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Working Paper Series
Labor-Market Uncertainty and Portfolio
Choice Puzzles
WP 14-13
Yongsung Chang
University of Rochester and Yonsei
University
Jay H. Hong
Seoul National University
Marios Karabarbounis
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Labor-Market Uncertainty
and Portfolio Choice Puzzles∗
Yongsung Chang
University of Rochester
Yonsei University
Jay H. Hong
Seoul National University
Marios Karabarbounis
Federal Reserve Bank of Richmond
June 16, 2014
Working Paper No. 14-13
Abstract
The standard theory of household-portfolio choice is hard to reconcile with the following
facts: (i) Households hold a small amount of equity despite the higher average rate of return.
(ii) The share of risky assets increases with the age of the household. (iii) The share of risky
assets is disproportionately larger for richer households. We develop a life-cycle model with
age-dependent unemployment risk and gradual learning about the income profile that can
address all three puzzles. Young workers, on average asset poor, face larger labor-market
uncertainty because of high unemployment risk and imperfect knowledge about their earnings
ability. This labor-market uncertainty prevents them from taking too much risk in the financial
market. As the labor-market uncertainty is gradually resolved over time, workers can take
more financial risks.
Keywords: Portfolio Choice, Labor-Market Uncertainty, Learning.
∗
Emails: yongsung.chang@rochester.edu, jayhong@snu.ac.kr and marios.karabarbounis@rich.frb.org. For
helpful suggestions we would like to thank seminar participants at ASU, the University of Virginia, Stony
Brook, Queens, Rochester, the Federal Reserve Bank of Philadelphia, the Bank of Greece, the University of
Piraeus, SED (Seoul), and the Federal Reserve Bank of Richmond. Any opinions expressed are those of the
authors and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve
System.
1
Introduction
Three stylized facts have been documented by Guiso, Haliassos, and Jappelli (2002), among
others: (i) Households hold a small amount of equity despite the higher average rate of
returns (equity premium puzzle). (ii) The share of risky assets increases with the age of the
household. (iii) The share of risky assets is disproportionately larger for richer households.
The standard life-cycle model of household portfolio choice is hard to reconcile with these
facts. The standard theory predicts that the risky share is negatively correlated with the age
and wealth of households, as a household with a constant relative-risk aversion should invest
aggressively when young and gradually move toward a safer portfolio.
We show that the age-dependent labor-market uncertainty helps us to address all three
portfolio choice puzzles. Young workers face much greater risk of being unemployed. According to the 2013 Current Population Survey, the average unemployment rate of male workers
ages 20-24 is as high as 14%, whereas that of workers ages 50-54 is 5.8%. According to Topel
and Ward (1992), the incidence of job turnover is highly concentrated among young workers.
In the first 10 years after entering the labor market, a typical worker holds 7 jobs (about
two-thirds of his career total). Moreover, young workers are uncertain about the shape of
their income profiles and this uncertainty is resolved over time (Guvenen (2007) and Guvenen
and Smith (2014)).
As a result, both actual and perceived labor-market uncertainty is much larger for young
workers, who on average have little wealth or are in debt. Since the labor-market outcome is
largely uninsurable, young investors, despite a longer investment horizon, would like to avoid
exposing themselves to too much risk in the financial market. By contrast, older investors,
who face less uncertainty in the labor market, can afford taking more risk in their financial
investments.
We quantitatively evaluate this link between labor-market uncertainty and financial risk in
an otherwise standard life-cycle model of portfolio choice (e.g., Cocco, Gomes, and Maenhout
(2005)). Our model features (i) age-dependent unemployment risk, (ii) gradual learning about
the income profile, (iii) noise in the Bayesian updating rule to achieve a plausible rate of
learning, (iv) a portfolio choice between a risk-free bond and a risky stock, and (v) a limited
ability for a household to insure against labor-market income risk.
Our model is calibrated to match the age profiles of unemployment risk, earnings volatility,
and consumption dispersion in the data. The age-dependent unemployment risk is based on
the estimates by Choi, Janiak, and Villena-Roldan (2014). The average productivity over
the life-cycle is taken from Hansen (1993), which is estimated from the CPS. The stochastic
process and cross-sectional dispersion of the life-cycle income profile are similar to those in
1
Guvenen and Smith (2014).
Our model helps us to address all three portfolio choice puzzles mentioned above. First,
our model matches the average risky share of 0.45, which we estimated based on the Survey
of Consumer Finances.1 This low risky share is achieved under the relative risk aversion of
5, much lower than the typical value required in standard models. Second, the age profile of
risky share closely tracks that in the data. According to the SCF, the average risky share
increases by 0.92 percentage point annually over ages 21-65. In our model, it increases by 0.36
percentage point. Third, the risky share and wealth are mildly positively correlated across
households in the SCF, 0.107. This correlation is mildly negative in our model: -0.147. While
our model fails to generate a positive correlation, it partially reconciles the large gap between
the data and the standard life-cycle model (our model without labor-market uncertainty),
which predicts a strongly negative correlation, -0.638.
In our model, workers update their priors about their earnings ability in a Bayesian fashion.
However, it is often found in various natural experiments that subjects are more conservative
in changing their priors than the standard Bayesian rule implies (e.g., Edwards (1968)). In
fact, under the standard Bayesian updating rule, while the model generates significant income
risk over longer horizons, the uncertainty over shorter horizons (e.g., one to five years ahead)
is resolved extremely fast. For example, within two years of entering the labor market, about
two-thirds of next period’s uncertainty (measured by the one-period forecast-error variance)
is resolved. We view this rate of learning to be unrealistically fast, especially if one considers
the intense career-search behavior of young workers. Typically, young workers shop around
for jobs before settling into a long-term career. Topel and Ward (1992) document that the
average number of jobs held by workers at the tenth year after entering the labor market is
approximately 7. Kambourov and Manovskii (2008) report that the average probability that
a young worker will switch occupations is approximately 25% on an annual basis. Frequent
job switching may diminish the informational context from previous labor-market experience,
especially if it takes place across completely different industries or occupations. We introduce
an additional noise in the Bayesian updating formula as a shortcut to reflect the reset of
priors due to moves between industries or occupations. Since it is difficult to obtain direct
estimates of how fast priors converge to the true earnings ability, we rely on the evidence on
occupational mobility such as Kambourov and Manovskii (2008) and Topel and Ward (1992).
We also verify that our parameterization matches the increasing age profile of consumption
dispersion in the data very well.
Our benchmark model introduces three new features into an otherwise standard life-cycle
model of portfolio choice (e.g., Cocco, Gomes, and Maenhout (2005)): (i) age-dependent
1
The detailed definition of risky share is explained in Section 2.
2
unemployment risk, (ii) imperfect information about the earnings profile, and (iii) noise in
the updating rule. In matching the average risky share in the data, the imperfect information
about the earnings profile contributes the most (64%), followed by the noise in the updating
rule (24%) and the age-dependent unemployment risk (12%).
Our theory predicts that workers in an industry (or occupation) with highly volatile earnings should take less risk in their financial portfolios. We empirically test this prediction using
the industry volatility of labor income estimated by Campbell, Cocco, Gomes, and Maenhout
(2001). We find that a household whose head is working in an industry where the labor-income
volatility is 10% larger than the mean makes safer financial investments, exhibiting a risky
share 2.2% lower than the average. This is consistent with Angerer and Lam (2009), who find
a negative correlation between labor-income risk and risky share of workers among the NLSY
1979 cohort.
Our work contributes to the existing literature on the life-cycle portfolio choice in several
ways. The closest paper to ours is Cocco, Gomes, and Maenhout (2005). We extend their
analysis in two important directions. First, we introduce age-dependent unemployment risk
and gradual learning about the age-earning profile. We show that perceived uncertainty may
be important for portfolio decisions, especially for young workers. Second, we test our model
over a wider ranger of portfolio statistics, most notably the correlations between the risky share
and financial wealth within and across age groups. Another closely related paper is Gomes
and Michaelides (2005), who show, among others, that heterogeneity in risk aversion and
Epstein-Zin preferences is not enough to account for the age profile of risky share. Wachter
and Yogo (2010) analyze the life-cycle profile of portfolios as we do. They match the age
profile in the data using non-homothetic utility and decreasing relative risk aversion, whereas
we match the portfolio profile using age-dependent labor-market uncertainty and constant
relative-risk-aversion preferences.
Our paper distinguishes itself from previous studies on the covariance between labor-market
risk and stock returns. Benzoni, Collin-Dufresne, and Goldstein (2007) show how labor income
and stock-market returns are likely to move together at a longer time horizon. As a result,
stocks are riskier for young workers than for old. Storesletten, Telmer, and Yaron (2007) show
that if labor income is perfectly correlated with stock returns, the age profile of risky share
can exhibit a hump shape. Lynch and Tan (2011) show that the countercyclical volatility
of labor-market income growth plays an important role in discouraging the stock-holdings
motive for poor and young households. Huggett and Kaplan (2013) decompose human capital
into the safe and risky components and find that the level of human capital and stock returns
have a small positive correlation.
We closely follow Guvenen (2007) and Guvenen and Smith (2014) in modeling the uncer-
3
tainty about earnings profile. Both papers examine the implications of imperfect information
about income profile for consumption over the life cycle. Consistent with their results, we
find that imperfect information coupled with heterogeneity in income profiles can match the
linearly increasing dispersion of consumption along the life cycle. However, we take a step
further and ask whether gradual learning about the income profile can also help to explain
the portfolio allocation puzzle between risky and riskless financial assets. Wang (2009) studies
portfolio choice with income heterogeneity and learning within an infinite horizon model. In
contrast, we employ a life-cycle model with a particular focus on the relation between risky
share and age. Finally, Campanale (2011) develops a life-cycle model in which investors learn
about stock-market returns. While uninformed investors can purchase information about the
stock market from informed investors, it is impossible to know a priori the unrealized path of
lifetime earnings. Hence, our model makes a more realistic assumption about the investor’s
earnings profile.
The paper is organized as follows. In Section 2, based on the extensive data from the SCF,
we document the stylized facts on household-portfolio profiles, including the three abovementioned puzzles. In particular, we provide detailed statistics across various age/wealth
groups. In Section 3 we present a simple 3-period model to illustrate the important interaction
between labor-market uncertainty and portfolio choice. Section 4 develops a fully specified
life-cycle model for our quantitative analysis. We then calibrate the model to match the age
profiles of unemployment risk, earnings volatility and consumption dispersion in the data. In
Section 5, we address three portfolio choice puzzles. We consider various specifications of the
model to evaluate the marginal contribution of each component of labor-market uncertainty
newly featured. Section 6 tests the prediction of our theory using the cross-industry variation
of income risks. Section 7 concludes.
2
Life-Cycle Profile of Households’ Portfolios
Using the 1998 Survey of Consumer Finances (SCF), we document some stylized facts on
the life-cycle profile of households’ portfolios. The SCF provides detailed information on the
households’ characteristics and their investment decisions. The survey is conducted every three
years and the basic pattern we document here is similar across the surveys.2 To be consistent
with our model (where households face a choice between risk-free and risky investment),
2
Benzoni and Chyruk (2009) show that the basic facts regarding the risky share of financial assets are the
same in the 2001, 2004, and 2007 surveys. Moreover, Ameriks and Zeldes (2004) use the available SCF studies
from 1983-1998. They find that both the unconditional and the conditional share weakly increase with age
(or exhibit a hump shape) if time effects are controlled for but increase strongly with age if they control for
cohort effects.
4
we classify assets in the SCF into two categories: “safe” and “risky” assets. (The detailed
description of how to classify assets into these two categories is discussed below.) Several facts
emerge:
1. Participation: Just a little over half (56.9%) of the population invests in risky assets.
This participation rate shows a hump shape over the life cycle, with its peak around the
average retirement age.
2. Conditional Risky Share: Households that invest in risky assets, on average, allocate
about half (45.5%) of their financial wealth to risky assets. This conditional risky share
increases monotonically over the life cycle.
3. Unconditional Risky Share: The participation rate and conditional risky share combined,
the unconditional risky share exhibits a hump shape over the life cycle.
4. Wealth Correlation: Wealthier households, on average, allocate a larger fraction of their
savings to risky assets.
In the SCF, some assets can be easily classified into one or the other. For example,
checking, savings, and money market accounts are safe investments, while direct holdings of
stocks is risky. However, other assets (e.g., mutual funds and retirement accounts) are invested
in a bundle of safe and risky instruments. Fortunately, the SCF provides information on how
these accounts are invested. The respondents are asked not only how much money they have
in each account but also where they are invested. If respondents report that most of the
money in those accounts is in bonds, money market, or other safe instruments, we classify
them as safe investments. If they report that the money is invested in some forms of stocks,
we categorize them as risky investment. If they report that the account involves investments
in both safe and risky instruments, we assign half of the money into each category.3
More specifically, the assets considered safe are checking accounts, savings accounts, money
market accounts, certificates of deposit, cash value of life insurance, U.S. government or state
bonds, mutual funds invested in tax-free bonds or government-backed bonds, trusts and annuities invested in bonds, and money market accounts. The assets considered risky are stocks,
brokerage accounts, mortgage-backed bonds, foreign and corporate bonds, mutual funds invested in stock funds, trusts and annuities invested in stocks or real estate and pension plans
3
While the SCF provides information on how individual retirement accounts are invested, it is not the case
with pension plans, e.g., 401(k), defined contribution plans and others. In this case, we classify half of the
money invested in these accounts as safe assets and the rest as risky assets (because the average risky share is
close to 50%). In Appendix B we recalculate the risky share with different split rules between safe and risky
assets such as 80-20 or 20-80, for example. The average of the risky share is affected by the split rule, but the
shape of the age profile is not.
5
Table 1: Household Savings by Account Types
Account Type
Total safe assets
Checking account
Savings account
Savings bond (safe)
Life insurance
Retirement accounts (safe)
Total risky assets
Stocks
Trust (risky)
Mutual funds (risky)
Retirement accounts (risky)
Total financial assets
Average (US $)
Participation
64,488
99.8%
3,483
4,746
5,916
8,955
12,103
86.9%
60.2%
22.4%
31.7%
38.1%
89,403
56.7%
32,898
6,121
12,574
26,376
20.6%
1.2%
16.3%
45.2%
153,891
100.0%
Note: Survey of Consumer Finances (1998).
that are a thrift, profit sharing or stock purchase plan. Also considered a risky investment is
the “share value of businesses owned but not actively managed excluding ownership of publicly
traded stocks.” We exclude the share value of actively managed business from our benchmark
definition of risky investments. Appendix A provides more detailed information on how we
construct our data from the SCF.
Table 1 summarizes the average amount (in 1998 dollars) held in specific accounts as well
as the participation rate (the fraction of households that have a positive amount in that
account). We restrict the sample to households that have a positive amount of assets. While
86.9% of households hold a checking account and 60.2% hold a savings account, only 20.6%
directly own stocks. Nearly every household (99.8%) owns some form of safe assets, while
only 56.7% invest in risky assets.
Risky Share by Age
We examine the risky share across different age and wealth groups. The risky share is defined
as the total value of risky assets divided by the total amount of financial assets. Figure 1
shows participation rates, conditional (on participation) risky share, and unconditional risky
share over the life cycle. The dotted line represents the average value by age and the solid line
6
Figure 1: Risky Share over the Life Cycle
Risky Share of Financial Assets
80
70
70
60
60
50
50
Percent
Percent
Fraction Owning a Risky Account
80
40
40
30
30
20
20
10
10
0
20
30
40
50
Age
60
0
70
Conditional Share
Unconditional Share
20
30
40
50
Age
60
70
Note: Survey of Consumer Finances (1998). The solid line represents the 5-year average. The
left panel shows the participation rate (the fraction of households that invest in risky assets).
The right panel shows the unconditional and conditional risky shares.
represents the 5-year average (e.g., 21-25, 26-30, etc.). In the left panel, the participation rate
(the fraction of households who participate in risky investment) exhibits a hump shape over
the life cycle with its peak just before the average retirement age. It increases from 24.7% in
the age group of 21-25, to 55.3% in ages 31-35, reaches its peak of 67.7% in ages 51-55 and
then decreases to 48.4% in ages 61-65.
The right panel shows the conditional and unconditional risky shares. The conditional
share—the share among the households that invest in risky assets—monotonically increases
over the life cycle. It increases from 40.6% in the 21-25 age group to 44.5% in the 41-45
age group, and then to 50.7% in the 61-65 age group. Since our model abstracts from the
participation decision, when we compare the model and data, we will focus on the conditional
risky share only. The average conditional risky share is 45.5%.4 The unconditional risky share
(participation rate times conditional risky share), also exhibits a hump shape. It rises from
10.5% at ages 21-25 to its peak of 34.3% at ages 51-55, and then decreases to 24.3% at ages
61-65. In sum, these life-cycle patterns of risky share clearly suggest that younger investors
are reluctant to take financial risks, despite longer investment horizons and higher average
4
The average risky share is defined as the average of risky shares across households, not the total amount
of risky assets divided by total amount of financial assets. Under the alternative definition, the average risky
89,403
share is 58.1% = 153,891
, much higher than ours. We prefer our definition because the latter definition is
sensitive to an outlier, i.e., a few extremely rich households that make extensive risky investments.
7
rates of return to risky investments.
Risky Share by Wealth
We turn our attention to the relationship between the risky share and wealth (total financial
assets). Table 2 reports the average risky share (both conditional and unconditional) across
5 quintile groups in the distribution of household wealth. Both measures of risky share show
strong positive correlations with the amount of household wealth. Wealthier households take
more risk in their financial investments. The unconditional risky share increases monotonically
from 5.3% in the 1st quintile to 38.1% in the 3rd, and 64.9% in the 5th. The conditional risky
share also increases from 35.9% in the 1st quintile to 44.4% in the 3rd, and 66.6% in the 5th.
The participation rate (not reported in the table) also monotonically increases with wealth.
In the 5th quintile of wealth, almost everyone (97%) makes risky investments.
Table 2: Risky Shares by Wealth
Risky Share
Wealth Quintile
Unconditional
Conditional
1st
2nd
3rd
4th
5th
5.3%
25.9%
38.1%
49.4%
64.9%
35.9%
40.5%
44.4%
51.7%
66.6%
Average
28.3%
45.5%
Note: Survey of Consumer Finances (1998)
Since older households are, on average, wealthier, one might suspect that the age effect
drives this positive correlation between the risky share and wealth. To check whether this is
the case in the data, Figure 2 plots the average conditional risky share across quintiles for
the 5-year age group. While it is not always monotonic (perhaps due to a small cell size), it
clearly shows a positive correlation between the risky share and wealth within each age group.
Robustness (1): Actively managed businesses and homeownership
So far, we have excluded two types of assets from household wealth: the amount of investment
in their own business and the value of house(s). To see whether the age profile of risky share is
robust to the inclusion of these assets, we re-calculate the conditional risky shares, including
8
Figure 2: Conditional Risky Share by Age and Wealth
Conditional Risky Share
80
70
60
50
40
30
20
80
5
60
Age
4
40
2
20
3
Wealth Quintile
1
each of these assets, in Table 3. The first column is our benchmark definition of risky shares
when these assets are not included in the household’s wealth.
In the second column (“Business included”), we include the net value of actively managed
own businesses as a part of risky assets.5 With the value of actively managed businesses
included, the average risky share increases to 50.8% (from 45.5% according to our benchmark
measure). However, the increasing pattern of the risky-share profile is unaffected. It increases
from 44.9% at ages 21-30 to 54.4% at ages 61-65.
In the third column (“House included”) we include the net worth of house(s) as a part
of the household’s risky assets. The net worth of house(s) is the sum of the house(s) value
minus the amount borrowed as well as other lines of credit or loans the investor may have.
We also add any investment in real estate such as vacation houses. With the net worth of
houses included, the average risky share increases significantly, to 69.7%. However, again, the
increasing pattern of the risky share profile remains unaffected. It increases from 61.4% at
ages 21-30 to 76.8% at ages 61-65.
Next, we compare the risky shares (using our benchmark measure) of the households that
own a business or a house. While the average risky share is slightly higher for business owners
(47.9% compared to 45.0%), the increasing pattern of the age profile remains the same. They
5
The net value of actively managed own business is the amount owed to the household by the business,
subtracting the money owed by the household.
9
Table 3: Conditional Risky Shares under Alternative Definitions and by Household Type
Benchmark
Age
Alternative
Business
House
included included
Household Type
Own Business?
No
Yes
Own Home?
No
Yes
21-30
31-40
41-50
51-60
61-65
40.2%
44.3%
45.9%
48.0%
50.1%
44.9%
50.2%
51.9%
52.7%
54.4%
61.4%
66.9%
70.7%
74.1%
76.8%
40.1%
44.7%
45.1%
47.6%
48.4%
41.5%
41.2%
49.7%
50.4%
56.4%
40.8%
37.0%
40.8%
38.0%
46.9%
42.3%
47.0%
47.1%
50.1%
50.7%
Average
45.5%
50.8%
69.7%
45.0%
47.9%
39.3%
47.9%
Note: Under “Business included” the net value of actively managed own businesses is included
as a part of risky assets. Under “House included” the net worth of house(s) is included as a
part of risky assets. The table also reports the conditional risky shares (using the benchmark
definition) by samples. To increase our sample size, we used ages 61-70 instead of 61-65 to
compute the average risky share for renters.
increase monotonically with age. There are many reasons for believing that homeownership
should affect the portfolio choice over the life cycle (e.g., marriage, having children, etc.).
For example, as noted by Cocco (2005), house price risk may crowd out stock holdings.
The average risky share is higher for homeowners (47.9% compared to 39.3% for renters).
However, the age profiles generally increase for both groups but not monotonically, especially
for renters, perhaps due to the small sample size. Our simple analysis does not suggest that
homeownership is the reason younger people choose to hold a smaller fraction of risky assets
in their portfolios.
Robustness (2): Net Savings and Retirement Accounts
In our benchmark definition, safe and risky assets are defined as gross savings in safe and
risky financial instruments, respectively. In our benchmark definition, consumer debt such as
credit card debt and other consumer loans is ignored. The rationale for this is that it is not
clear if liabilities should be subtracted from either the safe or the risky assets.6 In Table 4 we
calculate the risky share under the assumption that consumer debt is subtracted from the safe
6
Imagine an individual with $1,000 in her checking account, $1,000 in stocks, and $500 debt to a relative.
The individual may have placed the $500 in the checking account, in which case the net safe assets are $500
and the risky share is 66%. However, the individual may have invested the borrowed money in the stock
market, in which case the net risky assets are $500 and the risky share is 33%. Since there is no information
on how consumer debt is actually used, we choose to focus on gross savings.
10
Table 4: Conditional Risk Shares: Net Savings and Retirement Accounts
Age
Benchmark
Net Savings
Retirement Accounts
21-30
31-40
41-50
51-60
61-65
40.2%
44.3%
45.9%
48.0%
50.1%
48.8%
54.6%
51.8%
55.4%
51.6%
63.0%
61.9%
62.9%
65.4%
63.4%
Average
45.5%
53.0%
63.2%
Note: Under “Net Savings” the amount of net savings is used for the total amount of safe
assets. “Retirement Account Only” reflects the risky share of assets in the retirement accounts.
assets. Given the significant fraction of people with debt (45.5% in our sample), the average
risky share increases to 53.0% from 45.5% in Table 4. While the risky share still increases with
age (from 48.8% at ages group 21-30 to 55.4% at ages 51-60), it is not monotonic. Finally, we
examine how the savings in retirement accounts, which are considered long-term investments,
are allocated between safe and risky assets. Table 4 shows that the risky share still mildly
increases for ages 31-60 within the retirement accounts.
3
A Simple Portfolio-Choice Theory
Before developing a full-blown life-cycle economy, using a simple 3-period model, we illustrate how portfolio choice is affected by age, labor-market risk, and wealth.
A worker lives for three periods. Each period he receives income yt which is an i.i.d.
random variable with a probability function f (yt ). Preferences are given by
U =E
3
X
β t−1
t=1
ct 1−γ
1−γ
where γ is the coefficient of relative risk aversion and ct is consumption in period t. Two types
of financial assets are available for savings. One is a risk-free bond, bt , that pays a fixed gross
return, R and the other is a stock, st , that pays a stochastic gross return, Rs = R+µ+η, where
µ is the risk premium and η is excess return drawn from a normal distribution of N (0, ση2 ).
The probability density function associated with η is denoted by π(η). On average, the stock
yields a higher rate of return than the bond to compensate for the risk associated with η:
11
µ > 0.
Current income is divided between consumption, c, and savings, b0 + s0 . It is convenient
to collapse total wealth into a single state variable W = bR + sRs . Borrowing is not allowed
for each investment (b ≥ 0 and s ≥ 0). The present value of utility in period j, Vj , can be
written recursively when the next period’s value is denoted with a prime (0 ):
Vj (W, y) = max
0 0
c,s ,b
c1−γ
+β
1−γ
s.t.
Z Z
η0
Vj+1 (W , y )df (y )dπ(η )
0
0
0
0
j = 1, 2, 3
y0
c + s 0 + b0 = W + y
c ≥ 0,
s0 ≥ 0,
b0 ≥ 0
where V4 (·, ·) = 0.
Case 1: No labor income (Samuleson Rule).
Under the CRRA preferences, with no labor income in the future, a worker allocates savings
according to the constant share between risky and safe assets, the so-called Samuelson (1969)
Rule, so that the risky share is:7
1 µ
s0
≈
0
0
s +b
γ ση2
This rule is intuitive. The risky share (i) increases in the risk premium, µ, (ii) decreases in
the risk aversion, γ, and (iii) decreases with the risk of stock returns, ση . According to this
rule, the wealth, W , and investment horizon (age), j, are irrelevant for the portfolio decision,
inconsistent with advice often provided by financial analysts.8 The risky share is independent
of wealth because of CRRA preferences. While the longer horizon provides an opportunity to
weather the risk in stock returns, the variance of total returns also increases with the horizon.
With CRRA preferences and i.i.d. stock returns the two effects cancel each other so that the
risky share remains independent of the investment horizon.
Case 2: Deterministic labor income
We now illustrate how labor-market uncertainty affects the risky share in a three-period
example. In this example, the first period corresponds to “Young” worker, the second to the
7
“No labor income” refers to the case where tomorrow’s income y 0 = 0. Today’s labor income y is a part
of “cash in hand,” W + y.
8
Many financial analysts advise investors with longer investment horizons (i.e., young investors) to take
more risk. As Ameriks and Zeldes (2004) note, this advice is typically based on two observations: (i) Over
time stocks provide better returns than bonds. (ii) With a longer time horizon, investors have more time to
weather the ups and downs of the stock market.
12
Figure 3: Risky Share: Deterministic Labor Income
Risky Share of Financial Assets
Young
Old
100
90
Percent
80
70
60
50
40
0
1
2
3
Wealth
4
5
6
0
Note: Risky Share ( s0s+b0 ) for “Young” and “Old”.
“Old” worker, and the last to “Retired.”
First, consider the case where labor income is deterministic (y > 0, σy2 = 0) so that
0
there is no uncertainty in the labor-market outcome. Figure 3 plots the risky share ( s0s+b0 ) of
“Young” and “Old” for various levels of wealth. For both “Young” and “Old,” the risky share
decreases with wealth, completely opposite to what we saw in Section 2. When the labor
income is deterministic, having a job is equivalent to holding a risk-free fixed-income asset.
A worker with little wealth, because risk-free labor income makes up a large portion of total
wealth, would like to allocate most of his savings to risky investments. In fact, according to
the optimal policy function, whether young or old, a wealth-poor worker, whose W is close
to 0, allocates all his savings to stocks. As wealth increases, the risky share decreases. When
wealth is large relative to labor income (where labor income becomes a negligible portion of
total wealth), the risky share converges to the value implied by the Samuelson Rule.
The risky share decreases with age, again opposite to what we saw in the data. Since
the young anticipate a longer stream of deterministic labor income (for the two remaining
periods)–which is equivalent to holding a fixed-income asset, they are willing to take more
risk in their financial investments. Figure 3 shows that this is true for any given level of
wealth, unless the wealth is close to zero where the risky share is 100% for both young and
old. In sum, with no uncertainty in the labor market, the risky share decreases with age and
wealth, both of which are opposite to what we find in the SCF.
13
Figure 4: Risky Share: Stochastic Labor Income
No Labor−Market Risk
Young
Old
100
100
90
90
80
80
Risky Share (%)
Risky Share (%)
Low Labor−Market Risk
70
60
50
70
60
50
40
40
30
30
20
0
1
2
3
Wealth
4
5
20
0
6
1
100
100
90
90
80
80
70
60
50
30
3
Wealth
4
6
5
6
50
30
2
5
60
40
1
4
70
40
20
0
3
Wealth
High Labor−Market Risk
Risky Share (%)
Risky Share (%)
Medium Labor−Market Risk
2
5
20
0
6
1
2
3
Wealth
4
0
Note: Risky shares ( s0s+b0 ) of “Young” and “Old” for four values of labor-market variance (σy ).
Case 3: Stochastic labor income
We now consider the case where labor income is stochastic. Figure 4 shows the optimal risky
share for 4 different values of labor income risk, σy : zero (deterministic), low, medium and
high. As the labor-income risk increases, a worker becomes less willing to make risky financial
investments. The risky share declines for both young and old. Now, the large uncertainty in
the labor market discourages workers making further risky financial investments. With a high
enough labor-market risk (the last panel in Figure 4), (i) the young’s risky share is lower than
the old’s, and (ii) it is increasing in financial wealth. Young investors, on average wealth poor,
are highly exposed to risk, since their income consists mainly of highly volatile labor income.
This example clearly illustrates that labor-market uncertainty is crucial for the relationship
between the risky share and investors’ age and financial wealth.
14
4
4.1
Life-Cycle Model
Economic Environment
Demographics The economy is populated by a continuum of workers with total measure of
one. A worker enters the labor market at age j = 1, retires at age jR , and lives until age J.
There is no population growth.
Preferences
Each worker maximizes the time-separable discounted life-time utility:
U =E
J
X
δ
j=1
j−1 cj
1−γ
1−γ
(1)
where δ is the discount factor, cj is consumption in period j, and γ is the relative risk aversion.9 For simplicity, we abstract from the labor effort choice and assume that labor supply
is exogenous when employed.
Earnings Profile
We assume that the log earnings of a worker i with age j, Yji , is:
Yji = log zj + yji
with yji = ai + β i × j + xij + εij .
(2)
Earnings consist of common and individual-specific components. The component, zj , represents the average age-earnings profile, which is assumed to be the same across workers and
thus observable. The individual-specific component consists of deterministic (ai + β i × j)
and stochastic (xij + εij ) components. The deterministic component represents the individualspecific intercept (ai ) and growth rate (β i ) of the profile. The intercept of the earnings profile,
ai , is distributed across the population by ai ∼ N (0, σa2 ). The growth rate, β i is distributed
by β i ∼ N (0, σβ2 ). The stochastic component represents persistent (xij ) and purely transitory
(εij ) income shocks. The persistent component of the stochastic income shock, xij , follows an
AR(1) process:
(3)
xij = ρxij−1 + νji ,
with νji ∼ i.i.d. N (0, σν2 )
where the transition probability is represented by a common finite-state Markov chain Γ(xj |xj−1 ).
The transitory component follows εij ∼ i.i.d. N (0, σε2 ), where the probability distribution of
ε is denoted by f (ε). For simplicity, we will omit superscript i from now on. Workers are
9
Alternative preferences have also been proposed to address the portfolio choice puzzles. For example,
Gomes and Michaelides (2005) use Epstein-Zin preferences with heterogeneity in both risk aversion and intertemporal elasticity of substitution and Wachter and Yogo (2010) use non-homothetic preferences. We adopt
the standard preferences with constant relative risk aversion in order to highlight the role of labor-market
uncertainty.
15
assumed to have imperfect knowledge about their own individual-specific components. A
worker starts with a prior about a, β, and x, and updates these priors each period based on
the realized value of total earnings. Thus, young workers on average face larger uncertainty
about future earnings. This uncertainty is resolved over time as he gradually learns about his
earnings ability based on successive realizations of labor income.
Unemployment Risk
Each period workers face age-dependent unemployment risk. The
probability of being unemployed for the worker of age j is puj . For simplicity, we assume that
unemployment risk is not individual-history dependent.
Savings
Financial markets are incomplete in two senses. First, workers cannot borrow.
Second, there are only two types of assets for savings: a risk-free bond b (paying gross return
R in consumption units) and a stock s (paying Rs = R + µ + η) where µ (> 0) represents the
risk premium and η is the stochastic rate of return.10 Workers save to prepare for retirement
(life-cycle savings) and to ensure themselves against labor-market uncertainty (precautionary
savings).
Social Security The government runs a balanced-budget pay-as-you-go social security system. When a worker retires from the labor market at age jR , he receives a social security
benefit amount, ss, which is financed by taxing workers’ labor incomes at rate τss .11
Learning
In our benchmark model, workers do not have perfect knowledge about their
lifetime profile of earnings. While the non-deterministic part of labor income, y, is observed,
workers cannot perfectly distinguish between ability (a and β) and luck (x and ε). Instead,
they update their priors in a Bayesian fashion. Given the normality assumption, a worker’s
prior belief is summarized by the mean and variance of intercept, {µa , σa2 }, and those of slope,
{µβ , σβ2 }. Similarly, a worker’s prior belief about the persistent component of the income shock
is summarized by {µx , σx2 }. When the prior beliefs over the covariances are denoted by σax
,σaβ , and σβx , we can express the prior mean and variance as:
Mj|j−1
2
µa
σa σaβ σax
= µβ
Vj|j−1 = σaβ σβ2 σβx
µx j|j−1
σax σβx σx2 j|j−1
(4)
10
For simplicity, we abstract from the general equilibrium aspect by assuming exogenous average rates of
returns to both stocks and bonds, and the labor supply decision.
11
Ball (2008) analyzes financial investments for different levels of the social security benefit. He finds that
the generosity of the social security system has little effect on portfolio choice.
16
where the subscript j|j − 1 denotes the information at age j before the actual earning yj is
realized. The subscript j|j denotes the information after earnings yj is realized, i.e., posterior.
When a worker exactly knows his ability (a and β), the uncertainty about future earnings
reflects the uncertainty in the stochastic component (x and ε) only. However, when a worker
does not have perfect knowledge about his ability, uncertainty about labor income includes the
beliefs about his ability as well as the stochastic component. According to our calibration of
the income profile, which is based on the estimates in the literature, income uncertainty over 15 periods ahead is resolved extremely fast, under the standard Bayesian updating formula. For
example, within 3 years of a worker entering the labor market, almost 90% of the uncertainty
about the next period’s income is resolved. We argue that learning at this rate is not realistic,
especially considering the frequent job changes of young workers in search of a good match.
Typically, workers shop around for jobs before settling into a long-term career. According to
Topel and Ward (1992), in the first 10 years entering the labor market, a typical worker holds
7 jobs (about two-thirds of his career total). Kambourov and Manovskii (2008) report that the
average probability that a young worker will switch occupations is approximately 25% on an
annual basis. Frequent job switching may diminish the informational context from previous
labor-market experience, especially if it takes place across completely different industries or
occupations. To achieve a more realistic speed of learning, we extend the standard Bayesian
formula to include noise in the updating rule. One interpretation of such noise in the updating
rule is a “reset” of priors, reflecting possible information loss due to occupational/industrial
change.12
Specifically, when q denotes this noise in the priors updating rule, the posterior means and
variances at age j are given by:
Mj|j
= Mj|j−1 +
σa2 +σaβ +σax
σa2 +σβ2 j 2 +σx2 +σε2 +Γ
σaβ +σβ2 j+σβx
2
σa +σβ2 j 2 +σx2 +σε2 +Γ
σax +σxβ j+σx2
σa2 +σβ2 j 2 +σx2 +σε2 +Γ
Vj|j = Vj|j−1 −
a
qj
β
0
(yj − Hj Mj|j−1 ) + qj
qjx
σa2 +σaβ +σax
2
σa +σβ2 j 2 +σx2 +σε2 +Γ
σaβ +σβ2 j+σβx
σa2 +σβ2 j 2 +σx2 +σε2 +Γ
σax +σxβ j+σx2
2
σa +σβ2 j 2 +σx2 +σε2 +Γ
(5)
0
Hj Vj|j−1 + Qj
(6)
where Hj = [ 1 j 1 ]0 is a (3 × 1) vector and Γ = 2σaβ j + 2σax + 2σβx j. The parameter
12
It is often found in various natural experiments that subjects are more conservative in changing their
priors than the standard Bayesian rule implies. For example, according to Edwards (1968) “ ... it takes between
2 to 5 observations to make a subject change her prior beliefs to the same extent as what one observation
would do for a Bayesian learner.”
17
qjκ for κ ∈ {a, β, x} captures the additional noise in the updating rule. We assume qjκ ’s are
distributed as qj ∼ N (0, Qj ) where qj is the (3 × 1) vector on the right-hand side of (5),
and the variance-covariance matrix of noise Qj is a (3 × 3) matrix in which non-diagonal
elements are zero (mutually independent from each other). We assume that the noise shock
occurs after the update has taken place. Given that on expectation the shock will be zero,
qjκ will not affect the posterior of means at the time of the update, but only the posterior of
the variances. Our formulation is computationally convenient because we do not need qjκ as
an additional state variable: it is included in the age j. In Appendix C, we formally derive
these equations (5) and (6). Using the above formula, the next period’s income follows the
conditional distribution function:
F (yj+1 |yj ) = N (H0j+1 Mj+1|j , H0j+1 Vj+1|j Hj+1 + σε2j )
(7)
where
Mj+1|j = R Mj|j−1 +
σa2 +σaβ +σax
2
σa +σβ2 j 2 +σx2 +σε2 +Γ
σaβ +σβ2 j+σβx
σa2 +σβ2 j 2 +σx2 +σε2 +Γ
σax +σxβ j+σx2
2
σa +σβ2 j 2 +σx2 +σε2 +Γ
(yj − H0j Mj|j−1 )
Vj+1|j = RVj|j R0 + S
with R denoting a (3 × 3) matrix whose diagonal elements are (1, 1, ρ) and S denoting the
i
].
covariance matrix of a shock vector, [ 0 0 νj+1
Let k = {e, u} denote the employment status of a worker: employed or unemployed. As
in our illustrative example in Section 3, we collapse the total financial wealth into one state
variable W = bR + sRs . The value function of a worker at age j is:
Vjk (W, y, Mj|j−1 )
Z Z
e
k
u
= max u(c ) + δ(1 − pj )
Vj+1
(W 0 , y 0 , Mj+1|j )dFj (y 0 |y)dπ(η 0 )
ck ,s0 ,b0
η0
y0
(8)
+ δpuj
s.t.
Z Z
η0
y0
u
0 0
0
0
Vj+1 (W , y , Mj+1|j )dFj (y |y)dπ(η )
ck + s0 + b0 = (1 − τss ) expYj ×1{k = e} + ss × 1{j ≥ jR } + W
(9)
where 1{·} is an indicator function, and Yj = log zj + yj . Note that every period, both
employed and unemployed get a draw of y, which will affect the next period’s y 0 through
Fj (y 0 |y). However, a fraction puj of employed workers will lose their job (k = u). The state
variables include workers’ wealth, W , the individual-specific component of labor income, y,
18
and the prior mean (Mj|j−1 ). Note that the prior about the second moment (Vj|j−1 ) is not
explicitly included in the value function, since the age (j) is a sufficient statistic.13
Perfect Information Model
In order to evaluate the marginal contribution of each component of labor-market uncertainty on portfolio choice, we consider various specifications
differing with respect to assumptions about (i) the knowledge about income profile, (ii) the
age-dependent unemployment risk, and (iii) noise in the updating rule. The first alternative
specification we consider is the standard life-cycle model without any of these three features.
This specification is very similar to Cocco, Gomes, and Maenhout (2005). We will refer to
this specification as the perfect information model (PIM). In this case, the value function of
a j-year-old worker with an income profile of {a, β} is:
{a,β}
Vj
(W, x, ε)
s.t.
Z
= max
u(c) + δ
0 0
c,s ,b
η 0 ,x0 ,ε0
{a,β}
Vj+1 (W 0 , x0 , ε0 )df (ε0 )dΓ(x0 |x)dπ(η 0 )
(10)
c + s0 + b0 = (1 − τss ) expYj +ss × 1{j ≥ jR } + W.
The second alternative specification we consider is the perfect information model with agedependent unemployment risk, which is referred to as the PIM with unemployment risk.
Finally, we consider the benchmark model without noise in updating rule (i.e., benchmark
with the standard Bayesian updating rule): Benchmark without noise.
4.2
Calibration
We calibrate the model to be consistent with the income process and dispersion of consumption in the data. There are six sets of parameters: (i) life-cycle parameters, {jR , J}, (ii) preferences {γ, δ}, (iii) asset market structure {R, µ, ση2 }, (iv) labor income process {zj , ρ, σa2 , σβ2 , σε2 , σν2 },
(v) noise in the priors updating rule, {V1|0 , Qj }, and (vi) the social security system {τss , ss}.
Unless stated otherwise, we keep the parameter values constant across the models. Table 5
reports all parameter values for the benchmark case.
Life Cycle, Preferences, and Social Security
The model period is one year. Workers are
born and enter the labor market at age 21 and live for 60 periods, J = 60, which corresponds
to ages 21-80. Workers retire at age 65, jR = 45, when they start receiving the social security
benefit, ss. The social security tax rate τss = 20% targets a replacement ratio of 40% for an
13
For simplicity, the unemployment spell does not interrupt the learning process in our model; thus, the
speed of learning depends on age only, not individual-specific component. Making the learning individualemployment-history dependent will expand the state space enormously.
19
average productivity worker. The relative risk aversion, γ, is set to 5 to match the average
risky share in our benchmark specification. This value is much lower than those typically
adopted to match the average risky share in the literature. The discount factor, δ = 0.94,
is calibrated to match the wealth-to-income ratio of 3.5, the value commonly targeted in the
literature.14 Matching the wealth-to-income ratio is important for the quantitative analysis of
portfolio choice. As documented in previous studies such as Storesletten, Telmer, and Yaron
(2004), and Ball (2008), if the average asset holdings are too high relative to labor earnings,
the labor-income risk becomes negligible. As shown in our illustrative examples in Section 3,
the risky share converges to the value implied by the Samuelson Rule for an investor with a
sufficiently large amount of wealth.
Asset Returns
The rate of return to the risk-free bond R = 1.02 is based on the average
real rate of returns to 3-month US Treasury bills for the post-war period. Following Gomes
and Michaelides (2005), we set the equity premium µ to 4%. The standard deviation of the
innovations to the rate of return to stock ση is 18%, again, based on Gomes and Michaelides
(2005).15 We assume that the stock returns are orthogonal to labor-income risks.16
Unemployment Risk Based on the CPS for the period 1976-2013, Choi, Janiak, and VillenaRoldan (2014) estimate the transition rates from employment to unemployment over the life
cycle. Figure 5, reproduced based on their estimates, clearly shows that the annual probability of moving to being employed this period to unemployment next period monotonically
decreases with age. For example, a 21-year-old worker faces a 3.5% chance of moving from
being employed to unemployed next year, whereas a 64-year-old worker faces a much smaller
risk, less than 1%. We use these estimates for the age-dependent unemployment risk, puj .
Labor-Income Process
For the stochastic process of labor-income shocks, we use the estimates of Guvenen and Smith (2014). The dispersion of the individual-specific growth rate is
σβ2 = 0.03%. The AR(1) coefficient and the variance of innovation for the persistent income
shocks are ρ = 0.756 and σν2 = 5.15%. The variance of the i.i.d. component is σε2 = 1%.
We choose the dispersion of the intercept, σa2 = 13%, to match the cross-sectional variance of
14
In the perfect information model (PIM) we set δ = 1.101. In this case, the model requires a large discount
factor to match the wealth-to-income ratio observed in the data because (i) the precautionary savings motive
against labor-market uncertainty is small and (ii) an increasing profile of earnings induces workers to borrow
heavily early in life.
15
Jagannathan and Kocherlakota (1996) report that for the period between 1926 and 1990, the standard
deviation of annual real returns in the S&P stock price index was 21% as opposed to 4.4% in T-bills.
16
The empirical evidence on the correlation between labor-income risk and stock market returns is mixed.
While Davis and Willen (2000) find a positive correlation, Campbell, Cocco, Gomes, and Maenhout (2001)
find a positive correlation only for specific population groups.
20
Figure 5: Unemployment Risk over the Life Cycle
Unemployment Probability
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
20
30
40
50
60
70
Age
Note: Estimates from Choi, Janiak, and Villena-Roldan (2014).
log consumption of young workers (age 25) in the data, as calculated by Guvenen (2007).17
The deterministic earnings profile, which is common across workers, zj , is taken from Hansen
(1993). This parameterization leads to an increasing profile of variance in log-wages between
ages 25 to 55, by approximately 0.35 point, fairly consistent with Heathcote, Storesletten, and
Violante (2014) and Guvenen and Smith (2014). We assume that workers do not have any
prior knowledge regarding their income profile upon entering the labor market. We view this
assumption as a useful starting point to determine the effect of labor-market uncertainty on
investment behavior. In Section 5, we consider the model specification where workers know to
some degree their lifetime income profile and calibrate the amount of prior uncertainty based
on Guvenen (2007) and Guvenen and Smith (2014). We show that our results remain largely
intact, even with smaller amounts of prior uncertainty.
Speed of Learning We introduced additional noise in the priors updating rule to generate a
more realistic gradual resolution of uncertainty. We assume that the noise, q, is drawn from
the distribution qj ∼ N (0, Qj ). We also assume that the variance of noise Qj , is a linearly
decreasing function of age: Qj = λj V1|0 where λj = λ + JR−λ−1 (j − 1) and V1|0 is the initial
17
Studies like Deaton and Paxson (1994) and Storesletten, Telmer, and Yaron (2004) document a large
increase in the variance of log-consumption around 30 points. Other studies like Heathcote, Storesletten, and
Violante (2014) and Guvenen and Smith (2014) report a smaller rise, around 10 points, and a lower variance
at age 25. As a result Guvenen and Smith (2014) find a smaller value for σa2 . We choose to follow the data
estimates of Guvenen (2007) which basically fall between the aforementioned studies. We show that our results
are fairly robust to values for σa2 consistent with Guvenen and Smith (2014).
21
prior variance of ability parameters (a and β). The variance of noise starts from λ× V1|0 ,
decreases over time, and converges to 0 by the time a worker retires from the labor market.
We interpret q as a reset of priors about the earnings profile. Belief resetting may be
particularly frequent among young workers, who hold a number of short-term and part-time
jobs before settling into a long-term career. As a result, we link the frequency of belief resetting
to occupational mobility patterns. According to Topel and Ward (1992), the average number
of jobs held by workers within the first 10 years of entering the labor market is 7. Kambourov
and Manovskii (2008) estimate that the average probability that workers between the ages of
23-28 will switch occupations is 0.26 for non-college-educated workers and 0.23 for those with
some college education. Our parameterization suggests that λj is on average 0.25 during the
first 10 years in the labor market. This implies that for the first 10 years, workers reset their
priors by 25% of their initial priors (or switch occupations with probability 25%). This is
equivalent to workers having had 2.5 jobs in the first 10 years after entering the labor market.
One way to judge how fast income uncertainty is resolved over time is to check how
fast the cross-sectional dispersion of consumption increases over the life cycle. As we will
show in Section 5, our benchmark model closely tracks the age profile of the variance of log
consumption in the data, as documented by Guvenen (2007).
5
5.1
Results
Policy Functions
In order to understand the basic economic mechanism of the model, we first compare
the optimal portfolio choice (policy functions) of two specifications of the model: the perfect
information model (PIM) and the benchmark. Figure 6 shows the optimal financial portfolio
choice of a worker with median income at ages 25, 45, and 65, respectively. We show the
risky share for a wide range of financial wealth (from 0 to 20). In our benchmark case, the
average wealth is 7.02. In the PIM (left panel), consistent with our illustrative example in
Section 3, the risky share decreases with both wealth and age. When the earnings ability
is perfectly known, workers can predict future labor income (to the extent of the stochastic
variation due to income shocks) fairly well. Poor workers at all three ages—25, 45, and 65—for
whom (the relatively safe although stochastic) labor income makes up a large portion of total
wealth, would like to invest their financial wealth in risky assets. This is also true for younger
workers, who expect a long, relatively safe labor-income stream. For example, a 25-year-old
worker with median labor income and average wealth level (about 7 in our model) would like
to allocate almost all financial wealth to risky assets. In sum, a young and asset-poor worker
exhibits a higher risky share when the earnings profile is perfectly known.
22
Table 5: Benchmark Parameters
Parameter
Notation
Value
Target / Source
Life Cycle
Retirement Age
Risk Aversion
Discount Factor
Risk-free Rate
Equity-Risk Premium
Stock-Return Volatility
Social Security Benefit
Social Security Tax
Variance of Fixed Effect
Variance of Wage Growth
Variance of Transitory Component
Persistence Parameter
Variance of i.i.d. component
Common Age-Earnings Profile
Unemployment Risk
J
jR
γ
δ
R
µ
ση
ss
τss
σa2
σβ2
σν2
ρ
σε2
{zj }65
j=21
{puj }65
j=21
60
45
5
0.94
1.02
0.04
0.18
0.40
0.20
13%
0.030%
5.15%
0.756
1.0%
–
Figure 5
λ
0.3
–
–
Average Risky Share =0.455
Capital-Output Ratio=3.5
Gomes and Michaelides (2005)
Gomes and Michaelides (2005)
Gomes and Michaelides (2005)
Replacement Ratio
Balanced Social Security Budget
Consumption Variance for Age 25
Guvenen and Smith (2014)
Guvenen and Smith (2014)
Guvenen and Smith (2014)
Guvenen and Smith (2014)
Hansen (1993)
Choi, Janiak, and Villena-Roldan (2014)
Topel and Ward (1992),
Kambourov and Manovskii (2008)
Initial Variance of Noise
23
Figure 6: Optimal Portfolio Choice for a Worker with Median Income
Perfect Information Model
100
Age 65
Age 45
Age 25
90
80
80
70
70
60
50
40
60
50
40
30
30
20
20
10
10
0
5
10
Financial Assets
15
Age 65
Age 45
Age 25
90
Risky Share (%)
Risky Share (%)
Benchmark
100
0
20
5
10
Financial Assets
15
20
In our benchmark model (the right panel) where the income profile is uncertain, old workers
are willing to take a lot of risk in making financial investments. The optimal portfolio choice
of a 65-year-old worker, who retires next period, is similar to that of the PIM because the
labor-income risk is pretty much irrelevant for him. A 45-year-old worker with an average
amount of assets holds a risk share around 42%; hence, he is still cautious in making risky
financial investments. A 25-year-old worker, who entered the labor market 4 years ago, faces
two conflicting incentives for taking risk in financial investments. On the one hand, he would
like to hedge against the large labor-market uncertainty. On the other hand, he would like to
build up wealth quickly by taking advantage of the higher average rate of returns to stocks
(life-cycle savings motive). According to our benchmark calibration, the wealth effect on risky
share is not monotonic. The risky share increases with wealth when the wealth level is close
to 0, indicating that the life-cycle savings motive dominates the desire to hedge against labormarket uncertainty for wealth-poor workers. The risky share declines with assets after the
level of wealth exceeds 0.75 (one-tenth of the average wealth).
5.2
Comparison to Survey of Consumer Finances
We now generate statistics from a simulated panel of 10,000 workers from each model and
compare them to those from the SCF. Table 6 presents the basic statistics regarding the three
portfolio choice puzzles discussed in Section 2. The table reports the average risky share,
the growth rate of average risky share by age, and the correlation between the risky share
24
and wealth from two model specifications: the benchmark and the perfect information model
(PIM). In both models we use the same values for all other parameters.
Table 6: Risky Shares: Data vs. Models
Statistic
Average risky share
Slope of profile (pp)
Correlation with wealth
Data
PIM
Benchmark
45.5% 78.8%
0.92 –1.24
0.107 –0.638
45.1%
0.36
–0.147
Note: The slope of the age profile refers to the average increase of the risky share (in percentage
points) over the life cycle (ages 21 to 65). PIM refers to the perfect information model. The
0
correlation with wealth is based on Corr(W, s0s+b0 ). The data statistics are based on the SCF.
Our benchmark model matches the risky share in the data with the relative risk aversion
of 5: 45.5% in the data and 45.1% in the benchmark model. In the PIM, which is similar
to the standard life-cycle model without uncertainty about income profile and unemployment
risk, this ratio is 78.8%. To match the average risky share of 45.5% in the data, the PIM
requires a value of relative risk aversion above 15 under the same parameterization of the
income process. But even in this case, the PIM fails to generate an increasing profile of risky
share over the life cycle.
We next turn our attention to the age profile. Financial advisors often recommend that
young investors, facing a longer investment horizon, take more risk in financial investments.
However, our data based on the SCF show that the risky share on average increases by 0.92
percentage point each year between ages 21 and 65 (Table 6). Our benchmark model successfully reproduces an increasing—but to a lesser degree than that in the data—age profile of
risky share. In our benchmark model, on average, the risky share increases by 0.36 percentage
point. Young workers, faced with much larger uncertainty in the labor market, would like
to avoid too much risk in making financial investments. As the labor-market uncertainty is
gradually resolved over time, through updating priors about their ability and decreasing unemployment risk, they start taking more risks. By contrast, the PIM generates a risky-share
profile that decreases on average by 1.24 percentage points each year between ages 21 and 65.
Figure 7 plots the age profile of the risky share from both models. In the PIM, the risky
share (left panel) starts with 100% at age 21, monotonically declines to 77.2% at age 45, and
to 44.6% at age 65. In our benchmark model, however, the age profile of the risky share is
not monotonic. It starts with a low level of 36.3% at age 21, quickly increases to 52.4% at
age 24, decreases to 40.6% at age 38, and gradually increases to 52.0% at age 65. This is
25
Figure 7: Risky Share over the Life Cycle: Model
Perfect Information Model
Data
Model (PIM)
100
90
90
80
80
70
70
60
50
40
60
50
40
30
30
20
20
10
10
0
20
30
40
50
60
Data
Model (Bench)
100
Risky Share (%)
Risky Share (%)
Benchmark
0
20
70
Age
30
40
50
60
70
Age
Note: Left panel shows the total asset, bond, and stock holdings as well as risky shares for
the perfect information model (PIM). Right panel show the benchmark model.
because a young worker faces two conflicting incentives to take risks in making investments.
On the one hand, he would like to hedge against the large labor-market uncertainty. On the
other hand, he would like to build up his savings (life-cycle savings motive) quickly by taking
advantage of the risk premium. When the worker enters the labor market, the former effect
dominates, suppressing the risky share, then quickly the latter (life-cycle savings) effect comes
in, inducing him to take some risks for a while until he accumulates some assets. Overall, our
model roughly tracks the age profile of the risky share in the SCF. We view this as a partial
resolution in reconciling the tension between the data and theory about households’ portfolio
choice over the life cycle.
The third portfolio-choice puzzle is the correlation between the risky share and wealth.
According to the SCF, the total amount of financial assets and the risky share are weakly
positively correlated across households, with a correlation coefficient of 0.107: wealthy households tend to take more risk in making financial investments. In the PIM, this correlation is
strongly negative, with a correlation coefficient of -0.638. A decreasing age profile has contributed to this large negative correlation between wealth and risky share. Our benchmark
model partially corrects for this large discrepancy between the data and the standard model
and the correlation between the risky share and wealth becomes mildly negative: -0.147.
Young workers (on average, asset-poor) would like to hedge the labor-market uncertainty by
investing more conservatively. Table 7 reports the average risky shares for 5 wealth groups. In
26
the data, the average risky share gradually increases from 35.9% in the first quintile to 66.6%
in the 5th quintile. By contrast, in the PIM, the risky share monotonically decreases from
99% in the first quintile to 50% in the 5th. According to our benchmark model, the risky
share mildly decreases with wealth: 50% in the first to 41% in the 5th quintiles.
Table 7: Risky Share by Wealth
Wealth Quintile
Data
PIM
Benchmark
1st
2nd
3rd
4th
5th
35.9%
40.5%
44.4%
51.7%
66.6%
98.9%
95.1%
85.6%
67.2%
49.2%
50.0%
45.4%
44.6%
45.1%
41.0%
Average
45.5%
78.8%
45.1%
In the data, the positive correlation between the risky share and wealth still holds within
all age groups. Figure 8 shows the average risky shares across wealth quintiles conditional on
age (by age groups of 21-30, 31-40, 41-50, 51-60, and 61-65). In the data (solid), the risky
share increases with the wealth within each age group. By contrast, the PIM (with ♦) predicts
that the risky share monotonically decreases with wealth in all age groups. A poor worker—
for whom labor income makes up a large portion of total wealth—would like to invest a larger
portion of his financial wealth in risky assets. This is particularly true for young workers, ages
21-30, who allocate almost all of their financial wealth to risky assets. The benchmark model
(with ), however, cannot match this pattern, as the risky share conditional on age is largely
uncorrelated with wealth.
While we are mostly concerned about the first moment of the portfolio across households,
that is, the average risky share over the life cycle, it is also informative to see how the entire
distribution evolves with age. Figure 9 shows the cross-sectional distributions of the risky
share for three 15-year groups, ages 21-35, 36-50, and 51-65. In the data (left panels), while
the average risky share increases mildly, from 41.1% to 46.0%, and to 48.4%, respectively,
for ages 21-35, 36-50 and 51-65, the portfolio choice is widely dispersed across all three age
groups. The PIM (middle panels) generates a pattern of distribution by age that exhibits a
sharp change in the skewness of distribution. On the other hand, our benchmark model (right
panels) generates a much less skewed distribution along with a mild change in mean across
all three age groups. Obviously, the portfolio choices are much more widely dispersed in the
data: the risky share ranges from zero to one in all three age groups in the SCF. Even though
27
Figure 8: Risky Share and Wealth by Age Group
our model assumes heterogeneity in earnings profile, it fails to generate enough dispersions in
portfolio distribution because it abstracts from many other important sources of heterogeneity
such as inheritance, discount rate, and risk aversion, among others.
28
Figure 9: Distribution of Risky Share across Age Groups
Data − Age 21−35
PIM − Age 21−35
mean=97.9%
mean=45.5%
Distribution
mean=41.1%
Benchmark − Age 21−35
0
20
40
60
80
Risky Share (%)
100
0
20
Data − Age 36−50
40
60
80
Risky Share (%)
100
0
PIM − Age 36−50
40
60
80
Risky Share (%)
100
Benchmark − Age 36−50
mean=81.8%
mean=41.6%
Distribution
mean=46.0%
20
0
20
40
60
80
Risky Share (%)
100
0
20
Data − Age 51−65
40
60
80
Risky Share (%)
100
0
PIM − Age 51−65
40
60
80
Risky Share (%)
100
Benchmark − Age 51−65
mean=58.2%
mean=48.4%
Distribution
mean=48.4%
20
0
5.3
20
40
60
80
Risky Share (%)
100
0
20
40
60
80
Risky Share (%)
100
0
20
40
60
80
Risky Share (%)
100
Dispersion of Consumption by Age
It is a stylized fact that the cross-sectional dispersion of consumption increases over the
life cycle. For example, Deaton and Paxson (1994) find that the variance of log consumption
29
increases from 0.23 at age 25 to 0.5 at age 65. Storesletten, Telmer, and Yaron (2004) report a
similar profile. Other studies find a lower, but still increasing shape. Guvenen (2007) reports
that the variance of log consumption increases from 0.13 at age 25 to 0.4 at age 65. Guvenen
and Smith (2014) and Heathcote, Storesletten, and Violante (2014) find the variance to be
approximately 0.10 at age 25 to 0.20 at age 55. We follow the estimates of Guvenen (2007),
which basically fall between the aforementioned studies. As part of our sensitivity analysis,
we use a lower value of σa2 that matches the lower estimates of Guvenen and Smith (2014) and
Heathcote, Storesletten, and Violante (2014).
Figure 10 exhibits the age profile of the variance of log consumption in the data and those
from the two models (PIM and benchmark). As Guvenen (2007) points out, a model without
uncertainty about income profiles has difficulty generating the (linearly) increasing dispersion
in consumption. For example, a standard life-cycle model with stochastic income shocks
(with high persistence) produces a concave profile of consumption variance.18 By contrast, a
model where workers gradually learn their earnings abilities is able to generate an increasing
age-profile of consumption variance: the cross-sectional dispersion of consumption, which
depends on lifetime income, emerges as workers gradually learn about their earnings abilities.
According to Figure 10 , the age profile of consumption dispersion from the benchmark model
is in line with that in the data, whereas the PIM shows an increasing but concave profile.19
5.4
Decomposing the Contribution of Three Types of Uncertainty
We have introduced three types of labor-market uncertainty into the standard life-cycle
model: (i) age-dependent unemployment risk, (ii) imperfect information about earnings ability,
and (iii) noise in the Bayesian updating rule. In this section we evaluate the contribution of
each element by considering two additional model specifications. The first additional model
specification to consider is the PIM with age-dependent unemployment risk only, referred to
as PIM with unemployment risk. The comparison of this model with the PIM will isolate the
contribution of age-dependent unemployment risk. The second additional model to consider
is the benchmark model without noise in the updating rule, i.e., λ = 0. We call this version
as Benchmark without noise.The comparison of this specification with the benchmark will
provide a marginal contribution of noisy updating in matching the risky share in the data.
Table 8 summarizes types of labor-market uncertainty in our various specifications. Table 9
reports the average risky share as well as those by wealth quintile. Figure 11 shows the age
18
For example, the life-cycle model with an idiosyncratic income process similar to Storesletten, Telmer,
and Yaron (2004).
19
In our calibration we adopted the estimates of Guvenen and Smith (2014) for the slope of the profile, σβ2 ,
and stochastic income shocks σx2 and σε2 . We then chose σa2 = 0.13 to match the consumption variance of
young workers (initial dispersion in true ability).
30
Figure 10: Cross-Sectional Variance of Log Consumption by Age
Variance of Log−Consumption
0.45
0.4
Data
PIM
Benchmark
0.35
0.3
0.25
0.2
0.15
0.1
25
30
35
40
45
Age
50
55
60
65
Note: The age profile of the data is based on the estimate in Guvenen (2007).
profile of the risky share for all 4 model specifications.20
Table 8: Summary of Various Specifications
Unemployment Risk
Imperfect Information
Noise in Updating
(1): PIM
(2): PIM w/
unemp risk
No
No
No
Yes
No
No
(3): Benchmark (4): Benchmark
w/o noise
Yes
Yes
No
Yes
Yes
Yes
Adding the age-dependent unemployment risk to the PIM slightly decreases the average
risky share from 78.8% to 74.7% (the bottom row of Table 9). Figure 11 shows that the
impact of unemployment risk on risky share is most important for young workers (line with
“4”). For example, a 25-year-old worker who faces a 3% unemployment risk decreases the
risky share from 99.6% to 87.5%. The impact of unemployment risk on the portfolio choice
becomes negligible after age 40 when the annual unemployment risk becomes close to 1%.
Introducing uncertainty about the income profile into this specification (thus moving from
PIM with unemployment risk to Benchmark without noise) further decreases the average risky
20
We recalibrate the discount factor so that all model specifications match the same wealth-income ratio.
31
Table 9: Risky Shares: Various Specifications
Wealth
Quintile
Data
(1): PIM
(2): PIM w/
unemp risk
(3): Benchmark
w/o noise
(4): Benchmark
1st
2nd
3rd
4th
5th
35.9%
40.5%
44.4%
51.7%
66.6%
98.9%
95.1%
85.6%
67.2%
49.2%
80.1%
91.4%
84.7%
67.6%
50.3%
65.7%
54.6%
51.6%
49.9%
42.4%
50.0%
45.4%
44.6%
45.1%
41.0%
Average
45.5%
78.8%
74.7%
52.9%
45.1%
Figure 11: Risky Share Profiles from Models
Risky Share Profiles
100
Data
(1): PIM
(2): PIM w/ u risk
(3): Bench w/o noise
(4): Benchmark
90
80
Risky Share (%)
70
60
50
40
30
20
10
0
20
30
40
50
60
70
Age
Note: “Benchmark” features all three types of labor-market uncertainty: unemployment risk,
imperfect information about the income profile, and noise in updating formula (λ = 0.3).
“PIM w/ u” refers to PIM with unemployment risk. “Benchmark w/o noise” refers to the
benchmark without noise in updating formula (λ = 0). In each model specification, all other
parameters remain the same as those in Benchmark.
share to 52.9% in Table 9. The age profile of the risky share in this specification (with “5”)
shows that the impact of imperfect information is significant across all age ranges, except the
very young (ages 21-23). Large labor-market uncertainty affects the very young, borrowingconstrained investors in two conflicting ways. On the one hand, the uncertainty about labormarket outcomes discourages risky investment. On the other hand, it motivates workers to
32
accumulate assets faster in these early years to insure against adverse labor-income shocks
in the future. Relatively wealthier, very young investors can afford to invest in risky assets
that generate higher returns on average. This explains why imperfect information has a small
impact for ages younger than 25. Although sizable in its magnitude, the benchmark without
noise model cannot fully account for the low risky share. With standard Bayesian learning, the
long-run income risk perceived by workers can be significant, as Guvenen (2007) points out.
However, for portfolio choice, labor-market uncertainty over short horizons is more important
because workers can re-balance their portfolios every period. With standard Bayesian learning,
short-run income risk is quickly resolved, which explains why the benchmark without noise
model can only partially address the puzzles. Adding the noise in the updating rule reduces
the average risky share to a level consistent with the data (45.5%). The most impact takes
place for younger workers since noise is greater during the first years of one’s career. We discuss
more about the difference between short- and long-run income risk in the next subsection.
Looking across wealth quintiles, unemployment risk affects mostly wealth-poor workers.
Combining age-dependent unemployment risk and imperfect information about ability brings
down the risky share in all quintile groups significantly. Finally, the noise in the updating rule
has the most impact on the 1st and 2nd quintile of wealth groups.
Overall, in accounting for the total decrease in average risky share from 78.8% (PIM) to
45.1% (benchmark), (i) age-dependent unemployment risk has contributed 12%, (ii) imperfect
knowledge about the income profile has contributed the most, 65%, and (iii) noise in the
priors-updating rule has contributed about 23%.
5.5
Speed of Learning: Short- vs. Long-run Uncertainty
In order to better match the risky share of young workers in the data, our model requires an
additional noise in the Bayesian updating rule. Without this noise, the short-run uncertainty
about the income profile is resolved extremely fast. To distinguish between short-run and
long-run income risk, we use the forecast-error variance or minimum square error (MSE), as
a measure of uncertainty proposed by Guvenen (2007) and Guvenen and Smith (2014). The
forecast error variance is defined as:
MSEj+s|j =
H0j+s Vj+s|j Hj+s
+
σε2j
with
s
0s
Vj+s|j = R Vj|j R +
s−1
X
Ri SR0s .
i=0
The speed of learning about ability can be measured as the rate at which labor income
risk (measured by the MSE) under imperfect information converges to the one under perfect
information. We find that labor-market uncertainty at different horizons s (e.g., short-run vs.
long-run) resolves at different rates. In particular, we find that under the standard Bayesian
33
Figure 12: One-Period Forecast Error Variance of Income
One−period Forecast Variance
0.2
PIM
Bench w/o noise
Benchmark
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
20
25
30
35
40
45
50
55
60
65
Age
Note: The one-period forecast-error variance about income by age groups for three model
specifications: PIM, benchmark without noise, and benchmark.
updating rule, the short-run uncertainty resolves extremely fast, even though the long-run
uncertainty remains large.
Figure 12 shows the one-period forecast-error income variance, MSEj+1|j , by age, for three
model specifications: PIM, benchmark without noise, and benchmark. The one-period forecast error income variance of the PIM (line with diamonds) reflects the uncertainty due to
stochastic income shocks only (σν2 and σε2 ). Thus, it is not age dependent, by construction.
Without the noise in the updating rule (line with triangles), almost all uncertainty over a
one-period horizon is resolved within a few years. We do not view this as realistic, considering
the frequent job turnover at younger ages and the amount of time it takes for young workers
to settle into a long-term career (e.g., Topel and Ward (1992)). We introduced noise into the
Bayesian updating rule as a short cut to obtain a more realistic rate at which uncertainty
is resolved. The effect can be seen in the line with squares. Uncertainty over a one-period
horizon is significantly larger compared to the PIM and now is resolved at a much slower
rate.21
To elaborate more on this issue, we explore how the noise in updating affects workers’ belief
(perceived probability) about the next period’s income. Figure 13 plots the distributions of
this belief for each of three specifications: (i) small dispersion in the intercept term of the
21
As Guvenen (2007) points out, non-monotonicity occurs due to the growth component β × j. The
variance of this component increases exponentially with age, which can significantly slow down the resolution
of uncertainty. This also takes place with λ̄ = 0.0 but the effect is much smaller.
34
Figure 13: Distribution of Beliefs for Next Period’s Income
σ a2 = 0.08, λ = 0.0
σ a2 = 0.13, λ = 0.0
1.6
σ a2 = 0.13, λ = 0.3
1.6
PIM
Age 21
Age 25
Age 64
1.4
1.6
PIM
Age 21
Age 25
Age 64
1.4
1.2
1.2
1.2
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
−1
−0.5
0
0.5
Log−Income
1
0
−1
−0.5
0
0.5
Log−Income
PIM
Age 21
Age 25
Age 64
1.4
1
0
−1
−0.5
0
0.5
Log−Income
1
Note: Distribution of beliefs (perceived probability) for next period’s log income. The variance
of the intercept in the income profile is σa2 . The variance of noise relative to the initial prior
variance of the income profile is λ̄. PIM represents the probability distribution under perfect
information about the profile.
ability profile: σa2 = 0.08 and λ̄ = 0, (ii) benchmark without noisy updating: σa2 = 0.13
and λ̄ = 0, and (iii) benchmark: σa2 = 0.13 and λ̄ = 0.3. The reason we include the case
with small dispersion in a, σa2 = 0.08, is that this specification is identical to the income
process used in Guvenen and Smith (2014). Note that our benchmark specification adopted
a slightly larger value of σa2 = 0.13 to match a slightly larger initial variance of consumption,
i.e., cross-sectional variance of workers age 25. For comparison, we also plot the probability
distribution for the next period’s income in the PIM, which reflects the variance from the
stochastic income shocks only, and thus does not depend on age and iscommon across three
different specifications. We show the beliefs for three ages: 21 (new entrant), 25 (4 years of
labor-market experience), and 64 (retire next year).
The left panel plots the distribution for σa2 = 0.08 and λ̄ = 0, which corresponds to the
parameterization of Guvenen and Smith (2014). A 21-year-old worker who just entered the
labor market forecasts the next period’s income with the probability distribution shown by
the solid line. In 4 years, when he is 25 years old, his belief about the next period’s income has
shrunk significantly and is almost indistinguishable from that of the PIM. The middle panel
plots the same distributions for σa2 = 0.13 and λ̄ = 0 (benchmark without noise). Again,
without noise in the updating rule, the belief quickly converges to that of the PIM. By age 25
(within 4 years after entering the labor market) the probability distribution becomes almost
35
identical to that of the PIM where the income profile is perfectly known.22 When we introduce
noise into the updating rule (the right panel with σa2 = 0.13 and λ̄ = 0.3), learning becomes
much slower.
We have shown that income uncertainty over short horizons is resolved extremely fast
without the noise in updating. However, this is not true for uncertainty over a longer time
horizon. Figure 14 shows the forecast error variance for all horizons by three age groups: 25,
35, and 45. The left panel compares the forecast error between the benchmark without noise
model (plotted as triangles) and the PIM (plotted as diamonds). The right panel compares the
forecast error between the benchmark (plotted as squares) and the PIM (plotted as diamonds).
In the left panel the difference between the forecast variance is very small over a short time
horizon (1-5 years) but increases significantly over a long time horizon. Uncertainty about the
growth of the profile β translates into substantial income risk, but only over longer horizons.
In the right panel (benchmark) income risk is significantly larger under imperfect information
over both a short and a longer time horizon.
Figure 14: Forecast Error Variance of Future Income: Different Horizon
PIM vs. Benchmark w/o Noise
PIM vs. Benchmark
0.6
0.6
PIM
Bench w/o noise
PIM
Benchmark
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
MSE25+s|25MSE35+s|35
0
20
30
40
Age
MSE45+s|45
50
MSE25+s|25MSE35+s|35
0
20
60
30
40
Age
MSE45+s|45
50
60
Note: Left panel shows the MSEj+s|j for three age groups j = 25, 35, 45 for all horizons s
for two model specifications: PIM and benchmark without noise. Right panel plots the same
graphs for: PIM and benchmark.
This distinction between short- and long-run uncertainty is subtle but important for the
22
The uncertainty is resolved at a faster rate in our model compared to the one using the parameterization
by Guvenen and Smith (2014) because the signal-noise ratio is larger in our model (i.e., a larger σa2 relative
to σν2 and σε2 ).
36
portfolio choice. The lifetime uncertainty about the earnings ability is important for the total
savings, which is well illustrated by Guvenen (2007). However, for the portfolio choice, the
labor-market uncertainty over the short horizon (e.g., one-period forecast-error variance) is
more relevant because workers can adjust their portfolios frequently (e.g., every year in our
model). In other words, for the decision on the total amount of savings, uncertainty at the
longer horizon is important (because it takes a time to build up assets). For the decision on
portfolio choice, uncertainty at the short-run horizon is more relevant because workers are
able to adjust their portfolios frequently.
5.6
Sensitivity Analysis
We perform various sensitivity analyses to see whether our main results are robust with
respect to different parameterizations. In particular, we are concerned with the robustness
in four dimensions. First, we examine the case where workers have some private information
about their ability upon entering the labor market. Second, we consider two alternative values
of relative risk aversion: γ = 3 and γ = 4. Third, we see how the initial distribution of wealth
(the wealth distribution of 21-year-old workers) affects the results. Finally, we consider the
model with a smaller dispersion in the intercept of earnings profiles, σa2 = 0.08, the value used
in Guvenen and Smith (2014) for the direct comparison with Guvenen and Smith (2014). In
each sensitivity analysis, we keep all other parameters of the model the same as those in our
benchmark specification. Table 10 reports the results of these sensitivity analyses.
In our benchmark model we assumed that workers (or firms) are not informed about
their earnings ability upon entering the labor market. This might be too extreme given that
workers might have some private information that econometricians do not have access to.
Indeed, Guvenen (2007) and Guvenen and Smith (2014) find that workers know a significant
fraction of their lifetime income. In our model, the amount of prior knowledge is given by the
matrix:
(1 − ψa )σa2
σaβ
σax
V1|0 =
σaβ
(1 − ψβ )σβ2 σβx
σax
σβx
σx2
where all variances and covariances equal the population moments. In our benchmark we
assume that ψa = 0, ψβ = 0: workers have no more extra information than the econometricians
do, upon entering the labor market. Following Guvenen (2007) and Guvenen and Smith
(2014), we experiment with four sets of values: {ψa = 0, ψβ = 0.60}, {ψa = 0, ψβ = 0.80},
{ψa = 0.60, ψβ = 0.60}, {ψa = 0.80, ψβ = 0.80}.23 The more the worker knows about his
23
Guvenen (2007) and Guvenen and Smith (2014) examine prior uncertainty with respect to σβ2 . Since in
37
Table 10: Risky Shares: Sensitivity Analysis
0
Corr(W, s0s+b0 )
Model
Average Risky Share
Slope of Profile (pp)
Data
45.5%
0.96
0.106
Benchmark
45.1%
0.36
−0.147
ψa
ψa
ψa
ψa
46.0%
46.9%
47.4%
49.4%
0.36
0.38
0.31
0.26
−0.183
−0.206
−0.231
−0.287
γ=4
γ=3
60.0%
84.0%
0.18
−0.26
−0.228
−0.273
Initial Assets = 0.1 × W̄
45.2%
−0.12
−0.191
σa2 = 0.08, λ = 0.3
σa2 = 0.08, λ = 0.0
46.8%
54.3%
0.41
0.40
−0.178
−0.340
= 0, ψβ = 0.60
= 0, ψβ = 0.80
= 0.60, ψβ = 0.60
= 0.80, ψβ = 0.80
Note: The benchmark features ψa = 0, ψβ = 0, γ = 5, σa2 = 0.13, λ = 0.3, and zero initial assets.
income profile upon entering the labor market, the smaller the labor-market uncertainty he
faces. As a result, the average risky share increases compared to the benchmark case. When
a worker knows 80% of his profile ({ψa = 0.80, ψβ = 0.80}), the average risky share increases
to 49.6% compared to the benchmark 45.1%, while the risky share profile is still increasing
(but at a smaller rate). The correlation with wealth becomes more negative: -0.28 compared
to -0.14. In all other cases the results are close to those in our benchmark. As long as
labor-income uncertainty takes a while to resolve, even small amounts of prior uncertainty
can significantly deter young workers from taking financial risk.
The portfolio choice is obviously sensitive to the risk-aversion parameter. In our benchmark
model, the relative risk aversion is 5. As we lower the relative risk aversion to γ = 4 and γ = 3,
the risky share significantly increases to 60.0% and 84.0%, respectively. The increasing pattern
of the age profile remains the same with γ = 4: the risky share increases by 0.18 percentage
point on average. But the age profile shows a mild negative slope with γ = 3: the risky share
decreases with age by -0.26 percentage point on average. The correlation with wealth also
decreases to -0.228 and -0.273, respectively, with γ = 4 and γ = 3.
Young workers enter the labor market with zero assets in our benchmark model. While
our parameterization σa2 is set to a larger value, we also experiment with the prior uncertainty regarding this
parameter.
38
some young workers enter the labor market with significant amounts of debt in practice,
many can borrow or rely on family financing. To reflect this possibility, we assume that young
workers enter the labor market with a small amount of wealth, 10% of the economy-wide
average wealth. While it has little impact on the average risky share (45.2%), it does affect
the age profile of risk share. Now, the profile has a mild negative slope on average: it decreases
by 0.12 percentage point each year.
In our benchmark parameterization, we choose the initial dispersion of ability, σa2 = 0.13,
to match the cross-sectional variance of log consumption of 25-year-old workers in the data
(from Guvenen (2007)). We also experiment with σa2 = 0.08 following Guvenen and Smith
(2014), which targets a slightly smaller consumption dispersion. With the same amount of
relative variance for the noise in the priors-updating rule (i.e., σa2 = 0.08 and λ = 0.3), the
result is similar to that of the benchmark model. The average risky share slightly increases
to 46.8% and the average risky share increases over the life cycle by 0.40 percentage point,
similar to our benchmark. The correlation with wealth slightly decreases to -0.178. Without
noise in the updating rule (σa2 = 0.08 and λ = 0.0), the results change noticeably but they are
similar to our benchmark without noise (σa2 = 0.13 and λ = 0.0), reported in Table 9. Hence,
our choice of a slightly larger σa2 does not have a significant impact on the main results of our
analysis on the portfolio choice.
6
Industry Income Volatility and Risky Share
According to our model, the portfolio choice depends on uncertainty in the labor market
as well as that in the financial market. Our theory predicts that workers in jobs (e.g., industries or occupations) with highly volatile earnings should be conservative in making financial
investments. Testing this implication is not simple because workers also self-select into different industries and occupations with known volatility. Despite this limitation, we examine the
partial correlation between the risky share of household and the labor-income risk of industry.
The SCF provides information about households’ main job including the industry. For
labor-income risk measures, we use the estimates by Campbell, Cocco, Gomes, and Maenhout
(2001) based on the PSID.24 According to these estimates, workers in agriculture face the
largest uncertainty in income with a average variance of income shock of 31.7%, whereas
those in public administration face the smallest variance, 4.7%.25
24
The income specification used by Campbell, Cocco, Gomes, and Maenhout (2001) is log(Yit ) = f (t, Zi,t )+
νi,t +εi,t where f (t, Zi,t ) is a deterministic function of age and other characteristics, νi,t represents a permanent
shock that evolves based on νi,t = νi,t−1 + uit , with ui,t ∼ N (0, σu2 ) while εi,t is a temporary shock with
εi,t ∼ N (0, σε2 ). The variances reported here are the sum of the estimated variances for σu2 and σε2 for every
industry.
25
The variances of income shocks are 10.8% in construction, 8.9% in wholesale and retail trade, 9% in
39
Households’ risky shares are regressed on the income-risk measure of the industry and
other individual characteristics such as total income, age, education, number of children,
and marital status. Table 11 reports the estimated coefficients and their standard errors of
this regression. The statistically significant negative coefficient on labor-income risk confirms
that larger labor-market risk crowds out financial risk. As the risk in the labor market (the
variance of the labor-income shock) increases by 1%, the household’s risky share decreases by
0.22 percentage point (with a standard error of 0.06). This is consistent with Angerer and
Lam (2009), who find a negative correlation between labor-income risks and the share of risky
assets from the NLSY 1979 cohort. The other coefficients are consistent with our economic
priors. Workers with more education (a proxy for permanent income) and total income take
more risks in making financial investments. So do older workers.
Table 11: Regression of Risky Share on Income Risk of Industry
Dependent Variable = Household’s Risky Share
Constant
−0.8296∗∗∗
Industry’s income risk −0.2234∗∗∗
Log Income
0.0973∗∗∗
Age
0.0017∗∗∗
Education
0.1926∗∗∗
Number of children
−0.0121∗∗∗
Marriage dummy
0.0080
(0.0202)
(0.0601)
(0.0017)
(0.0002)
(0.0060)
(0.0016)
(0.0052)
Notes: The numbers in parentheses are standard errors. Measures of an industry’s laborincome risk are based on Campbell, Cocco, Gomes, and Maenhout (2001).
7
Conclusion
We develop a quantitative life-cycle model to examine whether labor-market uncertainty
helps us to understand the long-standing puzzles in household portfolio choice. (i) Households
hold a small amount of equity despite the higher average rate of returns (equity premium
puzzle). (ii) The share of risky assets increases with the age of the household. (iii) The share
of risky assets is disproportionately larger for richer households.
The model features three types of labor-market uncertainty: a gradual learning about true
transportation and finance, 11.8% in business services, 6.7% in transportation and communications, and 5.2%
in manufacturing.
40
earnings ability, age-dependent unemployment risk, and noise in the priors-updating rule.
When the model is calibrated to match the age profiles of earnings, unemployment risk, and
consumption dispersion in the data, the model provides a partial resolution to these puzzles.
The model is able to reproduce the average risk share (45 percent) in the data under the
relative risk aversion of 5, which is a much smaller value than that required in the standard
model. According to the SCF, the risky share increases on average by 0.92 percentage point
each year between ages 21 and 65. Our model also generates an increasing—but to a lesser
degree than that in the data—age profile of the risky share: it increases by 0.36 percentage
point each year. The standard life-cycle model predicts a strongly negative correlation between
the risky share and wealth, whereas it is mildly positive in the data. Our model partially
corrects for this failure and generates a mildly negative correlation between the risky share
and wealth.
In our model, young workers, on average asset poor, face larger labor-market uncertainty
because of high unemployment risk and imperfect knowledge about their earnings ability. To
hedge against this labor-market uncertainty, young investors are less willing to take risks in
making financial investments. As labor-market uncertainty resolves over time, they take more
risks in the financial market. Our theory predicts that workers in an industry with highly
volatile earnings should take less risk in financial portfolio. Based on the regression of the
household’s risky share on the industry’s labor-income risk, we find that a household working
in an industry where higher labor-income volatility is 10% larger than the mean exhibits a
risky share that is 2.2% lower than the average.
We argue that a complete theory of investment should consider the risk that investors face
not only in the financial market but also elsewhere, especially in the labor market.
41
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44
Appendix
A
Data: Survey of Consumer Finances
General Description The Survey of Consumer Finances is a cross-sectional survey conducted every 3
years. It provides detailed information on the finances of families in the U.S.A. Respondents are selected
randomly with a strong attempt to select families from all economic strata. The “primary economic unit”
consists of an economically dominant single individual or couple (married or living as partners) in a household
and all other individuals who are financially dependent on that individual or couple. In a household with
a mixed-sex couple the “head” is taken to be the male. One set of the survey cases was selected from a
multistage area-probability design and provides good coverage of characteristics broadly distributed in the
population. The other set of survey cases was selected based on tax data. This second sample was designed
to disproportionately select families that were likely to be relatively wealthy. Weights compensate for the
unequal probabilities of selection. To deal with respondents who were unable to provide a precise answer,
the survey gives the option of providing a range. While the number of respondents is 4,309, the number of
observations is 21,545. This is because variables that contained missing values have been imputed five times,
drawing repeatedly from an estimate of the conditional distribution of the data. Multiple imputation offers a
couple of advantages over singly imputed data.
Example of Survey We provide an example of the questionnaire related to checking accounts. The following questions are asked, among others. 1) Do you have any checking accounts at any institution? 2) How
many checking accounts do you have? 3) How much is in this account? (What was the average over the
last month?). In some other accounts like individual retirement accounts the respondent is asked specifically
how the money is invested. The questions are: 1) Do you have any individual retirement accounts? 2) How
much in total is in your IRA(s)? 3) How is the money in this IRA invested? Is most of it in certificates
of deposit or other bank accounts, most of it in stocks, most of it in bonds or similar assets or what? The
possible answers are: 1) CDs/ Bank accounts; money market; 2) Stock; Mutual funds; 3) Bonds/ Similar
assets; T-Bills; Treasury notes; 4) Combinations of 1, 2, 3; 5) Combinations of 2, 3; 6) Combinations of
1, 2; 7) Universal life policy or other similar insurance products, 8) Annuity, 9) Commodities, 10) Real estate/mortgages, 11) Limited partnership/ Other similar investments, 12) Brokerage accounts, 13) Split/ Other.
Construction of Variables In this section we explain the types of assets we categorize as safe and risky.
In all our definitions, we make use of weights, variable X42001.
• Checking accounts, Money Market Accounts: The variables X3506, X3510, X3514, X3518, X3522,
X3526 report the amount of money the respondent has in six different accounts. The respondent is
asked whether each of these accounts is a checking account or a money market account. Responses can
be found in variables X3507, X3511, X3515, X3519, X3523, X3527. We define Checking Accounts
(and respectively Money Market Accounts) as the sum of these accounts.
• Savings accounts: We define the sum of variables X3804, X3807, X3810, X3813, X3816 as Savings
Accounts.
• Certificates of Deposit: The variable X3721 gives the amount of money in certificates of deposits. We
define Certificates of Deposit to be equal to this variable as long as the account does not belong to
45
someone unrelated to the household (variable X7620 < 4).
• Saving bonds: We define as Savings Bonds(safe) the sum of variables X3902 (money saved in U.S.
government savings bonds), variable X3908 (face value of government bonds) and variable X3910 (money
in state and municipal bonds). We define as Savings Bonds (risky) the sum of variables X3906 (face
value of mortgage-backed bonds), variable X7934 (face value of corporate bonds) and variable X7633
(face value of foreign bonds).
• Life Insurance: Variable X4006 gives the cash value of life insurance policies, while variable X4010 gives
the amount currently borrowed using these policies. We define as Life Insurance the amount given
by X4006-X4010.
• Miscellaneous assets and debts: This category gives the amount of money the respondent is owed by
friends, relatives or others, money in gold or jewelry and others. Variable X4018 gives the total amount
owed and X4022, X4026, X4030 the dollar value in these types of assets. Variable X4032 is the amount
owed by the respondent. We define Miscellaneous Assets as X4018+ X4022 + X4026 + X4030X4032.
• Other Consumer Loans: Variables X2723, X2740, X2823, X2840, X2923, X2940 give the amount still
owed in loans like medical bills, furniture, recreational equipment or business loans. Using variables
X6842-X6847 we make sure these loans are not part of business loans and we define the variable Other
Consumer Loans to be equal to X2723 + X2740 + X2823 + X2840 + X2923 + X2940.
• Brokerage Accounts: Variable X3930 gives the amount the total dollar value of all the cash or call
money accounts, and the variable X3932 gives the current balance of margin loans at a stock brokerage.
We define Brokerage Accounts to be equal to X3039-X3932.
• Mutual Funds: Variable X3822 gives the total market value of all the stock funds, variable X3824
the total market value of all of the tax-free bond funds, variable X3826 the total market value of all
government-backed bonds, variable X3828 the total market value of other bond funds, and variable
X3830 the total market value of all of the combination funds or any other mutual funds of the respondent. We define as Mutual Funds (safe) the sum of variables X3824+X3826+X3828+0.5×X3830
and as Mutual Funds(risky) the sum of variables X3822+0.5×X3830.
• Publicly Traded Stocks: Variable X3915 gives the total market value of stocks owned by the respondent,
and variable X7641 the market value of stocks of companies outside the U.S. We define Stocks to be
equal to X3915+X7641.
• Annuities: Variable X6820 gives the total dollar value of annuities. Variable X6826 reports how the
money is invested. We define Annuities(safe) to be equal to X6820 if X6826=2 (Bonds/interest;
CDS/Money Market) and equal to 0.5×X6820 if X6826=5 (Split between Stocks/Interest; Combination
of Stocks, Mutual Fund, CDS). We define Annuities(risky) to be equal to X6820 if X6826=1 or =3
(Stocks; Mutual Funds or Real Estate) and equal to 0.5× X6820 if X6826=5.
• Trust: Variable X6835 gives the total dollar value of assets in a trust. Variable X6841 reports how
the money is invested. We define Trust(safe) to be equal to X6835 if X6841 = 2(Bonds/interest;
CDS/Money Market) and equal to 0.5×X6835 if X6841=5 (Split between Stocks/Interest; Combination
of Stocks, Mutual Fund, CDS). We define Trust(risky) to be equal to X6835 if X6841=1 or =3 (Stocks;
Mutual Funds or Real Estate) and equal to 0.5× X6835 if X6841=5.
46
• Individual Retirement Accounts: Variables X3610, X3620, X3630 report how much money in total is
in individual retirement accounts. Variable X3631 reports how the money is invested. We define
the variable IRA(safe) to be equal to X3610 + X3620 + X3630 if X3631 = 1 (money market) or
X3631 = 3 (Bonds/ Similar Assets; T-Bills) or X3631=11 (Universal life policy). IRA(safe) equals
2
3 (X3610 + X3620 + X3630) if X3631=4 (combination of money market-stock mutual funds-bonds
and T-bills), equal to 21 (X3610 + X3620 + X3630) if X3631=5 (combination of stock mutual fundsbonds and T-bills), and equal to 12 (X3610 + X3620 + X3630) if X3631=6 (combination of money
market-stock mutual funds) or X3631=-7 (split). Similarly we define the variable IRA(risky) to be
equal to X3610 + X3620 + X3630 if X3631 = 2 (stocks) or X3631 = 14 (Real Estate/Mortgages) or
X3631 = 15 (Limited Partnership) or X3631 = 16 (Brokerage account). IRA(risky) equals 13 (X3610 +
X3620 + X3630) if X3631 = 4 (combination of money market-stock mutual funds-bonds and T-bills),
equal to 12 (X3610 + X3620 + X3630) if X3631 = 5 (combination of stock mutual funds-bonds and
T-bills), and equal to 21 (X3610 + X3620 + X3630) if X3631 = 6 (combination of money market-stock
mutual funds) or X3631 = −7 (split).
• Pensions: The variables X4226, X4326, X4426, X4826, X4926, X5026 give the total amount of money
in present pension accounts. We subtract any possible loans against these accounts by using the variables
X4229, X4328, X4428, X4828, X4928, X5028. Variables X4216, X4316, X4416, X4816, X4916, X5016
provide information on how the money is invested. We define Pensions(risky) if any of the latter
variables equals 3 (Profit Sharing Plan) or 4 (Stock purchase plan). Other than these two options
the SCF does not provide many details regarding pension plans. For example, respondents can report
that the money is invested in a 401K without further information on how the money are invested. In
this case, we split the money half into Pensions(safe) and the other half into Pensions(risky). As
mentioned in the text, we experiment with other split rules and show our findings in Table B-1 of
Appendix B.
• Business: Variables X3129, X3229, X3329 report the net worth of businesses, variables X3124, X3224, X3324
and X3126, X3226, X3326 the amount owed to the business and the amount owed by the business, respectively. Finally, variable X3335 gives the share value of any remaining businesses. We define
Actively Managed Business to be equal to X3129 + X3229 + X3329 + X3124 + X3224 + X3324 −
X3126 − X3226 − X3326 + X3335. Similarly we define Non-Actively Managed Business as the
sum of X3408 + X3412 + X3416 + X3420 + X3424 + X3428.
• Housing: Variable X513, X526 gives the value of the land the respondent (partially) owns, variable
X604 the value of the site, and variable X614 the value of the mobile home the respondent owns.
Variable X623 gives the total value of home and site if he owns both. Variable X716 is the value of
home/apartment/property that the respondent owns (partially). We define Value of the Home as
the sum of the above variables. Variables X1706, X1806, X1906 give the total value of property such
as vacation houses or investment in real estate. Variables X1715, X1815, X1915 give the amount of
money owed in loans associated with this property. We define Net house worth to be equal to Value
of the Home +X1706 + X1806 + X1906 − X1715 − X1815 − X1915.
Definitions
Safe Assets = Checking Accounts + Money Market Accounts + Savings Accounts + Certificates of Deposit
+ Savings Bonds(safe) + Life Insurance + Miscellaneous Assets + Mutual Funds(safe) + Annuities(safe) +
Trust(safe) + IRA(safe)+ Pensions(safe)
47
Risky Assets= Savings Bonds(risky) + Brokerage Accounts + Stocks + Mutual Funds(risky) + Annuities(risky) + Trust(risky) + IRA(risky)+ Pensions(risky) + Non-Actively Managed Business
Risky Share=
B
Risky Assets
Safe Assets + Risky Assets
Division of Pension Plans Between Riskless and Risky
Assets
As mentioned in the main text, the SCF does not provide exact information on how pension plans, such
as 401(k)s, are invested. For our benchmark definition of the risky share, we categorized half of the money
invested in these accounts as safe asset holdings and half as risky assets. Our choice of an equal split related
to the average risky share is close to 50%. In this section we re-calculate the risky share of financial assets
using alternative split rules. In particular, we experiment with two extreme cases: a rule that allocates 80%
of the money in these accounts to safe assets (and 20% in risky), and a rule that allocates 20% of these money
in safe assets (and 80% in risky). We report our findings in Table B-1. The average risky share is sensitive to
our choice. Naturally, if we allocate most of the money to safe assets, the risky share will decrease to 38.2%. If
we allocate most of the money to risky assets, the risky share will increase to 52.6%. However, the increasing
age-pattern documented under our benchmark definition remains intact.
Table B-1: Portfolio Choice for Different Split Rules
C
Age group
Benchmark 50-50
80-20
20-80
21-30
31-40
41-50
51-60
61-65
39.9%
43.8%
46.0%
47.9%
50.7%
32.9%
34.6%
37.9%
40.9%
46.4%
47.0%
53.0%
54.2%
54.8%
55.0%
Average
45.5%
38.2%
52.6%
Updating Rule
We provide a brief description of our updating rule of priors about the earnings profile. Consider a simpler
case where income yt = zt + εt depends on a persistent component zt = ρzt−1 + η, with η ∼ N (0, ση2 ) and
an i.i.d. component ε ∼ N (0, σε2 ). Every period a worker observes income yt and updates his prior about the
true value of zt . After the update has taken place, the worker receives noise qt ∼ N (0, σq2 ). The prior belief is
summarized by two parameters: the mean of the distribution µzt|t−1 and the variance σz2t|t−1 . After observing
yt the worker will form a posterior belief, which in turn is summarized by µzt|t and σz2t|t . The posterior mean
will depend on the prior mean, the realized income and the realized shock. In particular
48
µzt|t = ωµzt|t−1 + (1 − ω)yt + qt
Our basic assumption is that only yt is known at the time of updating. The noise shock qt will occur at the
end of the period. Since the shock will be zero in expectation, the worker will just assign weights to the prior
mean and the income. To pick ω the worker minimizes the variance of µzt|t , so:
min E(µzt|t − zt )2 = min E(ωµzt|t−1 + (1 − ω)yt + qt − zt )2 =
ω
ω
min E(ωµzt|t−1 − ωzt + (1 − ω)ε + qt )2 = min E(ωµzt|t−1 − ω(ρzt−1 + η) + (1 − ω)ε + qt )2
ω
ω
= min E(−ωη + (1 − ω)ε + qt )2 == min E(−ωη + (1 − ω)ε)2 + σq2
ω
ω
The last equation made use of µzt|t−1 = ρzt−1 . We can see that parameter σq2 will not affect the optimal
weight ω. Minimizing this expression we have ω =
σε2
σε2 +ση2 .
Plugging ω back we get
µzt|t = ωµzt|t−1 + (1 − ω)yt + qt → µzt|t = µzt|t−1 +
ση2
yt + qt
σε2 + ση2
This is equivalent to equation (5) in the text. Doing the same with the variance:
E(−ωη + (1 − ω)ε)2 + σq2 = E(−
=
σε2
ση2
σε4 ση2 + σε2 ση4
σε2
2
2
η
+
ε)
+
σ
=
+ σq2 =
q
σε2 + ση2
σε2 + ση2
(σε2 + ση2 )2
ση2
σε2
ση2 + σq2 = ση2 − 2
σ 2 + σq2
2
+ ση
σε + ση2 η
This is equivalent to equation (6) in the text. Note that if σq2 = 0, we go back to the standard Bayesian
learning.
49
@FedFRASER