A Life-Cycle Model of Housing and Mortgage Choices with Labor Income and House Price Risks

Transcription

1 A Life-Cycle Model of Housing and Mortgage Choices with Labor Income and House Price Risks Wenli Li and Rui Yao This Version: February 24 Abstract We present a life-cycle model of housing and consumption choices with labor income and house price risks. In our model, the households make housing decisions along both the extensive margin of renting versus owning, and the intensive margin of house value. A long-term fixed-rate mortgage contract is introduced that allows mortgage refinance and default, both subject to significant costs. The setup allows us to explicitly distinguish between illiquid home equity and liquid financial assets. A calibrated version of the model generates housing tenure and consumption decisions largely consistent with empirical findings. In particular, home ownership rate and housing consumption are hump-shaped over the life cycle, while mortgage loanto-value ratio declines steadily with age, leading to a U-shaped home equity total networth ratio over life cycle. Additional comparative analysis reveals the importance of labor income and house price risks in affecting households housing and mortgage choices. Key Words: Life-cycle Model, Consumption, Saving, Housing, Mortgage JEL Classification Codes: E21, R21 We gratefully acknowledge Pittsburgh Supercomputing Center (PSC) for providing computing resources. Wenli Li can be reached at Department of Research, Federal Reserve Bank of Philadelphia, Ten Independence Mall, Philadelphia, PA 1916, wenli.li@phil.frb.org. Rui Yao can be reached at Department of Economics and Finance, Baruch College, Zicklin School of Business, One Bernard Baruch Way, Box B1-225, New York, New York 11, Tel: , Fax: , rui yao@baruch.cuny.edu.

2 1 Introduction For many households in the U.S., labor income is the most important source of income, and housing is the most important consumption and the largest component of assets. Households, however, are unable to perfectly trade away the risks associated with labor income and housing prices. This inability obviously complicates their housing and consumption choices. In this paper we present a life-cycle housing and consumption model with stochastic uninsurable labor income, both permanent and transitory, and housing prices. We allow permanent labor income and housing prices to co-vary, and consider housing purchases both on the extensive own-rent margin and on the intensive margin of value conditional on ownership. Our modelling strategy follows the buffer-stock savings model of Zeldes (1989), Deaton (1991), Blundell, Browning and Meghir (1994), Attanasio and Browning (1995), Carroll (1997), Carroll and Dunn (1997) and Fernandex-Villaverde and Krueger (21), among many others. The innovation is the explicitly modelling of housing as both a consumption and an investment good. Long-term mortgage contracts are written for the purchases, and home equity can be accessed via mortgage refinancing at a cost. This additional feature is important because empirically both home ownership and mortgage loan-to-value ratio exhibit strong life-cycle patterns. In particular, home ownership rates increase initially with household age, then decreases. Mortgage loan-to-value ratio decreases steadily with household age (Figure 1). The explicit modelling of a mortgage contract also separates this paper from the recent literature on housing and portfolio choices using dynamic life-cycle models. This literature includes, to name just a few, Cocco (23), Cocco, Gomes and Maenhout (22), Hu (23), and Yao and Zhang (23). In our model, same as in this literature a household holds his networth either in liquid asset or illiquid home equity. The assumption of costly mortgage refinancing and costly housing adjustment, however, implies that it is no longer optimal for the household to refinance every period so that he holds home mortgage to the maximum allowable amount. According to the Panel Study of Income Dynamics, the share of a household s illiquid home equity in his net worth exhibits a U-shape (Figure 2). A calibrated version of our model suggests that a household purchases a house when 2

3 his wealth on hand is high relative to his permanent labor income. A homeowner who experiences a negative labor income shock may choose to convert illiquid home equity to liquid asset to finance non-housing consumption in the absence of lower mortgage rate. In keeping with stylized empirical facts, in our simulation home ownership rates and house value exhibit hump-shaped life-cycle patterns, while the holding of illiquid home equity exhibits a U-shape over a household s age. Mortgage refinancing activities demonstrate a bimodal pattern young homeowners refinance to ease liquidity concerns, while old homeowners refinance to defer house selling expenses and avoid higher renting costs. Comparative static analysis further reveals the importance of labor income and house price risks in housing decisions. A higher transitory income risk delays households home purchases, and leads to frequent refinancing among young, liquidity-constrained homeowners. By contrast, a higher permanent income risk induces a household to save more. Although more of the savings are in liquid form, the household also purchases his house earlier, a result of the portfolio decision. Similar to the temporary labor income risk, a higher house price risk delays home ownership and increases the frequency of costly mortgage refinancing by a young household who tries to gain access to illiquid home equity. An increased comovement between permanent labor income risk and housing prices delays a household s house purchase and the value of the purchase given ownership. Refinancing activity for a young, liquidity-constrained household also declines in response. The rest of the paper is organized as follows. Section 2 discusses the model s assumptions and set-up. The calibration and parameterization is presented in section 3. Section 4 looks at the solution of our model in terms of the optimal housing and consumption rules. Section 5 reports the simulation results for the benchmark parameterization. Section 6 explores the separate effects on a household s life-cycle decisions of permanent versus transitory labor income risks, housing price risk, and the covariance between permanent labor income and housing prices. Finally, section 7 concludes. 3

4 2 The Model Economy The model considered here extends the standard life cycle model of buffer stock savings by incorporating a risky housing and mortgage contract. Optimal housing and mortgage choices have been studies in a life cycle setting in Campbell and Cocco (23), Cocco (24) and Yao and Zhang (24). Campbell and Cocco (23) study household risk management and optimal mortgage choice. While our fixed-rate mortgage contract follows closely that in Campbell and Cocco (23), we mode explicitly the housing decisions along both the extensive margin of owning versus renting choice, and the intensive margin of house value. Cocco (23) and Yao and Zhang (23) study the optimal portfolio choice (stock and bond/house equity) by a household in an environment with uncertain labor income and house prices, the focus of this paper is optimal household life-cycle housing and mortgage decisions. As a result, we abstract from household s portfolio choice within the liquid wealth (stock and bond mix). Instead we explicitly model the mortgage contract, and differentiate between liquid asset (bond) and house equity by introducing a cost in mortgage refinancing. 2.1 Demographics and Preferences The economy is discrete where households live for at most T periods, and T is a positive integer. The probability of being alive at time j for a household conditional on being alive at time j 1 is λ j. We assume that < λ j < 1 for all j < T and λ T =. The probability that a household is alive at period t is then given by F (t) = t λ j, (1) j= where F (t) 1 for all t T. We further assume that household size the number of adults and children vary over the life cycle, and denote the average household size at age t by N t. The series {N t } T t=1 is exogenously given. Households also differ in their educational achievements. In particular, we exogenously divide households into three categories, those with no high school diploma, graduates of high school, and graduates of college. A household of age t in the economy derives utility from consuming a numeraire good 4

5 C t and housing services H t. We denote the period utility function as U(C t, H t, N t ). The household also has a bequest motive represented by a function of total bequeathed wealth net of liquidation cost. Let B(.) denote the household s bequest function, we can write the household s life time utility as E T β t {F (t)u(c t, H t, N t ) + [F (t 1) F (t)]b(q t, N t )}, (2) t= where β is the time discount factor, and Q t defines the household s total wealth upon death at time t, net of housing liquidation costs. 2.2 Labor Income All households retire at age J (J < T ). Before retirement, households receive labor income, Y t, each period. As in Gomes and Michaelides (22), this labor income is exogenous and given by where P Y t income shock. Y t = P Y t ε t (3) P Y t = exp{f(t, Z t )}P Y t 1ν t for t =,..., J 1, (4) is the permanent component of the labor income, and ε t is a transitory labor The evolution of permanent labor income P Y t consists of a deterministic component f(t, Z t ) as a function of age and household characteristics Z t, and a persistent income shock ν t. We assume that ln ε t and ln ν t are independently identically distributed with mean {.5σ 2 ε,.5σ 2 ν}, and variances {σ 2 ε, σ 2 ν}, respectively. The ln P Y t is, therefore, a random walk with a deterministic drift f(t, Z t ), and the growth of a household labor income evolves according to ln Y t = f(t, Z t ) + ln ε t + ln ν t ln ν t 1, (5) with unconditional variance σ 2 ε + 2σ 2 ν. This process has a single Wold representation equivalent to the M A(1) process for household earnings growth estimated using household level data. 1 1 See MaCurdy (1989), Abowd and Card (1989), and Pischke (1995). 5

6 After retirement, a household receives a constant nonfinancial income each period, including pension, social security payment, and distribution from his retirement account, at a fraction (θ) of his labor income just before retirement (PJ 1 Y ). The parameters of the labor income process vary across education categories. 2.3 Housing and Mortgage Contracts A household can acquire housing services by either renting or owning a house. Owning a house in our model serves a dual purpose. It not only provides the household with housing services, but also allows the household to hold home equity. The household can separate the housing consumption choice from the housing investment choice by renting. If a household rents in period t 1, he can keep renting or buy a house and become a homeowner at period t. To rent, the household pays a fraction α of the market value of the rental house to the landlord. To buy a home, the household needs to pay at least a fraction δ of the house value as down payment and finance the rest of the purchase through a fixed-rate mortgage. Let P H t denote the price per unit of housing services at time t, then Pt H H t is the value of the house at time t. We assume that the house price growth follows a stochastic process with mean µ H, variance σ 2 H growth rate, and we denote the correlation ρ HY. and can be correlated with pre-retirement labor income There is a cost associated with the purchasing of new house in securing title and mortgage that is proportional to the house value, ρ. 2 refinance, M t, needs to satisfy the collateral constraint A newly initiated mortgage for purchase or M t (1 δ)p H t H t, (6) where δ captures the down payment requirement in mortgage contract. All mortgage loans are assumed to have a maturity of (T t + 1) such that it is paid off by the end of period T. This resembles the fixed-rate mortgage contract in Campbell and Cocco (23). 3 2 The cost is present no matter the investor borrows a mortgage or not. Mortgage refinance also involves the same cost, which is payable irrespective of mortgage balance borrowed. This reflects the observation that the same documentations required to secure a mortgage are also needed to complete a house purchase. 3 This specification of mortgage contract eliminates the yeas-to-maturity as a separable state variable and We 6

7 further assume that mortgage rate is the same as the lending rate that a household earns on its holding of bonds, r. Each period a homeowner entering into the economy can choose to stay in the house for the next period, sell his house, or default on his mortgage and have his house liquidated. If the household chooses to stay in the house, he has the option of convert some home equity to liquid wealth through a cash-out refinancing. Refinancing, however, entails a closing cost assumed to be a percentage ρ ( ρ 1) of the house value. The mortgage balance after refinancing has to satisfy the collateral constraint for a home buyer as in (6). 4 If the household chooses not to refinance, he is then required to pay down his mortgage balance according to the fixed-rate mortgage amortization schedule given below: M t = M t 1 (1 + r) M t 1 T j=t (1 + r)t j 1 = 1 (1 + r)t T 1 (1 + r) t T 1 M t 1. (7) If a household sells his house, he will incur the selling cost assumed to be a fraction φ ( φ 1) of the market value of the house. The full balance of mortgage is due upon selling of the house. After selling, the household faces the same decisions as a time t 1 renter: he can either rent or buy another house. Let l t be the household s beginning-of-period loan-to-value ratio, 5 l t = M t 1(1 + r). (8) Pt H H t 1 If a household defaults on his mortgage, his house will be liquidated and he will lose all his home equity and incur the default cost assumed to be a fraction κ (κ ) of the house greatly simplifies the computation. Campbell and Cocco (23) use a similar modelling strategy for their fixed-rate mortgages. They assume that the fixed-rate contract has a maturity of T periods initiated at time. When the mortgage is refinanced at time t, the household refinances the remaining principal with maturity T t such that the mortgage is paid down by the terminal date. 4 Under the assumptions of costly mortgage refinancing and equal lending and borrowing rates, a household, conditional on refinancing, will refinance to the highest loan-to-value ratio allowed by the lender. 5 We use l t to denote the mortgage balance at the beginning-of-the-period and l t to denote mortgage balance immediately after mortgage payment. The difference between the two, l t l t, represents the current period mortgage payment, principal and interests, if the investor does not refinance, or proceeds from cashout refinance. 7

8 value. Given that a household always has the option to sell the house, conditional on house liquidation, he will choose to default on the mortgage only if the gains from selling the house is lower than that from default, i.e., (1 l t φ)pt H H t 1 < κpt H H t 1, (9) or simply l t > 1 φ + κ. (1) Default can occur either because the house value has become too low relative to his mortgage balance (strategic default) or because the household receives a series of bad labor income shocks and has liquidity problem in financing his consumption and housing expenditures. 6 Note it is possible that a household s net worth upon house liquidation, after receiving current period labor income, falls below zero. In this case, we assume that the household receives a small subsidy at a level proportional to his current period permanent labor income, in the spirit of Laibson, Repetto and Tobacman (21). 7 All homeowners need to spend a fraction ψ ( ψ 1) of the house value on repair and maintenance to keep housing quality constant. At the beginning of each period, a household also receives an exogenous moving shock. Because moving is costless for renters, the moving shock does not impact on a renter s house service decision. A homeowner who experiences the moving shock, however, is forced to move out his house by selling or defaulting on the house. 2.4 Liquid Assets In addition to holding his assets in the form of house equity, a household can also save at the constant riskfree rate r. We denote these liquid savings as S t. For simplicity, we do not allow borrowing on liquid assets and rule out home equity loans, i.e., borrowing occurs only through mortgages: S t, for t =,..., T 1. (11) 6 Allowing for default effectively puts a ceiling on the equilibrium loan-to-value ratio and greatly simplifies our computation. In real life as well as our model simulation, since the default rate is very low, we argue that the wedge between the before-tax benefit mortgage rate and borrowing rate covers the default premium. 7 The subsidy is chosen to be small enough so that in simulation, it does not occur. 8

9 We view these savings (or cash on hand) as the liquid resources available for consumption and saving. 2.5 Wealth Accumulation and Budget Constraints A household s wealth accumulation process and intertemporal budget constraints at time t depends on his house tenure at time t 1, and his housing mortgage choices at time t. Specifically, let Q t be the household s spendable resources or wealth including labor income. 8 If the household is a renter at time t 1, then Q t = S t 1 (1 + r) + P Y t 1 exp{f(t, Z t )}ν t ε t. (12) If the household is a homeowner at t 1, then his spendable resources also include home equity (net of selling or default cost). Q t = S t 1 (1 + r) + P Y t 1 exp{f(t, Z t )}ν t ε t + max{1 φ l t, κ}p H t H t 1. (13) Additional notations are needed before we introduce households budget constraints. Let D o t denote house tenure that takes a value of 1 for homeowners and for renters; let D s t denote the selling decision for a homeowner at period t that takes a value of 1 if the homeowner sells his house and if not; Let D d t denote a homeowner s default decision that takes a value of 1 of a homeowner defaults and otherwise; let D r t denote the refinancing decision by a homeowner that takes a value of 1 if the homeowner refinances his mortgage and if not; let D m t denote the moving shock a household receives at the beginning of period t that takes a value of 1 if the household has to move and otherwise. A household s intertemporal budget constraint, therefore, can be written as follows. (1) If the household is a renter at t 1 (D o t 1 = ), or a homeowner but decides to sell his house in period t (D o t 1 = 1 and D s t = 1), and chooses to rent for the current period, C t + S t + αp H t H t = Q t. (14) 8 The advantage of defining household s net worth this way, i.e., including his home equity net of selling or defaulting cost, is that in our computation, we reduce the homeowner s problem to the renter s problem when he sells his house or has it foreclosed. 9

10 (2) If the household is a renter at t 1 (D o t 1 = ), or a homeowner but decides to sell his house or have it foreclosed in period t (D o t 1 = 1 and D s t = 1 or D d t = 1), and chooses to buy a house for time t, C t + S t + (δ + ψ + ρ)p H t H t = Q t. (15) (3) If the household is a homeowner at t 1 and decides to stay in the house for time t without refinancing the mortgage (D o t 1 = D o t = 1, D s t = D d t = D r t = ), where l t = C t + S t + {(l t l t ) + ψ}pt H H t 1 = Q t max{1 φ l t, κ}pt H H t 1, (16) 1 (1+r)t T l 1 (1+r) t T 1 t is the loan-to-value ratio at period t after mortgage principal and interest payment. The term {(l t l t )+ψ}pt H H t 1 captures the mortgage payment as well as house maintenance cost. 9 (4) If the household is a homeowner at t 1 and decides to stay in the house for time t but refinance his mortgage (D o t 1 = D o t = 1, D s t = D d t =, D r t = 1), C t + S t + {l t (1 δ) + ψ + ρ}pt H H t 1 = Q t max{1 φ l t, κ}pt H H t 1. (17) The term {(l t (1 δ) + ψ + ρ}pt H H t 1 represents the net sum of proceeds from cash-out refinance, house maintenance and mortgage refinance cost. 2.6 The Optimization Problem and Solution Method A household s problem is to maximize its discounted life-time utility subject to the intertemporal budget constraints given his initial endowment. We can write a representative households problem at time t = as follows. T max E β t {F (t)u(c t, H t, N t ) + [F (t 1) F (t)]b(q t, N t )}, A(t) t= where A(t) = {C t, H t, S t, Dt o, Dt r, Dt s, Dt d }, t =,..., T 1, subject to labor income process (equation 5), house down-payment constraint and mortgage amortization schedule (equations 9 If the household chooses not to liquidate his house, the household cannot access his home equity and does not incur the liquidation cost. We therefore subtract the home equity, max{1 φ l t, κ}pt H H t 1, from total net worth Q t. 1

11 6 and 7), borrowing constraint on liquid asset (equation 11), intertemporal budget constraints (equations 14 to 17), given initial house tenure D o ( 1), moving shock Dm, wealth after labor income Q, level of permanent labor income P Y, housing price P H, housing stock H ( 1), and mortgage M ( 1). Given the complexity of our problem, analytical solutions do not exist. We therefore use a numerical solution method based on value function iterations to derive optimal decision rules. The details are given in appendix A. Here we highlight a few transformations we did for the computation and then present the main idea behind our solution method. First, we assume that households within-period utility takes Cobb-Douglas form,modified to incorporate family size effect, U(C t, H t, N t ) = N t [( C t N t ) 1 ω ( H t N t ) ω ] 1 γ = N γ t (C 1 ω t 1 γ Ht ω ) 1 γ, (18) 1 γ where N t is the average effective household size at age t, following Laibson, Repetto, and Tobacman (23). The transformation of actual household size to its effective size captures the economies of scale in household consumption as argued in Lazear and Michael (198), among many others. We provide more detailed discussion when we parameterize our model economy. Following Damon, Spatt, and Zhang (21), we assume that upon its death, a household s liquidated wealth is used to purchase a L-period annuity to pay for his beneficiary s numeraire good consumption and housing services, with the annuity factor A L defined as A L = r(1+r)l (1+r) L 1. The beneficiary is assumed to have the same family size N t as that of the departing household. We further assume that the annuity income from bequest pays for the beneficiary s numeraire good consumption and rental costs. We set the beneficiary s numeraire good and housing service consumption at the fixed proportion of 1 ω, the optimal level for the Cobbω Douglas utility function when renting. The household s bequest function can then be defined as B(Q t ) t+l k=t+1 βt k t N γ t [(A L Q t )ω ω (1 ω) 1 ω ] 1 γ (1 γ)(αp H t ) ω(1 γ) N γ t β(1 β L )[(A L Q t )ω ω (1 ω) 1 ω ] 1 γ (1 β)(1 γ)(αp H t ) ω(1 γ) (19) 11

12 Second, given the recursive nature of the problem, we rewrite the household s intertemporal consumption and investment problem as follows V t (X t ) = max{λ t [N γ t A t +(1 λ t )N γ t where X t = {D o t 1, D m t, P Y t, P H t, H t 1, M t 1, Q t } (Ct 1 ω Ht ω ) 1 γ + βe t V t+1 (X t+1 )] 1 γ β(1 β L )[(A L Q t )ω ω (1 ω) 1 ω ] 1 γ }, (2) (1 β)(1 γ)(αpt H ) ω(1 γ) Finally, we further simply households optimization problems by exploiting the scaleindependence of the maximization problem and normalize all the variables by the household s permanent income Pt Y. The normalization reduces the number of dimensions by 1, i.e., Pt Y no longer serves as a state variable. In terms of computation, in the final period the value function corresponds to the bequest function at time T. We then use this value function to compute the policy rules for the previous period, and obtain the corresponding value function for this period. This procedure is repeated backwards until we reach period. We provide more additional details on model simplification and numerical solution in Appendix. 3 Model Parameterization 3.1 Demographics The model period is specified to be one year. Households enter the economy at age 2 and live for a maximum of 8 years (T = 6). The mandatory retirement age is 65 (J = 45). The conditional hazard rates of survival are taken from the 1998 life tables of the US National Center for Health Statistics (Anderson 21), and we assume that households liquid wealth after death is used to purchase a five-year annuity (L = 5). Put simply, a household wishes to provide his beneficiary with 5 years of numeraire good and housing services from the bequest. We use the 21 Survey of Consumer Finance (SCF) to calibrate the age-varying household size (the N t s), defined as the total number of head, spouse and children between the age of and 18. To construct the effective household size, we first calculate the average house- 12

13 hold equivalent size by head age using the 21 equivalence scale from the US Department of Health and Human Services (Federal Register 21), then fit a fifth-order polynomial to this equivalent size to obtain the profiles for our numerical simulation (the N t s). 3.2 Preferences The preference parameters are standard. We set the annual discount factor at β =.96, relative risk aversion coefficient at γ = 2, and risk-free rate r =.4. We pick the housing expenditure share (ω =.2) to match the average share of household housing expenditures in the 21 Consumer Expenditure Survey (US Department of Labor 21). 3.3 Labor Income The age-dependent deterministic labor income growth rate before retirement is based on the empirical estimation of Cocco, Gomes and Maenhout (21) by fitting a third-order polynomial to the labor income of households with different education achievements (high school dropout, high school graduate, and college graduate) using the Panel Study of Income Dynamics (PSID) data. 1 The profile exhibits a hump shape over the life cycle before retirement. For our baseline simulations, we use values of.1 for the standard deviation of the permanent shock (σ ν ) and.25 for the standard deviation of the transitory shock (σ ε ) for college graduates,.1 and.27 for high school graduates, and.13 and.32 for high school dropout. Campbell and Cocco (23) and Storesletten, Telmer, and Yaron (23) use similar numbers in their respective analysis. The replacement ratio after retirement (θ) is set to.68 of the household s labor income at age 64, within the range of those used in the literature. 3.4 House Prices We assume house price changes are uncorrelated, and set the mean of real house price appreciation µ H =, and the volatility σ H =.115. While our mean housing return falls 1 The measure of labor income here is broadly defined and includes unemployment compensation, welfare and transfers. 13

14 between the aggregate index level (Goetzmann and Spiegel 2) and the upper bound of the empirical estimates at the household level (Flavin and Yamashita 22), 11 the variance is close to those used by Campbell and Cocco (23), Case and Shiller (1989), and Poterba (1991). The correlation between house price appreciation and permanent labor income is set at.2 for the benchmark. Given the somewhat arbitrary nature of this number, we conduct conditional tests using higher correlation coefficients to test the robustness of our benchmark results. 12 We assume there is no correlation between house price appreciation and transitory labor income. 3.5 Rental Cost and Housing Expenditures Consistent with the historical real costs of renting and owning, we set the annual rental cost at α =.6, and the annual maintenance and depreciation cost at ψ =.2 (US Census Bureau 1992, Hu 22, and Campbell and Cocco 23). The cost of selling an existing house is set at φ =.6 of the market value of the house, the conventional fees charged by most real estate agents. The down payment requirement is set at δ =.2 of house value. The mortgage refinance is assumed to be ρ =.8 percent of house value. 13 Table 1 summarizes our model parameterization. 4 Decision Rules We report the decision rules for college graduates. The decision rules for high school dropouts and high school graduates are qualitatively similar, and we do not report them here to save 11 Based on 8 quarters of housing index data between March 198 and March 1999, Goetzmann and Spiegel (2) estimate real housing returns for 12 largest Metropolitan Statistical Areas (MSA) and the annualized arithmetic/geometric mean housing returns vary from -1.%/1.1% to 3.46%/3.26%. 12 Cocco and Campbell (23) used a correlation coefficient of.1 in their benchmark calibration. 13 According to Bennetto, Peach and Peristiani (1998), an industry standard for the transaction cost for a new mortgage excluding any up front points paid to the lender is between 1 percent to 1.5 percent of the mortgage amount. Assuming a loan-to-value ratio of.8, which is the optimal level after refinance in our setup, the refinance cost amounts to between.8 to 1.2 percent of the house value. 14

15 space. 4.1 Renters House Tenure and Mortgages Figure 3 depicts a renter s optimal house tenure choice as a function of his age and beginningof-period wealth after receiving labor income, normalized by his permanent income. The solid line represents the wealth trigger boundary of renting versus owning. At each given age, the household rents when his wealth is low relative to his labor income, and owns when his wealth is high relative to the labor income. The threshold decreases with the household s age initially, then levels off. It moves up sharply as the household reaches the end of his life cycle. Several factors explain the observed patterns. To own a house, the household first needs to meet the initial down payment requirement with his wealth on-hand, which is assumed to be 2 percent of the purchased house value, then the subsequent mortgage payment. With higher value of realized wealth, the household can afford a house closer to his desired size and stay in the house longer, and therefore reduce the cost of selling or defaulting. As a result, he is in a better position to benefit from home ownership. The level of the wealth relative to permanent labor income trigger bound for home ownership is primarily explained by the household s changing demographic structure, life cycle income profile, and bequest motive. A young household has a growing family size and a high present value of labor income. Hence, he wishes to own a large house relative to his current labor income so as to smooth intertemporal consumption of housing services and to avoid triggering house selling/default costs immediately. It would, therefore, require a higher relative wealth threshold to trigger home ownership. As the household ages, his family size and the present value of earnings decline, his desired house size also decreases. A lower level of relative wealth is enough to trigger home ownership. When the household approaches the end of his life cycle, his mortality rates increases and the bequest motive dominates his housing decision. Consequently, the household is reluctant to own unless his relative wealth is so high that the additional benefit of owning a house outweighs the cost of liquidation his 15

16 house at the time of his death. Under the current assumption of equal lending and borrowing rates, it is optimal for the homeowners to initiate a mortgage loan at the maximum allowed amount, i.e., the optimal loan-to-value ratio upon either mortgage refinancing or house purchases is 8 percent. The reason is twofold. First, negative shocks to labor income can force the household to draw upon his home equity, which involves significant refinancing charges or selling cost. Hence it is optimal to keep more assets in liquid form. Second, lower home equity commitment makes the option to default on one s mortgage more valuable Consumption and Liquid Asset Figure 4 plots a renter s housing expenditures relative to his permanent labor income as a function of his age and beginning of period wealth-income ratio. Conditional on either renting or owning, at a given age, a household spends more relative to his labor income on housing services as his wealth ratio increases. Across ages, at high wealth-income ratios, a household s current financial wealth is more important than his human wealth as measured by the permanent labor income. Consistent with the permanent income hypothesis, since a young household has many more years to live than an old one, he will spend less on housing. By contrast, at low wealth-income ratios, a household s current financial wealth is dominated by his human wealth, a young household will then consume more relative to his labor income. A renter s consumption of the numeraire good relative to his permanent income (Figure 6) follows the same pattern as that of housing services. His savings, wealth net of consumption, on the other hand, have the opposite pattern (Figure 7). 4.2 Homeowners At each period, after the realization of all shocks a homeowner is characterized by his age, mortgage LTV, wealth-income ratio, and his housing asset-income ratio. In the following subsections, we focus on the decision rules where we hold the homeowner s age and wealthincome ratio fixed. 16

17 4.2.1 House Tenure and Mortgages Figure 8 plots the discrete house tenure choice and associated mortgage decision as a function of his beginning-of-period LTV ratio and housing asset-income ratio for a homeowner at age 5 with wealth-income ratio taking the value 1.5. There are five different regions of action for a homeowner whose moving shock is negative: (1) the non-admissible region (N.A.) given mortgage LTV, housing asset-income ratio cannot take values in this region; (2) the stay and refinance region (Refi) the homeowner stays in his current house but refinances his mortgage; (3) the stay but no refinance region (Stay) the homeowner stays in his current house and does not refinance his mortgage; (4) the sell region (Sell) the homeowner sells his existing house, and either rent or purchase another one; (5) the default region (Default) the homeowner defaults on his mortgage and the house is foreclosed. For a homeowner with a positive moving shock, there are only three regions of action: (1) the non-admissible region (N.A.), same as above; (2) the sell region the area to the left of the dotted line and the N.A. boundary; and (3) the default region the area to the right of the dotted line. Note first the non-admissible region is convex, and within this region mortgage LTV and housing asset-wealth ratio satisfies (1 l t )Pt H H t 1 /Q t > Given that Q t Pt Y have (1 l t )Pt H H t 1 /Pt Y = 1.5, we then > 1.5. Intuitively, in order to finance a large house relative to his labor income, a homeowner has to borrow a large mortgage as well. We discuss the decisions of a homeowner with a negative moving shock first. The homeowner stays in the existing house when his housing asset-labor income ratio is not too far from the optimal level that he would have chosen absent of any transaction cost. While the lower housing asset-income ratio trigger boundary decreases in the LTV ratio, the upper boundary increases in the LTV ratio. This is due to the net effects of two competing forces: consuming the appropriate amount of housing services and maintaining the right amount of liquidity. The cost associated with selling a house is twofold. First it reduces liquidity especially when the LTV ratio exceeds 1 since the mortgage becomes due upon selling the 14 A homeowner cannot take on debt other than residential mortgage. His home equity, therefore, could not exceed his total wealth. 17

18 house. Second, the owner gives up the option to default in the future. Obviously, the cost is higher if the LTV ratio is high and the house is big. It is possible for a house value to fall significantly below its mortgage after a series of bad house price shocks. In this case, it will be optimal for the homeowner to default on the mortgage and have the house foreclosed. As can be seen from the figure, the default region is concave. This is intuitive because a homeowner is better at tolerating mortgage payments when his house size is about right, i.e., the utility of consuming the right among of housing services dominates the utility the default option is able to deliver. Within the Stay region, the homeowner converts a fraction of illiquid home equity into liquid asset through mortgage refinancing when the LTV ratio is low. Under our assumption of constant interest rate and zero spread between the savings rate and the borrowing rate, the refinancing decision is driven solely by the desire to smooth intertemporal consumption. In the case the homeowner receives a positive moving shock, he will have to move out of his house. His decision of selling versus defaulting is entirely dictated by the mortgage LTV ratio as indicated by the vertical dotted line. This results from our assumption that the cost of default is limited to the default period. The mortgage decisions for the homeowner is straightforward. If the homeowner buys another house or refinances his mortgage, his LTV ratio will be 8 percent for the same reasons we discussed for a renter. Otherwise, his LTV ratio will decline according to the payment schedule. Homeowners house tenure and mortgage decisions, not shown, also change with their wealth-income ratio. In particular, for a given mortgage LTV ratio, both the upper and the lower trigger bounds of the housing asset-income ratio decrease as the homeowner s wealthincome ratio increases. The intuition is that the homeowner spends more of his income on housing if his future labor income accounts for a smaller portion of his wealth. For a given housing asset-income ratio, the trigger boundary of the LTV ratio for refinancing is higher for lower wealth-income ratio, a result of the greater need for liquidity. The boundary of the LTV ratio for defaulting is lower for higher wealth-income ratio because of the reduced the marginal benefits of preserving additional liquidity when realized wealth dominates unreal- 18

19 ized future labor income Consumption and Liquid Asset Figure 9 plots the housing services for the same homeowner. The homeowner behaves as if he were a renter if he sells his existing house or has it foreclosed, and makes housing decisions accordingly. If the homeowner stays in his house, then he consumes the same amount of housing services as before. The homeowner s consumption of the numeraire good (Figure 1) follows the same pattern as the housing services except in the Stay region. In the Stay region, the numeraire consumption falls as the housing asset-income ratio increases. This reflects the non-separability of housing and non-housing consumption in the utility function and the intra-temporal substitution of the numeraire consumption for housing consumption. For a given housing asset-income ratio, the consumption exhibits a slight hump shape as the mortgage LTV ratio increases when the homeowner stays in his house without refinancing. This is due to the changing time value of the default of option and the fact that in our definition of wealth for homeowners (equation 13) we have already included the realized net benefit of the default but not the time value. The homeowner consumes more relative to his labor income when the time value of his default option increases, and vice versa. On the boundary of refinancing, the homeowner consumes more numeraire good upon refinancing, reflecting the role of cash-out refinancing in smoothing non-housing numeraire consumption. The homeowner s liquid savings (Figure 11) exhibit qualitatively the same pattern as his consumption of the numeraire good. We omit the discussion to save space. 5 Simulation Analysis We now examine an average household s optimal housing and consumption choices over the life cycle using the simulation method. To do that, we first simulate housing prices, permanent and transitory shocks to labor income, and the moving shock according to their respective stochastic process as specified in the model calibration. Then using the optimal decision rules discussed above, we calculate the household s optimal housing and consumption 19

20 choices. The state variables (house tenure, wealth-labor income ratio, housing asset-income ratio, and the mortgage LTV ratio) are updated accordingly each period. The time-series profiles of the optimal decisions are generated by taking the average of 5, simulations from t = 1 (age 2) to t = 6 (age 8). Figure 12 depicts the home ownership rate and mortgage LTV ratio (panel a), refinancing and default rates (panel b), housing and numeraire good consumption (panel c), and the respective shares of liquid savings and home equity in total wealth (panel d) against the household s age. The home ownership rate is hump shaped. It takes the value of before age 22, then rises sharply after that, reaching 8 percent by age 33 and almost 1 percent by age 47. The home ownership rate starts to decline at age 76, and eventually drops to 65 percent at the terminal age (8). Although these numbers are higher than those reported in the literature, the overall shape is in keeping with the empirical findings. The average mortgage LTV ratio decreases steadily with age. It starts at 8 percent at age 23, and declined to around 15 percent by age 8. The refinancing rate is bimodal. It reaches its first peak of about 1 percent at age 3, and the second peak of close to 14 percent at age 8. In our model economy, households refinance their mortgages to extract home equity for consumption purposes. When the household is young and has just purchased his house, he will not refinance since he does not have much home equity to justify the refinancing cost. When he reaches his early thirties, however, he has accumulated some home equity but remains liquidity constrained. As a result, he will refinance if receiving a negative income shock. As the household further ages, he gradually realizes his income potential and has sufficient savings to smooth consumption without accessing home equity. After retirement, because of the income drop, he starts to run down his liquid asset. Once the liquid asset is low enough, it will be worth the cost to cash out home equity. By doing so, the household defers the house selling and rental cost. The default rate peaks at slightly lower than 1 percent at age 3 and quickly goes down to zero at age 5. New homeowners are less likely to default because their house values have not changed and the 2 percent initial down payment makes default costly. Older households are also less likely to default because of the time-amortization of their fixed-rate mortgages that have reduced the mortgage balance. 2

21 Figure 6 panel c charts housing and numeraire good consumption against the household age. The consumption of both goods increase initially with the household age and then decreases tracking the average effective household size. The upward sloping consumption profile in the early part of the life cycle also reflects the precautionary savings motive as found in the previous literature, while the downward sloping portion captures the additional effect of the increasing mortality risk. Finally, Figure 6 panel d shows the composition of a household s total networth between liquid asset and illiquid home equity. We observe that the proportion of networth allocated to home equity has a U shape while the proportion allocated to liquid asset has an inverse-u shape. These results are intuitive. When the household is young, he has few assets and most of them are committed to his house. As he ages, his income increases and he accumulates more and more liquid assets until he reaches retirement (age 65). After that, because of the income drop the increasing mortality risk, the household begins to draw down his liquid assets first for consumption so as to defer refinancing and selling charges, and eventually accesses home equity as a last resort. 6 Comparative Static Analysis In this section, we explore the separate effect of permanent and transitory labor income risks on an average household s life-cycle housing and consumption choices. In particular, we simulate three economies that are identical to our benchmark model with the exception that we eliminate the transitory labor income risk in the first economy, and the permanent labor risk in the second. In the third economy, we assume that the labor income risk (permanent) and house prices risk are independent so as to further investigate the role of the correlation between the two risks in households life-cycle decisions. As in the benchmark economy, we generate the time-series profiles of the optimal decisions for each economy by taking the average of 5, simulations from t = 1 (age 2) to t = 6 (age 8). All the comparisons are done with respect to the benchmark simulation. 21

22 6.1 The Effect of Transitory Labor Income Risk Figure 7 plots the home ownership rate and mortgage LTV ratio (panel a), refinancing and default rates (panel b), housing and numeraire good consumption (panel c), and the respective share of liquid savings and home equity in total wealth (panel d) against the household s age for the economy with no transitory labor income risk. Without the transitory labor income risk, the home ownership rate over the life cycle changes in two ways in comparison to the benchmark case. First, the average age for firsttime home buyers is slightly older. This is because with additional transitory shocks, a few households can get rich relatively fast if they are lucky to receive a sequence of good shocks. As a result, these households will become homeowners. Once the household passes this initial stage and is older than 24, however, he is much more likely to own homes in the new economy. The reason The mortgage LTV ratio changes little from the benchmark case. On the one hand, households are becoming homeowners early and that should push up the average mortgage LTV ratio. On the other hand, since more of them own houses early, the average LTV ratio also moves down faster because of the payment schedule. Apparently, these two forces largely offset each other. As seen in Figure 7 panel 2, the household does not refinance his mortgages much in the new economy when young since he has little need to smooth consumption intertemporally. The default rate also comes down a tad. The consumption of both housing services and numeraire good, Figure 7 panel C,... Finally, homeowners on average hold much more of their net worth as home equity and much less as liquid asset (Figure 7 panel d), consistent with the reduced incentive to smooth consumption intertemporally and therefore the reduced desire to access home equity. 6.2 The Effect of Permanent Labor Income Risk Figure 8 plots the home ownership rate and mortgage LTV ratio (panel a), refinancing and default rates (panel b), housing and numeraire good consumption (panel c), and the respective shares of liquid savings and home equity in total wealth (panel d) against the 22

23 household s age for the economy without permanent labor income risk. The average home ownership rate remains hump-shaped over the household age. Interestingly, however, relative to the benchmark case, the rate starts taking a positive value at a slightly older age, increases more gradually after that reaching the value of 1 at around age 6 instead of 47, and declines more sharply beginning at age 76. By age 8, less than 65 percent of the households are homeowners. This suggests that the average household becomes homeowners later and stays in his house for a shorter period of time. The intuition for the result is as follows. The presence of permanent labor income risk creates a great need for precautionary savings in the sense that the household will start accumulating assets at a relatively younger age and in relatively larger amount. These assets could be either in the form of liquid savings or illiquid home equity. Due to the refinancing cost, the household will naturally allocate proportionally more to liquid savings especially when he is young. Under our parameterization, however, the return on the housing asset and the utility value provided by housing counter-balances the refinance cost so that the household diversifies his holdings by committing some networth to the house, i.e., he becomes homeowner at a younger age. In our new simulation, since we eliminated the permanent labor income risk, we observe the opposite: the household purchases his house when his older. Consistent with the observation that households on average are becoming households later and given our fixed-rate mortgage contract assumption, the mean mortgage LTV declines more slowly. At the terminal age, the household still holds close to 3 percent of his housing asset in mortgages. The refinancing rate (Figure 7 panel b) still exhibits a bimodal pattern, but overall the rate has moved up appreciably for all ages especially at the two peak ages. The reasoning is similar as above. Without permanent labor income risk, all the fluctuations in the household s income will be transitory and the household will not save as much as a buffer stock. As a result, the young household is more likely to be liquidity constrained and he is more comfortable at accessing his home equity when constrained since he expects his income to recover soon. The household will also have to rely more on mortgage refinancing to smooth consumption when old because the liquid asset is run down more quickly than 23

24 before. Default rate Figure 7 panel d confirms our early discussion: without permanent income risk, the household holds proportionally more of his net worth in home equity and proportionally less in liquid asset. 6.3 The Effect of the Permanent Income-House Price Correlation Figure 8 plots the home ownership rate and mortgage LTV ratio (panel a), refinancing and default rates (panel b), housing and numeraire good consumption (panel c), and the respective share of liquid savings and home equity in total wealth (panel d) against the household s age for the economy where the permanent labor income risk and the house price risk are independent. In the benchmark economy, consistent with the empirical work, the correlation was positive. 6.4 The Effect of House Price Risk 7 Conclusions In this paper we present a life-cycle model of optimal housing and consumption decisions with uninsurable labor income risk and fluctuating housing prices. The model consists three key features that distinguish it from the existing consumption literature. First, housing choices are explicitly modelled both along the own-rent extensive margin and the intensive margin of housing value conditional on ownership. Second, a long-term mortgage contract is required for house purchases. Lastly, homeowners can refinance at a cost their mortgages for consumption purposes. The last two features imply that liquid savings and illiquid home equity affect households consumption decisions differently. A calibrated version of the model generates simulation results concerning a household s optimal house tenure, mortgage and consumption decisions largely consistent with the empirical literature. In particular, home ownership rate increases initially with household age and then decreases. Mortgage loan-to-value ratio declines steadily with household age. We also find that refinance activity is bimodal. Young households tend to refinance to ease 24