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Working Paper Series

Stock Market Investment: The Role of
Human Capital

WP 15-07

Kartik Athreya
Federal Reserve Bank of Richmond
Felicia Ionescu
Federal Reserve Board
Urvi Neelakantan
Federal Reserve Bank of Richmond

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

Stock Market Investment: The Role of Human Capital∗
Kartik Athreya
FRB Richmond

†

Felicia Ionescu

‡

Federal Reserve Board

Urvi Neelakantan

FRB Richmond

June 12, 2015
Working Paper No. 15-07

Abstract
Portfolio choice models counterfactually predict (or advise) almost universal equity market
participation and a high share for equity in wealth early in life. Empirically consistent predictions have proved elusive without participation costs, informational frictions, or nonstandard
preferences. We demonstrate that once human capital investment is allowed, standard theory
predicts portfolio choices much closer to those empirically observed. Two intuitive mechanisms are at work: For participation, human capital returns exceed financial asset returns for
most young households and, as households age, this is reversed. For shares, risks to human
capital limit the household’s desire to hold wealth in risky financial equity.

JEL Codes: E21; G11; J24;
Keywords: Financial Portfolios; Human Capital Investment; Life-cycle

∗

We thank Nika Lazaryan for excellent research assistance on this project. We are grateful to Marco Cagetti,
Marios Karabarbounis, and seminar and conference participants at the 2015 Allied Social Science Association
Meeting, The Central Bank of Hungary, Computing in Economics and Finance, Econometric Society, and Midwest
Macro meetings for helpful comments and suggestions. The views expressed in this paper are those of the authors
and do not necessarily reflect the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.
All errors are ours.
†
Federal Reserve Bank of Richmond, P.O. Box 27622, Richmond, VA 23261, Kartik.Athreya@rich.frb.org,
Ph:804-697-8225
‡
Board of Governors of the Federal Reserve System, 2050 C St. Washington, D.C. 20551, felicia.ionescu@frb.gov,
Ph: 202-452-2504
§
Federal Reserve Bank of Richmond, P.O. Box 27622, Richmond, VA 23261, Urvi.Neelakantan@rich.frb.org,
Ph:804-697-8913

Introduction

Household investment in the stock market is generally limited. Not only is it true that a large
fraction of households avoid participation altogether, it is also the case that even for those who
do invest in equities, holdings as a share of their financial wealth are frequently modest. These
observations, particularly those on stock market participation, have proved extremely challenging
to explain within the context of models that avoid the imposition of nonstandard preferences, stock
market participation costs, or imperfect information. The contribution of this paper is to show
that once human capital investment is allowed for, stock market investment can be well understood
within an entirely standard setting.
Our approach is premised on a simple observation: Throughout life, households have human
capital as an investment opportunity and, in deciding how to invest over the life cycle, will take
this into account. Early in life, households’ initial human capital levels are low while the horizon
over which they will recoup any payoff from learning is long. This implies a high return to human
capital investment relative to stocks and a low opportunity cost in terms of forgone earnings. Early
investment in human capital therefore yields a sustained increase in expected future earnings. To
ensure that consumption remains smooth in the face of such an increase, all households who
invest in human capital early in life will desire, absent risk, to borrow and not save in financial
assets. However, the returns to both human capital and stocks are risky. This gives households a
countervailing incentive to build a buffer stock of riskless savings, leading those households who
save to avoid risky stocks. Thus, financial asset positions consistent with early investment in human
capital will likely involve low stock market participation when young, savings in both stocks and
risk-free assets in middle age as diversification becomes important, and dissaving in retirement.
To investigate this logic, we embed the classic Ben-Porath (1967) model of time allocation between working (“earning”) and human-capital accumulation (“learning”) into a life-cycle consumptionsavings model with uninsurable idiosyncratic labor income risk and financial portfolio choice. To
our knowledge, we are the first to study human capital investment and financial investment decisions in such a setting.
While intuitively appealing, there is no a priori guarantee that the option to invest in human
capital is capable of generating a quantitatively plausible account of observed household wealth
and portfolio behavior. A principal contribution of our paper is to demonstrate that it is. We will
show that when households have access to financial assets yielding empirically accurate returns,
a standard human capital investment process, disciplined to be consistent with observed earnings
data alone, implies household financial market behavior consistent with the data. Our model is
able to account extremely well for limited stock market participation, over the entire life cycle,
2

and improves on existing work in capturing the allocation of assets between stocks and bonds.
Importantly, our model’s predictions are broadly consistent with the observed path for household
wealth levels. This makes clear that the portfolio choices we derive describe empirically relevant
magnitudes for the size and division of cash flows that the household receives. It is useful to
note that our success in capturing financial investment over the life cycle comes from a model
parameterized to match earnings dynamics and dispersion alone, and in which there is no appeal
to stock-market participation costs, behavioral assumptions, or informational imperfections. This
strongly suggests the importance of human capital investment as an option over the life cycle.
A critical observation is that both total and marginal returns to human capital are individualspecific and depend on the household’s current holdings while, from the household’s perspective,
the marginal return to financial investments is invariant to any household-level characteristics.
Heterogeneity in the marginal returns to human capital arises from differences across agents not
only in their ability to learn or in their initial human capital, but also from any other source of variation in household productivity. One important source of additional heterogeneity is uninsurable
idiosyncratic wage risk. To ensure that the dispersion in the marginal payoffs to human capital
is of an empirically reasonable magnitude, we follow a huge body of existing work, and especially
that of Huggett, Ventura, and Yaron (2011), and allow for uninsurable idiosyncratic risk to the
payoffs from human capital.
The idea that human capital might play a significant role in how households invest their wealth
is not new (see, for example, the early work of Brito, 1978). Several papers study, as we do, portfolio
choice in a life-cycle setting with uninsurable, idiosyncratic labor income risk. However, in most of
these papers, human capital is only implicitly defined as the present value of exogenously imposed
labor income processes. Examples include Campbell, Cocco, Gomes, and Maenhout (2001), Gomes
and Michaelides (2003), Cocco (2005), Gomes and Michaelides (2005), Davis, Kubler, and Willen
(2006), Polkovnichenko (2007), and Chang, Hong, and Karabarbounis (2014).1 These papers,
building on earlier work of Jagannathan and Kocherlakota (1996), argue that it is the risk properties
of labor income that are likely to influence households’ investment in the stock market.
Despite the richness of the models employed by the work above, little work to date has studied
portfolios when households may also invest in their human capital. Indeed, we are only aware of
two papers that study financial portfolios in the presence of an option to invest in human capital.
In a theoretical contribution, Lindset and Matsen (2011) provide a stylized theory of investment
1

Chang, Hong, and Karabarbounis (2014) represents an innovation within the class of models with exogenous
human capital. They focus on understanding the share of wealth held in risky assets. Their model incorporates
front-loaded risk of unemployment into a model where agents must learn about the income-generating process that
they are endowed with. They show that data on shares can be interpreted as optimal behavior under a particular
specification of parameters, including one regulating the speed of Bayesian learning.

in financial wealth and education as “expansion options” in a complete markets infinite-horizon
economy, where the rental price of human capital is perfectly correlated with the risky financial
asset return. The paper provides insights into optimal portfolio weights when taking human
capital into account. It is, however, abstract and not aimed at confronting empirical regularities.
Roussanov (2010) is arguably the closest work to ours, as it studies portfolio choice in the presence
of a human capital investment option. Agents can exercise this option only once in their lifetime
and cannot work until it matures, which may take several periods. Since borrowing is disallowed in
this setting, nonparticipation is driven by agents’ need to save in order to to finance consumption
and education during the investment period. While Roussanov (2010) does not directly compare
model outcomes to data, he finds that allowing human capital investment can generate reasonable
implications for the share of equity in portfolios. Interestingly, he also shows that such a model
generates plausible average earnings paths even though it is not directly parameterized to do so. In
our model, by contrast, households may invest in human capital throughout life and, in particular,
after formal schooling is typically completed. They may also borrow. We obtain nonparticipation
even while allowing for borrowing because returns to human capital are higher than returns to
equity for the majority of young households. Our approach thus emphasizes financial investment
in a setting that explicitly captures a variety of empirical dimensions of human capital risk and
household heterogeneity over the entire life cycle. This is why, in terms of specifying the mechanism
for human capital accumulation, we follow Ben-Porath (1967) and Huggett, Ventura, and Yaron
(2011). The latter, in particular, not only endogenize human capital, but also capture both the
life-cycle and cross-sectional distribution of earnings.
As noted above, a common assumption in many papers aimed at understanding stock-market
behavior is that participation entails a cost, usually in the form of a fixed cost of entry; see,
for example, Campbell, Cocco, Gomes, and Maenhout (2001) and Cocco (2005).2 Some of the
preceding papers also make assumptions on preferences, such as allowing for habit formation
(Gomes and Michaelides, 2003; Polkovnichenko, 2007) or heterogeneous risk preferences (Gomes
and Michaelides, 2005). Along this dimension, our work is closest to that of Davis, Kubler, and
Willen (2006), who do not make additional assumptions on preferences or stock-market participation costs and obtain limited stock-market participation early in life via the presence of a wedge
between the borrowing rate and risk-free savings rate. However, they do not allow for human
capital investment, and, as we will show, this matters. Nevertheless, it is useful to keep in mind
2

Haliassos and Michaelides (2003) is an example of a paper that introduces a fixed cost in an infinite horizon
setting. However, once this entry cost is paid, households hold their entire financial wealth in stocks. In other words,
in their setting, the empirically observed coexistence of risky and risk-free asset holdings in household portfolios
remains a puzzle. For an assessment of the size of stock market participation costs, though exclusively in models
that abstract from human capital, see Khorunzhina (2013) and references therein.

that our model is indeed close to theirs: In the special case of our model where human capital investment is not permitted, we find results very similar to theirs. In essence, therefore, our
work can be seen as building most closely on the insights of three papers—Davis, Kubler, and
Willen (2006), Huggett, Ventura, and Yaron (2011), and Roussanov (2010)—to demonstrate that
household financial investment behavior can be quantitatively understood with standard tools.
While our model’s ability to closely account for participation (the “extensive margin” of stock
market investment) represents a contribution to one strand in the literature, our results for the
share of wealth invested in stocks (the “intensive margin”) connects our work to another strand of
the literature, starting with the classic work of Merton (1969) and Samuelson (1969). In general,
the studies that have examined the implications of labor income (even when it is endogenous) for
life-cycle portfolios concur that, in spite of labor income risk, a young investor should place much
of her financial wealth in the risky asset. This result holds in these models because labor income
shocks are assumed to be (nearly) independent from stock-market return innovations. Thus, a
young investor chooses to diversify away her human capital risk by holding a high fraction of her
liquid wealth in a well-diversified portfolio of stocks.3 However, as we show, once human capital
investment is disciplined to match observed earnings dispersion, the typical household’s share of
financial wealth held in stock-market equity is closer to the data—and far from 100%. Along this
dimension, our model shares with recent work the implication that shares should be hump shaped
over the life cycle (see, e.g. Benzoni, Collin-Dufresne, and Goldstein (2007) and the references
therein). The mechanism by which we obtain this result differs, however. While these authors
find that shares exhibit a hump shape if labor income and stock market returns are positively
correlated at long horizons, we show that positive correlation is not necessary. Moreover, we show
that once human capital investments are allowed, optimal behavior implies portfolio shares over
the life cycle that are closer to the data than predicted by prior work.
Before laying out the model in detail in Section 3, we describe some facts about household
3

For example, Cocco, Gomes, and Maenhout (2005) argue that as individuals age, the present value of their
labor income decreases because of the decrease in the number of remaining working years. Following the logic of
Jagannathan and Kocherlakota (1996), they further argue that labor income usually acts as a substitute for holding
a riskless asset and, as such, should encourage households to reduce the share of stocks in their portfolio as they
age. In the same spirit, Viceira (2001) shows that the fraction of savings optimally invested in stocks is larger for
employed investors than for retired investors when labor income risk is uncorrelated with stock return risk. Within
the class of models with exogenous human capital, recent work measures the extent to which earnings are bondlike or stock-like and studies the implications for the share of wealth held in equities (Benzoni, Collin-Dufresne,
and Goldstein, 2007; Huggett and Kaplan, 2015). Others examine the role of labor supply. For example, Gomes,
Kotlikoff, and Viceira (2008) endogenize the labor supply decision, thus allowing households who fare poorly on
the stock market to hedge their losses by working more to increase their labor income. They also conclude that
the optimal share of stocks in the household’s portfolio should decline with age. In important early work, Heaton
and Lucas (1997) find that households would want to allocate all of their savings to stocks under a variety of
assumptions, including the presence of transactions costs.

portfolios and earnings in the next section. The calibration is laid out in Section 4 and results are
provided in Section 5.

2
2.1

Data
Household Portfolios

We begin by describing salient facts about household financial portfolios from the Survey of Consumer Finances (SCF). The SCF is a survey of a cross section of U.S. families conducted every
three years by the Federal Reserve Board. It includes information about families’ finances as well
as their demographic characteristics. While the SCF provides us with rich detail about household
finances, it is not a panel, so it does not enable us to directly observe the evolution of finances over
the life cycle. To overcome this, we follow a methodology similar to Poterba and Samwick (1997)
to create life-cycle profiles.
Our goal is to construct life-cycle profiles of participation in the stock market and stockholdings
using cohort-level data. As Deaton (1985) describes, each successive cross-sectional survey of the
population will include a random sample of a cohort if the number of observations is sufficiently
large. Using summary statistics about the cohort from each cross section, a time series that
describes behavior as if for a panel can be generated. In particular, sample cohort means will be
consistent estimates of the cohort population mean.
To implement a procedure in this spirit, we begin by pooling households from all nine waves
of the 1989-2013 SCF into a single dataset. We assign a household to a cohort if the head of
the household is born within the three-year period that defines the cohort. We have 24 cohorts
in all, with the oldest consisting of households whose head was born between 1919 and 1921 and
the youngest consisting of households with heads born between 1988 and 1990. We include all
observations where the household head is between the ages of 23 and 79, to be consistent with
assumptions we make later in our theoretical model. For the same reason, we exclude from our
sample those households whose head has less than a high school diploma.
Except for the cohorts that are too young or too old to be represented in all waves of the survey,
we have at least a hundred observations of every cohort in each survey year. We use this data to
create life-cycle profiles of cohort participation in the stock market. We will define a household
as participating in the stock market if they have a positive amount of financial assets invested in
equity. The variable in the SCF that measures this includes directly held stocks as well as stocks
held in mutual funds, IRAs/Keoghs, thrift-type retirement accounts, and other managed assets.
In Figure 1, we plot the average participation of each of the 24 cohorts over their life cycle
6

(defining the cohort by the mid-point of the age range of the cohort). For example, we observe
the cohort born in 1943-45 from the time they are age 44–46 (in the 1989 wave of the SCF) to
the time they are age 68–70 (in the 2013 SCF). Figure 1 shows that participation for this cohort
increases from roughly 43% to 53%.
Figure 1: Household Stock Market Participation Rate by Cohort (SCF)

0.8

1919-1921
1922-1924

0.7

1925-1927

Participation Rate

1928-1930
1931-1933

0.6

1934-1936
1937-1939

0.5

1940-1942
1943-1945
1946-1948

0.4

1949-1951
1955-1957

0.3

1958-1960
1952-1954
1961-1963

0.2

1964-1966
1967-1969
1970-1972

0.1

1973-1975
1976-1978

1979-1981

60
Age

1982-1984
1985-1987
1988-1990

The differences in participation rates across cohorts may be the result of three factors: aggregate fluctuations experienced by all cohorts living in a particular year (time effects), lifetime
experiences that vary by year of birth (cohort effects), and getting older (age effects). Since we are
interested in participation over the life cycle—the changes in a household’s portfolio that result
from that household getting older—we need to distinguish age effects from cohort and time effects.
The three variables are perfectly collinear (age=year of birth–year of observation), which makes
separately identifying the three effects empirically challenging. Following Poterba and Samwick
(1997), we make the identifying assumption that time effects are zero. We recognize that making different identifying assumptions would generate different life-cycle estimates, particularly for
shares (Ameriks and Zeldes, 2004), and we discuss this later in the context of our results.

The decision to invest in stocks can be expressed using a standard probit model
Si∗

=α+

21
X

βn agei,n +

n=2

24
X

γm cohorti,m + ǫi

(1)

m=2

where Si = 1 if Si∗ > 0 and 0 otherwise. Si is the discrete dependent variable that equals 1 if
household i invests in stocks and zero otherwise. Si is determined by the continuous, latent variable
Si∗ , the actual amount invested in stocks. Si∗ , and thus Si , is specified in the above as a function of
agei,n and cohorti,m . We include 19 dummies for age categories ranging from 23–25 to 77–79, with
agei,n being the dummy variable that indicates whether the current age of the household head lies
in one of these intervals. We include 24 cohort dummies cohorti,m to represent cohorts born in one
of the three-year intervals in the range from 1919–21 to 1988–90.
The SCF oversamples wealthy households and therefore needs to be weighted to obtain estimates that are representative of the U.S. population. As in Poterba and Samwick (1997), we
estimate Equation (1) using year-specific sample weights normalized such that the sum of the
weights (which equals the population represented) remains constant over time. The results of the
estimation are reported in Table 2 in the Appendix.4 We use the coefficients to construct our
estimate of the life-cycle profile of stock-market participation. Figure 2 shows the results for the
cohort born in 1973–75.5 By our estimation, participation in the stock market increases till agents
reach age 60, after which it levels off.
We are also interested in portfolio allocation over the life cycle conditional on participation. In
other words, we want to know how the fraction of assets invested in stocks evolves over the life
cycle. As we will describe later, our model will have one risk-free asset b and one risky asset s, so
s
the measure in which we are interested is s+b
. As described earlier, the risky asset is the value of
equity that the household holds, which includes directly held stocks and stocks in mutual funds,
retirement accounts, and other managed assets.
s
for those who own equity. This measure lies between 0 and 1 by
We calculate the fraction s+b
construction, so we want our life-cycle estimate of it to lie between 0 and 1 as well. To ensure this,
s

we construct a logistic transformation to obtain the variable Yi = ln 1−s+bs . We run the following
s+b

We use all five implicates from the SCF in our estimation. While this provides accurate coefficients, the
statistical significance of the results may be inflated. We only need the values of the coefficients to construct
life-cycle profiles; therefore, we do not report the results of the significance tests.
5
Participation rates would be lower over the life cycle for older cohorts and higher for younger cohorts. We
choose to display results for this cohort to be consistent again with the assumptions in our theoretical model. As
we will see, debt in our model will be nondefaultable. Those born in 1973–75 would face a nondefaultable student
loan regime by the time they go to college.

Figure 2: Estimated Participation Rate over the Life Cycle (SCF, 1973–75 Birth Cohort)

0.9
0.8

Participation

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20

50
Age

Ordinary Least Squares (OLS) regression on this variable.6
Yi = α +

21
X

βn agei,n +

n=2

24
X

γm cohorti,m + ǫi

(2)

m=2

The results are reported in Table 3. As we did for participation, we use the reported coefficients
to estimate the life-cycle profile of portfolio allocation for the cohort born in 1973–75. Figure 3
shows the results. The estimated share of risky assets conditional on participation increases steadily
after age 25.

2.2

Earnings

Next, we compute statistics of age-earnings profiles from the CPS for 1969-2002 using a synthetic
cohort approach, following Ionescu (2009). To be precise, we use the 1969 CPS data to calculate
6

Note that, unlike Poterba and Samwick (1997), we do not use Tobit to estimate this equation. By construction,
our data is not censored— values below 0 and above 1 are infeasible. Moreover, since our variable of interest is the
share of risky assets in the household’s portfolio conditional on participation, it will always be strictly positive. It
is possible for it to exactly equal 1, but we have very few observations with this value, and in this instance we set
it to 0.999999.

Figure 3: Estimated Average Fraction of Stocks in Portfolio over the Life Cycle Conditional on
Participation for 1973–75 Birth Cohort (SCF)

Share of Stocks in Household Assets

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20

50
Age

the earnings statistics of 25-year-olds, the 1970 CPS data to compute earnings statistics of 26-yearolds, and so on. We include only those who have at least 12 years of education, to correspond with
our modeling assumption that agents start life after high school. To compute the mean, inverse
skewness, and Gini of earnings for households of age a in any given year, we average the earnings of
household heads between the ages of a − 2 and a + 2 to obtain a sufficient number of observations.
Life-cycle profiles for all three statistics are shown in Figure 18 in the Appendix.7
With these facts in hand, we turn to the description of the model.

Model

Our model is a standard model of life-cycle consumption and savings in the presence of uninsurable
risk (e.g. Gourinchas and Parker, 2002), but it contains two enrichments. First, households choose
7

We obtain real earnings in 2013 dollars using the Consumer Price Index. We convert earnings to model units
such that mean earnings at the end of working life, which equal $70,800, are set to 100.

their level of human capital, and second, households can invest in both risky and riskless assets.
The economy is populated by a continuum of agents who value consumption throughout a finite
life. Age is discrete and indexed by t = 1, ..., T , where t = 1 represents the first year after high
school graduation, and t = J represents the age of retirement. Agents enter the model endowed
with an initial level of human capital, h0 , which varies across the population.
In each period, households can divide their time between work and the accumulation of human
capital, as in the classic model of Ben-Porath (1967). Households consume and decide how to
allocate any wealth they have in period t between a risky asset st+1 and a risk-free asset bt+1 .
Households also have the option to borrow, that is bt ≥ −b, with b > 0, may be positive or
negative.
To capture risk and heterogeneity, we follow Huggett, Ventura, and Yaron (2011), and allow
for four potential sources of heterogeneity across agents — their immutable learning ability, a,
human capital stock, h, initial assets, x, and subsequent shocks to the yield on their holdings of
human capital, i.e., their earnings. The set of initial characteristics are jointly drawn according
to a distribution F (a, h, x) on A × H × X. Lastly, households are not subject to risks once they
retire, i.e., once t > J.

3.1

Preferences

All agents have identical preferences, with their within-period utility given by a standard CRRA
function with parameter σ, and with a common discount factor β. The general problem of an
individual is to choose consumption over the life cycle, {ct }Tt=1 to maximize the expected present
value of utility over the life cycle,
max

({ct }∈Π(Ψ0 ))

T
X
t=1

t−1

c1−σ
t
1−σ

(3)

Π(Ψ0 ) denotes the space of all feasible combinations {ct }Tt=1 , given initial state Ψ0 ≡ {a0 , h0 , x0 }.
Agents do not value leisure.

3.2

Financial Markets

There are two financial assets in which the agent can invest, a risk-free asset, bt , and a risky asset,
st , to be interpreted as stock-market equity.

Risk-free assets
An agent can borrow or save by taking negative or positive positions, respectively, in a risk-free
asset bt . Savings (bt ≥ 0) will earn the risk-free interest rate, Rf . Borrowing (bt < 0), however,
carries an additional (proportional) cost as in Davis, Kubler, and Willen (2006), denoted by φ,
to represent costs of intermediating credit. The borrowing rate, Rb , therefore, is higher than the
savings rate and given by: Rb = Rf + φ. As noted above, borrowing is subject to a limit b.
Risky assets
For ease of exposition, we will refer to the risky assets as “stocks” and denote the agent’s holdings
of these claims between period t and t + 1 by st+1 . Stocks yield their owners a stochastic gross
real return in period t + 1, Rs,t+1 whereby the excess return on stocks is given by:
Rs,t+1 − Rf = µ + ηt+1 ,

(4)

The first term µ is the mean excess return to stocks. The second, ηt+1 , represents the period t + 1
innovation to excess returns and is assumed to be independently and identically distributed (i.i.d.)
over time with distribution N(0, ση2 ).
Given asset investments at age t, bt+1 and st+1 , financial wealth at age t + 1 is given by
xt+1 = Ri bt+1 + Rs,t+1 st+1 , with Ri = Rf if b ≥ 0 and Ri = Rb if b < 0.

3.3

Human Capital

The key innovation of our work is to allow for human capital investment in a model of portfolio
choice. We do this by employing the workhorse model of Ben-Porath (1967), extended to allow for
risks to the payoff from human capital: In each period, agents can apportion some of their time to
acquiring human capital, or they may work and earn wages that depend on current human capital
and shocks. At any given date, an agent’s human capital stock summarizes their ability to turn
their time endowment into earnings. In this sense, it reflects earning ability and, critically, can be
accumulated over the life cycle. By contrast, learning ability, which governs the effectiveness of the
production function that maps time to human capital investment, is fixed at birth and does not
change over time. Both learning ability and initial human capital will be allowed to vary across
agents and, as we will demonstrate, heterogeneity in each is implied by earnings heterogeneity in
the data among the youngest cohorts and by the subsequent evolution of earnings dispersion.
Human capital investment in a given period occurs according to the human capital production
function, H(a, ht , lt ), which depends on the agent’s immutable learning ability, a, human capital,
12

ht , and the fraction of available time put into human capital production, lt . Human capital
depreciates at a rate δ. The law of motion for human capital is given by
ht+1 = ht (1 − δ) + H(a, ht , lt )

(5)

Following Ben-Porath (1967), the human capital production function is given by H(a, h, l) =
a(hl)α with α ∈ (0, 1). As demonstrated by Huggett, Ventura, and Yaron (2006), the Ben-Porath
model has the additional advantage of being able to match the dynamics of the U.S. earnings
distribution given the appropriate joint distribution of initial ability and human capital.

3.4

Labor Income

Human capital confers a return (i.e., its rental rate, wages) in each period that is subject to
stochastic shocks. Specifically, earnings are given by a product of the stochastic component, zt ,
the rental rate of human capital, wt , the agent’s human capital, ht , and the time spent in market
work, (1 − lt ).
Therefore, agent i’s earnings in period t are given by
log(yit ) = G(wt , ht , lt ) + zit

(6)

with G(wt , ht , lt ) representing the deterministic component as a function of rental rate wt , human
capital stock at age t, ht , and labor effort, 1 − lt , and zt representing the stochastic component.
The rental rate of human capital evolves over time according to wt = (1 + g)t−1 with the growth
rate, g.8
The stochastic component, zit , consists of an idiosyncratic temporary (i.i.d) shock ǫit ∼ N(0, σǫ2 )
and a persistent shock uit :
zit = uit + ǫit
where
uit = ρui,t−1 + νit
follows an AR(1) process as in Gourinchas and Parker (2002) and Hubbard, Skinner, and Zeldes
(1995), with νit ∼ N(0, σν2 ) representing an innovation to uit . The variables uit and ǫit are realized
at each period over the life cycle and are not correlated.
8

The growth rates for wages are estimated from data. See Section 4.1 for details.

3.5

Means-Tested Transfer and Retirement Income

To accurately capture the risk-management problem of the household, it is important to make
allowance for additional sources of insurance that may be present. In the United States, there is a
vast array of social-insurance programs that, if effective, bound households’ purchasing power away
from zero. Moreover, it is well known, since at least Hubbard, Skinner, and Zeldes (1995), that such
a system may be acting to greatly diminish savings among households who earn relatively little. In
our model, this will consist of unlucky households, households with low learning ability, or both.
To ensure that we confront households with an empirically-relevant risk environment in which
they choose portfolios, we specify a means-tested income transfer system, which, in addition to
asset accumulation, can provide another source of insurance against labor income risk (Campbell,
Cocco, Gomes, and Maenhout, 2001). Agents receive means-tested transfers from the government,
τt , which depend on age, t, income, yt , and net assets, xt . These transfers capture the fact that
in the U.S. social insurance is aimed at providing a floor on consumption. Following Hubbard,
Skinner, and Zeldes (1995), we specify these transfers by
τt (t, yt , xt ) = max{0, τ − (max(0, xt ) + yt )}

(7)

Total pre-transfer resources are given by max(0, xt ) + yt and the means-testing restriction is
represented by the term τ − max((0, xt ) + yt ). These resources are deducted to provide a minimal
income level τ . For example, if xt + yt > τ and xt > 0, then the agent gets no public transfer. By
contrast, if xt + yt < τ and xt > 0, then the agent receives the difference, case in which he has τ
units of the consumption good at the beginning of the period. Agents do not receive transfers to
cover debts, which requires the term max(0, xt ). Lastly, transfers are required to be nonnegative,
which requires the “outer” max.
After period t = J when agents start retirement, they get a constant fraction ψ of their income
in the last period as working adults, yJ , which they divide between risky and risk-free investments.

3.6

Agent’s Problem

The agent’s problem is to maximize lifetime utility by choosing asset positions in stocks and bonds
(or borrowing), and, in what is novel in our paper, time allocated throughout life to market work
and human capital investment
We formulate the problem recursively. Let any period t variable j be denoted by j and its
′
period t + 1 value by j . The household’s feasible set for consumption and savings is determined
by its age, t, ability, a, beginning-of-period human capital, h, net worth, x(b, s), current-period
14

realization of the persistent shock to earnings, u, and current-period transitory shock, ǫ.
In the last period of life, agents consume all available resources. The value function in the
last period of life is therefore simply their payoff from consumption in that period. Prior to this
terminal date, but following working life, agents are retired. Retired agents do not accumulate
1−σ

human capital and do not face human capital risk. Thus, we have VTR (a, x, yJ ) = c1−σ , where
c = x(b, s) + ψyJ . Notice that, when retired, human capital is irrelevant as a state, and in what
follows, is not part of the household’s state. Retired households face a standard consumptionsavings problem though, as in working life, they may invest in both risk-free and risky assets.
Indeed, in retirement, the only risk agents face comes from the uncertain return on stocks. Their
value function for retirees is given by
V R (t, a, b, s, yJ ) = sup{
b′ ,s′

c1−σ
′
′
t
+ βER′s V R (t + 1, a, b , s , yJ )}
1−σ

(8)

where
′

c+b +s

′

≤ ψyJ + Ri b + Rs s

b ≥ b
In the budget constraint, we remind the reader that Ri = Rf if b ≥ 0 and Ri = Rb if b < 0.
During working life, the agent faces uncertainty from the returns on human capital as well
as from any risk assumed in the portfolio they choose. The budget constraint makes clear that
current consumption c and total net financial wealth next period (b′ + s′) must not exceed the sum
of current labor earnings w(1 − l)hz, the value of the portfolio (Ri b + Rs s), and any transfers from
the social safety net τ (t, y, x).
V (t, a, h, b, s, u, ǫ) = sup {
l,h′ ,b′ ,s′

c1−σ
′
′
′
′
′
t
+ βEu′ |u, R′s V (t + 1, a, h , b , s , u , ǫ )}
1−σ

(9)

where
′

c+b +s

′

≤ w(1 − l)hz + Ri b + Rs s + τ (t, y, x) for t = 1, .., J − 1
′

s.t. l ∈ [0, 1], h = h(1 − δ) + a(hl)α , b ≥ b
The value function V (t, a, h, b, s, u, ǫ) thus gives the maximum present value of utility at age t from
states h, b, and s, when learning ability is a and the realized shocks are u and ǫ. The solution to this
problem is given by optimal decision rules lj∗ (t, a, h, b, s, u, ǫ), h∗ (t, a, h, b, s, u, ǫ), b∗ (t, a, h, b, s, u, ǫ),
15

and s∗ (t, a, h, b, s, u, ǫ), which describe the optimal choice of the fraction of time spent in human
capital production, the level of human capital, and risk-free and risky assets carried to the next
period as a function of age, t, human capital, h, ability, a, and current assets, b and s when the
realized shocks are u and ǫ.

Mapping the model to the data

There are four sets of parameters in the model: 1) standard parameters, such as the discount
factor and the coefficient of risk aversion; 2) parameters specific to asset markets; 3) parameters
specific to human capital and to the earnings process; and 4) parameters for the initial distribution
of characteristics. Our approach includes a combination of setting some parameters to values that
are standard in the literature, calibrating some parameters directly to data, and jointly estimating
those parameters that we do not directly observe in the data by matching moments for several
observable implications of the model. We summarize parameter values in Table 1 and describe in
detail below how we obtain them.
Table 1: Parameter Values: Benchmark Model
Parameter
T
J
β
σ
Rf
Rb
µ
ση
α
g
δ
ψ
τ
(ρ, σν2 , σǫ2 )
(µa , σa , µh , σh , ̺ah )

Name
Model periods (years)
Working periods
Discount factor
Coeff. of risk aversion
Risk-free rate
Borrowing rate
Mean equity premium
Stdev. of innovations to stock returns
Human capital production function elasticity
Growth rate of rental rate of human capital
Human capital depreciation rate
Fraction of income in retirement
Minimal income level
Earnings shocks
Parameters for joint distribution of ability
and initial human capital

Value
53
33
0.96
5
1.02
1.11
0.06
0.157
0.7
0.0013
0.0114
0.68
$17, 936
(0.951, 0.055, 0.017)
(0.246, 0.418, 87.08, 35.11, 0.57)

We follow agents from age 25 onward, as this captures the beginning of the portion of life
in which households make nontrivial investments in financial assets and in learning on the job.
Agents live T = 53 model periods, which corresponds to ages 25 to 78 and retire at age J = 58.
16

4.1

Preference and Financial Market Parameters
1−σ

t
The per period utility function is CRRA, u(ct ) = c1−σ
, with the coefficient of risk aversion σ = 5,
which is consistent with values chosen in the financial literature. Risk aversion is a key parameter

and so we conduct robustness checks on it, in particular we consider higher values up to the upper
bound of σ = 10 considered reasonable by Mehra and Prescott (1985). We also consider lower
values, such as σ = 3. The discount factor (β = 0.96) chosen is also standard in the literature.
We turn now to the parameters in the model related to financial markets. We fix the mean
equity premium to µ = 0.06, as is standard (e.g., Mehra and Prescott, 1985). The standard
deviation of innovations to the risky asset is set to its historical value, ση = 0.157. The risk-free
rate is set equal to Rf = 1.02, consistent with values in the literature (McGrattan and Prescott,
2000) while the wedge between the borrowing and risk-free rate is φ = 0.09 to match the average
borrowing rate of Rb = 1.11 (Board of Governors of the Federal Reserve System, 2014). Lastly,
we assume that innovations to excess returns are uncorrelated with innovations to the aggregate
component of permanent labor income.9

4.2

Human Capital and Earnings Parameters

The rental rate on human capital equals wt = (1 + g)t−1 where g is set to 0.0013, as in Huggett,
Ventura, and Yaron (2006). Given this growth rate, the depreciate rate is set to δ = 0.0114, so
that the model produces the rate of decrease of average real earnings at the end of the working life
cycle observed in the data. The model implies that at the end of the life cycle negligible time is
allocated to producing new human capital and, thus, the gross earnings growth rate approximately
equals (1 + g)(1 − δ). We set the elasticity parameter in the human capital production function,
α, to 0.7. Estimates of this parameter are surveyed by Browning, Hansen, and Heckman (1999)
and range from 0.5 to 0.9.
In the parametrization of the stochastic component of earnings, zit = uit +ǫit , we follow Abbott,
Gallipoli, Meghir, and Violante (2013) who use the National Longitudinal Survey of Youth (NLSY)
data using CPS-type wage measures to estimate the autoregressive coefficients for the transitory
9

Evidence on this correlation is mixed, ranging from negative to strongly positive. For instance, Lustig and
Van Nieuwerburgh (2008) show that innovations in current and future human wealth returns are negatively correlated with innovations in current and future financial asset returns, regardless of the elasticity of intertemporal
substitution, while Benzoni, Collin-Dufresne, and Goldstein (2007) argue that the correlation in labor income flows
and stock market returns is positive, and large in particular at long horizons. At the same time, prior studies that
have examined the relation between labor income and life-cycle financial portfolio choice assume that labor income
shocks are (nearly) independent from stock market return innovations (see Cocco, Gomes, and Maenhout, 2005;
Davis, Kubler, and Willen, 2006; Davis and Willen, 2013; Gomes and Michaelides, 2005; Haliassos and Michaelides,
2003; Roussanov, 2010 and Viceira, 2001)

and persistent shocks to wages. For the persistent shock, uit = ρui,t−1 + νit , with νit ∼ N(0, σν2 )
and for the idiosyncratic temporary shock, ǫit ∼ N(0, σǫ2 ), they report the following values for
high school graduates: ρ = 0.951, σν2 = 0.055, and σǫ2 = 0.017. We set retirement income to be
a constant fraction of labor income earned in the last year in the labor market. Following Cocco
(2005) we set this fraction to 0.682, the value for high school graduates. The income floor, τ
is expressed in 2013 dollars and is consistent with the levels used in related work (e.g. Athreya,
2008).10
Borrowing limits in the model will be allowed to vary across households. We introduce heterogeneity in these limits as follows: We first group agents in the model by quartiles of initial
human capital, then compute average earnings over the life cycle for each quartile. We then set
the borrowing limit for all agents within a quartile to be a given percentage of the average lifecycle earnings for that quartile. We obtain the relevant percentages from the SCF by dividing the
sample into income quartiles and calculating the average credit limit as a percentage of the average
income within each quartile. The resulting borrowing limits as a percentage of average earnings
by quartiles are: 55%, 48%, 35%, and 27%.11 Lastly, in our baseline model, we assume that the
returns to both risky assets (human capital and financial wealth) are uncorrelated.

4.3

The Distribution of Assets, Ability, and Human Capital

We turn now to parameters defining the joint distribution of initial heterogeneity in the unobserved
characteristics central to human capital accumulation. There are seven parameters, and using only
these, we are able to closely match the evolution, over the entire life cycle, of three functions of
moments of the earnings distribution: mean earnings, the ratio of mean to median earnings, and
the Gini coefficient of earnings.
To estimate the parameters of this distribution, we proceed as follows. First, for the asset
distribution, we use the SCF data described in Section 2 to compute the mean and standard
deviation of initial assets to be $22,568 and $24,256, respectively, in 2013 dollars. Second, we
calibrate the initial distribution of ability and human capital to match the key properties of the
life-cycle earnings distribution reported earlier using the CPS for 1969-2002.
Earnings distribution dynamics implied by the model are determined in several steps: i) we
compute the optimal decision rules for human capital using the parameters described above for an
initial grid of the state variable; ii) we simultaneously compute financial investment decisions and
10

The results turn out to be robust to the choice of this parameter; results are available upon request.
We extrapolate the first percentage from the other three rather than calculating it directly because of the large
numbers of zeros in the earnings data for the lowest quartile.
11

compute the life-cycle earnings for any initial pair of ability and human capital; and iii) we choose
the joint initial distribution of ability and human capital to best replicate the properties of U.S.
data.
To set values for these parameters, we search over the vector of parameters that characterize
the initial state distribution to minimize a distance criterion between the model and the data.
We restrict the initial distribution to lie on a two-dimensional grid spelling out human capital
and learning ability, and we assume that the underlying distribution is jointly log-normal. This
class of distributions is characterized by five parameters.12 We find the vector of parameters
γ = (µa , σa , µh , σh , ̺ah ) characterizing the initial distribution by solving
the minimization probPJ
2
2
2
lems minγ
j=5 |log(mj /mj (γ))| + |log(dj /dj (γ))| + |log(sj /sj (γ))| , where mj , dj , and sj are
mean, dispersion, and inverse skewness statistics constructed from the CPS data on earnings, and
mj (γ), dj (γ), and sj (γ) are the corresponding model statistics. Overall, we match 102 moments.13
Figure 4 illustrates the earnings profiles for individuals in the model versus CPS data when the
initial distribution is chosen to best fit the three statistics considered. We obtain
γ = (0.246, 0.418, 87.08, 35.11, 0.57)
The model performs well given riskiness of assets and stochastic earnings in the current paper.14
12

In practice, the grid is defined by 20 points in human capital and ability.
For details on the calibration algorithm see Huggett, Ventura, and Yaron (2006) and Ionescu (2009).
14
We obtain a fit of 9.4% (0% would be a perfect fit). As a matter of perspective, we note that a close relative
of this part of our model, Huggett, Ventura, and Yaron (2006), obtain a fit of 7% (for the same value of the
elasticity parameter α = 0.7). Theirs is a Ben-Porath model where the main choice is investment in human capital
to maximize lifetime earnings in a framework without investments
in financial assets, debt, and without earnings
1 PJ
uncertainty. As a measure of goodness of fit, we use 3J
j=5 |log(mj /mj (γ))| + |log(dj /dj (γ))| + |log(sj /sj (γ))|.
This represents the average (percentage) deviation, in absolute terms, between the model-implied statistics and the
data.
13

Figure 4: Life-cycle earnings
Mean of lifecycle earnings
200
180
160
140
120
100
80
60
40
Model
CPS data

20
0
25

Age
Mean/Median of lifecycle earnings
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
Model
CPS data

0.2
0
25

Age
Gini of lifecycle earnings
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Model
CPS data

0.1
0
25

Age

Results

Our paper aims to provide a quantitative explanation for the behavior of two observables over the
life cycle: stock-market participation and the share of financial wealth held in stocks. We will
therefore focus exclusively on these two objects and provide evidence from the model that helps
explain our findings. Before proceeding, we recall the basic mechanism at work. Early in life,
20

forgone wages are low while marginal returns to human capital investment are high. Moreover,
the horizon over which to reap the benefits of such high marginal rewards is long. This resolves
the competition between human and financial investment, especially that in risky equity, in favor
of the former. Later in life, this process reverses itself, making equity investment relatively more
attractive. The fact that acquiring human capital necessarily takes time away from working means
that, unlike financial investments, it alters the time path of earnings. This implies that prior to
making human capital investments, the household will consider how best to use financial assets
to ensure that consumption is smooth, given that earnings will not be. To illustrate this intuition
more explicitly, it is useful to consider a simplified, two-period version of our model.

5.1

A Simple Two-Period Model

Consider a setting in which there is only one financial asset, which is risk-free and can be used
for saving or borrowing. In the initial period, 0, agents choose how much to save or borrow using
this asset as well as how much of their time endowment to invest in human capital. There is
no uncertainty and we assume agents have standard CRRA preferences, with σ = 1 (log utility).
The remainder of the notation is as above, with h0 denoting initial human capital, Rb denoting
the interest rate on risk-free assets, δ the depreciation rate of human capital, α the elasticity of
investment in human capital, and a the ability to learn. w1 denotes first-period wages and b
denotes the exogenous limit on risk-free borrowing. Initial-period and first-period consumption
are given by c0 and c1 , respectively. Given this, the agent’s problem simplifies to:
max ln(c0 ) + β ln(c1 )
l0 ,b1

subject to
c0 + b1 ≤ w0 (1 − l0 )h0
c1 ≤ w1 h1 + Rb b1
h1 = h0 (1 − δ) + a(h0 l0 )α
b1 ≥ b
With w1 = w0 (1 + g), the optimal solution is:
l0∗

1
1 (1 + g)aα (1−α)
=
h0
Rb

and
b∗1 (l0∗ ) =

βRb w0 (1 − l0∗ )h0 − w0 (1 + g) [h0 (1 − δ) + a(h0 l0 )α ]
Rb (1 + β)

A first result is that the time invested in human capital increases in the growth rate of the rental
dl∗

dl∗

rate of human capital ( dg0 > 0) and in the elasticity of investment in human capital ( dα0 > 0).
Intuitively, as the returns to human capital grow, or as human capital accumulation becomes more
productive, individuals have a higher incentive to invest in human capital and respond accordingly.
Second, we see that to compensate for a low initial endowment of human capital, agents will
dl∗
invest more time in its accumulation. That is, we have that ( dh00 < 0). Investment in human capital
dl∗

also increases as the rental rate of human capital decreases ( dw00 < 0). Intuitively, a low h0 or w0
means a lower opportunity of investing in human capital because of a low market value of human
capital (w0 h0 ) in the first period. In a life-cycle setting, this explains why agents will front-load
educational investments, with investment later in life being too costly—precisely because of the
additional human capital having been accumulated earlier.
Lastly, and most central for our investigation, is the question of how the relative rates of return
dl∗
on human and financial assets matter for investment choices. We see that optimally, dR0b < 0.
This shows that individuals will invest less in human capital when financial investments are more
rewarding. More to the point, the simple model tells us that individuals who invest in human
capital early in life will lower their exposure to, or investment in, financial assets: We see that
db∗1
dl0∗

< 0.
When it comes to the implications of the ease with which households to accumulate human
capital for financial investment, the intuition is straightforward: As human capital depreciates
faster, or its rewards grow at a slower pace over the life cycle, individuals would rather invest
in financial assets. This intuition holds in the simple model: Optimal behavior implies that b∗1
increases in δ. That is, as the depreciation rate of human capital grows, financial assets become
more attractive, all else equal. Similarly, decreases in g, the growth rate of the wages (i.e., the
rental rate of human capital), increase the appeal of, and investment in, financial assets. With
these findings in mind, we now turn to the results from the full quantitative model.

5.2

Stock Market Investment

We begin by studying our model’s predictions for the stock-market participation rate. Figure 5
compares our results with their empirical counterparts from the SCF data. It captures one of our
paper’s two key results—stock-market participation in our model is completely consistent, not just
qualitatively, but quantitatively, with the data, over the entire life-cycle. Importantly, we see that

nonparticipation is not a pathology, but rather it is a direct implication of a standard model. As
the two-period model suggests, and as we will show shortly, this result is driven primarily by the
presence of human capital investment in our model. As a first step, we turn to Figure 6, which
shows the trajectory of time invested in human capital over the life cycle. As is clear, time spent on
human capital accumulation is at its highest early in life. For instance, at age 25, households spend
about a third of their time endowment on human capital accumulation. During the early part of
life, we see also that only around 30% of all households participate in the stock market. Diminishing
returns, and a shorter horizon to recoup the investment, imply that human capital accumulation
falls with age. As this occurs, we see that stock-market participation steadily increases, reaching
around 80% at retirement age. As retirement approaches, we see that the fraction of time allocated
to human capital falls sharply, reaching below 0.05 by retirement age.
Figure 5: Life-Cycle Stock Market Participation
Participation in stocks over the lifecycle
1
Model
SCF data
0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
25

45
Age

Figure 6: Time Allocated to Human Capital over the Life Cycle
Time allocated to human capital over the lifecycle
0.35

0.3

0.25

0.2

0.15

0.1

0.05

0
25

Age

Having shown that stock-market participation can be very well accounted for by the accommodation of human capital, we turn now to the “intensive” margin of stock market investment. As
seen in Figure 7, three things are salient. First, the model implies a higher share for wealth held
in equity than in our SCF data early in life, but this gap closes later in life. This is important
because the bulk of financial wealth is accumulated in this model, as in the data and in “standard”
theory, late in life. Households are buffer-stock savers for the early- to middle-age portions of the
life cycle, and only begin accumulating substantial wealth (as we will show further below) later in
life. As a result, our model accounts well for the proportion of wealth allocated to equity when
most relevant. Second, we see that the share implied by optimal behavior in the presence of human
capital remains far away from 100%. Importantly, this occurs despite the fact that households in
our model retain the ability to work to undo poor returns from the stock market. Third, the
hump-shaped profile for shares generated by our model is empirically more plausible than the
decreasing profile derived by much of the existing work. Recall that our data estimates for shares
are generated under the identifying assumption that time effects are zero and that cohort effects
matter. In fact, if we assumed the reverse, we know from Ameriks and Zeldes (2004) that the lifecycle profile of shares would be estimated to be flat or mildly hump shaped. If we abstracted from
time and cohort effects altogether and compared our estimates to life-cycle averages as in Gomes
and Michaelides (2005), our model’s predictions for shares would be very close to the data. Our
model implies that the conventional “100 minus age” rule of thumb often prescribed in financial
planning circles is not ideal in settings where human capital accumulation is an available decision.

Figure 7: Fraction of Stocks in Household Portfolio
Share of stocks over the lifecycle
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1
Benchmark
SCF data
0
25

Age

5.3

Implications for Wealth

While we have focused so far on stock market investment, our model also produces empirically consistent estimates of life-cycle wealth and its allocation between risky and risk-free assets. Figure 8
shows our model’s predictions for wealth accumulation over the life cycle. The trend in wealth
accumulation predicted by our model—as well as the trend of each of its components (risky and
risk-free assets)—is consistent with the data, despite not being targeted.15 Thus, our findings for
stock market investment arise from a model that captures the magnitude of household savings and
consumption throughout the life cycle.
15

For the data reported in this set of figures, we pool the 1989–2013 waves of the SCF and calculate age-specific
weighted means, following a process similar to Cagetti (2003).

Figure 8: Life-Cycle Wealth Accumulation
Mean of total assets over the lifecycle

×10 5

Model,
SCF data

0
25

Age

(a) Total Assets
4

×10

Mean of risky assets over the lifecycle

2.5

Mean of riskfree assets over the lifecycle

×10 5

Model
SCF data

Model,
SCF data

3.5
2
3

2.5

1.5

1
0.5
0.5

0
25

Age

(b) Risky Assets

5.4

The Role of Endogenous Human Capital

The most direct route to seeing that that our results are driven primarily by the presence of
endogenous human capital investment is to consider outcomes in which this channel is shut down.
To do this, we now study a setting in which agents exogenously obtain the labor income stream
generated by our benchmark model, but do not spend any time on human capital accumulation.
To study a case while retaining comparability to the benchmark model requires an additional
step, however. Specifically, we “assign” earnings to agents based on their initial endowment of
human capital.16 We retain all the other features of our model, including the shocks to earnings
16

Note that we still allow agents to differ in their initial endowments but assume that initial human capital and
ability are uncorrelated.

as well as the wedge between the interest rate on borrowing and savings. As such, this setting
is very close to that of Davis, Kubler, and Willen (2006). The result is a model in which the
distribution of earnings unfolds as in the benchmark model, but does so without requiring human
capital investment.
5.4.1

Exogenous Human Capital Investment and Stock-Market Investment

We first report results for participation in Figure 9. Observe now that the participation rate
is much higher than in the case with endogenous human capital accumulation. Indeed, we see
that it reaches 100% around age 55—a result similar to what Davis, Kubler, and Willen (2006)
obtain. This is an important observation because with exogenous human capital, our setting,
including its quantitative implications, becomes very similar to theirs, as well as papers cited
earlier (e.g., Gomes and Michaelides, 2005).17 Indeed, we are able to recover the result in these
papers that participation increases rapidly to a 100%, with the only deterrent being, just as in
Davis, Kubler, and Willen (2006), the presence of the borrowing wedge. By making borrowing
expensive—especially for those young households who would like to borrow—this wedge is helpful
in keeping households away from stock market participation early in life. However, this result
clarifies that the mechanism of high borrowing costs alone is not sufficient to explain limited stock
participation later in life, when households are less likely to borrow. The relative improvement
provided by our benchmark model drives home the relevance of households’ ability to augment
their human capital for their financial portfolio choices.
17

Note, however, that there are still quantitatively meaningful differences in our parameters and theirs, including
in risk-aversion, the interest rate on borrowing, and the share of income taken into retirement.

Figure 9: Life-Cycle Stock-Market Participation Under Exogenous Earnings
Participation in stocks over the lifecycle
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
Benchmark
SCF data
Exogenous earnings

0.1

0
25

Age

Turning next to shares, we see from Figure 10 that the exogeneity of earnings—as long as
it implicitly reflects the human capital accumulation undertaken in the benchmark economy—
does not strongly alter either the quantitative or qualitative properties of the model. As before,
the share of wealth held in stocks at any given age is strongly affected by households’ need for
diversification. In this particular case, the implicit path for human capital, as a quantitative
matter, exposes households to slightly less risk than in the benchmark, whereupon they increase
their exposure to equity markets.
Figure 10: Life-Cycle Stock-Market Shares Under Exogenous Earnings
Share of stocks over the lifecycle
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2
Benchmark
SCF data
Exogenous earnings

0.1

0
25

45
Age

5.4.2

Human Capital Accumulation Technology

An implication of our model is that the better the technology for learning, the less attractive
stock market investment will be. In other words, if the earnings that we observe in the data were
generated by a more productive human capital technology than in the benchmark, then we should
expect to see lower participation in the stock market than in the benchmark. To illustrate this,
consider a case in which the human capital technology is extremely productive: α = 0.9.18 To
preserve comparability, we recalibrate all the parameters needed to match earnings facts as in
the benchmark. The marginal densities for ability and initial human capital obtained from the
recalibration are to the left of those in the benchmark (Figure 11).
Figure 11: Comparison of Marginal Densities in Model with α = 0.7 and α = 0.9
Ability Density

Human Capital Density

0.018

Benchmark
Alpha 0.9

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) Ability

100

120

140

160

180

200

(b) Initial Human Capital

The main results are reported in Figure 12. Participation in the stock market is indeed much
lower than in the benchmark, particularly in the middle of the life cycle. This is consistent
with the idea that human capital competes with financial assets as an investment option. With
a high α, it competes favorably for longer because households encounter marginal returns to
human capital investment that diminish more slowly than in the benchmark model. As a result,
more households choose to forgo participation in the stock market in favor of human capital
accumulation. Conditional on investing in the stock market, households’ wealth allocation decisions
are driven primarily by other considerations such as their risk preferences. Shares, therefore, are
not markedly different in this experiment from those observed in the benchmark model.
18

The literature provides a range of estimates for this parameter (Browning, Hansen, and Heckman, 1999). While
this example reinforces one of the main mechanisms underlying our results, it is important to note that a value of
α = 0.9 is at the very high end of estimates in the literature and hence has less empirical plausibility.

Figure 12: Results with α = 0.9 in Recalibrated Model
Participation in stocks over the lifecycle

Share of stocks over the lifecycle

1
Model
SCF data

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.1
Model
SCF data

0
25

Age

(a) Participation

(b) Shares

Our model can also shed light on the effects of exogenous or policy-induced changes in the
learning technology. How would households in our model respond if they were to be confronted
with a change in the productivity of the learning technology? We address this case by considering
the effect of decreasing the value of α to 0.5 and increasing it to 0.9. To understand the implications
in this case, it is important to keep all other parameters as in the benchmark. The results are
reported in Figure 13.
First consider the case where the human capital technology is less productive (α=0.5). Two
opposing forces are at work here. On the one hand, because human capital is less productive,
agents have less incentive to invest time in it. On the other, to the extent that agents do want to
accumulate human capital, they need to invest more time to accumulate the same level of human
capital as in the benchmark. It turns out that the first effect dominates; agents invest less time
in human capital than in the benchmark, as the bottom left panel shows, with the effect that
their human capital levels are lower throughout working life than in the benchmark (bottom right
panel). This has two effects on participation. Less time invested in human capital leads to higher
participation early in life, while the slower growth rate of human capital over the life cycle (which
translates into a flatter path for earnings) leads to a flatter profile of participation over the life
cycle.
In the case where the human capital technology is more productive (α=0.9), the two opposing
forces described earlier also lead agents to invest less time in human capital accumulation. Despite
this, their human capital levels are higher and increasing much more steeply than in the benchmark.
30

The participation rate in the stock market is lower early in life but rises steeply to move past the
rate observed in the benchmark by age 50.
This experiment reveals a more general mechanism that is at work in our model. Agents have
two ways to move resources through time—using financial assets or human capital. The more
human capital pays off in the future, the steeper the earnings profile and the higher the incentive
to invest in human capital now. If agents can use financial assets to bring some of those future
earnings into the present to smooth consumption, they will, with the result that they do not invest
in stocks early in life and instead borrow to the extent possible. On the other hand, if earnings are
going to be flat, or if agents don’t expect high returns to human capital in the future, they will
enter financial markets early. The findings are similar if we change the growth rate of the rental
rate of human capital, g (results available upon request).

Figure 13: The Effect of the Elasticity of Human Capital Production on Investments
1
0.9

Participation in stocks over the lifecycle

Share of stocks over the lifecycle

Benchmark
Alpha=0.9
Alpha=0.5

0.9

0.8

0.7

0.6

0.5

0.4

0.3
0.3

0.2
0.2

0.1
0.1

0
25

Age

(a) Participation

(b) Shares
Time allocated to human capital over the lifecycle

0.35
Benchmark
Alpha=0.9
Alpha=0.5
0.3

0.25

0.2

0.15

0.1

0.05

0
25

Age

5.4.3

The Role of Initial Characteristics

In our setting, initial ability and initial human capital both influence the life-cycle earnings profile.
Specifically, initial human capital determines the initial level of earnings, while initial ability affects
the rapidity with which earnings grow from that level. We have already seen some of the effects
of the earnings profile on stock-market investment in the previous experiment; here we trace these
effects back to initial conditions. In our benchmark model, initial ability and human capital are
positively correlated. In order to describe their effects separately, the figures below are derived
from an experiment in which the conditional distributions of ability given human capital do not
vary with the level of human capital, and vice versa.
Figure 14 shows participation and human capital investment behavior by quartiles of ability
32

levels, with quartile 1 being the lowest. Agents with high ability accumulate human capital more
rapidly than agents with low ability. This is driven by the fact that investing time in human capital
is more productive for these agents, which increases their incentive to do so. Of course, these agents
do not have to invest as much time to accumulate the same amount of human capital as those with
lower ability, and as a result, will be able to enter retirement with a given wealth level with less
effort by virtue of their greater earnings capacity. These two forces work in opposite directions,
with the result that we observe that agents in the middle two quartiles invest the most time in
human capital investment, especially early in life (Figure 14c). Agents in the lowest quartile of
ability invest the least time in human capital accumulation, and their time investment remains
relatively flat over the life cycle.
When it comes to one of the main questions of interest to us, namely, stock-market participation,
we turn to Figure 14a. Recall that in the baseline model, a lower time investment in human capital
is associated with a higher stock-market participation rate. This is seen in stark terms here: The
lowest quartile participates at extremely high rates (80%). The intuition is simply that for lowability households, the effective rate of return from human capital is much lower than from equity
investment. Further, their earnings profile is relatively flat, which means that their participation
rate also remains flat over the life cycle. In contrast, the high initial investment in human capital,
particularly for quartiles 2 and 3, and the steeper earnings profile, particularly for quartile 4, is
associated with these groups exhibiting a steeply increasing stock market participation rate over
the life cycle. For these households, learning, especially when young, is a better investment than
earning and investing in equities. This analysis makes clear that once human capital investment
is allowed, the model suggests that learning ability, all else equal, should be inversely related to
equity investment. Of course, this fact and the fact that the model captures observed participation
suggests there are other forces at work. In terms of shares, we see (Figure 14b) that quite unlike
for participation, those who have chosen to invest in the stock market diversify in fairly consistent
manner: Shares are quite similar across ability quartiles, especially as households age. We will see
later that initial human capital as well as the risk properties of stock market returns and individual
risk aversion are the main drivers of shares and none of these are in play in this experiment.

Figure 14: Investment by Quartiles of Ability
Participation in risky assets over the lifecycle

Share of stocks over the lifecycle

0.9

0.8

1
quartile 1
quartile 2
quartile 3
quartile 4
all

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
25

quartile 1
quartile 2
quartile 3
quartile 4
all

0.9

0
25

Age

(a) Participation

(b) Shares
Time invested in human capital over the lifecycle

0.8
quartile 1
quartile 2
quartile 3
quartile 4

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
25

Age

Another dimension of initial heterogeneity is in the level of human capital with which households
enter the model. Figure 15 reports stock market investment (both participation and shares) as
well as the time allocated to human capital investment by quartiles of initial human capital. As
seen in panel 15c, time allocation as a function of initial human capital is inversely proportional
to its initial level: Those in quartile 1 (the lowest level of initial human capital) invest the most
time, while those in the highest quartile invest the least. The intuition is natural. Those with
high initial human capital face not only a high opportunity cost of additional accumulation, but
also stand to reap only low marginal returns. The reverse holds for those with low initial human
capital. For brevity we do not report the evolution of human capital levels but note that initial
differences in human capital levels persist over time, although with some “catch-up” due to those

with low initial human capital allocating higher amounts of time towards its accumulation.
What does this imply for the accompanying investment that households make in the stock
market? Those with the highest levels of initial human capital (quartile 4) participate in the stock
market at the highest rates, while those with the lowest levels participate at by far the lowest
rates. Specifically, participation within the top quartile is about 70% at age 25 and reaches 100%
participation by age 50 (Figure 15a). Quartiles 2 and 3 participate at around a 30% rate early
in life, and reach 100% participation after age 55. For the lowest quartile, participation starts
at around 15% and remains below 50% throughout working life. When we look at the shares of
financial wealth held by quartiles of initial human capital, as displayed in Figure 15b, we find that
all but the lowest quartile invest a fairly similar fraction of their wealth in stocks over the life cycle.
Stock market behavior in this case is influenced by two forces. First, households with high
initial human capital not only have relatively high earnings, but also do not expect earnings to
rise as rapidly over the life cycle as those with low initial human capital do. As a result, their
motivation to borrow early in life is limited, and the same force that leads to low time allocation
towards human capital investment encourages stock market participation. In other words, the
optimal overall portfolio for those with high initial human capital reflects the relative value of
savings, even early in life, and this leads to a relatively high rate of equity market participation.
By contrast, those with low human capital find it to be a far better investment and, moreover,
expect future earnings to be higher than present levels. Higher expected future earnings make
savings less attractive, as that would hinder the intertemporal smoothing of consumption. Indeed
some of these households would value borrowing (or, at the very least, not accumulating wealth).
Thus, saving via any financial asset, especially risky stocks, is less attractive. The individuals in
the lowest quartile also earn the least of all groups, and hence face significant uninsurable risk,
especially early in life. The riskiness of equity makes such investment unattractive. For households
in the middle quartiles of initial human capital, optimal investment behavior falls between these
two extremes.
While this case is instructive, it is important to note that it holds the correlation between
initial human capital and learning ability at zero. Overall participation will depend, in general,
on the joint distribution of ability and initial human capital. Indeed, in the baseline model, these
characteristics are positively correlated. Thus, those who face high costs of learning—and hence
wish to invest primarily in stocks—are frequently also those with low initial human capital—who
wish to invest in human capital instead. The net result is that participation rates in the baseline
model fall in between the levels implied by Figures 14 and 15.

Figure 15: Investment by Quartiles of Initial Human Capital
Participation in risky assets over the lifecycle

Share of stocks over the lifecycle

0.9

0.8

1
quartile 1
quartile 2
quartile 3
quartile 4
all

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
25

quartile 1
quartile 2
quartile 3
quartile 4
all

0.9

0
25

Age

(a) Participation

(b) Shares
Time invested in human capital over the lifecycle

0.8
quartile 1
quartile 2
quartile 3
quartile 4

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
25

Age

A common theme that emerges from the experiments described above is that higher human
capital accumulation, if achieved through a higher initial endowment of human capital and ability
or an improvement in its production technology, leads to an increase in earnings and stock market
participation. In these instances, the agent accumulates more human capital without necessarily
allocating additional time to it. On the other hand, any increase in human capital that comes
from households allocating more time to human capital investment leads to lower stock-market
participation.

5.5

The Role of Stock-Market Risk

The stock market, while it clearly offers a far higher average rate of return than risk-free savings,
may still not attract overwhelming participation due to the exposure that it creates for households.
To study the effect of the risk properties of stock returns on participation and shares, we examine
two cases in which equity market risk is different than in the baseline model. In Figure 16, we report
results under the assumptions that the standard deviation of stock market returns is low (50% less)
or high (50% more) compared to our benchmark (0.078 and 0.236, respectively). Interestingly, these
large differences in the risk properties of stocks have almost no effect on participation compared
to the benchmark. Rather, all the adjustment is on the intensive margin, and it is sizable. In the
case of higher-than-baseline riskiness of stock return, we find that household diversification pays a
significant role and leads to much lower proportions of wealth held in stocks than in the baseline.
Conversely, we observe that when stock market risk is cut, wealth shares balloon to nearly 80%
when averaged over the life cycle. Thus, an interesting implication of our analysis is that while
initial human capital levels and ability govern the decision to invest at all in the stock market, the
risk of stocks is what matters for the share of wealth held in equity.
Figure 16: Stock Market Investment with Low and High Risk of Stocks
1

Participation in stocks over the lifecycle
Benchmark
High risk of s
Low risk of s

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
25

Age

(a) Participation

5.6

Benchmark
High risk of s
Low risk of s

0.9

0.8

0
25

Share of stocks over the lifecycle

(b) Shares

Agents’ Risk Aversion

Finally, we study the effect of changing agents’ risk aversion in our setting. We consider two cases,
σ = 3 and σ = 10. The results are shown in Figure 17.
37

Figure 17: Effect of Changing Risk Aversion on Household Portfolios
Participation in stocks over the lifecycle

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Benchmark
Sigma=3
Sigma=10

0.9

0.8

0
25

Share of stocks over the lifecycle

1
Benchmark
Sigma=3
Sigma=10

0
25

Age

(a) Participation

(b) Shares

As seen clearly in the figures, the effect of changing risk aversion is qualitatively similar to
changing the riskiness of stock returns, in the sense that it does not have much effect on stock
market participation in the economy. Rather, households adjust the amount of their wealth that
they allocate to stocks in a completely natural manner, allocating a larger share to stocks when they
are less risk averse and a smaller share when they are more risk averse. One useful implication of
these results is that while we have employed a risk-aversion value that is standard in the portfoliochoice literature, it is higher than the value typically assumed in macroeconomics, which ranges
from 1 to 3 for example. Therefore, it is worth noting that neither stock-market participation nor
shares change substantially under lower risk aversion. This is only suggestive, however, as we do
not recalibrate the entire model when we change risk aversion.

Conclusion

Research on household portfolios frequently predicts that households will almost universally participate in equity markets and allocate a high share of financial wealth to equity, especially early
in life. These predictions are counterfactual and empirically consistent predictions have proved
hard to obtain without stock-market participation costs, informational frictions, or departures
from standard preferences. The central contribution of this paper is to demonstrate that once
human capital investment is allowed for, a standard model predicts stock-market participation
and equity-investment shares that are much closer to those empirically observed throughout the
38

life cycle.
Our approach is both novel and straightforward: We embed the classic human capital model
of Ben-Porath (1967) into a standard life-cycle model of portfolio choice where households face
uninsurable idiosyncratic shocks to productivity (e.g., Cocco, Gomes, and Maenhout, 2005)). Importantly, as in Huggett, Ventura, and Yaron (2006), households in our model are heterogeneous
with respect to characteristics governing initial human capital and their ability to acquire it.
Our findings flow from two simple and intuitive mechanisms: First, the returns to human
capital investment are highest early in life and exceed the constant returns on financial assets for
most households. As households age, this relationship reverses. Thus, stock-market participation
starts low and grows over the life-cycle, just as in the data. As for shares, the risks to human
capital limit the household’s desire to hold wealth in risky financial equity. Our results suggest
that the option to invest in human capital is important for understanding observed household
portfolio choices over the life cycle.

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Regression Tables
Table 2: Probit for Stock Market Participation (SCF), N=34,008
Age
23-25
26-28
29-31
32-34
35-37
38-40
41-43
44-46
47-49
50-52
53-55
56-58
59-61
62-64
65-67
68-70
71-73
74-76
77-79

Constant

Coefficient
(omitted)
0.3195
0.5079
0.5510
0.6580
0.8026
0.9430
0.9177
1.0862
1.1310
1.2002
1.2459
1.2166
1.1894
1.1660
1.1346
1.1051
1.1265
1.2015

Cohort
Coefficient
1919-1921 (omitted)
1922-1924
-0.0338
1925-1927
0.2211
1928-1930
0.3671
1931-1933
0.2360
1934-1936
0.3158
1937-1939
0.3858
1940-1942
0.4348
1943-1945
0.4710
1946-1948
0.6053
1949-1951
0.5457
1952-1954
0.6078
1955-1957
0.6222
1958-1960
0.6679
1961-1963
0.8107
1964-1966
0.7916
1967-1969
0.8857
1970-1972
0.9654
1973-1975
0.9716
1976-1978
1.0055
1979-1981
0.9860
1982-1984
0.9626
1985-1987
1.0282
1988-1990
0.9297

-1.4273

Table 3: OLS for Share of Risky Assets in Household Portfolio (SCF), N=21,778
Age
23-25
26-28
29-31
32-34
35-37
38-40
41-43
44-46
47-49
50-52
53-55
56-58
59-61
62-64
65-67
68-70
71-73
74-76
77-79

Constant

Coefficient
(omitted)
-0.0010
0.0353
0.1739
0.4163
0.5209
0.5531
0.6351
0.7963
0.8147
0.9260
0.8842
0.7891
0.9596
0.9803
0.9177
0.9793
1.1988
1.1405

Cohort
Coefficient
1919-1921 (omitted)
1922-1924
0.4470
1925-1927
0.5412
1928-1930
0.6711
1931-1933
0.5723
1934-1936
0.7014
1937-1939
0.8419
1940-1942
0.7739
1943-1945
0.9438
1946-1948
0.8771
1949-1951
1.0174
1952-1954
1.1772
1955-1957
1.0696
1958-1960
1.3184
1961-1963
1.3533
1964-1966
1.5286
1967-1969
1.4329
1970-1972
1.6140
1973-1975
1.4651
1976-1978
1.5958
1979-1981
1.4606
1982-1984
1.4250
1985-1987
1.1774
1988-1990
1.0460

-2.0059

Figures
Figure 18: Earnings Statistics (CPS)
Mean of lifecycle earnings
200

180

160

140

120

100

0
25

Age
Gini of lifecycle earnings
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
25

45
Age

Mean/Median of lifecycle earnings
2

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

0
25

45
Age

Full text of Working Papers (Federal Reserve Bank of Richmond) : Stock Market Investment : The Role of Human Capital, Working Paper 15-07

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