ESSAYS IN ASSET PRICING AND REAL ESTATE

Transcription

1 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

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3 2008 YU ZHANG ALL RIGHTS RESERVED

4 Abstract Essays in Asset Pricing and Real Estate Yu Zhang The first chapter of this dissertation introduces housing as a hedging asset in a lifecycle 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 covary 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-sizefits-all prescription is unsuitable for life-cycle investments. The second chapter explores the portfolio choice in a multi-asset setting. It considers 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 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.

5 Contents 1 Life-Cycle Portfolio Choice with Housing as a Hedging Asset Introduction The Economic Model Preferences The Labor Income Process Financial Assets House Price and Rent Wealth Accumulation The Investor's Optimization Problem Benchmark Parameterization Preference Parameters Labor Income Process Financial Assets Housing Parameters Correlations Numerical Results in the Benchmark Model What Drives Life-Cycle Portfolio Choice? Life-cycle Allocations Without Housing Housing's Hedging Function Rates of House Price and Rent Appreciation Low Down Payment Ratio vs. High Down Payment Ratio.. 21

6 1.5.5 Risk Aversion vs. Elasticity of Intertemporal Substitution (EIS) Utility Cost Calculations Life-cycle Allocations Across Housing Markets Correlation of Labor Income with House Price Correlation of House Price with Stock Price Correlation of Labor Income with Stock Price Conclusions 27 2 Cross-Sectional Portfolio Choice and the Hedging Functions of Housing Introduction Related Literature Data House Price, Rent, Labor Income, and Stock Price The PSID Family Files and Wealth Supplements Data Empirical Methodology Stock Participation and Home Ownership Decisions Asset Allocation Decision Results Heterogeneity of Correlations and Volatilities Likelihood of Participation Asset Allocation 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

7 List of Figures 1.1 Labor Income, Consumption, and Accumulated Wealth in the Benchmark Model Life-Cycle Portfolio Choice Profiles in the Benchmark Model Fraction of Stocks in Financial Assets Without Housing Portfolio Choice When Only Renting is Allowed Portfolio Choice When House Price and Rent Have Same Appreciation Rate Portfolio Choice When Ratio of House Price to Rent is Fixed Portfolio Choice with Different Down Payment Requirements Risk Aversion vs. Elasticity of Intertemporal Substitution Portfolio Choice with Labor Income and House Price Correlated Portfolio Choice with Stocks and House Price Correlated Portfolio Choice with Labor Income and Stocks Correlated Housing Investment vs. Age Stock Investment vs. Age 62 in

8 List of Tables 1.1 Benchmark Parameters Cross-Sectional Variations in Labor Income, House Price, Rent, and Stock Price Utility Cost Calculation (%) Cross-sectional Summary Statistics of p, a, and \i Summary Statistics of the PSID Data: Full Sample Summary Statistics of the PSID Data: Sub-sample Likelihood of Owning a House Likelihood of Owning Stocks Ratios of Housing Investments to Wealth Proportion of Wealth Invested to Stocks 61 IV

9 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 Association Meeting, Northern Finance Association Annual Meeting, Joint Statistical Meetings, Washington Area Finance Association Annual Meeting, the 2008 Southwestern Finance Association Annual Meeting and several universities for their helpful comments and suggestions. v

10 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

11 Chapter 1 Life-Cycle Portfolio Choice with Housing as a Hedging Asset

12 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 Maenhout (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 report 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 Consumer 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

13 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. Middleaged 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. Middleaged 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 literature. 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 experience 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

14 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 fluctuations 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 overlooked by the literature, which is that housing markets are local so that optimal portfolio choice varies cross-sectionally. Housing investment differs from other financial investments in that unlike the stock market, housing markets can not be accessed nationally by all investors. House tenure and portfolio choice differ across Metropolitan 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.

15 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).

16 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 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, H t ) from consuming the numeraire good Ct and a housing service flow which is assumed to be proportional to the housing shares H t. The housing shares H t 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 H t for both the quantity of housing service flow and the number of housing shares. The instantaneous utility u(ct, H t ) is of Cobb-Douglas form over Ct and H t : u(c u H t )^C}- e Hf, 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

17 7 wealth W t to his heirs. The investor obtains bequest utility Q t (W t ) 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 W t is used to purchase a house of size H t and numeraire consumption good C t for his beneficiary. That is, Q t {W t ) = V mvxcl- e H e t C t,h t l l subject to C t + P t H t = W t, where P t 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 «= 9 e (l - d) 1 ' 9. Intertemporally, the investor smooths out his numeraire consumption C t and housing consumption H t. 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: V t (X t ) = max{(l-/?) {At} I F{t)u{C u H t f-* + [l - F{t)]Q!-i +/3F(t)Et rl-j, v t \7(x t+ i) ^y-k (i.i) where j3 is the subjective discount factor, 7 is the risk aversion coefficient, ip is the

18 8 EIS, and X t and A t are the state variables and choice variables which will be defined in Section 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 Y t 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\ogY t = 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 deterministic, 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 Y K -\- That is, Y t = CY K -i, fort = K,... t T, where C is called the replacement ratio Financial Assets There are two financial assets: a risk-free asset (called bond), with gross real return rj; 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 -r f =fx s + ef,

19 9 where /x s 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 S t >0,B t >0,Vt, (1.2) where St and B t are stock holdings and bond holdings at time t, respectively House Price and Rent To receive housing service flows, the investor either rents or buys a house. I denote P t and R t as the house price and rent, respectively, per unit of housing H t. Again, the "unit" is in terms of both size and quantity, as defined in If the investor chooses to rent and consumes a housing service flow of size H t, he will need to pay RtH t per year. If he chooses to own, he will need to have a house of value PtH t. House price P t and rent R t are assumed to follow stochastic processes AlogPi = /x p + ef Alog# t - fi R + e? where // p and fi R 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.

20 Wealth Accumulation Let W t denote wealth, cash on hand to the investor at the beginning of time t. In each period t, W t 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 Y t, and iii) capital gains from his housing investment if he owned a house in the previous period. That is, W t = Bt^rj + St-irf + Y t + O t -i[(l - <l>)ptht-i - (1 - S)P t. 1 H t. 1 r f ], (1.3) where O t is a dummy variable denoting homeownership status. O t 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 (O t -i = 1) and sells it today, he obtains the current market value of his house P t H t -\ net of transaction cost. The transaction cost <j>p t H t -\ is a constant fraction <f> of the house value P t H t -\. 5 is the ratio of down payment to house value. Assume that at time t 1 the homeowner already paid 5P t _ih t -i for his house; he therefore carries mortgage balance (1 8)P t -ih t -\ to time t. Before he sells his house at time t, the investor needs to pay off (1 5)P t -.\Ht-\rf, 3 which is his mortgage balance outstanding plus interest. If the investor was not a homeowner in the previous period {O t -\ = 0), he receives zero income from the housing market. With W t received at the beginning of period t, the investor allocates this amount to the following three expenditures: i) numeraire good consumption C t, ii) investments in bonds and stocks, B t and St, and iii) purchasing housing service flows, which amount to H t. A renter (O t 0) pays RtH t and obtains his housing service flows from the rental market; a homeowner (0< = 1) pays down SP t H t to acquire a house of value PfH t and receives his housing service flows. We have W t = Ct + B t + St + {{l-o t )RtHt + O t 5PtHt]-<t>PtH t^ot-io t l {Ht _ l=h $A) 3 Note that 77 is a gross return, so the amount (1 5)P t -\H t -irf contains both mortgage balance and interest incurred.

21 11 Note that a homeowner who does not change his homeownership status (O t -i = O t = 1) and does not adjust his housing share (H t -i H t ) is not subject to transaction cost in housing. Therefore <f)p t H t -i cancels out from (1.3) and (1.4). Transaction cost in housing is incurred only by a homeowner who switches to a renter (O t ~i = 1 and O t = 0) or by a homeowner who adjusts his housing units {H t -\ ^ H t ). 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., M t = (1 S)P t H t. A 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 status (Ot-i), previous housing investment (H t -\), realized labor income (Y t ), house price (Pt), and market rent {Rt), the investor decides whether to rent or to own (O t ). He also decides on numeraire consumption C t and the size and composition of his portfolio, which consists of bond, stock, and housing (B t, St, H t ). Thus the state variables are X t = {O t -\, H t -i, Wt, Pt,Rt.,Y t } and the choice variables are A t = {O t, C t, B u S t, H t }. Then V t = V t (X t ). 4 This is a reasonable assumption, given the low down payments and cheap refinancing over the period of the late 1990's to 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.

22 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 W t as a normalizer to reduce the dimensions of the state space. After the normalization, the relevant state variables are: the investor's housing status O t -\, the labor-income-wealth ratio y t Y t /W t, the beginning-of-period house-value-wealth ratio h t = PtH t -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 c t C t /W t, the fraction of wealth allocated to bonds b t = B t /W tl the fraction of wealth allocated to stocks s t = St/Wt, the new house tenure choice Ot, and the new housing-investment-wealth ratio h t = H t P t /W t. 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.

23 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 and let the comparative statics exercises show the impacts on portfolio choice from 7 and ip separately 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 deterministic 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.

24 14 After retirement, the investor receives 68 percent of his labor income at age 64 (replacement ratio = 0.68), following the assumptions in Cocco, Gomes, and Maenhout (2005). Some of the literature estimate different labor income profiles for different education groups (e.g., college graduates, high school graduates, and those without a high school degree). According to Cocco, Gomes, and Maenhout (2005), however, education 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 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 fi s = 6.0%. The standard deviation of shocks to stock return is set at a s 15.7% 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 Expenditure 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 (O t -\ 1 and O t = 0), or when they change units of housing (O t -i = O t 1 and H t -i ^ H t ). 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 Indices (HPIs) from the Office of Federal Housing Enterprise Oversight (OFHEO) and

25 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 deviations 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 fi p = 1.8% and the mean rent appreciation rate is set at [i R = 0.4%. The volatilities of house price appreciation and rent appreciation are set at a p 5.5% and a R 3.0%, respectively 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 different 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 parameters 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 optimal 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

26 16 represent the average labor income and its standard deviation in the Quarterly Census 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 C t, labor income Y t, and accumulated wealth W t 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 before 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

27 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 ratio 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 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 B t, and numeraire consumptions C t - 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 conservative 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 afford 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.

28 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 investors 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. Conditional 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 C t is smooth but wealth W t 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.

29 What Drives Life-Cycle Portfolio Choice? 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 dominance 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 gradually 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 consideration, 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.

30 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 investments. 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 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

31 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 appreciation 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 withdraw 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 Low Down Payment Ratio vs. High Down Payment Ratio The benchmark model follows Yao and Zhang (2005a) in assuming a 20% down payment requirement for purchasing a house. In the decade before 2006, there had been

32 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 Risk Aversion vs. Elasticity of Intertemporal Substitution (EIS) Most life-cycle portfolio choice models use the power utility function. The power utility 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 7^ 1/ip- It can 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

33 23 in savings for future uncertainty. If we keep the level of risk aversion at 7 = 5 and increase the elasticity of intertemporal 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 investors 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 variations. That is, for each life-cycle portfolio choice rule, I compute the constant consumption 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 welfare losses resulting from completely excluding housing from the model is substantial:

34 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 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 variations 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