THE JOURNAL OF FINANCE • VOL. LX, NO. 2 • APRIL 2005
Optimal Life-Cycle Asset Allocation:
Understanding the Empirical Evidence
FRANCISCO GOMES and ALEXANDER MICHAELIDES∗
ABSTRACT
We show that a life-cycle model with realistically calibrated uninsurable labor income
risk and moderate risk aversion can simultaneously match stock market participation
rates and asset allocation decisions conditional on participation. The key ingredients of
the model are Epstein–Zin preferences, a fixed stock market entry cost, and moderate
heterogeneity in risk aversion. Households with low risk aversion smooth earnings
shocks with a small buffer stock of assets, and consequently most of them (optimally)
never invest in equities. Therefore, the marginal stockholders are (endogenously) more
risk averse, and as a result they do not invest their portfolios fully in stocks.
IN THIS PAPER, WE PRESENT A LIFE-CYCLE ASSET allocation model with intermediate
consumption and stochastic uninsurable labor income that provides an expla-
nation for two very important empirical observations: low stock market partic-
ipation rates in the population as a whole, and moderate equity holdings for
stock market participants.
Our life-cycle model integrates three main motives that have been identi-
fied as quantitatively important in explaining individual and aggregate wealth
accumulation. First, a precautionary savings motive driven by the presence of
undiversifiable labor income risk (Deaton (1991) and Carroll (1992, 1997)). Sec-
ond, pension income is lower than mean working-life labor income, implying
that saving for retirement becomes important at some point in the life cycle.
The combination of precautionary and retirement saving motives has recently
been shown to generate realistic wealth accumulation profiles over the life cy-
cle.1 Third, we explicitly incorporate a bequest motive that has recently been
∗Francisco Gomes is from the London Business School; Alexander Michaelides is from the London
School of Economics and CEPR. We thank Viral Acharya, Orazio Attanasio, Ravi Bansal, Michael
Brennan, Joao Cocco, Pierre Collin-Dufresne, Steve Davis, Karen Dynan, Bill Dupor, Lorenzo
Forni, Joao Gomes, Rick Green, Luigi Guiso, Michael Haliassos, Burton Hollifield, Urban Jermann,
Deborah Lucas, Pascal Maenhout, Monica Paiella, Valery Polkovnichenko, Ken Singleton, Nicholas
Souleles, Harald Uhlig, Raman Uppal, Annette Vissing-Jørgensen, Tan Wang, Paul Willen, Amir
Yaron, Steve Zeldes, Harold Zhang, two anonymous referees, and seminar participants at Carnegie
Mellon, Columbia, the Ente-Einaudi Center, the Federal Reserve Board, H.E.C., Leicester, LBS,
LSE, Tilburg, UCL, UNC Chapel Hill, Wharton, the 2002 NBER Summer Institute, MFS and SED
meetings for helpful comments. Previous versions of this paper have circulated with the title: “Life-
Cycle Asset Allocation: A Model with Borrowing Constraints, Uninsurable Labor Income Risk and
Stock Market Participation Costs.” We are responsible for any remaining errors.
1 See, for instance, Hubbard, Skinner, and Zeldes (1995), Carroll (1997), Gourinchas and Parker
(2002), Dynan, Skinner, and Zeldes (2002), and Cagetti (2003).
869
870 The Journal of Finance
shown to be important in matching the skewness of the wealth distribution (de
Nardi (forthcoming) and Laitner (2002)).
More recently, life-cycle models incorporating some (or all) of these motives
have been extended to include an asset allocation decision, both in an infinite
horizon2 and in a finite horizon, life-cycle setting.3 However, several important
predictions of these models are still at odds with empirical regularities. First,
low stock market participation in the population (Mankiw and Zeldes (1991))
persists. The latest Survey of Consumer Finances (2001) reports that only 52%
of U.S. households hold stocks either directly or indirectly (through pension
funds, for instance), while these models predict that, given the equity premium,
all households should participate in the stock market as soon as saving takes
place. Second, households in the model invest almost all of their wealth in
stocks, in contrast to both casual empirical observation and to formal empirical
evidence (see Poterba and Samwick (1999) or Ameriks and Zeldes (2001), for
instance).
We develop a life-cycle asset allocation model that tries to address these two
puzzles. We argue that it is possible to simultaneously match stock market par-
ticipation rates and asset allocation conditional on participation, with moderate
values of risk aversion (between one and five), and without extreme assump-
tions about the level of background risk. Our model has three key features.
First, we include a fixed entry cost for households that want to invest in risky
assets for the first time. A large literature has concluded that some level of
fixed costs seems to be necessary to improve the empirical performance of as-
set pricing models.4 Since the excessive demand for equities predicted by asset
allocation models is merely the portfolio-demand manifestation of the equity
premium puzzle, introducing a fixed cost in the model seems to be a natural ex-
tension. Moreover, recent empirical work suggests that small entry costs can be
consistent with the observed low stock market participation rates (see Paiella
(2001), Degeorge et al. (2002), and Vissing-Jørgensen (2002b)).
The other two key features are motivated by the (perhaps surprising) impli-
cation of the model that participation rates are an increasing function of risk
aversion, at least over a wide range of parameter values. Specifically, changing
risk aversion generates two opposing forces for determining the participation
decision. On the one hand, more risk-averse households optimally prefer to in-
vest a smaller fraction of their wealth in stocks. On the other hand, risk aversion
determines prudence and more prudent consumers accumulate significantly
2 See, for example, Telmer (1993), Lucas (1994), Koo (1998), Heaton and Lucas (1996, 1997,
2000), Polkovnichenko (2000), Viceira (2001), and Haliassos and Michaelides (2003).
3 See, for instance, Cocco, Gomes, and Maenhout (1999), Cocco (2000), Campbell et al. (2001), Hu
(2001), Storesletten, Telmer, and Yaron (2001), Davis, Kubler, and Willen (2002), Dammon, Spatt,
and Zhang (2001, forthcoming), Polkovnichenko (2002), Yao and Zhang (forthcoming), and Gomes
and Michaelides (2003). Bertaut and Haliassos (1997) and Constantinides, Donaldson, and Mehra
(2002) analyze three-period models where each period amounts to 20 years.
4 See, among others, Constantinides (1986), Aiyagari and Gertler (1991), He and Modest (1995),
Saito (1995), Heaton and Lucas (1996), Luttmer (1996, 1999), Basak and Cuoco (1998), and Vayanos
(1998).
Optimal Life-Cycle Asset Allocation 871
more wealth over the life cycle. We show that the higher wealth accumulation
motive dominates for moderate coefficients of relative risk aversion (RRA) (i.e.,
not greater than five). As a result, the less risk-averse investors have a weaker
incentive to pay the fixed cost. This explains why previous attempts to match
participation rates in the context of a life-cycle model were fairly unsuccessful. If
we try to match asset allocation decisions by assuming high values of risk aver-
sion, the implied participation rates are counterfactually high (e.g., Campbell
et al. (2001)). Motivated by this result, we allow for preference heterogeneity
in the population, the second key feature of the model. As argued before, since
the less risk-averse investors accumulate less wealth over the life cycle, the
majority optimally chooses not to pay the fixed cost. Therefore, endogenously
stock market participants tend to be the more risk-averse investors, and con-
sequently, even after paying the fixed cost, they do not invest their portfolios
fully in equities.
The final important feature of the model is the assumption of Epstein–Zin
preferences, which allows us to separate risk aversion from the elasticity of
intertemporal substitution (EIS). In the context of a life-cycle model with labor
income, wealth accumulation is a crucial determinant of both the stock mar-
ket participation and the asset allocation decision. Within the power utility
framework, households with low risk aversion also have a high EIS. Given that
the expected return from investing in the stock market is higher than the dis-
count rate, a higher EIS increases savings. As a result, even though the less
risk-averse agents would not save much for precautionary reasons, they would
have a strong incentive to save for retirement (and for a potential bequest mo-
tive). Thus, breaking the link between risk aversion and the EIS is crucial for
delivering predictions that are consistent with the observed empirical evidence.
Therefore, in our model, households with very low risk aversion and low
(moderate) EIS smooth idiosyncratic earnings shocks with a small buffer stock
of assets, and most of them never invest in equities (thus behaving as in the
Deaton (1991) infinite-horizon model).5 This seems to describe adequately the
behavior of a large fraction of the U.S. population that retires without signifi-
cant financial assets (and does not participate in the stock market). Within the
low EIS and low risk aversion group, only a small fraction owns stocks, and
they do so only as they get close to retirement. On the other hand, investors
with high prudence and high EIS are the ones who participate in the stock
market from early on, since they accumulate more wealth and therefore have
a stronger incentive to pay the fixed cost. Therefore, the marginal stockholders
are (endogenously) more risk averse and as a result they do not invest their
portfolios fully in stocks.
The heterogeneous agent model can simultaneously match the stock market
participation rate and the average equity allocation conditional on participa-
tion, from the Survey of Consumer Finances (SCF). The life-cycle profile of the
5 It is important to point out that we do not need heterogeneity in the EIS to obtain our results.
As we will show, the less risk-averse investors can have the same EIS as the more risk-averse, just
as long as this value is not too high (hence the need for Epstein–Zin preferences).
872 The Journal of Finance
participation rate is also very close to the one observed in the data. On the
negative side, the model still counterfactually predicts that young households
that have already paid the participation cost will invest most of their portfolio
in equities.6 Finally, the degree of heterogeneity in the wealth distribution is
quite comparable to the one observed in the data.
The rest of the paper is organized as follows. Section I summarizes results
from the existing empirical literature on life-cycle asset allocation, while Sec-
tion II outlines the model and calibration. In Sections III and IV, we discuss the
results in the absence and presence of the fixed entry cost, respectively. Finally,
Section V concludes.
I. Empirical Evidence on Life-Cycle Asset Allocation and Stock
Market Participation
In most industrialized countries, stock market participation rates have in-
creased substantially during the last decade. Nevertheless, a large percentage
of the population still does not own any stocks (either directly or indirectly
through pension funds). Moreover, even those households that do own stocks
still invest a significant fraction of their portfolios in alternative assets.
Figures 1A and B summarize evidence reported in Ameriks and Zeldes
(2001).7 The results are sensitive to the identifying assumptions regarding time
versus cohort effects. Time effects can arise, for example, from changes in mar-
ket structure (e.g., transaction costs or information) or because investors use
past returns to forecast future expected returns. Cohort effects can be due to dif-
ferences in lifetime earnings potential, or different institutional settings (e.g.,
the social security system). Since age (a), time (t), and cohort (c, birth year) are
linearly dependent (a ≡ t − c), when constructing age profiles, it is impossible
to simultaneously identify time and cohort effects.
Figure 1A plots the average life-cycle equity holdings for stock market par-
ticipants (as a share of total financial wealth), based on the 1989, 1992, 1995,
and 1998 samples of the SCF. Although the life-cycle profiles are very sensitive
to the inclusion of time dummies versus the inclusion of cohort dummies, the
average stock holdings are significantly below 100% in both cases.8 Figure 1B
plots the corresponding stock market participation rate, obtained by running
a probit regression on the same data. These results are less sensitive to the
choice of time versus cohort dummies. As expected, a very large fraction of the
population does not own equities. In both cases the participation rate gradually
6 Hu (2001) and Yao and Zhang (forthcoming) are able to reduce the equity demand of young
households by considering models with an explicit housing allocation decision.
7 Guiso, Haliassos, and Japelli (2002) obtain similar conclusions using cross-sectional informa-
tion for five different countries (United States, United Kingdom, The Netherlands, Germany, and
Italy).
8 The OLS regression with cohort effects predicts a share of financial wealth invested in stocks
above 100% for the oldest age groups. This is just the result of imposing the same cohort effects
on the full sample as in fact, in every individual cross-section, these age groups never invest more
than 60% of their wealth in equities.
Optimal Life-Cycle Asset Allocation 873
0
0.2
0.4
0.6
0.8
1
25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79
Age
Time dummies Cohort dummies
Figure 1A. Equity holdings as a fraction of total financial wealth for stock market
participants. The results are taken from Ameriks and Zeldes (2001), and they are obtained from
OLS regressions with age dummies and either time or cohort dummies. The data includes the 1989,
1992, 1995, and 1998 samples of the SCF.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79
Age
Time dummies Cohort dummies
Figure 1B. Stock market participation rate. The results are taken from Ameriks and Zeldes
(2001), and they are obtained from probit regressions with age dummies and either time or cohort
dummies. The data includes the 1989, 1992, 1995, and 1998 samples of the SCF.
874 The Journal of Finance
increases until approximately age 50. When including cohort dummies, the
profile is flat after age 50, while with time dummies it is decreasing. Ameriks
and Zeldes (2001) obtain the same results after redoing the analysis using
TIAA-CREF data from 1987 to 1996, and so do Poterba and Samwick (1999),
using SCF data.
We can summarize the existing evidence as follows.9 First, the stock mar-
ket participation rate in the U.S. population is close to 50%. Using the latest
numbers from the SCF, we compute it as 51.9% (details given in Appendix C).
Second, participation rates increase during working life and there is some ev-
idence suggesting that they might decrease during retirement, although this
might also be due to cohort effects. Third, conditional on stock market partici-
pation, households invest a large fraction of their financial wealth in alterna-
tive assets. According to the latest numbers from the SCF, the average equity
holdings as a share of financial wealth for stock market participants is 54.8%.
Fourth, there is no clear pattern of equity holdings over the life cycle.
II. The Model
A. Preferences
Time is discrete and t denotes adult age, which following the typical con-
vention in this literature, corresponds to effective age minus 19. Each period
corresponds to 1 year and agents live for a maximum of 81 (T) periods (age 100).
The probability that a consumer/investor is alive at time (t + 1) conditional on
being alive at time t is denoted by pt (p0 equal to 1).
Households have Epstein–Zin utility functions (Epstein and Zin (1989)) de-
fined over one single nondurable consumption good. Let Ct and Xt denote re-
spectively consumption level and wealth (cash on hand) at time t. Then, the
household’s preferences are defined by
Vt =

(1 − βpt)C1−1/ψt + βEt
[
pt
[
V 1−ρt+1
] + (1 − pt)b(X t+1/b)1−ρ1 − ρ
] 1−1/ψ
1−ρ


1
1−1/ψ
, (1)
where ρ is the coefficient of RRA, ψ is the EIS, β is the discount factor, and
b determines the strength of the bequest motive.10 Given the presence of a
bequest motive, the terminal condition for the recursive equation (1) is:
VT+1 ≡ b(X T+1/b)
1−ρ
1 − ρ . (2)
9 We must point out that several papers have contributed to this research. See, for example,
Guiso, Jappelli, and Terlizzese (1996) (who focus mostly on the impact of background risk on asset
allocation), King and Leape (1998), Heaton and Lucas (2000) and the papers in the volume edited
by Guiso, Haliassos, and Japelli (2002).
10 For more motivation and details on the modeling of bequest motives in life-cycle models see
Laitner (2002), or de Nardi (forthcoming).
Optimal Life-Cycle Asset Allocation 875
B. Labor Income Process
Following the standard specification in the literature, the labor income pro-
cess before retirement is given by
Yit = PitUit , (3)
Pit = exp( f (t, Zit))Pit−1Nit , (4)
where f (t, Zit) is a deterministic function of age and household characteristics
Zit, Pit is a permanent component with innovation Nit, and Uit is a transitory
component. We assume that ln Uit and ln Nit are independent and identically
distributed with mean {−0.5 ∗ σ 2u , −0.5 ∗ σ 2n }, and variances σ 2u and σ 2n , respec-
tively. The log of Pit evolves as a random walk with a deterministic drift, f (t, Zit).
For simplicity, retirement is assumed to be exogenous and deterministic, with
all households retiring in time period K, corresponding to age 65 (K = 46). Earn-
ings in retirement (t > K) are given by Yit = λPiK , where λ is the replacement
ratio (a scalar between zero and one). This specification, also standard in this
literature, considerably facilitates the solution of the model, as it does not re-
quire the introduction of an additional state variable (see Section II.E).
Durable goods, and in particular housing, can provide an incentive for higher
spending early in life. Modeling these decisions directly is beyond the scope of
the paper, but nevertheless we take into account these potential patterns in life-
cycle expenditures. Using the Panel Study of Income Dynamics, for each age (t)
we estimate the percentage of household income that is dedicated to housing
expenditures (ht) and subtract it from the measure of disposable income.11 More
details on this estimation are given below, when we discuss the calibration of
the model.
C. Financial Assets
The investment opportunity set is constant and there are two financial assets,
one riskless (Treasury bills or cash) and one risky (stocks). The riskless asset
yields a constant gross return, Rf , while the return on the risky asset (denoted
by R St ) is given by
RSt+1 − R f = µ + εt+1, (5)
where εt ∼ N(0, σ 2ε ).
We allow for positive correlation between stock returns and earnings shocks.
We let φN(φU) denote the correlation coefficient between stock returns and per-
manent (transitory) income shocks.
Before investing in stocks for the first time, the investor must pay a fixed
lump sum cost, F ∗ Pit. This entry fee represents both the explicit transaction
cost from opening a brokerage account and the (opportunity) cost of acquiring
information about the stock market. The fixed cost (F) is scaled by the level of
11 A similar approach is taken by Flavin and Yamashita (2002) in a model without labor income.
876 The Journal of Finance
the permanent component of labor income (Pit), as this significantly simplifies
the solution of the model. However, this specification is also motivated by the
interpretation of the entry fee as the opportunity cost of time.
D. Wealth Accumulation
We denote cash on hand as the liquid resources available for consumption
and saving. We define a dummy variable IP that is equal to one when the fixed
entry cost is incurred for the first time and is zero otherwise. The household’s
next period cash on hand (Xi,t+1) is given by
X i,t+1 = Sit RSt+1 + Bit R f + (1 − ht)Yi,t+1 − FIP Pi,t+1, (6)
where Sit and Bit denote respectively stock holdings, and riskless asset holdings
(cash) at time t, and ht is the fraction of income dedicated to housing-related
expenditures. Since the household must allocate cash on hand (Xit) between
consumption expenditures (Cit) and savings we also have
X it = Cit + Sit + Bit . (7)
Finally, we prevent households from borrowing against their future labor in-
come. More specifically we impose the following restrictions:
Bit ≥ 0, (8)
Sit ≥ 0. (9)
E. The Optimization Problem and Solution Method
The complete optimization problem is then
max
{Sit ,Bit }Tt=1
E(V0), (10)
where V0 is given by equations (1) and (2) and is subject to the constraints given
by equations (5) to (9), and to the stochastic labor income process given by (3)
and (4) if tK, and Yit = λPiK if t > K.
Analytical solutions to this problem do not exist. We therefore use a numerical
solution method based on the maximization of the value function to derive the
optimal decision rules. The details are given in Appendix A, and here we just
present the main idea. We first simplify the solution by exploiting the scale-
independence of the maximization problem and rewriting all variables as ratios
to the permanent component of labor income (Pit). The laws of motion and the
value function can then be rewritten in terms of the normalized variables, and
we use lowercase letters to denote them (for instance, xit ≡ X itPit ). This allows us
to reduce the number of state variables to three: age (t), normalized cash on
hand (xit) and participation status (whether the fixed cost has already been paid
or not). In the last period, the policy functions are determined by the bequest
motive and the value function corresponds to the bequest function. We can now
Optimal Life-Cycle Asset Allocation 877
use this value function to compute the policy rules for the previous period, and
given these, obtain the corresponding value function. This procedure is then
iterated backwards.
F. Computing Transition Distributions
After solving for the optimal policy functions, we can simulate the model to
replicate the behavior of a large number of households and compute, for ex-
ample, the corresponding average allocations. Here we propose an alternative
method of computing various statistics that is based on the explicit calcula-
tion of the transition distribution of cash on hand from one age to the next. The
computational details are given in Appendix B, but the intuitive idea is straight-
forward. Once we have solved for the policy functions, we can substitute those
in the budget constraint to obtain the distribution of xt+1 as a function of xt.
Doing this for every possible xt, we are effectively computing the full transition
matrix.12
Once we have these distributions, the unconditional mean consumption for
age t can then be computed as13
c¯t = θt
{
J∑
j=1
π It, j ∗ cI(x j , t)
}
+ (1 − θt)
{
J∑
j=1
πOt, j ∗ cO(x j , t)
}
, (11)
where J is the number of grid points used in the discretization of normalized
cash on hand, and π It, j and π
O
t, j are the probability masses associated with each
grid point at time t, for stockholders and nonstockholders, respectively. The
participation rate at age t (θt) is given by
θt = θt−1 + (1 − θt−1) ∗
∑
x j >x∗
πOt, j , (12)
where x∗ is the trigger point that causes participation, which is determined
endogenously through the participation decision rule.
Finally, if we use αt to denote the share of liquid wealth invested in the stock
market at age t, then the unconditional portfolio allocation is computed as:
α¯t =
θt ∗
{
J∑
j=1
π It, j ∗ α(x j , t) ∗
(
x j − cI(x j , t)
)}
θt ∗
J∑
j=1
[
π It, j ∗
(
x j − cI(x j , t)
)] + (1 − θt) ∗ J∑
j=1
[
πOt, j ∗
(
x j − cO(x j , t)
)] .
(13)
12 The results in the paper were computed both from the transition distributions and using Monte
Carlo simulations. The results were found to be identical, as long as the number of simulations is
not too small (2,000 or more).
13 Superscript I denotes households participating in the stock market, while superscript O de-
notes households out of the stock market.
878 The Journal of Finance
G. Parameter Calibration
G.1. Preference Parameters
We start by presenting results for a relatively standard choice, (risk aversion)
ρ = 5, (EIS) ψ = 0.2, and (discount factor) β = 0.96. However, later on we report
results for several different values of both the coefficient of RRA (ρ) and the
EIS (ψ), as these parameters have very important implications for our results.
We use the mortality tables of the National Center for Health Statistics to
parameterize the conditional survival probabilities.
The importance of the bequest motive (b) is set at 2.5. As we discuss below, this
parameter choice is motivated by the desire to match the wealth accumulation
profiles observed in the data, but we present some sensitivity analysis with
respect to this parameter.
G.2. Labor Income Process
The deterministic labor income profile ( f (t, Zit) reflects the hump shape
of earnings over the life cycle, and the corresponding parameter values, just
like the retirement transfers (λ), are taken from Cocco, Gomes, and Maenhout
(1999). With respect to standard deviations of the idiosyncratic shocks, the es-
timates range from 0.35 for σu and 0.12 for σn (Cocco, Gomes, and Maenhout) to
0.1 for σu and 0.08 for σn (Carroll (1992)). We use numbers similar to the ones in
Gourinchas and Parker (2002): σu = 0.15 and σn = 0.1. It is common practice to
estimate different labor income profiles for different education groups (college
graduates, high school graduates, households without a high school degree). In
our paper, we only report the results obtained with the parameters estimated
from the subsample of high school graduates, as the results for the other two
groups are very similar.
G.3. Asset Returns, Correlation and Fixed Cost
The constant net real interest rate (Rf − 1) is set at 2%, while for the stock
return process we consider a mean equity premium (µ) equal to 4% and a stan-
dard deviation (σε) of 18%. Considering an equity premium of 4% (as opposed
to the historical 6%) is a fairly common choice in this literature (e.g., Yao and
Zhang (forthcoming), Cocco (2001) or Campbell et al. (2001)). Even after hav-
ing paid the fixed entry cost, the average retail investor still faces nontrivial
transaction costs, mostly in the form of mutual fund fees. This adjustment is a
shortcut representation for those costs, since the dimensionality of the problem
prevents us from modeling them explicitly (as in Heaton and Lucas (1996), for
example).
The evidence on the magnitude of the correlation between stock returns and
permanent labor income shocks is mixed.14 Davis and Willen (2001) and Heaton
14 Moreover, it has been argued that these estimations suffer from a small sample bias, since the
time-series dimension is too short in micro data, and estimations using macro data usually yield
larger and more significant correlations (see, e.g., Jermann (1999)).
Optimal Life-Cycle Asset Allocation 879
and Lucas (2000) do not distinguish between the two components of labor in-
come (permanent and transitory) when computing the correlation coefficients.
For the purposes of calibrating our model, we need to know the magnitude of the
correlation coefficient for these two shocks separately. Campbell et al. (2001)
estimate the correlation between the permanent component of labor income
shocks and stock returns, and obtain a correlation coefficient of 0.15.15 They do
not estimate a correlation between transitory shocks and stock returns and just
assume it to be equal to zero. We use these numbers (φN = 0.15 and φU = 0.0)
for our benchmark calibration, and perform sensitivity analysis around these
values.
With respect to the fixed cost of participation we consider two limit cases:
one where the cost is zero, and one where it equals 0.025 (2.5% of the house-
hold’s expected annual income). This parameter reflects both the monetary cost
associated with the initial investment in the stock market, and the opportu-
nity cost associated with obtaining the necessary information for making such
investment.16
G.4. Housing Expenditures
We measure housing expenditures using data from the Panel Study of Income
Dynamics from 1976 until 1993.17 For each household, in each year, we compute
the ratio of annual mortgage payments and rent payments (housing-related
expenditures—H) relative to annual labor income (Y):
hit ≡ HitYit . (14)
We combine mortgage payments and rent together, since we are not model-
ing the housing decision explicitly. We identify the age effects by running the
following regression on the full panel:
hit = A + B1 ∗ age + B2 ∗ age2 + B3 ∗ age3 + time dummies + ζit , (15)
15 It is important to realize that in their tables, Campbell et al. (2001) actually report the cor-
relation of the aggregate component of permanent labor income shocks with stock returns. This
explains their high estimates: 45.6%. To obtain the correlation with the “total permanent shock,”
we need to adjust for the standard deviation of the aggregate component relative to the total, which
gives the 15% number.
16 Consider the average household that has an annual labor income of $35,000. If the time cost
were zero, then this value of F would imply a monetary cost of $875. If instead the monetary cost
were zero, then this would imply a time cost of 9.1 days (6.3 working days). More generally, any
convex combination of these two costs is acceptable, for example, a time cost of 1 (2) day(s) and a
monetary cost of $779 ($683). Paiella (2001) and Vissing-Jørgensen (2002b) used Euler equation
estimation methods to obtain implied participation costs from observed consumption choices. They
find values in the $75–200 range, but these are per-period costs, so our number is quite reasonable
when compared to their estimates.
17 Before 1976 there is no information on mortgage expenditures, and 1993 is the last year
available on final release from the PSID.
880 The Journal of Finance
Table I
Regression of the Ratio of Housing Expenditures to Labor Income
(heit), on Age Polynomials, and Time Dummies
The data are taken from the Panel Study of Income Dynamics from 1976 until 1993. For each
household, in each year, we compute the ratio of annual mortgage payments plus rent payments
relative to annual labor income, and regress this ratio against a constant, a cubic polynomial of age
(where age is defined as the age of the head of the household), and time dummies. We eliminate all
observations with age greater than 75.
Coefficient T-Stat
Constant 0.703998 5.47
Age −0.0352276 −3.70
Age2 0.0007205 3.17
Age3 −0.0000049 −2.84
Adj. R2 0.025
where age is defined as the age of the head of the household. We eliminate all
observations with age > 75.18 The estimation results are reported in Table I.
In the model we use
ht = max(A + B1 ∗ age + B2 ∗ age2 + B3 ∗ age3, 0), (16)
which, given our parameter estimates, truncates ht at zero for age80.
III. Results without the Fixed Participation Cost
A. Consumption and Wealth Accumulation
Figure 2A plots mean normalized consumption (c¯t), mean normalized wealth
(w¯t), and mean normalized income net of housing expenditures ((1 − ht) ∗ y¯t).
The preference parameters are ρ = 5 and ψ = 0.2, and the importance of the be-
quest motive (b) is set at 2.5. Early in life, the household is liquidity constrained
and saves only a small buffer stock of wealth. From approximately ages 30–35
onwards, she starts saving for retirement and bequests, and wealth accumu-
lation increases significantly. During retirement, consumption decreases as a
result of the very high effective discount rate (high mortality risk). Wealth does
not fall towards zero due to the presence of the bequest motive.19
Table II shows the mean consumption to wealth ratio for different values of
the preference parameters. We report results for values of risk aversion between
one and five and for values of the EIS between 0.2 and 0.8, since this is the
18 There are several reasons for eliminating these households. First, there are very few observa-
tions within each age group beyond age 75. Second, for most of these households, the values of hit
are equal to zero. Third, this is consistent with the estimation procedure used for the labor income
process.
19 Net income increases during the first years of retirement because the housing expenditures
(ht) are still positive and decreasing toward zero.
Optimal Life-Cycle Asset Allocation 881
0
2
4
6
8
10
12
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Age
Consumption Wealth Income
Figure 2A. Life-cycle profiles of consumption, income, and wealth for the baseline pref-
erence parameters: coefficient of RRA = 5, elasticity of intertemporal substitution equal to 0.2,
and bequest motive equal to 2.5.
Table II
Average Consumption–Wealth Ratio (C/X) Implied by the No-Fixed
Cost Model for Different Values of Both the Coefficient of Risk
Aversion (ρ) and the EIS (ψ), and for Different Age Groups
Age 20 until Age 35 Age 36 until Age 65 Age 66 until Age 100
ρ ψ = 0.8 ψ = 0.5 ψ = 0.2 ψ = 0.8 ψ = 0.5 ψ = 0.2 ψ = 0.8 ψ = 0.5 ψ = 0.2
1 98% 99% 99% 98% 99% 99% 100% 100% 100%
1.2 87% 92% 93% 43% 88% 94% 88% 100% 100%
2 76% 86% 90% 18% 35% 67% 25% 71% 97%
4 61% 67% 75% 14% 18% 27% 23% 29% 59%
5 55% 60% 66% 13% 16% 19% 25% 26% 47%
range that we consider in the remaining part of the paper, and it is consistent
with existing empirical evidence (see the discussion in Section IV.C.1). The top
panel considers the first adult years (20–35) during which wealth accumulation
is mostly driven by the precautionary savings motive. As a result, the optimal
consumption to wealth ratio is significantly more affected by prudence than by
the EIS. Since the more risk-averse investors are also the more prudent ones,
882 The Journal of Finance
the consumption to wealth ratio is a decreasing function of risk aversion. For
very low values of risk aversion (close to 1), C/X converges to the 100% limit
imposed by the borrowing constraint.
The second panel of Table II summarizes the remaining preretirement period
(36–65), during which savings are now determined by the preference for low-
frequency consumption smoothing, while the bottom panel reports the results
for the retirement period (66–100). The results are qualitatively identical in
both cases.20 The optimal consumption to wealth ratio is driven by the trade-off
between the (endogenous) expected return on invested wealth and the discount
rate, combined with the household’s sensitivity to these incentives (the EIS).
The less risk-averse households invest a larger fraction of their portfolio in
stocks, and therefore the expected return on their invested wealth is higher.
Thus, since for a given ψ < 1 the income effect dominates the substitution effect,
they have higher consumption to wealth ratios. The same intuition explains
why, for a given ρ , C/X is a decreasing function of the EIS. However, for both the
highest and lowest values of ρ that we consider, this pattern becomes weaker.
In the first case, the expected return on invested wealth is very close to the
discount rate and the consumption to wealth ratio is almost independent of
ψ .21 In the second case, C/X is close to the 100% limit given by the borrowing
constraint.
From the results in Table II, we can conclude that for the range of values that
we consider, the consumption to wealth ratio is a decreasing function of both ρ
and ψ at every stage of the life cycle.
B. Asset Allocation
Figure 2B graphs the unconditional mean asset allocation in equities (α¯t) for
the same preference parameters as in Figure 2A (ρ = 5, ψ = 0.2, and b = 2.5).
Even though earnings risk is uninsurable, cash is a closer substitute for future
labor income than are stocks (see Heaton and Lucas (1997)). Young households
are overinvested in their human capital and view this nontradeable asset as an
implicit riskless asset in their portfolio. Given that the holdings of this relatively
riskless asset are larger in the early part of the life cycle, all young households
participate in the stock market and they allocate most of their financial wealth
to stocks.22 As retirement approaches, and financial wealth increases relative
to the present value of future labor income, agents start investing in cash. When
retirement savings is at its peak, more than 50% of total wealth is now being
invested in the riskless asset.
During retirement, both future labor income (the present value of the pension
transfers) and financial wealth are falling, so that the optimal asset allocation
20 The results are qualitatively similar to the ones obtained by Campbell and Viceira (1999) in the
context of an infinite-horizon portfolio choice model without labor income and liquidity constraints.
21 As risk aversion increases further (not reported), the return on the portfolio falls below the
discount rate and the consumption to wealth ratio becomes an increasing function of the EIS.
22 During the very first years of adult life households hold a small fraction of their wealth in
cash, since the present value of future labor income is actually still increasing.
Optimal Life-Cycle Asset Allocation 883
0.00
0.20
0.40
0.60
0.80
1.00
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Age
Sh
ar
e
of
W
ea
lth
In
ve
st
ed
in
S
to
ck
s
Figure 2B. Life-cycle asset allocation for the baseline preference parameters: coefficient
of RRA = 5, elasticity of intertemporal substitution equal to 0.2, and bequest motive equal to 2.5.
is determined by the relative speed at which these two decrease. Naturally, this
depends both on the discount rate (adjusted for the survival probabilities) and
the strength of the bequest motive. Given our parameter values, during most of
the retirement period, future labor income and wealth decay at similar rates,
and as a result, the share of wealth invested in stocks remains approximately
constant.23
IV. Results with the Fixed Participation Cost
A. Baseline Case
We start by reporting the results for the baseline preference parameters (ρ =
5, ψ = 0.2, and b = 2.5). In the next sections, we consider different values.
A.1. Participation Decision and Asset Allocation
The participation decision is determined by four factors. First, it is an increas-
ing function of wealth accumulation. Intuitively, households that accumulate
more wealth over the life cycle have a stronger incentive to enter the stock
market. Second, for the same level of wealth accumulation, participation is a
23 This is true, except during the last years, when most households have very little financial
wealth left.
884 The Journal of Finance
0
0.2
0.4
0.6
0.8
1
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Age
Figure 3A. Stock market participation over the life cycle for the baseline preference
parameters: coefficient of RRA = 5, elasticity of intertemporal substitution equal to 0.2, and
bequest motive equal to 2.5.
positive function of the optimal share of wealth invested in stocks. Third, since
F is one-time cost, participation is also a positive function of the investment
horizon. Fourth, since the cost must be paid at the time of entry, the likelihood
of participating in the stock market is a negative function of current marginal
utility.
Figure 3A shows the stock market participation rate for the baseline pref-
erence parameters. Since young households are liquidity constrained, their
marginal utility is extremely high, and as a result they do not participate in the
stock market until sufficient wealth has been accumulated. As we can see from
Figure 3A, given these preference parameters, this happens very fast and by
age 25 the participation rate is almost 100%. As a result, the average life-cycle
profiles of wealth accumulation, consumption, and equity shares are almost
identical to the ones obtained without the fixed cost (reported in Figures 2A
and B), and are therefore omitted here.
A.2. Wealth Distributions
Figure 3B plots the evolution of the distributions of cash on hand for the
two types of agents: stock market participants and nonparticipants at age 30,
still with ρ = 5 and ψ = 0.2. There is a pronounced spike around the normal-
ized cash-on-hand level of 0.75; beyond that level, stock market participation
Optimal Life-Cycle Asset Allocation 885
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5 1.5 2.5 3.5 4.5 5.5
Normalized Cash on Hand
D
en
si
ty
Dis Age 30 out of Stock Market Dis Age 30 in Stock Market
Figure 3B. Distributions for normalized cash on hand, at age 30, for the baseline pref-
erence parameters: coefficient of RRA = 5, elasticity of intertemporal substitution equal to 0.2,
and bequest motive equal to 2.5.
becomes optimal and the two distributions overlap for a small interval, mostly
representing the incurrence of the fixed entry cost. Figure 3C plots the distri-
butions of cash on hand for ages 50 for both types of agents. Conditional on age,
the distribution of cash on hand for stockholders has a much higher variance
than the wealth distribution for the households that have not participated in
the stock market.
A.3. Sensitivity Analysis
Next, we perform some sensitivity analysis with respect to the importance of
the bequest motive. Figure 3D plots wealth accumulation for different values
of the parameter b, while Figure 3E plots the corresponding conditional asset
allocations. A stronger bequest motive increases wealth accumulation at every
stage of the life cycle, and the effect is strongest during retirement. The increase
in wealth accumulation leads to a modest reduction in the share of equity in-
vestment during working life. Since both of these effects have a fairly modest
impact until retirement, the implied participation rates are not significantly
affected, and therefore we do not report them. During retirement, an increase
in the bequest motive decreases the speed at which wealth is being drawn down,
and leads to a higher ratio of financial wealth to labor income. As a result, for a
given age, a stronger (weaker) bequest motive decreases (increases) the optimal
equity share.
886 The Journal of Finance
0
0.02
0.04
0.06
0.08
0 4 8 10 12 14 16 18 20
Normalized Cash on Hand
D
en
si
ty
Dis Age 50 In Stock Market Dis Age 50 out of Stock Market
2 6
Figure 3C. Distributions for normalized cash on hand, at age 50, for the baseline pref-
erence parameters: coefficient of RRA = 5, elasticity of intertemporal substitution equal to 0.2,
and bequest motive equal to 2.5.
0
2
4
6
8
10
12
14
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Age
b=3.5 b=2.5 b=1.5 b=0.0
Figure 3D. Wealth accumulation for different values of the bequest parameter (b), and
for the baseline preferences parameters: coefficient of RRA = 5 and elasticity of intertemporal
substitution equal to 0.2.
Optimal Life-Cycle Asset Allocation 887
0.00
0.20
0.40
0.60
0.80
1.00
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Age
Sh
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in
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s
b=3.5 b=2.5 b=1.5 b=0.0
Figure 3E. Asset allocation for stock market participants, for different values of the
bequest parameter (b), and for the baseline preference parameters: coefficient of RRA =
5, elasticity of intertemporal substitution equal to 0.2.
As mentioned before, the empirical evidence is mixed on the magnitude of
the correlation coefficients between stock returns and the different labor in-
come shocks (transitory and permanent). In our baseline calibration, we have
assumed φN = 0.15 and φU = 0.0, following the estimation of Campbell et al.
(2001). In Figure 3F, we now check whether the results are sensitive to these
values. We only report the asset allocation decisions, since the participation
decision is almost identical in all cases.
Campbell et al. (2001) do not actually estimate φU = 0.0; they just assume it.
Therefore, in our first experiment, we allow for a positive correlation between
stock returns and the transitory labor income shocks, in particular we consider
φN = 0.15 and φU = 0.1. The results are very similar to the ones obtained for the
baseline case. In the second experiment, we now set φN = 0.0 and assume that
the correlation is instead driven exclusively by the transitory shocks, thus set-
ting φU = 0.15. We again obtain results that are extremely close to our baseline
case. The share invested in equities is higher, but only marginally so. Finally,
we consider the case in which there is no correlation between stock returns and
labor income shocks. Only in this case do we find a visible difference relative to
the benchmark calibration, as investors now allocate a higher fraction of their
wealth to equities.
So far we have assumed that all households start at age 20 with zero ini-
tial wealth. Table III reports summary statistics for the wealth distribution for
888 The Journal of Finance
0.00
0.20
0.40
0.60
0.80
1.00
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
corrt=0.1/corrp=0.15 corrt=0.0/corp=0.15
corrt=0.15/corrp=0.0 corrt=0.0/corrp=0.0
Figure 3F. Asset allocation for stock market participants, for different values of the
correlation between stock returns and transitory (permanent) labor income shocks, de-
noted by corrt (corrp), with the baseline preference parameters: coefficient of RRA = 5,
elasticity of intertemporal substitution equal to 0.2, and bequest motive equal to 2.5.
Table III
The Wealth Distribution (Wealth-to-Income Ratios) for Households
with Head Aged 20 or Less
The data are taken from the 2001 SCF (details in Appendix C). The variable X defines liquid wealth
and Y denotes labor or pension income.
Decile (%) 10 20 30 40 50 60 70 80 90
X/Y 0.000 0.015 0.043 0.113 0.167 0.236 0.267 0.406 0.863
households of age 20 (or lower) from the SCF (details given in Appendix C).
When we use this distribution as the initial condition in our model, with the
exception of the first few years, the wealth profiles are virtually indistinguish-
able and therefore we do not report them.24 This result occurs because young
households are liquidity constrained and they therefore prefer to consume all
of this additional wealth rather than save it.
In our baseline calibration, we assume that housing expenditures constitute
a fixed proportion of labor income. We now allow for a stochastic component
24 In the implementation, we truncate the distribution from the SCF at its 90 percentile.
Optimal Life-Cycle Asset Allocation 889
in this ratio. More precisely, disposable income is now given by (1 − h˜t)Yi,t+1,
where
h˜t = ht ∗ exp
(
εht
)
, (17)
and εht follows a normal distribution with zero mean and variance σ
2
εh
.
In this experiment we set σεh = 0.25. This effectively corresponds to an in-
crease in the level of background risk, and it is equivalent to an increase in
the variance of the transitory labor income shocks. We find that the results are
quite similar to the baseline case and therefore we do not report them. The in-
crease in background risk reduces the willingness to invest in stocks, but since
these are transitory shocks, the effect is not very large. For the same reason,
the wealth accumulation and the participation rate are almost unaffected.
B. Changing Risk Aversion and the Impact of Background Risk
The stock market participation rate implied by the baseline parameters is
counterfactually high. In this section, we explore the model’s ability to produce
more realistic results by considering different preference parameter values.
As mentioned before, the participation decision is an increasing function of
both wealth (X) and the optimal share of wealth invested in risky assets (α).
Decreasing risk aversion increases the optimal share invested in stocks, but as
shown in Section III.A, it also decreases wealth accumulation at every stage of
the life cycle. Therefore, the impact on the participation decision resulting from
changes in risk aversion depends on which of these two effects dominates.
B.1. Wealth Accumulation
We start by decreasing ρ from 5 to 2, while maintaining the power utility
assumption, thus increasing the EIS (ψ) to 0.5. In Figure 4A, we plot the wealth
accumulation for this case and for the baseline parameter values (ρ = 5, ψ =
0.2, and b = 2.5). As expected, wealth accumulation is significantly reduced at
every stage of the life cycle. As previously shown in Section III.A (see Table II),
the average consumption to wealth ratio is now 86% for the age group 20–35,
and 35% for the age group 36–65, as opposed to 66% and 19% respectively.
However, from the results in Table II, we know that if we depart from power
utility and decrease both risk aversion (ρ) and the EIS (ψ) simultaneously, this
significantly reduces wealth accumulation. Consider then decreasing ρ to 2, but
now keeping ψ at 0.2. The consumption to wealth ratio for the first age group
is not significantly affected (90% instead of 86%) because, at this stage of the
life cycle, savings are essentially driven by prudence (which remains constant).
However, for the second age group, wealth accumulation is determined mostly
by the EIS. As a result, the average consumption to wealth ratio is now almost
doubled, increasing from 35% to 67%. As shown in Figure 4A, this leads to a
very significant reduction in life-cycle wealth accumulation.
890 The Journal of Finance
0
2
4
6
8
10
12
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Age
RRA=5 (EIS=0.2) RRA=2 (EIS=0.5) RRA=2 (EIS=0.2)
Figure 4A. Wealth accumulation over the life cycle, for different values of the prefer-
ence parameters (coefficient of RRA and elasticity of intertemporal substitution) with
bequest motive equal to 2.5.
B.2. Stock Market Participation Rates and Asset Allocation
Figure 4B plots the participation rates for different values of risk aversion,
with the EIS equal to 0.2. Given the large differences in wealth accumula-
tion, it is not surprising that the wealth effect dominates with respect to the
participation decision. The less prudent households save less, and as a result,
their participation rate is smaller. While almost all high prudent households
have already paid the fixed cost by age 25, only 75% of the households with
ρ = 2 have done so, although by age 35, even all of these investors have already
paid the fixed cost as well. However, from the less risk-averse households (i.e.,
ρ = 1.2), only a very small fraction (<20%) will ever invest in stocks. On the
other hand, as shown in Figure 4C, the reduction in risk aversion generates
counterfactually high equity holdings for those investors that have paid the
fixed cost.25
B.3. The Impact of Background Risk
The previous results illustrate one important trade-off generated by the level
of background risk. When faced with more background risk (e.g., due to more
25 For ρ = 1.2, the conditional equity share is always 100%, and therefore it is not included in
the figure.
Optimal Life-Cycle Asset Allocation 891
0
0.2
0.4
0.6
0.8
1
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Age
RRA=2 (EIS=0.2) RRA=5 (EIS=0.2) RRA=1.2 (EIS=0.2)
Figure 4B. Stock market participation rate for different values of the coefficient of
RRA, with elasticity of intertemporal substitution equal to 0.2, and bequest motive equal
to 2.5.
0.00
0.20
0.40
0.60
0.80
1.00
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Age
Sh
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in
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s
RRA=2 (EIS=0.2) RRA=5 (EIS=0.2)
Figure 4C. Asset allocation for stock market participants, for different values of the
coefficient of RRA, with elasticity of intertemporal substitution equal to 0.2, and bequest
motive equal to 2.5.
892 The Journal of Finance
0.00
0.20
0.40
0.60
0.80
1.00
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Age
Stocks (Baseline) Stocks (Higher risk) Participation (Baseline) Participation (Higher risk)
Figure 4D. Stock market participation rate and share of wealth invested in stocks for
different levels of background risk, with the baseline preferences: coefficient of RRA = 5,
elasticity of intertemporal substitution equal to 0.2, and bequest motive equal to 2.5.
labor income risk, consumption risk, or housing/mortgage risk) agents invest
a smaller fraction of their financial wealth in risky assets. However, they also
accumulate a larger buffer stock of wealth, and thus have a stronger incentive
to enter the stock market. We have considered three different experiments in
which we have increased the investor’s background risk. In the first two, we
have assumed a higher variance of respectively transitory and permanent labor
income shocks, and in the third, we have included a positive probability of a
disastrous labor income shock. Figure 4D shows the results for the case of the
first experiment, where the variance of the transitory labor income shocks was
increased by a factor of 3.26 As expected, background risk crowds out stock
holdings and households invest a smaller fraction of their portfolios in equities.
However, they also increase their buffer stock of wealth, and as a result the
stock market participation rate is higher than before.
C. Asset Allocation and Participation Rates with Preference Heterogeneity
C.1. Matching Participation Rates and Conditional Asset Allocations
Given our previous results, we can simultaneously match stock market par-
ticipation rates and asset allocation conditional on participation with moderate
26 The results for the other two cases are qualitatively identical, and they are available upon
request.
Optimal Life-Cycle Asset Allocation 893
degrees of risk aversion, if we allow for preference heterogeneity. Households
with very low risk aversion and low EIS smooth idiosyncratic earnings shocks
with a small buffer stock of assets, and most of them never invest in equities
(thus behaving as in the Deaton (1991) infinite-horizon model). This seems to
describe adequately the behavior of a large fraction of the U.S. population that
retires without significant financial assets (and does not participate in the stock
market). Within the low EIS and low risk-aversion group, only a small fraction
of them owns stocks, and they do so only as they get close to retirement. On the
other hand, investors with high prudence and high EIS are the ones who par-
ticipate in the stock market from early on, since they accumulate more wealth
and therefore have a stronger incentive to pay the fixed cost. Therefore, the
marginal stockholders are (endogenously) more risk averse, and as a result
they do not invest their portfolios fully in stocks.
In this final section, we try to evaluate how much heterogeneity we need to
match the data. In other words, can the model consistently explain the two facts
for a plausible distribution of preference parameters across the population?
Table IV reports participation rates and average equity shares for stock market
participants, for different distributions, and compares them with the empirical
evidence from the SCF. We first consider a 50% split between investors with
both low risk aversion and low EIS (ρ = 1.2 and ψ = 0.2), and investors with
moderate risk aversion and moderate EIS (ρ = 5 and ψ = 0.5). The model gives
a participation rate of 52.1% and an equity share of 54.5% for stock market
participants (Case 1 in Table IV), which matches fairly well with the empirical
evidence reported in Section I (and summarized in the first row of Table IV).
It is important to mention that this form of heterogeneity is consistent with
the existing empirical evidence. Attanasio, Banks, and Tanner (2002) show that
the CRRA coefficient is much higher (thus much lower EIS) for nonstockholders
Table IV
The Average Stock Market Participation Rate (P¯) and Average Stock
Holdings for Stock Market Participants (α¯P)
The first row reports data from the 2001 SCF (details in Appendix C), while the other four panels
report the results from the model for different distributions of investors. Case 1 assumes two
groups of agents, (ρ = 1.2 and ψ = 0.2) and (ρ = 5 and ψ = 0.5), with 50% weight each. Case 2 also
assumes two groups of agents, but now (ρ = 1.1 and ψ = 0.2) and (ρ = 5 and ψ = 0.5), with 50%
weight each, and with the initial wealth distribution calibrated from the SCF. In Case 3 we have
again two groups, (ρ = 1.07 and ψ = 0.5) and (ρ = 5 and ψ = 0.5), with 50% weight each. Finally,
Case 4 considers three groups of agents, (ρ = 1 and ψ = 0.2) and (ρ = 3 and ψ = 0.5) and (ρ = 5
and ψ = 0.5), with weights 40%, 30%, and 30%, respectively.
P¯ (%) α¯P (%)
Data 51.94 54.76
Model (Case 1) 52.14 54.48
Model (Case 2) 50.36 53.32
Model (Case 3) 54.42 56.24
Model (Case 4) 56.98 56.56
894 The Journal of Finance
than for stockholders. Vissing-Jørgensen (2002a) focuses on this distinction and
argues that “accounting for limited asset market participation is crucial for
obtaining consistent estimates of the EIS” (p. 827). Vissing-Jørgensen then ob-
tains estimates of the EIS greater than 0.3 for risky asset holders, while for the
remaining households the EIS estimates are small and insignificantly different
from zero. Vissing-Jørgensen and Attanasio (2003) further stress that loosen-
ing the link between risk aversion and intertemporal substitution can generate
implications about the covariance of stock returns and individual consumption
growth for stockholders that are not rejected in the data. They offer risk aver-
sion estimates for stockholders at around 5–10 and EIS estimates around 1.
Overall, the existing estimates of EIS and risk aversion are consistent with the
values that we use in this paper.
C.2. Life-Cycle Profiles
We now report the life-cycle profiles of stock market participation and asset
allocation implied by the model. As argued in Section I, in the data these profiles
are not very robust to specific assumptions about cohort or time effects (see
Figures 1A and B). As a result, in this paper, we have focused mostly on life-
cycle averages.
Figure 5A plots the stock market participation rate implied by the model
for different age groups, together with the corresponding numbers from the
SCF (see Appendix C for details), while Figure 5B does the same, but now for
0
0.2
0.4
0.6
0.8
1
[20,34] [35,44] [45,54] [55,64] [65,74] [75,100]
Age Groups
Model Data
Figure 5A. Stock market participation rate implied by the heterogeneous agent model
and stock market participation rate from the 2001 sample of the SCF.
Optimal Life-Cycle Asset Allocation 895
0
0.2
0.4
0.6
0.8
1
[20,34] [35,44] [45,54] [55,64] [65,74] [75,100]
Age Groups
Sh
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In
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Model Data
Figure 5B. Asset allocation for stock market participants rate implied by the hetero-
geneous agent model and asset allocation for stock market participants from the 2001
sample of the SCF.
the average asset allocation of stock market participants. The participation
rates are extremely similar, with the largest difference occurring at retirement
when the participation rate in the data declines, while it remains constant in
the model. However, as shown in Figure 1B, this is exactly one of the results
that is not robust to the assumption of cohort dummies versus time dummies.
With respect to the asset allocation decisions, we do observe a more significant
difference, in this case for young households. In the model, these agents invest
a significant fraction of their portfolio in equities, while in the data, regardless
of the controls, this does not happen.
C.3. Sensitivity Analysis and Robustness
So far we have assumed that households start at age 20 with zero initial
wealth, since we have seen in Section IV.A.3 that, for ρ = 5, using the initial
wealth distribution calibrated from the SCF does not affect the results. How-
ever, this might not be the case for investors with low risk aversion and low
EIS, since they save very little. By giving these households some positive initial
wealth, we are likely to see an increase in stock market participation rates. In
the third row of Table IV (Case 2) we show that this effect is not too large, and
we can replicate the previous results by considering a slightly lower value of
risk aversion (1.1).27
27 Alternatively, we could consider a lower value of the EIS. With ψ = 0.1 we would again obtain
very similar results.
896 The Journal of Finance
It is important to point out that we do not need to assume a very low value
of the EIS to generate large nonparticipation, since we can compensate for a
higher ψ by decreasing risk aversion even further. This is shown in the fourth
row of Table IV (Case 3), where we fix the EIS coefficient equal to 0.5 for both
types of investors. To reproduce the results in the first panel, we find that we
need to decrease ρ to 1.07 for the less risk-averse group.
Given our previous discussion, we know that households with risk aversion
between 1.5 and 4 tend to participate in the stock market from early on, and
invest almost all of their wealth in stocks. Naturally, it is not reasonable to as-
sume that the distribution of coefficients of risk aversion mysteriously collapses
to the two extremes that we have previously considered (1.2 and 5). In the fifth
row of Table IV (we now consider a smoother distribution, with ρ ranging from 1
to 5 (Case 4)). It is important to point out that this is not a uniform distribution,
as there is a slightly higher fraction of less risk-averse households. If we want
to match both facts simultaneously, with a (relatively) smooth distribution, we
need it to exhibit some negative skewness. As predicted, both the participation
rate and the equity share are now higher than before, but not significantly so.
The equity share is now 57%, while the participation rate is also equal to 57%,
numbers that are still extremely close to the empirical evidence (row 1).
D. Wealth Distribution
In this section, we compare the wealth accumulation predicted by the model
with the empirical evidence in the SCF. Given the (exogenous) differences in
the preference parameters and the (endogenous) differences in the participation
decision, our model generates a large degree of heterogeneity in wealth accu-
mulation. To illustrate, we compare both median wealth accumulation and the
extremes of the distribution (percentiles 10 and 90) to see if the model generates
the degree of heterogeneity observed in the SCF. We divide households into the
three usual age groups: buffer stock savers (20–35), retirement savers (36–65),
and retirees (over 66).
The results are shown in Table V. The model can replicate the low wealth ac-
cumulation patterns of the poorer households in the data. Households with the
lowest income realizations tend not to participate in the stock market and accu-
mulate very little wealth over the life cycle. This is consistent with the results
in Hubbard, Skinner, and Zeldes (1995), who illustrate in a similar model how
the presence of social insurance (pensions) can crowd out private saving over
the life cycle for the poorest quintile of the wealth distribution. Nevertheless,
in the SCF these households still accumulate some nonnegligible wealth dur-
ing retirement, something that does not happen in the model. For the median
household, the model does quite well early in life; it overshoots for the second
age group; and it undershoots at retirement. Finally, at the high end of the
distribution we can generate extremely large wealth accumulation, although
not quite as high as in the data. This difference is most significant during the
retirement period and early in life. Overall the degree of heterogeneity in the
wealth distribution is comparable to the one observed in the data. The model
Optimal Life-Cycle Asset Allocation 897
Table V
The Distribution of Wealth-to-Labor Income Ratios from the 2001
SCF for Different Age Groups
Appendix C provides more details. Results are provided for two different versions of the model:
with zero initial wealth and with the initial wealth distribution calibrated from the SCF. Results
are shown for different age groups.
Age Groups
20–35 36–65 65
Data
10th percentile 0.002 0.071 0.371
Median 0.287 2.170 7.931
90th percentile 2.702 10.648 33.363
Model (zero initial wealth)
10th percentile 0.006 0.005 0.006
Median 0.261 3.115 4.838
90th percentile 0.748 8.184 17.539
Model (initial wealth from the SCF)
10th percentile 0.006 0.005 0.006
Median 0.263 3.116 4.839
90th percentile 0.886 8.371 17.865
consistently generates low wealth accumulation at retirement, which would
suggest the presence of a stronger bequest motive, but as shown in
Section IV.A.3, a stronger bequest motive also increases wealth accumulation
at mid life (Figure 3D).
In the last panel of Table V, we simulate the model with the initial distri-
bution of wealth calibrated from the SCF. The results are almost identical to
the previous ones, with a minor increase in wealth accumulation at the 90
percentile.
V. Conclusion
In this paper, we present a life-cycle asset allocation model with realistically
calibrated uninsurable labor income risk that provides an explanation for two
very important empirical observations: low stock market participation rates
in the population as a whole, and moderate equity holdings for stock market
participants. We do not rely on high values of risk aversion, or on extreme
assumptions about background risk.
In our model, households with very low risk aversion and low EIS accumu-
late very little wealth and as a result, most of them never invest in equities.
On the other hand, the more prudent investors are the ones who participate in
the stock market from early on, as they accumulate more wealth and therefore
have a stronger incentive to pay the fixed entry cost. Therefore, the marginal
stockholders are (endogenously) more risk averse and as a result they do not
898 The Journal of Finance
invest their portfolios fully in stocks. On the negative side, the model still
counterfactually predicts that young households that have already paid the
participation cost invest most of their portfolio in equities.
Appendix A: Numerical Solution Method
We first simplify the solution by exploiting the scale-independence of the
maximization problem and rewriting all variables as ratios to the permanent
component of labor income (Pit). The laws of motion and the value function can
then be rewritten in terms of these normalized variables, and we use lowercase
letters to denote them (for instance, xit ≡ X itPit ). This allows us to reduce the
number of state variables to three: one continuous state variable (cash on hand,
Xit) and two discrete state variables (age, t, and participation status, whether
the fixed cost has been paid or not). We discretize the state space along the cash
on hand dimension (the only continuous state variable), so that the relevant
policy functions can now be represented on a numerical grid.
We solve the model using backward induction. For every age t prior to T, and
for each point in the state space, we optimize using grid search. So we need
to compute the value associated with each level of consumption, the decision
to pay the fixed cost, and the share of liquid wealth invested in stocks. From
the Bellman equation, these values are given as current utility plus the dis-
counted expected continuation value (EtVt+1(·, ·)), which we can compute once
we have obtained Vt+1. In the last period, the policy functions are determined
by the bequest motive and the value function corresponds to the bequest func-
tion, regardless of whether the fixed cost has been paid before or not. This gives
us the terminal condition for our backward induction procedure. We perform
all numerical integrations using Gaussian quadrature to approximate the dis-
tributions of the innovations to the labor income process and the risky asset
returns. For points that do not lie on state space grid, we evaluate the value
function using a cubic spline interpolation.
Once we have computed the value of all the alternatives, we just pick the
maximum, thus obtaining the policy rules for the current period (St and Bt).
At each point of the state space, the participation decision is computed by com-
paring the value function conditional on having paid the fixed cost (adjusting
for the payment of the cost itself ) with the value function conditional on non-
payment. Substituting these decision rules in the Bellman equation, we obtain
this period’s value function (Vt(·, ·)), which is then used to solve the previous
period’s maximization problem. This process is iterated until t = 1.
Appendix B: Computing the Transition Distributions
To find the distribution of cash on hand, we first compute the relevant opti-
mal policy rules: bond and stock policy functions for stock market participants
and nonparticipants and the {0, 1} participation rule as a function of cash on
hand. Let bI(x) and sI(x) denote respectively the bonds and stock policy rules
for individuals participating in the stock market, and let bO(x) be the savings
Optimal Life-Cycle Asset Allocation 899
decision for the individual out of the stock market. We assume that households
start their working lives with zero liquid assets. During working lives, for the
households that have not paid the fixed cost, the evolution of normalized cash
on hand is given by28
xt+1 =
[
bO(xt)R f
] { Pt
Pt+1
exp( f (t, Zt))
exp( f (t + 1, Zt+1))
}
+ (1 − ht+1)Ut+1
= w
(
xt
∣∣∣∣ PtPt+1 ,
exp( f (t, Zt))
exp( f (t + 1, Zt+1))
)
+ (1 − ht+1)Ut+1, (B1)
where w(x) is defined by the last equality and is conditional on { PtPt+1 } and the de-
terministically evolving exp( f (t, Zt ))exp( f (t+1, Zt+1)) . Denote the transition matrix of moving
from xj to xk,29 conditional on not having paid the fixed cost, as TOkj. Let denote
the distance between the equally spaced discrete points of cash on hand. The
random permanent shock PtPt+1 is discretized using Gaussian quadrature with
H points: PtPt+1 = {Nm}m=Hm=1 . T Ok j = Pr(xt+1=k | xt= j ) is found using30
m=H∑
m=1
Pr
(
xt+1 | xt , PtPt+1 = Nm
)
∗ Pr
(
Pt
Pt+1
= Nm
)
. (B2)
Numerically, this probability is calculated using
T Okjm = Pr
(
xk + 2 xt+1 xk −

2
∣∣∣∣ xt = x j , PtPt+1 = Nm
)
. (B3)
Making use of the approximation that for small values of σ 2u , U ∼ N(exp(µu +
0.5 ∗ σ 2u ), (exp(2 ∗ µu + (σ 2u )) ∗ (exp(σ 2u ) − 1)), and denoting the mean of
(1 − ht)U by U¯ and its standard deviation by σ , the transition probability con-
ditional on Nm equals
T Okjm =

xk +

2
− w(xt | Nm) − U¯
σ

 −

xk −

2
− w(xt | Nm) − U¯
σ

 , (B4)
where is the cumulative distribution function for the standard normal. The
unconditional probability from xj to xk is then given by
T Okj =
m=H∑
m=1
T Okjm Pr(Nm). (B5)
28 To avoid cumbersome notation, the subscript i that denotes a particular individual is omitted
in what follows.
29 The normalized grid is discretized between (xmin, xmax) where xmin denotes the minimum point
on the equally spaced grid and xmax denotes the maximum point.
30 The dependence on the deterministically evolving exp( f (t,Zt ))exp( f (t+1,Zt+1)) is implied and is omitted from
what follows for expositional clarity.
900 The Journal of Finance
Given the transition matrix TO (letting the number of cash-on-hand grid points
equal to J, this is a J by J matrix; T Okj represents the {kth, jth} element),
the next period probabilities of each of the cash-on-hand states can be found
using
πOkt =
∑
j
T Okj ∗ πOjt−1. (B6)
We next use the vector ΠOt (this is a J by 1 vector representing the mass of the
population out of the stock market at each grid point; πOkt represents the {kth}
element at time t) and the participation policy rule to determine the percentage
of households that optimally choose to incur the fixed cost and invest in risky
assets. This is found by computing the sum of the probabilities in ΠOt for which
x > x∗, x∗ being the trigger point that causes participation (x∗ is determined
endogenously through the participation decision rule). These probabilities are
then deleted from ΠOt and are added to Π
I
t , appropriately renormalizing both
{ΠOt , ΠIt} to sum to one. The participation rate (θt) can be computed at this
stage as
θt = θt−1 + (1 − θt−1) ∗
∑
x j >x∗
πOt, j . (B7)
The same methodology (but with more algebra and computations) can then be
used to derive the transition distribution for cash on hand conditional on having
paid the fixed cost, TIt . The corresponding normalized cash-on-hand evolution
equation is
xt+1 =
[
b(xt)R f + s(xt)RSt+1
] { Pt
Pt+1
exp( f (t, Zt))
exp( f (t + 1, Zt+1))
}
+ (1 − ht+1)Ut+1
= w
(
xt
∣∣ RSt+1, PtPt+1
)
+ (1 − ht+1)Ut+1, (B8)
where w(x) is now conditional on {RSt+1, PtPt+1 }.31 The random processes RSt+1 and
Pt
Pt+1
are discretized using Gaussian quadrature with H points: RSt+1 = {RSl }l=Hl=1
and PtPt+1 = {Nn}n=Hn=1 . T Ik j = Pr(xt+1=k | xt= j ) is obtained from
l=H∑
l=1
n=H∑
n=1
Pr
(
xt+1 | xt , RSt+1 = RSl ,
Pt
Pt+1
= Nn
)
∗ Pr (RSl ) ∗ Pr(Nn), (B9)
where Pr(RSl ) and Pr(Nn) stand respectively for Pr(R
S
t+1 = RSl ) and Pr( PtPt+1 =
Nn), and where the independence between PtPt+1 and R
S
t+1 was used.
32 Numeri-
cally, this probability is calculated using
31 The dependence on the nonrandom earnings component is omitted to simplify notation.
32 The methodology can be applied for an arbitrary correlation structure using the Choleski
decomposition of the variance-covariance matrix of the innovations.
Optimal Life-Cycle Asset Allocation 901
T Ik j ln = Pr
(
xk + 2 xt+1 xk −

2
∣∣∣∣ xt = x j , PitPit+1 = Nn, RSt+1 = RSl
)
. (B10)
The transition probability conditional on Nn, RSl , and R
B
m equals
T Ikjln =

xk +

2
− w(xt | Nn, RSl ) − U¯
σ


−

xk −

2
− w(xt | Nn, RSl ) − U¯
σ

 .
(B11)
The unconditional probability from xj to xk is then given by
T Ik j =
l=H∑
l=1
n=H∑
n=1
T Ik j ln Pr
(
RSl
)
Pr(Nn). (B12)
Given the matrix TI, the probabilities of each of the states are updated by
π Ikt+1 =
∑
j
T Ik j ∗ π Ij t . (B13)
Appendix C: SCF Data
The SCF is probably the most comprehensive source of data on U.S. house-
hold assets. The SCF uses a two-part sampling strategy to obtain a sufficiently
large and unbiased sample of wealthier households (the rich sample is chosen
randomly using tax reports). To enhance the reliability of the data, the SCF
makes weighting adjustments for survey nonrespondents; these weights were
used in computing the values reported in the tables. The specific names in the
codebook for the variables used are given below.
We construct a measure of labor income that matches as closely as possi-
ble the process for Yit (earnings) in the text. We therefore define labor income
as the sum of wages and salaries (X5702), unemployment or worker’s com-
pensation (X5716) and Social Security or other pensions, annuities, or other
disability or retirement programs (X5722). Liquid wealth is variable FIN in
the publicly available SCF data set, to which home equity was added. Variable
FIN is made up of LIQ (all types of transaction accounts—checking, saving,
money market, and call accounts), CDS (certificates of deposit), total directly
held mutual funds, stocks, bonds, total quasiliquid financial assets (the sum
of IRAs, thrift accounts, and future pensions), savings bonds, the cash value
of whole life insurance, other managed assets (trusts, annuities, and managed
investment accounts in which the household has equity interest), and other fi-
nancial assets: This includes loans from the household to someone else, future
proceeds, royalties, futures, nonpublic stock, and deferred compensation. We
902 The Journal of Finance
define home equity as the value of the home less the amount still owed on the
first and second/third mortgages and the amount owed on home equity lines
of credit. This definition of wealth is consistent with both the definitions in
Hubbard, Skinner, and Zeldes (1995) and in Heaton and Lucas (2000).
Financial assets invested in the risky asset can be directly held stock, stock
mutual funds, or amounts of stock in retirement accounts. We follow the pro-
cedures the SCF uses to construct this number for each household (variable
EQUITY). Specifically, this is done by computing the full value of stocks, adding
the full value if an asset is described as a stock mutual fund, and half the
value if the asset refers to a combination of mutual funds. For this purpose,
IRAs/Keoghs invested in stock are computed by adding the full value if mostly
invested in stock, half the value if split between stocks/bonds or stocks/money
market, and one-third of the value if split between stocks/bonds/money market.
We also add other managed assets with equity interest (annuities, trusts, and
MIAs) by adding the full value if mostly invested in stock, half the value if split
between stocks/MFs & bonds/CDs, or “mixed/diversified,” and one-third of the
value if “other.” We also add thrift-type retirement accounts invested in stock:
the full value if mostly invested in stock and half the value if split between
stocks and interest earning assets. Stock market participation is then deter-
mined by checking whether the full value of stocks (EQUITY) is greater than
zero (variable HEQUITY).
We construct the share of wealth in stocks conditional on HEQUITY being
positive as (EQUITY)/(FIN) where all the variables have been defined above.
REFERENCES
Aiyagari, Rao, and Mark Gertler, 1991, Asset returns with transactions costs and uninsured indi-
vidual risk, Journal of Monetary Economics 27, 311–331.
Ameriks, John, and Stephen Zeldes, 2001, How do household portfolio shares vary with age? Work-
ing paper, Columbia Business School.
Attanasio, Orazio, James Banks, and Sarah Tanner, 2002, Asset holding and consumption volatility,
Journal of Political Economy 110, 771–792.
Basak, Suleyman, and Domenico Cuoco, 1998, An equilibrium model with restricted stock market
participation, Review of Financial Studies 11, 309–341.
Bertaut, Carol, and Michael Haliassos, 1997, Precautionary portfolio behavior from a life cycle
perspective, Journal of Economic Dynamics and Control 21, 1511–1542.
Cagetti, Marco, 2003, Wealth accumulation over the life cycle and precautionary savings, Journal
of Business and Economic Statistics 21, 339–353.
Campbell, John, Joao Cocco, Francisco Gomes, and Pascal Maenhout, 2001, Investing retirement
wealth: A life cycle model, in John Campbell and Martin Feldstein, eds:. Risk Aspects of Social
Security Reform (University of Chicago Press, Chicago).
Campbell, John, and Luis Viceira, 1999, Consumption and portfolio decisions when expected re-
turns are time varying, Quarterly Journal of Economics 114, 433–495.
Carroll, Christopher, 1992, The buffer-stock theory of saving: some macroeconomic evidence, Brook-
ings Papers on Economic Activity 2, 61–156.
Carroll, Christopher, 1997, Buffer stock saving and the life cycle/permanent income hypothesis,
Quarterly Journal of Economics 112(1), 3–55.
Cocco, Joao, 2000, Portfolio choice in the presence of housing, Working paper, London Business
School.
Optimal Life-Cycle Asset Allocation 903
Cocco, Joao, Francisco Gomes, and Pascal Maenhout, 1999, Portfolio choice over the life cycle,
Working paper, Harvard University.
Constantinides, George, 1986, Capital market equilibrium with transaction costs, Journal of Po-
litical Economy 94, 842–862.
Constantinides, George, John Donaldson, and Rajnish Mehra, 2002, Junior can’t borrow: A new
perspective on the equity premium puzzle, Quarterly Journal of Economics 117, 269–296.
Dammon, Robert, Chester Spatt, and Harold Zhang, 2001, Optimal consumption and investment
with capital gains taxes, Review of Financial Studies 14, 583–616.
Dammon, Robert, Chester Spatt, and Harold Zhang, Optimal asset location and allocation with
taxable and tax-deferred investment, Journal of Finance 59, 999–1037.
Davis, Steven, Felix Kubler, and Paul Willen, 2002, Borrowing costs and the demand for equity
over the life cycle, Working paper, University of Chicago and Stanford University.
Davis, Steven, and Paul Willen, 2001, Occupation-level income shocks and asset returns: Their
covariance and implications for portfolio choice, Working paper, University of Chicago.
Deaton, Angus, 1991, Saving and liquidity constraints, Econometrica 59, 1221–1248.
de Nardi, Maria Cristina, Wealth inequality and intergenerational links, Review of Economic Stud-
ies (forthcoming).
Degeorge, Francois, Dirk Jenter, Alberto Moel, and Peter Tufano, 2002, Selling company shares
to reluctant employees: France Telecom’s experience, Working paper, HEC and Harvard
University.
Dynan, Karen, Jonathan Skinner, and Stephen Zeldes, 2002, The importance of bequests and life
cycle saving in capital accumulation: A new answer, American Economic Review 92, 274–278.
Epstein, Lawrence, and Stanley Zin, 1989, Substitution, risk aversion, and the temporal behavior
of consumption and asset returns: A theoretical framework, Econometrica 57, 937–969.
Flavin, Marjorie, and Takashi Yamashita, 2002, Owner-occupied housing and the composition of
the household portfolio, American Economic Review 92, 345–362.
Gomes, Francisco, and Alexander Michaelides, 2003, Portfolio choice with internal habit formation:
A life cycle model with uninsurable labor income risk, Review of Economic Dynamics 6, 729–
766.
Gourinchas, Pierre-Olivier, and Jonathan Parker, 2002, Consumption over the life cycle, Econo-
metrica 70, 47–89.
Guiso, Luigi, Michael Haliassos, and Tullio Japelli, 2002, Household portfolios: An international
comparison, in Luigi Guiso, Michael Haliassos, and Tullio Japelli, eds:. Household Portfolios
(MIT Press, Cambridge, MA).
Guiso, Luigi, Tullio Jappelli, and Daniele Terlizzese, 1996, Income risk, borrowing constraints and
portfolio choice, American Economic Review 86, 158–172.
Haliassos, Michael, and Alexander Michaelides, 2003, Portfolio choice and liquidity constraints,
International Economic Review 44, 144–177.
He, Hua, and David Modest, 1995, Market frictions and consumption-based asset pricing, Journal
of Political Economy 103, 94–117.
Heaton, John, and Deborah Lucas, 1996, Evaluating the effects of incomplete markets on risk
sharing and asset pricing, Journal of Political Economy 104, 443–487.
Heaton, John, and Deborah Lucas, 1997, Market frictions, savings behavior, and portfolio choice,
Macroeconomic Dynamics 1, 76–101.
Heaton, John, and Deborah Lucas, 2000, Portfolio choice and asset prices: The importance of en-
trepreneurial risk, Journal of Finance 55, 1163–1198.
Hu, Xiaoqing, 2001, Portfolio choices for home owners, Working paper, University of Illinois at
Chicago.
Hubbard, Glenn, Jonathan Skinner, and Stephen Zeldes, 1995, Precautionary saving and social
insurance, Journal of Political Economy 103, 360–399.
Jermann, Urban, 1999, Social security and institutions for intergenerational, intragenerational
and international risk-sharing: A comment, Carnegie-Rochester Conference Series on Public
Policy 50, 205–212.
King, Mervyn, and Jonathan Leape, 1998, Wealth and portfolio composition: Theory and evidence,
Journal of Public Economics 69, 155–193.
904 The Journal of Finance
Koo, Hyeng, 1998, Consumption and portfolio selection with labor income: A continuous time ap-
proach, Mathematical Finance 8, 49–65.
Laitner, John, 2002, Wealth inequality and altruistic bequests, American Economic Review 92,
270–273.
Lucas, Deborah, 1994, Asset pricing with undiversifiable risk and short sales constraints: Deepen-
ing the equity premium puzzle, Journal of Monetary Economics 34, 325–341.
Luttmer, Erzo, 1996, Asset pricing in economies with frictions, Econometrica 64, 1439–1467.
Luttmer, Erzo, 1999, What level of fixed costs can reconcile consumption and stock returns? Journal
of Political Economy 107, 969–997.
Mankiw, N. Gregory, and Stephen Zeldes, 1991, The consumption of stockholders and non-
stockholders, Journal of Financial Economics 29, 97–112.
Paiella, Monica, 2001, Transaction costs and limited stock market participation to reconcile asset
prices and consumption choices, IFS Working paper 01/06.
Polkovnichenko, Valery, 2000, Heterogeneous labor income and preferences: Implications for stock
market participation, Working paper, University of Minnesota.
Polkovnichenko, Valery, 2002, Life cycle consumption and portfolio choice with additive habit
formation preferences and uninsurable labor income risk, Working paper, University of
Minnesota.
Poterba, James, and Andrew Samwick, 1999, Household portfolio allocation over the life cycle,
NBER Working paper no. 6185.
Saito, Makoto, 1995, Limited market participation and asset pricing, Working paper, University
of British Columbia.
Storesletten, Kjetil, Chris Telmer, and Amir Yaron, 2001, Asset pricing with idiosyncratic risk and
overlapping generations, Working paper, Carnegie Mellon University.
Telmer, Chris, 1993, Asset-pricing puzzles and incomplete markets, Journal of Finance 48, 1803–
1832.
Vayanos, Dimitri, 1998, Transaction costs and asset prices: A dynamic equilibrium model, Review
of Financial Studies 11, 1–58.
Viceira, Luis, 2001, Optimal portfolio choice for long-horizon investors with nontradable labor
income, Journal of Finance 55, 1163–1198.
Vissing-Jørgensen, Annette, 2002a, Limited asset market participation and the elasticity of in-
tertemporal substitution, Journal of Political Economy 110, 825–853.
Vissing-Jørgensen, Annette, 2002b, Towards an explanation of household portfolio choice hetero-
geneity: Nonfinancial income and participation cost structures, Working paper, Northwestern
University.
Vissing-Jørgensen, Annette, and Orazio Attanasio, 2003, Stock-market participation, intertempo-
ral substitution and risk aversion, American Economic Review 93, 383–391.
Yao, Rui, and Harold Zhang, Optimal consumption and portfolio choice with risky housing and
borrowing constraints, Review of Financial Studies (forthcoming).