A Theory of Housing Collateral,
Consumption Insurance and Risk Premia
Hanno Lustig
University of Chicago
Stijn Van Nieuwerburghcurrency1
NYU Stern School of Business
May 14, 2004
currency1Hanno Lustig: hlustig@uchicago.edu, Stijn Van Nieuwerburgh: svnieuwe@stern.nyu.edu, 44 W Fourth Street,
New York, NY 10012. First version August 2002. The authors thank Thomas Sargent, Dave Backus, Andrew
Abel, Fernando Alvarez, Timothey Cogley, Harold Cole, Steven Grenadier, Robert Hall, Lars Peter Hansen, Urban
Jermann, Narayana Kocherlakota, Dirk Krueger, Martin Lettau, Sydney Ludvigson, Sergei Morozov, Lee Ohanian,
Monika Piazzesi, Martin Schneider, Kenneth Singleton, Laura Veldkamp, Pierre-Olivier Weill and Amir Yaron. We
also bene?ted from comments from seminar participants at Duke University, Stanford GSB, University of Iowa,
Universit? de Montreal, New York University Stern, University of Wisconsin, University of California at San Diego,
London Business School, London School of Economics, University College London, University of North Carolina,
Federal Reserve Bank of Richmond, Yale University, University of Minnesota, University of Maryland, Federal Reserve
Bank of New York, Boston University, University of Pennsylvania Wharton, University of Pittsburgh, Carnegie
Mellon University GSIA, Northwestern University Kellogg, University of Texas at Austin, Federal Reserve Board of
Governors, University of Gent, UCLA, University of Chicago, Stanford University, the Society for Economic Dynamics
Meeting in New York, and the North American Meeting of the Econometric Society in Los Angeles. For computing
support we thank NYU Stern, especially Norman White. Stijn Van Nieuwerburgh acknowledges ?nancial support
from the Stanford Institute for Economic Policy research and the Flanders Fund for Scienti?c Research.
ABSTRACT
In a model with housing collateral, the ratio of housing wealth to total wealth shifts the
conditional distribution of asset prices and consumption growth. A decrease in house prices
reduces the collateral value of housing, increases household exposure to idiosyncratic risk, and
increases the conditional market price of risk. The model quantitatively accounts for conditional
asset pricing moments, cross-sectional variation in value portfolio returns and key unconditional
asset pricing moments. The increase of the equity premium and Sharpe ratio when collateral is
scarce matches the increase observed in US data. The model also generates a return spread of
value ?rms over growth ?rms of the magnitude observed in the data. Assets with payo?s that
lay farther in the future, are less risky. Growth stocks are such long duration assets.
Introduction
Two aspects of ?nancial markets are often studied separately. One is their role in allocating
resources. The evidence suggests that countries with good institutions tend to reduce risk by
individuals (Gertler and Gruber (2002)). It also suggests that those same countries have better
aggregate economic performance (Levine (1997)). The other aspect of ?nancial markets is asset
returns. Arbitrage-free theory implies a pricing kernel that values all traded assets. In principle,
this common ingredient should a?ect the behavior of returns over time and generate connections
between returns on di?erent kinds of assets. Research along these lines has documented a number
of striking di?erences in expected returns: between equity and bonds (Mehra and Prescott (1985)),
between equity at di?erent points in time (Lettau and Ludvigson (2003)), between size- and value-
weighted portfolios (Fama and French (1992)), and between bonds of di?erent maturities (Backus
and Zin (1994)).
We argue that modelling these two aspects together leads to deeper insights about each. We
build a dynamic general equilibrium model that we think approximates the modest frictions in-
hibiting perfect risk-sharing in an advanced economy like the US. The model is based on two ideas:
that debts can be enforced only to the extent that they are collateralized, and that the primary
source of collateral is housing. Our emphasis on housing, rather than ?nancial assets, re?ects two
features of the US economy: the participation rate is much higher than for equity (two-thirds of
households own their own house), and the aggregate value of housing in the US is roughly dou-
ble the value of equity owned by households (Flow of Funds data). In addition, our earlier work
(Lustig and VanNieuwerburgh (2004b)) shows that asset returns are correlated with the amount of
available housing collateral, both in the time-series and in the cross-section, giving this approach
some empirical plausibility.
The ?rst insight is that amount of housing collateral signi?cantly shifts the risk-sharing pos-
sibilities of households. Risk-sharing patterns, explored empirically for metropolitan regions in
Lustig and VanNieuwerburgh (2004a), con?rm the collateral mechanism. The second insight is
that the same variable that indexes the risk-sharing possibilities among households, also indexes
the investment opportunity set. The main contribution of this paper is to show that time variation
in housing collateral can quantitatively account for three of the aforementioned di?erences in ex-
pected returns: between equity and the risk-free asset, between equity at di?erent points in time,
and between size and value portfolios.
1
More speci?cally, our model generates the following features. First, it predicts high and volatile
equity premia, as well as high Sharpe ratios, when collateral is scarce. The Sharpe ratio is volatile.
We document similar dynamics for the excess returns and the Sharpe ratio for US data. A direct
implication of the time-series variation in equity premia is that the housing collateral ratio ought to
predict future excess returns. The model quantitatively replicates the predictability pattern found
in the data.
Second, the collateral model generates a large value premium and the higher Sharpe ratios
for value stocks than growth stocks. Following Lustig and VanNieuwerburgh (2004b), we excess
regress returns on value decile portfolios on the empirical counterparts of the model's risk factors.
We simulate arti?cial book-to-market decile excess returns by using these empirical factor loadings
and the model's risk factors. The model simulation generates a return di?erence between the
extreme value and growth portfolios of 6 percent, as high as in the data. Moreover, the Sharpe
ratio on value portfolios is twice as high as the Sharpe ratio on growth portfolios. The model
generates a value premium because short duration assets, such as value stocks, are more risky than
long duration assets, such as growth stocks. Lettau and Wachter (2004) and Hansen, Heaton, and
Li (2004) take a similar approach to explain the value premium. The model generates a decreasing
term structure of consumption strips. The return spread between a basket of consumption strips
with a duration of 5 years (`value') and a basket with a duration of 40 years ('growth') is 6 percent.
Third, the model generates empirically plausible unconditional aggregate asset pricing moments.
The benchmark general equilibrium model that explains both risk-sharing dynamics and asset
prices is the Lucas (1978) model. It predicts perfect risk-sharing and implies that the only risk
factor relevant for asset prices is aggregate consumption growth Breeden (1979). There is a wealth
of empirical evidence against full consumption insurance at di?erent levels of aggregation: at the
household level (e.g. Attanasio and Davis (1996) and Cochrane (1991)), the regional level (e.g.
Hess and Shin (1998) and Lustig and VanNieuwerburgh (2004a)) and the international level (e.g.
Backus, Kehoe, and Kydland (1992)). The asset pricing predictions have been tested and rejected
(e.g. Hansen and Singleton (1983)). Its failure occurs both along the time-series and the cross-
sectional dimension. First, because aggregate consumption growth is approximately i.i.d., the
CCAPM implies a market price of risk that is approximately constant and there is very little
variation in the Sharpe ratio. Second, the covariance of asset returns with consumption growth
explains only a small fraction of the variation in the cross-section of stock returns of ?rms sorted in
2
portfolios according to size and value characteristics. It has been understood for more than a decade
that frictions are needed to bring the consumption-based model closer to the data. Incomplete
markets with exogenous borrowing constraints, short sales constraints or transaction costs (e.g.
Telmer (1993) and Heaton and Lucas (1996)), failed to deliver su?ciently large deviations from
the benchmark model. The goal of our paper is to introduce a di?erent kind of friction, associated
with the housing market, and to ask to what extent it can generate realistic asset price predictions.
We show that for a plausibly calibrated housing collateral process, we signi?cantly improve on the
predictions of the canonical CCAPM, both in terms of the time-series variation in conditional asset
pricing moments and the cross-sectional variation in returns.
Changes in the value of the housing stock perturb the risk sharing technology. The households
in this economy trade contingent claims to insure against labor income risk. These claims have to be
fully backed by the value of their housing wealth. The model relaxes the assumption that contracts
are perfectly enforceable (Alvarez and Jermann (2000)). As in Lustig (2003), we allow households
to forget their debts. The new feature of our model is that each household owns part of the housing
stock. Housing provides utility services and collateral services. When a household chooses to
forget its debts, it loses all its housing wealth but its labor income is protected from creditors. The
household is not excluded from trading.1 The lack of commitment gives rise to collateral constraints.
Their tightness depends on the abundance of housing collateral. We measure this by the housing
collateral ratio: the ratio of collateralizable housing wealth to total wealth. An increase in the
housing collateral ratio increases the scope for risk sharing. It decreases the conditional dispersion
of consumption growth across households.
Two channels in the model deliver time variation in the conditional market price of risk, at
di?erent frequencies. First, a drop in the housing collateral ratio adversely a?ects the risk sharing
technology that enables households to insulate their consumption from labor income shocks. This
makes households demand a higher price to bear risk in times with low housing collateral. This
is the source of low frequency variation in the market price of risk. Second, households are more
exposed to binding collateral constraints when the cross-sectional dispersion of labor income shocks
increases, typically when aggregate consumption growth is low (Constantinides and Du?e (1996)).
1we model the outside option as bankruptcy with loss of all collateral assets. In Kehoe and Levine (1993), Krueger
(2000), Krueger and Perri (2003), and Kehoe and Perri (2002), limited commitment is also the source of incomplete
risk-sharing across US households and across countries respectively. In contrast, the outside option upon default is
exclusion from future participation in ?nancial markets. In our model, all promises are backed by all collateral assets.
Geanakoplos and Zame (2000) and Kubler and Schmedders (2003) think of individual assets collateralizing individual
promises in an incomplete markets economy.
3
Because the aggregate weight shock increases after a series of large aggregate consumption growth
shocks, the conditional standard deviation of the stochastic discount factor rises. This is the source
of high frequency variation in the market price of risk. The combination of both ingredients can
explain the time-series variation in asset prices and the cross-section of value-portfolio returns.
The model includes a second channel that transmits housing shocks to asset prices: non-
separable preferences. If utility is non-separable in non-durable consumption and housing services,
households want to hedge against shocks to the share of housing consumption in total consump-
tion. This introduces composition risk, which is the focus of recent work by Piazzesi, Schneider,
and Tuzel (2004) and Yogo (2003). If housing services and consumption are complements and
rental price growth is positively correlated with returns, then households command a larger risk
premium.2 Our collateral e?ect does not hinge on the non-separability of preferences. Instead, it
relies on imperfect consumption insurance among heterogenous households induced by occasionally
binding collateral constraints. We use non-separable preferences to quantify the relative e?ects of
the composition risk mechanism and the collateral mechanism. We ?nd that only the latter can
quantitatively account for the observed volatility in conditional asset pricing moments and for the
value premium.
We organize the paper as follows. Section 1 describes the environment, characterizes equilib-
rium allocations and de?nes the pricing kernel. Section 2 discusses the model computation and
calibration. Asset pricing results from a simulation of the model are discussed in section 3. Section
5 concludes. Section 6 contains all ?gures and tables. The appendix contains some details of the
model (A.1, A.3), proofs of the propositions (A.2) and a description of the data (A.5). It also
discusses the results for an economy where preferences are recursive, rather than additive (A.4).
1. Model
This section starts with a complete description of the environment in section 1.1. The next section
(1.2), sets up the household problem in time zero trading environment and characterizes equilib-
rium allocations using stochastic consumption weight processes. The growth rate of an aggregate
consumption weight process drives the consumption growth of the unconstrained households; these
households price random payo?s. The stochastic discount factor captures the risk of binding sol-
2Dunn and Singleton (1986) and Eichenbaum and Hansen (1990) report substantial evidence against the null
of separability in a representative agent model with durable and non-durable consumption, but they conclude that
introducing durables does not help in reducing the pricing errors for stocks.
4
vency constraints. Section 1.3 explains how the dynamics of the collateral ratio a?ect equilibrium
allocations. To gain intuition, section 1.4 describes the dynamics in a simple two-agent economy.
Section 1.5 introduces sequential trading and discusses conditions under which these equilibria co-
incide with time zero trading equilibria. In section 1.6 we de?ne value and growth stocks in the
model and show that the model can only generate a value premium if the pricing kernel contains a
permanent component.
We consider the simplest model of housing markets that delivers variation in the amount of
housing collateral relative to total wealth. The housing market has e?cient rental markets or spot
markets for housing services. Ownership and consumption of housing are completely separated.
We calibrate the persistence of the consumption/housing expenditure ratio to the data. Variation
in the expenditure ratio changes the value of the housing tree relative to the value of the other,
non-durable consumption tree.
1.1. Environment
This economy is populated by a continuum of in?nitely lived households. The structure of uncer-
tainty is twofold. s = (y;z) is an event that consists of a household-speci?c component y 2 Y and
an aggregate component z 2 Z. These events take on values on a discrete grid S = Y ? Z. We use
st = (yt;zt) to denote the history of events. St denotes the set of possible histories up until time
t. s follows a Markov process with transition probabilities ? that obey:
?(z0jz) =
X
y02Y
?(y0;z0jy;z) 8z 2 Z;y 2 Y:
Because of the law of large numbers, ?z(y) denotes both the fraction of households drawing y when
the aggregate event is z and the probability that a given household is in state y when the aggregate
state is z.
We use fxg to denote an in?nite stream ?xt(st)?1t=0. There are two types of commodities in this
economy: a consumption good and housing services. These consumption goods cannot be stored.
We let fc(?0;s0)g denote the stream of consumption and we let fh(?0;s0)g denote the stream
of housing services of a household of type (?0;s0). The households rank consumption streams
5
according to the criterion:
U (fcg ;fhg) =
X
stjs0
1X
t=0
?t?(stjs0)u?ct(?0;st);ht(?0;st)?; (1)
where ? is the time discount factor. The households have power utility over a CES-composite
consumption good:
u(ct;ht) =
?
c
"?1
"t + ?h
"?1
"t
?(1??)"
"?1
1 ? ? :
? > 0 converts the housing stock into a service ?ow. " is the intratemporal elasticity of substitution
between non-durable consumption and housing services 3
The aggregate endowment of the non-durable consumption good is denoted feg. The growth
rate of the aggregate endowment depends only on the current aggregate state: et+1(zt+1) =
?(zt+1)et(zt). Aggregate consumption ca equals the aggregate endowment e. Each of the house-
holds is endowed with a claim to a labor income stream f?g. The labor income share ^(yt;zt)
only depends on the current state of nature. The level of labor income is given by ?(yt;zt) =
^(yt;zt)e(zt). The aggregate endowment is the sum of the individual endowments:
X
y02Y
?z(y0)^t(y0;z) = 1; 8z;t ? 0;
The aggregate endowment of housing services is denoted fhag. Rather than specifying a pro-
cess for fhag, we specify a process for the expenditure ratio of non-durable to housing services
consumption frg, where r(zt) = ca(zt)?(zt)ha(zt) and ?(zt) denote the relative price of a unit of housing
services. The natural logarithm log(r) is speci?ed as an autoregressive process and additionally
depends on the endowment growth rate ?(zt). Let R be the domain of r.
We use pt(stjs0) to denote the state price de?ator: it is the price of a unit non-durable consump-
tion to be delivered in state st, in units of time zero consumption. Finally, ?st [fdg] denotes the
price of claim to fdg in units of st consumption, ?st [fdg] = Ps?jst P1?=0 ?pt+? ?s?jst?dt+? ?s?jst?currency1.
Markets open only at time zero. Households purchase a complete, state-contingent consumption
3The preferences belong to the class of homothetic power utility functions of Eichenbaum and Hansen (1990).
Special cases are separability (" = ??1) and Cobb-Douglas preferences (" = 1).
6
plan fc(?0;s0);h(?0;s0)g subject to a single, time zero budget constraint:
?s0 [fc(?0;s0) + ?h(?0;s0)g] 6?0 + ?s0 [f?g] ; (2)
where ?0 is the initial non-labor wealth. We use ?0 to denote the initial distribution of non-labor
wealth holdings.
Solvency Constraints Households can default on their debts. When the household defaults, it
keeps its labor income in all future periods. The household is not excluded from trading, even in
the same period. However, all collateral wealth is taken away. As a result, the markets impose
a solvency constraints that keep the households from defaulting. There is one such constraint for
each node st on the household's trades:
?st [fc(?0;s0) + ?h(?0;s0)g] ? ?st [f?g] : (3)
The constraints depend on the state prices as well as the rental price. Changes in equilibrium prices
determine the tightness of the constraints, as we describe in the next section.
1.2. Equilibrium Prices and Allocations
We de?ne an equilibrium with time zero markets and characterize the equilibrium allocations.
These Kehoe and Levine (1993) equilibria are essentially Arrow-Debreu equilibria. Appendix A.1
?lls in the details.
De?nition. For given initial state z0 and for given distribution ?0, an equilibrium consists of
prices ?pt(stjs0);?(ztjz0)? and allocations ?ct(?0;st);ht(?0;st)? such that
? For given prices ?pt(stjs0)?, the allocations solve the household's problem of maximizing (1)
subject to (2) and (3) (except possibly on a set of measure zero).
? Markets clear for all t; zt:
X
yt
Z
ct(?0;yt;zt)d?0 ?(y
t;ztjy0;z0)
?(ztjz0) = c
a
t (zt) = et(z
t): (4)
X
yt
Z
ht(?0;yt;zt)d?0 ?(y
t;ztjy0;z0)
?(ztjz0) = h
a
t (zt) (5)
7
To determine the equilibrium consumption of households, it is helpful to examine the dual of
the household maximization problem. Let U0(fcg;fhg) denote the total utility from consuming fcg
and fhg: For given prices fp;?g a household with label (w0;s0) minimizes the cost C(?) of delivering
initial utility w0 to itself:
C(w0;s0) = min
fc;hg
(c0(w0;s0) + h0(w0;s0)?0(s0))
+
X
st
p(stjs0)?ct(w0;stjs0) + ht(w0;stjs0)?t(stjs0)?
subject to the initial promised utility constraint: U0(fcg;fhg) ? w0; and the solvency constraints
(3), one for each node st: The initial promised value w0 is determined such that the household spends
its entire initial wealth: C(w0;s0) = ?0 + ?s0 [f?g]: There is a monotone relationship between ?0
and w0.
Stochastic Consumption Weights Let f?(?0;s0)g denote the sequence of multipliers on the
solvency constraints (3) imposed on household (?0;s0). We de?ne ?t(?0;st) to be household (?0;s0)'s
cumulative Lagrange multiplier:
?t(?0;st) = `(?0;s0) +
tX
?=0
X
s??st
??(?0;s?):
We refer to ?t(?0;st) as the consumption weight in state st for household (?0;s0) in the dual
problem. The initial weight `(?0;s0) is the inverse of the Lagrange multiplier on the initial promised
utility constraint, ?0(?0;s0) = `. f?(?0;s0)g is a non-decreasing stochastic process.
The process ?at (zt) is the aggregate weight R ?t(yt;zt)1? d?t(zt). ?0 is the initial cross-sectional
distribution over `(?0;s0), implied by the initial wealth distribution ?0. ?t(zt) is the distribution
over weights after aggregate history zt. ?at (zt) summarizes to what extent constraints bind on
average.
If the solvency constraint does not bind, the household's weight remains unchanged. When the
constraint binds, its weight increases to a cuto? level `c(yt;zt) that depends only on the current
event yt.
?t = ?t?1 if ?t?1 > `c(yt;zt);
?t = `c(yt;zt) otherwise:
8
This imputes limited memory to the allocations: a household's individual history yt?1 is erased
whenever it switches to a state with binding constraints. This is the amnesia property, present in
many endogenously incomplete markets models (Ljungqvist and Sargent (2004)).
Risk-Sharing Rule There is a mapping from the multipliers at st to the equilibrium allocations
of both commodities. We refer to this mapping as the risk-sharing rule. This rule ?ows from the op-
timality conditions of the dual household problem and the market clearing conditions. Henceforth,
we express individual-speci?c variables as functions of (`;st) rather than (?0;st). We conjecture a
linear risk sharing rule: the consumption share and the housing services share is a function of the
household's own consumption weight and an aggregate sum of these weights:
ct(`;st) = ?t(`;s
t)1?
?at (zt) c
a
t (z
t) and ht(`;st) = ?t(`;st)
1
?
?at (zt) h
a
t (z
t): (6)
Indeed, this rule satis?es the ?rst order condition for non-durable and housing services consumption
and the market clearing conditions.
When a household switches to a state with a binding constraint, its consumption share increases.
Everywhere else, its consumption share is drifting downwards at the rate ? log ?at+1. Shocks to
?at (zt) re?ect aggregate shocks to the wealth distribution. Because they re?ect an inability to
insure against income shocks they can be interpreted as liquidity shocks.
The perfect commitment environment is a benchmark for understanding this risk sharing rule.
Because households are never constrained, the individual weight stays constant and is equal to
the initial consumption weight: ?t(st) = ?0(s0) = `. The aggregate weight process re?ects the
initial wealth distribution and is constant: ?a(z0) = R `(y0;z0)1? d?0(z0). Consumption shares are
constant; consumption levels only move with aggregate consumption. There is full insurance.
Rental Prices In equilibrium, all households equate the ratio of marginal utilities for these
commodities:
?t(zt) = ?
?h
t(?0;st)
ct(?0;st)
??1
"
= ?
?ha
t (zt)
cat (zt)
??1
"
:
The price of rental services is a function of the aggregate history zt only. As a result, all households
have the same equilibrium expenditure ratio rt(`;st) = rt(zt) = cat?
tha
. The price of a claim to the
aggregate housing dividend is ?zt [f?hag].
9
Stochastic Discount Factor For pricing purposes there is a stand-in consumer whose prefer-
ences are de?ned over \twisted" aggregate non-durable and housing services consumption processes:
Uw(fcag ;fhag) = U
??ca
?at
?
;
?ha
?at
??
The risk sharing rules (16) determines a household's intertemporal marginal rate of substitution
(IMRS). In each state, the payo?s are priced by the household with the highest IMRS. This maxi-
mum is attained for the unconstrained households. If not, there would be an arbitrage opportunity.
The implied stochastic discount factor (SDF) is
mt+1 = mat+1
??a
t+1
?at
??
; (7)
The SDF consists of two parts. First, without the collateral constraints, ours is a representative
agent economy. If utility is non-separable, the housing market introduces a novel risk factor: shocks
to the non-housing expenditure share.
mat+1 = ?
?ca
t+1
cat
??? ??a
t+1
?at
?"??1
"?1 :
where ?a is the aggregate non-durable expenditure share: ?at = catca
t +?tha
= rt1+rt . This is the IMRS
of the representative agent economy with non-separable preferences who consumes the aggregate
non-durable and housing services endowment. When preferences are separable (" = ??1), ma is
the SDF of the Lucas (1978) economy.
The second part of the SDF is the growth rate of the aggregate consumption weight process ?at+1.
It re?ects the risk of binding solvency constraints. When many households are severely constrained
in state zt+1, that state's price increases, because the unconstrained households experience high
marginal utility growth: ?a(zt+1) > ?a(zt). When nobody is constrained, the aggregate consump-
tion weight process stays constant (?a(zt+1) = ?a(zt),) and the Breeden-Lucas stochastic discount
factor re-emerges. The risk of binding solvency constraints endogenously creates heteroscedasticity
in the SDF.
The heteroscedasticity of the SDF, and with it the time-variation in the Sharpe ratio, comes
from two mechanisms. The ?rst is the wealth distribution dynamics induced by the solvency
constraints. A larger fraction of agents draws higher labor income shares b(y;z) when aggregate
10
consumption growth is low. As a result of the persistence of labor income shocks, cuto? levels
et(!;st) are higher. This increases the size of the aggregate weight shock ? log ?at+1 and the SDF
in low aggregate consumption growth states.4 The second source of heteroscedasticity relies on
endogenous movements in the collateral ratio; it is this paper's contribution. Movements in the
expenditure ratio and endogenous movements in the stochastic discount factor drive movements in
the housing collateral ratio. When collateral is scarce, households' solvency constraints bind more
frequently (see section 1.3).
No arbitrage implies that the return on a security j, Rjt+1, satis?es Et[mt+1Rjt+1] = 1. We think
of the return on the market as a levered claim to the aggregate non-durable endowment.
1.3. Collateral Supply
The tightness of the constraints depends on the ratio of aggregate housing wealth to total aggregate
wealth. To see this, we aggregate the solvency constraints across households and de?ne the housing
collateral ratio my ?zt?:
my(zt) = ?zt [fh
a?g]
?zt [fca + ha?g] =
?zt ??car ?currency1
?zt ??ca ?1 + 1r??currency1: (8)
Persistent variation in the non-durable expenditure ratio r gives rise to persistent variation in my
and in the amount of risk sharing that can be sustained. So, the housing collateral ratio indexes
the risk-sharing capacity in the economy. We make this more formal in the next propositions.
If the total consumption claim is priced high enough, then perfect risk sharing can be sustained.
Denote the price of a claim under perfect risk-sharing by ?currency1[f?g].
Proposition 1. Perfect risk sharing can be sustained if and only if
?currency1z
??
ca
?
1 + 1r
???
? ?currency1z;y [f?(y;z)g] for all (y;z;r)
Each household can consume the average endowment without violating its solvency constraint.
The following proposition states that an economy with more housing collateral (lower r) has
lower cuto? weights, allowing for more consumption smoothing. In the limit perfect risk-sharing
obtains. Conversely, a decrease in the supply of collateral brings the cuto? rules closer to their
4Constantinides and Du?e (1996) build a negative correlation between the dispersion of consumption growth
across households and aggregate stock returns in their model to generate large risk premia, drawing on earlier work
by Mankiw (1986). The model of Lustig (2003) is a di?erent version of this.
11
upper bound: the labor income shares. In the limit, as the collateral disappears altogether, the
households revert to autarky. The following proposition makes this point more formally.
Proposition 2. Assume utility is separable. Consider 2 economies with r1?(z?) < r2?(z?) for all
z? ? zt. Then the cuto? rules satisfy `1;c(yt;zt) ? `2;c(yt;zt). As r?(z?) ! 1 for all z? ? zt,
`c(yt;zt) ! ^(yt;zt). Conversely, as r?(z?) ! 0 for all z? ? zt; `c(yt;zt) ! 0
Perturbations of the r process also change the equilibrium aggregate weight process. An econ-
omy with a uniformly higher r process and less collateral as a result has higher liquidity shocks
and lower interest rates on average.
Corollary 1. Consider 2 economies with r1t (zt) < r2t (zt) for all zt. Fix the distribution of initial
multipliers across economies: ?10(z0) = ?20(z0): Then
n
?a;1t (zt)
o
?
n
?a;2t (zt)
o
and rf;1 ? rf;2.
The proposition and corollary illustrate the mechanism that underlies the time-variation in the
equilibrium market price of aggregate risk by comparing two economies with di?erent collateral
processes frg. In sections 2 and 3, we calibrate the evolution of the expenditure ratio r, simulate
the model and investigate the equilibrium changes in the conditional moments of the aggregate
weight process.
1.4. Two-agent Example
To provide more intuition for the collateral mechanism, we brie?y consider the case of two agents,
two aggregate states z = (re;ex) and two idiosyncratic income states y = (hi;lo). The vector
(!;r), where ! is the consumption share of agent 1, o?ers a complete description of the state space.
Suppose r is ?xed. The policy rule for next period's consumption share of agent 1 is character-
ized by a lower and an upper bound:
!0(!;y;r) = ! if !(y;r) > ! > !(y;r)
= !(y;r) if ! < !(y;r)
= !(y;r) if ! > !(y;r)
This agent's consumption share is constant as long as he stays in the same state. If she switches
to the high labor income state, then his consumption share jumps up, while the other agent's
12
consumption share jumps down, as in Alvarez and Jermann (2000).5
The Collateral E?ect The size of the jump depends on the level of the non-housing expenditure
ratio r. The cuto? value !(s;r) satis?es the solvency constraint for agent 1:
?(s;r) ??b1ca?currency1 = ?(s;r)
??
cat
?
1 + 1r
?
!
??
=
?
1 + 1r
?
?(s;r) [fcat !g]
From the solvency constraint for agent 2, we ?nd !(s;r):
?(s;r) ???1 ? b1?cat ?currency1 =
?
1 + 1r
?
?(s;r) [fcat (1 ? !t)g]
From the de?nition of the cuto? weights and keeping the pricing functional ? constant, the above
implies that
@!(s;r)
@r > 0 and
@!(s;r)
@r < 0:
The cuto? weights become tighter as the non-durable expenditure ratio r increases. Consequently,
there is less risk-sharing as the housing collateral ratio decreases.
Perfect risk sharing is feasible if solvency constraints are satis?ed when each agent consumes
half of the aggregate endowment in all states:
?currency1(s;r) ??b1ca?currency1 ? 12
?
1 + 1r
?
?currency1(s;r) [fcag] for all s
In the limit, as r decreases, the cuto? consumption shares no longer bind and perfect risk sharing
is feasible.
On the other hand as r increases, then housing contributes no collateral in the limit. Only
autarchy is feasible: !(hi;z;r) = b1hi;z; !(lo;z;r) = b1lo;z: To see why, note that ?currency1(s;r) ??b1?cacurrency1 +
?currency1(s;r) ??b2?cacurrency1 = ?currency1(s;r) [ca].
All previously mentioned insights still hold when r is stochastic, provided it is persistent enough.
The stochastic discount factor in this 2-agent economy is
mt+1 = mat+1
?
min !(st+1;rt+1)!(s
t;rt)
???
:
5The two-agent risk-sharing rule maps into the continuous-agent rule with ! = ?1=?1
?1=?1 +?1=?2 .
13
The size of the aggregate weight shock in the second part of the SDF depends on r. I.e. r governs
the size of the jump in consumption shares when an agent switched from the low to the high income
state.
When r is high enough, there is no collateral and
min !(st+1;rt+1)!(s
t;rt)
= ^?
1
lo;z0
^1hi;z or
1 ? ^1hi;z0
1 ? ^1lo;z ;
while, if r is low enough, consumption shares are constant:
min !(st+1;rt+1)!(s
t;rt)
= 1
Comparing one economy with low r to another with high r, low r means abundant collateral
and small liquidity shocks, while a high current expenditure ratio r means low collateral and large
liquidity shocks. The volatility of the SDF will be higher in the high r economy, because larger
jumps occur when the agents switch between hi and lo. The conditional volatility of the SDF is
essentially constant when r is low.
The average risk premium is large because we set b1hi;re > b1hi;ex and b1lo;re < b1lo;ex. This increase
in the dispersion of idiosyncratic shocks when aggregate consumption growth is low delivers larger
liquidity shocks when aggregate consumption growth is low.
We exploit the ?rst mechanism to impute the right low frequency variation to the market price
of risk. The high frequency variation comes from the second mechanism.
1.5. Sequential Trading
This section describes a sequential trading arrangement. It illustrates the nature of the collateral
constraints in a more intuitive way. We then argue that the equilibrium with sequential trading
can be mapped into the time zero trading equilibrium, de?ned earlier.
The ?nancial markets are complete. Households trade a complete set of contingent claims
a in forward markets. at(`;st;s0) is a promise made by agent (`;s0) to deliver one of unit the
consumption good if event s0 is realized in the next period. These claims trade at a price qt(st;s0).
All prices are quoted in units of the non-durable consumption good. ?t denotes the rental price;
pht (zt) denotes the (asset) price of the housing stock.
14
Household Problem The household problem is to maximize utility over non-durable consump-
tion and rental services (1) subject to the following collateral constraints and wealth constraints.
At the start of the period, the household purchases goods in the spot market ct(`;st), rental services
in the rental market hrt(`;st), contingent claims in the ?nancial market and shares in the housing
stock hot+1(`;st) subject to a wealth constraint:
ct(`;st) + ?t(zt)hrt(`;st) +
X
s0
qt(st;s0)at(`;st;s0) + pht (st)hot+1(`;st) ? Wt(`;st):
Next period wealth is:
Wt+1(`;st;s0) = ?t+1(st;s0) + at(`;st;s0) + hot+1(`;st)
h
pht+1(st;s0) + ?t+1(st;s0)
i
:
All of a household's state-contingent promises are backed by the cum-dividend value of its housing
hot+1, owned at the end of period t. In each node st; households face a separate collateral constraint
for each event s0:
?at(`;st;s0) ? hot+1(`;st)
h
pht+1(st+1) + ?t+1(st+1)
i
; for all st;s0: (9)
The collateral constraints prevent bankruptcy because households are not allowed to borrow more
in a given state than the cum-dividend value of the housing stock in that state.6
Competitive Equilibrium
De?nition. Given a distribution over initial wealth and endowments ?0, a competitive equilibrium
is a feasible allocation ?c(`;st);hr(`;st);at?1(`;st);ho(`;st)? and a price vector ?qt?1;ph;?? such
that (1) for given prices and initial wealth, the allocation solves each household's maximization
problem and (2) the markets for the consumption good, the housing services, the contingent claims
and housing shares clear.
The equilibria in the economy with sequential trading are equivalent to the time zero Kehoe
and Levine (1993) equilibria, if the equilibrium interest rates are high enough.
Proposition 3. If the interest rates are high enough, the sequential equilibrium allocations can be
supported as a Kehoe-Levine equilibrium.
6O? the equilibrium path, default results is the loss of housing collateral. This mimics foreclosure under Chapter
7 of US bankruptcy legislation.
15
The proof is in appendix A.3 and follows Alvarez and Jermann (2000).
To show the equivalence, we de?ne the market state price pt(zt) as the product of the Arrow
prices for the events along a path zt:
pt(zt) = qt?1(zt?1;z0)qt?2(zt?1):::q0(z1);
where pt(zt) is the price at time 0 of a unit of consumption to be delivered at node zt.
By iterating forward on the collateral constraints in (9), substituting for the time 0 budget
constraint, and imposing a no-arbitrage condition on ?ph?, the sequence of collateral constraints
can be restated as a non-negativity constraint on net wealth in every history:
?st [fc(`;s0) + ?h(`;s0)g] ? ?st [f?g] ; 8st;t ? 0: (10)
1.6. Value Premium
Growth stocks (value stocks) can be thought of as a basket of consumption strips that is heavily
weighted towards longer (shorter) maturities (Dechow, Sloan, and Soliman (2002) and Lettau and
Wachter (2004)). Consumption strips are claims to period t + k aggregate consumption (ct+k),
where k is the horizon in years.
Formally, the multiplicative (one year) equity premium on a non-levered claim to the stream of
aggregate consumption fckg1k=1, E0Re0;1[fckg], can be written as a weighted sum of expected excess
returns on consumption strips:
1 + ?0 = 1 + E0[Re0;1[fckg]] = E0M1E0
?P1
k=1 E1MkckP1
k=1 E0Mkck
?
=
1X
k=1
E0MkckP
1
k=1 E0Mkck
E1Mkck
E0Mkck
1
E0M1
=
1X
k=1
!k E0R0;1 [ck]R
0;1 [1]
=
1X
k=1
!kE0Re0;1 [ck] ;
with weights
!k = E0Mkck1
k=1 E0Mkck
:
The second term in the sum is the expected return on a period k consumption strip E0R0;1 [ck] in
excess of the risk-free rate R0;1 [1]. The weights can be interpreted as the value of the period k
consumption strip relative to the total value of all consumption strips. Mk is the pricing kernel in
period k. It is linked to the stochastic discount factor m by Mk = m1 ? ? ? ? ? mk.
16
Value stocks can then be modelled as a claim to a weighted stream of aggregate consumption
ffv(k)ckg1k=1, where the function f(?) puts more weight on the consumption realizations in the
near future. For example, fv(k) = Ceak, where a is a negative number and C is a normalization
constant, C =
P1
k=1 ckP1
k=1 eakck
. Likewise, growth stocks can be thought of as a claim to a weighted
stream of aggregate consumption ffg(k)ckg1k=1, where the function f(?) puts more weight on the
consumption realizations in the far future. For example, fg(k) = Ceak, where a is a positive
number. The multiplicative equity premium on such a claim is
1 + ~0 =
1X
k=1
~k E0R0;1 [ck]R
0;1 [1]
;
with modi?ed weights
~k = f(k)E0Mkck1
l=1 f(l)E0Mlcl
:
The following proposition shows that the properties of the pricing kernel determine the sign of
the value spread. In particular, if the pricing kernel has no permanent component, then the model
generates a growth premium.
Proposition 4. If ? > 1 and f(k) = Ceak;a > 0 then
lim
k!1
Et+1Mt+k
EtMt+k = 1 ) lim 1 + e?0 = lim R
c
t+1;k ? 1 + ?0;
for any other sequence of weights f!kg
Proof: see appendix. The proof relies on insights in Alvarez and Jermann (2001).
This implies that the highest equity premium is the one on the farthest out consumption strip.
In the absence of a permanent component in the pricing kernel, there is a growth premium. The
pricing kernel in our model contains a permanent component through the multiplicative component
stemming from the risk of binding solvency constraints: The aggregate weight shock (?a)? is a non-
decreasing stochastic process. This is a necessary condition for generating a value premium.
In the representative agent economy, the equity premia on consumption strips do not change
with the horizon. This is easy to show for additive preferences that are separable in both com-
modities and aggregate endowment growth that is i.i.d with mean ?. The pricing kernel is
simply a function of the aggregate consumption growth rate between period 1 and period k:
Mk = ???k ??k?1 ? ? ? ???1 . Because the aggregate endowment grows every period at the rate ?,
17
M1c1 = ???1 ?1c0. For the period k strip, Mkck = ?1??k ?1??k?1 ? ? ? ?1??1 c0. Hence, the expected
return on a period k strip is:
E0R0;1 [ck] = E1 (Mkck)E
0 (Mkck)
=
E1
?
?1??k ?1??k?1 ? ? ? ?1??1 c0
?
E0
?
?1??k ?1??k?1 ? ? ? ?1??1 c0
? =
??
1
??
?1??
The expression does not depend on the horizon k. This shows that equity premia are constant across
strips of di?erent horizons in the representative agent economy. The term structure of consumption
strips is ?at.7
2. Computation
To solve the model numerically, we rely on an approximation of g; the growth rate of the aggregate
weight process using a truncated history of aggregate shocks. This is discussed in section 2.1. In
2.2, we fully calibrate the model. We simulate the model and discuss the results in section 3.
2.1. Approximating Stationary Equilibria
In general, the aggregate weight process depends on the entire history of shocks z1. To avoid
the curse of dimensionality, we truncate aggregate histories (Lustig (2003)). Households do not
keep track of the entire aggregate history, only the last k lags: zkt = (zt;zt?1;? ? ? ;zt?k) and the
current expenditure ratio rt(zt). The current expenditure ratio rt contains additional information
not present in the truncated history zk, namely rt?k.
For a household starting the period with weight ? 2 L, the policy function l(y0;z0;?;r;zk) :
L ? R ? Zk ! R produces the new individual weight in state (y0;z0). There is one policy function
l(?) for each pair (y0;z0) 2 Y ? Z. The policy function gcurrency1(z0;r;zk) : R ? Zk ! R forecasts the
aggregate weight shock when moving to state z0 after history (zk;r).
De?nition. A stationary stochastic equilibrium is a joint distribution over individual weights,
individual endowments, current housing - endowment ratio, truncated aggregate histories, a time
invariant distribution ?currency1(r;zk) (?;y), and updating rules l(?) and gcurrency1(?). For each ?zk0;zk? with zk0 =
7A similar result obtains if preferences are non-separable and aggregate expenditure share growth is i.i.d., even
when aggregate expenditure share growth is correlated with aggregate consumption growth.
18
?z0;zk?
?currency1(r;zk0) =
X
zk
?(zk0jzk)
Z
Q
?
?;y;r;zk
?
?currency1(r;zk)(d? ? dy)
where Q??;y;r;zk? is the transition function induced by the policy functions.
The forecast of the aggregate weight shock satis?es
gcurrency1(z0; r;zk) =
X
y02Y
Z
l(y0;z0; ?;r;zk)1? ?currency1r;zk(d? ? dy)?(y
0;z0jy;z)
?(z0jz) ; (11)
for each z0. Prices are determined using the stochastic discount factor in equation (7), and using
gcurrency1(?) as an approximation to the actual g(?).
For any given realization fzg, the actual aggregate weight shock g(?) di?ers from the forecast
gcurrency1(?) because the distribution over individual weights and endowments ?currency1(?) di?ers from the actual
distribution ?(?), which depends on z1. The de?nition of stationary equilibrium implies that, on
average across aggregate histories, ?currency1(?) = ?(?), and markets clear. That is, for every aggregate
state z0, the allocation error
ca(z0;r;zk) ? ca(z0; z1) = g
currency1(z0; r;zk) ? g(z0; r;z1)
gcurrency1(z0; r;zk) (12)
is on average zero.8 As k increases, the approximation error decreases because market clearing
holds on average in long histories.
Algorithm We compute the approximating equilibrium as follows. The aggregate weight shock
process is initialized at the full insurance value (gcurrency1 = 1) and the corresponding stochastic discount
factor is computed. The cuto? rule for the individual weight shocks ensure that the solvency
constraints holding with equality. Then the economy is simulated by drawing fztgTt=1 for T = 10;000
and fytgTt=1 for a cross-section of 5;000 households. For each truncated history, we compute the
sample mean of the aggregate weight shock fgcurrency1t (z0;r;zk)gTt=1 and the resulting stochastic discount
factor fmcurrency1t (z0;r;zk)gTt=1. A new cut-o? rule is computed with these new forecasts. These two steps
are iterated on until convergence.
Throughout we use k = 5 and report percentage allocation errors as a measure of closeness to
the actual equilibrium. The average error in equation 12 in a simulation of 10,000 periods is 0.0011
8There is an exact aggregation result if aggregate uncertainty is i.i.d., with k=0. See Lustig (2003) for a proof in
a model without housing.
19
with standard deviation .0035. The largest error in absolute value is 0.0282.
2.2. Calibration
In this section we fully calibrate the model and report the approximation errors. Table 2 summarizes
our choices for benchmark and alternative parameter values.
Income Process The ?rst driving force in the model is the Markov process for the non-durable
endowment. It contains an aggregate and an idiosyncratic component.
The aggregate endowment growth process is taken from Mehra and Prescott (1985) and repli-
cates the several moments of aggregate consumption growth in the 1871-1975 data. The growth
rate of the aggregate endowment, ?, follows an autoregressive process:
?t(zt) = ???t?1(zt?1) + "t;
with ?? = ?:14;E(?) = :0183 and ?(?) = :0357. We discretize the AR(1) process with two
aggregate growth states z = (zexp;zrec) = [1:04;:96] and an aggregate state transition matrix
[:83;:17;:69;:31]. The implied ratio of the probability of a high aggregate endowment growth state
to the probability of a low aggregate endowment growth state is 2.65. The unconditional probability
of a low endowment growth state is 27.4 percent.
The calibration of a heteroscedastic labor income process is taken from Storesletten, Telmer,
and Yaron (2004). They conclude that the volatility of idiosyncratic labor income shocks in the US
more than doubles in recessions. Log labor income shares follow an AR(1) with autocorrelation of
.92 and a conditional variance of .181 in low and .0467 in high aggregate endowment growth states.
Again the AR(1) process is discretized into a two-state Markov chain. The resulting individual
income states are ??hi;?lo? = [:6578;:3422] in the high and [:7952;:2048] in the low aggregate
endowment growth state.9 We refer to the counter-cyclical labor income share dispersion as the
Constantinides and Du?e (1996) e?ect.
Expenditure Ratio Following Piazzesi, Schneider, and Tuzel (2004), we specify an autoregres-
sive process for the expenditure ratio r. The aggregate expenditure depends on the aggregate
9The only di?erence with the Storesletten, Telmer, and Yaron (2004) calibration is that recessions are shorter in
our calibration. In their paper the economy is in the low aggregate endowment growth state 50 percent of the time.
That implies that the unconditional variance of our labor income process is lower.
20
endowment growth:
log rt+1 = ?+ ?r log rt + br?t+1 + ?r?t+1; (13)
where ?t+1 is an i.i.d. standard normal process with mean zero, orthogonal to ?t+1. In our
benchmark calibration we set ?r = :96, br = :93 and ?r = :03. We discretize the process for log(r)
as a ?ve-state Markov process.
The parameter values are close to the estimates of (13) we ?nd in US National Income and
Products Accounts Data. Panel A of table 1 shows estimates for ?r and br that are consistent across
samples and data sources. In periods of high aggregate consumption growth, the expenditure ratio
increases.10
A second calibration switches o? the e?ect of ? on log(r): ?r = :96, br = 0. Both calibrations ?x
?r = :03. We choose the constant ? to match the average housing expenditure share of 19 percent
in the National Income and Product Accounts data for 1929-2002.
Average Housing Collateral ratio We scale up aggregate income. This scaling allows us
to simultaneously have an average expenditure share of housing services of 19 percent and an
average ratio of housing wealth to total wealth 5 percent (benchmark). In the model, the ratio
of the aggregate non-durable endowment et to the aggregate non-durable consumption cat is 1.
The empirical counterpart to et is compensation of employees. The empirical counterpart to cat
is consumption expenditures on non-durables and services excluding housing services. On average
between 1929-2003, the ratio of the former to the latter is 1.17. We use this factor to scale up labor
income ? in the model. The rescaling implies an average housing collateral ratio of 5 percent. We
investigate the sensitivity of our results by also considering an economy with ten percent collateral.
Preference Parameters In the benchmark calibration we use additive utility with ? = :95,
? = f5;8g, " = :05. We ?x ? = 1 throughout.11 We also compute the model for ? 2 [2;10] and
" 2 [:05;:75]. A choice for the parameter " implies a choice for the volatility of rental prices:
?(?log ?t+1) =
??
?? 1
" ? 1
??
???(?log rt+1)
10Alternatively, we could have calibrated a persistent process for the rental price log(?
t). Panel B shows that rental
prices increase in response to a positive aggregate consumption growth in the post-war sample.
11Note that ?cucc
uc = ?t? + (1 ? ?t)
1
". It is a linear combination of ? and " with weights depending on thenon-durable expenditure share ?
t = ct?tht+ct . In all calibrations ?t = :81 on average.
21
In NIPA data (1930-2002), the left hand side is .046 and the right-hand side is .041. The implied
" is .098. A choice for " too close to one implies excessive rental price volatility. We take " = :05
as our benchmark and explore parameter values " ? :75.
Our benchmark parametrization, as well as the other parameters we consider for sensitivity
analysis are summarized in table 2.
Market Return We assume that ?nancial assets are in zero net supply.12 Because of complete
markets we price them as redundant securities. The de?ne the market return as the return on a
levered claim to the aggregate consumption process fcat g. In the data, dividends are more volatile
than aggregate consumption. For the period 1930-2001, the leverage parameter ? in ? log dt+1 =
??log cat+1, is 4.4. We denote the return on a levered claim to aggregate consumption growth
Rl and choose leverage parameter ? = 3. We also price a non-levered claim on the aggregate
consumption stream. We denote the corresponding return Rc.
3. Results
Our empirical targets are twofold: (1) to quantitatively assess the variation in conditional asset
pricing moments, conditional on the housing collateral ratio and compare it to the data (section 3.2),
and (2) to generate a return spread between value and growth portfolio returns of the magnitude ob-
served in the data (section 3.3). We also match the unconditional moments for the equity premium,
its unconditional volatility and Sharpe ratio, and the risk-free rate (section 3.1). Throughout, we
contrast the results with those of a representative agent economy with non-separable preferences.
3.1. Unconditional Asset Pricing Moments
Collateral Model Table 3 summarizes the unconditional ?rst and second moments of asset
returns for the collateral model with non-separable preferences. Table 4 reports unconditional
asset pricing moments for the representative agent model with non-separable preferences.
Because consumption growth is less volatile in the data than dividend growth, the relevant
comparison of the excess stock market return in the data is with a leveraged consumption claim
12Allowing for ?nancial assets in positive net supply will not qualitatively a?ect the dynamics of our model. It
will merely alter the ratio of collateralizeable wealth to total wealth. In the data, mortgages and home equity lines
of credit represent 75 percent of all collateral assets (household sector balance sheet, Flow of Funds data). For these
two reasons, the omission of ?nancial wealth as collateral does not seem critical.
22
in the model. The benchmark model with ?ve percent collateral is able to generate a high and
volatile levered equity risk premium. The calibration with ? = 5 and " = :05 in panel 1 of table
3 generates a 5.1 percent equity premium. The standard deviation is 21.3 percent. In the data
(panel 6), the excess return on the market portfolio is 7.5 percent with a volatility of 19.8 percent.
To understand the e?ect of the leverage, we also price a non-levered consumption claim. Its equity
premium is 3.0 percent for ? = 5 and 7.1 percent for ? = 8. The Sharpe ratio in the model with
? = 8 is 0.42, equal to the Sharpe ratio observed for 1927-2002.
The model with ? = 8 (? = 5) generates an average risk free rate of 2.9 (5.7) percent, close
to the 3.9 percent in the data. For a higher ?, the aggregate weight shocks are bigger on average.
This increases the conditional expectation of the stochastic discount factor and pushes down the
risk-free rate. The one failure of the model with additive utility is the unconditional volatility of
the risk-free rate. It is 7.3 percent in the economy with ? = 5, 12.5 percent in the economy with
? = 8, but only 3.2 percent in the data.13
Panel 2 of table 3 explores the e?ect of varying the value of the intratemporal elasticity pa-
rameter ". The e?ect of a higher intratemporal elasticity of substitution is to increase the equity
premium and the market price of risk and to lower the risk-free rate. In the case of ? = 8, the
equity premium for " = :75 is 4.5 percent higher than for " = :05. This is largely due to a decline in
the risk-free rate of 4.4 percent. Apart from making the collateral ratio more volatile, changing "
mainly changes the representative agent component ma of the stochastic discount factor. Because
the average amount of collateral is held ?xed at 5 percent, this is largely a `representative agent
economy' e?ect.
Increasing the coe?cient of relative risk aversion ? from 2 to 10 in panel 3 increases the equity
premium on a levered (non-levered) consumption claim from 1.3 to 15.1 (.5 to 11) percent. The
unconditional Sharpe ratio increases from .08 to 0.53. The increase in ? decreases the risk-free rate
from 7 to 0 percent. The ?rst reason for this risk-free rate e?ect is that households cannot borrow
as much as they would like because of binding collateral constraints. Second, the risk-free asset
provides insurance against binding constraints, and this lowers the risk-free rate even further. This
is a precautionary savings e?ect coming from the constraints. If households are more risk averse,
they want to insure better against the risk of binding constraints. The main e?ect of a higher
13Risk-free rates were more volatile prior to 1927. The unconditional standard deviation of the real risk-free rate is
6.5 percent for the 1889-1979 period (data from Shiller's web site). The model with recursive preferences, described
in appendix A.4, generates a volatility of the risk-free rate of 6.5 percent. It still generates a large equity premium
(6 percent), a low risk-free rate, and volatile stock market returns.
23
coe?cient of relative risk aversion is to amplify the collateral e?ect, coming through the second
part of the stochastic discount factor, g?t+1.
The economy with ten percent collateral is closer to the representative agent economy because
the collateral constraints are not as tight. The expected excess return on a levered consumption
claim is still high. For (? = 8;" = :05), the equity premium is 7.8 percent (panel 4). With 10
percent collateral, the risk-free rate is higher on average, but less volatile.
Lastly, panel 5 shows that the results are not very sensitive to a change in the expenditure share
process. This table displays the results for an AR(1) speci?cation for log rt without consumption
growth term on the right-hand side (br = 0). Under this speci?cation, the housing collateral ratio
is less volatile. The equity premium and the risk-free rate are one percent lower on average. The
volatility of Rl;e is 5 percent lower than in the benchmark economy. Changes in " have a smaller
e?ect on the unconditional asset pricing moments.
Representative Agent Model We contrast the results from the collateral model with the
unconditional asset pricing moments in the representative agent economy. Preferences are non-
separable between non-durable and housing services consumption, but the collateral a?ect is shut
down. The equity premium is compensation for aggregate consumption growth risk (as in Lucas
(1978)) and aggregate composition risk (as in Piazzesi, Schneider, and Tuzel (2004)).
Table 4 shows that the equity premium in the representative agent economy is substantially
smaller than in the collateral model. The levered consumption claim has an expected excess return
of 2.4 percent for (? = 5;" = :05) and 4.5 percent for (? = 8;" = :05). This is less than half as
big as for the collateral model with ?ve percent collateral. The equity premium on a non-levered
claim is one-third the magnitude of the collateral model. In addition, the levels of the risk-free
rate and the stock return are much too high in the representative agent economy. For ? = 8, the
risk-free rate is 15.8 percent and the stock return is 20.3 percent on average. This compares to
4 percent and 11.4 percent in the data for 1927-2002. Moreover, the risk-free rate increases with
?. A more risk-averse representative agent wants to borrow increasingly against her labor income
and this drives up the risk-free rate. Note that this is the opposite risk-free rate e?ect as in the
collateral model.
An increase in the intratemporal elasticity of substitution " from .05 to .75 increase the equity
premium by 4 percent (1.7) percent in a representative agent economy with ? = 8 (? = 5). The
risk-free rate goes down by 6 (1.9) percent. The average Sharpe ratio for the representative agent
24
economy with " = :75 is .47 (.27), in line with the historical average. However, the empirically
plausible equity premium, risk-free rate and Sharpe ratio for " = :75 come at the expense of an
implausibly high rental price growth volatility. For " = :75, the unconditional standard deviation
of rental price growth is 19 percent per annum. In the data, rental price growth volatility is
below 5 percent. Driving " even closer to one leads to exponentially increasing rental price growth
volatility.14 This is the reason we choose " = :05 as our benchmark economy. In addition, the
volatility of the risk-free rate increases sharply with ".
Housing Market Statistics The model also has predictions for the return on housing in own-
ership. In the collateral model, the expected excess return on a claim to the aggregate housing
dividend stream is similar to the return on the aggregate consumption stream. Panel 1 of table
5 shows that the housing equity premium is 6.8 (2.6) percent for ? = 8 (? = 5). The standard
deviation of this return is 19.2 (12.7) percent, leading to a Sharpe ratio of .35 (.20). The Sharpe
ratio on a non-levered consumption claim in table 3 was .40 (.23). Using PSID data for 1968-1992,
Flavin and Yamashita (2002) estimate the expected excess return and its standard deviation on
home ownership to be 6.6 percent and 14.2 percent. This implies a Sharpe ratio for housing of .46.
Our benchmark model with ? = 8 generates expected returns and Sharpe ratios for the housing
market that are broadly consistent with the Flavin and Yamashita (2002) numbers. In the repre-
sentative agent economy (panel 3 of 5), the expected excess return on home ownership is too low,
the expected return too high and not su?ciently volatile to match the data.
No Conditional Heteroscedasticity in Income To gauge the relative importance of the two
sources of time-variation in risk premia, we shut down the Constantinides and Du?e (1996) mech-
anism. The labor income share in the low idiosyncratic income state is the same whether the
economy experiences low or high aggregate consumption growth. More precisely, we set the income
share ? equal to [.6935,.6935,.3065,.3065] instead of [.6578,.7952,.3422,.2048] in the benchmark cal-
ibration. This implies the same unconditional labor income volatility, but the labor income share
process is conditionally homoscedastic. It is constant across aggregate growth states.
Table 6 shows that the housing collateral mechanism alone generates a sizeable equity premium
of 7.4 percent (? = 8, 5 percent collateral). The standard deviation of the excess return is 23.5
percent and the Sharpe ratio is 0.31. This amounts to 70 percent of the magnitudes we found for
14For " > 1 the representative agent model generates a negative equity premium.
25
our benchmark model in table 3. The mean risk-free rate is slightly higher, but less volatile.
3.2. Conditional Asset Pricing Moments
The ?rst main prediction of the model is that asset prices behave di?erently in episodes of high
collateral and in periods with collateral scarcity. Expected excess returns are low and the risk-free
rate stable in periods when my is high. However, when collateral is scarce, the equity premium and
the Sharpe ratio are high. Movement in the housing collateral ratio induces substantial variation
in the Sharpe ratio. In this section we do a quantitative assessment of this time-variation and
show that is consistent with the data. The unconditional asset pricing moments obscure this time-
variation because they average over di?erent collateral regimes. The failure of the representative
agent model goes beyond unconditional asset pricing moments; it misses the time-variation in the
Sharpe ratio completely.
Shocks to the Risk Sharing Technology The shocks to the collateral ratio come from shocks
to the housing endowment. Panel 1 of ?gure 1 shows the housing collateral ratio my together with
the ratio of housing services consumption to total consumption 1 ? ? = 11+r. It is a typical two
hundred period window of a long simulation of the benchmark model. The housing collateral ratio
is the closely correlated with the housing expenditure share. It is also a persistent process.
Panel 2 illustrate the collateral mechanism. It plots the cuto? consumption share, which is the
consumption share at which the solvency constraint holds with equality (dotted line). This is the
consumption share of a constrained household. The household's average income share is normalized
to one. The consumption share jumps to the cuto? level when the household runs into a binding
constraint. This happens when its income share switches from the low to the high idiosyncratic
state. The graph shows that the consumption share for constrained households is bigger when
collateral is scarce (my is low, full line). For example, in period 195, the consumption share is 15
percent above its mean of one, whereas in period 50, the consumption share is only 7 percent above
its mean.
When collateral is scarce, the solvency constraints bind more severely. The consumption share of
the constrained households takes a large jump up, while the unconstrained households' consumption
share decreases precipitously. As a result, the cross-sectional standard deviation of consumption
growth increases. In times of collateral scarcity, there is less risk-sharing. Panel 3 of ?gure 1 plots
the consumption growth dispersion (dashed line, left axis) against the housing collateral ratio my
26
(full line, right axis).
The aggregate weight shock gt+1 = ?log ?at+1, which we refer to as the liquidity shock, governs
the rate at which the consumption share of the unconstrained agents decreases. Panel 4 plots the
aggregate weight shock g? (dotted line, left axis) against the housing collateral ratio (full line, right
axis). In times of collateral scarcity, the constraints bind more tightly and this is re?ected in a
large liquidity shock. For example, in period 50 or 110, the liquidity shock is close to one, whereas
in period 195 it is 1.07. The stochastic discount factor is high and more volatile in such periods.
When housing collateral is abundant, the aggregate weight shock is close to 1 and our model's
stochastic discount factor reduces to the one in the representative agent economy. We turn to the
e?ects on asset prices next.
Conditional Asset Pricing Moments The expected return on stocks in excess of the risk-free
rate is higher in periods of collateral scarcity. Zooming in on the same 200 simulation periods
of the benchmark calibration, the ?rst panel of ?gure 2 displays an expected excess return on a
non-levered claim to aggregate consumption (dotted line, left axis). The equity premium is below 4
percent when the housing collateral ratio is high (for example in period 110), and almost 11 percent
when my is low (for example in period 190).
Likewise, the second panel of ?gure 2 shows that the conditional volatility of the excess return
on the consumption claim (dotted line, left axis) is 10 percent when collateral is abundant (period
110) and doubles to 20 percent when collateral is scarce (period 195). Excess returns are much
more stable when collateral is abundant.
The net result of the collateral mechanism is a Sharpe ratio that is higher in times of collateral
scarcity. The third panel of ?gure 2 plots the Sharpe ratio on the stock return against the housing
collateral ratio (dotted line, left axis). It is 0.3 in period 110 and almost 0.6 in period 195.
We estimate the Sharpe ratio on annual data from 1927-1992 and compare it to the variation
in the Sharpe ratio generated by the model. The conditional mean return is the projection of the
excess return on the housing collateral ratio, the dividend yield and the ratio of aggregate labor
income to consumption, all of which have been shown to forecast annual returns. Likewise, the
conditional volatility is the projection of the standard deviation of intra-year monthly returns on
the same predictors. Using the projection coe?cient estimates we form the Sharpe ratio as the
ratio of the predicted excess returns and predicted volatility. Table 7 shows the estimation results
for 1 year returns (column 1), but also for 5 year and 10-year cumulative excess returns. The last
27
three lines indicate the mean and standard deviation of the Sharpe ratio as well as its correlation
with the housing collateral ratio. In the estimation, the standard deviation of the Sharpe ratio
on 1, 5 and 10 year cumulative excess returns is .10, .18, and .20. Lettau and Ludvigson (2003)
do a similar exercise for quarterly excess returns between 1952:4 and 2000:4.15 Their estimate of
the unconditional standard deviation of the Sharpe ratio is .45. In the model, the unconditional
standard deviation of the Sharpe ratio is .40, .42, .40 for 1, 5 and 10 year cumulative excess returns
on a non-levered consumption claim. Our model easily generates a highly volatile Sharpe ratio.
Other models have a hard time generating this volatility. The unconditional standard deviation of
the Sharpe ratio is .29 in the collateral model, compared to .09 for the Campbell and Cochrane
(1999) model and the consumption volatility model of Lettau and Ludvigson (2003). The volatility
of the Sharpe ratio in the representative agent model is even smaller.
Furthermore, the correlation between the Sharpe ratio and the measure of collateral scarcity
f is positive in the data and equal to .25, .32, and .50 for 1, 5 and 10 year cumulative excess
returns. The corresponding correlations in the model are large and positive (.50, .59 and .39).
Figure 3 plots the Sharpe ratio on 5-year and 10-year cumulative returns for the collateral model.
Figure 4 does the same for the 5-year cumulative return in the data.
Figure 5 summarizes the ?ndings for conditional asset pricing moments. The top row plots
the conditional mean, standard deviation and Sharpe ratio on a claim to aggregate consumption,
averaged over histories of the aggregate state zk, against the housing collateral ratio. In each graph,
the solid line plots the conditional moments, conditional on observing a low aggregate consumption
growth rate tomorrow (?(z0) = :96), the dashed line is conditional on observing a high aggregate
consumption growth rate (?(z0) = 1:04). On average, the equity premium is 9 percent higher when
collateral is scarce (my = .04) than when it is abundant (my = .10), conditional on being in a boom
tomorrow. It is 6 percent higher conditional on being in a recession. Stock returns are up to ten
percent more volatile when collateral is scarce. The conditional Sharpe ratio, conditional on being
in a boom tomorrow, is .6 when collateral is scarce and .3 when collateral is abundant. The bottom
row displays the conditional market price of risk ?t[mt+1]=Et[mt+1], an upper bound on the Sharpe
ratio, the conditional price dividend ratio and the risk-free rate. The price-dividend ratio is high
when collateral is scarce. The demand for insurance against binding solvency constraints drives
15They ?nd that the dividend yield, the default spread, the term spread, the relative risk-free rate and the con-
sumption wealth ratio plus two lags of volatility jointly explain roughly 30 percent of the time series variation in the
conditional second moment and 9 percent of the time series variation in the conditional ?rst moment of the excess
stock market return. They using this set of variables to form conditional moments and to compute the Sharpe ratio.
28
up the price stocks. So, the model simultaneously generates a high equity premium and a high
price-dividend ratio because the risk-free rate is very low when collateral is scarce (last panel). The
model with recursive preferences shows the same conditional asset pricing patterns (see appendix
A.4).
Long Horizon Predictability One of the implications of the time-variation in the equity pre-
mium is that the housing collateral model should predict returns. We explore this predictability
in depth in our empirical paper (Lustig and VanNieuwerburgh (2004b)). Panel 1 of table 8 sum-
marizes the predictability results of the housing collateral ratio for 1 to 8 year ahead cumulative
excess returns (data for 1927-2003 and 1945-2003). The housing collateral ratio we use in this table
is based on the outstanding value of mortgage debt (see appendix A.5 for a detailed description of
its construction). The data are supportive of the collateral e?ect: excess returns are higher when
collateral is scarce. The e?ect becomes larger and statistically more signi?cant with the horizon.
The R2 increases.16
Our model replicates the pattern of predictability for the housing collateral ratio. The second
panel of table 8 reports regression results inside the model of excess returns on our measure of
housing collateral ratio scarcity. When housing collateral is scarce (my is low), the excess return
is high. The magnitude of the slope coe?cients is close to the one we ?nd in the data. Moreover,
the R2 of the predictability regression increase with the predictability horizon, just as in the data.
We ?nd this negative relationship between myt and the excess return on a non-levered claim to
aggregate consumption, as well as for a levered claim to aggregate consumption.
Other variables, such as the price-dividend ratio and the risk-free rate have also been shown
to predict excess returns. In the data, a higher dividend yield and a lower risk-free rate forecast
higher future excess returns. The former becomes more important at longer horizons, whereas the
latter is more important at short horizons. We also explore the predictability of the price-dividend
ratio for returns inside the model. The theory predicts a negative sign for returns and a positive
sign for excess returns, whereas the data show a negative sign in both regressions.17 Lastly, the
model predicts a negative relationship between current risk-free rate and future excess returns but
a positive relationship with future returns. The data show a negative relationship in either case.
16Predictability results for the other two collateral measures we consider are reported in Lustig and VanNieuwer-
burgh (2004b).
17Because the risk-free rate is not very volatile in the data, the empirical results for regressions where the left-hand
side variable is the real returns instead of the excess return are very similar.
29
This pattern of predictability of the price-dividend ratio and the risk-free rate is mainly driven
by the risk-free rate dynamics. When collateral is scarce, the price-dividend ratio is high and the
risk-free rate is low. The theory predicts that the equity premium is high (see ?gure 5). Because of
the persistence in the housing collateral ratio, future equity premia are also high. This must mean
that future realized excess returns are high on average. However, the high excess returns are the
result of lower realized returns and even lower future risk-free rates.18
3.3. Value Premium
Value ?rms, with a high ratio of book equity to market equity, historically pay higher returns than
growth ?rms, with a low book-to-market ratio. We use annual return data for 1927-2003 for the
US for 10 value portfolios from Fama and French (1992). The decile portfolios are formed every
year by sorting the universe of stocks on the ratio of book value to market value of equity. Table
9 reports sample means for the excess return, its unconditional volatility and the Sharpe ratio on
the ten book-to-market deciles. The annual excess return on a zero-cost investment strategy that
goes long in the highest book-to-market decile and short in the lowest decile is 5.5 percent for
1927-2003. The value premium is 5.7 percent for quintile portfolios. Similar value premia are found
for monthly and quarterly returns. Using quarterly data for 1951-2002, Lettau and Wachter (2004)
document that the unconditional Sharpe ratio for value stocks (.64) is twice as large as for growth
stocks (.32). In table 9, we ?nd a similar increase for annual data from 1945-2003 (from .37 to .56),
but a smaller increase over the entire period 1927-2003 (.32 to .42).
Lustig and VanNieuwerburgh (2004b) show that value stocks command higher expected returns
because their returns covary more strongly with aggregate consumption growth when collateral is
scarce. This mechanism accounts for more than 80 percent if the cross-sectional variation in the
book-to-market portfolio returns in the data. The second main exercise of this paper is to show
the collateral model can endogenously generate a value premium of the magnitude observed in the
data.
Decile Return Processes in Data In a ?rst step, we use the data on the decile value portfolio
returns to describe the return-generating process for each of the book-to-market decile portfolios.
We specify return processes for value and growth stocks as linear functions of the state variables
in our model. These are aggregate consumption growth, aggregate expenditure share growth, the
18Results for the data and model are available upon request.
30
housing collateral ratio and the interaction terms of the housing collateral ratio with aggregate
consumption growth and expenditure share growth. The return in excess of a risk-free rate on the
jth book-to-market decile portfolio is:
Re;jt+1 = ?0 + ?jmy f t+1 + ?jc?log cat+1 + ?jc;my f t+1?log cat+1 +
?j??log ?at+1 + ?j?;my f t+1?log ?at+1 + ?jt+1; (14)
where f t+1 = mymax?myt+1mymax?mymin is rescaled so that all realizations are between 0 and 1. It is a measure
of collateral scarcity. The beta vector in equation (14) is estimated by ordinary least squares and
reported in table 10. The ?rst panel uses a housing collateral ratio based on the market value of
outstanding mortgages, the second panel uses a measure based on residential real estate wealth and
the third panel employs a measure based on residential ?xed asset values.
Returns on value stocks (decile 1) are high in recessions, i.e. they covary negatively with
aggregate consumption growth. Growth stocks are much less sensitive to aggregate consumption
growth; j?cj increases monotonically from decile 1 to decile 10. This pattern is robust across
di?erent collateral measures. As for the collateral e?ect, value stocks covary more strongly with
aggregate consumption growth when collateral is scarce (?c;my > 0). For growth stocks this e?ect
is much less pronounced; ?c;my increases monotonically from decile 1 to decile 10. This pattern
is robust across di?erent collateral measures. Lastly, value stocks are more sensitive to aggregate
expenditure share shocks; ?? increases monotonically for all three collateral measures.
Decile Return Processes in Model In a second step, we generate ten excess return processes
as the product of the factor loadings ??jmy;?jc;?jc;my;?j?;?j?;my? estimated from the data and the
aggregate state variables (factors) generated in the model. For each excess return, the intercept
?j0 is determined such that the Euler equation Et[mt+1Rj;et+1] = 0 is satis?ed. This is a consistency
requirement that the SDF of the model prices these assets correctly. We then simulate the model
for 10,000 periods and compute unconditional mean and standard deviation of each of the decile
portfolio returns.
Table 11 reports the excess returns on the ten value portfolios predicted by the collateral model,
ordered from growth (B1) to value (B10) for ? = 5 and ? = 8. In each panel we use three sets of
empirical factor loadings, corresponding to the three housing collateral measures. These are the
three sets of betas from table 10.
31
For the ?xed asset measure, the value spread is 6 percent for ? = 8 and 4 percent for ? = 5. For
the other two collateral measure, the value spread is 4 percent for ? = 8 and 3 percent for ? = 5.
For ? = 8, the model is able to replicate the 5.2 percent value spread in the data. Furthermore,
the model predicts a noticeable increase the Sharpe ratio between the ?rst decile (growth) and the
tenth decile (value). The Sharpe ratio doubles for all three collateral measures and ? = 8. In the
data there is similar increase in the higher Sharpe ratio for post-war data (see table 9).
In contrast, the representative agent economy generates no value premium. We estimate the
consumption betas from an equation like 14, but with aggregate consumption growth and aggregate
expenditure share growth as the factors. These are the only two risk factors in the representative
agent economy. There is much less of a pattern in the betas than what we found for the collateral
model. The estimated betas are used together with aggregate consumption growth shocks in the
model to generate 10 excess return processes. There is no pattern in the excess returns. The value
premium is zero. The Sharpe ratios on growth stocks are higher than the ones on value stocks, the
opposite pattern as found in the data.19
Duration Value stocks are stocks whose cash-?ow pro?les are tilted towards the present whereas
growth stocks' payo?s are farther in the future. Dechow, Sloan, and Soliman (2002) estimate a 6
year duration for value stocks and a 20 year duration for growth stocks.
Lettau and Wachter (2004) provide insight into the properties that an asset pricing model must
satisfy to generate a lower expected return for long duration assets (growth stocks) than for short
duration assets (value stocks). The correlation between aggregate consumption growth and the
Sharpe ratio on equity must be non-negative. Then, long duration assets are less risky because a
negative consumption growth innovation leads to lower risk premia in the future. They point out
that many asset pricing models, such as the Campbell and Cochrane (1999), generate a negative
correlation instead.
Our model endogenously generates this positive correlation. Figure 6 plots the Sharpe ratio for
a simulation of the model and indicates periods with negative consumption growth by shaded bars.
The Sharpe ratio increases during periods of high aggregate consumption growth and falls when
consumption growth is negative.20 There is history dependence. The decline is bigger the longer was
19Detailed results available upon request.
20We ?nd the same positive correlation in the data. We form a conditional Sharpe ratio as the ratio of predicted
excess returns and predicted volatility, based on regressions with an aggregate consumption growth dummy, a lagged
aggregate consumption growth dummy and the housing collateral ratio as independent variables. The dummy vari-
ables capture boom times, they are 1 when aggregate consumption growth is above the sample mean minus one
32
the preceding boom. The left picture on the second row of ?gure 5 shows the same relationship.
The conditional market price of risk is higher in periods of high aggregate consumption growth
(dashed line) than in periods of low aggregate consumption growth (full line).
The key to this positive correlation is the combination of history dependence in the wealth
dynamics and the persistence of the housing collateral ratio. In every successive period of high
aggregate consumption growth, the wealth distribution becomes more condensed because fewer
households are constrained and the unconstrained households' consumption share drifts down. A
low consumption growth shock after a long period of expansion leads to a very large aggregate
weight shock and high SDF (?gure 7). Many households are constrained and their consumption
share jumps up. This shock wipes out the left tail of the wealth distribution. In anticipation of
such large shock, households demand a large excess return per unit of standard deviation. This
generates the positive correlation between aggregate consumption growth and the Sharpe ratio.
The wealth dynamics are not enough. The persistence of the housing collateral ratio is essential
to generate enough persistence in the Sharpe ratio so as to make the correlation positive.21 The
collateral model's ability and the representative agent model's failure to price the value portfolios
goes back to the properties of their respective stochastic discount factors.
Consumption Strips As we showed in section 1.6, the building blocks of the equity premia on
value and growth stocks are equity premia on consumption strips. Ultimately, the collateral model
generates a higher expected return and a higher Sharpe ratio for value stocks than growth stocks
because short term assets are more risky than long term assets. Therefore, we compute risk premia
on consumption strips of various maturities.
Figure 8 plots expected excess returns and ?gure 9 Sharpe ratios on consumption strips of
horizons 2 to 30 years for the benchmark model with additive utility (? = 3 ? 8). Risk premia and
Sharpe ratios on consumption strips are lower for long horizon strips. The duration e?ect is even
stronger in the collateral model with recursive utility (see appendix A.4).
Table 12 reports equity premia on claims to fCeakckg (see section 1.6). These are baskets of
consumption strips of di?erent maturities, where the constant a governs the duration of the basket.
We vary a from -.5 to .5. The corresponding baskets have a duration between 2.3 years and 43
years. We think of the basket with duration of 5 years as the value stock and the basket with
sample standard deviation.
21We computed the model without the collateral e?ect and found that the term structure of risk premia was ?at.
33
duration of 40 years as the growth stock (Dechow, Sloan, and Soliman (2002)). The value spread
is 3.9 percent for ? = 5 and 5.7 percent for ? = 8. The latter matches the value spread in the
data of 5.7 percent. In addition, Sharpe ratios on value portfolios are much higher than on growth
portfolios. For ? = 8, the Sharpe ratio on the 5-year duration portfolio is .44 and the Sharpe ratio
on the 40-year duration portfolio is .08.
4. Conclusion
This paper speci?es, calibrates and solves a general equilibrium asset pricing model with housing
collateral. Agents write state-contingent promises backed by the value of the housing stock. Time
variation in the price of housing induces time variation in the economy's ability to share labor
income risk. When the ratio of housing wealth to total wealth is low, households with binding
collateral constraints experience a larger change in the consumption share. In such periods of
limited risk-sharing possibilities, agents demand a higher risk premium on ?nancial assets. The
housing collateral mechanism endogenously generates time-varying volatility in the Sharpe ratio on
equity. This is a novel feature of the model. This quantitatively matches the dynamics of the Sharpe
ratio on equity in US data: It is high in periods of collateral scarcity and volatile. In contrast, the
representative agent model delivers virtually no variation in the Sharpe ratio. Other equilibrium
models have similar di?culties generating enough volatility in the Sharpe ratio. The model also
explains cross-sectional variation in excess returns along the value dimension. Excess returns on
short duration assets (such as value stocks) are higher than returns on long duration assets (such
as growth stocks). The reason for this duration e?ect is that a negative consumption growth shock
leads to lower future Sharpe ratios. Lastly, the model quantitatively matches key unconditional
asset pricing moments. To the best of our knowledge, this is one of the few models that starts from
?rst principles and quantitatively matches the unconditional and conditional moments of aggregate
asset prices, and generates a meaningful variation in the cross-section of returns.
Why does the collateral model work better than the standard consumption CAPM? The model
suggest that the answer lies in allowing for time-variation in risk-sharing among heterogeneous
agents. The standard CCAPM implies that risk-sharing is always perfect. In Lustig and Van-
Nieuwerburgh (2004a), we provide direct empirical support for the underlying time-variation in
risk-sharing. Using US metropolitan area data, we ?nd that the degree of insurance between re-
gions decreases when the housing collateral ratio is low. This is consistent with evidence from
34
Blundell, Pistaferri, and Preston (2002), who ?nd evidence for time-variation the economy's risk
sharing capacity. Regional income and consumption data provide direct support for the existence
and importance of the collateral mechanism, whose asset pricing implications where the focus of
this paper.
In future work, we plan to further exploit the link between asset prices and risk-sharing by
using asset prices to measure the cost of consumption uncertainty, along the lines of Alvarez and
Jermann (2003).
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37
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38
5. Figures and Tables
Figure 1. Risk Sharing and Collateral Ratio
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion is 8. The ?rst panel plots
the non-durable expenditure share ?, the ratio of non-durable consumption to non-durable plus housing services consumption
(dotted line). The second panel is the cuto? level consumption share at which the solvency constraints hold with equality
(dotted line). The third panel is the cross-sectional standard deviation of consumption growth across households (dotted line).
The fourth panel is the aggregate weight shock gt+1 (dotted line). The full line in each panel is the collateral ratio my, the
ratio of housing wealth to total wealth (right axis). The graphs display a two hundred period model simulation.
39
Figure 2. Conditional Asset Pricing Moments and Collateral Ratio
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion is 8. The ?rst panel
plots the expected excess return Rc;et on a non-levered claim to aggregate consumption (dotted line), the second panel is the
conditional standard deviation of the excess return Rc;et (dotted line), and the third panel is the Sharpe ratio on a non-levered
claim to aggregate consumption (dotted line). The full line in each panel is the collateral ratio my, the ratio of housing wealth
to total wealth (right axis). The graphs display a two hundred period model simulation.
40
Figure 3. Housing Collateral Ratio and Long-Horizon Sharpe Ratio in Model.
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion is 8. This the Sharpe
ratio on a 10 year and 5 year cumulative excess return on a non-levered consumption claim (dotted line), and the collateral
ratio my is the ratio of housing wealth to total wealth (full line) for a one hundred period model simulation.
Figure 4. Housing Collateral Ratio and Long- Horizon Sharpe Ratio in Data.
This is the Sharpe ratio on 5-year cumulative stock market returns in the data for 1928-1997. See table 7 for details on its
construction. The housing collateral ratio my is zero on average by construction (see appendix A.5 for the estimation procedure).
41
Figure 5. Summary Conditional Asset Pricing Moments.
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion is 8. This the expected
excess return on a claim to aggregate consumption, its conditional standard deviation and its Sharpe ratio (top row), and
the conditional market price of risk, conditional price-dividend ratio and the risk-free rate, averaged over histories and plotted
against the housing collateral ratio. The full line denotes the conditional moments, conditional on observing a low aggregate
consumption growth rate tomorrow, whereas the dotted line denotes the moments conditional on observing a high aggregate
consumption growth rate.
42
Figure 6. Aggregate Consumption Growth Shocks and Sharpe Ratio on Equity.
Benchmark model calibration with risk aversion 8 and 5 percent collateral. One hundred period model simulation. Shaded bars
indicate periods with low aggregate consumption growth. The dotted line denotes the Sharpe ratio on a non-levered claim to
aggregate consumption Et[Rc;e]=?t[Rc;e].
Figure 7. Aggregate Consumption Growth Shocks and The Stochastic Discount Factor.
Benchmark model calibration with risk aversion 8 and 5 percent collateral. One hundred period model simulation. Shaded bars
indicate periods with low aggregate consumption growth. The dotted line denotes the Stochastic Discount Factor mt+1.
43
Figure 8. Term Structure of Risk Premia on Consumption Strips.
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion ? is between 3 and 8.
The ?gure plots the expected excess return on a claim to aggregate consumption k periods from now, k = 2;3;:::;30.
Figure 9. Term Structure of Sharpe Ratios on Consumption Strips.
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion ? is between 3 and 8.
The ?gure plots the conditional Sharpe ratio on a claim to aggregate consumption k periods from now, k = 2;3;:::;30.
44
Table 1
Expenditure Share and Rental Price Regression Results.
Panel A reports regression results for log(rt+1) = ?log(rt)+??log(ct+1)+?t+1, where r is the expenditure share of nondurable
consumption. Panel B reports results for the regression log(?t+1) = ? ? rlog(?t) + br?log(ct+1) + ?t+1, where ? is the rental
price. Below the OLS point estimates are HAC Newey-West standard errors. The left panel reports the results for the entire
sample, while the right panel reports the results for the post-war sample. The variables with superscript 1 are available for
1926-2002. The variables with superscript 2 are only available for 1929-2002. The data appendix contains detailed de?nitions
and data sources for these variables.
Expl. Var. ?r br ?r br
1926/9-2002 1945-2002
Panel A: Expenditure Share
log(r1) .925 .950
(.039) (.033)
log(r1) .890 .824 .957 .824
(.033) (.141) (.033) (.180)
log(r2) .940 .936
(.037) (.026)
log(r2) .940 .816 .952 .816
(.032) (.159) (.027) (.181)
Panel B: Rental Price
log(?1) .953 .851
(.027) (.056)
log(?1) .955 .102 .817 .261
(.027) (.181) (.054) (.240)
log(?2) .941 .911
(.023) (.046)
log(?2) .932 -.321 .896 .259
(.023) (.158) (.047) (.172)
45
Table 2
Parameter Calibration
Parameter Benchmark Sensitivity Analysis
? 8 [2,3,...,9,10]
" .05 [.15,.25,...,.65,.75]
? 1 ?
? 4.26 ?
?r .96 ?
br .93 0
?r .03 ?
E[my] 19 (5%) 9 (10 %)
? [1.04,.96] ?
? [.6578,.7952,.3422,.2048] [.6935,.6935,.3065,.3065]
? 1 3
46
Table 3
Unconditional Asset Pricing Moments for Collateral Model.
Averages from a simulation of the model for 5,000 agents and 10,000 periods. In the ?rst column, Rl;e denotes the excess
return on a levered claim to aggregate consumption growth, with leverage parameter ? = 3. Rc;e denotes the excess return
on a non-levered claim to aggregate consumption growth. The third column reports the unconditional mean of the risk-free
rate. Columns four to six report unconditional standard deviations of levered and non-levered consumption claims and risk-free
rate. The last two columns report Sharpe ratios on levered and non-levered consumption claims. Panel 1 is the benchmark
calibration with 5 percent collateral. The expenditure share process is an AR(1) with an aggregate consumption growth term:
log rt = ? + ?r log rt?1 + br?(log(ct+1)) + ?r?t. Panel 2 reports results for di?erent parameters of intratemporal elasticity
of substitution between non-durable and housing services consumption ". All other parameters are held constant at their
benchmark level and there is 5 percent collateral. Panel 3 varies the coe?cient of relative risk aversion ? (5 percent collateral).
Panel 4 reports the moments for an economy with 10 percent collateral. Panel 5 is the benchmark calibration with 5 percent
collateral but with an expenditure share process is an AR(1) without consumption growth term: log rt = ?+?r log rt?1 +?r?t.
Panel 6 reports historical averages for 1927-2001 (data from Kenneth French) and for 1889-1979 (data from Shiller).
(?;") E(Rl;e) E(Rc;e) E(rf) ?(Rl;e) ?(Rc;e) ?(rf) E(Rl;e)?(Rl;e) E(Rc;e)?(Rc;e)
Panel 1: Benchmark
(5;0:05) 0:051 0:030 0:057 0:213 0:129 0:073 0:238 0:230
(8;0:05) 0:108 0:071 0:029 0:257 0:179 0:125 0:421 0:398
Panel 2: Varying "
(8;0:15) 0:110 0:091 0:024 0:258 0:175 0:128 0:425 0:516
(8;0:35) 0:114 0:097 0:016 0:256 0:176 0:128 0:443 0:550
(8;0:55) 0:125 0:110 0:001 0:261 0:186 0:134 0:479 0:593
(8;0:75) 0:153 0:123 ?0:015 0:274 0:188 0:136 0:558 0:657
Panel 3: Varying ?
(2;0:05) 0:013 0:005 0:069 0:162 0:067 0:022 0:081 0:079
(4;0:05) 0:036 0:019 0:067 0:194 0:105 0:052 0:183 0:176
(6;0:05) 0:068 0:042 0:050 0:227 0:145 0:089 0:300 0:293
(10;0:05) 0:151 0:110 ?0:003 0:285 0:213 0:159 0:530 0:514
Panel 4: 10 percent Collateral
(5;0:05) 0:033 0:028 0:101 0:159 0:090 0:051 0:207 0:318
(5;0:75) 0:053 0:050 0:068 0:177 0:118 0:077 0:301 0:427
(8;0:05) 0:078 0:056 0:075 0:207 0:150 0:107 0:376 0:373
(8;0:75) 0:127 0:098 0:025 0:233 0:189 0:137 0:547 0:517
Panel 5: AR(1) process for log(r)
(5;0:05) 0:041 0:026 0:048 0:165 0:108 0:071 0:252 0:239
(5;0:75) 0:042 0:028 0:048 0:166 0:112 0:069 0:255 0:247
(8;0:05) 0:091 0:066 0:012 0:207 0:154 0:119 0:440 0:427
(8;0:75) 0:104 0:080 ?0:001 0:220 0:176 0:124 0:475 0:456
Panel 6: Data
1927-2002 0:075 0:039 0:198 0:032 0:419
1889-1979 0:060 0:014 0:192 0:065 0:313
47
Table 4
Unconditional Asset Pricing Moments for Representative Agent Model.
Averages from a simulation of the representative agent model with non-separable preferences for 10,000 periods. The rows in
each block are for di?erent values for the coe?cient of relative risk aversion ? and the intratemporal elasticities of substitution
between non-durable and housing services consumption ". All other parameters are held constant at their benchmark level. The
expenditure share process is an AR(1) with an aggregate consumption growth term: log rt = ?+?r log rt?1 +br?(log(ct+1))+
?r?t. In the ?rst column, Rl;e denotes the excess return on a levered claim to aggregate consumption growth, with leverage
parameter ? = 3. Rc;e denotes the excess return on a non-levered claim to aggregate consumption growth. The third column
reports the unconditional mean of the risk-free rate. Columns four to six report unconditional standard deviations of levered
and non-levered consumption claims and risk-free rate. The last two columns report Sharpe ratios on levered and non-levered
consumption claims.
(?;") E(Rl;e) E(Rc;e) E(rf) ?(Rl;e) ?(Rc;e) ?(rf) E(Rl;e)?(Rl;e) E(Rc;e)?(Rc;e)
(5;0:05) 0:024 0:010 0:129 0:127 0:057 0:031 0:186 0:177
(5;0:35) 0:023 0:012 0:127 0:132 0:061 0:034 0:173 0:189
(5;0:75) 0:041 0:023 0:110 0:150 0:084 0:049 0:271 0:275
(8;0:05) 0:045 0:023 0:158 0:146 0:074 0:055 0:309 0:310
(8;0:35) 0:051 0:026 0:151 0:152 0:080 0:061 0:334 0:325
(8;0:75) 0:085 0:055 0:098 0:179 0:114 0:085 0:472 0:480
48
Table 5
Unconditional Asset Pricing Moments for Housing Market.
Averages from a simulation of the collateral model with 5,000 agents for 10,000 periods. In the left panel the coe?cient of relative
risk aversion ? is 5; in the right panel it is 8. The rows are for di?erent intratemporal elasticities of substitution between non-
durable and housing services consumption ". All other parameters are held constant at their benchmark level. The expenditure
share process is an AR(1) with an aggregate consumption growth term: log rt = ? + ?r log rt?1 + br?(log(ct+1)) + ?r?t. In
the ?rst column, ?(?log(?t)) denotes the volatility of rental price growth. In the second column Rh;e denotes the return on a
claim to the aggregate housing endowment in excess of the risk-free rate. The third column denotes the unconditional standard
deviation of Rh;e. The fourth column gives the average housing collateral ratio, the ratio of housing wealth to total wealth my.
Panel 1 is the collateral model with 5 percent collateral, panel 2 is the collateral model with 10 percent collateral and panel 3
is the representative agent economy.
? ?(?log(?t)) E(Rh;e) ?(Rh;e) myt ?(?log(?t)) E(Rh;e) ?(Rh;e) myt
Panel 1: 5 percent Housing Collateral
Risk Aversion 5 Risk Aversion 8
:05 0:050 0:026 0:127 0:056 0:050 0:068 0:192 0:057
:15 0:055 0:026 0:123 0:056 0:056 0:070 0:196 0:058
:35 0:073 0:027 0:128 0:056 0:073 0:072 0:195 0:058
:55 0:105 0:032 0:134 0:056 0:105 0:082 0:206 0:059
:75 0:190 0:044 0:151 0:058 0:189 0:103 0:221 0:062
Panel 2: 10 percent Housing Collateral
Risk Aversion 5 Risk Aversion 8
:05 0:050 0:016 0:088 0:105 0:050 0:048 0:156 0:106
:15 0:056 0:016 0:089 0:104 0:055 0:048 0:157 0:108
:35 0:073 0:015 0:085 0:056 0:073 0:056 0:168 0:108
:55 0:105 0:019 0:095 0:107 0:105 0:060 0:167 0:109
:75 0:189 0:028 0:112 0:108 0:189 0:088 0:201 0:114
Panel 3: representative Agent Economy
Risk Aversion 5 Risk Aversion 8
0:05 0:050 0:006 0:053 0:055 0:050 0:015 0:079 0:056
0:15 0:056 0:006 0:054 0:055 0:056 0:015 0:080 0:056
0:35 0:073 0:006 0:057 0:056 0:073 0:018 0:085 0:057
0:55 0:105 0:009 0:061 0:056 0:104 0:023 0:095 0:058
0:75 0:188 0:016 0:078 0:058 0:189 0:041 0:122 0:059
Table 6
Unconditional Asset Pricing Moments - No CD E?ect.
Same as Table 3, but without conditional heteroscedasticity in the individual income process (Constantinides and Du?e (1996)
e?ect). In particular, the vector of income shares ? is [.6935,.6935,.3065,.3065] as opposed to [.6578,.7952,.3422,.2048] in table
3.
? E(Rl;e) E(Rc;e) E(rf) ?(Rl;e) ?(Rc;e) ?(rf) E(Rl;e)?(Rl;e) E(Rc;e)?(Rc;e)
Panel 1: 5 percent Housing Collateral
5 0:036 0:022 0:078 0:193 0:136 0:066 0:187 0:165
8 0:074 0:055 0:059 0:235 0:181 0:111 0:315 0:305
Panel 2: 10 percent Housing Collateral
5 0:024 0:011 0:129 0:126 0:056 0:031 0:193 0:194
8 0:057 0:036 0:123 0:179 0:116 0:080 0:318 0:306
49
Table 7
Long-Term Sharpe Ratios in Data.
Parameter estimates for Ret+1 = b0 +b1Rt +b2dpt +b3lct +b4 f t +"t+1 and V olt+1 = a0 +a1dpt +a2lct +a3 f t +a4V olt +
a5V olt?1. The variables dp, lc and f are the dividend yield, the labor-consumption ratio and the housing collateral ratio
(based on value of mortgages). Re denotes the value weighted market return in excess of a 1 month T-bill return. V olt is the
standard deviation of the 12 monthly returns in year t. R1;R5;R10 denote the 1-year, 5-year and 10-year ahead cumulative
excess returns. The estimation is by GMM with the OLS normal conditions as moment conditions. Standard errors are Newey-
West with lag length 3. The estimation period is the longest common sample: 1927-1992. The last three rows indicate the
sample mean, sample standard deviation of the predicted Sharpe ratio and sample correlation between the Sharpe ratio and the
scarcity of housing collateral. The predicted Sharpe ratio is the predicted mean excess return divided by its predicted standard
deviation.
R1 Vol1 R5 Vol5 R10 Vol10
constant -.24 .09 .71 -.004 .76 .05
(s.e.) (.34) (.05) (0.28) (.04) (.42) (.04)
lag ret .04 .74 .76
(s.e.) (.13) (.13) (.08)
dp 1.07 .33 -1.20 .40 .26 .12
(s.e.) (2.00) (.27) (3.16) (.22) (3.00) (.21)
lc .22 -.07 -.76 .03 -.92 -.03
(s.e.) (.32) (.05) (.33) (.03) (.45) (.04)
f .02 -.01 .48 -.04 .70 .01
(s.e.) (0.20) (.02) (.20) (.02) (.23) (.02)
lag vol .51 .96 .80
(s.e.) (.20) (.12) (.12)
2 lag vol -.18 -.19 .03
(s.e.) (.13) (.17) (.11) (.10)
E[Sharpe] .40 .73 1.12
?[Sharpe] .10 .18 .20
?[Sharpe; f ] .26 .32 .50
50
Table 8
Predictability of K-Year Excess Returns.
Results of regressing log K-horizon excess returns on the housing collateral ratio. The ?rst panel reports the results in the data.
The t-stats in brackets are computed using the Newey West covariance matrix with K lags. The returns are cum-dividend
returns on the value-weighted CRSP index. The housing collateral measure is based on the market value of outstanding
mortgages ~ t and is rescaled to lie between 0 and 1: ~ t = max(myt) ? myt=(min(myt) ? myt). The risk-free rate is
the average 3-month yield from CRSP's Fama-Bliss risk-free rates and in?ation is constructed from the CPI (BLS). The long
sample contains annual data from 1930-2003. The post-war sample is from 1945-2003. The second panel reports the same
regressions inside the model. The regressions were obtained by simulating the model for 10,000 periods under the benchmark
parametrization with risk aversion 5. The expenditure share process is an AR(1) with an aggregate consumption growth term:
log rt = ?+ ?r log rt?1 + br?(log(ct+1)) + ?r?t. The housing collateral ratio myt is scaled to lie between 0 and 1 for all t.
b0 b1 R2 b0 b1 R2
Panel 1: Data
Horizon Entire Sample Post-War Sample
1 0:01 0:15 0:02 0:02 0:15 0:04
[0:24] [1:25] [0:63] [1:73]
2 0:07 0:25 0:03 0:06 0:31 0:07
[0:66] [1:11] [0:83] [1:75]
3 0:16 0:31 0:03 0:12 0:46 0:09
[1:02] [0:97] [1:04] [1:80]
4 0:29 0:32 0:02 0:22 0:56 0:09
[1:50] [0:80] [1:42] [1:82]
5 0:38 0:47 0:03 0:34 0:66 0:08
[1:83] [1:16] [1:67] [1:77]
6 0:41 0:79 0:07 0:43 0:89 0:10
[1:70] [1:73] [1:58] [1:86]
7 0:41 1:20 0:12 0:49 1:26 0:14
[1:32] [2:11] [1:30] [2:02]
8 0:37 1:80 0:18 0:50 1:79 0:19
[0:91] [2:71] [1:01] [2:26]
Panel 2: Model
Horizon Leverage =1 Leverage=4
1 ?0:00 0:05 0:00 ?0:00 0:09 0:00
2 ?0:07 0:18 0:02 ?0:06 0:23 0:01
3 ?0:15 0:33 0:04 ?0:14 0:41 0:02
4 ?0:26 0:51 0:06 ?0:24 0:62 0:03
5 ?0:39 0:72 0:09 ?0:37 0:85 0:04
6 ?0:55 0:96 0:11 ?0:51 1:10 0:06
7 ?0:72 1:22 0:14 ?0:69 1:40 0:07
8 ?0:92 1:51 0:17 ?0:88 1:73 0:09
51
Table 9
Annual Value-Weighted Excess Returns in B/M deciles
Excess return in excess of a risk-free rate, volatility and Sharpe ratio. The risk-free rate is the average yield on a 3-month
T-Bill. The stock returns on the book-to-market deciles are from Kenneth French's web site, annual data for 1930-2003 and
1945-2003 (source CRSP).
Decile 1 2 3 4 5 6 7 8 9 10
Panel 1: Sample 1930-2003
E(Re) 0:071 0:084 0:080 0:081 0:100 0:099 0:108 0:127 0:131 0:139
?(Re) 0:222 0:194 0:196 0:229 0:228 0:238 0:250 0:274 0:291 0:332
E(Re)=?(Re) 0:321 0:431 0:411 0:355 0:437 0:417 0:433 0:464 0:449 0:419
Panel 2: Sample 1945-2003
E(Re) 0:078 0:087 0:086 0:084 0:106 0:107 0:110 0:130 0:126 0:143
?(Re) 0:209 0:175 0:175 0:178 0:182 0:178 0:194 0:214 0:212 0:257
E(Re)=?(Re) 0:372 0:497 0:492 0:473 0:580 0:599 0:566 0:604 0:592 0:558
52
Table 10
Beta Estimates for Book-to-Market Decile Returns in Data - 5 Factors.
OLS regression of excess returns of the 10 book-to-market deciles on a constant, a scaled version of the collateral measure f t,
the aggregate consumption growth rate ?(log(ct+1)), the interaction term f t?(log(ct+1)), the aggregate expenditure share
growth rate ?log(?t+1), and the interaction term f t?(log(ct+1)). These are the ?ve risk factors in the collateral model.
The ?rst line of each panel is for the lowest book-to-market decile (growth), the last line for the highest book-to-market decile
(value). The number is brackets are OLS t-statistics. In the ?rst panel the housing collateral ratio is based on the value of
outstanding mortgages, in the second panel, the housing collateral ratio is based on the value of residential real estate wealth,
and in the third panel it is based on the value of residential ?xed assets. The data are annual for the period 1930-2003.
?0 ?my ?c ?c;my ?? ??;my
Panel 1: Mortgage-Based Collateral
1 0:61 12:94 ?0:50 3:40 15:63 ?34:80
[5:44] [12:38] [1:70] [3:89] [6:16] [15:09]
2 3:59 10:09 ?0:86 3:20 10:85 ?22:54
[4:90] [11:16] [1:53 [3:51] [5:55] [13:60]
3 4:74 6:76 ?1:83 5:16 13:62 ?26:97
[4:89] [11:13] [1:52] [3:50] [5:54] [13:56]
4 3:64 10:73 ?2:69 6:44 24:46 ?40:95
[5:35] [12:18] [1:67] [3:83] [6:06] [14:85]
5 3:00 16:99 ?3:16 7:51 22:36 ?40:71
[5:41] [12:31] [1:69] [3:87] [6:12] [15:00]
6 0:32 18:41 ?0:90 5:49 21:52 ?37:33
[5:45] [12:41] [1:70] [3:90] [6:18] [15:12]
7 6:22 9:06 ?3:55 9:47 20:90 ?33:64
[6:03] [13:72] [1:88] [4:31] [6:82] [16:71]
8 3:90 19:38 ?3:08 8:67 24:21 ?42:34
[6:55] [14:92] [2:04] [4:69] [7:42] [18:18]
9 5:94 18:50 ?4:41 9:45 23:25 ?33:85
[7:07] [16:10] [2:21] [5:06] [8:01] [19:62]
10 4:78 21:63 ?4:36 10:45 25:72 ?37:27
[8:04] [18:29] [2:51] [5:75] [9:10] [22:29]
Panel 2: Residential Wealth-Based Collateral
1 1:24 11:30 ?0:78 4:34 15:88 ?36:85
[5:64] [12:75] [1:84] [4:17] [6:23] [15:75]
2 6:05 3:78 ?0:89 3:42 9:61 ?19:25
[5:12] [11:57] [1:67] [3:79] [5:65] [14:30]
3 6:77 1:92 ?1:76 4:95 12:23 ?23:24
[5:11] [11:55] [1:67] [3:78] [5:64] [14:27]
4 7:22 1:93 ?2:65 6:43 22:09 ?34:30
[5:65] [12:76] [1:84] [4:18] [6:24] [15:77]
5 5:70 10:28 ?3:23 7:71 20:58 ?36:54
[5:69] [12:86] [1:86] [4:21] [6:28] [15:89]
6 2:66 12:08 ?1:28 6:62 19:70 ?32:80
[5:72] [12:93] [1:87] [4:23] [6:32] [15:98]
7 9:74 0:18 ?3:99 10:43 18:55 ?26:68
[6:27] [14:18] [2:05] [4:64] [6:93] [17:52]
8 5:72 14:60 ?3:54 9:79 22:61 ?38:87
[6:82] [15:41] [2:23] [5:05] [7:53] [19:04]
9 8:86 10:78 ?4:89 10:54 20:70 ?26:95
[7:35] [16:62] [2:40] [5:44] [8:12] [20:53]
10 7:85 13:46 ?4:94 11:81 23:54 ?31:84
[8:34] [18:85] [2:72] [6:17] [9:21] [23:30]
Panel 3: Fixed Assets-Based Collateral Measure
1 ?3:96 20:12 ?0:75 4:15 15:29 ?33:73
[10:03] [21:13] [2:20] [5:08] [7:84] [20:58]
2 6:82 0:43 ?0:92 3:57 5:82 ?7:92
[8:83] [18:60] [1:94] [4:47] [6:90] [18:11]
3 5:59 2:33 ?2:17 5:97 10:53 ?17:72
[8:96] [18:89] [1:97] [4:54] [7:01] [18:39]
4 4:84 3:68 ?3:67 9:07 22:76 ?35:40
[9:85] [20:75] [2:16] [4:99] [7:70] [20:20]
5 1:15 15:52 ?4:44 10:66 21:69 ?38:43
[10:03] [21:14] [2:20] [5:08] [7:85] [20:58]
6 ?1:21 16:16 ?2:14 8:72 20:36 ?33:43
[10:12] [21:33] [2:22] [5:13] [7:92] [20:77]
7 8:97 ?2:04 ?5:32 13:78 17:53 ?23:52
[10:95] [23:08] [2:40] [5:55] [8:57] [22:47]
8 1:76 17:16 ?5:10 13:68 24:15 ?41:89
[12:06] [25:40] [2:65] [6:11] [9:43] [24:73]
9 7:38 8:51 ?6:67 15:12 21:10 ?27:91
[12:84] [27:06] [2:82] [6:51] [10:05] [26:35]
10 ?3:95 32:00 ?7:74 18:53 29:21 ?48:55
[14:23] [29:99] [3:12] [7:21] [11:13] [29:20]
53
Table 11
Model-Generated Value Premium in Collateral Model
The table reports expected returns, standard deviation and Sharpe ratio on an arti?cial asset generated with a set of betas
listed in Table 10, but with intercept ?0 chosen so that the Euler equation is satis?ed for this asset in the model. Each panel
corresponds to a set of betas for a di?erent collateral measure. The parametrization is the benchmark one with expenditure
share process is an AR(1) without aggregate consumption growth term.
Decile 1 2 3 4 5 6 7 8 9 10
Panel 1: Risk Aversion 5
Mortgage-Based Collateral Measure
E(Re) 0:01 0:01 0:01 0:01 0:02 0:03 0:03 0:03 0:02 0:03
?(Re) 0:06 0:04 0:05 0:07 0:07 0:10 0:10 0:10 0:09 0:12
E(Re)=?(Re) 0:18 0:23 0:23 0:23 0:23 0:28 0:26 0:26 0:26 0:27
Residential Wealth-Based Collateral Measure
E(Re) 0:01 0:01 0:01 0:02 0:02 0:03 0:03 0:03 0:03 0:04
?(Re) 0:07 0:04 0:05 0:07 0:07 0:11 0:12 0:11 0:11 0:14
E(Re)=?(Re) 0:15 0:23 0:23 0:24 0:24 0:28 0:26 0:27 0:26 0:27
Fixed Assets-Based Collateral Measure
E(Re) 0:01 0:02 0:02 0:02 0:02 0:04 0:04 0:04 0:04 0:05
?(Re) 0:07 0:06 0:07 0:09 0:10 0:13 0:17 0:15 0:16 0:19
E(Re)=?(Re) 0:17 0:26 0:25 0:24 0:23 0:26 0:25 0:25 0:25 0:25
Panel 2: Risk Aversion 8
Mortgage-Based Collateral Measure
E(Re) 0:01 0:01 0:02 0:02 0:02 0:04 0:04 0:03 0:03 0:05
?(Re) 0:06 0:04 0:05 0:06 0:07 0:10 0:10 0:10 0:09 0:12
E(Re)=?(Re) 0:16 0:24 0:30 0:31 0:26 0:38 0:40 0:35 0:36 0:38
Residential Wealth-Based Collateral Measure
E(Re) 0:01 0:01 0:02 0:02 0:02 0:04 0:05 0:04 0:04 0:05
?(Re) 0:07 0:04 0:05 0:07 0:07 0:11 0:12 0:11 0:11 0:14
E(Re)=?(Re) 0:15 0:33 0:35 0:38 0:31 0:40 0:42 0:37 0:39 0:40
Fixed Assets-Based Collateral Measure
E(Re) 0:01 0:03 0:03 0:04 0:03 0:05 0:07 0:06 0:07 0:07
?(Re) 0:07 0:06 0:07 0:10 0:10 0:14 0:17 0:15 0:16 0:19
E(Re)=?(Re) 0:15 0:43 0:41 0:40 0:34 0:40 0:43 0:38 0:41 0:36
Table 12
Risk Premia on Portfolios of Consumption Strips.
This table reports the expected excess return, the conditional standard deviation and the Sharpe ratio on baskets of consumption
strips. The consumption strips are computed for the baseline model with additive utility and ? = 5;8. The ?rst row denotes
the duration of each basket in years. The baskets are weighted combinations of consumption strips of di?erent horizons k, with
weights governed by Ceak. Each column reports the basket for a value of parameter a ranging from -.5 to .5.
Panel 1: Risk Aversion 5
Duration 2:4 3:0 4:2 5:6 8:2 14:3 25:6 35:2 39:5 41:4 43:4
E[Re] 0:031 0:032 0:034 0:035 0:036 0:035 0:025 0:008 ?0:004 ?0:011 ?0:020
?[Re] 0:106 0:112 0:119 0:123 0:128 0:132 0:131 0:125 0:120 0:117 0:113
E[Re]=?[Re] 0:290 0:288 0:285 0:285 0:283 0:266 0:190 0:065 ?0:033 ?0:095 ?0:180
Panel 1: Risk Aversion 8
Duration 2:3 2:9 4:1 5:4 7:9 13:9 25:7 35:7 40:0 41:7 43:4
E[Re] 0:059 0:063 0:068 0:071 0:074 0:072 0:052 0:027 0:014 0:007 0:002
?[Re] 0:131 0:141 0:154 0:161 0:168 0:174 0:172 0:166 0:161 0:159 0:157
E[Re]=?[Re] 0:451 0:447 0:443 0:442 0:438 0:411 0:304 0:164 0:084 0:047 0:013
54
A. Appendix
A.1. Arrow-Debreu Equilibrium
This appendix spells out the household problem in an economy where all trade takes place at time zero.
Household Problem A household of type (?0;s0) purchases a complete contingent consumption plan fc(?0;s0);h(?0;s0)g
at time-zero market state prices fp;p?g. The household solves:
sup
fc;hg
U(c(?0;s0);h(?0;s0))
subject to the time-zero budget constraint
?s0 [fc(?0;s0) + ?h(?0;s0)g] 6?0 + ?s0 [f?g] ;
and an in?nite sequence of collateral constraints for each t
?st [fc(?0;s0) + ?h(?0;s0)g] ? ?st [f?g];8st:
Dual Problem Given Arrow-Debreu prices fp;?g the household with label (?0;s0) minimizes the cost C(?) of
delivering initial utility w0 to itself:
C(w0;s0) = min
fc;hg
(c0(w0s0) + h0(w0;s0)?0(s0))
+
X
st
p(stjs0) ?ct(w0;stjs0) + ht(w0;stjs0)?t(stjs0)?
subject to the promise-keeping constraint
U0(fcg;fhg; w0;s0) ? w0
and the collateral constraints
?st [fc(w0;s0) + ?h(w0;s0)g] ? ?st [f?g] ;8st:
The initial promised value w0 is determined such that the household spends its entire initial wealth:
C(w0;s0) = ?0 + ?[f?g] :
There is a monotone relationship between ?0 and w0.
The above problem is a convex programming problem. We set up the saddle point problem and then make it
recursive by de?ning cumulative multipliers (Marcet and Marimon (1999)). Let ` be the Lagrange multiplier on the
promise keeping constraint and ?t(w0;st) be the Lagrange multiplier on the collateral constraint in history st. De?ne
a cumulative multiplier at each node: ?t(w0;st) = 1 ? Pst ?t(w0;st). Finally, we rescale the market state price
^t(st) = pt(zt)=?t?t(stjs0). By using Abel's partial summation formula and the law of iterated expectations to the
55
Lagrangian, we obtain an objective function that is a function of the cumulative multiplier process ?i :
D(c;h;?;w0;s0) =
X
t?0
X
st
8
<
:?
t?(stjs0)
2
4 ?t(w0;s
tjs0)^pt(st)?ct(w0;st) + ?
t(s
t)ht(w0;st)?
+?t(w0;st)?st [f?g]
3
5
9
=
;
such that
?t(w0;st) = ?t?1(w0;st?1) ? ?t(w0;st); ?0(w0;s0) = 1
Then the recursive dual saddle point problem is given by:
inf
fc;hg
sup
f?g
D(c;h;?; w0;s0)
such that X
t?0
X
st
?t?(stjs0)u(ct(w0;st);ht(w0;st)) ? w0
To keep the mechanics of the model in line with standard practice, we re-scale the multipliers. Let
?t(`;st) = `?
t(w0;st)
;
The cumulative multiplier ?(`;st) is a non-decreasing stochastic sequence (sub-martingale). If the constraint for
household (`;s0) binds, it goes up, else it stays put.
First Order Necessary Conditions The f.o.c. for c(`;st) is :
^(st) = ?t(`;st)uc(ct(`;st);ht(`;st)):
Upon division of the ?rst order conditions for any two households `0 and `00, the following restriction on the joint
evolution of marginal utilities over time and across states must hold:
uc(ct(`0;st);ht(`0;st))
uc(ct(`00;st);ht(`00;st)) =
?t(`00;st)
?t(`0;st) : (15)
Growth rates of marginal utility of non-durable consumption, weighted by the multipliers, are equalized across agents:
?t+1(`0;st+1)
?t(`0;st)
uc(ct+1(`0;st+1);ht+1(`0;st+1))
uc(ct(`0;st);ht(`0;st)) =
^t+1(st+1)
^t(st) =
?t+1(`00;st+1)
?t(`00;st)
uc(ct+1(`00;st+1);ht+1(`00;st+1))
uc(ct(`00;st);ht(`00;st)) :
There is a mapping from the multipliers at st to the equilibrium allocations of both commodities. We refer to
this mapping as the risk-sharing rule.
ct(`;st) = ?t(`;s
t)1?
?at (zt) c
a
t (z
t) and ht(`;st) = ?t(`;st)
1
?
?at (zt) h
a
t (z
t): (16)
This rule ?ows from the optimality conditions and the market clearing conditions.
The time zero ratio of marginal utilities is pinned down by the ratio of multipliers on the promise-keeping
constraints. For t > 0; it tracks the stochastic weights ?. From the ?rst order condition w.r.t. ?t(`;st) we obtain a
56
reservation weight policy:
?t = ?t?1 if ?t?1 > `c(yt;zt); (17)
?t = `c(yt;zt) otherwise: (18)
and the collateral constraints hold with equality at the bounds:
?st
hn
ct(`;st; ?t(`;st)) + ?hi(`;st;?(`;st))
oi
= ?st [f?g] :
A.2. Collateral E?ect
Proof of Proposition 1 Denote the price of a claim under perfect risk-sharing by ?currency1[f?g]. Perfect risk sharing
can be sustained if and only if
?currency1z
??
ca
?
1 + 1r
???
? ?currency1z;y [f?(y;z)g] for all (y;z;r)
If this condition is satis?ed, each household can get a constant and equal share of the aggregate non-durable and
housing endowment at all future nodes. Perfect risk-sharing is possible. q.e.d.
Proof of Proposition 2 Assume utility is separable. Let C(`;yt;zt) denote the cost of claim to consumption
in state ?yt;zt? for a household who enters the period with weight ? . The cuto? rule `c(yt;zt) is determined such
that the solvency constrain binds exactly: ?y;zt [f?g] = C(?;yt;zt), where C(?;yt;zt) is de?ned recursively as:
C(?;yt;zt) = `
c(yt;zt)
?at (zt)
?
1 + 1r
t
?
+ ?
X
zt+1
?(zt+1jzt)
X
y0
?(yt+1;zt+1jyt;zt)
?(zt+1jzt) mt+1(z
t+1)C(?0;yt+1;zt+1);
and ?0 is determined by the cuto? rule. Note that the stochastic discount factor mt+1(zt+1) does not depend on rt(zt)
because we assumed that utility is separable. This also implies that the cost of a claim to labor income ?y;zt [f?g]
does not depend on r.
We ?rst proof the result for a ?nite horizon version of this economy. In the last period T, the cuto? rule is
determined such that:
?(yT?1;zT?1) = `
c(yT?1;zT?1)
?aT (zT?1)
?
1 + 1r
T?1
?
+ ?
X
zt+1
?(zT jzT?1)
X
y0
?(yT ;zT jyT?1;zT?1)
?(zT jzT?1) mT (z
T jzT?1) ?01=?
?aT (zT );
where ?01=??a
T (zT )
? ? ?yT ;zT ?. Given r1T?1 < r2T?1 , this implies that `1;c(yT?1;zT?1) < `2;c(yT?1;zT?1) for all
(yT?1;zT?1). By backward induction we get that, for a given sequence of ??at (zt)?, `1;c(yt;zt) < `2;c(yt;zt) for all
nodes (yt;zt) in the ?nite horizon economy. This in turn implies that ??a;1t (zt)? ? ??a;2t (zt)? for all zt. This follows
57
directly from the de?nition of
?at (zt) =
X
yt
Z
?t(`;yt;zt)?(z
t;ytjz0;y0)
?(ztjz0) d?0 (19)
=
X
yt
Z
`c(yt;zt)
?t?1(`;yt;zt)?(z
t;ytjz0;y0)
?(ztjz0) d?0 (20)
+
X
yt
Z `c(yt;zt)
`c(yt;zt)?(z
t;ytjz0;y0)
?(ztjz0) d?0 (21)
?at (zt) is non-decreasing in `c(yt;zt).
The proof extends to the in?nite horizon economy if the transition matrix has no absorbing states. The reason
is that limT!1Et ??T?tmT (zT jzt)?zT ;yT currency1 does not depend on the current state (yt;zt). q.e.d.
Proof of Corollary 1 Follows from the de?nition of the cuto? level in the previous proof. For a given
sequence of ??at (zt)?, it is obvious that `1;c(yt;zt) < `2;c(yt;zt) for all nodes (yt;zt). This in turn implies that
??a;1
t (z
t)? ? ??a;2
t (z
t)?. This follows directly from the de?nition of the aggregate weight shock (21). As a result,
?at (zt) is non-decreasing in `c(yt;zt). q.e.d.
Proof of Proposition 3 The proof follows Alvarez and Jermann (2000). Appendix A.3 explains the relation-
ship between the static and sequential budget constraints and solvency constraints.
Proof of Proposition 4 Following the de?nition of Alvarez and Jermann (2001), the pricing kernel M has
no permanent component if
lim
k!1
Et+1Mt+k
EtMt+k = 1:
If, in our model, this condition is satis?ed, then it is also true that
lim
k!1
Et+1Mt+kct+k
EtMt+kct+k = 1
since Mt+k = ?t+kc??t+k(?at+k)? and if
lim
k!1
Et+1?t+kc??t+k(?at+k)?
Et?t+kc??t+k(?at+k)? = 1
then it has to be true for ? > 1 that
lim
k!1
Et+1?t+kc1??t+k (?at+k)?
Et?t+kc1??t+k (?at+k)? = 1
because 0 ? ?t+kc1??t+k (?at+k)? ? ?t+kc??t+k(?at+k)? for ? > 1. As a result the new random variable ?t+kc1??t+k (?at+k)? is
bounded above by the old random variable, for each state of nature. In addition each realization of the new random
variable is drawn from the same probability measure as the old random variable, both at t and t + 1.
Let the one period holding return on a period-k consumption strip be given by:
Rct+1;k = MtM
t+1
Et+1Mt+kct+k
EtMt+kct+k ;
58
then we know, from the derivation above, that
lim
k!1
Rct+1;k = MtM
t+1
:
Furthermore, for any return Et[Mt+1Mt Rt+1] = 1, we know that Et[log(Mt+1Mt Rt+1)] ? log Et[Mt+1Mt Rt+1] = 0 by
Jensen's inequality. This implies that Et log( MtMt+1 ) ? Et log(Rt+1) or
Et log lim
k!1
Rct+1;k = log MtM
t+1
? Et log(Rt+1) for any asset return Rt+1
This implies that the expected log excess return exceeds that any other asset:
Et log lim
k!1
Rct+1;k
Rt+1;1 ? Et log
(Rt+1)
Rt+1;1
Let f(k) = Ceak with a > 0 for growth stocks. In the absence of a permanent component in the pricing kernel:
lim
a!1
1 + e0 = lim
k!1
Rct+1;k ? 1 + ?0 for any other sequence of weights f!kg
This implies that the highest equity premium is the one on the farthest out consumption strip. In the absence of a
permanent component in the pricing kernel, there is a growth premium. q.e.d.
A.3. Sequential versus Time-Zero Constraints
We show under which conditions the sequence of budget constraints and collateral constraints in the sequential
market setup can be rewritten as one time-zero budget constraint and the collection of collateral constraints shown
in equation (10). The proof strategy follows Sargent (1984) (Chapter 8).
Budget Constraint First, we show how the Arrow-Debreu budget constraint obtains from aggregating suc-
cessive sequential budget constraints. The sequential budget constraint is:
ct(`;st) + ?t(zt)hrt(`;st) +
X
s0
qt(st;s0)at(`;st;s0) + pht (st)hot+1(`;st) ? Wt(`;st):
Next period wealth is:
Wt+1(`;st;s0) = ?t+1(st;s0) + at(`;st;s0) + hot+1(`;st)
h
pht+1(st;s0) + ?t+1(st;s0)
i
:
Multiply the second equation by qt+1(s0) and sum over states. Then substitute the expression for Pqt+1(s0)at+1(s0)
into the ?rst equation.
ct + ?thrt +
X
s0
qt+1(s0)Wt+1(s0) 6Wt +
X
s0
qt+1(s0)?t+1(s0) +
hot+1
?X
s0
qt+1(s0)
h
pht+1(s0) + ?t+1(s0)
i
? pht
!
:
59
Similarly, for period t + 1:
ct+1 + ?t+1hrt+1 +
X
s00
qt+2(s00)Wt+2(s00) 6 Wt+1 +
X
s00
qt+2(s00)?t+2(s00) +
hot+2
0
@X
s00
qt+2(s00)
h
pht+2(s00) + ?t+2(s00)
i
? pht+1
1
A:
Substituting the expression for t + 1 into the expression for t by substituting out Wt+1, we get:
ct + ?thrt +
X
s0
qt+1(s0) ?ct+1 + ?t+1hrt+1currency1 +
X
s0
X
s00
qt+1(s0)qt+2(s00)Wt+2(s00) 6
Wt +
X
s0
qt+1(s0)?t+1(s0) +
X
s0
X
s00
qt+1(s0)qt+2(s00)?t+2(s00) + hot+1
?X
s0
qt+1(s0)
h
pht+1(s0) + ?t+1(s0)
i
? pht
!
+
X
s0
qt+1(s0)hot+2(s0)
0
@X
s00
qt+2(s00)
h
pht+2(s00) + ?t+2(s00)
i
? pht+1
1
A:
Let ?st be the value of a dividend stream fdg starting in history st priced using the market state prices fpg :
?st [fdg] =
X
j?0
X
st+jjst
pt+j(st+j)dt+j(st+j);
where for a given path st+j following history st; p is de?ned as
pt+j(st+jjst) = qt+j
?
st+jjst+j?1
?
qt+2(st+2jst+1):::qt+1(st+1jst):
Repeating the successive substitutions, the budget set is given by
?st [fc + ?hrg] 6Wt ? ?t + ?st [f?g] (22)
under 2 assumptions: (1) the transversality condition
lim
j!1
X
st+j
pt+j(st+j)Wt+j(st+j) = 0; (23)
is satis?ed and (2) there are no arbitrage opportunities:
pht+j?1(st+j?1) =
X
st+jjst+j?1
qt+j(st+j)
h
pht+j(st+j) + ?t+j(st+j)
i
; 8j ? 0;8st+j (24)
If the latter condition were not satis?ed, a household could achieve unbounded consumption by investing suf-
?ciently high amounts in housing shares ho and ?nancing this by borrowing. This is a feasible strategy because
ownership shares in the housing tree are collateralizable.
Because W0 = ?0 + ?0, and relabelling hrt = ht, we obtain the Arrow-Debreu budget constraint
?s0 [fc + ?hg] 6?0 + ?s0 [f?g]
60
Collateral Constraints Second, we show the equivalence between the collateral constraints of the sequential
markets setup and the solvency constraint in the static economy. The sequential collateral constraints are:
h
pht (zt) + ?t(zt)
i
hot?1(st?1) + at?1(st?1;st) ? 0;
and the collateral constraints in a history st:
?st [fc + ?hg] ? ?st [f?g] : (25)
The equivalence follows if and only if
at?1(st?1;st) + hot?1(st?1)
h
pht (zt) + ?t(zt)
i
= ?st [fc + ?h ? ?g]:
But this follows immediately from the budget constraint (22) holding with equality and the de?nition of W:
Wt(st) ? ?t(s) = at?1(st?1;st) + hot?1(st?1)
h
pht (zt) + ?t(zt)
i
:
Under conditions (23) and (24) an allocation that is feasible and immune to the threat of default in sequential
markets is feasible and immune to the threat of default in time-zero markets.
The equivalence implies that the allocation of home-ownership ho is indeterminate in the sequential economy.
A.4. Recursive Preferences
In the main text, we assume additive utility. In this section, we show how the model's stochastic discount factor
changes when preferences are of the Kreps and Porteus (1978) type. We show that this type of preferences is important
to generate low risk premia on long horizon assets.
Preferences The household's utility at time t, Vt, is given by a composite of the utility it derives from current
consumption and its future expected utility:
Vt =
2
4(1 ? ?)
?
c
"?1
"t + ?h
"?1
"t
?(1??)"
"?1 + ? (R
tVt+1)1??
3
5
1
1??
;
where future expected utility is de?ned by
RtVt+1 = ?E[V 1??t+1 ]? 11??
The coe?cient ? is the inverse of the intertemporal elasticity of substitution, ? measures the risk aversion and "
measures the intratemporal elasticity of substitution between non-durable and housing services consumption. Additive
utility is a special case with ? = ?.
61
Risk-Sharing Rule The risk-sharing rule with recursive preferences takes the same form as equation (16), but
the stochastic consumption weights are di?erent:
ct(`;st) =
^?t(`;st) 1
^?at (zt) c
a
t (z
t) and ht(`;st) = ^?t(`;st)
1
?
^?at (zt) h
a
t (z
t): (26)
The new stochastic consumption weight ^t(`;st) equals the old stochastic consumption weight ?t(`;st) multiplied by
the utility gradient between period 0 and period t, M0;t(`;st):
M0;t(`;st) =
Y
s??st
M?; M? =
? V
?
R??1V?
????
:
The new consumption weights still have a recursive structure:
^?t+1 =
??
t+1
?t Mt+1
?
^?t:
While the individual consumption weights f?g were non-decreasing processes, this is no longer true for the new
stochastic weight shocks f^g, because Mt+1 may be less than one. Furthermore, even if the solvency constraints
never bind, the new consumption weights change over time.
As before, the aggregate weight shock is the cross-sectional average of the individual stochastic consumption
weights: ^at (zt) = E[^
1
?
t (`;s
t)].
Stochastic Discount Factor The stochastic discount factor still equals the marginal utility growth of the
unconstrained households. It is of the same form as in the additive utility case
mt+1 = ?
?ca
t+1
cat
??? ??a
t+1
?at
?"??1
"?1
?^
?at+1
^?at
!?
: (27)
The ?rst two terms re?ect aggregate consumption growth risk and composition risk. The last term however embodies
both long-run consumption growth risk and the risk of binding solvency constraints. The long-run consumption
growth risk is the last term of the representative agent's SDF:
mat+1 = ?
?ca
t+1
cat
??? ??a
t+1
?at
?"??1
"?1
? V e
t+1
RtV et+1
????
? ;
where V et denotes the continuation utility of a representative agent who consumes the aggregate non-durable and
housing services endowment. Epstein and Zin (1991) show that the representative agent SDF can be rewritten as
a function of the gross return on an asset paying the aggregate non-durable and the aggregate housing endowment
stream, Rc;h.
mat+1 = ?1??1??
?ca
t+1
cat
????1??
1??
? ?
?at+1
?at
?"(1?? )?1
"?1 ?Rc;h
t+1
????
1?? ;
The SDF of the economy is still the product of the representative agent economy's SDF and a liquidity shock.
mt+1 = mat+1~?: (28)
62
Table 13
Unconditional Asset Pricing Moments - Recursive Utility.
Same as Table 3, but preferences are of the recursive utility type with inverse intertemporal elasticity of substitution ? = 5 and
? = :05. The risk aversion parameter ? varies between 6 and 8.
? E(Rl;e) E(Rc;e) E(rf) ?(Rl;e) ?(Rc;e) ?(rf) E(Rl;e)?(Rl;e) E(Rc;e)?(Rc;e)
6 0:044 0:028 0:061 0:178 0:121 0:067 0:248 0:235
7 0:050 0:036 0:062 0:178 0:120 0:070 0:284 0:298
8 0:060 0:041 0:068 0:179 0:118 0:061 0:337 0:347
Contrary to the aggregate weight shock g = ?
a
t+1
?at in the case of additive utility, the new liquidity shock ~g is no longer
theoretically restricted to be greater than or equal to one.
Calibration For the economy with recursive preferences, we use ? = 5, where ? is the inverse of the intertemporal
elasticity of substitution.
Unconditional Asset Pricing Moments Table 13 shows the unconditional asset pricing statistics for the
collateral model under recursive preferences. The equity premium on a levered consumption claim for the economy
with 5 percent collateral and ? = 8 is 6 percent, excess stock returns have a volatility of 18 percent and the Sharpe
ratio of the stock return is 0.33. The risk-free rate is 6.8 percent on average. These moments are of the same
magnitude as the ones we found for additive preferences. However, the volatility of the risk-free rate is only half as
large: 6.1 percent versus 12.5 percent under additive preferences.
Conditional Asset Pricing Moments All relationships between the housing collateral ratio and the
conditional asset pricing moments carry over to the model with recursive preferences. Figure 10 shows a similar
pattern for conditional moments as ?gure 5.
Consumption Strip Risk Premia Figure 11 and 12 plot the equity premia and Sharpe ratios on the same
strips for an inverse intertemporal elasticity of substitution ? = 5. We recall that when ? = ?, we are back in the case
of additive preferences. If ? is su?ciently above ?, the decline in equity premia and Sharpe ratios for long horizon
strips is large. The equity premium on a claim to aggregate consumption two periods from now is 4 percent, while
the equity premium on a claim to aggregate consumption 30 years from now is -2 percent. Claims on payo?s far
in the future have a lower expected excess return than claims on payo?s in the near future. Likewise, the Sharpe
ratio falls from 0.4 to -0.1. Claims on payo?s far in the future have a lower conditional Sharpe ratios than claims on
payo?s in the near future.
Betas on Consumption Strips Finally, the model generates the same pattern of betas for consumption
strips as was found in the data for value portfolios. A regression inside the model of excess returns on the consumption
strips (in excess of the zero coupon bonds of the same maturity) on the asset pricing factors in equation (14) is
consistent with the pattern in the betas as the one estimated for the 10 value decile portfolios in the data.
63
Figure 10. Summary Conditional Asset Pricing Moments - Recursive Utility
The average collateral share is 5 percent, the discount factor is .95, the coe?cient of risk aversion is 7, the inverse intertemporal
elasticity of substitution is 5. This the expected excess return on a claim to aggregate consumption, its conditional standard
deviation and its Sharpe ratio (top row), and the conditional market price of risk, conditional price-dividend ratio and the
risk-free rate, averaged over histories and plotted against the housing collateral ratio. The full line denotes the conditional
moments, conditional on observing a low aggregate consumption growth rate tomorrow, whereas the dotted line denotes the
moments conditional on observing a high aggregate consumption growth rate.
Figure 11. Term Structure of Risk Premia on Consumption Strips - Recursive Utility
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion ? 2 f6;7;8g. Preferences
are of the Epstein-Zin type with inverse intertemporal elasticity of substitution of ? = 5. The ?gure plots the expected excess
return on a claim to aggregate consumption k periods from now, k = 2;3;:::;30.
In the data (table 10), we found that excess returns on value stocks (short duration) had a higher correlation
with consumption growth than growth stocks (long duration). The consumption growth beta, ?c, is higher for value
stocks. Figure 13 shows the same pattern for the consumption strips in the model with recursive preferences. The
consumption beta for a 2 year consumption strip is higher than the consumption beta for a 30 year consumption
strip.
64
Figure 12. Term Structure of Sharpe Ratios on Consumption Strips - Recursive Utility
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion ? 2 f6;7;8g. Preferences
are of the Epstein-Zin type with inverse intertemporal elasticity of substitution of ? = 5. The ?gure plots the conditional
Sharpe ratio on a claim to aggregate consumption k periods from now, k = 2;3;:::;30.
In the data (table 10), we also found strong evidence that excess returns on value stocks (short duration) have a
much higher correlation with consumption growth when collateral is scarce than growth stocks. I.e. the conditional
correlation, conditional on my is higher. The composite consumption beta, or `collateral beta' ?c+my?c;my measures
this conditional correlation. When collateral is scarce ( f = 1 or my = mymin), the collateral beta for value stocks
is 6.09 and 2.90 for growth stocks in the data. The model with recursive preferences generates a higher collateral
beta for short duration assets (`value stocks') than for long duration assets (`growth stocks'). The full line in ?gure
14 illustrates how the conditional correlation decreases with rising maturity for the model with ? = 5 and ? = 8.
Figure 13. Consumption Growth Betas - Recursive Utility
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion ? 2 f6;7;8g. The inverse
elasticity of substitution is ? = 5. The ?gure plots the consumption growth betas ?c, obtained from a regression inside the
model of excess returns of consumption strips of di?erent horizons on the risk factors of the model (equation ??).
65
Figure 14. Collateral Betas - Recursive Utility
The average collateral share is 5 percent, the discount factor is .95 and the coe?cient of risk aversion ? is 8. The inverse
elasticity of substitution is ? = 5. The ?gure plots the consumption growth betas ?c + my?c;my, obtained from a regression
inside the model of excess returns of consumption strips of di?erent horizons on the risk factors of the model (equation ??).
The betas are plotted for periods in which the housing collateral ratio is high (full line), average (dash-dotted line) and low
(dashed line).
A.5. Data Appendix
We use two sets of variables: ?nancial variables and aggregate macroeconomic variables. All variables are annual and
for the United States.
Financial Data
Aggregate Dividends Aggregate dividends are the dividends on the Standard and Poor's composite stock
price index. The data are available for the period 1889-2001 from Robert Shiller's web site. The standard deviation
of real dividend growth is .11 for 1930-2001.
Market Return We also include the market return Rvw, the value-weighted return on all NYSE, AMEX and
NASDAQ stocks.
Size and Book-to-Market Portfolios We use ten portfolios of NYSE, NASDAQ and AMEX stocks,
grouped each year into ten value (book-to-market ratio) bins. Book-to-market is book equity at the end of the prior
?scal year divided by the market value of equity in December of the prior year. Portfolio returns are value-weighted.
All returns are expressed in excess of an annual return on a one-month Treasury bill rate (from Ibbotson Associates).
The returns are available for the period 1926-2002 from Kenneth French's web site and are described in more detail
in Fama and French (1992).
66
Aggregate Macroeconomic Data
Consumption and Income Consumption is non-durable consumption C, measured by total expenditures
minus apparel and minus rent and imputed rent. The housing expenditure ratio, r, is the ratio of non-durable
expenditures to rent expenditures.
The income endowment in the model corresponds to an after-government income concept; it includes net transfer
income. Aggregate income Y is labor income plus net transfer income. Nominal data are from the National Income
and Product Accounts for 1930-2002. Consumption and income are de?ated by the consumer price index and divided
by the number of households N.
Price Indices Aggregate rental prices ?t are constructed as the ratio of the CPI rent component pht and the
CPI food component pct. Data are for urban consumers from the Bureau of Labor Statistics for 1926-2001. The
price of rent is a proxy for the price of shelter and the price of food is a proxy for the price of non-durables. We
use the rent and food components because the shelter and non-durables components are only available from 1967
onwards. Two-thirds of consumer expenditures on shelter consists of owner-occupied housing. The BLS uses a rental
equivalence approach to impute the price of owner-occupied housing. Because ?t is a relative rental price, our theory
is conceptually consistent with the BLS approach. We also use the all items CPI, pat , which goes back to 1889. All
indices are normalized to 100 for the period 1982-84.
Housing Collateral We use three distinct measures of the housing collateral stock HV : the value of outstand-
ing home mortgages (mo), the market value of residential real estate wealth (rw) and the net stock current cost value
of owner-occupied and tenant occupied residential ?xed assets (fa). The ?rst two time series are from the Historical
Statistics for the US (Bureau of the Census) for the period 1889-1945 and from the Flow of Funds data (Federal
Board of Governors) for 1945-2001. The last series is from the Fixed Asset Tables (Bureau of Economic Analysis)
for 1925-2001.
We use both the value of mortgages HV mo and the total value of residential ?xed assets HV rw to be robust to
changes in the extent to which housing can be used as a collateral asset. We use both HV rw, which is a measure of
the value of housing owned by households, and HV fa which is a measure of the value of housing households live in,
to be robust to changes in the home-ownership rate over time. Real per household variables are denoted by lower
case letters. The real, per household housing collateral series hvmo;hvrw;hvfa are constructed using the all items
CPI from the BLS, pa, and the total number of households, N, from the Bureau of the Census.
Housing Collateral Ratio Log, real, per household real estate wealth (log hv) and labor income plus transfers
(log y) are non-stationary. According to an augmented Dickey-Fuller test, the null hypothesis of a unit root cannot
be rejected at the 1 percent level. This is true for all three measures of housing wealth (hv = mo;rw;fa).
If a linear combination of log hv and log y, log (hvt) + $ log (yt) + ? , is trend stationary, the components log hv
and log y are said to be stochastically cointegrated with cointegrating vector [1;$;?]. We additionally impose the
restriction that the cointegrating vector eliminates the deterministic trends, so that log (hvt) + $ log (yt) + #t + ? is
stationary. A likelihood-ratio test (Johansen and Juselius (1990)) shows that the null hypothesis of no cointegration
relationship can be rejected, whereas the null hypothesis of one cointegration relationship cannot. This is evidence
67
for one cointegration relationship between housing collateral and labor income plus transfers. We estimate the
cointegration coe?cients from vector error correction model:
2
4 ?log (hvt)
?log (yt)
3
5 = ?[log (hvt) + $ log (yt) + #t + ?] +
KX
k=1
Dk
2
4 ?log (hvt?k)
?log (yt?k)
3
5 + "t: (29)
The K error correction terms are included to eliminate the e?ect of regressor endogeneity on the distribution
of the least squares estimators of [1;$;#;?]. The housing collateral ratio my is measured as the cointegration
relationship:
myt = log (hvt) + ^ log (yt) + ^ + ^
The OLS estimators of the cointegration parameters are superconsistent: They converge to their true value at rate
1=T (rather than 1=pT). The superconsistency allows us to use the housing collateral ratio my as a regressor without
need for an errors-in-variables standard error correction.
68