R f u
[
, Rkt+1  ,
]
R2f (ωt Rt+1 − R f )2
′
Et u [Wt+2 ]
1 2 ′′′
2Rf u


, Rkt+1  ,
[ ]
R2f (ωt+1 Rt+2 − R f )2
′
Et u [Wt+2 ]

, Rkt+1  .
Therefore, neglecting the remaining cross-product terms in (B15) allows to write the risk premium on asset
k as

Et (Rkt+1 − R f ) = −Covt 
′′
1
1! R f u
[
]
R2f (ωt Rt+1 − R f )

, Rkt+1 
′
Et u [Wt+2 ]
[ ]


2
1 2 ′′′
2 (ω R
R
u
R
−
R
)
t
t+1
f
f
f
2!
−Covt 
, Rkt+1 
Et u′ [Wt+2 ]
[ ]


2
1 2 ′′′
2 (ω
R
u
R
R
−
R
)
t+1
t+2
f
f
2 f
−Covt 
, Rkt+1 
Et u′ [Wt+2 ]
6
(B16)
Since, ωt Rt+1 = RMt+1 , (B16) simplifies to
Et (Rkt+1 − R f ) = −
[ ]
′′
R f u R2f
Covt (RMt+1 , Rkt+1 )
Et u′ [Wt+2 ]
[ ]
2 u′′′ R2
(
)
R
f
1 f
−
Covt (RMt+1 − R f )2 , Rkt+1
′
2! Et u [Wt+2 ]
[ ]
2 u′′′ R2
)
(
R
f
f
1
2
−
(R
−
R
)
,
R
Cov
Mt+2
f
t
kt+1 .
2! Et u′ [Wt+2 ]
(B17)
I use (B7), the approximation Et+1 (RMt+2 − R f )2 ≈ σ2Mt+1 , and rewrite the risk premium (B17) as
Et (Rkt+1 − R f ) = −
[ ]
′′
R f u R2f
Covt (RMt+1 , Rkt+1 )
′
Et u [Wt+2 ]
[ ]
2 u′′′ R2
(
)
R
f
1 f
−
Covt (RMt+1 − R f )2 , Rkt+1
′
2! Et u [Wt+2 ]
[ ]
′′′
2
2
(
)
1 Rf u Rf
−
Covt σ2Mt+1 , Rkt+1 .
′
2! Et u [Wt+2 ]
(B18)
′′′
R2 u [R2 ]
Since the co-skewness and the market volatility factors have identical prices, − 2!1 E fu′ [W f ] , it is straightfort
t+2
ward to show that, the co-skewness factor and the market volatility factor have the same factor loading in
the pricing kernel specification.
7
Appendix C: Pricing Kernel Derived in a One-Period Model with Two Dates
Using the Small Noise Expansion Series
In this section, I show that when the small noise assumption (A6) is used, existing asset pricing models
can be derived in a one-period model with two dates. I consider the representative agent’s optimization in
a one-period model with two dates
max Et (u [Wt+1 ]) ,
ωt
(C1)
where the investor’s wealth is
(
)
e
e
Wt+1 = Wt R f + ωt Rt+1
and Rt+1
= Rt+1 − R f .
(C2)
Without loss of generality, I assume that Wt = 1. The first-order conditions for (C1) are
( ′
)
Et u [Wt+1 ] (Rkt+1 − R f ) = 0, for k = 1, 2, ...
(C3)
and the expected excess return can be obtained by applying the covariance decomposition Covt (X,Y ) =
Et (XY ) − Et (X) Et (Y ) to (C3):
(
Et (Rkt+1 − R f ) = −Covt
)
′
u [Wt+1 ]
, Rkt+1 .
Et u′ [Wt+1 ]
(C4)
I use the small noise assumption
Rkt+1 − Et Rkt+1 = εYkt+1 ,
(C5)
Wt+1 − Et Wt+1 = (ωt Yt+1 ) ε,
(C6)
and write
where Yt+1 is a vector whose components are Ykt+1 for k = 1, 2, .... The Taylor expansion series of
′
u [Wt+1 ] around ε = 0 produces
( ′
)[1]
( ′
)[Q]
′
′
1
u [Wt+1 ] = lim u [Wt+1 ] + lim u [Wt+1 ]
ε + ... +
lim u [Wt+1 ]
εQ ,
ε→0
ε→0
Q! ε→0
(C7)
( ′
)[ j]
′
where u [Wt+1 ]
represents the jth derivative of u [Wt+1 ] with respect to ε. I notice that the term
8
′
limε→0 u [Wt+1 ] is constant and use (C7) to express the risk premium (C4) as
Et (Rkt+1 − R f ) = ε2 akt [ε] ,

(C8)

)[ j]
′
u
[
W
]
t+1
1
1


akt [ε] = ∑
Covt − lim
,Ykt+1  ε j−1 .
′
ε→0 j Et u [Wt+1 ]
j=1 ( j − 1)!
with
(
Q
(C9)
(C9) simplifies to
Q
akt [ε] =
1
∑ ( j − 1)! akt
[ j−1]
[0] ε j−1 .
(C10)
j=1

where
[ j−1]
akt

[0] = Covt − lim
( ′
)[ j]
u
[
W
]
t+1
1
ε→0
′
j Et u [Wt+1 ]


,Ykt+1 
(C11)
For j = 1, 2, and 3,
( ′
)[1]
′′
′
u [Wt+1 ]
= u [Wt+1 ] Wt+1 ,
( ′
)[2]
( ′ )2
′′′
′′
′′
u [Wt+1 ]
= u [Wt+1 ] Wt+1 + u [Wt+1 ] Wt+1 ,
( ′ )3
( ′
)[3]
′′′′
′′′
′′
′
′′
′′′
= u [Wt+1 ] Wt+1 + 3u [Wt+1 ] Wt+1 Wt+1 + u [Wt+1 ] Wt+1 ,
u [Wt+1 ]
(C12)
(C13)
(C14)
where
′
Wt+1 =
′′
′′′
∂2 Wt+1
∂3 Wt+1
∂Wt+1
, Wt+1 =
,
and
W
=
,
t+1
∂ε
∂2 ε
∂3 ε
(C15)
and
′
′′
′′′
lim Wt+1 = ωt Yt+1 and lim Wt+1 = lim Wt+1 = 0.
ε→0
ε→0
ε→0
(C16)
I ignore high-order terms Q > 3 in (C8)-(C9) and use (C12)–(C14) to show that
(
)
1 [1]
1 [2]
[0]
Et (Rkt+1 − R f ) = ε2 akt [ε] = ε2 akt [0] + akt [0] ε + akt [0] ε2 ,
1!
2!
9
(C17)
which simplifies to
Et (Rkt+1 − R f ) =
with
1
Covt ((ωt Rt+1 ) , Rkt+1 )
℘
(
)
ρ
− 2 Covt ((ωt Rt+1 ) − Et (ωt Rt+1 ))2 , Rkt+1
℘
(
)
κ
+ 3 Covt ((ωt Rt+1 ) − Et (ωt Rt+1 ))3 , Rkt+1 ,
2℘
′
℘= −
′′′
(C18)
′′′′
u [R f ]
℘2 u [R f ]
℘3 u [R f ]
,
ρ
=
−
,
and
κ
=
.
2 u′ [R f ]
3 u′ [R f ]
u′′ [R f ]
(C19)
I recall that the risk premium on the risky asset can be expressed as the negative of the covariance of the
pricing kernel with the return on the risky asset
Et (Rkt+1 − R f ) = −R f Covt (Mt+1 , Rkt+1 ) ,
(C20)
and I compare the risk premium (C18) to (C20) and recover the functional form of the pricing kernel Mt+1 :
)
)
κ ( 3
1
ρ ( 2
1
2
3
− 3
,
−
rMt+1 + 2
rMt+1 − Et rMt+1
rMt+1 − Et rMt+1
R f ℘R f
℘ Rf
2℘ R f
Mt+1 =
(C21)
where
rMt+1 = RMt+1 − Et RMt+1 where RMt+1 = ϖt Rt+1 and ϖt = Wt ωt .
(C22)
When ρ = κ = 0, (C21) reduces to the pricing kernel in the CAPM model. When κ = 0, (C21) reduces to
the pricing kernel in the market co-skewness models of ? and ?. When ρ ̸= 0 and κ ̸= 0, (C21) reduces to
the pricing kernel in ?. The price of the market co-skewness factor is − ℘2ρR f < 0, while the price of risk of
the market co-kurtosis is
κ
2℘3 R f
> 0.
To summarize, the small noise expansion produces existing pricing kernels in a one-period model with
two dates. The pricing kernel is a polynomial function of the market return. In Appendix D, I show that
when the time interval is extended to a two-period model with three dates, the pricing kernel as opposed
to (C21) is a function of the market return, the market volatility, the market skewness, and the market kurtosis. I show that the prices of these risk factors are related to risk-aversion, skewness preference, kurtosis
preference, and high-order preferences. These prices are not to be confused with the price of risk of powers
of the market return.
10
Appendix D: Pricing Kernels in a Two-Period Model with Three Dates Using
the Small Noise Expansion Series
D.1. Expected Return Decomposition
As opposed to Appendix C, I derive the pricing kernel when the representative agent maximizes his exe =R
pected utility over a two-period [t,t + 2] interval with three dates t, t +1, and t +2. I define Rt+1
t+1 −R f
as the vector of excess return on the risky assets. The representative agent optimization problem is
(
max Et
{ωt }
with
)
max Et+1 (u [Wt+2 ])
(D1)
{ωt+1 }
(
)(
)
e
e
Wt+2 = Wt R f + ωt Rt+1
R f + ωt+1 Rt+2
,
(D2)
Rkt+1 − Et Rkt+1 = εYkt+1 and Rkt+2 − Et+1 Rkt+2 = εYkt+2 .
(D3)
where
Without loss of generality, I assume that Wt = 1 and solve (D1) in two steps. I first solve
Vt+1 = max Et+1 (u [Wt+2 ]) .
(D4)
max Et (Vt+1 ) .
(D5)
{ωt+1 }
Second, given Vt+1 (in D4), I solve
{ωt }
STEP 1: The first-order conditions of (D4) are
( ′
)
Et+1 u [Wt+2 ] (Rkt+2 − R f ) = 0, for k = 1, 2, ...
(D6)
Rkt+2 − Et+1 Rkt+2 = εYkt+2 .
(D7)
where
I apply the covariance definition, Covt+1 (X,Y ) = Et+1 (XY ) − Et+1 (X) Et+1 (Y ), to (D6) and show that the
11
risk premium on the risky asset is
(
Et+1 (Rkt+2 − R f ) = −Covt+1
)
′
u [Wt+2 ]
, Rkt+2 .
′
Et+1 u [Wt+2 ]
(D8)
′
The Taylor expansion series of u [Wt+2 ] around ε = 0 produces
(
( ′
)[ j] )
1
lim u [Wt+2 ]
ε j,
j! ε→0
Q
′
u [Wt+2 ] =
∑
j=0
(D9)
( ′
)[ j]
′
where u [Wt+2 ]
represents the jth derivative of u [Wt+2 ] with respect to ε. I replace (D9) in (D8) and
obtain
Et+1 (Rkt+2 − R f ) = ε2 akt+1 [ε] ,
(D10)
where
Q
akt+1 [ε] =
1
∑ ( j − 1)! akt+1 [0] ε j−1
[ j−1]
(D11)
j=1

and

[ j−1]
akt+1 [0] = Covt+1 − lim
( ′
)[ j]
u
[
W
]
t+2
1
ε→0
j Et+1 u′ [Wt+2 ]


,Ykt+2  .
(D12)
For j = 1, 2, and 3,
( ′
)[1]
u [Wt+2 ]
( ′
)[2]
u [Wt+2 ]
( ′
)[3]
u [Wt+2 ]
( ′
)[4]
u [Wt+2 ]
′′
′
= u [Wt+2 ] Wt+2 ,
( ′ )2
′′′
′′
′′
= u [Wt+2 ] Wt+2 + u [Wt+2 ] Wt+2 ,
( ′ )3
′′′′
′′′
′′
′
′′
′′′
= u [Wt+2 ] Wt+2 + 3u [Wt+2 ] Wt+2 Wt+2 + u [Wt+2 ] Wt+2 ,
( ′ )4
( ′ )2 ′′
′′′′′
′′′′
′′
′′′′
= u [Wt+2 ] Wt+2 + 6u [Wt+2 ] Wt+2 Wt+2 + u [Wt+2 ] Wt+2
( ′′ )2
′′′
′′′
′
′′′
+4u [Wt+2 ] Wt+2 Wt+2 + 3u [Wt+2 ] Wt+2 ,
(D13)
(D14)
(D15)
(D16)
(D17)
where
′
Wt+2 =
′′
′′′
′′′′
∂2 Wt+2
∂3 Wt+2
∂4 Wt+2
∂Wt+2
=
=
, Wt+2 =
,
W
,
and
W
,
t+2
t+2
∂ε
∂2 ε
∂3 ε
∂4 ε
12
(D18)
and
∂Wt+2
∂ε
∂2 Wt+2
lim
ε→0
∂2 ε
3
∂ Wt+2
lim
ε→0
∂3 ε
∂4 Wt+2
lim
ε→0
∂4 ε
lim
ε→0
= R f (ωt Yt+1 ) + R f (ωt+1Yt+2 ) ,
(D19)
= 2 (ωt Yt+1 ) (ωt+1Yt+2 ) ,
(D20)
= 0,
(D21)
= 0.
(D22)
Combining (D10) and (D7) produces
Rkt+2 = R f + ε2 akt+1 [ε] + εYkt+2 .
(D23)
I use assumption (D7) and then expand (D12) for j = 1, ..., 4, ignoring the cross-product terms to show:
1
R f Covt+1 (ωt+1Yt+2 ,Ykt+2 ) ,
℘
(
)
ρ
[1]
akt+1 [0] = − 2 R2f Covt+1 (ωt+1Yt+2 )2 ,Ykt+2 ,
℘
(
)
κ 3
[2]
3
akt+1 [0] =
R
Cov
(ω
Y
)
,Y
t+1
t+1 t+2
kt+2 ,
℘3 f
(
)
δ
[3]
akt+1 [0] = − 4 R4f Covt+1 (ωt+1Yt+2 )4 ,Ykt+2 ,
℘
[0]
akt+1 [0] =
with
′
′′′
u [R2f ]
′′′′
(D24)
(D25)
(D26)
(D27)
′′′′′
2
2
2
℘2 u [R f ]
℘3 u [R f ]
℘4 u [R f ]
℘= − ′′ 2 , ρ =
,
κ
=
−
,
δ
=
.
2 u′ [R2f ]
3 u′ [R2f ]
4 u′ [R2f ]
u [R f ]
(D28)
[ j−1]
STEP 2: Now, I assume (D23) where akt+1 [0] is defined in (D24)–(D27) and solve (D5)
max Et (Vt+1 ) .
(D29)
( ′
)
Et u [Wt+2 ] (Rkt+1 − R f ) = 0,
(D30)
Rkt+1 − Et Rkt+1 = εYkt+1
(D31)
{ωt }
The first-order conditions are
where
13
and
Rkt+2 = R f + ε2 akt+1 [ε] + εYkt+2 .
(D32)
I recall that the proof of (D32) is given in STEP 1. I apply the definition of the covariance, Covt (X,Y ) =
Et (XY ) − Et (X) Et (Y ), to (D30) and show that the risk premium on the risky asset is
(
Et (Rkt+1 − R f ) = −Covt
)
′
u [Wt+2 ]
, Rkt+1 .
′
Et u [Wt+2 ]
(D33)
′
The Taylor expansion series of u [Wt+2 ] around ε = 0 produces
(
( ′
)[ j] )
1
lim u [Wt+2 ]
ε j,
j! ε→0
Q
′
u [Wt+2 ] =
∑
j=0
(D34)
( ′
)[ j]
′
where u [Wt+2 ]
represents the jth derivative of u [Wt+2 ] with respect to ε. I replace (D34) in (D33)
and show that
Et (Rkt+1 − R f ) = ε2 akt [ε] ,
(D35)
with
Q
akt [ε] =
1
∑ ( j − 1)! akt
[ j−1]
[0] ε j−1
(D36)
j=1


)[ j]
1 u [Wt+2 ]


[ j−1]
akt [0] = Covt − lim
,Ykt+1  .
′
ε→0 j Et u [Wt+2 ]
and
I notice that
(
(
′
)(
(D37)
)
e
e
Wt+2 = R f + ωt Rt+1
R f + ωt+1 Rt+2
.
(D38)
Since I show in STEP 1 that
Rkt+2 = R f + ε2 akt+1 (ε) + εYkt+2 .
(D39)
The investor’s wealth (D38) can be expanded as follows
(
)(
(
e
Wt+2 = R f + ωt Rt+1
R f + ωt+1 ε2 at+1 (ε) + εYt+2
14
))
(D40)
Therefore,
∂Wt+2
∂ε
∂2 Wt+2
lim
ε→0
∂2 ε
3
∂ Wt+2
lim
ε→0
∂3 ε
∂4 Wt+2
lim
ε→0
∂4 ε
lim
ε→0
= R f (ωt Yt+1 ) + R f (ωt+1Yt+2 ) ,
(D41)
= 2R f ωt+1 at+1 (0) + 2 (ωt Yt+1 ) (ωt+1Yt+2 ) ,
(D42)
′
= 6R f ωt+1 at+1 (0) + 6 (ωt Yt+1 ) (ωt+1 at+1 (0)) ,
(D43)
(
)
′
′′
= 24 (ωt Yt+1 ) ωt+1 at+1 (0) + 12R f ωt+1 at+1 (0) .
(D44)
[ j−1]
I use equations (D13)–(D17) and (D41)–(D44) to expand akt
[0], ignore the cross-product terms, and
show:
Rf
Covt (ωt Yt+1 ,Ykt+1 ) ,
℘
(
) (1 − ρ)
(
)
ρ
[1]
2
2
akt [0] = − 2 R2f Covt (ωt Yt+1 )2 ,Ykt+1 +
R
Cov
(ω
Y
)
,Y
t
t+1 t+2
kt+1 ,
f
℘
℘2
(
(
) (κ − 2ρ)
)
κ 3
[2]
3
3
3
akt [0] =
R
Cov
(ω
Y
)
,Y
R
Cov
(ω
Y
)
,Y
+
,
t
t
t+1
t
t+1
t+2
kt+1
kt+1
f
f
℘3
℘3
(
(
) (δ − 3κ)
)
δ
[3]
4
4
akt [0] = − 4 R4f Covt (ωt Yt+1 )4 ,Ykt+1 −
R
Cov
(ω
Y
,Y
.
)
t
t+1
t+2
kt+1
f
℘
℘4
[0]
akt [0] =
(D45)
(D46)
(D47)
(D48)
I combine (D36) and (D31) and show
Rkt+1 = R f + ε2 akt [ε] + εYkt+1 ,
[ j−1]
where akt [ε] is defined in (D36) with akt
(D49)
[0] defined in (D45)–(D48).
D.2. Pricing Kernels
In this section, I use the relation between the pricing kernel and the risk premium on risky assets
Et (Rkt+1 − R f ) = −R f Covt (Mt+1 , Rkt+1 )
to recover the functional form of the pricing kernel.
Pricing Kernel with Co-skewness and Volatility Risk When Q=1
15
(D50)
I derive the pricing kernel when the order in (D36) is Q = 1. I recall that, from (D49) and (D23),
Rkt+1 = R f + ε2 akt [ε] + εYkt+1 and Rkt+2 = R f + ε2 akt+1 [ε] + εYkt+2 .
(D51)
When Q = 1, the risk premium (D35) is
Et (Rkt+1 − R f ) =
(
)
1
ρ
R f Covt (ωt Yt+1 ,Ykt+1 ) ε2 − 2 R2f Covt (ωt Yt+1 )2 ,Ykt+1 ε3
℘
℘
)
(
(1 − ρ) 2
2
3
(ω
Y
)
,Y
R
Cov
+
t+1 t+2
t
kt+1 ε ,
f
℘2
(D52)
which simplifies to
Et (Rkt+1 − R f ) =
( 2
)
1
ρ
, Rkt+1
R f Covt (rMt+1 , Rkt+1 ) − 2 R2f Covt rMt+1
℘
℘
)
(
(1 − ρ)
2
(r
)
,
R
,
+
Cov
Mt+2
t
kt+1
℘2
(D53)
where
rMτ+1 = RMt+1 − Et RMτ+1 , RMτ+1 = ϖτ Rτ+1 , and ϖτ = lim Wτ ωτ with τ = t,t + 1.
ε→0
(D54)
I notice that
(
)
(
)
(
)
Covt (rMt+2 )2 , Rkt+1 = Covt Et+1 (rMt+2 )2 , Rkt+1 = Covt σ2Mt+1 , Rkt+1 ,
(D55)
where σ2Mt+1 is the variance of the market return. I compare (D53) to (D50) and recover the pricing kernel
Mt+1 =
( 2
) (1 − ρ) ( 2
)
1
1
ρ
2
− rMt+1 + 2 R f rMt+1
− Et rMt+1
− 2
σMt+1 − Et σ2Mt+1 .
Rf ℘
℘
℘ Rf
(D56)
Pricing Kernel with Co-skewness and Volatility Risk When Q=2
I derive the pricing kernel when the order in (D36) is Q = 2. The risk premium (D35) is
Et (Rkt+1 − R f ) =
( 2
)
1
ρ
R f Covt (rMt+1 , Rkt+1 ) − 2 R2f Covt rMt+1
, Rkt+1
℘
℘
( 2
)
( 3
)
(1 − ρ)
κ
+
Covt σMt+1 , Rkt+1 + 3 R3f Covt rMt+1
, Rkt+1
2
℘
2℘
(κ − 2ρ)
+
Covt (sMt+1 , Rkt+1 ) ,
2℘3
16
(D57)
[ 3 ]
where sMt+1 = Et+1 rMt+1
is the skewness of the market return. I compare (D57) to (D50) and recover
the pricing kernel
Mt+1 =
( 2
)
( 3
)
1
ρ
κ
1
2
3
− rMt+1 + 2 R f rMt+1
− Et rMt+1
− 3 R2f rMt+1
− Et rMt+1
Rf ℘
℘
2℘
)
(1 − ρ) ( 2
(κ
−
2ρ)
− 2
σMt+1 − Et σ2Mt+1 −
(sMt+1 − Et sMt+1 ) .
℘ Rf
2℘3 R f
(D58)
Pricing Kernel with Co-skewness and Volatility Risk When Q=3
I derive the pricing kernel when the order in (D36) is Q = 3. The risk premium (D35) is
Et (Rkt+1 − R f ) =
( 2
)
1
ρ
R f Covt (rMt+1 , Rkt+1 ) − 2 R2f Covt rMt+1
, Rkt+1
℘
℘
( 2
)
( 3
)
(1 − ρ)
κ
+
Covt σMt+1 , Rkt+1 + 3 R3f Covt rMt+1
, Rkt+1
2
℘
2℘
( 4
)
(κ − 2ρ)
δ 4
+
Cov
(s
,
R
)
−
R
Cov
r
,
R
t
Mt+1
t
kt+1
kt+1
f
Mt+1
2℘3
3!℘4
(3κ − δ)
+
Covt (kMt+1 , Rkt+1 ) ,
3!℘4
(D59)
[ 4 ]
is the kurtosis of the market return. I compare (D59) to (D50) and recover the
where kMt+1 = Et+1 rMt+1
pricing kernel
Mt+1 =
( 2
)
( 3
)
1
1
ρ
κ
2
3
− rMt+1 + 2 R f rMt+1
− Et rMt+1
− 3 R2f rMt+1
− Et rMt+1
Rf ℘
℘
2℘
)
) (1 − ρ) ( 2
δ 3( 4
4
+
R f rMt+1 − Et rMt+1
σMt+1 − Et σ2Mt+1
− 2
4
3!℘
℘ Rf
(κ − 2ρ)
(δ − 3κ)
−
(sMt+1 − Et sMt+1 ) +
(kMt+1 − Et kMt+1 ) .
2℘3 R f
3!℘4 R f
17
(D60)
Appendix E: An Example of Log Pricing Kernel with Fundamental Factors
and Volatility Risk: Bansal and Yaron (2004)
I follow ? and consider in log form the SDF in the ? model
(
Mt+1
log
Mt
with θ ≡
1−γ
1− ψ1
)
= θ log β −
θ
gt+1 + (θ − 1) log (RMt+1 ) ,
ψ
(E1)
where γ is the coefficient of relative risk aversion, ψ is the elasticity of intertemporal substi-
tution, β is the subjective discount factor and gt+1 is the log consumption growth. ? specify the evolution
of (log) consumption growth gt+1 as
gt+1 = µ + xt + σt ηt+1 ,
(
)
2
σt+1
= σ2 + ν1 σt2 − σ2 + σξ ξt+1 ,
xt+1 = ρ xt + φe σt et+1 ,
ξt+1 , et+1 , ηt+1 ∼ i.i.d N (0, 1) .
(E2)
(E3)
In the model, xt is a persistent component of the expected consumption growth rate, and σt2 is the conditional variance of consumption with unconditional mean σ2 . The time-series dynamics (E2)–(E3), combined with the SDF (E1), produces a log SDF that can be expressed as a function of market variance.
Because the expressions are derived using standard techniques, I intend to be brief. The return on a
consumption claim can be approximated as
log RM,t+1 = κ0 + κ1 zt+1 + gt+1 − zt ,
with
(
)
κ0 = log 1 + ez −
ez
z,
(1 + ez )
κ1 =
(E4)
ez
,
(1 + ez )
where zt is the log price-consumption ratio. The approximate solution for the log price-consumption ratio
is
zt = A0 + A1 xt + A2 σt2 .
(E5)
To show that the log SDF is a function of the innovation in the market variance, I write the innovation in
the log market portfolio return as
(
)
log (RM,t+1 ) − Et log (RM,t+1 ) = κ1 A1 φe σt et+1 + κ1 A2 σξ ξt+1 + gt+1 .
18
(E6)
I derive the conditional variance of the market portfolio return
which implies
(
)
σ2M,t = 1 + κ21 A21 φ2e σt2 + κ21 A22 σ2ξ ,
(E7)
(
) (
)
σ2M,t+1 − Et σ2M,t+1 = 1 + κ21 A21 φ2e σξ ξt+1 .
(E8)
The variance risk premium is
(
Et σ2M,t+1
)
(
)
− EtQ σ2M,t+1
(
)
Mt+1 2
= −Covt log
, σM,t+1
Mt
(
)
= λm,ξ 1 + κ21 A21 φ2e σ2ξ , with
(E9)
λm,ξ = (1 − θ) κ1 A2 .
(E10)
The log pricing kernel (E1) simplifies to
(
)
θ
Mt+1
= θ log β + (θ − 1) (κ0 − zt ) + (θ − 1) κ1 zt+1 + (θ − 1) −
gt+1 .
log
Mt
ψ
(E11)
Under the time-series assumptions (E2)-(E3), I replace gt+1 , zt+1 and zt in equation (E11) and use (E8) to
simplify the log pricing kernel:
log
))
(
(
Mt+1
= ζt + D1 (gt+1 − Et (gt+1 )) + D2 (xt+1 − Et (xt+1 )) + D3 σ2M,t+1 − Et σ2M,t+1 ,
Mt
(E12)
where D1 , D2 , and D3 are
θ
ψ
(1 − θ) κ1 A2
),
1 + κ21 A21 φ2e
D1 = (θ − 1) − , D2 = − (1 − θ) κ1 A1 , D3 = − (
with
ζt
= θ log β + (θ − 1) (κ0 − zt ) + (θ − 1) κ1 A0 + (θ − 1) κ1 A1 ρ xt
(
)
(
(
))
θ
+ (θ − 1) −
(µ + xt ) + (θ − 1) κ1 A2 σ2 + ν1 σt2 − σ2 .
ψ
This was the final step.
19
(E13)
Appendix F: The Pricing Kernel When the Utility Function Depends on the
Market Return and Volatility Factors
I consider a representative agent who maximizes his expected utility over wealth. The parsimonious way to
allow the representative agent’s utility function to be a function of the market volatility is to assume that the
agent can only invest in the market return and the variance swap contract. A return variance swap has zero
net market value at entry (?). At maturity, the payoff to the long side of the swap is equal to the difference
between the realized variance over the life of the contract and a constant called the variance swap rate
V SMt+1 = σ2Mt+1 − EtQ σ2Mt+1 ,
(F1)
(
)
where σ2Mt+1 denotes the proxy for the realized variance between t and t + 1. EtQ σ2Mt+1 denotes the fixed
variance swap rate that is determined at time t. The representative agent chooses the portfolio weights ωM
and ωv to maximize his expected utility
max Et u [Wt+1 ] ,
(F2)
{ωM ,ωv }
where
Wt+1 = R f + ωM (RMt+1 − R f ) + ωv (V SMt+1 − R f ) .
(F3)
The first-order conditions are
( ′
)
( ′
)
Et u [Wt+1 ] (V SMt+1 − R f ) = 0 and Et u [Wt+1 ] (RMt+1 − R f ) = 0,
and the pricing kernel is
(F4)
′
u [Wt+1 ]
.
Mt+1 =
Et (u′ [Wt+1 ])
(F5)
′
The first-order Taylor expansion series of the marginal utility u (Wt+1 ) around (Et RMt+1 , Et V SMt+1 ) produces a pricing kernel
(
)
Mt+1 = G0 + G1 (RMt+1 − Et RMt+1 ) + G2 σ2Mt+1 − Et σ2Mt+1 ,
with
′
G0 =
′′
(F6)
′′
u [Et Wt+1 ]
u [Et Wt+1 ]
u [Et Wt+1 ]
, G1 = ωM
, and G2 = ωv
.
′
′
′
Et (u [Wt+1 ])
Et (u [Wt+1 ])
Et (u [Wt+1 ])
20
(F7)
′′
Since u [Et Wt+1 ] < 0, the price of the market factor is positive if ωM is positive, and the price of volatility
risk is negative if ωv is negative. Bondareko (2007) considers an investor who maximizes the expected
value of the constant (CRRA) utility function where the investor’s wealth is defined by (F3) and shows (in
Table 6, page 47) that ωm < 0 and ωv < 0 when the investor’s risk aversion takes the values 1, 2, 3, 5, 10,
20, and 50. Hence, G1 > 0 and G2 > 0.
Furthermore, in a one-period model, any high-order Taylor expansion series of the marginal utility
′
u [Wt+1 ] in (F5) will not produce a pricing kernel function of the market volatility, market skewness, and
market kurtosis as presented in Appendix D.
21
Table I: Preference Parameters and Implied Prices of Risk Using Industry Portfolio Returns (Robustness to Size, Book-to-Value, and
Momentum Factors):
Table I presents results of GMM tests of the Euler equation condition, E Mt+1 Rt+1 = 1 using the pricing kernel derived in Proposition 1 when the
investment horizon T − t = 2, augmented with ? size and book-to-market factors. I estimate the preference parameters by using the ? weighting
′
matrix ERt+1 Rt+1 . Column (1) presents the mean of the pricing kernel and Columns (2) and (3) present the risk aversion and skewness preference,
respectively. Column (4) presents the Hansen and Jagannathan distance measure with p-values for the test of model specification. Columns (5)–
(7) present the annualized price of market, co-skewness, and market volatility risk, using the estimated preference parameters. The set of returns
I use in my estimations are those of 30 industry-sorted portfolios augmented by the return on a one-month Treasury bill, covering the sample
periods 01/1986–12/2000, 01/1990–12/2000, 01/1996–12/2006, 01/1990–12/2000, and 01/1986–12/2006. For the market portfolio, I use the
value-weighted NYSE/AMEX/NASDAQ index, also known as the value-weighted index of the Center for Research in Security Prices (CRSP).
As my proxy for the volatility of the market return, I use the Chicago Board Options Exchange (CBOE)s VXO, the VIX implied volatilities, and
the Realized Volatility RV, respectively. In Panel A, I present the results when I use the VXO. Panel B presents the results when I use the VIX.
Panel C presents the results when I use the realized volatility RV.
Panel A: Market Volatility: VXO
Subperiod: 01/1986–12/2000
Coefficient
p-value
Subperiod: 01/1996–12/2006
Coefficient
p-value
Subperiod: 01/1986–12/2006
Coefficient
p-value
1
Rf
1
℘
ρ
HJ Dist
λM (%)
λSKD (%)
λVOL (%)
0.996
(0.000)
4.211
(0.032)
1.095
(0.002)
0.116
(0.162)
10.082
–2.072
–0.424
0.997
(0.000)
4.009
(0.169)
1.827
(0.182)
0.175
(0.002)
9.701
–3.185
–1.033
0.996
(0.000)
4.318
(0.023)
1.050
(0.001)
0.085
(0.081)
9.961
–1.976
–0.187
1
Rf
1
℘
ρ
HJ Dist
λM (%)
λSKD (%)
λVOL (%)
0.996
(0.000)
3.403
(0.268)
1.741
(0.270)
0.199
(0.037)
6.931
–1.689
–0.896
0.997
(0.000)
3.755
(0.206)
1.913
(0.264)
0.177
(0.003)
9.086
–2.926
–0.939
0.997
(0.000)
4.608
(0.061)
1.482
(0.027)
0.113
(0.006)
9.457
–2.667
–0.920
1
Rf
1
℘
ρ
HJ Dist
λM (%)
λSKD (%)
λVOL (%)
0.996
(0.000)
3.990
(0.201)
1.118
(0.000)
0.208
(0.016)
7.705
–1.376
–0.774
0.997
(0.000)
3.636
(0.286)
1.138
(0.000)
0.172
(0.003)
8.174
–1.461
–1.487
0.997
(0.000)
3.623
(0.195)
1.215
(0.001)
0.097
(0.076)
7.027
–1.241
–1.519
Panel B: Market Volatility: VIX
Subperiod: 01/1990–12/2000
Coefficient
p-value
Subperiod: 01/1996–12/2006
Coefficient
p-value
Subperiod: 01/1990–12/2006
Coefficient
p-value
Panel C: Market Volatility: RV
Subperiod: 01/1990–12/2000
Coefficient
p-value
Subperiod: 01/1996–12/2006
Coefficient
p-value
Subperiod: 01/1990–12/2006
Coefficient
p-value
22
Table II: Descriptive Statistics on Dow 30 stocks
This table presents summary statistics for the monthly returns on Dow 30 Stocks. Maximum, minimum, mean, standard deviation (Std), skewness,
and kurtosis are reported for each stock. The descriptive statistics are computed for the sample period from January 1990 to December 2006.
Dow Jones Stocks
Microsoft
Honeywell
AT&T Inc
Coca Cola
E.I. DuPont de Nemours
Exxon Mobil
General Electric
General Motors
International Business Machines
Altria (was Philip Morris)
United Technologies
Procter and Gamble
Caterpillar
Boeing
Pfizer
Johnson & Johnson
3M Corporation
Merck
Alcoa
Walt Disney Co.
Hewlett-Packard
McDonalds
JP Morgan Chase
Wal-Mart Stores
Intel Corp
Verizon Communications
Home Depot
American Int’l Group
CitiGroup
American Express IBM
(International Business Machines)
Minimum
Maximum
Mean
Std
Skewness
Kurtosis
–0.3435
–0.3840
–0.1876
–0.1910
–0.1699
–0.1165
–0.1765
–0.2403
–0.2619
–0.2656
–0.3202
–0.3570
–0.2146
–0.3457
–0.1707
–0.1601
–0.1578
–0.2577
–0.2387
–0.2678
–0.3199
–0.2567
–0.3468
–0.2080
–0.4449
–0.2099
–0.2059
–0.2310
–0.3401
–0.2933
0.4078
0.5105
0.2900
0.2228
0.2174
0.2322
0.1924
0.2766
0.3538
0.3427
0.2461
0.2509
0.4079
0.1949
0.2655
0.1881
0.2580
0.2276
0.5114
0.2415
0.3539
0.1826
0.3257
0.2643
0.3382
0.3901
0.3023
0.2387
0.2608
0.2031
0.0251
0.0141
0.0097
0.0115
0.0091
0.0127
0.0134
0.0072
0.0123
0.0164
0.0154
0.0135
0.0157
0.0120
0.0151
0.0142
0.0108
0.0114
0.0115
0.0100
0.0176
0.0114
0.0161
0.0136
0.0223
0.0076
0.0193
0.0129
0.0214
0.0139
0.1024
0.0901
0.0717
0.0660
0.0673
0.0465
0.0626
0.0951
0.0890
0.0830
0.0711
0.0631
0.0836
0.0788
0.0735
0.0629
0.0581
0.0773
0.0907
0.0775
0.1099
0.0697
0.1004
0.0730
0.1213
0.0721
0.0849
0.0665
0.0889
0.0754
0.3978
–0.0367
0.2064
–0.2103
0.1367
0.6183
0.1297
0.1984
0.3456
–0.2708
–0.6543
–0.7790
0.4065
–0.5865
0.1404
0.0878
0.4040
–0.0737
0.7795
–0.0686
0.0874
–0.2540
–0.1720
0.1671
–0.3005
0.8210
0.2564
0.0223
–0.1032
–0.8057
4.6155
9.3331
4.1836
4.0537
2.8528
5.4267
3.5668
3.1706
4.3209
5.0895
6.1925
8.7933
4.7004
4.4729
2.9849
3.1813
4.8393
3.3652
6.9529
3.9232
3.6069
3.5148
4.8161
3.4579
3.6254
6.7551
3.5188
4.2054
4.3008
4.6244
23
Table III: Preference Parameters and Implied Prices of Risk Using the Dow 30 Returns:
Table III presents results of GMM tests of the Euler equation condition, E Mt+1 Rt+1 = 1 using the pricing kernel derived in Proposition 1 when the
investment horizon T − t = 2 is augmented with ? size and book-to-market factors. I estimate the preference parameters by using the ? weighting
′
matrix ERt+1 Rt+1 . Column (1) presents the mean of the pricing kernel and Columns (2) and (3) present the risk aversion and skewness preference
respectively. Column (4) presents the Hansen and Jagannathan distance measure with p-values for the test of model specification. Columns (5)–(7)
present the annualized price of market, co-skewness, and market volatility risk, using the estimated preference parameters. The set of returns I
use in my estimations are those of 30 Dow Jones returns augmented by the return on a one-month Treasury bill, covering the sample period
01/1990–12/2006. For the market portfolio, I use the value-weighted NYSE/AMEX/NASDAQ index, also known as the value-weighted index of
the Center for Research in Security Prices (CRSP). As my proxy for the volatility of the market return, I use the Chicago Board Options Exchange
(CBOE)s VXO, the VIX implied volatilities, and the Realized Volatility RV, respectively. In Panel A, I present the results when I use the VXO,
the VIX, and the realized volatility RV. Panel B presents the results when I control for the Fama and French and the momentum factors.
1
Rf
1
℘
ρ
HJ Dist
λM (%)
λSKD (%)
λVOL (%)
Coefficient
p-value
VIX
0.997
(0.000)
3.921
(0.039)
1.603
(0.045)
0.121
(0.003)
8.047
–2.088
–0.897
Coefficient
p-value
RV
0.997
(0.000)
3.782
(0.053)
1.705
(0.082)
0.120
(0.003)
7.761
–2.066
–0.906
Coefficient
p-value
0.997
(0.000)
5.750
(0.008)
1.003
(0.000)
0.101
(0.319)
11.154
–2.582
–0.148
Coefficient
p-value
VIX
0.997
(0.000)
4.771
(0.046)
1.394
(0.008)
0.115
(0.004)
9.791
–2.689
–0.869
Coefficient
p-value
RV
0.997
(0.000)
4.608
(0.061)
1.482
(0.028)
0.113
(0.006)
9.457
–2.667
–0.920
Coefficient
p-value
0.9967
(0.000)
5.4074
(0.065)
1.0277
(0.000)
0.0913
(0.410)
10.4890
–2.3396
–1.2963
Panel A
VXO
Panel B
VXO
24
Table IV: Preference Parameters and Implied Prices of Risk When the Pricing Kernel Depends on Both Stochastic Volatility and Stochastic
Skewness:
Table IV presents results of GMM tests of the Euler equation condition, E Mt+1 Rt+1 = 1 using the pricing kernel derived in Proposition 2 when
′
the investment horizon T − t = 2. I estimate the preference parameters by using the ? weighting matrix ERt+1 Rt+1 . Column (1) presents the
mean of the pricing kernel and Columns (2), (3), and (4) present the risk aversion, skewness preference, and kurtosis preference respectively.
Column (5) presents the Hansen and Jagannathan distance measure with p-values for the test of model specification. Columns (6)–(9) present
the annualized price of market, co-skewness, market co-kurtosis, market skewness risk, and market volatility risk, using the estimated preference
parameters. In Panel A, the set of returns I use in my estimations are those of 30 industry returns augmented by the return on a one-month Treasury
bill, covering the sample period 01/1990–12/2006. For the market portfolio, I use the value-weighted NYSE/AMEX/NASDAQ index, also known
as the value-weighted index of the Center for Research in Security Prices (CRSP). As my proxy for the volatility of the market return, I use the
Chicago Board Options Exchange (CBOE)s VIX implied volatilities. As my proxy for the market skewness, I use the skewness measure implied
from options prices. I use the ? formula to compute the skewness measure. In Panel B, I use the Dow 30 returns as the test asset.
1
Rf
1
℘
ρ
κ
HJ Dist
λM (%)
λSKD (%)
λKUR (%)
λVOL (%)
λSM (%)
0.997
2.958
1.264
1.913
0.1138
6.174
–3.781
2.152
–0.961
–2.130
(0.000)
(0.107)
(0.010)
(0.061)
(0.077)
0.997
3.390
1.197
1.848
0.109
6.687
–3.707
2.875
–1.099
–2.850
(0.000)
(0.010)
(0.000)
(0.000)
(0.102)
Panel A: 30 Industry Returns
Coefficient
p-value
Panel B: Dow 30 returns
Coefficient
p-value
25
Risk Aversion
Skewness Preference
Skewness Preference
Risk Aversion
6
2.4
RA
SP
2.2
5.5
2
Skewness Preference
Risk Aversion
5
4.5
4
3.5
1.8
1.6
1.4
1.2
1
0.8
3
0.6
2.5
1996
1998
2000
2002
2004
0.4
1996
2006
1998
2000
years
2002
2004
2006
years
Price of Volatility Risk
Price of coskewness Risk
Price of Variance Risk
Price of Coskewness Risk
1.5
0
−0.5
1
−1
Price (%)
Price (%)
0.5
0
−1.5
−2
−0.5
−2.5
−1
−1.5
1996
−3
1998
2000
2002
2004
−3.5
1996
2006
years
1998
2000
2002
2004
2006
years
Figure I. Preference Parameters and Prices of Risk
Figure I depicts the risk aversion, skewness preferences, price of the volatility risk, and the price of co-skewness risk when I estimate the pricing
kernel with constant preference parameters for different sample periods [1986 + j, 1986 + 10 + j] when j = 0, ..., 10. I report the preference
parameters for different years j. I estimate the preference parameters of the pricing kernel via GMM utilizing the Euler equation condition
Et Mt+1 Rkt+1 = 1, where Mt+1 represents the pricing kernel. I estimate the parameters by using the ? weighting matrix. The sets of returns I use
in my estimations are those of 30 industry-sorted portfolios covering the period January 1986 through December 2006, augmented by the return
on a 30-day Treasury bill. I use the Chicago Board Options Exchange (CBOE)’s VOX as my proxy for the market volatility.
26
01/1996–12/2006
01/1986–12/2000
Pricing Kernel (PK)
Pricing Kernel (PK)
2.5
3
2.5
2
PK
PK
2
1.5
1.5
1
1
0.5
0.2
0.5
0.2
0.1
0.02
0.02
−0.02
0
−0.1
−0.01
−0.2
0.01
0
0
−0.1
rm
0.1
0.01
0
−0.2
rm
∆σ2m
−0.01
−0.02
∆σ2m
01/1986–12/2006
Pricing Kernel (PK)
2.5
PK
2
1.5
1
0.5
0.2
0.1
0.02
0.01
0
0
−0.1
r
m
−0.01
−0.2
−0.02
∆σ2m
Figure II. Estimated Pricing Kernels
Figure II depicts point estimates of the pricing kernels estimated with constant preference parameters. The support for the graphs is the range
of the return on the value-weighted index and the implied volatility difference. I estimate the preference parameters of the pricing kernel via
GMM utilizing the Euler equation condition Et Mt+1 Rkt+1 = 1, where Mt+1 represents the pricing kernel. I estimate the parameters by using the ?
weighting matrix. The sets of returns I use in my estimations are those of 30 industry-sorted portfolios covering the period January 1986 through
December 2006, augmented by the return on a 30-day Treasury bill. I use the Chicago Board Options Exchange (CBOE)’s VXO as my proxy for
the market volatility.
27
01/1996–12/2006
01/1986–12/2000
Projected Pricing Kernel
3.5
2.5
3
2
2.5
PK
PK
Projected Pricing Kernel
3
1.5
2
1
1.5
0.5
1
0
−0.1
−0.08 −0.06 −0.04 −0.02
0
0.02
0.04
0.06
0.5
−0.1
0.08
−0.05
0
r
r
m
0.05
0.1
m
01/1986–12/2006
Projected Pricing Kernel
3.5
3
PK
2.5
2
1.5
1
0.5
−0.1
−0.05
0
r
0.05
0.1
m
Figure III. Projected Pricing Kernels
Figure III depicts the projection of the estimated pricing kernel estimated with constant preference parameters (see Figure 2) on a polynomial
j
function of the market return, P M t+1 = ∑5j=0 b j rMt+1 . The support for the graphs is the observed range of the return on the value-weighted index.
28
1988−1995: 1/(℘)=4.6, ρ=1.52, 1/Rf=0.98
2
PK Model
PK Data
Upper CI
Lower CI
1.8
1.6
Pricing Kernel
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.94
0.96
0.98
1
Returns
1.02
1.04
1.06
Figure IV. Model Implied Projected Pricing Kernel and Observed Pricing Kernel: 1988–1995
Figure IV depicts the model implied projected pricing kernel and the observed pricing kernel. The support for the graphs is the observed range
1
of the return on the S&P 500 index. Each year, we find the set of parameters (1/R f , ℘
, ρ) that generates a projected pricing kernel close to the
observed pricing kernel in terms of the distance measure (23).
29
1988−1995: 1/(℘)=2.6, ρ=1.73, 1/Rf=0.95
4
ARA Model
ARA Data
Upper CI
Lower CI
2
ARA
0
−2
−4
−6
−8
0.96
0.98
1
Returns
1.02
1.04
1.06
Figure V. Model Implied “projected” Absolute Risk Aversion and Observed Absolute Risk Aversion:
1988–1995
Figure V depicts the model implied “projected” absolute risk aversion and the observed absolute risk aversion. The support for the graphs is the
1
observed range of the return on the S&P 500 index. Each year, we find the set of parameters (1/R f , ℘
, ρ) that generates a “projected” absolute
risk aversion close to the observed absolute risk aversion in terms of the distance measure (27).
30