Appendix to “Long-Run Risk, the Wealth-Consumption
Ratio, and the Temporal Pricing of Risk”
By Ralph S.J. Koijen, Hanno Lustig,
Stijn Van Nieuwerburgh and Adrien Verdelhan
∗
In this appendix, we first derive four risk premia: the expected excess
returns on a consumption claim, on equity and on real and nominal
bonds. We then obtain the Alvarez and Jermann (2005) decomposition
of the SDF in the long-run risk model. We present the parameter values
used in our calibration and our simulation results. Finally, we report
robustness checks and empirical variance ratios.
∗ Koijen: The University of Chicago Booth School of Business 5807 South Woodlawn Avenue, Chicago,
IL 60637; [email protected]. Lustig: Anderson School of Management, University of California at Los Angeles, Box 951477, Los Angeles, CA 90095; [email protected]. Van Nieuwerburgh: Department of Finance, Stern School of Business, New York University, 44 W. 4th Street, New
York, NY 10012; [email protected]. Verdelhan: Department of Finance, MIT Sloan School of Management and Bank of France, E52-436, 50 Memorial Drive, Cambridge, MA 02142; [email protected]. We
thank Rustom Irani for excellent research assistance and Ravi Bansal, Jaroslav Borovicka, Mike Chernov,
Itamar Drechsler, John Heaton, Lars Hansen, Jose Scheinkman, Ivan Shaliastovich and George Tauchen
for comments.
1
2
THE AMERICAN ECONOMIC REVIEW
I.
MAY 2010
Wealth-Consumption Ratio and Consumption Risk Premium
We start from the aggregate budget constraint:
c
Wt+1 = Rt+1
(Wt − Ct ).
(1)
The beginning-of-period (or cum-dividend) total wealth Wt that is not spent on aggregate
c
consumption Ct earns a gross return Rt+1
and leads to beginning-of-next-period total
wealth Wt+1 . The return on a claim to aggregate consumption, the total wealth return,
can be written as
Wt+1
Ct+1 W Ct+1
c
=
Rt+1
=
.
Wt − Ct
Ct W Ct − 1
We use the Campbell (1991) approximation of the log total wealth return rtc = log(Rtc )
around the long-run average log wealth-consumption ratio µwc ≡ E[wt − ct ]:
c
rt+1
= κc0 + ∆ct+1 + wct+1 − κc1 wct ,
where the linearization constants κc0 and κc1 are non-linear functions of the unconditional
mean log wealth-consumption ratio µwc :
κc1 =
eµwc
eµwc
> 1 and κc0 = − log (eµwc − 1) + µwc
µwc .
−1
e
−1
eµwc
Throughout the paper, we use lower letters to denote logs.
The Euler equation for any asset i with lognormal return Ri implies:
(2)
i 1
i 1
i
0 = Et [sdft+1 ] + Et rt+1
+ Vart [sdft+1 ] + Vart rt+1
+ Covt sdft+1 , rt+1
2
2
2
We conjecture that the wealth-consumption ratio is linear in the state variables xt , σgt
2
and σxt
:
2
2
wct = µwc + Wx xt + Wgs σgt
− σg2 + Wxs σxt
− σx2
We first compute the different components of equation 2:
c
rt+1
=
c Et rt+1
=
c c
rt+1 − Et rt+1 =
c Vt rt+1
=
r0c
=
2
2
r0c + [1 + Wx (ρ − κc1 )] xt + Wgs (νg − κc1 ) σgt
− σg2 + Wxs (νx − κc1 ) σxt
− σx2
+ σgt ηt+1 + Wx σxt et+1 + Wgs σgw wg,t+1 + Wxs σxw wx,t+1
2
2
− σx2
r0 + [1 + Wx (ρ − κc1 )] xt + Wgs (νg − κc1 ) σgt
− σg2 + Wxs (νx − κc1 ) σxt
σgt ηt+1 + Wx σxt et+1 + Wgs σgw wg,t+1 + Wxs σxw wx,t+1
2
2
2 2
2 2
σgt
+ Wx2 σxt
+ Wgs
σgw + Wxs
σxw
κc0 + µg + (1 − κc1 ) µwc
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
3
Epstein and Zin (1989) show that the log real stochastic discount factor is
sdft+1
=
=
sdft+1 − Et [sdft+1 ] =
Et [sdft+1 ] =
Vt [sdft+1 ] =
µs
c
Covt rt+1
, sdft+1
=
θ
c
θ log δ − ∆ct+1 + (θ − 1) rt+1
ψ
θ
µs + − + (θ − 1) [1 + Wx (ρ − κc1 )] xt
ψ
2
2
+ {Wgs (νg − κc1 ) (θ − 1)} σgt
− σg2 + {Wxs (νx − κc1 ) (θ − 1)} σxt
− σx2
1
+ θ 1−
− 1 σgt ηt+1 + (θ − 1) {Wx σxt et+1 + Wgs σgw wg,t+1 + Wxs σxw wx,t+1 }
ψ
1
− 1 σgt ηt+1 + (θ − 1) {Wx σxt et+1 + Wgs σgw wg,t+1 + Wxs σxw wx,t+1 }
θ 1−
ψ
θ
µs + − + (θ − 1) [1 + Wx (ρ − κc1 )] xt
ψ
2
2
+ {Wgs (νg − κc1 ) (θ − 1)} σgt
− σg2 + {Wxs (νx − κc1 ) (θ − 1)} σxt
− σx2
2
1
2
2
2
2 2
2 2
− 1 σgt
+ (θ − 1) Wx2 σxt
+ Wgs
σgw + Wxs
σxw
θ 1−
ψ
θ
θ log δ − µg + (θ − 1) r0c
ψ
c
c = Et (rt+1
− Et rt+1
)(sdft+1 − Et [sdft+1 ])
1
2
2
2
2
2
2
=
θ 1−
− 1 σgt
+ Wx2 (θ − 1) σxt
+ Wgs
(θ − 1) σgw
+ Wxs
(θ − 1) σxw
ψ
Plugging these different components into equation (2) evaluated at i = c yields:
(
)
2
θ2
1
c
2
2 2
2 2
2 2
(3)
0 = r0 + µs +
1−
σg + Wx σx + Wgs σgw + Wxs σxw
2
ψ
1
c
(4)
+θ − + [1 + Wx (ρ − κ1 )] xt
ψ
(
2 )
θ
1
c
2
(5)
+
2Wgs (νg − κ1 ) + θ 1 −
σgt
− σg2
2
ψ
(6)
+
2
θ
2Wxs (νx − κc1 ) + θWx2 σxt
− σx2
2
4
THE AMERICAN ECONOMIC REVIEW
MAY 2010
Then setting all coefficients equal to zero we obtain:
(4)
=⇒ Wx =
(5)
=⇒ Wgs =
(6)
=⇒ Wxs
1−
1
ψ
κc1 − ρ
θ 1−
1
ψ
2
2 (κc1 − νg )
θ
=
2 (κc1 − νx )
1−
1
ψ
κc1 − ρ
!2
If the IES exceeds 1, then Wx > 0, Wgs < 0, and Wxs < 0.
Plugging these coefficients back into equation (3) implicitly defines a nonlinear equation
in one unknown (µwc ), which can be solved for numerically, characterizing the average
wealth-consumption ratio.
According to (2), the risk premium (expected excess real return corrected for a Jensen
term) on the consumption claim is given by1 :
c,e c
, sdft+1
Et rt+1
= −Covt rt+1
1
2
2
2
2
2
2
=
1−θ 1−
σgt
+ Wx2 (1 − θ) σxt
+ Wgs
(1 − θ) σgw
+ Wxs
(1 − θ) σxw
ψ
=
2
2
2
2
λη σgt
+ Wx λe σxt
+ Wgs λgw σgw
+ Wxs λxw σxw
with the market price of risk vector Λ = [λη , λe , λgw , λxw ] given by:
λη
=
1
− θ 1−
−1 = γ >0
ψ
λe
=
(1 − θ) Wx =
λgw
=
(1 − θ) Wgs = −
λxw
=
γ−
1
ψ
κc1 − ρ
(1 − θ) Wxs = −
(γ − 1) (γ −
1
ψ)
2 (κc1 − νg )
(γ − 1) (γ −
1
ψ)
2 (κc1 − νx ) (κc1 − ρ)2
If the IES is sufficiently large (γ > 1/ψ), then λe > 0, λgw < 0, and λxw < 0.
II.
Equity Risk Premium
We log-linearize return on portfolio: rt+1 = κ0 + ∆dt+1 + pdt+1 − κ1 pdt , and conjecture that the price-dividend ratio
is linear in the state variables: pdt = µpd + Dx xt +
2
2
Dgs σgt
− σg2 + Dxs σxt
− σx2
As we did for the return on the consumption claim, we compute innovations in the
1 Recall
that the log riskfree rate is yt (1) = −Et [sdft+1 ] − 12 Vart [sdft+1 ].
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
5
dividend claim return, and its conditional mean and variance:
rt+1
=
rt+1 − Et [rt+1 ] =
Et [rt+1 ] =
Vart [rt+1 ] =
r0
=
2
2
r0 + {φx + Dx (ρ − κ1 )} xt + Dgs (νg − κ1 ) σgt
− σg2 + Dxs (νx − κ1 ) σxt
− σx2
+ϕd σgt ηd,t+1 + Dx σxt et+1 + Dgs σgw wg,t+1 + Dxs σxw wx,t+1
ϕd σgt ηd,t+1 + Dx σxt et+1 + Dgs σgw wg,t+1 + Dxs σxw wx,t+1
2
r0 + {φx + Dx (ρ − κ1 )} xt + Dgs (νg − κ1 ) σgt
− σg2
2
+Dxs (νx − κ1 ) σxt
− σx2
2
2
2 2
2 2
ϕ2d σgt
+ Dx2 σxt
+ Dgs
σgw + Dxs
σxw
κ0 + µpd (1 − κ1 ) + µd
2
2
2
2
Covt [rt+1 , sdft+1 ] = (θ − 1) Wgs Dgs σgw
+ Wxs Dxs σxw
− γϕd τgd σgt
+ (θ − 1) Wx Dx σxt
Plug these different components into equation (2):
0 =
µs + r0 +
1 2
1
1
2
γ − 2γϕd τdg + ϕ2d σg2 + [Wx (θ − 1) + Dx ]2 σx2 + [Wgs (θ − 1) + Dgs ]2 σgw
2
2
2
1
2 2
+ (7)
[Wxs (θ − 1) + Dxs ] σxw
2
1
+ (8)
− + [φx + Dx (ρ − κ1 )] xt
ψ
1 2
2
+ (9) γ − 2γϕd τdg + ϕ2d + Wgs (κc1 − νg ) (1 − θ) + Dgs (νg − κ1 ) σgt
− σg2
2
1
2
2
+ (10)[Wx (θ − 1) + Dx ] + Wxs (κc1 − νx ) (1 − θ) + Dxs (νx − κ1 ) σxt
− σx2
2
Then setting all coefficients equal to zero we get:
(8)
(9)
=⇒ Dx =
=⇒ Dgs =
φx −
κ1 − ρ
2
1
2
2 γ − 2γϕd τdg + ϕd −
κ1 − νg
1
2
(10)
=⇒ Dxs =
1
ψ
hφ
1
x− ψ
κ1 −ρ
−
1 i2
γ− ψ
c
κ1 −ρ
−
1
2
γ−
1
ψ
(γ − 1)
1
1 (γ−1)(γ− ψ )
c −ρ 2
2
κ
( 1 )
κ1 − νx
Plugging these into (7) implicitly defines a nonlinear equation in one unknown (i.e., µpd ),
which can be solved for numerically, characterizing the mean price-dividend ratio.
The D coefficients are the betas of the equity market portfolio with respect to the four
fundamental consumption growth shocks.
6
THE AMERICAN ECONOMIC REVIEW
MAY 2010
The equity risk premium is equal to:
e Et rt+1
= −Covt [rt+1 , sdft+1 ]
G0
Ggs
Gxs
2
2
2
2
= (ϕd τgd ) λη σgt
+ Dx λe σxt
+ Dgs λgw σgw
+ Dxs λxw σxw
2
2
= G0 + Ggs σg2 + Gxs σx2 + Ggs σgt
− σg2 + Gxs σxt
− σx2
2
2
= Dgs λgw σgw
+ Dxs λxw σxw
= ϕd τgd γ
= Dx λe
III.
Real Bond Returns and Risk Premium
We start off the expression for the real stochastic discount factor derived in the first
sub-section above. Let define the following three parameters: sx ≡ − ψ1 , sgs ≡ − 12 (γ −
1
1)(γ − ψ1 ), and sxs ≡ − 21 (γ − 1)(γ − ψ1 ) (κc −ρ)
2 . Using notation defined above and in the
1
previous sub-sections, the real stochastic discount factor is:
2
2
sdft+1 = µs + sx xt + sgs σgt
− σg2 + sxs σxt
− σx2
−λη σgt ηt+1 − λe σxt et+1 − λgw σgw wg,t+1 − λxw σxw wx,t+1
Let pbt (n) = log Ptb (n) be the log price and ytb (n) = − n1 pbt (n) the yield of an n-period
real bond.
We conjecture that the log prices of real
linear in the state variables: pt (n) =
bonds are
2
2
−B0 (n) − Bx (n)xt − Bgs (n) σgt
− σg2 − Bxs σxt
− σx2
The coefficients are initialized at zero and satisfy the following recursions:
B0 (n)
Bx (n)
Bgs (n)
Bxs (n)
1
2 2
= B0 (n − 1) − µs − [λgw + Bgs (n − 1)] σgw
2
o
1n
2 2
−
[λxw + Bxs (n − 1)] σxw
+ λ2η σg2
2
1
2
− [λe + Bx (n − 1)] σx2
2
1
= ρBx (n − 1) +
ψ
1
1
1
= νg Bgs (n − 1) + (γ − 1)(γ − ) − γ 2
2
ψ
2
"
#2
1
1
(γ − ψ )
1
1 γ−ψ
= νx Bxs (n − 1) + (γ − 1) c
−
+ Bx (n − 1) .
2
(κ1 − ρ)2
2 κc1 − ρ
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
7
These recursions imply the following limit values:
Bx (∞)
=
Bgs (∞)
=
Bxs (∞)
=
1
ψ(1 − ρ)
1
2 (γ − 1)(γ −
1
ψ)
− 12 γ 2
1 − νg
1
2 (γ
(γ− 1 )
ψ
− 1) (κc −ρ)
2 −
1
1
2
h γ− 1
ψ
κc1 −ρ
1 − νx
i2
+ Bx (∞)
.
We define B(∞) ≡ [Bx (∞), Bgs (∞), Bxs (∞)]′ .
The real bond risk premium on monthly holding period returns is equal to:
b
rt+1
(n) ≡
b
b
rt+1 (n) − Et rt+1 (n) =
h
i
b,e
Et rt+1
(n)
=
=
F0 (n) =
Fgs (n) =
Fxs (n) =
b
nytb (n) − (n − 1)yt+1
(n − 1)
−Bx (n − 1)σxt et+1 − Bgs (n − 1)σgw wg,t+1
−Bxs (n − 1)σxw wx,t+1
b
−Covt rt+1
, sdft+1
2
2
F0 (n) + Fgs (n)σg2 + Fxs (n)σx2 + Fgs (n) σgt
− σg2 + Fxs (n) σxt
− σx2
2
2
−Bgs (n − 1)λgw σgw
− Bxs (n − 1)λxw σxw
,
0,
−Bx (n − 1)λe .
We now define some vectors and matrices to present results in a more compact way.
2
2
Let the vector Xt summarize all real state variables: Xt ≡ [xt , σgt
− σg2 , σxt
− σx2 ]′ .
Let εt+1 denote the corresponding gaussian, i.i.d shocks: εt+1 ≡ [et+1 , wg,t+1 , wx,t+1 ]′ .
2
2
2
We define Σt ≡ diag[σxt
, σgw
, σxw
]. The law of motion of the state vector Xt is Xt+1 =
1
ΓXt + Σt2 εt+1 , where Γ is a 3 by 3 diagonal matrix with ρ, ϕzg , and ϕzx on the diagonal.
Let B(n) denote all the n-period real bond parameters: B(n) ≡ [Bx (n), Bgs (n), Bxs (n)]′ .
Using this notation, we can rewrite the real bond risk premium as:
h
i
b,e
Et rt+1
(n) = −B(n − 1)′ Σt Λ.
IV.
Nominal Bond Returns and Risk Premium
We start off the expression for the real stochastic discount factor derived above. We use
a $ superscript to denote nominal variables. The nominal stochastic discount factor is
then:
$
sdft+1
≡
=
sdft+1 − πt+1
2
2
− σx2 − (π̄t − µπ )
µs − µπ + sx xt + sgs σgt
− σg2 + sxs σxt
− (λη + ϕπg ) σgt ηt+1 − (λe + ϕπx ) σxt et+1 − λgw σgw wg,t+1 − λxw σxw wx,t+1 − σπ ξt+1
Let p$t (n) = log Pt$ (n) be the log price and yt$ (n) = − n1 p$t (n) the yield of an n-period
8
THE AMERICAN ECONOMIC REVIEW
MAY 2010
nominal bond.
We conjecture that the log prices of nominal bonds are linear in the state variables:
$
2
$
2
p$t (n) = −B0$ (n) − Bx$ (n)xt − Bgs
(n) σgt
− σg2 − Bxs
σxt
− σx2 − Bπ$ (n) (π̄t − µπ )
The coefficients are initialized at zero and satisfy the following recursions:
i2 h
i2
1 h
$
$
$
$
2
σπ + Bπ (n − 1)σz + λgw + Bgs (n − 1) σgw
B0 (n) = B0 (n − 1) − µs + µπ −
2
h
i2
h
i2 1
$
2
$
−
λxw + Bxs (n − 1) σxw + ϕπg + λη + ϕzg Bπ (n − 1) σg2
2
i2
1h
− ϕπx + λe + Bx$ (n − 1) + ϕzx Bπ$ (n − 1) σx2
2
Bx$ (n) = ρBx$ (n − 1) + αx Bπ$ (n − 1) − sx
i2
1h
$
$
Bgs
(n) = νg Bgs
(n − 1) − sgs −
λη + ϕπg + ϕzg Bπ$ (n − 1)
2
i2
1h
$
$
λe + ϕπx + Bx$ (n − 1) + ϕzx Bπ$ (n − 1)
Bxs
(n) = νx Bxs
(n − 1) − sxs −
2
$
$
Bπ (n) = απ Bπ (n − 1) + 1.
These recursions imply the following limit values:
Bx$ (∞) =
$
Bgs
(∞) =
$
Bxs
(∞) =
Bπ$ (∞) =
αx Bπ$ (∞) − sx
1−ρ
2
−sgs − 12 λη + ϕπg + ϕzg Bπ$ (∞)
1 − νg
2
1
−sxs − 2 λe + ϕπx + Bx$ (∞) + ϕzx Bπ$ (∞)
1 − νx
1
.
1 − απ
$
$
We define B $ (∞) ≡ [Bx$ (∞), Bgs
(∞), Bxs
(∞), Bπ$ (∞)]′ .
The nominal bond risk premium on monthly holding period returns is equal to:
b,$
rt+1
(n) ≡
h
i
b,$
b,$
rt+1
(n) − Et rt+1
(n) =
h
i
b,$,e
Et rt+1
(n) =
=
F0$ (n) =
$
Fgs
(n) =
$
Fxs
(n) =
$
nyt$ (n) − (n − 1)yt+1
(n − 1)
$
− Bx$ (n − 1) + Bπ$ (n − 1)ϕzx σxt et+1 − Bgs
(n − 1)σgw wg,t+1
$
−Bxs
(n − 1)σxw wx,t+1 − Bπ$ (n − 1) (ϕzg σgt ηt+1 + σz ξt+1 )
h
i
b,$
$
−Covt rt+1
, sdft+1
i
h
2
$
2
$
$
$
− σx2
− σg2 + Fxs
(n) σxt
(n)σg2 + Fxs
(n)σx2 + Fgs
(n) σgt
F0$ (n) + Fgs
o
n
$
2
$
2
− λgw Bgs
(n − 1)σgw
+ λxw Bxs
(n − 1)σxw
+ σπ σz Bπ$ (n − 1)
− (λη + ϕπg ) ϕzg Bπ$ (n − 1)
− (λe + ϕπx ) Bx$ (n − 1) + Bπ$ (n − 1)ϕzx
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
9
Define the following vector and matrix objects:
b$
b$
Λ
B (n)
bt
Σ
εbt+1
≡ [λη + ϕπg , λe + ϕπx , λgw , λxw , σπ ],
$
$
(n), Bπ$ (n)σz ],
≡ [Bπ$ (n)ϕzg , Bx$ (n) + Bπ$ (n)ϕzx , Bgs
(n), Bxs
2
2
2
2
≡ diag[σgt
, σxt
, σgw
, σxw
, 1],
≡ [ηt+1 , et+1 , wg,t+1 , wx,t+1 , ξt+1 ]
Then we can write the nominal bond risk premium compactly as:
h
i
b,$,e
b $′ (n − 1)Σ
b tΛ
b $.
Et rt+1
(n) = −B
V.
Decomposition of the Real SDF
The following proposition shows how to decompose the SDF of the long-run risk model
into a martingale component and the dominant pricing component.
Proposition 1. The stochastic discount factor of the long-run risk model can be decomposed into a martingale component and the dominant pricing component:
T
Mt+1
MtT
P
Mt+1
MtP
=
=
1
′
′
β exp −B∞
(I − Γ) Xt + B∞
Σt2 εt+1 ,
1
′
′
β −1 exp µs + [S ′ + B∞
(I − Γ)] Xt − (Λ′ + B∞
) Σt2 εt+1 − λη σgt ηt+1 .
To show this, we start from the definition of the dominant pricing component of the
pricing kernel:
β t+n
MtT = lim b
,
n→∞ P (n)
t
Recall that log real bond prices are affine in the state vector:
2
2
pbt (n) = −B0 (n) − Bx (n)xt − Bgs (n) σgt
− σg2 − Bxs σxt
− σx2
= −B0 (n) − B(n)′ Xt .
We can then write the dominant pricing component of the SDF as:
MtT
=
lim β t+n exp (B0 (n) + B(n)′ Xt ) .
n→∞
The constant β is chosen in order to satisfy Assumption 1 in Alvarez and Jermann
(2005):
P b (n)
0 < lim t n < ∞.
n→∞ β
Recall that B0 (n) is defined recursively:
B0 (n)
=
−
o
1n
2 2
B0 (n − 1) − µs −
[λgw + Bgs (n − 1)] σgw
2
o
1n
2 2
2
[λxw + Bxs (n − 1)] σxw
+ λ2η σg2 + [λe + Bx (n − 1)] σx2
2
10
THE AMERICAN ECONOMIC REVIEW
MAY 2010
Because of the affine term structure of the model and the stationarity of the state
vector X, the limit limn→∞ B(n) = B(∞) is finite. Taking limits on both sides of the
equation above leads to:
o
1n
2 2
lim B0 (n) − B0 (n − 1) = −µs −
[λgw + Bgs (∞)] σgw
n→∞
2
o
1n
2 2
2
−
[λxw + Bxs (∞)] σxw
+ λ2η σg2 + [λe + Bx (∞)] σx2
2
The limit of B0 (n) − B0 (n − 1) is finite, so that B0 (n) grows at a linear rate in the
limit. We choose the constant β to offset the growth in B0 (n) as n becomes very large.
Setting
o
1n
2 2
2 2
2
β = exp µs +
[λgw + Bgs (∞)] σgw
+ [λxw + Bxs (∞)] σxw
+ λ2η σg2 + [λe + Bx (∞)] σx2
2
guarantees that Assumption 1 in Alvarez and Jermann (2005) is satisfied.
We can now write the dominant pricing component of the SDF as:
T
1
Mt+1
′
′
2
=
β
exp
−B
(I
−
Γ)
X
+
B
Σ
ε
,
t
t+1
∞
∞
t
MtT
To derive the martingale component of the SDF, let us go back to the SDF itself. Let
S and Λ denote the parameters of the real SDF: S ≡ [sx , sgs , sxs ]′ , Λ ≡ [λe , λgw , λxw ]′ .
Then the real SDF is:
1
Mt+1
SDFt+1 =
= exp µs + S ′ Xt − Λ′ Σt2 εt+1 − λη σgt ηt+1 .
Mt
As a result, the martingale component of the SDF is:
P
Mt+1
MtP
=
=
1.
T −1
Mt+1
MT
t
1
′
′
β −1 exp µs + [S ′ + B∞
(I − Γ)] Xt − (Λ′ + B∞
) Σt2 εt+1 − λη σgt ηt+1 .
Mt+1
Mt
P
We need to verify that the martingale component is a martingale, i.e that Et [Mt+1
/MtP ] =
To do this, recall that the bond parameters evolve as:
Bx (n)
Bgs (n)
Bxs (n)
= ρBx (n − 1) − sx
1
= νg Bgs (n − 1) − sgs − λ2η
2
1
= νx Bxs (n − 1) − sxs − [λe + Bx (n − 1)]2 .
2
Taking limits as n → ∞ leads to:
1
1
2
B(∞)′ (I − Γ) = −S ′ + [0, − λ2η , − [λe + Bx (∞)] ]′ .
2
2
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
11
To check the martingale condition, plug the definition of β in the following expression:
Et
P
Mt+1
MtP
= β
−1
1 ′
1 2 2
′
′
′
exp µs + [S + B∞ (I − Γ)] Xt + (Λ + B∞ ) Σt (Λ + B∞ ) + λη σgt .
2
2
The term in front of Xt is equal to [0, − 12 λ2η , −
1
2
2
2
[λe + Bx (∞)] ]′ . Terms in σgt
and
2
σxt
cancel out. We next plug in the expression for β and check that Et [
P
Mt+1
]
MtP
= 1.
We now turn to the conditional variances of the log SDF and its dominant pricing and
T
P
] and Vart [sdft+1
].
martingale components, Vart [sdft+1 ], Vart [sdft+1
2
Vart [sdft+1 ] = Λ′ Σt Λ + λ2η σgt
T
′
Vart [sdft+1
] = B∞
Σt B∞
P
′
2
Vart [sdft+1
] = (Λ′ + B∞
)Σt (Λ + B∞ ) + λ2η σgt
.
P
The conditional variance ratio Vart [sdft+1
]/Vt [sdft+1 ] equals
′
′
P
−B∞
Σt Λ − 12 B∞
Σt B∞
Vt [sdft+1
]
=1−
1
1
2
′
2
Vt [sdft+1 ]
2 Λ Σt Λ + 2 λη σgt
′
The first term in the numerator corresponds to the bond risk premium (−B∞
Σt Λ).
1 ′
It includes the Jensen term ( 2 B∞ Σt B∞ ). As a result, the numerator corresponds to
the bond risk premium without the Jensen term. The denominator corresponds to the
maximum risk premium (also without the Jensen term).
Note that the maximal Sharpe ratio in the model is:
M axSRt
=
=
=
VI.
σt (log SDFt+1 )
q
2 + λ2 σ 2 + λ2 σ 2 + λ2 σ 2
λ2e σxt
gw gw
xw xw
η gt
1
2 2
Λ′ Σt Λ + λ2η σgt
Decomposition of the Nominal SDF
The following proposition shows how to decompose the nominal SDF of the long-run
risk model into a martingale and a dominant pricing component. To avoid confusion, we
use M N to denote the nominal pricing kernel.
Proposition 2. The stochastic discount factor of the long-run risk model can be decomposed into a martingale component and the dominant pricing component:
T
M Nt+1
M NtT
P
M Nt+1
M NtP
=
=
1
$′
b $′ Σ
b 2 bt+1 ,
β̃ exp −B∞
(I − Γ̃)X̃t + B
∞ t ε
h
i
1
$′
$ ′b 2
b$ + B
b∞
β̃ −1 exp µs − µπ + S̃ ′ + B∞
I − Γ̃ X̃t − (Λ
) Σt εbt+1 .
To show this, we start from the definition of the dominant pricing component of the
12
THE AMERICAN ECONOMIC REVIEW
MAY 2010
pricing kernel:
β̃ t+n
,
n→∞ P $b (n)
t
M NtT = lim
Recall that log real bond prices are affine in the state vector:
p$b
t (n)
$
2
$
2
= −B0$ (n) − Bx$ (n)xt − Bgs
(n) σgt
− σg2 − Bxs
σxt
− σx2 − Bπ$ (π̄t − µπ )
= −B0$ (n) − B $ (n)′ X̃t ,
2
2
where we define X̃t = [xt , σgt
− σg2 , σxt
− σx2 , π̄t − µπ ].
We can then write the dominant pricing component of the SDF as:
M NtT = lim β̃ t+n exp B0$ (n) + B $ (n)′ X̃t .
n→∞
The constant β̃ is chosen in order to satisfy Assumption 1 in Alvarez and Jermann
(2005):
P $b (n)
0 < lim t n < ∞.
n→∞
β
Recall that B0$ (n) is defined recursively:
B0$ (n)
i2 h
i2
1 h
$
2
= B0$ (n − 1) − µs + µπ −
σπ + Bπ$ (n − 1)σz + λgw + Bgs
(n − 1) σgw
2
i2
h
i2 1 h
$
2
−
λxw + Bxs
(n − 1) σxw
+ ϕπg + λη + ϕzg Bπ$ (n − 1) σg2
2
i2
1h
− ϕπx + λe + Bx$ (n − 1) + ϕzx Bπ$ (n − 1) σx2
2
Because of the affine term structure of the model and the stationarity of the state
vector X̃, the limit limn→∞ B $ (n) ≡ B $ (∞) is finite. Taking limits on both sides of the
equation above leads to:
i2 h
i2
1 h
$
2
lim B0$ (n) − B0$ (n − 1) = −µs + µπ −
σπ + Bπ$ (∞)σz + λgw + Bgs
(∞) σgw
n→∞
2
h
i2
h
i2 1
$
2
−
λxw + Bxs
(∞) σxw
+ ϕπg + λη + ϕzg Bπ$ (∞) σg2
2
i2
1h
− ϕπx + λe + Bx$ (∞) + ϕzx Bπ$ (∞) σx2
2
The limit of B0$ (n) − B0$ (n − 1) is finite, so that B0$ (n) grows at a linear rate in the
limit. We choose the constant β̃ to offset the growth in B0$ (n) as n becomes very large.
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
13
Setting
β̃
=
i2 h
i2
1 h
$
2
exp µs − µπ +
σπ + Bπ$ (∞)σz + λgw + Bgs
(∞) σgw
2
h
i2
h
i2 1
$
2
+
λxw + Bxs
(∞) σxw
+ ϕπg + λη + ϕzg Bπ$ (∞) σg2
2
i2 1h
+ ϕπx + λe + Bx$ (∞) + ϕzx Bπ$ (∞) σx2
2
guarantees that Assumption 1 in Alvarez and Jermann (2005) is satisfied.
We can now write the dominant pricing component of the SDF as:
T
M Nt+1
M NtT
=
=
where
$′
β̃ exp −B∞
(I − Γ̃)X̃t + Bπ$ (∞)ϕzg σgt ηt+1 + Bπ$ (∞)σz ξt+1
$
$
+[Bx$ (∞) + Bπ$ (∞)ϕzx ]σxt et+1 + Bgs
(∞)σgw wg,t+1 + Bxs
(∞)σxw wx,t+1
$′
b $′ Σ
b .5 bt+1 ,
β̃ exp −B∞
(I − Γ̃)X̃t + B
∞ t ε
Γ̃

ρ
 0
= 
0
αx
0
νg
0
0
0
0
νx
0

0
0 
0 
απ
To derive the martingale component of the SDF, let us go back to the SDF itself. Let
S̃ ≡ [sx , sgs , sxs , −1]′ . Then the nominal SDF is:
M Nt+1
M Nt
=
=
exp µs − µπ + S̃ ′ X̃t − (λη + ϕπg ) σgt ηt+1
− (λe + ϕπx ) σxt et+1 − λgw σgw wg,t+1 − λxw σxw wx,t+1 − σπ ξt+1 )
b $′ Σ.5
exp µs − µπ + S̃ ′ X̃t − Λ
ε
b
t+1
t
As a result, the martingale component of the SDF is:
P
M Nt+1
M NtP
=
=
M Nt+1
M Nt
T
M Nt+1
M NtT
−1
h
i
$′
β̃ −1 exp µs − µπ + S̃ ′ + B∞
I − Γ̃ X̃t − [λη + ϕπg + Bπ$ (∞)ϕzg ]σgt ηt+1
−[λe + ϕπx + Bx$ (∞) + Bπ$ (∞)ϕzx ]σxt et+1
=
$
$
−[λgw + Bgs
(∞)]σgw wg,t+1 − [λxw + Bxs
(∞)]σxw wx,t+1 − [σπ + Bπ$ (∞)σz ]ξt+1
h
i
$′
b$ + B
b $ )′ Σ
b .5 εbt+1 .
β̃ −1 exp µs − µπ + S̃ ′ + B∞
I − Γ̃ X̃t − (Λ
∞
t
P
We need to verify that the martingale component is a martingale, i.e that Et [Mt+1
/MtP ] =
14
THE AMERICAN ECONOMIC REVIEW
MAY 2010
1. To do so, recall that the bond parameters evolve as:
Bx$ (n) =
$
Bgs
(n) =
$
Bxs
(n) =
Bπ$ (n) =
ρBx$ (n − 1) + αx Bπ$ (n − 1) − sx
i2
1h
$
νg Bgs
(n − 1) − sgs −
λη + ϕπg + ϕzg Bπ$ (n − 1)
2
i2
1h
$
λe + ϕπx + Bx$ (n − 1) + ϕzx Bπ$ (n − 1)
νx Bxs (n − 1) − sxs −
2
απ Bπ$ (n − 1) + 1.
Taking limits as n → ∞ leads to:
i2
i2 ′
1h
1h
B(∞)$′ (I−Γ̃)+S̃ ′ = 0, −
λη + ϕπg + ϕzg Bπ$ (∞) , −
λe + ϕπx + Bx$ (∞) + ϕzx Bπ$ (∞) , 0 .
2
2
To check the martingale condition, we plug in the definition of β̃ in the expression for
the martingale component of the nominal SDF, and use the above equation for B(∞)$′ (I−
Γ̃) + S̃ ′ . After some algebra, we indeed find that
Et
P
M Nt+1
M NtP
=
1.
We now turn to the conditional variances of the log SDF and its dominant pricing and
$,T
$,P
$
martingale components, Vart [sdft+1
], Vart [sdft+1
] and Vart [sdft+1
].
$
Vart [sdft+1
] =
=
$,T
Vart [sdft+1
] =
2
2
2
2
2
2
+ λ2gw σgw
+ λ2xw σxw
+ σπ2
(λη + ϕπg ) σgt
+ (λe + ϕπx ) σxt
b $′
b b$
Λ
∞ Σt Λ ∞
2
2
Bπ$ (∞)2 ϕ2zg σgt
+ [Bx$ (∞) + Bπ$ (∞)ϕzg ]2 σxt
$
2
$
2
+ Bxs
(∞)2 σxw
+ Bπ$ (∞)2 σz2
+Bgs
(∞)2 σgw
=
$,P
Vart [sdft+1
] =
$′ b b $
b∞
B
Σt B∞
2
2
+ [λe + ϕπx + Bx$ (∞) + Bπ$ (∞)ϕzx ]2 σxt
[λη + ϕπg + Bπ$ (∞)ϕzg ]2 σgt
$
2
$
2
+[λgw + Bgs
(∞)]2 σgw
+ [λxw + Bxs
(∞)]2 σxw
+ [σπ + Bπ$ (∞)σz ]2
=
$ ′ b b$
$
b$ + B
b∞
b∞
(Λ
) Σt (Λ + B
)
P
The conditional variance ratio Vart [sdft+1
]/Vt [sdft+1 ] equals
$,P
Vt [sdft+1
]
$
Vt [sdft+1
]
=1−
b $′ Σ
b b $ 1 b $′ b b $
−B
∞ t Λ − 2 B∞ Σt B∞
1 b $′ b b $
Λ Σt Λ
2
∞
∞
The first term in the numerator corresponds to the nominal bond risk premium of an
infinite horizon bond, which includes a Jensen term. The second term in the numerator is that Jensen term. As a result, the numerator corresponds to the nominal bond
risk premium without the Jensen term. The denominator corresponds to the maximum
nominal risk premium, also without the Jensen term.
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
VII.
15
Calibration
Table 1 reports the model parameter values we use; they are the ones proposed in
Bansal and Shaliastovich (2007). Table 2 reports the model loadings on state variables.
The model is simulated for 60,000 months and aggregated up to quarterly frequency
2
for comparison with our quarterly data. In the simulation, negative values for σg,t+1
and
2
σx,t+1 are replaced by very small positive values in simulation.
Table 4 reports the mean, standard deviation and autocorrelation of the stochastic discount factor (SDF ), its martingale (SDF P ) and dominant pricing (SDF T ) components,
the conditional variance ratio ω, the maximum risk premium without Jensen adjustment
(M ax RP ) and the risk premium of an infinite maturity bond without Jensen adjustment
(BRP (∞)). Table 3 reports the mean and standard deviations of the real and nominal
yields and bond risk premia in the model and compare them to the same moments in
the actual nominal data. Table 5 reports moments of quarterly inflation in the model
and in the data. Quarterly inflation is obtained as the sum of three consecutive monthly
inflation rates.
The Bansal and Shaliastovich (2007) calibration generates an annual consumption
growth rate of 2.12 percent with a standard deviation of 3.52 percent. It generates
an annual inflation rate of 3.52 percent with a standard deviation of 2.49 percent.
VIII.
Robustness Checks
As robustness checks, we considered both changes on the real and on the nominal side
of the economy.
On the real side, we conduct two experiments. First, we find that a slight decrease in
the persistence of the long-run component in consumption growth ρx could decrease the
long-horizon consumption variance ratios and the real variance ratio significantly, and
increase the long term real yield from negative to positive values. As a result, the model
would need to rely less on a large inflation risk premium in order to match the nominal
yield curve, thus lowering the variation of MtT in the nominal pricing kernel. However, if
all the other parameters are maintained at their previous values, the model would then
imply too much volatility of the wealth-consumption ratio and an equity risk premium
that is much too low. Second, we shut down the heteroscedasticity in consumption
growth by calibrating σxw and σgw to very low values. We keep all the other parameters
at their previous values. In this case, the real and nominal conditional variance ratios
are respectively 1.20 and 0.63 (see Table 6). They are closer to 1, but equity and bond
risk premia are constant.
On the nominal side, we first check the robustness of our results to a slightly different
calibration of the inflation dynamics. First, we vary each inflation parameter independently in either direction. We report in Table 7 the mean maximum risk premium (M RP ),
the mean bond risk premium BRPJ (including the Jensen term) and the mean variance
ratio ω for different values of the inflation parameters. We simulate the model for a low
and a high value of each parameter (25 percent above and below the benchmark value
reported in Table 1). The only exception is the parameter απ , which we cannot increase
by 25 percent without running into stationarity issues. The high value is a 10 percent
increase for that parameter. We find that ωt only changes noticeably with αx and απ .
To further investigate the sensitivity to these two parameters, Figure 1 in the appendix
plots ωt (left axis) and the five-year nominal bond risk premium (right axis) against αx
(horizontal axis). As we vary αx away from its benchmark value of -0.35, we simultaneously vary απ to match the observed persistence of quarterly inflation. We also choose
µπ and σπ to keep the mean and volatility of inflation at their benchmark values. The
16
THE AMERICAN ECONOMIC REVIEW
MAY 2010
figure shows that ωt is essentially unchanged over a wide range of values for αx and never
comes close to the desired value of one.
Next, we consider a calibration that matches the observed mean, variance, and persistence of inflation, the 5-1-year yield spread, and the persistence of the 5-year nominal
bond risk premium. This calibration delivers a nominal variance ratio ωt that is much
too high.
Finally, we ask whether we can find inflation parameters that deliver a nominal variance
ratio of 1. We find that we can, while matching the mean inflation, the slope of the
nominal term structure, and the persistence of the nominal BRP, but inflation ends up
being 2.5 times too volatile and not persistent enough.
5
ω
BRP
ω
4.5
0.8
4
0.7
3.5
0.6
3
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.1
0.5
0
−0.9
−0.8
−0.7
−0.6
−0.5
α
−0.4
−0.3
−0.2
5Yr Nominal Bond Risk Premium
1
0.9
0
−0.1
x
Figure 1. Variance Ratio and Nominal Bond Risk Premium: Sensitivity Analysis
The figure plots the conditional variance ratio ωt (against the left axis) and the five-year nominal bond risk premium (against
the right axis) for different values of the parameter αx (on the horizontal axis). As we vary αx away from its benchmark value
of -0.35, we simultaneously vary απ to match the observed persistence of quarterly inflation. We also choose µπ and σπ to
keep the mean and volatility of inflation at their benchmark values.
IX.
Empirical Variance Ratios
Alvarez and Jermann (2005) show that – assuming that the process Xt satisfies the same
regularity conditions as above and that Xt+1 /Xt is strictly stationary and limk→∞ k1 Var(Et+k [Xt ]) =
0 – then
P Xt+1
1
Xt+k
Var
=
lim
Var
,
k→∞ k
Xt
XtP
Note that the entropy measure used by Alvarez and Jermann (2005) collapses to the halfvariance since all variables are conditionally normal. This result implies that long-horizon
variance ratios are informative about the variance of the martingale component. We now
turn to the empirical variance ratios of the two components of the SDF, e.g consumption
growth and the wealth consumption ratio.
If changes in log consumption or changes in the log wealth-consumption ratio are i.i.d,
then the variance of long-horizon changes in each variable should grow with the horizon.
Ph
We compute variance ratios at horizon h as V R(h) = Var[ j=0 ∆xt+j ]/[hVar(∆xt )],
for x = c and x = wc. We simulate the model at monthly frequency. Table 1 in the
appendix reports the model parameters. We start from the parameter values in Bansal
and Shaliastovich (2007).
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
17
Figure 2 reports these variance ratios for consumption growth, the change in the wealthconsumption ratio, and inflation. The left panel corresponds to actual data; the right
panel uses simulated series. Let us first focus on actual data. The variance ratio of
the wealth-consumption ratio clearly decreases with the horizon. It is below 0.6 within
five years. Consumption growth exhibits a very different pattern: its variance ratio
first increases for horizons up to 5 years; it then decreases, but even after 15 years, the
variance ratio is still above one. As a result, there is strong evidence of persistence and
mean-reversion in the wealth-consumption ratio, but not in consumption growth.
Let us now turn to simulated data. The variance ratios of the wealth-consumption
ratio are in line with the data. They decrease linearly with the horizon, from 1 to
approximately 0.5 at the 30-year horizon. In the data, the variance ratio decreases from
1 to 0.6. Consumption growth, however, exhibits a very different pattern. At long
horizons, it displays more persistence in the model than in the data. The bottom panel
shows that the inflation persistence is similar in model and data, with a slight divergence
maybe at longer horizons.
Model:Consumption Growth
20
15
15
VR(h)
VR(h)
Data: Consumption Growth
20
10
5
0
5
0
20
40
0
60
1
0.5
0
20
40
0
60
0.5
0
60
0
20
40
60
Model: Inflation
30
VR(h)
30
VR(h)
40
1
Data: Inflation
20
10
0
20
Model: Change in Log Wealth−Consumption Ratio
1.5
VR(h)
VR(h)
Data: Change in Log Wealth−Consumption Ratio
1.5
0
10
0
20
40
Horizon (h) in quarters
60
20
10
0
0
20
40
60
Horizon (h) in quarters
Figure 2. Variance Ratios for Consumption Growth, the Change in the Log Wealth Consumption Ratio and Inflation in the Data and in the Model.
P
The variance ratio of ∆xt is equal to V R(h) = V ar[ h
j=0 ∆xt+j ]/[hV ar(∆xt )]. The left panel corresponds to actual data.
The right panel corresponds to simulated data. Data are quarterly. Actual data come from Lustig et al. (2009). The sample is
1952:II-2008:IV.
18
THE AMERICAN ECONOMIC REVIEW
MAY 2010
REFERENCES
Alvarez, Fernando and Urban Jermann, “Using Asset Prices to Measure the Measure the Persistence of the Marginal Utility of Wealth.,” Econometrica, 2005, (4), 1977–2016.
Bansal, Ravi and Ivan Shaliastovich, “Risk and Return in Bond, Currency, and Equity Markets,”
June 2007. Working Paper Duke University.
Campbell, John Y., “A Variance Decomposition for Stock Returns,” Economic Journal, 1991, 101,
157–179.
Epstein, Larry and Stan Zin, “Substitution Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework,” Econometrica, 1989, 57, 937–968.
Lustig, Hanno, Stijn Van Nieuwerburgh, and Adrien Verdelhan, “The Wealth-Consumption
Ratio,” August 2009. Working Paper.
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
19
Table 1—Model Parameter Values
Parameter
Preference Parameters:
Subjective discount factor
Intertemporal elasticity of substitution
Risk aversion coefficient
BS(2007)
δ
ψ
γ
0.9987
1.5
8
Consumption Growth Parameters:
Mean of consumption growth
Long-run risk persistence
News volatility level
News volatility persistence
News volatility of volatility
Long run-risk volatility level
Long run-risk volatility persistence
Long run-risk volatility of volatility
µg
ρ
σg
νg
σgw
σx
νx
σxw
0.0016
0.991
0.004
0.85
1.15e − 6
0.004σg
0.996
0.062 σgw
Dividend Growth Parameters:
Mean of dividend growth
Dividend leverage
Dividend loading on news volatility
Dividend loading on long-run risk volatility
Volatility loading of dividend growth
Correlation of consumption and dividend news
µd
φx
φgs
φxs
ϕd
τgd
0.0015
1.5
0
0
6.0
0.1
Inflation Parameters:
Mean of inflation rate
Inflation leverage on news
Inflation leverage on long-run news
Inflation shock volatility
Expected inflation AR coefficient
Expected inflation loading on long-run risk
Expected inflation leverage on news
Expected inflation leverage on long-run news
Expected inflation shock volatility
µπ
ϕπg
ϕπx
σπ
απ
αx
ϕzg
ϕzx
σz
0.0032
0
−2.0
0.0035
0.83
−0.35
0
−1.0
4.0e − 6
This table reports the calibrated parameters values for our simulation. We take them from
Table IV and Table C.I in Bansal and Shaliastovich (2007).
MAY 2010
Table 2—Model Loadings on State Variables
constant
wc
µwc
6.4
pd
µpd
2
σgt
− σg2
x
1
1− ψ
Wx =
κc
1 −ρ
Wgs =
31
Dx =
1
φx − ψ
κ1 −ρ
Dgs =
THE AMERICAN ECONOMIC REVIEW
1
ERP
BRP
(Real)
BRP
(Nominal)
66
2
(1 − θ) Wgs Dgs σgw
2
+ (1 − θ) Wxs Dxs σxw
2
+ϕd τgd γσg
+Dx (θ − 1) Wx σx2
1 −1 2
θ ψ
2(κc
1 −νg )
Wxs =
2
1 γ 2 −2γϕ τ
d dg +ϕd
2
κ1 −νg
i
1 (1−γ)+φ
γ− ψ
gs
[
h
κ1 −νg
θ
2(νx −κc
1)
1 −1
ψ
κc
1 −ρ
2
−1.8 × 106
−7.7
+2
5.6
2
σxt
− σx2
]
Dxs =
"
 γ− 1
ψ
1
2
1 −φ
x
+ψ
ρ−κ1
ρ−κc
1
#2
+
(
(1−γ) γ− 1
ψ
2
ρ−κc
1
(
)
κ1 −νx
xs
+ κ1φ−ν
x
1.3 × 102
−4.3 × 106
Ggs = γϕd τgd
Gxs = (1 − θ) Wx Dx
0.003
0
4.8
4.6 × 104
2
(θ − 1) Wgs Bgs (n − 1)σgw
2
+ (θ − 1) Wxs Bxs (n − 1)σxw
+ (θ − 1) Wx Bx (n − 1)σx2
Fx (n) = 0
Fgs (n) = 0
Fxs (n) = (θ − 1) Wx Bx (n − 1)
−0.0014
0
0
−2.1 × 104
$
2
(θ − 1) Wgs Bgs
(n − 1)σgw
$
2
+ (θ − 1) Wxs Bxs (n − 1)σxw
Fx$ (n) = 0
$
Fgs
(n) = − (γ + ϕπg )
×ϕzg Bπ$ (n − 1)
$
Fxs
(n) = [(θ − 1) Wx − ϕπx ] × Bx$ (n − 1) + Bπ$ (n − 1)ϕzx
0
−0
+σπ σz Bπ$ (n − 1)
$
2
− (γ + ϕπg
) ϕzg Bπ (n − 1)σg
+ ((θ − 1) Wx − ϕπx ) Bx$ (n − 1) + Bπ$ (n − 1)ϕzx σx2
4.3 × 104
20
0.0015

)
This table reports the model loadings on a constant and the state variables. We consider the log wealth-consumption ratio (wc), the log
price-dividend ratio (pd), the equity risk premium (ERP ), the real and nominal bond risk premia (BRP ) at the n-year horizon.

VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
21
Table 3—Real and Nominal Yield Curves
1
2
3
M ean Y ields
5.33
5.52
5.69
5.80
5.89
Std
2.81
2.77
2.70
2.69
2.65
M aturity
4
5
30
200
Nominal Bonds - Data
Nominal Bonds - Model
M ean Y ields
5.19
5.46
5.75
6.06
6.38
12.82
20.02
Std
2.92
2.79
2.65
2.53
2.43
1.60
0.36
M ean BRP
0.33
0.93
1.59
2.27
2.97
16.81
24.43
Std
0.07
0.18
0.28
0.38
0.46
1.13
1.18
Real Bonds - Model
M ean Y ields
1.26
1.05
0.83
0.61
0.39
−4.71
−13.63
Std
1.39
1.35
1.32
1.30
1.29
1.10
0.25
−0.39
−0.83
−1.28
−1.73
−2.19
−11.14
−16.21
0.05
0.10
0.15
0.19
0.23
0.52
0.55
M ean BRP
Std
The top panel reports the mean and standard deviation of nominal bond yields in the Fama-Bliss data. The data are for 1952
until 2008, and only bond yields of maturities one through five years are available. The maturity is in years. The yields and
returns are annualized and reported in percentage points. The middle panel does the same for nominal bond yields for a 60,000
month simulation of the LRR model. It also reports the mean and standard deviation of the nominal bond risk premia. The
bottom panel reports the same model-implied moments for real bonds.
Table 4—Conditional Variance Ratio
M ean
Std
AR(1)
Nominal SDF
SDF $
SDF $,P
SDF $,T
ωt$
M ax RP
BRP (∞)
0.99
1.00
0.98
0.37
30.62
18.72
SDF
SDF P
SDF T
ωt
M ax RP
BRP (∞)
1.00
1.00
1.02
1.65
30.69
−19.05
0.23
0.14
0.10
0.06
2.52
1.04
−0.01
−0.01
−0.01
0.98
0.99
0.99
Real SDF
0.23
0.30
0.07
0.11
2.54
0.58
−0.01
−0.01
−0.01
0.98
0.99
0.99
This table reports the mean, standard deviation and autocorrelation of the stochastic discount factor (SDF ), its martingale
(SDF P ) and dominant pricing (SDF T ) components, the conditional variance ratio ω, the maximum risk premium without
Jensen adjustment (M ax RP ) and the risk premium of an infinite maturity bond without Jensen adjustment (BRP (∞)). The
table reports the autocorrelation of each monthly variable in logs. The top (bottom) panel focuses on the nominal (real)
stochastic discount factor. The numbers are computed from a 60,000 month simulation.
22
THE AMERICAN ECONOMIC REVIEW
MAY 2010
Table 5—Inflation: Model vs Data
Data
πt
Model
M ean
Std
AR(1)
M ean
Std
AR(1)
0.85
0.62
0.86
0.88
1.25
0.76
This table reports the mean, standard deviation and autocorrelation of the quarterly inflation rate. The left panel corresponds
to actual data, from Lustig, Van Nieuwerburgh and Verdelhan (2009). The right panel corresponds to simulated data, from the
model. The mean and standard deviation are in percentage.
Table 6—Conditional Variance Ratio: No Heteroscedasticity
M ean
Std
AR(1)
Nominal SDF
SDF $
SDF $,P
SDF $,T
ωt$
M ax RP
BRP (∞)
SDF
SDF P
SDF T
ωt
M ax RP
BRP (∞)
1.00
1.00
1.00
1.20
8.74
−1.74
0.99
1.00
0.99
0.63
8.70
3.18
0.12
0.13
0.01
0.00
0.00
0.00
−0.01
−0.01
−0.01
1.00
1.00
1.00
Real SDF
0.12
−0.01
0.10
−0.01
0.03
−0.01
0.00
1.00
0.00
1.00
0.00
1.00
This table reports the mean, standard deviation and autocorrelation of the stochastic discount factor (SDF ), its martingale
(SDF P ) and dominant pricing (SDF T ) components, the conditional variance ratio ω, the maximum risk premium without
Jensen adjustment (M ax RP ) and the risk premium of an infinite maturity bond without Jensen adjustment (BRP (∞)). The
table reports the autocorrelation of each monthly variable in logs. The top (bottom) panel focuses on the nominal (real)
stochastic discount factor. The numbers are computed from a 60,000 month simulation.
VOL. 100 NO. 2
THE WEALTH-CONSUMPTION RATIO
23
Table 7—Sensitivity to Inflation Specification
M ax RP
BRP (∞)
ω
Low
High
Low
High
Low
High
µp
30.62
30.62
18.72
18.72
0.37
0.37
ϕπg
30.62
30.62
18.72
18.72
0.37
0.37
ϕπx
30.64
30.60
18.70
18.74
0.37
0.37
σπ
30.62
30.62
18.72
18.72
0.37
0.37
απ
30.62
30.62
5.61
26.42
0.81
0.10
αx
30.62
30.62
14.54
21.84
0.51
0.27
ϕzg
30.62
30.62
18.72
18.72
0.37
0.37
ϕzx
30.62
30.62
18.63
18.81
0.38
0.37
σz
30.62
30.62
18.72
18.72
0.37
0.37
This table reports the mean maximum risk premium (M ax RP ) , the mean bond risk premium BRP (∞) (including the Jensen
term) and the mean variance ratio ω. We vary one parameter at a time, and simulate the model for a low and a high value of
each parameter (25 percent above and below the benchmark value reported in the first column of Table 1). The only exception
is the parameter απ , which we cannot increase by 25 percent without running into stationarity issues. The high value is a 10
percent increase for that parameter.