Nominal Mortgage Contracts and the Effects of Inflation on Portfolio Allocation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Nominal Mortgage Contracts and the Effects of Inflation on Portfolio Allocation Joseph B. Nichols 2007-67 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Nominal Mortgage Contracts and the Effects of Inflation on Portfolio Allocation Joseph B. Nichols∗ JELClassificationCodes: E21,G11,G21,R21,C61 March2007 Abstract Householdswhowishtoextracthomeequitythroughrefinancingtheirmortgagefaceahidden transaction cost. The real value of the fixed nominal mortgage payment declines over time with inflation. Thechangeintherealvalueofthemortgagepaymentsfromtakingonanewmortgage is positive and an increasing function of inflation; higher inflation thus discourages households from re-balancing their portfolio as frequently as they would otherwise. The life cycle model developedinthispaperdemonstrateshowtheshareoftotalwealthheldinhousingissensitiveto the rate of inflation, even when perfectly anticipated. Households hold larger positions in home equity earlier in the life cycle and smaller positions later in the life cycle as the rate of inflation increases. ∗Comments are welcome at: Joseph.B.Nichols@frb.gov. I would like to thank John Rust, JohnShea, AnthonyYezer, MichaelPries, ShaneSherlund, KathleenJohnson, AndreasLehnert, MichaelHighfield,GreggForte,KarenDynan,JulieHolt-Baker,andCarolynAlerfortheircomments. I would also like to thank seminar participants at the American Real Estate and Urban EconomicsAssociation, EconometricSociety, FinancialManagementAssociation, andBoardof Governors of the Federal Reserve System. Financial support from the University of Maryland PopulationCenterSeedGrant,EconomicClubofWashingtonDoctoralResearchFellowship,and Department of Housing and Urban Development Doctoral Dissertation Grant is gratefully acknowledged, asiscomputingresourcesfromthePittsburgSupercomputingCenter. Theanalysis andconclusionsexpressedhereinarethoseoftheauthoranddonotnecessarilyrepresenttheviews oftheBoardofGovernorsoftheFederalReserveSystemorothermembersofitsstaff.

1 Introduction Housingwealthrepresentsasignificantshareofmosthouseholdstotalwealth. Flavin and Yamashita (2002) used the Panel Study of Income Dynamics to show that among homeowner households with a head between 18 and 30 years old, 67.8% of their portfolio is in their home. A common explanation for this concentration is that private homes are “lumpy” assets and households face significant transaction costs in extracting home equity through refinancing their mortgages. Generally, these transaction costs represent loan fees, realtor fees, and the time cost associated with the transaction. This paper explores a hidden, yet significant potential further source of transaction costs associated with mortgages with fixed nominal payments, and then shows how this cost is sensitive to inflation, even whenthatinflationisperfectlyanticipated. Those who examine the effects of inflation on the real macroeconomy commonlyassumethatpricesare“sticky.” Stickypricesareusuallytakentobewages controlled by long-term contracts and prices of goods and services. These examples overlook perhaps one of the stickiest of prices: the recurring payments specified in many mortgage contracts, which are fixed in nominal terms for up to thirty years,regardlessofchangesinthelevelofinflation. A key price held constant for so long has important effects on the investment behavior of homeowners over the life cycle. Given inflation, the real value of a fixed nominal mortgage payment declines over the life of a mortgage contract, whilethepaymentonanewcomparablemortgagepaymentincreasesinlinewith 2

inflation. The gap between the old and new real mortgage payment does not depend on higher mortgage interest rates or larger real mortgage balances, but are a direct result of the declining real value of the fixed nominal mortgage payment fromtheoriginalmortgage. Thishiddentransactioncostsdiscourageshouseholds from more frequent re-balancing of their total portfolio by shifting assets from homeequityintofinancialassets. With about 70 percent of homeowners having a mortgage on their primary residence (Bucks, Kennickell, and Moore, 2006), housing wealth cannot be realistically modeled using a single state variable, as can holdings of a standard risky or risk-free financial asset. The model must also allow for the terms of the mortgage contract. The second section of this paper describes the household’s choice where housing wealth is assumed to consist of three connected components: an asset, a liability, and a contract for a stream of recurring payments. In a standard fixed-rate self-amortizing mortgage, the recurring payments for the principal and interest are fixed at a nominal level for the life of the loan, even as the real value falls. I show that under certain specific, but not terribly binding, assumptions, a household who later refinances at the same mortgage interest rate, will face an increaseintherealvalueoftherecurringmortgage payment. Starting in the third section of this paper, a detailed partial-equilibrium life cycle model with an explicit mortgage contract is developed. Themodel captures householdsavingandconsumptiondecisionsoverthelifecycleinanenvironment with several important features of the U.S. homeownership system. In particular, it allows for a transition in tenure from renter to owner, and for households to 3

increase or decrease their consumption of housing by trading up to a larger home or trading down to a smaller home. Households also have access to a standard thirty-year fixed-rate self-amortizing mortgage to purchase their homes and have theopportunitytotaptheirhomeequitythroughrefinancingtheirmortgages. This paperdoesnotrepresentaseriousattempttoformallycalibrateamodelofhousing wealth or to estimate the maximum likelihood parameters of such a model. The goal is rather to see how closely the model can match certain stylized facts while usingfairlystandardandcommonparametervaluesusedintherelatedliterature. The model’s solution is used to demonstrate how an inflation rate that is perfectlyanticipatedandunchangingoverthelifecycleaffectsthehousehold’sportfolio allocation by increasing the transaction costs associated with shifting assets from housing to financial assets. The model is solved for different inflation rates andalsoforacaseinwhichthemortgagecontractisinrealandnotnominalterms. The results suggest that, the higher rate of inflation (that is, when the hidden cost to refinancing is higher), households hold larger positions in home equity earlier in the life cycle, and smaller positions later in the life cycle. The results also help explains why retired households hold such a significant portion of their wealth in housing. 2 Theoretical Model This section lays out the theoretical justification behind the main conclusion of the paper: the nature of a standard fixed-rate mortgage contract implies that 4

the cost of re-balancing a household’s portfolio between housing and financial wealth is an increasing function of inflation, even when that inflation is perfectly anticipated. The assumptions underlying the theorems presented in this section are that the household invests α of their total wealth at time 0, W , in a financial 0 0 assetA ,andusestheremainder,1−α ,asadownpaymentonahouse. Givena 0 0 requireddownpaymentrateofµ,thevalueofthehouse,P is (1−α0)W0. 0 µ Theorem 2.1. As long as the total return on housing over the holding period is positive, the share of wealth held in the financial asset will decline over the holding period. Proof. Aftertyearsthevalueofthefinancialportfolio,thehouse,andtheremainingmortgagebalancearerespectively, A = α W (1+η )t t 0 0 s (1−α )W P = 0 0 (1+η )t t h µ (1−α )(1−µ)W 1−(1+ν)t−30 0 0 D = t (cid:18) µ (cid:19)(cid:18)(1−(1+ν)−30)(1+π)t(cid:19) where (1 + η )t and (1 + η )t are the total rates of return on the financial asset s h and housing over the holding period, ν is the mortgage interest rate, and π is the inflation rate. Note that the mortgage balance is in nominal terms, and must be adjustedusingtherateofinflation. 5

Thenewvalueoftheportfolio shareofthefinancialassetis A t α = t A +P −D t t t α W (1+η )t 0 0 s = α W (1+η )t + (1−α0)W0(1+η )t − (1−α0)(1−µ)W0 1−(1+ν)t−30 0 0 s µ h µ (1−(1+ν)−30)(1+π)t (cid:16) (cid:17)(cid:16) (cid:17) (1+η )t s = (1+η )t + 1−α0 (1+η )t −(1−µ) 1−(1+ν)t−30 s α0µ (cid:16) h (1−(1+ν)−30)(1+π)t (cid:17) 1 = 1+ 1−α0 (1+η )t −(1−µ) 1−(1+ν)t−30 α0µ(1+ηs)t (cid:16) h (1−(1+ν)−30)(1+π)t (cid:17) Thetheoremholdsif, α < α t 0 1 < α 1+ 1−α0 (1+η )t −(1−µ) 1−(1+ν)t−30 0 α0µ(1+ηs)t (cid:16) h (1−(1+ν)−30)(1+π)t (cid:17) 1−α 1−(1+ν)t−30 α + 0 (1+η )t −(1−µ) > 1 0 µ(1+η )t (cid:18) h (1−(1+ν)−30)(1+π)t(cid:19) s 1−(1+ν)t−30 α µ(1+η )t +(1−α ) (1+η )t −(1−µ) > µ(1+η )t 0 s 0 (cid:18) h (1−(1+ν)−30)(1+π)t(cid:19) s 1−(1+ν)t−30 (1−α ) (1+η )t −(1−µ) > (1−α )µ(1+η )t 0 (cid:18) h (1−(1+ν)−30)(1+π)t(cid:19) 0 s 1−(1+ν)t−30 µ(1+η )t +(1−µ) < (1+η )t s (1−(1+ν)−30)(1+π)t h The balance on a self-amortizing mortgage begins at the starting value of the loanandslowlytrendstozerooverthelifeofthemortgage. Therefore, 1−(1+ν)t−30 = 1, when t = 0 1−(1+ν)−30 6

andisdecreasingint. Therefore, 1−(1+ν)t−30 (1+η )t > µ(1+η )t +(1−µ) → α > α h s (1−(1+ν)−30)(1+π)t 0 t The first term on the right-hand side of the inequality above is the total return on the financial asset discounted by the fraction of the initial home value the household puts up as a down payment. The second term is bounded above by the initial loan-to-value ratio, and is decreasing over time. If the household forgoes mortgage financing, and pays the full home value up-front (µ = 1), the above condition holds only if the total return on housing is greater than the total return on financial assets. At the other extreme, if the household puts no-money down (µ = 0), the condition holds if the total return on housing does not fall below the rateofamortization. Thisconclusionishardlycounter-intuitive. Giventhatpower of leveraging granted by the mortgage combined with the ongoing paying down of the mortgage balance, households will over time slowly decrease the share of their wealth in the financial asset and increase the share held in home equity over time. Thekeycontributionofthispaper,aslaidoutinthefollowingtheorem,isthat the costs associated with extracting accumulated home equity is an increasing function ofinflation. Theorem2.2. Ahouseholdthatrefinancesitsexistinghomeinordertore-balance 7

its portfolio between home equity and financial assets will face higher real mortgage payments as long as the total return on the financial asset does not exceed theleveragedtotalreturnonhousing. Proof. After t years the household refinances its house, and puts (1 − α )(A + 0 t P −D )backintohomeequity. Thenewmortgagebalanceis t t D = P −(1−α )(A +P −D ) t+1 t 0 t t t (1−α )W (1+η )t 0 0 h D = t+1 µ 1−α (1−µ)(1−(1+ν)t−30) − (1−α )W α (1+η )t + 0 (1+η )t − 0 0 (cid:18) 0 s µ (cid:18) h ((1−(1+ν)−30)(1+π)t)(cid:19)(cid:19) Therealvalueofthepreviousmortgagepaymentis νD 0 M = t (1−(1+ν)−30)(1+π)t andtherealvalueofthenewmortgagepaymentis νD t+1 M = t+1 (1−(1+ν)−30) 8

Theratioofthenewmortgagepaymenttotheoldcanbewritten as M D (1+π)t t+1 t+1 = M D t 0 (1−α0)W0 (1−α )(1−µ)(1−(1+ν)t−30) −α µ(1+η )t +α (1+η )t (1+π)t µ 0 (1−(1+ν)−30)(1−π)t 0 s 0 h = (cid:16) (cid:17) (1−α0)(1−µ)W0 µ α (1+π)t((1+η )t −µ(1+η )t) 1−(1+ν)t−30 0 h s = +(1−α ) (1−µ) 0 1−(1+ν)−30 Thetheoremholdsif, M t+1 > 1 M t α (1+π)t((1+η )t −µ(1+η )t) 1−(1+ν)t−30 0 h s +(1−α ) > 1 (1−µ) 0 1−(1+ν)−30 α (1+π)t((1+η )t −µ(1+η )t) 1−(1+ν)t−30 0 h s > 1−(1−α ) (1−µ) 0 1−(1+ν)−30 1−(1+ν)t−30 1−(1+ν)t−30 (1−α ) < 1 → 1−(1−α ) > 0 0 1−(1+ν)−30 0 1−(1+ν)−30 α (1+π)t((1+η )t −µ(1+η )t) 0 h s > 0 (1−µ) (1+η )t −µ(1+η )t > 0 h s (1+η )t h > (1+η )t → M > M s t+1 t µ The term, (1+ηh)t , represents the leverage-adjusted total return on housing. µ These theorems do not hold when housing generates a return significantly below that of the risky asset or when the amount of leveraging is very small. If when extractingtheirhomeequity,thehouseholdputslessbackintohomeequity,either because they are trading up to a larger home or they which to keep their LTV 9

ratio, µ, constant, the gap between the real value of their new and old mortgage paymentsisevengreater. Corrolary2.3. Thesizeoftheincreaseintherealmortgagepaymentsuponequity extraction is an increasing function of inflation as long as the total return on the financialassetdoesnotexceedtheleveragedtotalreturnonhousing. When we differentiate the change in the mortgage payments by the inflation rate,π,wefindthat, d M t(1+π)t−1α ((1+η )t −µ(1+η )t) t+1 0 h s = dπ (cid:20) M (cid:21) 1−µ t Therefore, (1+η )t d M h > (1+η )t → t+1 > 0 s µ dπ (cid:20) M (cid:21) t The theorems laid out in this section support the argument that households faceasignificanttransactioncostofextractingtheirhomeequitythatistiedtothe fixed nominal nature of the mortgage payment. In addition, this transaction cost isanincreasingfunctionofinflation. Thisresultholdswhetherornottheinflation isexpectedorunexpected. 10

3 Simulation Model The structure of the model was chosen to highlight the effects of mortgage contracts on the evolution of housing wealth, via its effects on the transaction costs for extracting home equity. Theaddition of an explicit mortgage contract to the standard model creates an additional layer of complexity in the model, which is embedded in the wealth transition rules. To accommodate this additional complexity and keep the model tractable, several important assumptions are required and are discussed in detail below. This section deals only with the aspects of the model and the underlying assumptions that differ from those of the standard model. AppendixAprovidesamoreextensivediscussionofthemodel,andTable B-2inAppendixBliststhemodelparametersandtheirdefinitions. One approach to modeling the role of housing wealth over a household’s life cycle is to develop a simple model that captures only a few of the most important aspects of housing as an investment good. Papers such as Martin (2001) and Ferna´ndez-Villaverde and Krueger (2001) follow this approach. The advantage of such models is that many can be solved analytically or embedded in a general equilibrium framework and solved numerically. The primary disadvantage is that many have relatively narrow scopes. A second approach, as in Li and Yao (2004) andHu(2002),istosacrificesimplicityforamorecomplicatedpartialequilibrium modelthatcanbesolvednumericallyusingstochasticdynamicprogramming. An advantage of this more complex type of model is that it presents a more realistic picture of the role of housing wealth over the life cycle. The disadvantage is an upper bound on the model’s complexity, beyond which the solution times are no 11

longer tractable, although parallel processing in a grid-cluster or super-computer environment can extend this upper bound. The complexity of the model requires that great care be exercised in presenting the results and currently precludes the optionofembeddingthemodelinageneralequilibrium framework. This paper pursues the latter approach, specifically using a dynamic stochastic optimizing framework for the household based on that in Rust and Phelan (1996). Rust and Phelan set up and solve a dynamic programming problem of labor supply with incomplete markets, Social Security, and Medicare. The dynamic programming problem in their paper is solved by making the continuous state spaces discrete and then using backward recursion to solve for the optimal value of the continuous choice variable at each point on the state-space grid. The detailed rules governing the determination of Social Security and Medicare benefits are embedded in the income-transition matrix. The model in this paper has a structure similar to those in Rust and Phelan and in Li and Yao, in that it embedsthedetailedcharacteristicsofthemortgagecontractintheincome-transition matrix;thesignificantinnovationofthemodelisitsinclusionofanexplicitmortgagecontract. Inthemainversionofthemodelinflationispresentbutisconstant and perfectly anticipated. The goal of the paper is to show that even under this strict assumption about inflation, optimal portfolio allocation differs under differentlevelsofconstantinflation. Theremainderofthissectiondiscussesthecomplicationsintroducedbyinclusion of the mortgage contract. The sections of the model that do not significantly differfrom thoseinotherpapersarediscussedinAppendixA. 12

As noted, households can store their wealth in a real asset by purchasing a house. Inthemodelitisonlythroughthepurchaseofahouse,andtheacquisition of a mortgage loan, that households can borrow against their future income. The model’s use of durable goods as collateral is in the same spirit as in Ferna´ndez- VillaverdeandKrueger(2001). Theonlymortgagecontractavailabletothehouseholdinthismodelrequiresa20percentdownpayment,hasatermofthirtyyears, and requires mortgage payments based on a fixed interest rate and the size of the original mortgage. The mortgage balance and the mortgage payment are both in nominalterms,whiletherestofthemodelisinrealterms. Householdspurchasing ahomearealsorequiredtopayatransactioncostequalto10percentofthevalue ofthehome. Thiscostrepresentsrealtor’s’fees,creditchecks,andotherexpenses associatedwiththepurchase. To completely model the effects of a fixed-rate mortgage over the life cycle one must keep track of four additional continuous state variables: (1) the current value of the home, (2) the current balance of the mortgage, (3) the level of the fixed mortgage payment, and (4) the share of the fixed mortgage payment that is deducted from the outstanding mortgage balance in a given year. Because the value of home equity is the difference between the value of the home minus the remainingbalanceontheoutstandingmortgagetomeasureequity,onemusttrack boththevalueofthehomeandthemortgagebalance. Thenatureofthemortgage contract complicates what would be a logical approach for tracking the mortgage balance, the addition of a continuous state variable. In particular, the principal paidonaself-amortizingmortgageisafunctionoftheageofthemortgage. Initial 13

payments are almost completely composed of interest, and the final payments on a thirty year mortgage, on the other hand, are almost completely principal. Thisprocessismodeledbyincludingthemortgageageasadiscretestatevariable with thirty-one discrete values and imposing a strict structure on the evolution of the price of housing. The fact that the mortgage balance and mortgage payment are in nominal terms provides an additional motivation for including the age of the mortgage in the state space, as the real values of the mortgage balance and paymentdeclinesteadilyoverthelifeofthemortgagebecauseofinflation. Many factors in the model are conditional on current housing tenure, such as thecostofhousingserviceswhichincludesrentormortgagepaymentsandmaintenance costs, the level of utility derived from housing, and the change in wealth associated with the appreciation in home values. To reduce the number of state spaces,householdschoosingtoownarelimitedtobuyingeitherasmalloralarge home. The size of a large home is assumed to be twice that of a small home. The real price of housing has a positive trend over time. It is assumed that the price of rental and owner-occupied housing evolve at the same rate. Home prices are further governed by two assumptions: (1) the purchase prices of both small and large homes increase deterministically by the average increase in market price in eachperiodand(2)thevalueofhomesthathavealreadybeenpurchasedchanges according to a stochastic process, with the expected increase equal to the nonstochastic increase in market price. A significant result of these two assumptions is that a household that has had a series of periods of above-average rates of appreciation will own a home worth more than the market value of a comparable 14

home, whereas a household that has had a series of periods with below-average appreciation will own a home worth less than the market value of a comparable home. Theprimary motivation for the twoassumptions isaneedto control thenumber of state spaces. The mortgage payment paid by a household will vary according to the size of the house and its purchase price. By assuming that the price of new housing is deterministic and by including as state variables the age of the mortgageandsizeoftheexistinghouse,weareabletodefineboththelevelofthe mortgagepaymentandtheremainingmortgagebalanceasafunctionofthesestate variables. Another advantageof using thisprocessto model theevolution of both marketandhousehold-levelhomepricesisthatthereissignificantcross-sectional variation in home appreciation. In effect, the model forces households always to buytheaverage-pricehome,regardlessoftheirownrealizedpriceappreciation. The effect of steadily increasing home prices provides another motivation for the inclusion of the age of the mortgage as a state variable. For example, because of the steady increase in home prices, the initial mortgage and the related mortgagepaymentsonagivenhometodaywouldbesignificantlygreaterthanthe mortgage on asimilar home twenty years ago. Asthe model shows, this provides adisincentiveforolderhouseholdstomoveorrefinance. The model’s design allows households to choose their current consumption, their savings, their savings allocated to risky assets as opposed to a risk-free asset, the type of housing they occupy, and whether to refinance their mortgages. Households face uncertainty in the returns on risky assets and housing, the prob- 15

ability of surviving, and income through a transitory shock. Income also has a deterministic component that is a function of age. The model includes moving, maintenance,andtransactioncosts. Themodelalsoincludestheoptiontodefault onamortgage andthecostsofdoingso. Themodelissolvedgiventhetermsofatraditionalthirtyyearfixed-ratemortgagecontract. Thevaluesofnonstructuralparameters,suchasreturnsondifferent types of assets, the survival probability, mortgage terms, and the income process, aretakenfromhistorical dataandarediscussedinAppendixB. 3.1 Consumption of Housing The housing choice in period t, can take values associated with a rental unit, i ,asmallhomei ,andalargehomei . Householdsmayincreasetheirmortgage r s l balances through the use of cash-out refinancing. The model does not currently include home equity lines of credit as an option for withdrawing equity. The number of housing units available to rent is continuous, whereas the number of housingunitsprovidedbysmallandlargehomesisfixed. Thenumberofhousing unitsassociatedwiththehousingchoicei isdefinedbythefunctionh(i ). t t Renters choose the number of housing units represented by the size of the rental unit so that the intra-period marginal utility of housing is equal to the marginal utilityofnondurableconsumption. ∂U(c ,h(i )) ∂U(c ,h(i )) t t t t (3.1) = ∂c ∂h(i ) t t 16

Theoptimalsizeofarentalunitthatequalizedtheintra-periodmarginalutilityof housingtothemarginalutilityofnondurableconsumptionmaynowbedefinedas afunctionofconsumption: (3.2) h(i ) = (φ/(1−φ))c r t The parameter φ is the measure of preference between a unit of housing and consumption,asdefinedintheutilityfunctionpresentedinAppendixA.Asshownin the equation above, the model implies that φ will equal the share of total householdexpendituresassociatedwithhousing. 3.2 Price of Housing The price per housing unit is the same across all types of housing. Large homes cost more than small homes because they provide more units of housing. Rent is proportional to the current market value of the home that renters choose. The value of owner-occupied units evolves stochastically, and the value of newly purchasedandrentalunitsaresetequaltothecurrentdeterministicmarketprice. The formulas for the market value of home type i , P (i ), and the housing wealth, t t t H ,transition ruleareasfollows: t+1 (3.3) P (i ) = (1+η )tP h(i ) t t h 0 t 17

H (1+r ), i = i  t h t+1 t   (3.4) H t+1 =     P t (i t ), e i t+1 6= i t i t+1 ∈ i s ,i l    0, i t+1 = i r      (3.5) r ∼ N(η ,σ2) h h h where P is the price of a single unit of housing in period 0; P (i ) is the price 0 t t in period t of the number of housing units associated with housing choice i ; r t h is the realized rate of appreciation on housing in period t; η is the expected raete h of appreciation on housing; and σ2 is the variance of house price growth. Note h that the price of owner-occupied housing is allowed to evolve differently from the market price of housing overall in order to capture the idiosyncratic aspect of housingreturns. Notealsothathomepricesareinrealterms,sotheincreaseinthe marketpriceofhousingisnotduetogeneralinflationbutrathertoarealincrease inthevalueofthehouseovertime. 3.3 The Mortgage Theinterestrateandinitialtermtomaturityareassumedtobeconstantovertime and across households. The only variable determining the mortgage payment is the price of the home price when purchased. The homeowner’s payment changes only when a new mortgage is entered into and that occurs only when the household refinances the mortgage or sells the house. A cash-out refinancing resets the 18

numberofyearsleftonthemortgage. Theformulafortherealvalueofamortgage paymentattimetafterκ yearsonahouseoftypei is: t t ν(1−µ)P (3.6) M (i ,κ ) = t−κt t t t (1−(1+ν)−30)(1+π)κt where ν is the nominal mortgage interest rate, π is the inflation rate, and µ is the requireddownpayment. Alsothecostofhousingservicesreflectsthemaintenancecostspaidbyhomeowners. Asaresult,theformulafortherealcostofhousingservicesis: M (i ,κ )+δH , i ∈ i ,i  t t t t t s l (3.7) X (i ,κ ) = t t t    0.06P (i ), i = i t r t r    whereδH ismaintenancecosts,whichareassumedtobeapercentageofcurrent t home value. Rent is equal to 6 percent of the current market value of the unit beingrentedandrenterspaynoneofthemaintenancecostsfortheproperty. The present value of the homeowner’s home equity is the current value of the house minus the amount of the outstanding mortgage balance. The value of the houseincreasesordecreasesaccordingtothestochasticreturnonhousing,andthe outstanding mortgage balance is a monotonically declining function of the age of the mortgage. The formula for the real value of the mortgage balance at time t 19

afterκ yearsonahouseoftypei is: t t (3.8) M (i ,κ )1−(1+ν)κt−30 , i ∈ i ,i and κ ≤ 30  t t t ν t s l t D (i ,κ ) = t t t    0, (i ∈ i ,i and κ > 30) or (i = i ) t s l t t r    The formula for the real mortgage payment is used to calculate the amount of mortgage interest paid for tax purposes. The real values must be adjusted back to nominal terms because the deduction is in nominal terms. The formula for the mortgage interestdeductionis: (3.9) I (i ,κ ) = M (i ,κ )(1−(1+ν)κt−30)(1+π)κt t t t t t t 3.4 Net Change in Liquid Assets from Sale or Refinancing The net change in liquid assets after paying transaction costs and down payments for a homeowner moving in the next period, i ∈ {i ,i ,i }, is given t+1 r s l by H −D (i ,κ )−µP (i )−τH −χ, i 6= i  t t t t t t+1 t t+1 t (3.10) G (i ,i ,κ ) = t t t+1 t    0, i = i t+1 t    where τH is the transaction cost, µ is the down payment rate, and χ is a fixed t moving cost paid regardless of which type of housing is being purchased. When thehouseholdchoosesnottomove,(i = i ),ithasazeronetgain. t+1 t 20

The net gain in liquid assets after choosing to refinance a mortgage is defined as the sum of the difference between the mortgage balances after and before the refinancingandafeeforthetransaction: (1−ζ)D (i ,κ )−D (i ,κ ), κ 6= κ +1  t t t+1 t t t t+1 t (3.11) Z (κ ,κ ) = t t t+1    0, κ = κ +1 t+1 t    whereζ representstheshareofthenewmortgageaccountedforbythetransaction costs associated with refinancing. Interest rates are constant in this model, so there is no possibility of refinancing at a lower interest rate. The only benefit of refinancing is to extract home equity and use the proceeds to invest in financial assets or to smooth consumption. When no refinancing occurs, the net gain is zero. When the household extracts cash by refinancing, Z (κ ,κ ) > 0. Only t t t+1 householdsthatchoosenottomoveinagivenperiodmaychoosetorefinance. 3.5 Default Penalties The model contains a default penalty. In any period, the household must be able to cover its housing expenses, which, in the case of rentals, consists only of the rent, and in the case of homeowners consists of the mortgage payment and maintenance costs. If a homeowning household fails to cover its housing expense,itmustmoveinthenextperiodintorentalhousingandforfeitallitshome equity and all its financial equity above some small nominal amount. Defaulting 21

rentersontheotherhandmustsimplymovetoanewrentalunit. Householdsthat can cover their expenses by selling their current house and extracting their home equityareallowedtodoso. Ahouseholdthatcanaffordtheassociatedtransaction costsmayalsoavoiddefaultthroughacash-outrefinancingandthereforekeepits housingequity. Current consumption of households that default is constrained to equal the smallnominalamountofequitylefttothemafterdefault. TherestrictionthatA t+1 may not be negative, combined with the definitions of X (i ,κ ), Z (κ ,κ ), t t t t t t+1 G (i ,i ,κ )andthebudgetconstraint,createsanupperboundonpossiblelevels t t t+1 t of nondurable consumption and also rules out some possible choices of housing tenure. Ifthehouseholdcannotaffordthedownpaymentforalargehomewithout incurring negativewealth,itisnotallowedtomovetosuchahome. 4 Baseline Simulation Model Results The parameter values for the model calibration are chosen to be consistent with other models in the literature as cited above. The parameter values for the sizes of small and large homes are set so that they represent, respectively, 80 percent and 120 percent of the size of a median-priced home in 1990. The share of total household expenditures, including both renters and owners, allocated to housing expenditures, (φ) is set to 0.2, which is the value in the 2001 Consumer Expenditure Survey from the U.S. Department of Labor. Appendix B contains more information on the values of the market parameters and preference parame- 22

terschosen. The model is used to generate 10,000 simulations. Households, represented byasinglehouseholdhead,beginasrenterswithnoassetsatage20. Households retire at age 65 and have a maximum potential age of 80. The simulations track householdsaccumulationofhousingandfinancialwealthovertheirlifetimes. Income among surviving households drops sharply at retirement whereas the path of consumption over the life cycle is much smoother, (figure 4.1, panel A). Younger households who are aggressively saving for a down payment consume the smallest share of their wealth (panel B). Once households become homeowners, their consumption as a share of total wealth climbs, peaking near 16 percent aroundage30. Ashouseholdsapproachretirement,theystarttoaccumulatemore wealth,andconsumptionasashareoftotalwealthstartstofall,reachingalowof 9 percentatage65 (panel C).In retirement, households drawdown their savings, and consumption as a share of total wealth climbs again. At the outset of retirementtheaveragehouseholdhasroughlyforty-fivetimesitsannualpost-retirement incomesavedinbothhousingandfinancialwealth. Housing wealth is hump-shaped, on average, over the life cycle of the household,reachingapeakatage65anddecliningafterage70(figure4.2panelA).The brief plateau in the growth of housing wealth at age 50 is caused by many householdseithertradingdowntosmallerhomesorrefinancingtheirexistingmortgage to ensure that their nominal mortgage payments are fixed for the rest of their expected lives.1 Financial wealth over the life cycle is more sharply humped and 1This behavior becomes more apparent under several alternative scenarios presented later in 23

FIGURE 4.1: Consumption andIncome (a) Wages and Consumption 3.5 3 2.5 2 1.5 1 0.5 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS rof )000,0$( eulaV naeM Wages Consumption (b) Consumption as Share of Total Wealth 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 20 30 40 50 60 70 80 Age htlaeW latoT/noitpmusnoC egarevA (c) Ratio of Total Wealth to Income 45 40 35 30 25 20 15 10 5 0 20 30 40 50 60 70 80 Age emocnI/htlaeW latoT egarevA 24

FIGURE 4.2: WealthandPortfolio Choice (a) Simulated Housing Wealth 7 6 5 4 3 2 1 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS rof )000,0$( eulaV naeM (b) Simulated Financial Wealth 14 12 10 8 6 4 2 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS rof )000,0$( eulaV naeM (c) Simulated Share of Housing Assets for Owners 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS rof dleH stessA fo erahS naeM (d) Simulated Risky Share of Financial Assets 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 20 30 40 50 60 70 80 Age trohoC ni srovivruS rof dleH stessA fo erahS naeM 25

peaksatage65(panelB). PanelArevealsthathouseholdsuseaccumulatedhomeequitytofinancetheir consumption of nondurables only late in retirement after their reserves of financial wealth have been largely depleted. These results are fairly consistent with some previous empirical work. Venti and Wise (2000) found that housing wealth wasnotinfactusedtosupportnon-housingconsumption. Theyfoundthathouseholds resort to their home equity only when faced by a significant shock such as the death of a spouse or a serious illness. Similarly, Sheiner and Weil (1992) found that anticipation of illness or death significantly increases the probability that households will reduce their home equity. However, the present model does resultinamorerapiddeclinelateinlifeinhousingwealththanpreviousempirical studieshaveshown. Themodel’somissionandretireelaborsupplymightexplain thisfailure. Inparticularhouseholdsmightviewtheirhomeequityasanimportant source of savings to tap when faced with a serious health shock and be unwilling toextractthatequityintheabsenceofsuchashock. Theycouldavoidconsuming theirhousingwealthbycontinuing toworkafterretirement. The simulated mean share of assets held in housing for surviving household heads between the ages of about 35 and 65 is consistently near 40 percent, (figure 4.2 panel C). The housing share is high among young households, who must invest a large portion of their savings in a down payment. As financial wealth grows faster than housing wealth, the housing share falls but somewhat climbs thepaper. Itistoalargedegreeaproductofthespecificmodeldesign,specificallytheinteraction ofthethirty-yearmortgagewiththemaximumageofthehouseholdssetat80. Intheabsenceof these restrictions, households would still attempt to “lock-in” their nominal mortgage payments priortoretirement,butthisactivitywouldnotbeclumpedtogetheratage50. 26

FIGURE 4.3: HousingTenureChoice (a) Simulated Homeownership Rate 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS fo noitcarF (b) Simulated Share of Homeowners in Large Homes 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS fo noitcarF (c) Simulated Loan−to−Value Ratio 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 30 40 50 60 70 80 Age emoH fo eulaV−ot−naoL fo oitaR egarevA (d) Simulated Refinancing Rate 0.025 0.02 0.015 0.01 0.005 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS fo noitcarF 27

again. The jagged nature of the curve reflects a comparatively small number of simulations. The housing share climbs in retirement, as households draw down financial wealth before they extract home equity because of the transaction costs. This behavior can explain some of the “over investment” in housing seen in the empirical data, as reported by Flavin and Yamashita (2002). The implication is that some degree of over investment in housing can arise as a result of the mortgage contract rather than any suboptimal behavior by irrational consumers. Note, though, that the degree of over-investment implied by the model is below that seen in the data; Flavin and Yamashita find the mean share of total assets held in housingtobe67percent. Regarding the allocation of financial assets over the life cycle, young households – who are focused on saving for a down payment or already have a large share of their wealth in housing – invest less in risky financial assets and more in therisk-freeasset(figure4.2panelD).Olderhouseholdsalsohavetheirallocation weighted to less risky financial assets, but in their case because they have drawn downtheirfinancialwealthrelativetotheirhousingwealth. Theshareofthefinancial portfolio invested in risky assets peaks around age 50, just when households starttoactivelyshifttheirtotalportfolio awayfrom homeequity. The model results also document other housing decisions over the life cycle. Homeownership increases rapidly for younger households and declines very slightly in retirement (figure 4.3 panel A). The share of homeowners living in larger homes has a similar contour with a considerable drop at age 50 (panel B), as households trade down in retirement to access housing wealth to finance con- 28

sumption. The sharpness of the decline at age 50 results largely from the combination of the thirty year term of the mortgage and the assumed maximum age of 80 and thus should not be taken too literally. In particular, at 50 years old, many households take advantage of the thirty year mortgage term to hold constant their nominal mortgage payments for the rest of their lives. These households will continue to receive a constant stream of utility from their home, the real value of the mortgage payments will fall because of inflation. In effect, in buying a home at age 50, households are purchasing an annuity from which the stream of real payments – the difference between the implicit rent and the real mortgage cost – will increase with time and be at its highest during retirement when income is at its lowest. This hypothesis is supported by results from simulations where the term of the mortgage is varied (figure 4.4). When the mortgage term is shortened to twenty years and the retirement age remains at 65, homeowners delay the shift to smaller homes to age 60. The results are not sensitive to the changing of the retirementage;inparticular,whentheretirementageis75andthemortgageterm isthirtyyears,homeownersstillshifttosmallerhomesatage50. Thusproximity toretirementcanberuledoutasafactor. Panel C shows the path of the loan-to-value ration over the life-cycle. At the time they purchase their homes, households are required to have a loan-to-value ratioof80percent(panelC).Theythenpaydowntheirmortgagethroughtheregular amortization schedule and the average loan-to-value ratio falls. The average loan-to-value ratio seems to stabilize at 10% before climbing late in retirement in 29

FIGURE 4.4: WhyTradeDownat50? Simulated Share of Homeowners in Large Homes 0.25 0.2 0.15 0.1 0.05 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS fo noitcarF Baseline 20−year mortgage term Retire at 75 response to a surge in cash-out refinancing. This surge can be seen in Panel D. In addition,youngerhouseholdsandthosethathavejustpurchasedtheirhomestake advantageofrefinancingtore-balancetheirportfolios andsmooththeirincome. 5 The Effects of Inflation on Portfolio Allocation Thissectionshowshowportfolioallocationacrossthelifecyclechangeswith different levels of inflation. The model is re-solved for different levels and corresponding sets of simulations are generated. Setting the inflation parameter to a higher level simulates theeffectsof high inflation, andsetting it to zerosimulates the effect of mortgage contracts that are real rather than nominal. The levels of wealth accumulation, housing demand, refinancing activity, and portfolio allocationundereachalternativeassumptionarethencomparedtothebasecase. 30

FIGURE 5.1: RentandMortgagePayments 500 450 400 350 300 250 200 0 5 10 15 20 25 30 Years s$ Rent Nominal Mortgage Real Mortgage Thepresenceofnominalmortgagecontractseffectivelyshiftstherealcostsof homeownershipforwardoverthetermofthemortgage(figure5.1). Thefactthat the real value of the mortgage payment declines over the life of the mortgage is, ofcourse,factoredintotheoriginalmortgage rate. Figure 5.2 and 5.3 show how portfolio allocation and housing demand differ under different inflation rates. Under the zero-inflation scenario (the reddottedlines)thetransactioncostsfacinghouseholdsinrebalancingtheirportfolios are significantly reduced. Households, especially younger households, hold less housing wealth. A common strategy in the presence of large transaction costs for durablegoodsisforahouseholdto“over-buy,”thatisbuyalargerhousethanthey wouldotherwise,knowingthattheywouldbeunabletoeasilyre-adjusttheirlevel of housing consumption later in the life-cycle when they might actually want a larger home. With zero inflation reducing the transaction costs, there is signifi- 31

FIGURE 5.2: WealthandPortfolio Choice (a) Simulated Housing Wealth 7 6 5 4 3 2 1 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS rof )000,0$( eulaV naeM (b) Simulated Share of Housing Assets 0.8 Baseline High Inflation 0.7 No Inflation 0.6 0.5 0.4 0.3 0.2 0.1 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS rof dleH stessA fo erahS naeM Baseline High Inflation No Inflation FIGURE 5.3: HousingTenureChoice (a) Simulated Share of Homeowners in Large Homes 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 20 30 40 50 60 70 80 Age trohoC ni srovivruS fo noitcarF (b) Simulated Loan−To−Value Ratio 0.8 Baseline 0.7 High Inflation No Inflation 0.6 0.5 0.4 0.3 0.2 0.1 0 20 30 40 50 60 70 80 Age emoH fo eulaV−ot−naoL fo oitaR egarevA Baseline High Inflation No Inflation 32

cantly less of this “over-buying.” In addition, we do not observe the sudden shift towards smaller homes at age 50 seen in the base case, since the real value of the mortgagepaymentisnolongerdeclining,andhouseholdsgainlessofabenefitof lockingtheirmortgagesprior toretirement. Theresultsfromthehighinflationscenario(thegreenlines)areamirrorimage ofthosefromthelowinflationscenario. Nowthatthetransactioncostsassociated with extracting home equity are higher, households rebalance their mortgage less frequently, resulting in an increase in housing wealth held by the young and very oldandadecreaseinhousingwealthheldbythemiddle-aged andearlyretirees. The sudden shift towards smaller homes at age 50 is even more pronounced under the high inflation scenario, as the benefit of locking their mortgages prior to retirement is greater with higher inflation. Many households attempt to tradeup just prior at age 50, knowing that if they get several positive income shocks they would be able to afford the higher real mortgage payments, which would graduallydeclineoverthelifeofthemortgage. Whentheydonotreceivepositive income shocks, households in this model trade down because they fear they will not be able to make the mortgage payments once their income falls in retirement. Duringperiodsofhighinflation,mortgagecontractswithfixednominalpayments are especially attractive to households. During the prime of their earnings years (their 50s) the real mortgage payment is high. The real mortgage payment then falls rapidly during retirement, just when their income also falls. As discussed above,nominalmortgagepaymentsallowolderhouseholdstoshifttheexpenseof housing forward to earlier years to ensure their ability to consume housing later; 33

thehighertherateofinflation,thegreaterthebenefittodoso. 6 Conclusion The model developed in this paper demonstrates the way in which perfectly anticipated inflation, even when reflected in the nominal interest rate on mortgages, can distort the household’s portfolio allocation over the life cycle. A standardthirty-yearfixed-ratenominalmortgagecontractresultsindecliningreal mortgage payments. The longer a mortgage is held, the greater the difference between the household’s mortgage payment and the payment for a mortgage refinanced in the current period. This widening gap discourages households from shifting assets from home equity to financial portfolios. When calibrated using commonlyacceptedparameters,theresultscanexplainatleastsomeofthe“overinvestment in housing” documented in the earlier literature and also help explain why retired households hold such a significant portion of their wealth in housing. Inflation distorts the household’s portfolio allocation by introducing a hidden transactioncostthroughitseffectontherealvalueofthefixednominalmortgage payment. When inflation is high, the gap in the real value of a new mortgage and an existing mortgage is larger, and households rebalance their portfolio between homeequityandfinancialassetslessfrequently. Wheninflationislow,household find extracting their home equity to be cheaper, and therefore more frequently rebalancetheirportfolio. 34

Appendix A - Detailed Partial Equilibrium Life Cycle Model This appendix provides a detailed description of the partial equilibrium life-cycle model used in this paper. Aspects of the model that significantly differ from the basic model used in the literature are discussed in the main body of the paper. The household’s optimization problem is to maximize lifetime utility, defined as: (A-1) 80 E βtρ U(c ,h(i ))+βt(1−ρ )(θ U (A )+θ U (H )−θ U (D )) t t t t A B t H B t D B t tX=20 s.t. c > 0,∀t = 20,...,80 t (c1−φh(i )φ)1−λ (A-2) U(c ,h(i )) = t t t t 1−λ b1−λ (A-3) U (b) = B 1−λ where • c representstheconsumption ofnondurables; t • h(i ) represents the number of units of housing services consumed, given t 35

the housing tenure choice in period t (note that while the number of units of housing services consumed varies with tenure choice, the utility gained fromaunitofservicesdoesnotvary); • A ,H , and D are the values of financial assets, home, and mortgage debt t t t leftasbequests,respectively; • β representsthediscountrate; • ρ isthesurvivalprobability; t • φ represents the measure of preference between a unit of housing and consumption; • λrepresentsameasureofriskaversion;and • θ ,θ , and θ represent parameters associated with the utility of leaving A H M bequests. A household lives at most 60 years from age 20 to a maximum of age 80. The utility of bequests is treated as separable to capture the disutility associated with negative as well as positive bequests. It faces uncertainty about its survival, temporary income shocks, and the rate of return on both housing and risky assets. In addition to the stochastic elements for income and the rate of return on risky assets,thehouseholdsmayexperienceadditionalshocks. Asmallprobabilityexists thatthehouseholdwillexperienceunemploymentinoneperiod,reducingincome to zero. Also, a small independent probability exists of a stock market crash, in 36

which case the household will lose 100 percent of its investment in the risky financial asset. The probability of a stock market crash is in addition to the regular standard deviation associated with the stochastic rate of return on risky assets. Theinclusionofthisadditionalshockisrequiredinordertopersuadehouseholds nottoholdfinancialportfoliosconsistingentirelyofriskyassets. Thepriceofthe consumptiongoodissetequaltounityandtherentalpriceofhousingissetequal toaconstantratiooftheunderlyingpriceofthehousingunit. Theinflationrateis constant,positive,andknown. Households begin at age 20 as renters with no savings. Thus, they have no financial wealth and no housing wealth. In each period, a household receives a draw from an age-dependent income process. The model contains only transitory shocks. In retirement, pension income is set to 60 percent of the deterministic portion of income at age 65. Pension income is still subject to transitory shocks representinguncertaintyregarding medicalcosts. Households can store their wealth in two different classes of assets: financial and real. The household’s financial assets are held in a portfolio of risk-free and risky assets. The household can costlessly rebalance its financial portfolio between risk-free and risky assets in every period. Households with zero wealth face a binding liquidity constraint for financial assets in that they cannot borrow against their future income. The only way to effectively borrow against future income is through a mortgage for which the household meets the down payment and income requirements. Households also cannot purchase leveraged portfolios, inwhichtheyborrowattherisk-freeratetoinvestmoreinriskyassets. Inaddition 37

tomovingtooneofthethreetypesofhousing,arentalunit(i ),asmallhouse(i ), r s and a large house (i ), the household can also decide to stay in its current home, l {i = i }. Households may add to their mortgage balance through cash-out t+1 t refinancing. Thetransitionrulefortheleveloffinancialwealthisdefinedas: (A-4) A = (1+(1−γ)(α r +(1−α )r)) t+1 t st t ×(A −c −X (if,κ )+G (i ,i ,κ )+Z (κ ,κ )) t t t t t t t t+1 t t t t+1 +(1−γ)e +γI (i ,κ ) t+1 t t t s.t. A g≥ 0 t+1 0 ≤ α ≤ 1 t where • A istheleveloffinancialassetsinperiodt; t • A is a random variable that depends on the stochastic rate of return on t+1 riskyassets(r )inperiodtandtherealizationsofearnings(e )inperiod st t+1 t+1; f g • α istheshareinvestedinriskyassetsintimet; t • r isthedeterministicrateofreturnonrisk-freeassets; • X (i ,κ ) is the housing cost, that is the rent, or the mortgage plus maint t t tenance and taxes, incurred in period t for a household currently choosing 38

tenuretypei withamortgageκ yearsold; t t • I (i ,κ )isthemortgage interestpaid; t t t • G (i ,i ,κ ) is the net gain, that is, the proceed or cost from a housing t t t+1 t transaction including the realized capital gain or loss from a sale and the down payment and transaction costs from a new purchase, for a household choosingi thisperiodandi nextperiod; t t+1 • Z (κ ,κ ) is the net gain, that is, the value of home equity extracted after t t t+1 transactioncosts,from cash-outrefinancing;and • γ is the tax rate on income and capital gains (note that both income and capital gains have the same tax rate and that taxes on capital gains are paid immediately). The net gain from a home sale is tax free and the mortgage interest paid is deducted from taxable income. Both housing expenses and the amount of the mortgage interest deduction are functions of the current housing choice and the age of the mortgage. Refinancing is modeled as a choice to increase the remaining number of years on the mortgage, or inversely, to shorten the current age of the mortgage. The model allows only cash-out refinancing and does not allow prepayments. Theageofamortgageforarentalunitandofamortgagethathasbeen paidoffiszero. Households receive their wages at the same time they realize the returns on their investment from the previous period. As a result, the state variable A rept resents all available cash on hand, consisting of previous financial wealth and 39

current income. The income process is defined as a deterministic function of age plusatransitory shock: (A-5) log(e ) = ψ +ψ t+ψ t2 +εe t 0 1 2 t εe ∼ N(0,σ ) t e Therealrateofreturnonriskyassetsisarandom variablewiththedistribution (A-6) r ∼ N(η ,σ2) st s s whereη istheexpectedrealrateofreturnontheriskyassetandσ2isthevariance. s s Thehousehold’soptimizationproblemistochoosevariablesc ,α ,i ,κ given t t t+1 t+1 a series of state variables t,κ ,i ,A ,H to optimize equation (A-1) given equat t t t tions (A-2) through (A-6). The household has only one choice of mortgage contract, with a fixed down payment rate. The choice variable κ captures the t+1 ability of a household to cash-out home equity by refinancing and thereby reduce theeffectiveageofthemortgage. Thevaluefunctionofthehouseholdisthemaximumutility,subjecttothedefaultconstraintsofthevaluefunctionsforthehouseholdsthatchoosenextperiod’s 40

tenuretypei ∈ {i ,i ,i ,i }: t+1 r s l t (A-7) A −X (i ,κ ) < 0 & t t t t A −X (i ,κ )+ max (G (i ,i )+Z (κ ,κ )) > 0 ⇒ t t t t t t t+1 t t t+1 it+1,κt+1 V (i ,A ,H ,κ ) = max V it+1(i ,A ,H ,κ ) t t t t t t t+1 t t t it+16=itorκt+16=κt+1,ct,αt (A-8) A −X (i ,κ ) < 0 & t t t t A −X (i ,κ )+ max (G (i ,i )+Z (κ ,κ )) < 0 ⇒ t t t t t t t+1 t t t+1 it+1,κt+1 V (i ,A ,H ,κ ) = U(ω,h(i ))+βρ V (i ,ω,0,0)+β(1−ρ )θ U (ω) t t t t t t t t r t A B (A-9) A −X (i ,κ ) > 0 ⇒ t t t t V (i ,A ,H ,κ ) = max V it+1(i ,A ,H ,κ ) t t t t t t m t t t it+1∈{ir,is,il},ct,αt,κt+1 whereω istheamountofconsumptionandwealthprotectedindefaultfromcreditors. Equation (A-7) is the value function when the household’s recurring housing expenses, X (i ,κ ), are greater than its available liquid assets, A and its t t t t net equity after selling or refinancing its home is positive, A − X (i ,κ ) + t t t t max (G (i ,i ) + Z (κ ,κ )) > 0. Faced with this constraint, the it+1,κt+1 t t t+1 t t t+1 household must either move, i 6= i , or refinance, κ 6= κ + 1. Equation t+1 t t+1 t (A-8)isthevaluefunctionwhenthehouseholdcannotcoveritsrecurringhousing expenses out of its liquid assets and its net equity after selling or refinancing the 41

home is negative. This household must move to a rental unit, i = i , and have t+1 r both its consumption and remaining wealth limited to ω. Equation (A-9) is the value function when the household can cover its recurring housing expenses out of their liquid assets. The only limits to their choices are those embedded in the constraintsinequation(A-4). Thevaluefunctionconditional onnextperiod’stenurechoicei is t+1 (A-10) maxU(c ,h(i ))+βρ V (i ,A ,H ,1) t t t t t+1 t+1 t+1  ct,αt for i ∈ {i ,i ,i }  t+1 r s l    +β(1−ρ )(θ U (A )+θ U (H )−θ U (D )), V it+1(i ,A ,H ,κ ) =    t A B t H B t D B t t t t t t    max U(c ,h(i ))+βρ V (i ,A ,H ,1) t t t t t+1 t+1 t+1 ct,αt,κt+1 for i = i  t+1 t    +β(1−ρ )(θ U (A )+θ U (H )−θ U (D )),   t A B t H B t D B t     suchthatequations(A-2) through(A-9) hold. ThecodeusedtosolvethisproblemwaswritteninC.Onesolutionoftheproblem initially took roughly two weeks on a dual processor Pentium Xeon 1.8GHz with 512K L2 cache and 1GB of RAM running Linux. In order to improve upon the run-time, the code was rewritten to take advantage of parallel processing, usingtheMessagePassingInterface(MPI)standard. Inthisversionofthecodeone processorisdesignatedthemasterwhileapoolofotherprocessorsaredesignated slaves. As the model is solved recursively by year, the master distributes the current value function for all previous years to the slaves. Each slave then solves for the optimal value function for a subset of state spaces for the given year. The slaves then return the new value function values to the master. The master then 42

combinesthenewvalueswiththevaluefunctionforthepreviousyear,completing the recursion for one year. The problem was solved using 61 high-performance Digital Alpha 64-bit microprocessors running at 450MHz each on a scalable parallel Cray T3E at the Pittsburgh Supercomputing Center. One solution involved roughly 1.3 billion evaluations of the value function and took roughly eight and a halfhours. Appendix B - Baseline Model Parameter Values The parameter values for the baseline model are chosen to be consistent with other models in the relevant literature. As was discussed in appendix A, the income process consists of a deterministic factor and a transitory factor. The income process is based on the results of regressions of Social Security earnings on age and age-squared. The dependent variable is the log of the wage income in constant 1990 dollars. The transitory factor of wage is reflected in the estimated standard error of the regression. The wage is converted from log to level terms in the model. At age 65 the level of the deterministic wage falls to a flat level equalto60%ofthelastperiod’sincomebeforeanytransitoryshocks,acondition representingasystemofforcedretirementandadefinedbenefitpensionplan. The coefficients and standard deviation used in this version of the model are in Table B-1. The market price of a housing unit is the result of setting the deterministic home price at age 60 with the National Association of Realtors’ 1990 median home price. It is assumed that a median home consists of 10 housing units. The 43

TABLE B-1: LogIncomeRegressionResults Constant ψ 7.28626 0 CoefficientAge ψ 0.10278 1 CoefficientofAge2 ψ -0.00098 2 Std. Dev. σ 0.80778 w R2 15.5% ProbabilityofUnemployment υ 1% home prices are converted to constant 1990 dollars and the deterministic home price series are calculated using the historical average return. The average and standard deviation of the return on housing are at taken from Li and Yao (2004) and are consistent with Campbell and Cocco (2003). The mortgage interest rate used is the average rate on loans with 80 percent loan-to-value ratios as reported by Freddie Mac from 1969 to 2001, adjusting for the inflation rate. The percent required for the down payment represents the minimum needed to avoid paying mortgageinsurance. Thetransaction,maintenance,andmovingcostsarebasedon survey data provided by the National Association of Realtors. The values chosen forthecurrentversionofthemodelareinTableB-2. Theriskandreturnonrisky assetsfollowsYaoandZhang(2004). ThevaluesforthepreferenceparametersshowninTableB-3belowwerechosen to replicate certain stylized facts about the role of owner-occupied housing in portfolios, specifically the large share of total wealth held in home equity. An λ value of 2 represents a relatively low, but realistic, level of risk aversion. An β value of 0.96 is a commonly used discount rate. The φ value of 0.2 reflects theshareoftotalhouseholdexpendituresallocatedtohousingexpendituresinthe 2001ConsumerExpenditureSurveyfromtheU.S.DepartmentofLabor. Thedis- 44

TABLE B-2: ValuesofMarketParameters ParameterNameandDefinition Symbol Value Realriskfreerateofreturn r 2% Priceof1housingunit,atage60 P (1) 1.003 60 Sizeofsmallhomes h(i ) 8 s Sizeoflargehomes h(i ) 12 l Meanofrealreturnonhousing η 1% h Standarddeviationofhousingreturn σ 11.5% h Meanofrealreturn onriskyasset η 6% s Standarddeviationofriskyassetreturn σ 15.7% s Probabilityof100%lossonriskyasset ς 1% Mortgageinterestrate ν 5% Percentrequiredasdownpayment µ 20% Percentofhomepricelosttotransactioncosts τ 10% Maintenancecosts δ 0.7% Movingcosts χ 0.3 TaxRate γ 30% RefinancingCosts ζ 3% Inflation π 2% Note: Unitsnotinpercentarein$10,000s. 45

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