essays in asset pricing and real estate...ann arbor, ml 48106-1346 ©2008 yu zhang all rights...
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ESSAYS IN ASSET PRICING AND REAL ESTATE
YU ZHANG
Submitted in partial fulfillment of the
requirements for the degree
of Doctor of Philosophy
under the Executive Committee
of the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2008
UMI Number: 3333475
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YU ZHANG
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Abstract
Essays in Asset Pricing and Real Estate
Yu Zhang
The first chapter of this dissertation introduces housing as a hedging asset in a life-
cycle portfolio choice model and addresses the empirically documented hump-shaped
life-cycle stock investment pattern. I show that the life-cycle pattern of housing
investment has a crucial influence on investments in stocks. House tenure choice is
endogenized and an investor uses housing investment to hedge against both labor
income risk and rent risk when labor income, house price, rent, and stock price co-
vary with each other. The "U-shaped" life-cycle housing investment profile helps
to explain the equity allocation puzzle. This paper also demonstrates that optimal
portfolio choice varies across local housing markets and industries, so that a one-size-
fits-all prescription is unsuitable for life-cycle investments.
The second chapter explores the portfolio choice in a multi-asset setting. It consid
ers a more realistic portfolio which contains not only financial assets but also housing
investment and human capital. I look at the covariance structure of labor income,
house price, rent, and stock price and examine the possibility of households using
these multiple assets in hedging. I obtained the data of these four time series and
computed the correlations and volatilities over the period 1980 and 2004. In addition,
the PSID data with its Geocode data are used to conduct cross-sectional portfolio
choice analysis. I find evidence that the PSID households use housing investment to
hedge against labor income risk and rent risk, consistent with the findings by Davidoff
(2006) and Sinai and Souleles (2005) and in the real estate literature.
Contents
1 Life-Cycle Portfolio Choice with Housing as a Hedging Asset 1
1.1 Introduction 2
1.2 The Economic Model 6
1.2.1 Preferences 6
1.2.2 The Labor Income Process 8
1.2.3 Financial Assets 8
1.2.4 House Price and Rent 9
1.2.5 Wealth Accumulation 10
1.2.6 The Investor's Optimization Problem 11
1.3 Benchmark Parameterization 12
1.3.1 Preference Parameters 13
1.3.2 Labor Income Process 13
1.3.3 Financial Assets 14
1.3.4 Housing Parameters 14
1.3.5 Correlations 15
1.4 Numerical Results in the Benchmark Model 15
1.5 What Drives Life-Cycle Portfolio Choice? 19
1.5.1 Life-cycle Allocations Without Housing 19
1.5.2 Housing's Hedging Function 20
1.5.3 Rates of House Price and Rent Appreciation 20
1.5.4 Low Down Payment Ratio vs. High Down Payment Ratio . . 21
1.5.5 Risk Aversion vs. Elasticity of Intertemporal Substitution (EIS) 22
1.6 Utility Cost Calculations 23
1.7 Life-cycle Allocations Across Housing Markets 24
1.7.1 Correlation of Labor Income with House Price 25
1.7.2 Correlation of House Price with Stock Price 25
1.7.3 Correlation of Labor Income with Stock Price 26
1.8 Conclusions 27
2 Cross-Sectional Portfolio Choice and the Hedging Functions of Hous
ing 35
2.1 Introduction 36
2.2 Related Literature 39
2.3 Data 42
2.3.1 House Price, Rent, Labor Income, and Stock Price 42
2.3.2 The PSID Family Files and Wealth Supplements Data . . . . 44
2.4 Empirical Methodology 45
2.4.1 Stock Participation and Home Ownership Decisions 45
2.4.2 Asset Allocation Decision 47
2.5 Results 48
2.5.1 Heterogeneity of Correlations and Volatilities 48
2.5.2 Likelihood of Participation 49
2.5.3 Asset Allocation 52
2.6 Conclusion 54
Bibliography 63
Appendices 67
A Numerical Solution 67
B Utility Loss Metric 68
C The Panel Study of Income Dynamics Data 69
ii
List of Figures
1.1 Labor Income, Consumption, and Accumulated Wealth in the Bench
mark Model 29
1.2 Life-Cycle Portfolio Choice Profiles in the Benchmark Model 31
1.3 Fraction of Stocks in Financial Assets Without Housing 32
1.4 Portfolio Choice When Only Renting is Allowed 33
1.5 Portfolio Choice When House Price and Rent Have Same Appreciation
Rate 33
1.6 Portfolio Choice When Ratio of House Price to Rent is Fixed . . . . 33
1.7 Portfolio Choice with Different Down Payment Requirements . . . . 33
1.8 Risk Aversion vs. Elasticity of Intertemporal Substitution 33
1.9 Portfolio Choice with Labor Income and House Price Correlated . . . 34
1.10 Portfolio Choice with Stocks and House Price Correlated 34
1.11 Portfolio Choice with Labor Income and Stocks Correlated 34
2.1 Housing Investment vs. Age 62
2.2 Stock Investment vs. Age 62
in
List of Tables
1.1 Benchmark Parameters 29
1.2 Cross-Sectional Variations in Labor Income, House Price, Rent, and
Stock Price 30
1.3 Utility Cost Calculation (%) 30
2.1 Cross-sectional Summary Statistics of p, a, and \i 55
2.2 Summary Statistics of the PSID Data: Full Sample 56
2.3 Summary Statistics of the PSID Data: Sub-sample 57
2.4 Likelihood of Owning a House 58
2.5 Likelihood of Owning Stocks 59
2.6 Ratios of Housing Investments to Wealth 60
2.7 Proportion of Wealth Invested to Stocks 61
IV
Acknowledgements
I would like to express my deepest appreciation to my advisor Chris Mayer for his
guidance and support. I am grateful to John Donaldson for his expert advice and
friendly encouragement. I sincerely thank the other members of my proposal and
defense committees, Stefania Albanesi, Andrew Ang, Sharon Harrison and Tomasz
Piskorski for their invaluable help. I also thank seminar participants at Columbia
Business School, the Federal Reserve Board, the 2007 Financial Management As
sociation Meeting, Northern Finance Association Annual Meeting, Joint Statistical
Meetings, Washington Area Finance Association Annual Meeting, the 2008 South
western Finance Association Annual Meeting and several universities for their helpful
comments and suggestions.
v
To my parents, Yunsheng Zhang and Yachu Chen,
for teaching me the importance of hard work.
To my wife, Yanhua Zhang, my soulmate,
for her love and support.
VI
Chapter 1
Life-Cycle Portfolio Choice with
Housing as a Hedging Asset
2
1.1 Introduction
Financial advisors recommend that young investors allocate most of their liquid
wealth to stocks and suggest that the fraction of their liquid wealth invested in stocks
should decrease as they age. Malkiel (1996) gives a heuristic suggestion that the
percentage held in equities should be equal to 100 minus the investor's age, so that
a 30-year-old investor should hold 70 percent of his financial wealth in stocks, while
a 70-year-old investor should hold only 30 percent in stocks. The financial literature
provides rationales for this popular wisdom. For example, Cocco, Gomes, and Maen-
hout (2005) argue that since non-tradable labor income is a substitute for risk-free
assets and decreases over a lifetime, an investor should optimally shift his financial
wealth from stocks to bonds as he ages. They suggest that the optimal share invested
in stocks should roughly decrease over the life cycle and that young investors should
hold only stocks and no bonds.
Empirically, however, the data show that young investors hold a lot of bonds and
their stock holdings are less than those of middle-aged investors, contradicting what
is suggested by industry recommendations and previous models. Several studies re
port that stock holding is typically low in young adulthood and that it peaks a few
years before retirement. For example, Ameriks and Zeldes (2004) use data from the
Survey of Consumer Finances and TIAA-CREF and find that risky asset ownership
and risky asset shares are hump-shaped functions of age. Bertaut and Starr-McCluer
(2002) analyze the Federal Reserve Flow of Funds Accounts and the Survey of Con
sumer Finances and also find hump-shaped risky asset allocations over the life cycle.
This discrepancy between the conventional financial advice and the actual choices
of individuals when allocating their liquid wealth is known as the equity allocation
puzzle.
In this paper, I introduce housing investment as a hedging asset and address
the equity allocation puzzle. I show that the life-cycle housing investment profile
influences bond and stock allocations. Early in the life cycle, risky stock investment
3
represents only a small fraction of liquid wealth due to young investors' intentions to
become homeowners. The consumption role of housing and the possibility of using
home equity to hedge against adverse shocks to labor income induce investors to invest
a large portion of their wealth in housing. Prior to owning a house, an investor seeks
to become a homeowner as soon as possible to take advantage of housing's hedging
utility. Young investors are low on labor income and thus favor bonds, avoiding
the riskiness of stocks. Immediately after purchasing a house, a young homeowner
has little capital to invest in stocks; high leverage in home equity crowds out stock
investment due to the substitutability as assets of stock and home equity. Middle-
aged investors, however, are better able to allocate liquid wealth to stocks: they have
higher labor income, and they have accumulated more wealth. Relative to young
investors, the middle-aged have a low ratio of housing investment to wealth. Middle-
aged investors consequently choose to increase their relative stock holdings in order to
take advantage of the equity premium. The inverse-hump-shaped profile of housing
investment over an investor's lifetime thus helps to explain the hump-shaped equity
investments pattern.
The two hedging functions of housing have been documented in the real estate
literature but have not yet been emphasized in the life-cycle portfolio choice litera
ture. The first objective of my paper is to at least partially fill in this gap. Firstly,
housing provides a hedge against labor income risk, as Davidoff (2006) argues. In
an incomplete market, investors can neither sell nor contract on their labor income.
Housing investment may offset the adverse shocks to labor income. Davidoff (2006)
finds that the co-movement of house price growth and labor income growth has a
negative impact on both the probability of homeownership and the size of housing
investment. Homeownership is especially attractive to investors who are likely to ex
perience negative shocks to labor income and house price at different times. Taking
into account the correlation between the labor income growth rate and house price
appreciation, investors can use housing as a hedging asset for labor income risk. The
4
second hedging function of housing is to hedge against rent risk, as Sinai and Souleles
(2005) point out. They argue that renting has risk, as renters are exposed to fluctu
ations in rent. Housing is a hedging asset against rent risk because homeowners do
not pay rent.
The second objective of my paper is the identification of an important fact over
looked by the literature, which is that housing markets are local so that optimal
portfolio choice varies cross-sectionally. Housing investment differs from other finan
cial investments in that unlike the stock market, housing markets can not be accessed
nationally by all investors. House tenure and portfolio choice differ across Metropoli
tan Statistical Areas (MSAs) due to variations in house prices and rents in those
areas, everything else being equal. Even within an MSA, individual decisions differ
since investors' labor incomes have different co-movements with the local house price
and rent. I find large cross-sectional variations of correlations among labor income
growth rate, house price appreciation, rent appreciation, and stock return. This paper
explicates that house tenure choice and portfolio choices differ cross-sectionally.
The academic literature on asset allocation is extensive.1 Among papers that
focus on life-cycle portfolio choice, early attempts were undertaken by Merton (1969)
and Samuelson (1969). Assuming complete markets and ignoring labor income, they
show that the optimal portfolio rule for an investor with a power utility function and a
constant investment opportunity set is to invest a constant fraction of wealth in risky
stocks, independent of wealth and age. In a realistic life-cycle setting, however, risky
labor income cannot be capitalized. Cocco, Gomes, and Maenhout (2005) calibrate
the non-tradable labor income streams, and their model suggests a decreasing optimal
share invested in equities over the life cycle.
Given the unsatisfactory performance of previous models in matching the data's
life-cycle portfolio choice patterns, incorporating housing into a life-cycle portfolio
choice model has received some attention. Housing has the rare quality of playing
1 See Curcuru, Heaton, Lucas, and Moore (2004) for a comprehensive review.
5
a dual role for investors: providing a flow of consumption services and being an
investment. Housing not only serves as a major source of utility, but it is also the
largest investment for most US households. The 2001 Survey of Consumer Finances
shows that home value accounts for 55% of the average homeowner's total assets. In
contrast, stock investments account for just 12% of household assets.2 Considering
the magnitude of housing investment, we should include housing in a portfolio choice
model.
Cocco (2005) looks at the life-cycle optimization problem of homeowners and
shows that house price risk has a crowding-out effect on stock holdings. He ignores
the effects of the renting-versus-owning decision, however, by excluding renters from
his model. Yao and Zhang (2005a) examine the optimal dynamic portfolio decision for
both owners and renters. They assume away rent risk by denning rent as a constant
fraction of house price. Their model predicts that all investors become homeowners
after age 40, and that the percentage of liquid assets held as stock is roughly decreasing
over time. Both predictions are inconsistent with the empirical findings.
My model takes into account the fact that house price deviates from rent. In the
model, housing's utility as a hedge against both labor income risk and rent risk is
time-varying with fluctuations in house-price-to-rent ratio and labor income growth
rate. The model predicts a hump-shaped life-cycle stock investment pattern while
matching the life-cycle pattern of homeownership.
The rest of this paper is organized as follows: Section 1.2 describes the model and
the method of solving for policy functions. Section 1.3 calibrates a benchmark model,
while the corresponding results are presented in Section 1.4. Section 1.5 discusses
what drives house tenure and portfolio choice over the lifetime. Section 1.6 further
examines the importance of housing by computing the welfare losses with different
assumptions about housing. Section 1.7 conducts comparative statics exercises and
explores optimal portfolio choice across housing markets. Section 1.8 concludes the
2 See Yao and Zhang (2005a).
6
paper.
1.2 The Economic Model
The model is developed on the basis of Cocco (2005) and Yao and Zhang (2005a) with
the new features of having endogenous house tenure choice and deviations of rent to
house price, in order to emphasize housing's hedging functions to labor income risk
and rent risk.
1.2.1 Preferences
Time is discrete, and an investor lives for at most T years. For simplicity T is assumed
to be exogenous and deterministic. Following convention in academic literature, I use
F(t) to denote the probability that the investor is alive at time (t + 1) conditional on
being alive at time t. Given that T is exogenous and deterministic, we have F(0) = 1
and F(T) = 0.
In each period t 6 { 0 , 1 , . . . , T} , if the investor is alive, he obtains instantaneous
utility u(Ct, Ht) from consuming the numeraire good Ct and a housing service flow
which is assumed to be proportional to the housing shares Ht. The housing shares Ht
are interpreted as units of housing in terms of both size and quantity. Without loss
of generality, I assume the quantity of housing service flow is as large as the number
of housing shares. Hereafter I use the same Ht for both the quantity of housing
service flow and the number of housing shares. The instantaneous utility u(Ct, Ht) is
of Cobb-Douglas form over Ct and Ht:
u(CuHt)^C}-eHf,
where 9 is a housing preference parameter which determines the consumption shares
in a static model.
If the investor dies at time t, all of his assets are liquidated and he passes down his
7
wealth Wt to his heirs. The investor obtains bequest utility Qt(Wt) from the bequest.
The utility function applied to the bequest is assumed to be identical to the utility
function when the investor is alive. For simplicity, I assume that upon the investor's
death, the liquidated wealth Wt is used to purchase a house of size Ht and numeraire
consumption good Ct for his beneficiary. That is,
Qt{Wt) = mvxCl-eHet
V Ct,Ht l l
subject to
Ct + PtHt = Wt,
where Pt is price per unit of housing at time t.
Due to the simple assumption of a Cobb-Douglas form for the bequest function,
we can derive the indirect utility from bequest as
QtW) = po = K-pe>
where « = 9e(l - d)1'9.
Intertemporally, the investor smooths out his numeraire consumption Ct and hous
ing consumption Ht. The power utility function is widely used as the intertemporal
utility function in life-cycle portfolio choice papers. It implies that the risk aversion
parameter is the inverse of the Elasticity of Intertemporal Substitution (EIS). In this
case, it is not clear whether results are driven by difference in risk aversion or by
difference in EIS. To answer this question, I use the Epstein-Zin preference instead:
Vt(Xt) = max{(l-/?) {At} I
F{t)u{CuHtf-* + [l - F{t)]Q !-i
+/3F(t)Et rl-J,
vt\7(xt+i) ^y-K ( i . i )
where j3 is the subjective discount factor, 7 is the risk aversion coefficient, ip is the
8
EIS, and Xt and At are the state variables and choice variables which will be defined
in Section 1.2.6.
1.2.2 The Labor Income Process
Of the T periods of his life, the investor works in the first K periods, after which
he retires. I assume an inelastic supply of labor and incomplete markets. At each
time t before retirement, he receives labor income Yt against which he cannot borrow.
Similar to Cocco (2005) and Yao and Zhang (2005a), I assume the labor income
process before retirement is stochastic and given by
A\ogYt = f{t) + eJ fort = 0,...,K-l,
where /(£) is a deterministic function of age t, and ej is a shock to the labor-income
growth rate.
For simplicity, retirement age is assumed to be exogenous and deterministic, with
the investor retiring at time K. After retirement, the investor's income is determinis
tic, representing payments from pensions and Social Security, and is assumed to be a
constant fraction £ (a scalar between zero and one) of his pre-retirement labor income
YK-\- That is,
Yt = CYK-i, fort = K,...tT,
where C is called the replacement ratio.
1.2.3 Financial Assets
There are two financial assets: a risk-free asset (called bond), with gross real return
r j ; and a risky asset (called stock), with gross real return rf. rj is assumed to be
constant over time, rf follows the stochastic process
rf -rf=fxs + ef,
9
where /xs is the equity premium and ef is the shock to equity returns.
I assume that the investor can short sell neither bonds nor stocks, so we have
St>0,Bt>0,Vt, (1.2)
where St and Bt are stock holdings and bond holdings at time t, respectively.
1.2.4 House Price and Rent
To receive housing service flows, the investor either rents or buys a house. I denote
Pt and Rt as the house price and rent, respectively, per unit of housing Ht. Again,
the "unit" is in terms of both size and quantity, as defined in 1.2.1. If the investor
chooses to rent and consumes a housing service flow of size Ht, he will need to pay
RtHt per year. If he chooses to own, he will need to have a house of value PtHt.
House price Pt and rent Rt are assumed to follow stochastic processes
AlogPi = /xp + ef
A l o g # t - fiR + e?
where / /p and fiR are the mean appreciation rates of house price and rent, respectively,
and ef and ef are shocks to house price appreciation rate and rent appreciation rate.
I assume these shocks are correlated with those to labor income and stock return.
Notice that my paper differs from Cocco (2005), Cocco, Gomes, and Maenhout
(2005), and Yao and Zhang (2005a) in that I allow for endogenous house tenure choice
throughout the investor's life cycle and I allow for time-varying deviations of rent from
house price. These two relaxations of the assumptions in the literature emphasize the
two hedging functions of housing investment.
10
1.2.5 Wealth Accumulation
Let Wt denote wealth, cash on hand to the investor at the beginning of time t. In
each period t, Wt is composed of three sources: i) the proceeds from his investments
in bonds and stocks in the previous period Bt-iTf + St-irf, ii) his stochastic labor
income Yt, and iii) capital gains from his housing investment if he owned a house in
the previous period. That is,
Wt = Bt^rj + St-irf + Yt + Ot-i[(l - <l>)PtHt-i - (1 - S)Pt.1Ht.1rf], (1.3)
where Ot is a dummy variable denoting homeownership status. Ot takes value 1 if
the investor is an owner in period t and 0 otherwise. If the investor owned a house in
the last period (Ot-i = 1) and sells it today, he obtains the current market value of
his house PtHt-\ net of transaction cost. The transaction cost <j>PtHt-\ is a constant
fraction <f> of the house value PtHt-\. 5 is the ratio of down payment to house value.
Assume that at time t — 1 the homeowner already paid 5Pt_iHt-i for his house; he
therefore carries mortgage balance (1 — 8)Pt-iHt-\ to time t. Before he sells his house
at time t, the investor needs to pay off (1 — 5)Pt-.\Ht-\rf ,3 which is his mortgage
balance outstanding plus interest. If the investor was not a homeowner in the previous
period {Ot-\ = 0), he receives zero income from the housing market.
With Wt received at the beginning of period t, the investor allocates this amount to
the following three expenditures: i) numeraire good consumption Ct, ii) investments
in bonds and stocks, Bt and St, and iii) purchasing housing service flows, which
amount to Ht. A renter (Ot — 0) pays RtHt and obtains his housing service flows
from the rental market; a homeowner (0< = 1) pays down SPtHt to acquire a house
of value PfHt and receives his housing service flows. We have
Wt = Ct + Bt + St + {{l-Ot)RtHt + Ot5PtHt]-<t>PtHt^Ot-iOtl{Ht_l=H$A)
3 Note that 77 is a gross return, so the amount (1 — 5)Pt-\Ht-irf contains both mortgage balance and interest incurred.
11
Note that a homeowner who does not change his homeownership status (Ot-i =
Ot = 1) and does not adjust his housing share (Ht-i — Ht) is not subject to transac
tion cost in housing. Therefore <f)PtHt-i cancels out from (1.3) and (1.4). Transaction
cost in housing is incurred only by a homeowner who switches to a renter (Ot~i = 1
and Ot = 0) or by a homeowner who adjusts his housing units {Ht-\ ^ Ht).
To limit the number of state variables and to maintain tractability, I follow Yao
and Zhang (2005a) and assume that the after-tax mortgage rate is the same as the
after-tax rate of return on the riskfree bond. Under this assumption, there is an
indeterminacy with respect to bond and mortgage holdings. In order to pin down
the bond holding, I assume that the investor always carries the maximum mortgage
balance allowed, i.e., Mt = (1 — S)PtHt.A
1.2.6 The Investor's Optimization Problem
In this model, the investor is assumed to be a price taker. He maximizes his discounted
life-time utility by solving the following optimization problem:
max EfVbl {MLo
where Vo is given by (1.1) and is subject to (1.2), (1.3), and (1.4).
In each period, after observing his carried-in wealth (Wt), previous housing sta
tus (Ot-i), previous housing investment (Ht-\), realized labor income (Yt), house
price (Pt), and market rent {Rt), the investor decides whether to rent or to own
(Ot ). He also decides on numeraire consumption Ct and the size and composi
tion of his portfolio, which consists of bond, stock, and housing (Bt, St, Ht). Thus
the state variables are Xt = {Ot-\, Ht-i, Wt, Pt,Rt.,Yt} and the choice variables are
At = {Ot, Ct, Bu St, Ht}. Then Vt = Vt(Xt).
4 This is a reasonable assumption, given the low down payments and cheap refinancing over the period of the late 1990's to 2006. Letting mortgage rate differ from the riskfree rate introduces an additional state variable, namely, mortgage balance. Due to the curse of dimensionality, this is beyond the scope of this paper.
12
As well known in the portfolio choice literature, analytic solutions to this problem
do not exist. I solve the problem numerically based on maximization of the value
function to derive the optimal decision rules. It is shown in the Appendix A that
the above problem can be simplified using the investor's wealth Wt as a normalizer
to reduce the dimensions of the state space. After the normalization, the relevant
state variables are: the investor's housing status Ot-\, the labor-income-wealth ratio
yt — Yt/Wt, the beginning-of-period house-value-wealth ratio ht = PtHt-i/Wt, and
the house-price-rent ratio Pt/Rt- In the investor's optimization problem, the relevant
choice variables are: the numeraire-good-wealth ratio ct — Ct/Wt, the fraction of
wealth allocated to bonds bt = Bt/Wtl the fraction of wealth allocated to stocks st =
St/Wt, the new house tenure choice Ot, and the new housing-investment-wealth ratio
ht = HtPt/Wt. I derive the policy functions numerically using backward induction.
In the last period, the policy functions are trivial: CT = (1 — 0)WT and HfPr =
6WT by the Cobb-Douglas rule, and no savings as the investor dies at time T. I
then substitute this value function in the Bellman equation and compute the policy
rules for the previous period. To optimize, I use standard grid search procedure and
discretize the state-space for the continuous state variables. For off-grid points, I
use interpolations for approximations. Four-point quadrature is used to evaluate the
integrals. Given these policy functions, I can obtain the corresponding value function.
I iterate backwards until t = 0. The details are given in Appendix A.
1.3 Benchmark Parameterization
This section calibrates the benchmark model. Table 1.1 summarizes the baseline
parameters. The following subsections describe how these numbers are estimated
from the data or chosen from the literature.
13
1.3.1 Preference Parameters
The investor enters the model at the age 20 and makes decisions annually until the
last period, at age 80 (T = 60)5 . The conditional survival probabilities F(t) are
calibrated using the 1998 Life Table for the total US population from the National
Center for Health Statistics (Anderson (2001)).
Annual discount factor is f3 — 0.96, with risk aversion parameter 7 = 5 and EIS
i[) = 0.2. In the benchmark model I set 7 = 1/ijj, so the Epstein-Zin utility function
collapses to the standard CRRA power utility function. To explore how 7 and if)
separately drive the results, I break the link so that 7 ^ 1/tp in Section 1.5.5 and let
the comparative statics exercises show the impacts on portfolio choice from 7 and ip
separately.
1.3.2 Labor Income Process
In the benchmark model, I assume the investor retires at age 65 (K = 45). The labor
income process calibrations before and after retirement follow the specifications in the
literature. Before retirement, labor income grows at a deterministic rate fit) with a
random shock. The deterministic growth rate f(t) depends on age. Cocco, Gomes,
and Maenhout (2005) calibrate the age-dependent deterministic labor income growth
rate by fitting a third-order polynomial to the labor income of high school graduates
using the Panel Study of Income Dynamics (PSID) data. I calibrate the determin
istic mean growth rate of labor income before retirement based on their empirical
estimation. Similar to Yao and Zhang (2005a), I consider only transitory shocks e(
to the labor-income growth rate. The standard deviation of the labor-income shocks
is set to be 12%, based on estimates in Chapter 2 of this dissertation using quarterly
data from 1980Q1 to 2004Q4 from the Quarterly Census of Employment and Wages
(QCEW).
5 According to the National Vital Statistics Reports (Vol 56, 9) by the National Center for Health Statistics, US life expectancy was 77.8 years in 2004.
14
After retirement, the investor receives 68 percent of his labor income at age 64 (re
placement ratio £ = 0.68), following the assumptions in Cocco, Gomes, and Maenhout
(2005).
Some of the literature estimate different labor income profiles for different educa
tion groups (e.g., college graduates, high school graduates, and those without a high
school degree). According to Cocco, Gomes, and Maenhout (2005), however, edu
cation groups do not differ much in their life-cycle portfolio choice profiles. In this
paper I report results obtained only with parameters estimated from the subsample
of high school graduates.
1.3.3 Financial Assets
Following the standard parameterization of financial assets, the risk-free rate is set at
2% so that the gross return on bonds rf — 1.02 and the risk premium is fis = 6.0%.
The standard deviation of shocks to stock return is set at as — 15.7%.
1.3.4 Housing Parameters
The housing preference parameter is set at 6 = 0.2. This represents the average
proportion of household expenditures devoted to housing in the 2001 Consumer Ex
penditure Survey. Transaction cost in housing is set at <f> = 6%. Here, transaction
cost is strictly monetary and does not include time cost of search or psychological
stress of moving. Only homeowners pay transaction cost when they sell their house;
that is, when they transition from owning to renting (Ot-\ — 1 and Ot = 0), or
when they change units of housing (Ot-i = Ot — 1 and Ht-i ^ Ht). Renters can
adjust their units of housing frictionlessly. The down payment ratio is assumed to
be 5 = 0.2; therefore a homeowner always puts down 20% of house value as a down
payment and assumes 80% of house value as a mortgage.
The next Chapter of this dissertation obtains quarterly data on House Price In
dices (HPIs) from the Office of Federal Housing Enterprise Oversight (OFHEO) and
15
the Rent Indices (RIs) from Reis, Inc., a real estate consulting firm. Using a quarterly
sample from 1980Q1 to 2004Q4, he estimates the mean growth rates and standard de
viations of house prices appreciation and rent growth. Following the estimates in that
paper, I set the mean house price appreciation rate in this paper at fip = 1.8% and
the mean rent appreciation rate is set at [iR = 0.4%. The volatilities of house price
appreciation and rent appreciation are set at ap — 5.5% and aR — 3.0%, respectively.
1.3.5 Correlations
As pointed out in the Introduction, the literature neglects an important aspect of
housing: the fact that housing markets are local. Previous papers have assumed
either zero or constant correlations among labor income growth rate, stock return,
house price appreciation, and rent appreciation. However, as shown in Chapter 2
and will be discussed in Section 1.7, the data show huge variations in the correlations
across local housing markets and across labor income groups.
With the hedging functions of housing investment in mind, investors will have dif
ferent optimal portfolio choice over their lifetime when they face different correlations.
To explore the impacts of variations in these correlations, I choose the cross-sectional
mean values in the data, according to the estimates in Chapter 2, as the parame
ters in the benchmark model. Table 1.1 lists these correlations. Section 1.7 contains
the comparative statics exercises which demonstrate how changing these correlations
impact the portfolio choice.
1.4 Numerical Results in the Benchmark Model
This section presents the simulated life-cycle portfolio choice profiles using the opti
mal policy functions derived in Section 1.2. I simulate life cycles for 4000 households.
The households are randomly given a starting labor income level which is uniformly
distributed with a mean $30,000 and standard deviation $20,000. These numbers
16
represent the average labor income and its standard deviation in the Quarterly Cen
sus of Employment and Wages (QCEW) sample. The initial house value is set at
$120,000. I then simulate the labor income growth rate, house price appreciation,
rent appreciation, and stock return according to the parameters in Table 1.1. Using
the policy functions derived in 1.2.6, the investors optimally choose to own or to rent
and they determine their portfolio choices over the life cycle. Figure 1.1 and Figure
1.2 present the mean profiles across all investors in the benchmark model.
Figure 1.1 plots the mean numeraire consumption Ct, labor income Yt, and accu
mulated wealth Wt in the benchmark model. Early in life, an investor starts to save
for retirement, and wealth accumulation increases significantly. After the investor
reaches middle age, he begins to access his savings and decumulate wealth to smooth
out consumption. His numeraire consumption peaks around age 60, several years be
fore retirement. After retirement, his labor income is only a constant fraction of his
pre-retirement labor income. The investor reduces his consumption and his wealth
decreases rapidly. Due to the bequest motive, the investor optimally saves a small
portion of his accumulated wealth for his heirs. After the age of 80, the investor is
no longer alive, so that both consumption and savings are zeros in the last period of
the life cycle.
The optimal house tenure choice is shown in the top-left panel of Figure 1.2. This
figure plots the percentage of investors who choose to own rather than rent a house at
various points in their life cycle. Housing's dual role as a consumption good and an
investment makes it attractive, so that an investor seeks to become a homeowner as
early as possible. In the early years of the life cycle, not many investors have enough
money to purchase a house. These investors are mostly renters. Eventually, as they
accumulate enough wealth, more and more investors switch to being homeowners.
Homeownership rates reach about 78% around age 40 and remain stable around 80%
until age 76. These homeownership rates match those observed in the 2005 Housing
Vacancies and Homeownership Annual Statistics by the U.S. Census Bureau, which
17
show that homeownership rates are close to 80% for investors between the age of 45
to 80. As investors reach the end of their lives, some find it optimal to withdrawn
their wealth from home equity for consumption or to switch back to being a renter.
Consequently, homeownership rates drop after age 76.
The top-right panel of Figure 1.2 presents the mean ratio of housing investment
to wealth. There are not many homeowners before age 30. Those who do become
homeowners in the very early stages of their lives can afford only small houses; those
who wait longer can afford to purchase larger houses. Therefore the average ratio
of housing investment to wealth increases for some years. Around age 32, this ra
tio reaches about 270%, reflecting the fact that young investors have little wealth
accumulated and that their home equities are highly leveraged. As investors age,
accumulated wealth increases sharply. However, due to transaction costs, housing
adjustments are rare. Housing investment drops relative to wealth to about 100% at
ages 50-60. After that, investors decumulate wealth and the ratio climbs up again for
the remainder of an investor's lifetime.
Conditional on housing investment, investors optimally choose how to allocate
their wealth to stocks St, bonds Bt, and numeraire consumptions Ct- The ratio of
financial assets allocated to stocks, S/(S + B), is presented in the middle-left panel of
Figure 1.2. Consistent with the empirical findings, we observe a hump-shaped ratio
of financial assets allocated to stocks as a function of age. Young investors save for
housing investment. Due to concerns about liquidity, they adopt a relatively con
servative investment strategy and allocate more money to bond investments relative
to stocks. As they age, they receive higher labor income and they accumulate more
wealth. Most of these investors have already become homeowners. They now can af
ford a riskier investment style. As a result, the percentage of stock in financial assets
grows. After middle age, stock's riskiness starts to dominate, which again induces a
conservative investment strategy. Therefore S/(S + B) drops gradually until the last
year of the life cycle.
18
Previous models in the literature have suggested that it is optimal to hold a lot
of stocks when a person is young and the fraction of wealth invested in stocks should
decrease as the investor ages. Their argument is that equity premium is high and
over the long-run stocks are not very risky. However, these predictions conflicts with
the hump-shaped stock holdings observed in the data.
My model predicts a hump-shaped stock investment profile over the life cycle,
taking housing investment into account. Housing's consumption role implies that in
vestors will try to maintain smoothness of housing consumption. Homeowners thus
smooth out housing investments over time, due to the inseparability of consumption
of and investment in housing. For most investors, how much to invest in housing has
to be considered before the decisions on portfolio choices for other assets. Condi
tional on housing investment, stock and bond investment are determined optimally.
Since housing investment and stocks are substitutes in terms of financial investments,
housing investment crowds out stocks. Early in the life cycle, an investor has a high
house-value-to-wealth ratio. This implies that he should invest less in stocks when he
is young. When he arrives at middle age, he has a lower present value of future labor
income and his house-value-to-wealth ratio falls. At this stage he would like to hold
more stock to take advantage of the high equity premium. However, as the investor
approaches the end of his life cycle, he decreases his risky stock investment. The
"U-shaped" housing investment profile implies the "hump-shaped" life-cycle profile
of stock in a liquid portfolio.
The last three panels of Figure 1.2 present the mean profiles of investments in
numeraire consumption, stocks, and bonds as fractions of wealth. Similar to the
housing investment profile, the ratio of numeraire consumption to wealth is inverse
hump-shaped. This is because numeraire consumption Ct is smooth but wealth Wt
is hump-shaped. Compared with young investors and elderly investors, middle-aged
investors are more aggressive and hold more stock investments. Bond investments
are relatively smooth over the life-cycle, with an increase before retirement age.
19
1.5 What Drives Life-Cycle Portfolio Choice?
1.5.1 Life-cycle Allocations Without Housing
To illustrate the role housing investment plays in optimal asset allocations over the
life cycle, this section removes housing H from the benchmark model. In the reduced
model, the investor obtains utility only from his numeraire consumption C and he
has access to only stocks S and bonds B for investment purposes.
Consistent with the findings in previous papers, the reduced model predicts dom
inance of stocks over bonds. Figure 1.3 presents the optimal percentage of stocks in
the liquid portfolio predicted by the reduced model. Without housing, the optimal
allocation to stocks is roughly a decreasing function of age, with 100% allocation of
liquid wealth to stocks for an investor under age 40. For this young investor, the
present value of labor income is very high. Bond-like labor income makes it optimal
to reduce bond holdings, while a high equity premium leads to popularity of stocks
over bonds. If without the short-selling constraint on bonds, the investor would have
borrowed to invest more in stocks. Due to the non-short-selling constraint, a young
investor will hold no bonds, optimally allocating 100% of his liquid wealth to stocks.
After age 40, the investor has lower present value of labor income, therefore he grad
ually transfer his wealth from stocks to bonds; the percentage of stocks in his liquid
portfolio decreases as he ages.
Figure 1.3 demonstrates the importance of incorporating housing in a life-cycle
portfolio choice model. When housing is present, as it is in the benchmark model, the
life-cycle equity allocation is hump-shaped. Without housing-investment considera
tion, the optimal risky asset holding is 100% for young investors and decreases with
age. What drive the difference are the motive to save for down payments on home
equity and the crowding-out effect of housing investment to stocks.
20
1.5.2 Housing's Hedging Function
In the benchmark model, housing serves as a hedge against adverse shocks to labor
income and against rent fluctuations. If owning is not allowed, housing is merely a
consumption good and it loses its hedging utility. In Figure 1.4, all investors are
renters. When there is only housing consumption and no housing investment, the
investor must rely on stocks to hedge his labor income risk. As Figure 1.4 shows,
optimal investments in stocks are now higher for all ages in the life cycle. The impact
is most significant a few years before retirement, where there is uncertainty in labor
income and an investor requires more cushion-offs from stocks, now the only hedging
asset.
Depending on age, housing's hedging function has mixed effects on bond invest
ments. Before retirement, bond investment is reduced when housing investment is
not allowed, since in this age range labor income is stochastic and the investor needs
to invest more in stock, which is the only hedging asset now. After retirement, the
annuities the investor receives are risk-free, just like bonds. At this point there is no
more labor income risk. With less risk, the investor has less need to hold stock for
hedging purposes. The decrease in stock investment after retirement allows for higher
investment in bonds.
1.5.3 Rates of House Price and Rent Appreciation
Arguably, the mean house price appreciation rate 1.8% used in the benchmark model
might be too high compared to the mean rent appreciation rate 0.4%. In equilibrium,
house price should be the present value of all future rents, just as stock price should
be the present value of future dividends. In this scenario, house price and rent should,
appreciate at the same rate. However, this may not be true out of equilibrium. The
past two decades saw run-up in house prices in the US, partly due to the lowering
of interest rates by the Federal Reserve and the proliferation of structured finance
products. The higher appreciation rate of house price relative to that of rent makes
21
it attractive to own a house, as documented by the big increases in homeownership
rates across the US in recent decades.
To explore the impacts on portfolio choice of the mismatch between the apprecia
tion rate of house price and that of rent, in this section I lower the mean appreciation
rate of house price to 0.4%, the same as the mean rent growth rate in the past twenty
years. Figure 1.5 displays the impacts on portfolio choices.
When house price appreciation rate is lowered to equal that of rent, there are
two impacts. On one hand, the slow increase in house prices means that houses are
not as expensive as before, as a result owning a house is easier; on the other hand,
from an investor's perspective, housing is less attractive with a low appreciation rate.
The overall effect is that housing investment now represents a smaller ratio to the
accumulated wealth. With more money left due to the lowered housing investment,
an investor will increase both stock and bond investments. Ownership rates increase
for young investors, since with lower house prices, housing is more affordable to more
young investors. However, due to the loss of investment incentive, homeownership
rates drop significantly for middle-aged investors. After retirement, investors with
draw their money from the stock markets. Some renters find it optimal to invest in
a house using the equity proceeds. This leads to a slight increase in homeownership
rates after retirement.
If we fix the ratio of house price to rent, as assumed in Yao and Zhang (2005a),
then housing investment lost its hedging function against rent risk. In Figure 1.6 we
can see that with a fixed house price to rent ratio, the investor optimally holds less
in housing and his stock investment is higher relative to bonds.
1.5.4 L o w D o w n P a y m e n t R a t i o v s . H i g h D o w n P a y m e n t
R a t i o
The benchmark model follows Yao and Zhang (2005a) in assuming a 20% down pay
ment requirement for purchasing a house. In the decade before 2006, there had been
22
a lot of easy credit extended in home mortgage. The possibility of obtaining second
mortgages such as piggy-back mortgages makes the effective down payment as low as
5%. On the other hand, given the recent turmoil in the subprime mortgage markets,
creditors are now tightening lending standards. In the future we should expect higher
down payment requirements.
Figure 1.7 considers two different assumed down payment ratios: 5% and 50%. A
5% down payment ratio represents easy credit. In this scenario, an investor has more
money left for stock and bond investments. Middle-aged investors are more likely to
use this money to buy stocks. In comparison, elderly investors are more likely to buy
more bonds. This is not surprising since middle-aged investors need to hedge against
risky labor income, while the elderly do not. The opposite patterns are observed with
a hard-credit case as proxied by a 50% down payment requirement: when a hight
down payment ratio is required, investors have less to invest in stocks and bonds.
1.5.5 Risk Aversion vs. Elasticity of Inter temporal Substi
tut ion (EIS)
Most life-cycle portfolio choice models use the power utility function. The power util
ity function implies 7 = 1/tp, that is, an increase in the risk aversion is accompanied
by a decrease in the elasticity of intertemporal substitution. It is therefore hard to
tell whether the changes in asset allocations are due to the changes in risk aversion or
changes in degrees of intertemporal substitution. The Epstein-Zin preference breaks
the link and lets 7 7̂ 1/ip- It c a n answer the question of how investors of various risk
aversions and EIS have different portfolio choices. Figure 1.8 presents the results.
When we hold the elasticity of intertemporal substitution constant at ift = 0.2
and increase the risk aversion parameter to 7 = 10, investors are more risk averse.
They prefer riskfree bonds to stocks and home equities. The consequences of this
are: lower homeownership rates, less housing investment, and higher bond holdings.
Stock investments are lower as well except for young investors, who hold more stock
23
in savings for future uncertainty.
If we keep the level of risk aversion at 7 = 5 and increase the elasticity of intertem
poral substitution to ip — 5.0, investors have a stronger motive to save for retirement.
They are likely to invest more in stocks, which bring higher returns. They are also
more eager to become homeowners as quickly as possible. However, since young in
vestors do not have much accumulated wealth, investors who become homeowners
early in life do not have much wealth to invest as home equity. Thus, the ratio of
housing investment to wealth is lower, though homeownership rates are higher. With
a stronger motive to save for the future, young investors invest more in bonds. As the
investors approach the end of their life cycle, they invest less in bonds and consume
more of their accumulated wealth.
1.6 Utility Cost Calculations
To further analyze housing investment, we can measure the economic importance of
the optimal portfolio profiles obtained under different situations. One meaningful
metric to measure the differences among portfolio strategies is the utility cost of each
portfolio rule relative to the optimal rule in the benchmark model.
The welfare calculations are done using the standard consumption-equivalent vari
ations. That is, for each life-cycle portfolio choice rule, I compute the constant con
sumption stream that gives the investor the same level of expected lifetime utility as
the consumption stream that can be financed by the portfolio rule. Relative utility
losses of adopting a certain portfolio rule relative to following the optimal rule are
then calculated by measuring the deviation of the equivalent consumption stream of
the portfolio rule from the optimal-rule-derived equivalent consumption stream. The
computations are similar to those in Cocco, Gomes, and Maenhout (2005). More
details are provided in Appendix B.
Table 1.3 reports the results for these utility cost calculations. The size of the wel
fare losses resulting from completely excluding housing from the model is substantial:
24
the investor loses about 28% of annual consumption in the reduced model relative
to the benchmark model. If we include housing but shut down its hedging function,
the utility loss is about 8% of annual consumption. When we lower the appreciate
rate of house price to that of rent, as in Section 1.5.3, the investor incurs a utility
cost of about 16% of annual consumption which is consistent with the findings in the
academic literature that house price appreciation stimulates consumption.6
1.7 Life-cycle Allocations Across Housing Markets
To date, the life-cycle portfolio choice literature has focused only on predictions at
the aggregate level. That is, given the parameterization of labor income, house price,
rent, and stock returns, the literature has explored the optimal allocations of wealth
to equities, bonds, and housing investment, etc., for a representative agent.
However, housing markets differ from equity markets in that there are huge vari
ations cross-sectionally. San Francisco's house price, rent, and their correlations with
labor income and stock return differ from those in Houston. Even within a particular
housing market, investors with different labor income profiles differ in the corre
25
1.7.1 Corre la t ion of L a b o r I n c o m e w i t h H o u s e P r i c e
As an investment, home equity can be used to hedge against the adverse shocks to
labor income. An asset is more popular when its shocks have a low correlation with
the shocks to labor income. Cross-sectionally there is a huge variation of correlations
of labor income with house price.
Figure 1.9 shows how this correlation affects the optimal asset allocation. Here I
consider two cases in addition to the benchmark model: when the correlation between
labor income growth and house price appreciation is 0.8, and when it is -0.8. When
house price and labor income are less correlated, such as in the -0.8 correlation case,
home equity is a useful hedge against adverse shocks to labor income. In this case,
more investors are willing to own a house. Before retirement, an investor allocates
more wealth to housing because of its hedging function; homeownership rates and
housing investment are both higher among non-retirees. After retirement, however,
an investor receives a fixed income and home equity's hedging function to labor income
is no longer of use. Homeowners need not hold more home equity after retirement.
As a substitute investment to home equity, stock investment is crowded out when
house price and labor income are less correlated. Investors also need less riskless
investment in bonds because they already hedge labor income risk using home equity.
The investors have higher numeraire consumption; they are better off than those in
the benchmark case.
In the case of highly correlated house price and labor income, we observe the
opposite patterns: lower homeownership rates, less housing investment, higher stock
investment, and greater allocation to bonds. The intuition mirrors that of the low-
correlation case.
1.7.2 Correlation of House Price with Stock Price
Aside from its consumption role, housing competes with stock in terms of invest
ment, especially when their correlations are low. Figure 1.10 presents two situations
26
when the two investments have a low correlation (-0.3) and when they have a high
correlation (0.3).
When home equity and stock are highly correlated, the two investments are more
like each other. House equity loses its popularity as an investment due to the higher
substitutability of stocks. Homeownership rates drop and owners invest less in hous
ing. At the same time, investors hold less stocks as the two assets now have less
diversification; it is unwise to invest heavily in highly correlated assets. The third
asset, bonds, now becomes the most popular investment due to its low correlations
with the other two assets. Investors tilt their investments to bonds at all times in the
life cycle. Thus bonds represent a high proportion of liquid assets.
We see different results in the case of a low correlation between house price appre
ciation and stock return. When the two assets are less correlated, an investor enjoys
greater diversification. Investors tend to invest more money in these two assets as a
whole. Bond holdings decrease since investors transfer their money from the bond
markets to stock and housing investment. Whether to invest more in stocks or in
home equity depends on the expected returns on these two investments. Stocks have
a higher mean return than home equity. Therefore, investors withdraw money from
the bond markets and invest in the stock markets. An investor also decrease his
housing investment and shift his money to invest in stocks. Financial assets are now
mostly stocks and the ratio remains flat for investors at most ages.
1.7.3 Correlation of Labor Income with Stock Price
Benzoni, Collin-Dufresne, and Goldstein (2004) point out that labor income and
stock return are significantly positively correlated in the long run, although they
have almost zero correlation in the short run. They show that with some degrees of
correlation, young investors find it optimal to short sell stocks. It is useful to explore
the attractiveness of stock relative to home equity when we change its correlation
with labor income. We can see the implications in Figure 1.11, where we consider
27
the case in which labor income growth has a correlation of 0.8 with the stock returns,
and a second case where the correlation is -0.8.
Investors whose labor income is highly correlated with stocks should not invest
heavily in the stock market. These investors optimally favor the bond market. Sur
prisingly, they also have less investment in home equity. A reason might be that a
high correlation of stocks with labor income implies a less advantageous investment
opportunity set for the investors; i.e., they have less wealth. They prudently invest in
the safe asset, bonds, more than in the risky assets, namely, stocks and home equity.
1.8 Conclusions
Regarding the life-cycle allocation of liquid assets to stocks, previous theoretical pre
dictions are at odds with empirical findings. Common wisdom suggests that the
optimal ratio of stocks to liquid assets should be roughly a decreasing function of age.
In the data, however, we observe a hump-shaped pattern.
In this paper, I introduce housing investment as a hedging asset in a life-cycle
portfolio choice model and address the stock allocation puzzle. I show that the "U-
shaped" life-cycle pattern of housing investment influences the hump-shaped stock
investment pattern. Given the co-movements of labor income, house price, and rent,
the house tenure choice is endogenized and housing investment is used to hedge against
both labor income risk and rent risk. The attractiveness of homeownership motivates
investors to become homeowners as early as they can. For young investors with little
savings and low labor income, riskless assets are popular since they guarantee down
payments on housing. With housing investments already made, middle-aged investors
can afford the risk of holding more stocks. My model considers housing investment
and predicts a hump-shaped pattern of stock holding in liquid portfolio over the life
cycle, consistent with the data.
This paper also explores optimal portfolio choice across housing markets. Unlike
the stock market, which is accessible to all investors, housing markets are restricted by
28
locality. In areas where house prices and stock prices move together, the results pre
sented suggest that investors should invest less in stocks and more in bonds. Within a
particular housing market, investors whose labor incomes are poorly correlated with
house price should favor housing investment over stocks.
29
Table 1.1: Benchmark Parameters Parameter Life span Retirement time Risk aversion coefficient Discount factor Elasticity of intertemporal substitution Replacement ratio Housing preference Down payment requirement House selling cost Gross riskfree rate Mean house price appreciation Mean rent appreciation Mean stock return Std. Dev. of labor income shock Std. Dev. of house price appreciation Std. Dev. of rent appreciation Std. Dev. of stock return Correlation (House Price Appreciation, Labor Income Shock) Correlation(Rent Appreciation, Labor Income Shock) Correlation(Stock Return, Labor Income Shock) Correlation(House Price Appreciation, Rent Appreciation) Correlation (House Price Appreciation, Stock Return) Correlation(Rent Appreciation, Stock Return) Minimum House-Price-to-Rent Ratio Maximum House-Price-to-Rent Ratio
Symbol T K
7 0 v> c e 5
<t> r / - l UP-I
PR-1
Ps - 1 Oy
Up
<TR
OS
PYP
PYR
PYS
PPR
PPS
PRS m i n { | } m a x { | }
Value 60 45
5 0.96 0.20 0.68 0.20 0.20 0.06 2%
1.8% 0.4% 6.0%
12.0% 5.5% 3.0%
15.7% 0.14 0.13 0.00 0.30 0.04 0.10
10 30
30
Table 1.2: Cross-Sectional Variations in Labor Income, House Price, Rent, and Stock Price This table reports summary statistics of mean growth rates, standard deviations, and correlations of labor income, house price, rent, and stock price.
PYP
PYR PYS
PPR
PPS PRS
Oy
aP
<7R
C"S
PY UP
PR Ps
Mean 0.16 0.16 0.02 0.42 0.04 0.12
4.7% 6.4% 4.2%
15.7% 0.8% 3.7% 0.2% 8.1%
Stdev 0.21 0.21 0.21 0.13 0.09 0.10
1.6% 2.3% 1.9% 0.0% 1.0% 2.6% 0.9% 0.0%
Min -0.65 -0.42 -0.78 0.16
-0.13 -0.10 1.9% 3.3% 2.9%
15.7% -1.2% 0.1%
-1.1% 8.1%
Max 0.73 0.85 0.56 0.73 0.33 0.36
9.9% 10.2% 13.9% 15.7% 3.8% 8.0% 2.1% 8.1%
Source: Author's calculations using quarterly data from QCEW, OFHEO, Reis, and CRSP. Sample period: 1980Q1 - 2004Q4.
Table 1.3: Utility Cost Calculation (%) This table reports the welfare calculations in the form of standard consumption-equivalent variations. For each rule, I compute the constant consumption stream that gives the investor the same level of utility as the stream that can be financed by the portfolio choice rule. Relative utility losses are measured by the percentage change of this equivalent composite consumption stream when deviating from the optimal portfolio choice rule to the rule considered. The rules considered are: *The portfolio choice rule when housing is excluded completely in the reduced model; **the one when housing is included but its hedging function is shut down; ***the one when house appreciation rate is lowered to the level of the rent appreciation.
Rule Cost
Without Housing* 28.31%
No Renting** 7.94%
p r>*** U — un.
15.84%
31
ouu
250
200
150
100
50
" i •
Wealth
— — Labor Income
i ' • • ' • ' " • ' i > > • •
. K * " * •
/ / / / /
t i
t 1
1 1
f i
t t
t t
1 t
t 1
1 1
t t
t 1
1 t / / /
/ / /
1 1 • 1 1 |
"* *. ^ - W
\ \ \
\
\ \ \ \ \
•
•
--
•
\ \ \ \ \ \ I
\ \ \ \ \ \ \ \ \ I
t
1 .
* J
Figure 1.1: Labor Income, Consumption, and Accumulated Wealth in the Benchmark Model
32
O
(%) M/Hd
(%)d;qsjeuMoeLUOH
t o ^ CM
(%) (8+S)/S s-i
fcuO
33
90
80
70
GO
^ 50
5" 40
30
20
10
1 ——— Wilh H gu sing (Benchmark) j | — ™ - fjo Housing |
I
"
-
"
1
\ t
-
-
J. • •
t-
i \
Figure 1.3: Fraction of Stocks in Financial Assets Without Housing
Fig
ure
1.4:
P
ortf
olio
Cho
ice
Whe
n O
nly
Ren
ting
is
All
owed
co
100 80
60
40
20
0 20
I 1
^^
—
Be
nch
ma
rk
• u
p = u
R
,-
'"S
" m
m
—
""
H
3 .
..
. \
30
40
50
Ag
e 60
70
80
Figu
re 1
.5:
Port
folio
Cho
ice
Whe
n H
ouse
Pri
ce a
nd R
ent
Hav
e Sa
me
App
reci
atio
n R
ate
co
Or
100 80
| 60
a 40
s
20
01—
2
0
Fig
ure
1.6:
P
ortf
olio
Cho
ice
Whe
n R
atio
of
Hou
se P
rice
to
Ren
t is
Fix
ed
^^
—
Ben
chm
ark
- —
•» 5
% D
own
. <
•".«
• 50
% D
own
/ —
' —̂
f N̂
20
30
40
50
A
ge
Fig
ure
1.7:
P
ortf
olio
Cho
ice
wit
h D
iffe
rent
D
own
Pay
men
t R
equi
rem
ents
oo
OS
2.
C I £ o
I
100
80
60
40
20
0 20 —
—
Ben
chm
ark
. _
_-
y=
5
y=1
0
. 1
/ I
j
1 /-
*'
i ,-
• /
\
,.,,
• 1-
"i 1.
1 1 %,]
30
40
50
Age
60
70
80
100
80
60
40
20
\ ' * t "
'•-
-.
H.
. .
i <
#*
^ *'
, 1
1 / *
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41
Chapter 2
Cross-Sectional Portfolio Choice
and the Hedging Functions of
Housing
42
2.1 Introduction
The life-cycle portfolio choice problem has attracted attention from both the academia
and the industry. As the Social Security system becomes less reliable nowadays, it is
increasingly important for investors to understand how to make their own investment
for retirement and to allocate their money in different asset classes over the life cycle.
The literature has proposed various theories and quantitative recommendations
have been made about optimal portfolios at different stages of investors' life cycles.
However, empirical findings are at odds with the popular investment wisdom.1 An
anomaly in the data is the low participation rate and low investment in stocks, despite
the historically high returns on stocks. This anomaly is especially significant for young
investors. Another puzzle is the substantial cross-sectional heterogeneity in portfolio
choice.2
Different forms of market imperfections have been proposed as explanations for
the odds of empirical findings with theoretical recommendations. However, to my best
knowledge most of the papers in the literature to date abstract from investigating a
more realistic portfolio, which consists of not only financial assets but also housing
investment and non-traded human capital. The latter two assets are especially criti
cal. As Curcuru, Heaton, Lucas, and Moore (2004) points out, "[T]o understand the
portfolio allocations of households it is important to examine their financial positions
beyond investment" in marketable securities....[N]on-traded or background risks in the
form of housing, ... human capital and the like, are predicted to have an impact
on portfolio choice." Human capital is like a naturally endowed asset. In incom
plete markets, we have no way to divest it from our portfolio, nor can we write a
contract on the adverse shocks to it. Therefore investors are subject to undiversi-
fiable background risks in labor income. Housing investment, if any, is of a much
1 Curcuru, Heaton, Lucas, and Moore (2004) provides a comprehensive review of the theories in the portfolio choice literature.
2 For a review of the empirical findings, see Ameriks and Zeldes (2004), Curcuru, Heaton, Lucas, and Moore (2004), and Vissing-Jorgensen (2002) for example.
43
larger magnitude than any other asset in a portfolio for the majority of investors.
According to the 2001 Survey of Consumer Finances, housing investment accounts
for about 55% of the average household's wealth, dwarfing stock investment, which
is the second largest investment but accounts for only 12% of wealth for an average
household.3 In addition, housing investment differs from other risky investment in
that housing has the dual role of being a consumption and an investment. Due to the
inseparability of its role as consumption and investment, housing investment accounts
for a big fraction of wealth and has a crowding-out effect on other investments such
as stock and bond holdings.
Another oversight of most empirical papers investigating the life-cycle portfolio-
choice problem is their failure to incorporate the possibility of hedging using different
asset classes in the cross-sectional portfolio choice analysis. When constructing a port
folio, an investor keeps in his mind the diversification effects of investing in different
assets. An investor could utilize the correlations of background and market risks to
construct the optimal portfolio and diversify out the impacts of adverse shocks. For
example, if two investors are identical except that investor A's labor income correlates
less with the local house price than investor B's labor income, then optimally investor
A invests more in housing relative to investor B, since housing is good hedge against
adverse shocks to his labor income. As this paper demonstrates, there is a large cross-
sectional heterogeneity in the covariance matrix of different assets. The differences
in the correlations and volatilities help to explain the cross-sectional heterogeneity in
portfolio choice.
The purpose of this paper is to examine empirically whether households utilize
the hedging functions of different assets in a more realistic portfolio. Unlike the
empirical literature which focused on portfolio choice only at the national aggregate
level, this paper considers heterogeneity across industries and across housing markets
in exploring the Panel Study of Income Dynamics (PSID) data. In particular, I
3 See Yao and Zhang (2005a) for details.
44
explore the portfolio choice problem over the life cycle, taking into consideration the
cross-sectional differences in correlations and volatilities of house price, rent, labor
income and stock price. I obtained a Geocode dataset from the PSID which allows
me to conduct empirical portfolio choice analysis across housing markets.
First, I show that cross-sectionally there are large dispersions in the correlations
and volatilities. For example, households in the petroleum refining and related in
dustries in Columbus, Ohio, had a correlation of labor income shock and house price
appreciation rate of -0.65 over the period of 1980 to 2004. In contrast, households in
the real estate industry residing in Orange County, California, had a correlation of
+0.73. The volatilities in rents and house prices are also very different across local
housing markets, ranging from about 3% per year to about 10%. Other things being
equal, investors facing different correlations and volatilities would invest differently
in housing and other financial assets.
Secondly, I find empirical support that investors use two hedging functions of
housing in forming their portfolio. These two hedging functions of housing are pro
posed in the real estate literature but have not yet been applied to the portfolio choice
problem in finance. The first hedging function is to hedge against labor income risk,
as proposed by Davidoff (2006). Using the University of Minnesotas Integrated Pub
lic Use Microdata (IPUMS) 1990 Census data, Davidoff shows that the co-movement
of house price growth and labor income growth has a negative impact on both the
probability of homeownership and the size of housing investment. The second hedging
function of housing is used to hedge against rent risk, as first documented by Sinai
and Souleles (2005). These two authors point out that renting has risk, as renters are
exposed to annual fluctuations in rent. They develop a house tenure choice model
to show that the probability of homeownership increases with net rent risk. These
authors find empirical evidence from the 1990 and 1999 Current Population Survey
(CPS) March Annual Demographic Supplements data to show this hedging function.
These two papers investigate the two risks separately. In my paper, with both labor
45
income risk and rent risk in existence, the PSID data support both hedging func
tions of housing, while labor income risk seems to outweigh rent risk. I find that the
hedging-against-labor-income function is not only important for house tenure choice
but also for asset allocation: the lower the correlation between labor income shocks
and house price growth rates, the higher the possibility of owning a house and the
higher fraction of wealth is allocated to housing investment.
The rest of this paper is organized as follows: Section 2.2 discusses related papers
in the portfolio literature. Section 2.3 describes the data used. This section also
explains the computations of the key variables in this paper, which are the cross-
sectional correlations and volatilities of labor income, house price, rent, and stock.
Section 2.4 turns to the empirical methodology. Empirical regression results are
reported in Section 2.5. I examine both the determinants of the likelihoods of entering
the housing market and the stock market and the determinants of asset compositions.
Section 2.6 concludes the paper.
2.2 Related Literature
Early papers by Merton (1969) and Samuelson (1969) provide theoretical support
for portfolio choice in complete markets. In a world with complete markets and
no friction, the asset allocation problem does not matter: the two-fund-separation
theorem would tell us that everybody should hold a portfolio consisting of only the
market and the risk-free asset, and that the portfolio is independent of age or wealth;
how each individual investor differs in their portfolio would be determined by the
differences in their risk aversions.
In real life, however, markets are incomplete and there are undiversifiable risks
to the investors. In a more realistic setting, the impacts of undiversifiable labor
income risk have been considered by the theoretical frameworks in Benzoni, Collin-
Dufresne, and Goldstein (2004), Cocco, Gomes, and Maenhout (2005), Gomes and
Michaelides (2004), and Gomes and Michaelides (2005). Empirical investigations of
46
labor income risk include Heaton and Lucas (2000a), which finds weak evidence t ha t
income risk exerts a downward influence on risky financial asset holdings. Heaton and
Lucas (2000b) investigates the 1970-1990 Panel of Individual Tax Returns and finds
extensive heterogeneity in both the volatility of individual income and the correlation
of individual income with stock returns. Vissing-Jorgensen (2002) uses the PSID data
and finds that mean nonfinancial income positively impacts the stock participation
and the share of financial wealth allocated to stocks conditional on participation,
and that the volatility of nonfinancial income has a negative impact on these two
quantities. She finds no evidence of an effect of the correlation of nonfinancial income
with the stock return on portfolio choice.
Real estate exposure is another important background risk, since housing invest
ment outweighs all other investments for the majority of the households in the US.
This important source of background risk is not explored until recently. Theoreti
cal papers include Cocco (2005), Hu (2005), Yao and Zhang (2005a), and Cauley,
Pavlov, and Schwartz (2005). Cocco (2000), Yao and Zhang (2005b), and Li and Yao
(2006). Empirically, Kullmann and Siegel (2005) focuses on real estate crowding-out
effects. They find that the volatility of house price has a negative impact on risky
asset holdings.4
In the finance literature, some recent papers started to look at the effects of
covariances among multiple risky assets on portfolio choice. For example, Davis and
Willen (2000a) and Davis and Willen (2000b) research the covariance of occupation-
level income shocks and asset returns and its implications for portfolio choice. Using
a simple graphical approach to portfolio selection, the authors show that covariance
between income shocks and asset returns indeed affect portfolio choice over the life
cycle. Using repeated cross sections of the Current Population Survey from 1967
to 1994, they group households into different levels of education, occupation and
sex, and measure the correlation between market returns and labor income for each
4 Some other examples are Ameriks and Zeldes (2004), Bertaut and Starr-McCluer (2002), Friend and Blume (1975), and Poterba and Samwick (1997).
47
group. They find significant differences in portfolio choice between these groups.
Another paper that considers covariance of background risk and market risk is by
Massa and Simonov (2006). They use a unique panel of Swedish data and consider
how the covariance could help households hedge non-tradable risks. Surprisingly, they
find little evidence that Swedish people use individual stocks to hedge non-financial
income risk. Their conclusion is that these households have a tendency to invest in
stocks that they are familiar with in terms of geography or professional proximity.
Though similar to my paper, the two papers above focus themselves to stocks only
and ignore housing investment.
There are two hedging functions of housing are proposed in the real estate litera
ture but have not yet been applied to the portfolio choice problem in finance. Davidoff
(2006) argues that housing investment has a hedging function against labor income
risk. Davidoff shows that the co-movement of house price growth and labor income
growth has a negative impact on both the probability of homeownership and the size
of housing investment. Sinai and Souleles (2005), on the other hand, point out that
renting has risk, as renters are exposed to fluctuations in rent. They develop a house
tenure choice model to show that the probability of homeownership increases with
net rent risk. These two papers investigate the two risks separately.
My paper combines the two hedging functions of housing in the real estate lit
erature and the portfolio choice literature and explore the covariances among labor
income, house price, rent, and stock price. I then investigate whether households
utilize the correlation structure in hedging and how the correlations and volatilities
influence the portfolio choice over the life cycle.
In my paper, with both labor income risk and rent risk in existence, the PSID
data support both hedging functions of housing. I find that the hedging-against-
labor-income function is not only important for house tenure choice but also for asset
allocation: the lower the correlation between labor income shocks and house price
growth rates, the higher the possibility of owning a house and the higher fraction of
48
wealth is allocated to housing investment.
2.3 Data
Two sets of data are used in this paper. The first dataset contains time series of
house price, rent, labor income, and stock price for ten industries and forty-seven
Metropolitan Statistical Areas (MSAs) over the period 1980 and 2004. For each
industry and each MSA, I compute the correlations among the four variables as
well as their volatilities and mean growth rates. The cross-sectional correlations and
volatilities will then be used in the empirical investigation of portfolio choice.
The second dataset is the PSID data which contains information about households'
investments in housing, stocks and other financial assets, along with information
on the household's income and demographic characteristics. The availability of a
Geocode data allows for analysis of cross-sectional portfolio choice.
2.3.1 House Price, Rent , Labor Income, and Stock Price
As explained in Introduction, an optimal portfolio will take into consideration hedging
among different assets. Therefore the key variables in this empirical cross-sectional
analysis include the correlations of house price, rent, stock price, and labor income for
each industry and for each MSA. In addition, the volatilities of these four time series
are also important determinants for the heterogeneity of portfolio choice. This paper
pulls data on house price, rent, stock price, and labor income from various sources
and estimates the above key variables.
House prices at the MSA level are the House Price Indexes (HPIs) obtained from
the Office of Federal Housing Enterprise Oversight (OFHEO). The OFHEO HPI is
a broad measure of the movement of single-family house prices. OFHEO reviews
repeat mortgage transactions on single-family properties, of which these mortgages
were purchased or securitized by Fannie Mae or Freddie Mac. Since it has a good
49
control of the quality of the houses measure over time, the OFHEO HPI is one of the
most popular house price indexes used. Although OFHEO provides MSA-level HPI
starting from 1975, I only use the period 1980 to 2004 due to the availability of rent
data and labor income data, which are described below.
Following Himmelberg, Mayer, and Sinai (2005), rent is proxied by the Rent
Index (RI) from Reis, Inc. Reis is a real estate consulting firm that provides quarterly
average rent for a "representative" two-bedroom apartment in a metro area. Similar to
the OFHEO's quality control of house price indexes, Reis attempts to hold the quality
of the rental unit constant over time such that their rent data are representative. I
obtained from Reis rent indexes of 47 MSAs over the period 1980Q1 to 2004Q4.
Labor income is proxied by the mean wage data from the Bureau of Labor Statis
tics (BLS). The BLS provides mean wage data at the industry level and at the MSA
level. Quarterly data are available for the period of 1989 to 2000. Prior to 1988,
only annual data are available, starting from 1975. I use annual data up to 1988 and
quarterly data thereafter. Due to the small sample I obtained from the PSID, I group
households according to their SIC codes to the 10 SIC divisions as defined by the US
Department of Labor.5 I compute the mean wage for these 10 SIC divisions and use
them to proxy for labor income in these industries.
I use the S&P500 index from the Center for Research in Security Prices (CRSP)
to compute stock returns. In matching the range and frequency of the other three
time series, I use quarterly stock data from 1980Q1 to 2004Q4.
The sample has in total 47 MSAs and 10 industries. For each industry and each
MSA, I use the full period 1980Q1 to 2004Q4 to compute the correlations among
house price, rent, labor income and stock. The same full sample is used to compute
the mean growth rates and volatilities of these four time series. Table 2.1 displays the
dispersions of these correlations, mean growth rates, and volatilities across industries
and across MSAs. Notations Y, P, R, S stand for labor income, house price, rent,
5 http://www.osha.goy/pls/imis/sic_manual.html
50
and stock, respectively. I use pXy to denote the correlation of two time series X and
Y. For example, pyp indicates the sample correlation of labor income in a certain
industry with the house price in an MSA. I use ax to denote the volatility of a time
series X and nx to indicate its mean growth rate.
2.3.2 The PSID Family Files and Wealth Supplements Data
This paper uses household-level data from the Panel Study of Income Dynamics
(PSID) data in the cross-sectional portfolio choice analysis. In particular, this paper
uses the most recent Family Files and the Wealth Supplements in 1984, 1989, 1994,
1999, 2001, 2003 and 2005. The PSID provides data on the household's portfolio
choice, including stock participation, amount of stock investment and other risky
financial investments, home ownership status, value of house, mortgage, and other
assets and debts. Besides these, the PSID data also provides information about the
respondent's demographics such as income, age, sex, marital status, education, health
situation, number of children in the household, etc. The cross-sectional sample in this
paper is the Survey Research Center sample of the PSID which was representative of
households from the 48 contiguous states of the US.
From the 1984-2005 family files I construct a data set containing financial and
demographic information for each of the households in the sample. Since the PSID
data only provides wealth information at the household level, I choose to use the
family files rather than the individual files. For each household, I obtain information
on household income collected in the previous years to the survey years. Wealth
Supplements files are used to obtain information on the households' positions in
cash, bonds, stocks, other business wealth, housing investment and other real estate
investments. These wealth files are imputed from the family files. To avoid data
imputation errors, variables in the family files are compared to the imputed values in
wealth files for corrections. Following Vissing-Jorgensen (2002), I leave topcoding of
51
wealth or income variables at their topcodes since they are rare.
In addition, I obtained a Geocode data from the PSID which provides information
on where the respondents live at the Metropolitan Statistical Areas (MSA) level and
also on the respondents' occupational industries. This makes it feasible to conduct
the cross-sectional analysis in this paper.
The appendix describes the PSID data in details.
2.4 Empirical Methodology
Following Kullmann and Siegel (2005), I investigate the participation decision for all
households but the asset allocation decision only for participating households.
2.4.1 Stock Participation and Home Ownership Decisions
I estimate the likelihood of stock ownership using
OwnSij = l{LevSlt>0} (2.1)
LevS*, = {3sX*t + e?t (2.2)
where OwnSij is the binary selection rule of whether the household owns stock di
rectly or indirectly through IRAs. It takes value one if the household owns stocks and
zero otherwise. OwnSij depends on an unobserved desired level of stock investment,
LevS*t) which is assumed to be a linear function of predetermined and exogenous
variables Xft. Household i participates in the stock market in period t only if his
desired level of stock investment LevS*t is nonnegative. eft denotes the error term.
Similarly I model the likelihood of house ownership using
OwnHitt = l{LevHlt>0} (2 .3)
LevH*t = PHX» + e% (2.4)
52
where OwnHi<t is a dummy variable for the home ownership of the household, which
is one if the household owns a house and zero otherwise. OwnHiit depends on an
unobserved desired level of housing investment, LevH*t, which is assumed to be a
linear function of predetermined and exogenous variables Xft. Household i owns a
house in period t if the desired level of housing investment LevH*t is nonnegative.
eft denotes the error term.
I estimate the likelihood of owning a house and that of owning stocks using Probit
regressions. In the explanatory variable vectors Xft and Xft) I include some of the
correlations and volatilities estimated in 2.3.1. For example, in examining whether
households use housing investment to hedge against labor income risk, I include pyp
and investigate if this correlation negatively affects the probability of purchasing a
house. To explore the hypothesis that households use homeownership to hedge against
rent risk, I examine if up is positively associated with higher chances of owning a house
and more wealth invested in housing.
Following Kullmann and Siegel (2005) and Vissing-Jorgensen (2002), I include the
following control variables in Xft and Xft in the participation regressions as well as
in the asset allocation regressions:
Ageiit, Age2it: Age and age squared of the household's head. With a limited-
time investment horizon and in incomplete markets, investment opportunities are
time varying, so optimal portfolio choice depends on age. Previous studies have
confirmed that portfolio choice is a life-cycle problem. Investors of different ages have
participation rates and they allocate different shares of their financial wealth to stocks
and housing.
Incomeij: Total income of the household. In the PSID questionnaires, the re
spondents are asked about their incomes in the previous year.
Male^t' A dummy variable whether the household head is male. It takes value 1
if the household head is male and 0 otherwise.
Whiten'. A dummy variable indicating whether the household head is white. A
53
value of one indicates yes.
Healthy^: A dummy variable indicating whether the household head is in healthy
condition.
NumKidSi/. The number of children in this household.
Marriedij. Dummy variables whether the household head is married. A value of
one indicates yes and zero indicates no.
College^, Graditf A dummy variable whether the household head has a col
lege degree or a postgraduate degree. Education could be a significant factor if risk
aversion differ across education groups.
Busiitt: A dummy variable whether the household has other business income. It
takes value one if the answer is yes.
REitt: A dummy variable whether the household has other real estate investment.
It takes value one if the answer is yes.
BusiToNWi/. For those who invest in other business, this is the share of networth
allocated to other business investments.
REToNWitt'. For those who invest in other real estate, this is the share of networth
allocated to other real estate investments.
Table 2.2 and 2.3 provide summary statistics of the variables used in this pa
per. All dollar values are deflated to 1984 dollars using the Consumer Price Indexes
(CPI). Most of the variables used in the participation regressions here will be used in
estimating asset allocation decisions in the next section.
2.4.2 Asset Allocation Decision
I denote s^t as the share of wealth invested in stocks. si)t is only observed for those
households who participates in the stock market (OwnSij — 1), therefore I only
keep stockholders in the regressions of stock allocation. I denote hi>t as the ratio of
housing investment to the household' net worth. Similarly, only for the homeowners
54
(OwnHij — 1) do we observe /ii>t. I model si)t and hitt as follows:
Si,t = lSXlt + 4 (2.5)
hit = yHX% + e?t (2.6)
To deal with the sample selection problem, I use the two-step Heckman estimations
for both housing investment and stock investment.
2.5 Results
2.5.1 Heterogeneity of Correlations and Volatilities
Across industries and MS As, the dispersions in the correlations and volatilities are
large. For example, households in the petroleum refining and related industries in
Columbus, Ohio, had a correlation of labor income shock and house price appreciation
rate is -0.65 over the period of 1980 to 2004. In contrast, households in the real
estate industry residing in Orange County, California, had a correlation of +0.73. In
comparison, the correlations of house price with stock are not as volatile as those of
house price with labor income. The lowest correlation of house price and stock is -0.13
(households in the real estate industry in Sacramento, California) and the highest is
0.33 (households in the electronic and other electrical equipment and components
industries in Memphis, Tennessee).
The large differences have an important implication on asset allocations. An im
portant feature of housing investment is its locality. Unlike stocks, which can be
accessed by investors nationwide, housing investment is restricted by locality. The
majority of investors face difficulty when investing in housing markets where they
do not reside. For a typical investor who can only invest in his own housing mar
ket, the comovements of housing markets with the stock markets and with the local
labor markets influence the determination of portfolio choice. Other things being
55
equal, these investors would invest differently in housing and other financial assets.
Prom the perspectives of hedging uninsurable labor income risks, we should expect
that investors who have high correlations of labor income shocks and house price
appreciations would tend to invest less in housing, other things being equal, and vice
versa.
2.5.2 Likelihood of Participation
Table 2.4 and Table 2.5 present the results in the Probit regressions for home owner
ship and stock market participation, respectively. The dependent variables in these
two tables are dummy variables which take values of one if the household owns a
house (or stock) and zero otherwise. The coefficients are the marginal effects of the
independent variables evaluated at the sample means. I use maximum likelihood es
timation and the lines below reported estimated coefficients are t-statistics robust to
heteroscedasticity and autocorrelation in addition to controlling for time effect and
individual effect. Table 2.4 and Table 2.5 both have estimations of four models.
Home Ownership
The biggest investment decision for most households is housing investment. Hous
ing investment has a hedging function against labor income risk, as proposed by
Davidoff (2006). Using the University of Minnesotas Integrated Public Use Micro-
data (IPUMS) 1990 Census data, Davidoff shows that the co-movement of house price
growth and labor income growth has a negative impact on both the probability of
homeownership and the size of housing investment.
In the PSID data, the households also seem to use housing investment to hedge
against their non-tradable background labor income risk. Model II and Model IV in
Table 2.4 show that the coefficients of pyp are -0.275 and -0.257, respectively, and these
two coefficients are statistically significant at 5% degree of confidence. This implies
that people whose labor income shocks have lower correlation with house price growth
56
rates have a higher tendency to own a house. Decreasing the correlation by 1 increases
the likelihood of owning a house by about 27.5%. This is not surprising, since when
investors construct their investment portfolio, they keep the diversification effect in
their minds. If labor income is naturally endowed and we cannot divest it from our
portfolio, we will be in favor of a second asset, housing investment in this case, which
will hedge the risks in our first asset, namely, human capital.
The second hedging function of housing is used to hedge against rent risk, as
first documented by Sinai and Souleles (2005). These two authors point out that
renting has risk, as renters are exposed to fluctuations in rent. They develop a house
tenure choice model to show that the probability of homeownership increases with
net rent risk. These authors find empirical evidence from the 1990 and 1999 Current
Population Survey (CPS) March Annual Demographic Supplements data to show
that rent risk encourages home ownership.
Here I find similar patterns using the PSID data. In Model III and Model IV of
Table 2.4, the coefficients on rent volatility are both statistically significant around
0.1%. This means that in areas where the volatilities of rent are higher, people have
a higher tendency to own a house in order to hedge themselves against the rent risk.
Rent risk is still significant in the present of labor income risk, as Model IV shows.
Compared with the labor income risk, however, rent risk is of smaller magnitude.
Increasing the volatility of rent by one only increases the likelihood of owning a house
by 0.1%.
The hedging functions to labor income risk and rent risk mentioned above are
after controlling for other demographics. Over the life cycle, the likelihood of owning
a house displays a humped-shape pattern, with the coefficient on Age being positive
(0.087) and the coefficient on Age2 being negative (-0.0006). Moreover, homeowners
are more likely to be people with higher income, healthy, married, having more kids,
female or white household heads. Having other business income and other real estate
investment also increases the probability of owning a house. However, conditional on
57
having other business income or other real estate investment, the shares of networth
allocated to other business or other real estate investment clearly decreases the likeli
hood, reflecting the crowding-out effects of other investments to housing investment.
Stock Ownership
After the consideration of investing in housing, the investors consider whether they
should participate in the stock market to take advantages of its high returns over the
long rum.
Homeowners are more likely to participate in the stock markets than renters,
with the difference of likelihood being about 70%. In general, homeowners are more
wealthier than renters. With higher savings, homeowners have higher capacity in
dealing with adverse shocks to their labor incomes. They are less risk averse and can
take higher risks in investing their money. Stocks, which are risky in the short run
but pay off nicely in the long run, are popular to homeowners.
Surprisingly the PSID respondents do not seem to use stock as a hedging asset for
their labor income risk. If they did, stock would be more popular to those investors
whose labor income shocks are less correlated to stock returns. However, in Table
2.5, we see the opposite. The coefficients on pya are statistically positive at about
0.532, which means an increase of the correlation by one increase the probability of
owning stocks by 53%. Massa and Simonov (2006)'s familiarity story might explain
this finding. In their paper, Massa and Simonov use a unique panel of Swedish data
and investigate whether the Swedish investors use individual stocks to hedge non-
tradable risks. Their dataset have detailed time series information on income and
portfolio composition. Their finding is that Swedish investors do not use individual
stocks to hedge against labor income risks. Instead, the stock investments are to
particular stocks which are familiar to the investors in terms of geography or profes
sional proximity. Here, the PSID data also shows little evidence for using stocks for
hedging.
58
People who invest in stocks are more likely to be those with higher income, fe
male, white, healthy, with less children, married, have higher education. Again, hav
ing other business income or other real estate investment are associated with higher
probability of owning stocks. Conditional on participation in these two areas, how
ever, more investments in these two assets lead to lower likelihood of participating in
the stock market.
2.5.3 Asset Allocation
In this section I look at investors' allocation of their wealth to housing investment and
stock investment. Table 2.6 and Table 2.7 present the results. The dependent vari
ables are the ratios of the dollar amounts of housing investment and stock investment,
respectively, to networth. Dependent variables are those used in the Probit regres
sions in Section 2.5.2. Since only investors who have participated have investments,
in this section regressions are conducted for participants only, that is, homeowners or
stock owners.
Figure 2.1 shows that for a typical investor the housing investment is a "U"-shaped
function of age, with the lowest ratio of housing investment to age around retirement
age of 65. In contrast, stock investment is a humped-shaped function of age, as Figure
2.2 indicates. Investors around age of 60 have the highest fraction of wealth in the
stock markets.
Housing Investment
First let us consider how investor invest in the biggest and most important asset,
housing. Table 2.6 shows the results. Over the life cycle, housing investment displays a
"U" shaped pattern, with the coefficient on age significant at -0.165 and the coefficient
on age2 as 0.0012. Costly housing adjustment explains this first-decline-then-rise life-
cycle pattern of housing investment relative to wealth. When it is costly to move,
investors tend to stick to their houses for a longer time, which means the house values
59
stay relatively constant for a long period. In contrast, wealth accumulation exhibits
a humped-shape pattern over the life cycle: Young investors just start their career
and have less saving. Middle-aged investors in general earn more and they have
accumulated years of savings. Elderly investors are approaching the end of their lives
so they start to decumulate their wealth. Therefore, the ratio of house investment to
wealth exhibits a "U" shaped pattern, first going down then climbing up.
People who have a lower ratio of housing investment to wealth are more likely to
be those with higher income, white, have more children, having other business income
or other real estate income. The more allocation of wealth to other business or other
real estate investment, the less is left for housing investment.
pyp has a negative coefficient of about -0.767, indicating a hedging motive to invest
in housing against adverse shocks to labor income. Rent volatility ar is not significant
here. This comparison further confirms that housing's hedging-against-labor-income-
risk function is more important than the function to hedge against rent risk.
Stock Investment
Allocation to stock investment is another big decision for the investors. The regression
results are presented in Table 2.7. Similar to the findings in the determinants for stock
market participation, the correlation between stock returns and labor income shocks is
positively associated with wealth allocation to stock investment. With the correlation
increased by one, there is about 14.6% increase of wealth invested in stocks. From the
hedging prospective, this is surprising, since rational investors should decrease their
exposure to stocks if stock returns are highly correlated with their labor income. An
explanation is again Massa and Simonov (2006) 's familiarity story.
The life-cycle pattern of wealth allocation to stock is humped-shaped, with the
coefficient on age significantly at 0.0045 and the coefficient on age2 as -0.00003. Higher
allocation to stock is associated with higher income. Households who invest higher
fraction of wealth in the stock markets are more likely to have less children and higher
60
education.
2.6 Conclusion
This paper revisits the heterogeneity in cross-sectional portfolio choice. It investigates
a realistic portfolio consisting of not only financial assets but also housing investment
and non-traded human capital. In particular, it incorporates the impacts of covari-
ances of labor income, house price, rent, and stock price on portfolio choice over the
life cycle. The PSID Data and its Geocode data are used to conduct empirical anal
ysis across industries and across MSAs. I find large dispersions in the correlations of
labor income, house price, rent and stock price. The cross-sectional variations allows
for the possibility to use multiple assets as hedge for labor income risk and rent risk.
The data show that households seem to use housing investment as a hedge against
labor income risk and rent risk. The lower the correlation between labor income
shocks and house price appreciation, or the higher the rent volatility, the more likely
a household owns a house and the greater the fraction of wealth allocated to housing
investment. I find little evidence for households using stock investment as a hedge
against labor income risk.
61
Table 2.1: Cross-sectional Summary Statistics of p, a, and p,
This table presents the summary statistics of correlations (p), mean growth rates (p), and volatilities (a) of labor income (Y), house price (JP), rent (R), and stock (S) over the period of 1980 to 2004. For each industry and each MSA, these variables are computed using the full sample. The numbers are across all industries and MSAs.
Variable
PYP
PYR
PYS
PPR
PPS
PRS
uY
Op
°R crs HY
PP
PR PS
Obs 365 365 365 376 376 376 365 376 376 376 365 376 376 466
Mean 0.16 0.16 0.02 0.42 0.04 0.12
4.7% 6.4% 4.2%
15.7% 0.8% 3.7% 0.2% 8.1%
Std. Dev. 0.21 0.21 0.21 0.13 0.09 0.10
1.6% 2.3% 1.9% 0.0% 1.0% 2.6% 0.9% 0.0%
Min -0.65 -0.42 -0.78 0.16
-0.13 -0.10 1.9% 3.3% 2.9%
15.7% -1.2% 0.1%
-1.1% 8.1%
Max 0.73 0.85 0.56 0.73 0.33 0.36
9.9% 10.2% 13.9% 15.7% 3.8% 8.0% 2.1% 8.1%
Source: Author's calculations.
62
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Table 2.4: Likelihood of Owning a House
This table presents the marginal effects of Probit regressions evaluated at sample means. In each column, the first row are the parameter estimates and the second row are the t-statistics.
OwnH busi_nw
re_nw
busi
realestate
income
income2
age
age2
male
white
healthy
numkids
married
college
PYP
OR
Year Effect N
pseudo B?
(1) -0.492** (-2.78)
-1.602** (-9.23)
0.161** (2.33)
0.700** (8.03)
0.0234** (11.82)
-0.0000936** (-6.65)
0.0871** (10.43)
-0.000636** (-6.86)
-0.435** (-8.45) 0.324** (7.04)
0.203** (2.93)
0.0879** (4.68)
0.871** (18.28) 0.0219 (0.62)
Yes 7888 0.280
(2) -0.496** (-2.80)
-1.606** (-9.25) 0.162** (2.34)
0.709** (8.12)
0.0235** (11.82)
-0.0000932** (-6.61)
0.0857** (10.25)
-0.000622** (-6.70)
-0.437** (-8.48)
0.315** (6.81)
0.197** (2.84)
0.0869** (4.62)
0.873** (18.32) 0.0232 (0.65)
-0.275** (-2.90)
Yes 7888 0.281
(3) -0.492** (-2.78)
-1.613** (-9.29)
0.161** (2.33)
0.706** (8.10)
0.0235** (11.86)
-0.0000940** (-6.68)
0.0869** (10.42)
-0.000634** (-6.85)
-0.431** (-8.37) 0.323** (7.01)
0.201** (2.90)
0.0874** (4.65)
0.869** (18.24) 0.0222 (0.63)
0.00112** (2.23) Yes 7888 0.281
(4) -0.495** (-2.80)
-1.615** (-9.30) 0.162** (2.34)
0.714** (8.17)
0.0235** (11.86)
-0.0000935** (-6.64)
0.0857** (10.25)
-0.000621** (-6.70)
-0.433** (-8.40)
0.314** (6.80)
0.195** (2.82)
0.0865** (4.60)
0.872** (18.28) 0.0233 (0.66)
-0.257** (-2.69)
0.000982* (1.94) Yes 7888 0.281
Table 2.5: Likelihood of Owning Stocks
Note: Marginal Effects of Probit regression evaluated at sample mean, rows are parameter estimates and the second rows are t-statistics.
OwnS vh_nw
busi_nw
r e j r w
ownhouse
busi
rea les ta te
income
income 2
age
age 2
male
whi te
hea l thy
numkids
marr ied
college
PYS
age*pYS
ay
age*o-y
Year Effect N
pseudo .ft2
(1) -0.215** (-16.23) -0.552** (-3.50)
-0.843** (-5.40)
0.713** (14.99) 0.246** (4.32)
0.342** (5.10)
0.0272** (14.68)
-0.0000954** (-7.43)
0.00806 (1.00)
-0.0000696 (-0.79)
-0.0897* (-1.72)
0.278** (6.16)
0.253** (3.66)
-0.0651** (-3.94) 0.0711 (1.48)
0.284** (9.03)
Yes 7888 0.205
(2) -0.214** (-16.13)
-0.553** (-3.50)
-0.840** (-5.37)
0.706** (14.84) 0.255** (4.47)
0.340** (5.06)
0.0271** (14.62)
-0.0000950** (-7.40)
0.00767 (0.95)
-0.0000746 (-0.85)
-0.0877* (-1.68)
0.280** (6.19)
0.255** (3.69)
-0.0659** (-3.98) 0.0735 (1.53)
0.280** (8.87)
0.532** (2.04)
-0.00592 (-0.97)
Yes 7888 0.207
(3) -0.215** (-16.24) -0.558**
(-3.54) -0.845**
(-5.40) 0.708** (14.88) 0.246** (4.33)
0.343** (5.11)
0.0273** (14.72)
-0.0000959** (-7.46)
0.0000535 (0.01)
-0.0000746 (-0.85)
-0.0905* (-1.74)
0.280** (6.20)
0.254** (3.67)
-0.0657** (-3.97) 0.0715 (1.49)
0.285** (9.04)
-1.051** (-2.87)
0.0216** (2.58)
Yes 7888 0.206
(4) -0.214** (-16.13) -0.558**
(-3.53) -0.841**
(-5.38) 0.702** (14.74) 0.255** (4.47)
0.340** (5.06)
0.0272** (14.65)
-0.0000954** (-7.42)
-0.000553 (-0.06)
-0.0000801 (-0.91)
-0.0879* (-1.68)
0.282** (6.23)
0.256** (3.70)
-0.0665** (-4.01) 0.0738 (1.54)
0.280** (8.87)
0.545** (2.08)
-0.00629 (-1.02)
-1.047** (-2.86)
0.0221** (2.63)
Yes 7888 0.207
Table 2.6: Ratios of Housing Investments to Wealth
Note: Robust regression estimates. The first rows are parameter estimates and the second rows are t-statistics.
h_nw busi-nw
re_nw
busi
real estate
income
income2
age
age2
male
white
healthy
numkids
married
college
PYP
&ge*pyp
<7R
age*(Tjj
Inverse Mills Ratio
Year Effect N
R2
(1) -1.216** (-5.63) 0.188 (0.76)
-0.266** (-3.85)
-0.684** (-7.57)
-0.00948** (-2.86)
0.0000161 (0.89)
-0.165** (-10.70)
0.00122** (8.47) -0.115 (-1.12)
-0.184** (-2.55)
-0.211** (-2.28)
0.0892** (3.98) 0.0510 (0.39) 0.0210 (0.51)
-2.085** (-13.08)
Yes 7888 0.122
(2) -1.209** (-5.58) 0.212 (0.86)
-0.271** (-3.90)
-0.692** (-7.63)
-0.0101** (-3.07)
0.0000193 (1.07)
-0.167** (-10.90)
0.00124** (8.65)
-0.0978 (-0.96)
-0.198** (-2.75)
-0.216** (-2.33)
0.0861** (3.85) 0.0212 (0.16) 0.0197 (0.48)
-0.761* (-1.77)
0.0162* (1.70)
-2.069** (-13.05)
Yes 7888 0.123
(3) -1.212** (-5.59) 0.198 (0.80)
-0.267** (-3.86)
-0.689** (-7.60)
-0.00964** (-2.91)
0.0000169 (0.93)
-0.165** (-10.75)
0.00123** (8.51) -0.112 (-1.10)
-0.186** (-2.58)
-0.211** (-2.29)
0.0889** (3.96) 0.0454 (0.35) 0.0207 (0.51)
-0.000222 (-0.08)
-0.00000126 (-0.02)
-2.092** (-13.15)
Yes 7888 0.123
(4) -1.210** (-5.57) 0.221 (0.89)
-0.271** (-3.91)
-0.695** (-7.64)
-0.0103** (-3.10)
0.0000199 (1.10)
-0.167** (-10.94)
0.00125** (8.68)
-0.0971 (-0.96)
-0.198** (-2.76)
-0.216** (-2.33)
0.0860** (3.84) 0.0181 (0.14) 0.0191 (0.47)
-0.777* (-1.80)
0.0163* (1.71)
-0.000853 (-0.32)
0.0000105 (0.18)
-2.076** (-13.11)
Yes 7888 0.124
67
Table 2.7: Proportion of Wealth Invested to Stocks
Note: Robust regression estimates. The first rows are parameter estimates and the second rows are t-statistics.
SJ1W
vh_nw
busi-nw
re_nw
ownhouse
busi
rea les ta te
income
income 2
age
age 2
male
whi te
hea l thy
numkids
marr ied
college
PYS
&ge*pYS
CTy
age* cry
Inverse Mills Ra t io
Year Effect N
R2
(1) -0.0503**
(-10.02) -0.134**
(-7.03) -0.260** (-10.23)
0.0905** (5.28)
0.0173** (2.03)
0.0560** (5.40)
0.00656** (10.19)
-0.0000235** (-7.96)
0.00451** (4.04)
-0.0000316** (-2.61)
-0.00775 (-1.05)
0.0746** (8.63)
0.0638** (5.88)
-0.0170** (-6.49)
-0.00352 (-0.51)
0.0581** (7.93)
0.194** (5.90)
Yes 7888 0.162
(2) -0.0491**
(-9.96) -0.131**
(-6.95) -0.256** (-10.15.)
0.0860** (5.12)
0.0178** (2.09)
0.0540** (5.25)
0.00641** (10.12)
-0.0000229** (-7.86)
0.00428** (3.83)
-0.0000327** (-2.70)
-0.00675 (-0.92)
0.0736** (8.55)
0.0633** (5.85)
-0.0169** (-6.46)
-0.00327 (-0.48)
0.0558** (7.79)
0.141** (3.72)
-0.00202** (-2.37)
0.187** (5.79)
Yes 7888 0.162
(3) -0.0512**
(-10.21) -0.138** (-7.21)
-0.264** (-10.36)
0.0921** (5.42)
0.0186** (2.18)
0.0571** (5.51)
0.00669** (10.39)
-0.0000241** (-8.15)
0.00287** (2.44)
-0.0000330** (-2.73)
-0.00862 (-1.17)
0.0761** (8.78)
0.0650** (5.99)
-0.0174** (-6.64)
-0.00324 (-0.47)
0.0592** (8.10)
-0.241** (-4.48)
0.00453** (3.77)
0.199** (6.10)
Yes 7888 0.163
(4) -0.0500**
(-10.17) -0.135**
(-7.14) -0.260** (-10.29)
0.0878** (5.27)
0.0191** (2.24)
0.0552** (5.37)
0.00656** (10.33)
-0.0000235** (-8.06)
0.00262** (2.22)
-0.0000343** (-2.83)
-0.00754 (-1.03)
0.0752** (8.71)
0.0646** (5.97)
-0.0174** (-6.62)
-0.00297 (-0.43)
0.0570** (7.97)
0.146** (3.83)
-0.00214** (-2.50)
-0.236** (-4.41)
0.00457** (3.79)
0.193** (6.00)
Yes 7888 0.163
68
To 40 60 80 100 Age
Figure 2.1: Housing Investment vs. Age
20 4o" 3) llQ 100 Age
Figure 2.2: Stock Investment vs. Age
Bibliography
Ameriks, J., and S. P. Zeldes, 2004, "How Do Household Portfolio Shares Vary With Age?," working paper, Columbia Business School.
Anderson, R. N., 2001, "United State Life Tables 1998," National Vital Statistics Reports, 48, 1-40.
Barberis, N., 2000, "Investing for the Long Run When Returns Are Predictable," The Journal of Finance, 55, 225-264.
Benzoni, L., P. Collin-Dufresne, and R. S. Goldstein, 2004, "Portfolio Choice over the Life-Cycle in the Presence of Trickle Down Labor Income," working.
Bertaut, C , and M. Starr-McCluer, 2002, "Household portfolios in the United States," in Household Portfolios, ed. by J. T. Guiso L, Haliassos M. The MIT Press.
Bertaut, C. C , 1998, "Stockholding Behavior Of U.S. Households: Evidence From The 1983-1989 Survey Of Consumer Finances," The Review of Economics and Statistics, 80, 263-275.
Bertaut, C. C , and M. Haliassos, 1997, "Precautionary Portfolio Behavior from a Life-Cycle Perspective," Journal of Economic Dynamics and Control, 21, 1511-1542.
Brandt, M. W., 2004, "Portfolio Choice Problems," .
Brueckner, J. K., 1997, "Consumption and Investment Motives and the Portfolio Choices of Homeowners," Journal of Real Estate Finance and Economics, 15:2, 159-180.
Campbell, J., and J. Cocco, 2005, "How Do House Prices Affect Consumption? Evidence From Micro Data," working.
Campbell, J. Y., and L. M. Viceira, 1999, "Consumption and Portfolio Decisions When Expected Returns Are Time Varying," The Quarterly Journal of Economics, 114, 433-495.
Canner, N., N. G. Mankiw, and D. N. Weil, 1997, "An Asset Allocation Puzzle," The American Economic Review, 87.
Cauley, S. D., A. D. Pavlov, and E. S. Schwartz, 2005, "Homeownership as a Constraint on Asset Allocation," .
69
70
Cocco, J., 2000, "Hedging House Price Risk With Incomplete Markets," working.
, 2005, "Portfolio Choice in the Presence of Housing," The Review of Financial Studies, 18, 535-567.
Cocco, J. F., F. J. Gomes, and P. J. Maenhout, 2005, "Consumption and Portfolio Choice over the Life Cycle," Review of Financial Studies, 18.
Curcuru, S., J. Heaton, D. Lucas, and D. Moore, 2004, "Heterogeneity and Portfolio Choice: Theory and Evidence," working.
Davidoff, T., 2006, "Labor Income, Housing Prices and Homeownership," Journal of Urban Economics, 59, 209-235.
Davis, S. J., and P. Willen, 2000a, "Occupation-Level Income Shocks and Asset Returns: Their Covariance and Implications for Portfolio Choice," working.
, 2000b, "Using Financial Assets to Hedge Labor Income Risks: Estimating the Benefits," working.
Elmendorf, D. W., and M. S. Kimball, 2000, "Taxation of Labor Income and the Demand for Risky Assets," International Economic Review, 41.
Faig, M., and P. Shum, 2002, "Portfolio Choice in the Presence of Personal Illiquid Projects," Journal of Finance.
Flavin, M., and T. Yamashita, 1998, "Owner-Occupied Housing and the Composition of the Household Portfolio over the Life Cycle," .
, 2002, "Owner-Occupied Housing and the Composition of the Household Portfolio," The American Economic Review, March.
Fratantoni, M., 1998, "Homeownership and Investment in Risky Assets," Journal of Urban Economics, 4A, 27-42.
, 2001, "Homeownership, Committed Expenditure Risk, and the Stockholding Puzzle," Oxford Economic Papers, 53, 241-59.
Friend, I., and M. E. Blume, 1975, "The Demand for Risky Assets," The American Economic Review, 65, 900-922.
Gomes, F., and A. Michaelides, 2004, "A Human Capital Explanation for an Asset Allocation Puzzle," working.
, 2005, "Optimal Life-Cycle Asset Allocation: Understanding the Empirical Evidence," Journal of Finance, LX:2.
Greene, W., 2000, Econometric Analysis. Prentice Hall.
Haliassos, M., and C. C. Bertaut, 1995, "Why do so Few Hold Stocks?," Economic Journal, 105, 1110-1129.
Han, L., 2004, "The Effects of Price Uncertainty on Housing Demand in the Presence of Lumpy Transaction Costs," working.
Heaton, J., and D. Lucas, 2000a, "Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk," The Journal of Finance.
71
, 2000b, "Portfolio Choice in the Presence of Background Risk," The Economic Journal, pp. 1-26.
, 2005, "Market Frictions, Savings Behavior, And Portfolio Choice," Macroe-conomic Dynamics, 1, 76-101.
Himmelberg, C , C. Mayer, and T. Sinai, 2005, "Assessing High House Prices: Bubbles, Fundamentals and Misperceptions," Journal of Economic Perspectives, 19, 6792.
Hu, X., 2005, "Portfolio Choices for Homeowners," Journal of Urban Economics, 58, 114-136.
Kullmann, C , and S. Siegel, 2005, "Real Estate and its Role in Household Portfolio Choice," working.
Li, W., and R. Yao, 2006, "The Life-Cycle Effects of House Price Changes," working.
Luigi Guiso, M. H., and T. Jappelli (eds.), 2001, Household Portfolios. The MIT Press.
Luttmer, E. G. J., 1999, "What Level of Fixed Costs Can Reconcile Consumption and Stock Returns?," Journal of Political Economy, 107, 969-997.
Malkiel, B. G., 1996, A Random Walk Down Wall Street: Including a Life-Cycle Guide to Personal Investing. Norton.
Marshall, D. A., and N. G. Parekh, 1999, "Can Costs of Consumption Adjustment Explain Asset Pricing Puzzles?," Journal of Finance, 54, 623-654.
Massa, M., and A. Simonov, 2006, "Hedging, Familiarity and Portfolio Choice," The Review of Financial Studies, 19, 2.
Merton, R., 1969, "Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Case," Review of Economics and Statistics, 51, 247-257.
, 1971, "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, 3, 373-413.
Mossin, J., 1968, "Optimal Multiperiod Portfolio Policies," Journal of Business, 41, 215-229.
Ortalo-Magne, F., and S. Rady, 2006, "Housing Market Dynamics: On the Contribution of Income Shocks and Credit Constraints," The Review of Economic Studies, 73, 459-485.
Polkovnichenko, V., 2004, "Limited stock market participation and the equity premium," Finance Research Letters, 1, 24-34.
Poterba, J. M., and A. A. Samwick, 1997, "Household Portfolio Allocation Over the Life Cycle," working.
Samuelson, P., 1969, "Lifetime Portfolio Selection by Dynamic Stochastic Programming," Review of Economics and Statistics, 51, 239-246.
Silos, P., 2005, "Housing, Portfolio Choice and the Macroeconomy," working.
72
Sinai, T., and N. Souleles, 2005, "Owner-Occupied Housing as a Hedge Against Rent Risk," Quarterly Journal of Economics, 120, 763-789.
Viceira, L. M., 2001, "Optimal Portfolio Choice for Long-Horizon Investors with Nontradable Labor Income," Journal of Finance, 56, 433-470.
Vissing-Jorgensen, A., 2002, "Towards an Explanation of Household Portfolio Choice Heterogeneity: Nonfinancial Income and Participation Cost Structures," working.
Wachter, J. A., 2002, "Portfolio and Consumption Decisions under Mean-Reverting Returns: An Exact Solution for Complete Markets," The Journal of Financial and Quantitative Analysis, 37, 63-91.
Yao, R., and H. Zhang, 2005a, "Optimal Consumption and Portfolio Choices with Risky Housing and Borrowing Constraints," The Review of Financial Studies, 18(4), 197-239.
Yao, R., and H. H. Zhang, 2005b, "Optimal Life-Cycle Asset Allocation with Housing as Collateral," working.
73
Appendices
A Numerical Solution
The investor's high-dimensional optimization problem can be simplified using a trick following Yao
and Zhang (2005a):
Recall that the Bellman equation is:
Vt(Xt) = max((l - (3) \ vt+\..(n. u.^-i _i_ n _ EY+un1-? {At} {
F{t)u{CuHty-v + [1 - F(t)]Qt
+0F(t)Et\ viVTc^+i) -p; (A-l)
subject to the wealth constraint
Wt = Bt.1rt + St-irf + Yt + Ot-i[{l-4>)PtHt-i-(l-6)Pt-iHt-irf] (A-2)
and the budget constraint
Wt = Ct + Bt + St + [(l-Ot)RtHt+OtSPtHt]-<l,PtHt.1Ot.iOtl{Ht_1=Ht}, (A-3)
where Xt = {Ot-i, Ht.u WUPU Rt, Yt} and At = {Ot,Ct, BuSt, Ht}.
I simplify the above problem using the investor's wealth Wt as a normalizer. I define Ct = Ct/Wt,
bt = Bt/Wt, st = St/Wt, ht = HtPt/Wt , ht = Ht^Pt/Wt, and yt = Yt/Wt. The Cobb-Douglas
assumption of the utility function implies that the numeraire consumption, housing consumption,
and portfolio rules, {ct, bt,st,ht}, are independent of the investor's wealth level Wt- With the above
normalization, the relevant state variables for the investor's maximization problem can be written
as xt = {Ot-i,yt,ht, j^-} and the relevant choice variables are at = {Ot,Ct,bt,St,ht}.
Define
vt(xt) = m
74
Then
where
vt(xt) = max<(l - /3)
Wt+i Wt
F{t)[c\-"hl]^+[l-F{t)}K i - ;
+/3F(t)Et \(Wt+l P? l - 7 n ^ ^ )TT ^
Btrt + Strf+1 + Yt+1 + Ot[(l - <t>)Pt+iHt - (1 - S)PtHtrf] Wt
= 6 tr ; + Strf+1 + W % i + Otht[(l - 0 ) % i - (1 - 5) r /]
(A-4)
(A-5)
The budget constraint becomes
.R, l = ct+bt+st + [{l- Ot)-± + Ot8}ht - Ot-XOt<t>htl{Ht_1=Ht} (A-6)
The state variables in time t + 1 are xt+i = {Ot,yt+i,ht+i, -^±L}, where
Yt+1 Yt+i Wt
- -yt i
Pt+i Wt
yt+i =
ht+i
Pt+i Rt+i
Wt+i HtPt+i
Yt Wt+l°l
HtPtPt+iWt
Wt+i WtPtWt+i Pt+i Rt Pt
Pt Rt+i Rt
= ht Pt Wt+i
(A-7)
(A-8)
(A-9)
The above problem can be solved numerically using backward recursion. Given the presence of a
bequest motive, the terminal condition for the recursive equation is VT = K. I discretize the labor-
income-wealth share yt = Yt/Wt into a grid of 50 over the interval [0,2], and the house-value-wealth
ratio Jit = PtHt-i/Wt into a grid of 80 over the interval [0, 4], the house-price-to-rent ratio Pt/Rt
into a grid of 80 over the interval [10, 30], respectively. Interpolations are used to approximate
off-grid points. The value function at date T is used to solve for the optimal decision rule at date
T — 1. The procedure is repeated recursively until the solution for date t = 0 is found.
B Utility Loss Metric
The utility losses are calculated using the standard consumption-equivalent variations: For each
life-cycle portfolio choice rule, I compute the constant consumption stream that gives the investor
75
the same level of life-time expected utility as the consumption stream financed by the portfolio rule.
This constant consumption stream is the so-called "equivalent consumption stream". To measure
relative utility loss, I compute the percentage of deviation of the equivalent consumption stream of
the candidate portfolio rule from the benchmark one.
Note that this paper differs from Cocco, Gomes, and Maenhout (2005). Here the investor obtains
utility from a "composite consumption" good, which is a function of numeraire consumption good
C and housing consumption H, while in their paper there is no H. In computing the equivalent
consumption stream, I compute only a constant "composite consumption" that makes the investor as
well off in expected utility as the consumption stream financed from the (C, i/)-combination portfolio
rule. I compute utility losses by comparing the equivalent "composite consumptions" across portfolio
rules.
C The Panel Study of Income Dynamics Data
The PSID is a longitudinal study of a representative sample of the US individuals and the family
units in which they reside. The PSID study began in 1968 with a sample size of 4800 families. In the
2001 sample it had more than 7000 families. The PSID interviewed and reinterviewed individuals
from families every year from 1968 to 1996. In 1997, the PSID switched from annual sampling to
biennial data collection. The data provides a broad range of information about the respondents.
The central focus of the study is economic and demographic, with substantial details about income
sources and amounts, employment, family structure, residential locations. My paper is particularly
interested in housing related variables in the data.
To keep the sample representative of the civilian noninstitutional population of the US, I excluded
the poverty sample and the Latino sample and only used the Survey Research Center sample. For
more detailed descriptions of the PSID data, please refer to Kullmann and Siegel (2005) and PSID's
website: http://psidonline.isr.umich.edu/.
The PSID has a "Sensitive Dataset" which contains Geocode Match Files that allows researchers to
link the main PSID data to Census data. With the addition of data on neighborhood characteristics
for the geographic areas in which the respondents live, researchers can conduct economic analysis
across areas.