Endogenous Life-Cycle Portfolio Allocation in the Presence of both ...
Endogenous Life-Cycle Portfolio Allocation in the Presence of both Housing Investment and Tax-Deferred Accounts Denis Pelletier North Carolina State University∗ and Cengiz Tunc Central Bank of the Republic of Turkey† First version: February, 2012 Current version: May, 2015 Abstract: This paper develops a life-cycle model to address the joint effects of housing investment tax-deferred accounts on the portfolio allocation of households. Besides the distinction between the taxable accounts and tax-deferred accounts, the model employs a comprehensive housing investment structure and Epstein-Zin recursive preferences. The results of the model shows that both tax-deferred accounts and housing investment have sizable crowding out effect on the risky asset investment of households and the effect is found to be larger for housing investment than for tax-deferred accounts.
JEL classification: G11, D91, R21, D14 Keywords: Portfolio Choice, Life-cycle Models, Housing, Household Saving.
∗
North Carolina State University, Box 8110, College of Management, NCSU, Raleigh, NC 27695-8110, USA, de-
nis [email protected]. † Central Bank of the Republic of Turkey, Istiklal Cad. 10 Ulus Ankara, Turkey, [email protected].
1
Introduction
Tax-deferred retirement accounts (TDAs) have shown considerable growth for more than two decades. The share of defined-contribution plans in the US households retirement assets has increased from about 10% in 1980 to more than 25% as of 2013 (see Figure 1). These accounts provide households with the opportunity of investing in tax-exempt assets to help accumulate enough wealth for their retirement. Taxes on these accounts are only effective upon withdrawals of funds. As more households rely on TDAs, they face both an optimal asset location problem (i.e. deciding how much to hold in taxable and tax-deferred accounts) and an optimal asset allocation problem (i.e. deciding how to distribute funds between risky and riskless assets within each account). Hence understanding the effects of the TDAs on households’ life cycle portfolio allocation becomes increasingly important.
28 26 24 22 20 18 16 14 12 10
1985
1990
1995
2000
2005
2010
Figure 1: The Share of DC plans in the US Retirement Assets (Source: Financial Accounts of the United States) Similar to TDAs, housing investment plays a critical role on household life-cycle wealth allocation decisions because housing investment constitutes a significant share of homeowners’ wealth. According to the Bureau of Census, the mean homeownership rate in the US between 1980-2014 is 2
65.8% with standard deviation of 1.72. Furthermore, Figure 2 shows that the average sales price of houses sold in the US is more than five times of per capita income.
8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Figure 2: The Ratio of Average Sales Price of Houses Sold to Per Capita GDP (Source: Bureau of Census) The contribution of this study to the portfolio allocation literature is to combine these two important components of wealth into a fairly comprehensive life-cycle model in order to analyze the portfolio allocation of households in a more realistic framework. So far the standard life-cycle models are not sufficiently comprehensive in the sense that they exclude either TDAs (Grossman and Laroque (1990), Cocco (2004), Yao and Zhang (2005), Hu (2005), Campanale, Fugazza, and Gomes (2014), Flavin and Yamashita (2002)) or a comprehensive housing investment (Gomes, Michaelides, and Polkovnichenko (2009), Zhou (2012), Dammon, Spatt, and Zhang (2004), Garlappi and Huang (2006), Shoven and Sialm (2003)) and therefore can’t fully explain life-cycle portfolio allocation profiles of households. In order to capture the effects of TDAs and housing investment on households life-cycle wealth and portfolio allocation decisions more precisely, we build a fairly rich structural life-cycle model pa-
3
rameterized to be consistent with empirical data. In this model, we consider finitely living households facing mortality risk, borrowing and short-sale constraints, and receiving stochastic labor income. In every period, households make consumption, housing investment, and asset location and allocation decisions. Regarding financial assets, households can invest in both TDAs and TAs (asset location) and decide on the allocation of wealth between the risky and the riskless assets within each account (asset allocation). Both households and employers make contributions to TDAs during the working period. TDAs provide higher returns than taxable accounts since taxes are paid only upon withdrawal of funds whereas returns on taxable assets are taxed once these returns accrued. Hence households can accumulate funds in TDAs faster than the accumulation of funds in taxable accounts. Besides the distinction between TDAs and TAs, we have modeled housing investment in a fairly comprehensive way. Housing investment is different from other financial investments because it is not only an asset but also a durable consumption good from which households derive utility. It is also, contrary to liquid financial assets, illiquid and often highly leveraged. Furthermore, the wealth allocation of households depends on their homeownership status. While homeowners are committed to pay a flow of payments so as to pay back their mortgage they also enjoy homeownership. Renters, on the other hand, have no commitments and only pay rent but they don’t get the benefits of being homeowners. This model is also specified using Epstein-Zin type recursive preferences [Epstein and Zin (1989)] where the relative risk aversion (RRA) is disentangled from the elasticity of intertemporal substitution (EIS). While RRA gives information about how agents deal with uncertainty across possible states of the world, (EIS) is just a time preference parameter. The problem with the commonly used constant relative risk aversion (CRRA) utility function is that RRA is the inverse of EIS. In other words, CRRA imposes two different roles on the same parameter while EZ provides the flexibility to relax this constraint and have two separate values for these two parameters. The results of our model suggest the presence of TDAs and housing investment have significant effects on consumption and financial wealth allocation of households. Households begin with higher shares of risky assets in each account but decrease this share significantly throughout the life-cycle. Therefore, to some extent, the model is able to mimic the portfolio allocation profiles observed in the empirical data. The presence of both TDAs and housing investment provides households with 4
alternative investment opportunities with different features. As households have more investment alternatives, they are able to distribute risk accordingly. They do not need to rely heavily on one investment in which they need to have higher share of risky asset investment so as to accumulate enough funds. In the absence of either TDAs or housing investment, however, we observe higher shares of risky asset investment. Therefore, the presence of both TDAs and housing investment have clear crowding out effect on risky asset investment and the effect is larger in the presence of housing investment than the presence of TDAs. The reminder of the paper proceeds as follows. Section 2 goes over the previous studies and discuss the contribution of this study to the existing literature. Section 3 introduces the model and its assumptions. Section 4 presents the parameterization and calibration of the model. The simulation results of the model with the baseline parameters, various extensions of the baseline model, and a comparison of the model to the data are given in Section 5. Finally, Section 6 gives concluding remarks and possible extensions of the model for future work.
2
Literature Review
This paper is related to two separate strands of the literature on portfolio allocation, analyzing either (1) the effects of TDAs or (2) housing investment on household portfolio allocation profiles. In the first strand, Gomes, Michaelides, and Polkovnichenko (2009) use a life-cycle model with both TDAs and TAs to analyze the portfolio allocation of both direct stockholders (those investing in both TDAs and TAs) and indirect stockholders (those investing in only TDAs). They find that TDAs increase wealth accumulation which is generally transformed into consumption for retirement. Furthermore, households with lower saving incentives respond less to TDAs. Zhou (2012) finds that differential costs of stock market participation in TDAs and TAs can explain the higher participation rate in TDAs early in life relative to TAs. He also shows that the decline in the participation rate in TDAs late in life is due to the differential tax treatments between these two accounts. Bergstresser and Poterba (2004) study household behaviors from the Survey of Consumer Finances (SCF). The findings of their paper suggest that about one-third of households with both TAs and TDAs are following asset location
5
strategies that are tax inefficient1 . They conclude that non-tax considerations may play an important part in households’ portfolio allocation decisions. Dammon, Spatt, and Zhang (2004) show that taxes play an important role in asset location and allocation decisions. They find that households prefer to hold bonds in TDAs and equities in TAs. However, households’ decisions also depend on other factors such as liquidity. The predictions of their model substantially deviate from the empirical observations that households hold a mix of both stocks and bonds in both their TDAs and TAs. Amromin (2003) claims the interaction between labor income risk and accessibility restriction to TDAs is a key factor in explaining the portfolio allocation of households. The second strand of the literature is on the effects of housing investment on households portfolio allocations. Although housing was ignored by the portfolio allocation literature, recently a growing literature treats housing as an important component of portfolio allocation. Using an asset allocation model where infinitely lived households derive utility from a single indivisible durable good, Grossman and Laroque (1990) argue that illiquid assets could answer the equity premium puzzle. Extending the Grossman and Laroque (1990) model by including both durable and nondurable consumption goods into the utility function, Flavin and Nakagawa (2008) compares a housing model to a habit persistence model and finds that while both deliver many of the same implications, empirical tests using household level data strongly favor the housing model. Furthermore, Villaverde and Krueger (2010) employ a general equilibrium model of life cycle asset allocation to analyze the effects of consumer durable goods on consumption and asset allocations. Piazzesi, Schneider, and Tuzel (2007) develop a housing-included consumption based asset pricing model. Housing is incorporated to the model both as an asset and as a consumption good. Their paper focuses on the effects of housing-consumption asset pricing models on the predictability of the return on stocks. A recent work by Campanale, Fugazza, and Gomes (2014) focuses on the role of transaction cost between liquid and illiquid financial assets on matching the empirical portfolio allocation profiles. There are some life-cycle portfolio allocation studies that focus directly on the role of housing investment with the finding of crowding out effect of housing investment on risky asset investment (Hu (2005), Cocco (2004), Yao and Zhang (2005), and Tunc and Pelletier (2013)). Although the 1
Tax-efficiency means putting the least tax efficient funds (funds that have higher tax-rate) in TDAs and most tax-
efficient in TAs.
6
focus of Amromin, Huang, and Sialm (2007) is different than the focus of our paper, it is worth mentioning this paper as it analyze the trade-off between mortgage prepayments and TDA retirement savings. Using SCF data, the paper concludes that at least 38% of households who prepay their mortgages could benefit from increasing their TDAs contributions instead of prepaying mortgages. A recent study by Marekwica, Schaefer, and Sebestian (2013) also studies the effects of both TDA and housing investment on consumption portfolio problem of households. However, their study focuses more on the tax-arbitrage opportunity from the existence of both housing investment and TDA. Unlike these studies, this paper combines these two strands of the literature in order to analyze life cycle portfolio allocation decisions of households in the presence of both housing investment and TDAs. It makes an explicit comparison between the crowding out effect of TDA and the housing investment on households portfolio allocation decision. In other words, it investigates how households asset location and allocation decision varies in each cases so as to assess the effect of each features on the portfolio allocations of households. Furthermore, in various comparative statics, this paper analyze the effects of the down payment, the size of TDAs contribution rate on the portfolio allocation of households as well as the differences between the portfolio allocation of both homeowners and renters.
3 3.1
Model Setup Household Preferences
The model is a discrete time life-cycle portfolio allocation model. Each period in the model corresponds to one year and as a general convention in the life-cycle literature each year is actually the real age of a household minus 19. Households live for a maximum of T years. The probability that a household is alive at time t + 1 conditional on being alive at time t is equal to qt . Households derive utility from a constant elasticity of substitution (CES) utility function with non-durable consumption goods (C), and housing investment (H). Preferences are in the form of Epstein-Zin where the RRA is
7
disentangled from the EIS: Vt
θ n θ1 o 1−γ 1−γ 1−γ 1−γ = u(Ct , Ht ) θ + β Et qt Vt+1 + (1 − qt )Wt+1
θ =
1−γ 1 − 1/ψ
(1) (2)
1
u (Ct , Ht ) = [Ctv + Htv ] v ,
(3)
where β is the time discount factor, γ is the RRA parameter, and ψ is the EIS parameter. The intratemporal elasticity of substitution between nondurable consumption goods and housing investment is 1/(1 − υ). Wt+1 is the households’ total wealth at time t + 1 that would be bequested in the case households pass away.
3.2
Labor Income Process
Households supply labor inelastically during working life and receive a stochastic labor income Yit which is composed of both deterministic and stochastic components. The deterministic component is a function of households’ age, education, marital status, and gender. Labor income received by household i at age t is denoted by Yit . Define yit = log(Yit ) where yit has the following process: yit = f (t, Zit ) + ulit ,
(4)
f (t, Z) = β0 + β1 age + β2 age2 + β3 gender + β4 marital status + β5 educ,
(5)
where shocks to the log of labor income ulit composes of both aggregate (ηtl ) and idiosyncratic (εlit ) components. We assume that the idiosyncratic component is transitory and i.i.d normally distributed with mean 0 and variance σε2l . The aggregate shock has an AR(1) process: l l ηtl = φηt−1 + wi,t ,
(6)
l where φ is the persistence parameter and wi,t is i.i.d. normally distributed with mean 0 and variance
σw2 l . Households retire at age of K where K corresponds to real age of 65 (K = 46). During the retirement, households receive a constant fraction of their last working period labor income Yit = ξYiK where 0 < ξ < 1 is replacement rate 2 . 2
During the whole retirement period uit becomes uiK .
8
3.3
Housing Investment
All households are renters in the first period. They endogeneously decide to be homeowners or stay as renters in each of the following periods. A typical homeowner could maintain his homeownership status by either staying in his current house or moving to another house. He could also decide to be renter by selling his current house and moving to a renter property. Similarly, a typical renter could decide to buy a house and become homeowner or stay as a renter. Households are required to pay a fraction d of the market value of the house as down payment and to finance the rest through a mortgage. In order to capture the illiquidity of housing investment, when households decide to sell their house, they incur a liquidation cost equal to a fraction κ of the house’s market value. Homeowners also pay in each period for the maintenance and depreciation expenses an amount equal to a proportion δ of the house’s market value. However, renters only pay annual rent which is equal to a proportion α of the house’s market value. We define the per unit price of housing is Pth . Therefore the value of a house of size Hi is Pth Hi . Define pht = log(Pth ), and assume that pht has the following stochastic form: ∆pht = µh + εht ,
εht ∼ N (0, σε2h ),
(7)
where µh as the average growth rate of house prices. The downpayment of a house size Hi at time t is dPth Hi . Hence the mortgage loan RMt that typical households can take against their house is RMt ≤ (1 − d)Pth Hi .
3.4
Financial Assets and Wealth Accumulation
Households can invest in only two financial assets: a risky asset and a riskless asset. Investment in TAs and TDAs have different returns due to the tax treatments. Return on investment in TAs are taxed once these returns accrued. However, investment on TDAs are taxed when households withdraw funds from them. Throughout their working life, households contribute to TDAs a fraction kl of before-tax earnings. Employers also contribute a fraction ke of before-tax earnings to households’
9
TDAs. Households cannot withdraw funds from TDAs before retirement.3 Once households retire in period K, they face a withdrawal rate equal to the inverse of their life expectancy. The tax rate on these withdrawals is the same as the income tax rate, τy . Real gross return on investment in the riskless asset is Rb . After-tax gross return is b 1 + R (1 + π) − 1 (1 − τb ) b e = , R 1+π
(8)
where π is a constant inflation rate and τb is the tax-rate on nominal return for riskless asset investments. Gross real return on the risky asset is s Rt+1 = Rb + µs + εst+1 ,
(9)
where µs is the real, before-tax equity premium and εst+1 is a random shock that is i.i.d.N (0, σε2s ). The gross return on the risky asset at time t is comprised of a constant nominal dividend yield dy and a stochastic nominal capital gain cgt . Dividend yields are taxed at the rate τdy and capital gains are taxed at the rate τcg . The before and after-tax real gross return on the risky asset investment takes the following forms: s Rt+1 =
1 + cgt + dy , 1+π
es = 1 + cgt (1 − τcg ) + dy (1 − τdy ) . R t+1 1+π 3.4.1
(10)
(11)
Renter Wealth Accumulation
Depending on the homeownership status, there are two types of households in this modei, homeowners and renters. We start first by considering a typical renter’s wealth accumulation and budget constraint. The labor income of renters is taxed after the contribution to TDAs is made. Once a renter receives his taxed labor income and taxed return on investment made in TAs at time t − 1, he allocates the total cash-on-hand (Xt ) to (1) consumption expenditure, (2) housing investment, and finally (3) investment in the TAs and the allocation of funds within this account to risky and riskless assets. Regarding housing investment, renters decide to pay the down payment and become homeowners if it is 3
In real life, it is possible to withdraw funds from TDAs with a penalty. Due to computational concerns we do not
incorporate this feature into the model.
10
optimal or stay as renters and wait until the next period. Wealth accumulation and budget constraints of renters have the following form: TD b TD s R + k ∗ Yt Rt + Bt−1 WtT D = St−1
es + Bt−1 R eb LWt = St−1 R t
(12) (13)
Xt ≡ LWt + (1 − τy ) [Yt − kl Yt ]
(14)
k ∗ = kl + ke
(15)
Xt = St + Bt + Ct + (1 − HRt ) αPth Ht + HRt Mt + dPth Ht
(16)
Mt = M Pt + M It
(17)
RMt = RMt−1 − M Pt ,
(18)
where WtT D is the total funds of a typical households in TDAs at time t. The investment made in the TD TD ). Likewise, St−1 (Bt−1 ) denotes (Bt−1 risky (riskless) asset in TDAs at time t − 1 is denoted by St−1
the investment made in the risky (riskless) asset in TAs at time t − 1. Mt is the annual mortgage payment that homeowners pay. Mortgage payments consist of both principal payments (M Pt ) and interest payments (M It ). The remaining balance of the mortgage loan at time t is RMt . HRt is a dummy variable that equals 1 if a household is a homeowner at time t, and 0 otherwise. The wealth allocation and budget constraints of renters during the retirement period is different than the working period since households don’t make contributions to TDAs but withdraw funds from it. During the retirement period, renters have the following form for wealth allocation and budget
11
constraint: TD b TD s R − Qt Rt + Bt−1 WtT D = St−1
Qt = ςt WtT D
(19) (20)
es + Bt−1 R eb LWt = St−1 R t
(21)
Xt ≡ LWt + (1 − τy ) Yt + (1 − τy ) Qt Xt = St + Bt + Ct + (1 − HRt ) αPth Ht + HRt Mt + dPth Ht
(22)
Mt = M Pt + M It
(23)
RMt = RMt−1 − M Pt
(24)
where ςt is the withdrawal rate of funds from TDAs at time t and Qt is the amount of funds households withdraw from TDAs at this time. 3.4.2
Homeowner Wealth Accumulation
Homeowners’ wealth in TDAs (WtT D ), the contributions to this account, and the form of liquid wealth (LWt ) are the same as for renters. However, the amount of cash-on-hand, Xt , is different since homeowners make mortgage payments. The interest component of this payment is tax-exempt. Labor income of homeowners is taxed after the TDAs contribution and interest component of mortgage payments are deducted. Combining this deducted and taxed labor income with taxed returns on investment made in TAs at time t − 1, homeowners allocate this total cash-on-hand (Xt ) to (1) consumption expenditures, (2) housing investment and expenditures, and finally (3) investment in TAs. Furthermore they allocate their funds between risky and riskless assets within each account. Therefore homeowners wealth accumulation and budget constraints before retirement take the following
12
structure: TD b TD s R + kt∗ Yt Rt + Bt−1 WtT D = St−1
(25)
es + Bt−1 R eb LWt = St−1 R t
(26)
Xt ≡ LWt + (1 − τy ) [Yt − kl Yt − M It ]
(27)
Xt = Ct + St + Bt + τh Pth Ht−1 + δPth Ht−1 + HRt (1 − M St ) [Mt ] + HRt M St dPth Ht + Mt − ((1 − κ) Pt Ht−1 − RMt ) + (1 − HRt ) M St αPth Ht − (1 − κ) Pth Ht−1 − RMt ,
(28)
where τh denotes the property tax for housing, δ denotes the rate for maintenance and depreciation expenses of houses, κ is the rate of liquidation cost in the case of selling the house, and M St is a dummy variable which is 1 if a homeowner decides to move from the current house and is 0 otherwise. The after-retirement wealth accumulation and budget constraints, however, are slightly different than the before-retirement case because homeowners are withdrawing funds from their TDAs and making no contribution to this account. Furthermore, these withdrawals are subject to income tax. TD s TD b WtT D = St−1 Rt + Bt−1 R − Qt
(29)
Qt = ςWkT D
(30)
es + Bt−1 R eb LWt = St−1 R t
(31)
Xt ≡ LWt + (1 − τy ) [Yt − M It ] + (1 − τy ) Qt
(32)
Xt = Ct + St + Bt + τh Pth Ht−1 + δPth Ht−1 + HRt (1 − M St ) [Mt ] + HRt M St dPth Ht + Mt − (1 − κ) Pth Ht−1 − RMt + (1 − HRt ) M St αPth Ht − (1 − κ) Pth Ht−1 − RMt .
(33)
Household total wealth at time t (Wt ) is the sum of investment made in both TDAs and TAs at time t − 1 plus the house market value minus the remaining mortgage debt. Households bequest this total wealth to inheritors if they pass away before the terminal period T. If a household lives up to the final period T then he bequests WT +1 . We also impose short-sale and non-negativity constraints on consumption, housing investment, and financial asset investments: Ct ≥ 0, St ≥ 0, BtT D ≥ 0, StT D ≥ 0, Bt ≥ 0, Ht ≥ 0, ∀t, 13
(34)
Wt = WtT A + Pth Ht−1 − RMt−1 + WtT DA .
3.5
(35)
Optimization Problem
Before defining the optimization problem in a more concrete form, we list the state and control variables. The state variables of this problem are age, t, liquid wealth (LWt ), wealth in TDAs (WtT D ), homeownership status at time t − 1 (Ot−1 ), the size of house owners have at time t − 1 (Ht−1 ), and remaining mortgage debt at time t − 1 (RMt−1 ). We denote the state variables by Ω where Ωt = Ωt t, LWt , WtT D , Ot−1 , Ht−1 , RMt−1 . The control variables are consumption (Ct ), the size of house to be chosen at time t (Ht ), the homeownership decision at time t (Ot ), asset allocation decision within TDAs (st,T DAs ), and asset allocation decision in TAs (st,T As ). We define the control variables by Ψ where Ψt = Ψt (Ct , Ht, Ot , st,T DAs , st,T As ). After defining the state and control variables in a more compact form, the household optimization problem is given by θ n θ1 o 1−γ 1−γ 1−γ Vt (Ωt ) = max u(Ct , Ht ) θ + β Et qt Vt+1 (Ωt+1 )1−γ + (1 − qt )Wt+1 ,
{Ψt }
(36)
subject to Equations 1 to 35. This model has no analytical solution. Hence we use a numerical approximation method and solve the model using backward induction starting from time T to t = 1. Please see the details of the solution method in Appendix A.
4 4.1
Parameterization Preference Parameters
The baseline parameters of the model are presented in Table 1. The annual discount factor β is 0.95. Thanks to Epstein and Zin (1989) recursive preferences we are able to disentangle relative risk aversion from the elasticity of intertemporal substitution. The values for both relative risk aversion, γ, and the elasticity of intertemporal substitution, ψ, are 4 and 0.3, respectively. We parameterized
14
the conditional survival probabilities from the national Vital Statistics Reports of the National Center for Health Statistics (NCHS).4
4.2
Labor Income Process
The age-dependent labor income process is based on the PSID data from 1980-2007. We exclude families that were part of the Survey of Economic Opportunities in order to obtain a random sample of US households5 . The permanent component of labor income, equation (5), is a function of households’ age and age squared, as well as dummy variables for education (1 if the head of the household has at least 12 years of education, 0 if not), marital status (1 if married and 0 if not), and gender (1 if head of household is male and 0 if female). It captures the hump-shape of earnings during the working period. We define labor income as the sum of wage income, unemployment compensations, social security, total transfers, child supports, and other welfare of both head of the family and his wife if present. During the retirement period, households receive a constant fraction (ξ) of their last working period income as non-financial income. We set this replacement ratio to 50% instead of 66% in Cocco (2004) because households with TDAs have lower replacement ratio in general. We follow Campbell, Cocco, Gomes, and Maenhout (2001) to estimate the persistency of labor income and the standard deviations of both permanent and transitory shocks to labor income. The estimated persistency parameter φ is 0.80 which is very similar to Guvenen (2009a). The estimated standard deviation of the permanent shock (εlit1 ) and transitory shock (εlit2 ) are 0.0292% and 0.3487% respectively. These estimates are in line with the estimates in Campbell, Cocco, Gomes, and Maenhout (2001) and Cocco (2004).
4.3
Housing Investment
The mean real log-return on housing investment (µh ) calculated from the Case-Shiller index is 1.9% with a standard deviation (σεh ) equal to 5.7%. We assume the increase in house price is partly due to 4 5
http://www.cdc.gov/nchs/data/nvsr/nvsr58/nvsr58_21.pdf Tabel 1. The original PSID sample was drawn from two independent sources: an over-sample of roughly 2000 poor fami-
lies selected from the Survey of Economic Opportunities (SEO) and a nationally-representative sample of roughly 3000 families from all states. Since the SEO over samples low income families, it isn’t a random sample of the US population.
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Table 1: Baseline Parameters
Description
Parameter
Value
Time discount factor
β
0.95
Relative risk aversion
γ
4
Elasticity of intertemporal substitution
ψ
0.3
Elasticity of intratemporal substitution parameter
υ
0.25
Retirement labor income factor
ξ
0.50
Mean return on housing investment
µh
1.5%
Down-payment rate
d
20%
b
Before-tax gross return on riskless asset investment
R
Inflation rate
π
2.5%
Tax rate on riskless asset investment
τb
25%
Equity premium
µs
4%
Nominal dividend yield
dy
3.05%
Tax rate on capital gains
τcg
20%
Tax rate on dividend yields
τdy
25%
Employer’s contribution to retirement account
ke
2%
1.03
Employee’s contribution to retirement account
kl
2%
Tax rate on labor income
τy
20%
Annual Rent
α
7.5%
Property tax (housing)
τh
1%
House maintenance and depreciation expenses
δ
1%
Liquidation cost of selling house
κ
10%
Persistence parameter of labor income shocks
φ
0.80
Std. of persistent shock to labor income
σl2
0.0292
Std. of temporary shock to labor income
σl1
0.1487
Std. of shocks to return on housing inv.
σh
0.057
Std. of shocks to return on risky asset inv.
σs
0.20
- return on housing and return on risky asset inv.
ρsh
0.18
- return on housing investment and labor income
ρhl
0.075
- return on risky asset inv. and labor income
ρsl
0.10
Correlations between shocks
16
the quality increase, we set the mean real log return as 1.5% over a year. We set the down-payment rate at 20% in the baseline model and analyze its effects in the robustness check by imposing different down-payment rates. Renters pay 7% of the house’s market value as annual rent6 . Homeowners pay 1% of house’s market value as the cost for maintenance and depreciation expenses and 1% of house’s value as property tax. Similar to Cocco (2004) Hu (2005) and Yao and Zhang (2005), homeowners bear a liquidation cost (κ) of 10% of house value when they sell their houses. We set υ to 0.25 which brings about a intratemporal substitution between housing and consumption, 1/ (1 − ν), as 1.33 which is in line with the parameter utilized by Piazzesi, Schneider, and Tuzel (2007). The mortgage rate, in general, is higher than the return on the riskless asset because it bears a longterm interest risk and a default risk. However, in this model there is no risk associated with mortgages hence we set the loans on mortgage loan at 3% which is the same with the return on before-tax return on riskless asset investment (i.e. bond) Homeowners pay back the mortgage loan over 25 years.
4.4
Asset Returns, Taxes, and Correlations
The before-tax real gross return on riskless assets (Rb ) is 1.03. For the risky assets, we set the equity premium (µs ) to 4% contrary to the historical 6% to account for transaction costs as in e.g. Gomes and Michaelides (2005), Campbell, Cocco, Gomes, and Maenhout (2001), and Yao and Zhang (2005). Return on the risky assets is decomposed into a constant dividend yield and a stochastic capital gain. The constant nominal dividend yield (di) is 3.05% calculated from S&P500 between 1968-2010. The standard deviation of returns on the risky asset (σεs ) is 20%. We set the tax on dividend yields (τdy ) at 25% and the tax on capital gains (τcg ) at 20% as in Gomes, Michaelides, and Polkovnichenko (2009). In order to reflect the average income tax of a typical household, we set the proportional tax on labor income (τy ) at 25%. The inflation rate (π) is 2,5% which is the average inflation rate for the last 25 years. We assume that households make a constant contribution (kl ) to their TDAs which is equal to 2% of their annual labor income and employer contribution is also 2%. Gomes, Michaelides, and Polkovnichenko (2009) have estimated the contribution rate for different investment durations. As the 6
According to American Housing Survey, in 2013 the median price specified housing units in the United States is
$160,000 and monthly median rent is $919. Then a rough calculation shows an annual rent of 6.89% of house value, close to 7% used in this paper.
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Table 2: Financial Wealth Income Ratios
Age
20-35
36-50
51-65
65+
Model
1.74
2.23
4.8
7.64
Data
0.44
1
1.93
4.48
duration of the investment increases the contribution rate decreases. Since we assume the contribution is compulsory throughout the working period, a low rate is in line with their estimation. During their retirement period, households withdraw funds from their TDAs based on the withdrawal rate. The withdrawal rate is the inverse of the maximum remaining years a household can live. In the baseline model, we assume a 0.18 correlation between shocks to return on risky asset and shocks to housing investment return (ρsh = 0), 0.09 percent correlation between persistent shocks to labor income and shocks to the return on investment in the risky asset ( ρsl ) and 0.075 correlation between persistent shocks to labor income and shocks to the return on housing investment (ρhl ).
5
Results
Table 2 reports the financial wealth-to-income ratios for different age groups both in the model and in the SCF 2010 data. We compare the median ratios instead mean ratios because the financial wealth displays significant skewness. Both the model and the data show that the ratio of financial wealth to income increases over time as the increase in households’ financial wealth is higher than the increase in their labor income. However there exist a discrepancy between the model and the data in terms of the size of the ratios. The model predicts higher ratios than the empirical counterpart. One important reason for the discrepancy stems from the mandatory 4% of labor income contribution to the TDA (2% from employer and 2% from households) during the working period. Due to computational burden we do not estimate optimal contribution rate. However, with a lower or flexible contribution structure, the accumulated wealth in the TDA and hence in total financial assets would be lower, which would imply lower financial wealth-to-income ratios in the model. 18
The life cycle asset location and allocation profiles of households, and homeownership status for the baseline case are displayed in Table 3. The first two rows show the risky asset share within each account while the third row shows the share of all risky assets in the whole financial investment. According to the table households begin with higher share of risky asset within each accounts and gradually decrease these shares over time. The result is in line with the professional financial advice that the fraction of wealth that people should hold in the risky asset should decline with age.7 Young households can hold more risky asset than retired households because they can have enough time horizon to compensate any big negative shock to their investment. Households labor income can be considered as an implicit riskless asset, the present value of future labor income is high for young households. Therefore they can As households get older they are left with relatively hold relatively larger proportion of risky asset in their portfolio. However, as retirement approaches, the present value of their labor income decreases and they change their portfolio allocation towards riskless one. The results are in line with the findings of Blake, Wright, and Zhang (2014) in which optimal weighting in equities is initially very high and declines subsequently as the retirement approaches. One important distinction between the risky asset shares in these accounts is that this share is higher in TDA than in TA throughout the life-cycle. The tax-exempt status of TDA encourage households to be more aggressive in risky asset investment in this account than the risky asset investment in the TA because the mean real return on risky asset investment in TDA is 7% while the return on the same investment in TA is less than 5%. As we will show later, the presence of TDAs partially crowds out risky asset investment in TA because households do not need to be aggressive on the investment in TA when they already have TDA investment for their retirement. The last row shows the homeownership rate over the life-cycle. Housing is modeled both an asset that generates return and a durable consumption good that provides utility to households. Therefore, homeownership is encouraged in the model and the model somewhat over-predicts the homeownership rate. While during the first 15 years the homeownership rate is about 66%, it increases further 7
Two of famous asset allocation rules are 100 Rule and 120 Rule. 100(120) Rule means 100(120) - your current
age is the percentage you should be investing in risky asset and the balance should be in the riskless asset. Furthermore, the following website from Vanguard gives similar advice for portfolio allocation during the life-cycle : https:// personal.vanguard.com/us/funds/vanguard/TargetRetirementList
19
Table 3: Life-Cycle Profiles for the Baseline Case
Model
Data
Age
20-35
36-50 51-65
65+
20-35
36-50 51-65
65+
Share of Risky Asset in TAs
0.66
0.31
0.22
0.25
0.27
0.28
0.30
0.35
Share of Risky Asset in TDAs
0.97
0.72
0.47
0.38
0.57
0.54
0.52
0.42
All Risky Assets /All Financial Assets
0.83
0.70
0.47
0.34
0.30
0.26
0.28
0.45
Homeownership Rate
0.66
0.87
0.96
0.79
0.33
0.67
0.85
0.92
for the next 30 years to 96%. It is only during the retirement period that the homeownership rate decreases to 79%. The second part of Table 3 display the portfolio allocation profiles generated from the SCF data for 2010. When comparing the baseline case results with the SCF data of 2010, the model over-estimate the risky asset share in TA for young households, however it generates low and reasonable shares for the subsequent periods8 . In TDA, similar to TA, the model over-estimates the risky asset share for the first two period, but have reasonable shares for the subsequent periods. One reason for the large over estimation in TDA in early periods is possibly due to the mandatory contribution to the TDA over the working period. When young households make contribution to TDA, they could take more risk in early period by heavily investing in risky asset in TDA (i.e about 97%) because they can more comfortably offset any big negative shock to their investment than retired people. The model also over-estimates the homeownership rate over the life-cycle but the discrepancy between the model and the data decreases over time. Next, we compare the baseline model to the models where we omit either TDA or housing investment. These comparisons will show the effects of each of these components on households portfolio location and allocation decisions. For the first comparison, we exclude TDA in the model. So, when TDAs are omitted, households have housing investment decision and the decision regarding the allocation of financial wealth between risky and the riskless assets only in TA. The portfolio allocation of households in the absence of TDA displayed in Table 4 indicates higher 8
The real data are from 2010 Survey of Consumer Finances as SCF provides a detailed survey of households financial
investment information. A detailed description of the construction of the variables is presented in Appendix B.
20
Table 4: Life-Cycle Profiles in the Absence of TDAs
Age
20-35 36-50
51-65
65+
Share of Risky Asset in TAs
0.84
0.69
0.53
0.32
Homeownership Rate
0.86
0.90
0.95
1.00
shares of risky asset in TA throughout the life-cycle compared to the baseline case. Households allocate more than 80% of their financial wealth to risky asset early on and similar to the baseline model, they gradually decrease this share over time to about 30% during the retirement. However, the share of risky asset in TA in the baseline model starts with 66% and gradually decrease to 25%. The difference between the two cases is intuitive in the sense that when there is no retirement account, households perceive TA as a partial substitute for TDA and have to devote more funds towards the risky asset in this account in order to accumulate more wealth to use for retirement consumption. Therefore, the presence of TDA in the baseline case have partial crowding out effect on the risky asset investment in TA. Finally, the homeownership rate in the absence of TDA is higher than the baseline case throughout the life cycle albeit the difference is strong during the first fifteen years of working life and during the retirement period. Housing in this model is not only a durable consumption good but it also serves as an investment tool. In the absence of TDA, households have a stronger incentive to invest more in housing in order to enjoy the return on housing so the homeownership rate increases. In another perspective, the difference between these two models regarding the homeownership rate displays the crowding out effect of TDA on homeownership. One should note that households do not invest all of their financial wealth in the risky asset in the absence of TDA because they enjoy the return on housing investment as well. In a recent paper, Gomes, Michaelides, and Polkovnichenko (2009) find that in the absence of TDAs households invest 100% of their financial wealth in the risky asset while in the presence of TDAs households invest about 70% of their investment in the risky asset. The reason for 100% investment in the risky asset in the Gomes, Michaelides, and Polkovnichenko (2009)’s model is most likely due to the lack of housing investment. In the second comparison, we exclude the housing investment from the model but keep TDAs. 21
Table 5: Life-Cycle Profiles in the Absence of Housing Investment
Age
20-35 36-50 51-65
65+
Share of Risky Asset in TAs
0.71
0.84
0.83
0.53
Share of Risky Asset in TDAs
0.95
0.71
0.48
0.26
All Risky Assets /All Financial Assets
0.81
0.73
0.52
0.28
Table 5 displays the portfolio allocation profiles of households in the absence of housing investment. As households do not have housing as an investment, they need to increase their risky asset investment in TA so as to compensate the lack of housing investment and its returns. Hence, we observe higher share of risky asset investment in TA than the baseline case throughout the life-cycle. In the baseline case, when households become homeowner, they substitute risk equity with risky housing (Yao and Zhang (2005), Marekwica, Schaefer, and Sebestian (2013), and Cocco (2004)). While this share follows a decreasing trajectory throughout the life-cycle in the baseline case, in the absence of housing investment households increase this share for the working period and decrease during the retirement. On the other hand, the risky asset share in TDA in both the baseline case and no-housing case are very similar. Both follow a decreasing trajectory over the life-cycle.
5.1 5.1.1
Comparative Statics The Size of Contribution to TDA
In the baseline model, both households and employers make 2% of labor income as constant contribution (in total 4%) to the TDA during the working period. In this subsection, we analyze the effects of different contribution rates on the portfolio allocation of households. Gomes, Michaelides, and Polkovnichenko (2009) and Blake, Wright, and Zhang (2014) show that contribution rate is agedependent, not constant. However, estimating the size of contribution rate is computationally burdensome for this comprehensive model. Therefore, instead of estimating the size of contribution rate, we do the analysis for pre-determined different contribution rates and left the optimal contribution rate estimation in this setup for future work.
22
Table 6: Life-Cycle Profiles for Different Contribution Rates
Age
20-35 36-50
51-65
65+
kl = 0.01 & ke = 0.01 Share of Risky Asset in TAs
0.76
0.56
0.43
0.32
Share of Risky Asset in TDAs
1.00
0.91
0.71
0.54
All Risky Assets /All Financial Assets
0.82
0.83
0.66
0.46
Homeownership Rate
0.80
0.88
0.98
1.00
0.66
0.32
0.22
0.23
0.97
0.72
0.48
0.39
kl = 0.02 & ke = 0.02 Share of Risky Asset in TAs Share of Risky Asset in TDAs All Risky Assets /All Financial Assets
0.83
0.70
0.47
0.35
Homeownership Rate
0.66
0.88
0.97
1.00
0.58
0.12
0.09
0.17
Share of Risky Asset in TDAs
0.92
0.54
0.35
0.32
All Risky Assets /All Financial Assets
0.80
0.54
0.35
0.29
Homeownership Rate
0.61
0.92
0.97
1.00
kl = 0.03 & ke = 0.03 Share of Risky Asset in TAs
kl = 0.04 & ke = 0.04 Share of Risky Asset in TAs
0.48
0.04
0.05
0.16
Share of Risky Asset in TDAs
0.84
0.42
0.29
0.29
All Risky Assets /All Financial Assets
0.75
0.42
0.29
0.27
Homeownership Rate
0.59
0.94
0.95
1.00
As the size of the contribution rate increases households have less incentive to be aggressive in risky asset investment in both accounts. Households aim to smooth their consumption over the life-cycle. Therefore, in the case of higher contribution rates, if they are aggressive in risky asset investment, then their consumption would not be smooth over the life-cycle as their retirement consumption would be quite high. Therefore, as the size of the contribution rate increases, households decrease their risky asset investment in both accounts. Homeownership rate also decreases slightly during the first fifteen years as the size of the contribution rates increases. We do not observe any significant effect on the homeownership rates for other periods.
23
5.1.2
The Size of the Down-Payment Rate
The size of the down-payment has some implication for both the homeownership rate and the allocation of financial wealth. In the baseline case, households are required to pay 20% of house value as down-payment and finance the rest through mortgage for 25 years. Table 7 displays the portfolio allocation and the homeownership rates for different down-payment rates. As the size of the downpayment rate increases we observe lower homeownership rates throughout the life-cycle albeit the the effect is strong for early period. One interesting result is that as the size of down-payment rate increases we observe higher shares of risky asset investment in both TA and TDA while the effect is stronger for the investment in TA. One channel for the effect is that as the size of the down-payment rate increases households need to be more aggressive to be able to accumulate more fund to pay the down-payment. Furthermore, the observed lower homeownership rate due to higher down-payment rate could lead households to be more relatively aggressive in risky asset investment as it is the only way to compensate the lack of return on housing investment. 5.1.3
Homeowners vs Renters
The portfolio allocation of homeowners and renters are expected to be different as homeowners enjoy their investment on housing apart from their investment in other financial assets while renters have no housing investment. In the model, all households begin as renter in the first period. Starting from the second period, homeownership decision is endogenously made in each period. Table 8 displays the portfolio allocation of these two types of agents. Note that a typical household can be a homeowner for a while and then can decide to be renter for another time period. The homeownership part of the table shows the portfolio allocation of households during the period in which they are homeowner and similarly the renter part of the table displays the portfolio allocation profiles of households for the time period in which they are renter. The table shows that renters have higher shares of risky asset investment in both TA and TDA almost through out the life-cycle. As renters are required to pay 20% of house value as down-payment, they need to accumulate this payment through investing in the risky asset. Once they pay the down-
24
Table 7: Life-Cycle Profiles for Different Down-Payment Rates
Age d=0
d = 0.02
d = 0.05
d = 0.10
d = 0.15
d = 0.20
d = 0.25
Share of Risky Asset in TAs
20-35 36-50 51-65
65+
0.40
0.05
0.06
0.03
Share of Risky Asset in TDAs
0.89
0.65
0.37
0.28
All Risky Assets /All Financial Assets
0.74
0.64
0.37
0.24
Homeownership Rate
0.84
0.97
0.99
0.99
Share of Risky Asset in TAs
0.46
0.07
0.04
0.06
Share of Risky Asset in TDAs
0.90
0.65
0.38
0.29
All Risky Assets /All Financial Assets
0.75
0.64
0.38
0.25
Homeownership Rate
0.82
0.97
0.99
0.99
Share of Risky Asset in TAs
0.45
0.10
0.06
0.08
Share of Risky Asset in TDAs
0.91
0.66
0.39
0.31
All Risky Assets /All Financial Assets
0.75
0.64
0.39
0.26
Homeownership Rate
0.76
0.96
0.99
0.99
Share of Risky Asset in TAs
0.61
0.16
0.10
0.12
Share of Risky Asset in TDAs
0.96
0.67
0.41
0.33
All Risky Assets /All Financial Assets
0.79
0.66
0.41
0.29
Homeownership Rate
0.75
0.95
0.99
1.00
Share of Risky Asset in TAs
0.61
0.20
0.15
0.16
Share of Risky Asset in TDAs
0.97
0.69
0.43
0.36
All Risky Assets /All Financial Assets
0.82
0.67
0.43
0.31
Homeownership Rate
0.74
0.92
0.98
1.00
Share of Risky Asset in TAs
0.66
0.32
0.22
0.23
Share of Risky Asset in TDAs
0.97
0.72
0.48
0.39
All Risky Assets /All Financial Assets
0.83
0.70
0.47
0.35
Homeownership Rate
0.66
0.88
0.97
1.00
Share of Risky Asset in TAs
0.65
0.37
0.29
0.35
Share of Risky Asset in TDAs
0.98
0.74
0.52
0.45
All Risky Assets /All Financial Assets
0.82
0.72
0.51
0.43
Homeownership Rate
0.64
0.84
0.96
1.00
25
Table 8: Homeowner vs Renter Distinction
Age
20-35 36-50 51-65
65+
Share of Risky Asset in TAs
0.63
0.25
0.21
0.23
Share of Risky Asset in TDAs
0.97
0.71
0.47
0.39
All Risky Assets /All Financial Assets
0.88
0.69
0.47
0.35
Share of Risky Asset in TAs
0.82
0.75
0.49
0.56
Share of Risky Asset in TDAs
0.91
0.85
0.61
0.54
All Risky Assets /All Financial Assets
0.84
0.81
0.62
0.50
Homeowners
Renters
payment, then they reduce the share of risky asset investment. Furthermore, in the absence of housing investment which is less risky, renters are expected to have more risky asset investment in order to offset the absence of housing investment return. With the similar motivation, renters have slightly more of their TDA investment in the risky asset than homeowners’ TDA risky asset share. As renters have no housing investment, they prefer to have more risky asset in TDA to accumulate more wealth for the retirement. However, for homeowners, housing investment is an alternative investment choice for the retirement apart from TDA. Hence they don’t necessarily need to have relatively more of their TDAs investment in the risky asset than renters.
6
Conclusion
This paper analyzes the portfolio allocation of households in a fairly realistic life-cycle model in which housing investment and TDAs are jointly integrated. In the presence of both TDAs and housing investment, households reduce risky asset investment quite significantly and the results are, to some extent, match the empirical counter parts. The crowding out effect of housing investment on the risky asset investment is larger than the crowding out effect of the presence of TDAs. The model also shows as the size of contribution rate increases households reduce their risky asset investment. Furthermore, the size of down payment rate has sizable impact on homeownership rate and risky asset investment. Although the model is quite comprehensive, there are several extensions for future work. First of 26
all, instead of imposing constant contribution rate to TDAs, a better and more realistic approach would be to allow households to make the contribution decisions endogeneously. Another extension would be to distinguish households according to their perception to risk as in Gomes, Michaelides, and Polkovnichenko (2009), Attanasio, Banks, and Tanner (2002), Guvenen (2009b) in order to address higher participation and investment rates in the risky assets. Finally, instead of applying a flat tax-rate to everyone, replacing it with a more realistic progressive tax-rate would likely increase the benefits from TDAs as retired people are subject to lower tax-rate than working people.
References Amromin, G. (2003). ‘Households Portfolio Choices in Taxable and Tax-Deferred Accounts: Another Puzzle?’, European Finance Review, (7): 547–582. Amromin, G., Huang, J., and Sialm, C. (2007). ‘The Tradeoff Between Mortgage Prepaymants and Tax-Deferred Retirement Savings’, Journal of Public Economics, (91): 2014–2040. Attanasio, O. P., Banks, J., and Tanner, S. (2002). ‘Asset Holding and Consumption Volatility’, Journal of Political Economy, 110(4): 771–792. Bergstresser, D., and Poterba, J. (2004). ‘Asset Allocation and Asset Location: Household Evidence from the Survey of Consumer Finance’, Journal of Public Economics, 88: 1893–1915. Blake, D., Wright, D., and Zhang, Y. (2014). ‘Age-dependent Investing: Optimal Funding and Investment Strategies in Defined Contribution Pension Plans When Members are Rational Life-cycle Financial Planners’, Journal of Economic Dynamics and Control, 38: 105–124. Campanale, C., Fugazza, C., and Gomes, F. (2014). ‘Life-cycle Portfolio Choice with Liquid and Illiquid Financial Assets’, Journal of Monetary Economics. Campbell, J. Y., Cocco, J. F., Gomes, F. J., and Maenhout, P. J. (2001). ‘Investing Retirement Wealth: A Life-Cycle Model’, in J. Y. Campbell and M. Feldstein (eds.), Risk Aspects of Investment-Based Social Security Reforms, pp. 439–482. University of Chicago Press.
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Cocco, J. F. (2004). ‘Portfolio Choice in the Presence of Housing’, The Review of Financial Studies, 18(2): 535–567. Dammon, R. M., Spatt, C. S., and Zhang, H. H. (2004). ‘Optimal Asset Location and Allocation with Taxable and Tax-Deferred Investing’, Journal of Finance, 59: 999–1037. Epstein, L. G., and Zin, S. (1989). ‘Substitution, Risk Aversion, and the Temporal Behavior of Consumption Growth and Asset Returns I: A Theoretical Framework’, Econometrica, 57(4)(4): 937– 969. Flavin, M., and Nakagawa, S. (2008). ‘A Model of Housing in the Presence of Adjustment Costs: A Structural Interpretation of Habit Persistence’, American Economic Review, 98(1): 474–495. Flavin, M., and Yamashita, T. (2002). ‘Owner-Occupied Housing and the Composition of the Household Portfolio’, American Economic Review, 92(1): 345–362. Garlappi, L., and Huang, J. (2006). ‘Are Stocks Desirable in TAx-deferred Accounts?’, Journal of Public Economics, 90: 2257–2283. Gomes, F. J., and Michaelides, A. (2005). ‘Optimal Life Cycle Asset Allocation: Understanding the Empirical Evidence’, Journal of Finance, 60(2): 869–904. Gomes, F. J., Michaelides, A., and Polkovnichenko, V. (2009). ‘Optimal Saving with Taxable and Tax-Deferred Accounts’, Review of Economic Dynamics, 12: 718–735. Grossman, S. J., and Laroque, G. (1990). ‘Asset Pricing and Optimal Portfolio Choice in the Presence of Illiquid Durable Consumption Goods’, Econometrica, 58(1): 25–51. Guvenen, F. (2009a). ‘An Empirical Investigation of Labor Income Process’, Review of Economic Dynamics, 12: 58–79. (2009b). ‘A Parsimonious Macroeconomic Model for Asset Pricing’, Econometrica, 77(6): 1711–1740. Hu, X. (2005). ‘Portfolio Choice for Homeowners’, Journal of Urban Economics, 58: 114–136. 28
Marekwica, M., Schaefer, A., and Sebestian, S. (2013). ‘Life Cycle Asset Allocation in the Presence of Housing and Tax-deferrd Investing’, Journal of Economic Dynamics and Control, 37: 1110–1125. Piazzesi, M., Schneider, M., and Tuzel, S. (2007). ‘Housing, Consumption and Asset Pricing’, Journal of Financial Economics, 83: 531–569. Shoven, J. B., and Sialm, C. (2003). ‘Asset Location in Tax-deferred and Conventional Savings Accounts’, Journal of Public Economics, (88): 23–38. Tauchen, G., and Hussey, R. (1991). ‘Quadrature-Based Methods for Obtaining Approximate Solution to Nonlinear Asset Pricing Models’, Econometrica, 59(2): 371–396. Tunc, C., and Pelletier, D. (2013). ‘Endogeneous Housing Investment and Portfolio Allocation’, CBRT Working Paper 1345. Villaverde, J. F., and Krueger, D. (2010). ‘Vintage Article: Consumption and Saving over the Life Cycle: How Important are Consumer Durables’, Working Paper. Yao, R., and Zhang, H. H. (2005). ‘Optimal Consumption and Portfolio Choices with Risky Housing and Borrowing Constraints’, The Review of Financial Studies, 18(1): 197–239. Zhou, J. (2012). ‘Life-cycle Stock Market Participation in Taxabe and Tax-Deferred Accounts’, Journal of Economic Dynamics and Control, (36): 1814–1829.
Appendix A: Numerical Solution of the Model We begin by discretizing the state space and variables over which the choices are made with equally spaced grids. The density functions of the random variables (i.e. shocks to labor income process, shocks to return on risky asset, and shocks to return on housing investment) are approximated by the Gaussian quadrature method of Tauchen and Hussey (1991). In period T + 1, the policy functions are determined by the bequest motive. The value function in this period coincides with the utility function, which is the bequest function. In every period prior to T + 1, we obtain the utility function for different combinations of housing, consumption, and other 29
state and choice variables. Then the value function for a typical time t is equal to the utility function of that period plus the discounted expected continuation value (Et [Vt+1 ]). If the continuation value doesn’t lie on the state space grid, we compute the value function using cubic spline interpolation. This backward induction process is iterated from age T to 1. Once we compute the value function of all the alternatives, we choose the one that maximizes the value function over all choice variables. The optimum policy rules for consumption, housing, and investment in financial assets correspond to ones that maximize the value function. At each point in the state space, the risky asset participation decision is done by comparing the value function conditional on having paid the fixed cost with the value function conditional on having not yet paid the fixed cost. Similarly, the homeownership decision (e.g. house buying or selling decision) is done by comparing the value function conditional on being a renter with the value function conditional on being a homeowner. In both comparisons, adjustments for the payment of the fixed cost of risky asset participation and costs accrued from buying/selling a house (e.g. down payment, annual mortgage payment, liquidation cost etc.) are taken into account respectively.
Appendix B: The SCF Data The SCF gives detailed information about financial wealth of US households. In order to appropriately address the distribution of wealth it combines two random samples. While in one is geographically based random sample, the other sample includes the wealthy families disproportionally. Using the SCF we construct the riskless and risky assets in both taxable account and tax-deferred account.9 . In the data there is no asset as riskless. However, some assets are less risky than others. We define the riskless assets in the TA consists of CDS (certificate of deposits), GBMUTF (government bond mutual funds), TFBMUTF (tax-free bond mutual funds, SAVBND (saving bonds), CHECKING ( checking accounts), CALL (total value of cash or call accounts). The risky assets in TA, on the other hand consists of STOCKS (publicly traded stocks), STMUTF (stock mutual funds), MORTBND 9
Although in the model we have only riskless and risky assets. It is actually more realistic to have more assets with
different degrees of riskiness. However, due to the computational issues and the general convention in the literature we divide assets as risky and the riskless only
30
(mortgage backed bonds) and OBND (corporate or other type of bonds including foreign bonds). Furthermore, we treated the following assets either risky, riskless, or half risky and half riskless depending on the answer of the respondents on the allocation: TRUST (trust) SAVING and MMA (saving and money market accounts). Regarding the allocation of assets in the TDAs between the risky and the riskless one, we use households allocation decisions between risky and riskless ones on IRA/KEOGH (IRA and KEOGH accounts), ANNUITY (annuities), in other future pensions (FUTPEN), and all other account-type retirement plans.
31
JEL classification: G11, D91, R21, D14 Keywords: Portfolio Choice, Life-cycle Models, Housing, Household Saving.
∗
North Carolina State University, Box 8110, College of Management, NCSU, Raleigh, NC 27695-8110, USA, de-
nis [email protected]. † Central Bank of the Republic of Turkey, Istiklal Cad. 10 Ulus Ankara, Turkey, [email protected].
1
Introduction
Tax-deferred retirement accounts (TDAs) have shown considerable growth for more than two decades. The share of defined-contribution plans in the US households retirement assets has increased from about 10% in 1980 to more than 25% as of 2013 (see Figure 1). These accounts provide households with the opportunity of investing in tax-exempt assets to help accumulate enough wealth for their retirement. Taxes on these accounts are only effective upon withdrawals of funds. As more households rely on TDAs, they face both an optimal asset location problem (i.e. deciding how much to hold in taxable and tax-deferred accounts) and an optimal asset allocation problem (i.e. deciding how to distribute funds between risky and riskless assets within each account). Hence understanding the effects of the TDAs on households’ life cycle portfolio allocation becomes increasingly important.
28 26 24 22 20 18 16 14 12 10
1985
1990
1995
2000
2005
2010
Figure 1: The Share of DC plans in the US Retirement Assets (Source: Financial Accounts of the United States) Similar to TDAs, housing investment plays a critical role on household life-cycle wealth allocation decisions because housing investment constitutes a significant share of homeowners’ wealth. According to the Bureau of Census, the mean homeownership rate in the US between 1980-2014 is 2
65.8% with standard deviation of 1.72. Furthermore, Figure 2 shows that the average sales price of houses sold in the US is more than five times of per capita income.
8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Figure 2: The Ratio of Average Sales Price of Houses Sold to Per Capita GDP (Source: Bureau of Census) The contribution of this study to the portfolio allocation literature is to combine these two important components of wealth into a fairly comprehensive life-cycle model in order to analyze the portfolio allocation of households in a more realistic framework. So far the standard life-cycle models are not sufficiently comprehensive in the sense that they exclude either TDAs (Grossman and Laroque (1990), Cocco (2004), Yao and Zhang (2005), Hu (2005), Campanale, Fugazza, and Gomes (2014), Flavin and Yamashita (2002)) or a comprehensive housing investment (Gomes, Michaelides, and Polkovnichenko (2009), Zhou (2012), Dammon, Spatt, and Zhang (2004), Garlappi and Huang (2006), Shoven and Sialm (2003)) and therefore can’t fully explain life-cycle portfolio allocation profiles of households. In order to capture the effects of TDAs and housing investment on households life-cycle wealth and portfolio allocation decisions more precisely, we build a fairly rich structural life-cycle model pa-
3
rameterized to be consistent with empirical data. In this model, we consider finitely living households facing mortality risk, borrowing and short-sale constraints, and receiving stochastic labor income. In every period, households make consumption, housing investment, and asset location and allocation decisions. Regarding financial assets, households can invest in both TDAs and TAs (asset location) and decide on the allocation of wealth between the risky and the riskless assets within each account (asset allocation). Both households and employers make contributions to TDAs during the working period. TDAs provide higher returns than taxable accounts since taxes are paid only upon withdrawal of funds whereas returns on taxable assets are taxed once these returns accrued. Hence households can accumulate funds in TDAs faster than the accumulation of funds in taxable accounts. Besides the distinction between TDAs and TAs, we have modeled housing investment in a fairly comprehensive way. Housing investment is different from other financial investments because it is not only an asset but also a durable consumption good from which households derive utility. It is also, contrary to liquid financial assets, illiquid and often highly leveraged. Furthermore, the wealth allocation of households depends on their homeownership status. While homeowners are committed to pay a flow of payments so as to pay back their mortgage they also enjoy homeownership. Renters, on the other hand, have no commitments and only pay rent but they don’t get the benefits of being homeowners. This model is also specified using Epstein-Zin type recursive preferences [Epstein and Zin (1989)] where the relative risk aversion (RRA) is disentangled from the elasticity of intertemporal substitution (EIS). While RRA gives information about how agents deal with uncertainty across possible states of the world, (EIS) is just a time preference parameter. The problem with the commonly used constant relative risk aversion (CRRA) utility function is that RRA is the inverse of EIS. In other words, CRRA imposes two different roles on the same parameter while EZ provides the flexibility to relax this constraint and have two separate values for these two parameters. The results of our model suggest the presence of TDAs and housing investment have significant effects on consumption and financial wealth allocation of households. Households begin with higher shares of risky assets in each account but decrease this share significantly throughout the life-cycle. Therefore, to some extent, the model is able to mimic the portfolio allocation profiles observed in the empirical data. The presence of both TDAs and housing investment provides households with 4
alternative investment opportunities with different features. As households have more investment alternatives, they are able to distribute risk accordingly. They do not need to rely heavily on one investment in which they need to have higher share of risky asset investment so as to accumulate enough funds. In the absence of either TDAs or housing investment, however, we observe higher shares of risky asset investment. Therefore, the presence of both TDAs and housing investment have clear crowding out effect on risky asset investment and the effect is larger in the presence of housing investment than the presence of TDAs. The reminder of the paper proceeds as follows. Section 2 goes over the previous studies and discuss the contribution of this study to the existing literature. Section 3 introduces the model and its assumptions. Section 4 presents the parameterization and calibration of the model. The simulation results of the model with the baseline parameters, various extensions of the baseline model, and a comparison of the model to the data are given in Section 5. Finally, Section 6 gives concluding remarks and possible extensions of the model for future work.
2
Literature Review
This paper is related to two separate strands of the literature on portfolio allocation, analyzing either (1) the effects of TDAs or (2) housing investment on household portfolio allocation profiles. In the first strand, Gomes, Michaelides, and Polkovnichenko (2009) use a life-cycle model with both TDAs and TAs to analyze the portfolio allocation of both direct stockholders (those investing in both TDAs and TAs) and indirect stockholders (those investing in only TDAs). They find that TDAs increase wealth accumulation which is generally transformed into consumption for retirement. Furthermore, households with lower saving incentives respond less to TDAs. Zhou (2012) finds that differential costs of stock market participation in TDAs and TAs can explain the higher participation rate in TDAs early in life relative to TAs. He also shows that the decline in the participation rate in TDAs late in life is due to the differential tax treatments between these two accounts. Bergstresser and Poterba (2004) study household behaviors from the Survey of Consumer Finances (SCF). The findings of their paper suggest that about one-third of households with both TAs and TDAs are following asset location
5
strategies that are tax inefficient1 . They conclude that non-tax considerations may play an important part in households’ portfolio allocation decisions. Dammon, Spatt, and Zhang (2004) show that taxes play an important role in asset location and allocation decisions. They find that households prefer to hold bonds in TDAs and equities in TAs. However, households’ decisions also depend on other factors such as liquidity. The predictions of their model substantially deviate from the empirical observations that households hold a mix of both stocks and bonds in both their TDAs and TAs. Amromin (2003) claims the interaction between labor income risk and accessibility restriction to TDAs is a key factor in explaining the portfolio allocation of households. The second strand of the literature is on the effects of housing investment on households portfolio allocations. Although housing was ignored by the portfolio allocation literature, recently a growing literature treats housing as an important component of portfolio allocation. Using an asset allocation model where infinitely lived households derive utility from a single indivisible durable good, Grossman and Laroque (1990) argue that illiquid assets could answer the equity premium puzzle. Extending the Grossman and Laroque (1990) model by including both durable and nondurable consumption goods into the utility function, Flavin and Nakagawa (2008) compares a housing model to a habit persistence model and finds that while both deliver many of the same implications, empirical tests using household level data strongly favor the housing model. Furthermore, Villaverde and Krueger (2010) employ a general equilibrium model of life cycle asset allocation to analyze the effects of consumer durable goods on consumption and asset allocations. Piazzesi, Schneider, and Tuzel (2007) develop a housing-included consumption based asset pricing model. Housing is incorporated to the model both as an asset and as a consumption good. Their paper focuses on the effects of housing-consumption asset pricing models on the predictability of the return on stocks. A recent work by Campanale, Fugazza, and Gomes (2014) focuses on the role of transaction cost between liquid and illiquid financial assets on matching the empirical portfolio allocation profiles. There are some life-cycle portfolio allocation studies that focus directly on the role of housing investment with the finding of crowding out effect of housing investment on risky asset investment (Hu (2005), Cocco (2004), Yao and Zhang (2005), and Tunc and Pelletier (2013)). Although the 1
Tax-efficiency means putting the least tax efficient funds (funds that have higher tax-rate) in TDAs and most tax-
efficient in TAs.
6
focus of Amromin, Huang, and Sialm (2007) is different than the focus of our paper, it is worth mentioning this paper as it analyze the trade-off between mortgage prepayments and TDA retirement savings. Using SCF data, the paper concludes that at least 38% of households who prepay their mortgages could benefit from increasing their TDAs contributions instead of prepaying mortgages. A recent study by Marekwica, Schaefer, and Sebestian (2013) also studies the effects of both TDA and housing investment on consumption portfolio problem of households. However, their study focuses more on the tax-arbitrage opportunity from the existence of both housing investment and TDA. Unlike these studies, this paper combines these two strands of the literature in order to analyze life cycle portfolio allocation decisions of households in the presence of both housing investment and TDAs. It makes an explicit comparison between the crowding out effect of TDA and the housing investment on households portfolio allocation decision. In other words, it investigates how households asset location and allocation decision varies in each cases so as to assess the effect of each features on the portfolio allocations of households. Furthermore, in various comparative statics, this paper analyze the effects of the down payment, the size of TDAs contribution rate on the portfolio allocation of households as well as the differences between the portfolio allocation of both homeowners and renters.
3 3.1
Model Setup Household Preferences
The model is a discrete time life-cycle portfolio allocation model. Each period in the model corresponds to one year and as a general convention in the life-cycle literature each year is actually the real age of a household minus 19. Households live for a maximum of T years. The probability that a household is alive at time t + 1 conditional on being alive at time t is equal to qt . Households derive utility from a constant elasticity of substitution (CES) utility function with non-durable consumption goods (C), and housing investment (H). Preferences are in the form of Epstein-Zin where the RRA is
7
disentangled from the EIS: Vt
θ n θ1 o 1−γ 1−γ 1−γ 1−γ = u(Ct , Ht ) θ + β Et qt Vt+1 + (1 − qt )Wt+1
θ =
1−γ 1 − 1/ψ
(1) (2)
1
u (Ct , Ht ) = [Ctv + Htv ] v ,
(3)
where β is the time discount factor, γ is the RRA parameter, and ψ is the EIS parameter. The intratemporal elasticity of substitution between nondurable consumption goods and housing investment is 1/(1 − υ). Wt+1 is the households’ total wealth at time t + 1 that would be bequested in the case households pass away.
3.2
Labor Income Process
Households supply labor inelastically during working life and receive a stochastic labor income Yit which is composed of both deterministic and stochastic components. The deterministic component is a function of households’ age, education, marital status, and gender. Labor income received by household i at age t is denoted by Yit . Define yit = log(Yit ) where yit has the following process: yit = f (t, Zit ) + ulit ,
(4)
f (t, Z) = β0 + β1 age + β2 age2 + β3 gender + β4 marital status + β5 educ,
(5)
where shocks to the log of labor income ulit composes of both aggregate (ηtl ) and idiosyncratic (εlit ) components. We assume that the idiosyncratic component is transitory and i.i.d normally distributed with mean 0 and variance σε2l . The aggregate shock has an AR(1) process: l l ηtl = φηt−1 + wi,t ,
(6)
l where φ is the persistence parameter and wi,t is i.i.d. normally distributed with mean 0 and variance
σw2 l . Households retire at age of K where K corresponds to real age of 65 (K = 46). During the retirement, households receive a constant fraction of their last working period labor income Yit = ξYiK where 0 < ξ < 1 is replacement rate 2 . 2
During the whole retirement period uit becomes uiK .
8
3.3
Housing Investment
All households are renters in the first period. They endogeneously decide to be homeowners or stay as renters in each of the following periods. A typical homeowner could maintain his homeownership status by either staying in his current house or moving to another house. He could also decide to be renter by selling his current house and moving to a renter property. Similarly, a typical renter could decide to buy a house and become homeowner or stay as a renter. Households are required to pay a fraction d of the market value of the house as down payment and to finance the rest through a mortgage. In order to capture the illiquidity of housing investment, when households decide to sell their house, they incur a liquidation cost equal to a fraction κ of the house’s market value. Homeowners also pay in each period for the maintenance and depreciation expenses an amount equal to a proportion δ of the house’s market value. However, renters only pay annual rent which is equal to a proportion α of the house’s market value. We define the per unit price of housing is Pth . Therefore the value of a house of size Hi is Pth Hi . Define pht = log(Pth ), and assume that pht has the following stochastic form: ∆pht = µh + εht ,
εht ∼ N (0, σε2h ),
(7)
where µh as the average growth rate of house prices. The downpayment of a house size Hi at time t is dPth Hi . Hence the mortgage loan RMt that typical households can take against their house is RMt ≤ (1 − d)Pth Hi .
3.4
Financial Assets and Wealth Accumulation
Households can invest in only two financial assets: a risky asset and a riskless asset. Investment in TAs and TDAs have different returns due to the tax treatments. Return on investment in TAs are taxed once these returns accrued. However, investment on TDAs are taxed when households withdraw funds from them. Throughout their working life, households contribute to TDAs a fraction kl of before-tax earnings. Employers also contribute a fraction ke of before-tax earnings to households’
9
TDAs. Households cannot withdraw funds from TDAs before retirement.3 Once households retire in period K, they face a withdrawal rate equal to the inverse of their life expectancy. The tax rate on these withdrawals is the same as the income tax rate, τy . Real gross return on investment in the riskless asset is Rb . After-tax gross return is b 1 + R (1 + π) − 1 (1 − τb ) b e = , R 1+π
(8)
where π is a constant inflation rate and τb is the tax-rate on nominal return for riskless asset investments. Gross real return on the risky asset is s Rt+1 = Rb + µs + εst+1 ,
(9)
where µs is the real, before-tax equity premium and εst+1 is a random shock that is i.i.d.N (0, σε2s ). The gross return on the risky asset at time t is comprised of a constant nominal dividend yield dy and a stochastic nominal capital gain cgt . Dividend yields are taxed at the rate τdy and capital gains are taxed at the rate τcg . The before and after-tax real gross return on the risky asset investment takes the following forms: s Rt+1 =
1 + cgt + dy , 1+π
es = 1 + cgt (1 − τcg ) + dy (1 − τdy ) . R t+1 1+π 3.4.1
(10)
(11)
Renter Wealth Accumulation
Depending on the homeownership status, there are two types of households in this modei, homeowners and renters. We start first by considering a typical renter’s wealth accumulation and budget constraint. The labor income of renters is taxed after the contribution to TDAs is made. Once a renter receives his taxed labor income and taxed return on investment made in TAs at time t − 1, he allocates the total cash-on-hand (Xt ) to (1) consumption expenditure, (2) housing investment, and finally (3) investment in the TAs and the allocation of funds within this account to risky and riskless assets. Regarding housing investment, renters decide to pay the down payment and become homeowners if it is 3
In real life, it is possible to withdraw funds from TDAs with a penalty. Due to computational concerns we do not
incorporate this feature into the model.
10
optimal or stay as renters and wait until the next period. Wealth accumulation and budget constraints of renters have the following form: TD b TD s R + k ∗ Yt Rt + Bt−1 WtT D = St−1
es + Bt−1 R eb LWt = St−1 R t
(12) (13)
Xt ≡ LWt + (1 − τy ) [Yt − kl Yt ]
(14)
k ∗ = kl + ke
(15)
Xt = St + Bt + Ct + (1 − HRt ) αPth Ht + HRt Mt + dPth Ht
(16)
Mt = M Pt + M It
(17)
RMt = RMt−1 − M Pt ,
(18)
where WtT D is the total funds of a typical households in TDAs at time t. The investment made in the TD TD ). Likewise, St−1 (Bt−1 ) denotes (Bt−1 risky (riskless) asset in TDAs at time t − 1 is denoted by St−1
the investment made in the risky (riskless) asset in TAs at time t − 1. Mt is the annual mortgage payment that homeowners pay. Mortgage payments consist of both principal payments (M Pt ) and interest payments (M It ). The remaining balance of the mortgage loan at time t is RMt . HRt is a dummy variable that equals 1 if a household is a homeowner at time t, and 0 otherwise. The wealth allocation and budget constraints of renters during the retirement period is different than the working period since households don’t make contributions to TDAs but withdraw funds from it. During the retirement period, renters have the following form for wealth allocation and budget
11
constraint: TD b TD s R − Qt Rt + Bt−1 WtT D = St−1
Qt = ςt WtT D
(19) (20)
es + Bt−1 R eb LWt = St−1 R t
(21)
Xt ≡ LWt + (1 − τy ) Yt + (1 − τy ) Qt Xt = St + Bt + Ct + (1 − HRt ) αPth Ht + HRt Mt + dPth Ht
(22)
Mt = M Pt + M It
(23)
RMt = RMt−1 − M Pt
(24)
where ςt is the withdrawal rate of funds from TDAs at time t and Qt is the amount of funds households withdraw from TDAs at this time. 3.4.2
Homeowner Wealth Accumulation
Homeowners’ wealth in TDAs (WtT D ), the contributions to this account, and the form of liquid wealth (LWt ) are the same as for renters. However, the amount of cash-on-hand, Xt , is different since homeowners make mortgage payments. The interest component of this payment is tax-exempt. Labor income of homeowners is taxed after the TDAs contribution and interest component of mortgage payments are deducted. Combining this deducted and taxed labor income with taxed returns on investment made in TAs at time t − 1, homeowners allocate this total cash-on-hand (Xt ) to (1) consumption expenditures, (2) housing investment and expenditures, and finally (3) investment in TAs. Furthermore they allocate their funds between risky and riskless assets within each account. Therefore homeowners wealth accumulation and budget constraints before retirement take the following
12
structure: TD b TD s R + kt∗ Yt Rt + Bt−1 WtT D = St−1
(25)
es + Bt−1 R eb LWt = St−1 R t
(26)
Xt ≡ LWt + (1 − τy ) [Yt − kl Yt − M It ]
(27)
Xt = Ct + St + Bt + τh Pth Ht−1 + δPth Ht−1 + HRt (1 − M St ) [Mt ] + HRt M St dPth Ht + Mt − ((1 − κ) Pt Ht−1 − RMt ) + (1 − HRt ) M St αPth Ht − (1 − κ) Pth Ht−1 − RMt ,
(28)
where τh denotes the property tax for housing, δ denotes the rate for maintenance and depreciation expenses of houses, κ is the rate of liquidation cost in the case of selling the house, and M St is a dummy variable which is 1 if a homeowner decides to move from the current house and is 0 otherwise. The after-retirement wealth accumulation and budget constraints, however, are slightly different than the before-retirement case because homeowners are withdrawing funds from their TDAs and making no contribution to this account. Furthermore, these withdrawals are subject to income tax. TD s TD b WtT D = St−1 Rt + Bt−1 R − Qt
(29)
Qt = ςWkT D
(30)
es + Bt−1 R eb LWt = St−1 R t
(31)
Xt ≡ LWt + (1 − τy ) [Yt − M It ] + (1 − τy ) Qt
(32)
Xt = Ct + St + Bt + τh Pth Ht−1 + δPth Ht−1 + HRt (1 − M St ) [Mt ] + HRt M St dPth Ht + Mt − (1 − κ) Pth Ht−1 − RMt + (1 − HRt ) M St αPth Ht − (1 − κ) Pth Ht−1 − RMt .
(33)
Household total wealth at time t (Wt ) is the sum of investment made in both TDAs and TAs at time t − 1 plus the house market value minus the remaining mortgage debt. Households bequest this total wealth to inheritors if they pass away before the terminal period T. If a household lives up to the final period T then he bequests WT +1 . We also impose short-sale and non-negativity constraints on consumption, housing investment, and financial asset investments: Ct ≥ 0, St ≥ 0, BtT D ≥ 0, StT D ≥ 0, Bt ≥ 0, Ht ≥ 0, ∀t, 13
(34)
Wt = WtT A + Pth Ht−1 − RMt−1 + WtT DA .
3.5
(35)
Optimization Problem
Before defining the optimization problem in a more concrete form, we list the state and control variables. The state variables of this problem are age, t, liquid wealth (LWt ), wealth in TDAs (WtT D ), homeownership status at time t − 1 (Ot−1 ), the size of house owners have at time t − 1 (Ht−1 ), and remaining mortgage debt at time t − 1 (RMt−1 ). We denote the state variables by Ω where Ωt = Ωt t, LWt , WtT D , Ot−1 , Ht−1 , RMt−1 . The control variables are consumption (Ct ), the size of house to be chosen at time t (Ht ), the homeownership decision at time t (Ot ), asset allocation decision within TDAs (st,T DAs ), and asset allocation decision in TAs (st,T As ). We define the control variables by Ψ where Ψt = Ψt (Ct , Ht, Ot , st,T DAs , st,T As ). After defining the state and control variables in a more compact form, the household optimization problem is given by θ n θ1 o 1−γ 1−γ 1−γ Vt (Ωt ) = max u(Ct , Ht ) θ + β Et qt Vt+1 (Ωt+1 )1−γ + (1 − qt )Wt+1 ,
{Ψt }
(36)
subject to Equations 1 to 35. This model has no analytical solution. Hence we use a numerical approximation method and solve the model using backward induction starting from time T to t = 1. Please see the details of the solution method in Appendix A.
4 4.1
Parameterization Preference Parameters
The baseline parameters of the model are presented in Table 1. The annual discount factor β is 0.95. Thanks to Epstein and Zin (1989) recursive preferences we are able to disentangle relative risk aversion from the elasticity of intertemporal substitution. The values for both relative risk aversion, γ, and the elasticity of intertemporal substitution, ψ, are 4 and 0.3, respectively. We parameterized
14
the conditional survival probabilities from the national Vital Statistics Reports of the National Center for Health Statistics (NCHS).4
4.2
Labor Income Process
The age-dependent labor income process is based on the PSID data from 1980-2007. We exclude families that were part of the Survey of Economic Opportunities in order to obtain a random sample of US households5 . The permanent component of labor income, equation (5), is a function of households’ age and age squared, as well as dummy variables for education (1 if the head of the household has at least 12 years of education, 0 if not), marital status (1 if married and 0 if not), and gender (1 if head of household is male and 0 if female). It captures the hump-shape of earnings during the working period. We define labor income as the sum of wage income, unemployment compensations, social security, total transfers, child supports, and other welfare of both head of the family and his wife if present. During the retirement period, households receive a constant fraction (ξ) of their last working period income as non-financial income. We set this replacement ratio to 50% instead of 66% in Cocco (2004) because households with TDAs have lower replacement ratio in general. We follow Campbell, Cocco, Gomes, and Maenhout (2001) to estimate the persistency of labor income and the standard deviations of both permanent and transitory shocks to labor income. The estimated persistency parameter φ is 0.80 which is very similar to Guvenen (2009a). The estimated standard deviation of the permanent shock (εlit1 ) and transitory shock (εlit2 ) are 0.0292% and 0.3487% respectively. These estimates are in line with the estimates in Campbell, Cocco, Gomes, and Maenhout (2001) and Cocco (2004).
4.3
Housing Investment
The mean real log-return on housing investment (µh ) calculated from the Case-Shiller index is 1.9% with a standard deviation (σεh ) equal to 5.7%. We assume the increase in house price is partly due to 4 5
http://www.cdc.gov/nchs/data/nvsr/nvsr58/nvsr58_21.pdf Tabel 1. The original PSID sample was drawn from two independent sources: an over-sample of roughly 2000 poor fami-
lies selected from the Survey of Economic Opportunities (SEO) and a nationally-representative sample of roughly 3000 families from all states. Since the SEO over samples low income families, it isn’t a random sample of the US population.
15
Table 1: Baseline Parameters
Description
Parameter
Value
Time discount factor
β
0.95
Relative risk aversion
γ
4
Elasticity of intertemporal substitution
ψ
0.3
Elasticity of intratemporal substitution parameter
υ
0.25
Retirement labor income factor
ξ
0.50
Mean return on housing investment
µh
1.5%
Down-payment rate
d
20%
b
Before-tax gross return on riskless asset investment
R
Inflation rate
π
2.5%
Tax rate on riskless asset investment
τb
25%
Equity premium
µs
4%
Nominal dividend yield
dy
3.05%
Tax rate on capital gains
τcg
20%
Tax rate on dividend yields
τdy
25%
Employer’s contribution to retirement account
ke
2%
1.03
Employee’s contribution to retirement account
kl
2%
Tax rate on labor income
τy
20%
Annual Rent
α
7.5%
Property tax (housing)
τh
1%
House maintenance and depreciation expenses
δ
1%
Liquidation cost of selling house
κ
10%
Persistence parameter of labor income shocks
φ
0.80
Std. of persistent shock to labor income
σl2
0.0292
Std. of temporary shock to labor income
σl1
0.1487
Std. of shocks to return on housing inv.
σh
0.057
Std. of shocks to return on risky asset inv.
σs
0.20
- return on housing and return on risky asset inv.
ρsh
0.18
- return on housing investment and labor income
ρhl
0.075
- return on risky asset inv. and labor income
ρsl
0.10
Correlations between shocks
16
the quality increase, we set the mean real log return as 1.5% over a year. We set the down-payment rate at 20% in the baseline model and analyze its effects in the robustness check by imposing different down-payment rates. Renters pay 7% of the house’s market value as annual rent6 . Homeowners pay 1% of house’s market value as the cost for maintenance and depreciation expenses and 1% of house’s value as property tax. Similar to Cocco (2004) Hu (2005) and Yao and Zhang (2005), homeowners bear a liquidation cost (κ) of 10% of house value when they sell their houses. We set υ to 0.25 which brings about a intratemporal substitution between housing and consumption, 1/ (1 − ν), as 1.33 which is in line with the parameter utilized by Piazzesi, Schneider, and Tuzel (2007). The mortgage rate, in general, is higher than the return on the riskless asset because it bears a longterm interest risk and a default risk. However, in this model there is no risk associated with mortgages hence we set the loans on mortgage loan at 3% which is the same with the return on before-tax return on riskless asset investment (i.e. bond) Homeowners pay back the mortgage loan over 25 years.
4.4
Asset Returns, Taxes, and Correlations
The before-tax real gross return on riskless assets (Rb ) is 1.03. For the risky assets, we set the equity premium (µs ) to 4% contrary to the historical 6% to account for transaction costs as in e.g. Gomes and Michaelides (2005), Campbell, Cocco, Gomes, and Maenhout (2001), and Yao and Zhang (2005). Return on the risky assets is decomposed into a constant dividend yield and a stochastic capital gain. The constant nominal dividend yield (di) is 3.05% calculated from S&P500 between 1968-2010. The standard deviation of returns on the risky asset (σεs ) is 20%. We set the tax on dividend yields (τdy ) at 25% and the tax on capital gains (τcg ) at 20% as in Gomes, Michaelides, and Polkovnichenko (2009). In order to reflect the average income tax of a typical household, we set the proportional tax on labor income (τy ) at 25%. The inflation rate (π) is 2,5% which is the average inflation rate for the last 25 years. We assume that households make a constant contribution (kl ) to their TDAs which is equal to 2% of their annual labor income and employer contribution is also 2%. Gomes, Michaelides, and Polkovnichenko (2009) have estimated the contribution rate for different investment durations. As the 6
According to American Housing Survey, in 2013 the median price specified housing units in the United States is
$160,000 and monthly median rent is $919. Then a rough calculation shows an annual rent of 6.89% of house value, close to 7% used in this paper.
17
Table 2: Financial Wealth Income Ratios
Age
20-35
36-50
51-65
65+
Model
1.74
2.23
4.8
7.64
Data
0.44
1
1.93
4.48
duration of the investment increases the contribution rate decreases. Since we assume the contribution is compulsory throughout the working period, a low rate is in line with their estimation. During their retirement period, households withdraw funds from their TDAs based on the withdrawal rate. The withdrawal rate is the inverse of the maximum remaining years a household can live. In the baseline model, we assume a 0.18 correlation between shocks to return on risky asset and shocks to housing investment return (ρsh = 0), 0.09 percent correlation between persistent shocks to labor income and shocks to the return on investment in the risky asset ( ρsl ) and 0.075 correlation between persistent shocks to labor income and shocks to the return on housing investment (ρhl ).
5
Results
Table 2 reports the financial wealth-to-income ratios for different age groups both in the model and in the SCF 2010 data. We compare the median ratios instead mean ratios because the financial wealth displays significant skewness. Both the model and the data show that the ratio of financial wealth to income increases over time as the increase in households’ financial wealth is higher than the increase in their labor income. However there exist a discrepancy between the model and the data in terms of the size of the ratios. The model predicts higher ratios than the empirical counterpart. One important reason for the discrepancy stems from the mandatory 4% of labor income contribution to the TDA (2% from employer and 2% from households) during the working period. Due to computational burden we do not estimate optimal contribution rate. However, with a lower or flexible contribution structure, the accumulated wealth in the TDA and hence in total financial assets would be lower, which would imply lower financial wealth-to-income ratios in the model. 18
The life cycle asset location and allocation profiles of households, and homeownership status for the baseline case are displayed in Table 3. The first two rows show the risky asset share within each account while the third row shows the share of all risky assets in the whole financial investment. According to the table households begin with higher share of risky asset within each accounts and gradually decrease these shares over time. The result is in line with the professional financial advice that the fraction of wealth that people should hold in the risky asset should decline with age.7 Young households can hold more risky asset than retired households because they can have enough time horizon to compensate any big negative shock to their investment. Households labor income can be considered as an implicit riskless asset, the present value of future labor income is high for young households. Therefore they can As households get older they are left with relatively hold relatively larger proportion of risky asset in their portfolio. However, as retirement approaches, the present value of their labor income decreases and they change their portfolio allocation towards riskless one. The results are in line with the findings of Blake, Wright, and Zhang (2014) in which optimal weighting in equities is initially very high and declines subsequently as the retirement approaches. One important distinction between the risky asset shares in these accounts is that this share is higher in TDA than in TA throughout the life-cycle. The tax-exempt status of TDA encourage households to be more aggressive in risky asset investment in this account than the risky asset investment in the TA because the mean real return on risky asset investment in TDA is 7% while the return on the same investment in TA is less than 5%. As we will show later, the presence of TDAs partially crowds out risky asset investment in TA because households do not need to be aggressive on the investment in TA when they already have TDA investment for their retirement. The last row shows the homeownership rate over the life-cycle. Housing is modeled both an asset that generates return and a durable consumption good that provides utility to households. Therefore, homeownership is encouraged in the model and the model somewhat over-predicts the homeownership rate. While during the first 15 years the homeownership rate is about 66%, it increases further 7
Two of famous asset allocation rules are 100 Rule and 120 Rule. 100(120) Rule means 100(120) - your current
age is the percentage you should be investing in risky asset and the balance should be in the riskless asset. Furthermore, the following website from Vanguard gives similar advice for portfolio allocation during the life-cycle : https:// personal.vanguard.com/us/funds/vanguard/TargetRetirementList
19
Table 3: Life-Cycle Profiles for the Baseline Case
Model
Data
Age
20-35
36-50 51-65
65+
20-35
36-50 51-65
65+
Share of Risky Asset in TAs
0.66
0.31
0.22
0.25
0.27
0.28
0.30
0.35
Share of Risky Asset in TDAs
0.97
0.72
0.47
0.38
0.57
0.54
0.52
0.42
All Risky Assets /All Financial Assets
0.83
0.70
0.47
0.34
0.30
0.26
0.28
0.45
Homeownership Rate
0.66
0.87
0.96
0.79
0.33
0.67
0.85
0.92
for the next 30 years to 96%. It is only during the retirement period that the homeownership rate decreases to 79%. The second part of Table 3 display the portfolio allocation profiles generated from the SCF data for 2010. When comparing the baseline case results with the SCF data of 2010, the model over-estimate the risky asset share in TA for young households, however it generates low and reasonable shares for the subsequent periods8 . In TDA, similar to TA, the model over-estimates the risky asset share for the first two period, but have reasonable shares for the subsequent periods. One reason for the large over estimation in TDA in early periods is possibly due to the mandatory contribution to the TDA over the working period. When young households make contribution to TDA, they could take more risk in early period by heavily investing in risky asset in TDA (i.e about 97%) because they can more comfortably offset any big negative shock to their investment than retired people. The model also over-estimates the homeownership rate over the life-cycle but the discrepancy between the model and the data decreases over time. Next, we compare the baseline model to the models where we omit either TDA or housing investment. These comparisons will show the effects of each of these components on households portfolio location and allocation decisions. For the first comparison, we exclude TDA in the model. So, when TDAs are omitted, households have housing investment decision and the decision regarding the allocation of financial wealth between risky and the riskless assets only in TA. The portfolio allocation of households in the absence of TDA displayed in Table 4 indicates higher 8
The real data are from 2010 Survey of Consumer Finances as SCF provides a detailed survey of households financial
investment information. A detailed description of the construction of the variables is presented in Appendix B.
20
Table 4: Life-Cycle Profiles in the Absence of TDAs
Age
20-35 36-50
51-65
65+
Share of Risky Asset in TAs
0.84
0.69
0.53
0.32
Homeownership Rate
0.86
0.90
0.95
1.00
shares of risky asset in TA throughout the life-cycle compared to the baseline case. Households allocate more than 80% of their financial wealth to risky asset early on and similar to the baseline model, they gradually decrease this share over time to about 30% during the retirement. However, the share of risky asset in TA in the baseline model starts with 66% and gradually decrease to 25%. The difference between the two cases is intuitive in the sense that when there is no retirement account, households perceive TA as a partial substitute for TDA and have to devote more funds towards the risky asset in this account in order to accumulate more wealth to use for retirement consumption. Therefore, the presence of TDA in the baseline case have partial crowding out effect on the risky asset investment in TA. Finally, the homeownership rate in the absence of TDA is higher than the baseline case throughout the life cycle albeit the difference is strong during the first fifteen years of working life and during the retirement period. Housing in this model is not only a durable consumption good but it also serves as an investment tool. In the absence of TDA, households have a stronger incentive to invest more in housing in order to enjoy the return on housing so the homeownership rate increases. In another perspective, the difference between these two models regarding the homeownership rate displays the crowding out effect of TDA on homeownership. One should note that households do not invest all of their financial wealth in the risky asset in the absence of TDA because they enjoy the return on housing investment as well. In a recent paper, Gomes, Michaelides, and Polkovnichenko (2009) find that in the absence of TDAs households invest 100% of their financial wealth in the risky asset while in the presence of TDAs households invest about 70% of their investment in the risky asset. The reason for 100% investment in the risky asset in the Gomes, Michaelides, and Polkovnichenko (2009)’s model is most likely due to the lack of housing investment. In the second comparison, we exclude the housing investment from the model but keep TDAs. 21
Table 5: Life-Cycle Profiles in the Absence of Housing Investment
Age
20-35 36-50 51-65
65+
Share of Risky Asset in TAs
0.71
0.84
0.83
0.53
Share of Risky Asset in TDAs
0.95
0.71
0.48
0.26
All Risky Assets /All Financial Assets
0.81
0.73
0.52
0.28
Table 5 displays the portfolio allocation profiles of households in the absence of housing investment. As households do not have housing as an investment, they need to increase their risky asset investment in TA so as to compensate the lack of housing investment and its returns. Hence, we observe higher share of risky asset investment in TA than the baseline case throughout the life-cycle. In the baseline case, when households become homeowner, they substitute risk equity with risky housing (Yao and Zhang (2005), Marekwica, Schaefer, and Sebestian (2013), and Cocco (2004)). While this share follows a decreasing trajectory throughout the life-cycle in the baseline case, in the absence of housing investment households increase this share for the working period and decrease during the retirement. On the other hand, the risky asset share in TDA in both the baseline case and no-housing case are very similar. Both follow a decreasing trajectory over the life-cycle.
5.1 5.1.1
Comparative Statics The Size of Contribution to TDA
In the baseline model, both households and employers make 2% of labor income as constant contribution (in total 4%) to the TDA during the working period. In this subsection, we analyze the effects of different contribution rates on the portfolio allocation of households. Gomes, Michaelides, and Polkovnichenko (2009) and Blake, Wright, and Zhang (2014) show that contribution rate is agedependent, not constant. However, estimating the size of contribution rate is computationally burdensome for this comprehensive model. Therefore, instead of estimating the size of contribution rate, we do the analysis for pre-determined different contribution rates and left the optimal contribution rate estimation in this setup for future work.
22
Table 6: Life-Cycle Profiles for Different Contribution Rates
Age
20-35 36-50
51-65
65+
kl = 0.01 & ke = 0.01 Share of Risky Asset in TAs
0.76
0.56
0.43
0.32
Share of Risky Asset in TDAs
1.00
0.91
0.71
0.54
All Risky Assets /All Financial Assets
0.82
0.83
0.66
0.46
Homeownership Rate
0.80
0.88
0.98
1.00
0.66
0.32
0.22
0.23
0.97
0.72
0.48
0.39
kl = 0.02 & ke = 0.02 Share of Risky Asset in TAs Share of Risky Asset in TDAs All Risky Assets /All Financial Assets
0.83
0.70
0.47
0.35
Homeownership Rate
0.66
0.88
0.97
1.00
0.58
0.12
0.09
0.17
Share of Risky Asset in TDAs
0.92
0.54
0.35
0.32
All Risky Assets /All Financial Assets
0.80
0.54
0.35
0.29
Homeownership Rate
0.61
0.92
0.97
1.00
kl = 0.03 & ke = 0.03 Share of Risky Asset in TAs
kl = 0.04 & ke = 0.04 Share of Risky Asset in TAs
0.48
0.04
0.05
0.16
Share of Risky Asset in TDAs
0.84
0.42
0.29
0.29
All Risky Assets /All Financial Assets
0.75
0.42
0.29
0.27
Homeownership Rate
0.59
0.94
0.95
1.00
As the size of the contribution rate increases households have less incentive to be aggressive in risky asset investment in both accounts. Households aim to smooth their consumption over the life-cycle. Therefore, in the case of higher contribution rates, if they are aggressive in risky asset investment, then their consumption would not be smooth over the life-cycle as their retirement consumption would be quite high. Therefore, as the size of the contribution rate increases, households decrease their risky asset investment in both accounts. Homeownership rate also decreases slightly during the first fifteen years as the size of the contribution rates increases. We do not observe any significant effect on the homeownership rates for other periods.
23
5.1.2
The Size of the Down-Payment Rate
The size of the down-payment has some implication for both the homeownership rate and the allocation of financial wealth. In the baseline case, households are required to pay 20% of house value as down-payment and finance the rest through mortgage for 25 years. Table 7 displays the portfolio allocation and the homeownership rates for different down-payment rates. As the size of the downpayment rate increases we observe lower homeownership rates throughout the life-cycle albeit the the effect is strong for early period. One interesting result is that as the size of down-payment rate increases we observe higher shares of risky asset investment in both TA and TDA while the effect is stronger for the investment in TA. One channel for the effect is that as the size of the down-payment rate increases households need to be more aggressive to be able to accumulate more fund to pay the down-payment. Furthermore, the observed lower homeownership rate due to higher down-payment rate could lead households to be more relatively aggressive in risky asset investment as it is the only way to compensate the lack of return on housing investment. 5.1.3
Homeowners vs Renters
The portfolio allocation of homeowners and renters are expected to be different as homeowners enjoy their investment on housing apart from their investment in other financial assets while renters have no housing investment. In the model, all households begin as renter in the first period. Starting from the second period, homeownership decision is endogenously made in each period. Table 8 displays the portfolio allocation of these two types of agents. Note that a typical household can be a homeowner for a while and then can decide to be renter for another time period. The homeownership part of the table shows the portfolio allocation of households during the period in which they are homeowner and similarly the renter part of the table displays the portfolio allocation profiles of households for the time period in which they are renter. The table shows that renters have higher shares of risky asset investment in both TA and TDA almost through out the life-cycle. As renters are required to pay 20% of house value as down-payment, they need to accumulate this payment through investing in the risky asset. Once they pay the down-
24
Table 7: Life-Cycle Profiles for Different Down-Payment Rates
Age d=0
d = 0.02
d = 0.05
d = 0.10
d = 0.15
d = 0.20
d = 0.25
Share of Risky Asset in TAs
20-35 36-50 51-65
65+
0.40
0.05
0.06
0.03
Share of Risky Asset in TDAs
0.89
0.65
0.37
0.28
All Risky Assets /All Financial Assets
0.74
0.64
0.37
0.24
Homeownership Rate
0.84
0.97
0.99
0.99
Share of Risky Asset in TAs
0.46
0.07
0.04
0.06
Share of Risky Asset in TDAs
0.90
0.65
0.38
0.29
All Risky Assets /All Financial Assets
0.75
0.64
0.38
0.25
Homeownership Rate
0.82
0.97
0.99
0.99
Share of Risky Asset in TAs
0.45
0.10
0.06
0.08
Share of Risky Asset in TDAs
0.91
0.66
0.39
0.31
All Risky Assets /All Financial Assets
0.75
0.64
0.39
0.26
Homeownership Rate
0.76
0.96
0.99
0.99
Share of Risky Asset in TAs
0.61
0.16
0.10
0.12
Share of Risky Asset in TDAs
0.96
0.67
0.41
0.33
All Risky Assets /All Financial Assets
0.79
0.66
0.41
0.29
Homeownership Rate
0.75
0.95
0.99
1.00
Share of Risky Asset in TAs
0.61
0.20
0.15
0.16
Share of Risky Asset in TDAs
0.97
0.69
0.43
0.36
All Risky Assets /All Financial Assets
0.82
0.67
0.43
0.31
Homeownership Rate
0.74
0.92
0.98
1.00
Share of Risky Asset in TAs
0.66
0.32
0.22
0.23
Share of Risky Asset in TDAs
0.97
0.72
0.48
0.39
All Risky Assets /All Financial Assets
0.83
0.70
0.47
0.35
Homeownership Rate
0.66
0.88
0.97
1.00
Share of Risky Asset in TAs
0.65
0.37
0.29
0.35
Share of Risky Asset in TDAs
0.98
0.74
0.52
0.45
All Risky Assets /All Financial Assets
0.82
0.72
0.51
0.43
Homeownership Rate
0.64
0.84
0.96
1.00
25
Table 8: Homeowner vs Renter Distinction
Age
20-35 36-50 51-65
65+
Share of Risky Asset in TAs
0.63
0.25
0.21
0.23
Share of Risky Asset in TDAs
0.97
0.71
0.47
0.39
All Risky Assets /All Financial Assets
0.88
0.69
0.47
0.35
Share of Risky Asset in TAs
0.82
0.75
0.49
0.56
Share of Risky Asset in TDAs
0.91
0.85
0.61
0.54
All Risky Assets /All Financial Assets
0.84
0.81
0.62
0.50
Homeowners
Renters
payment, then they reduce the share of risky asset investment. Furthermore, in the absence of housing investment which is less risky, renters are expected to have more risky asset investment in order to offset the absence of housing investment return. With the similar motivation, renters have slightly more of their TDA investment in the risky asset than homeowners’ TDA risky asset share. As renters have no housing investment, they prefer to have more risky asset in TDA to accumulate more wealth for the retirement. However, for homeowners, housing investment is an alternative investment choice for the retirement apart from TDA. Hence they don’t necessarily need to have relatively more of their TDAs investment in the risky asset than renters.
6
Conclusion
This paper analyzes the portfolio allocation of households in a fairly realistic life-cycle model in which housing investment and TDAs are jointly integrated. In the presence of both TDAs and housing investment, households reduce risky asset investment quite significantly and the results are, to some extent, match the empirical counter parts. The crowding out effect of housing investment on the risky asset investment is larger than the crowding out effect of the presence of TDAs. The model also shows as the size of contribution rate increases households reduce their risky asset investment. Furthermore, the size of down payment rate has sizable impact on homeownership rate and risky asset investment. Although the model is quite comprehensive, there are several extensions for future work. First of 26
all, instead of imposing constant contribution rate to TDAs, a better and more realistic approach would be to allow households to make the contribution decisions endogeneously. Another extension would be to distinguish households according to their perception to risk as in Gomes, Michaelides, and Polkovnichenko (2009), Attanasio, Banks, and Tanner (2002), Guvenen (2009b) in order to address higher participation and investment rates in the risky assets. Finally, instead of applying a flat tax-rate to everyone, replacing it with a more realistic progressive tax-rate would likely increase the benefits from TDAs as retired people are subject to lower tax-rate than working people.
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Cocco, J. F. (2004). ‘Portfolio Choice in the Presence of Housing’, The Review of Financial Studies, 18(2): 535–567. Dammon, R. M., Spatt, C. S., and Zhang, H. H. (2004). ‘Optimal Asset Location and Allocation with Taxable and Tax-Deferred Investing’, Journal of Finance, 59: 999–1037. Epstein, L. G., and Zin, S. (1989). ‘Substitution, Risk Aversion, and the Temporal Behavior of Consumption Growth and Asset Returns I: A Theoretical Framework’, Econometrica, 57(4)(4): 937– 969. Flavin, M., and Nakagawa, S. (2008). ‘A Model of Housing in the Presence of Adjustment Costs: A Structural Interpretation of Habit Persistence’, American Economic Review, 98(1): 474–495. Flavin, M., and Yamashita, T. (2002). ‘Owner-Occupied Housing and the Composition of the Household Portfolio’, American Economic Review, 92(1): 345–362. Garlappi, L., and Huang, J. (2006). ‘Are Stocks Desirable in TAx-deferred Accounts?’, Journal of Public Economics, 90: 2257–2283. Gomes, F. J., and Michaelides, A. (2005). ‘Optimal Life Cycle Asset Allocation: Understanding the Empirical Evidence’, Journal of Finance, 60(2): 869–904. Gomes, F. J., Michaelides, A., and Polkovnichenko, V. (2009). ‘Optimal Saving with Taxable and Tax-Deferred Accounts’, Review of Economic Dynamics, 12: 718–735. Grossman, S. J., and Laroque, G. (1990). ‘Asset Pricing and Optimal Portfolio Choice in the Presence of Illiquid Durable Consumption Goods’, Econometrica, 58(1): 25–51. Guvenen, F. (2009a). ‘An Empirical Investigation of Labor Income Process’, Review of Economic Dynamics, 12: 58–79. (2009b). ‘A Parsimonious Macroeconomic Model for Asset Pricing’, Econometrica, 77(6): 1711–1740. Hu, X. (2005). ‘Portfolio Choice for Homeowners’, Journal of Urban Economics, 58: 114–136. 28
Marekwica, M., Schaefer, A., and Sebestian, S. (2013). ‘Life Cycle Asset Allocation in the Presence of Housing and Tax-deferrd Investing’, Journal of Economic Dynamics and Control, 37: 1110–1125. Piazzesi, M., Schneider, M., and Tuzel, S. (2007). ‘Housing, Consumption and Asset Pricing’, Journal of Financial Economics, 83: 531–569. Shoven, J. B., and Sialm, C. (2003). ‘Asset Location in Tax-deferred and Conventional Savings Accounts’, Journal of Public Economics, (88): 23–38. Tauchen, G., and Hussey, R. (1991). ‘Quadrature-Based Methods for Obtaining Approximate Solution to Nonlinear Asset Pricing Models’, Econometrica, 59(2): 371–396. Tunc, C., and Pelletier, D. (2013). ‘Endogeneous Housing Investment and Portfolio Allocation’, CBRT Working Paper 1345. Villaverde, J. F., and Krueger, D. (2010). ‘Vintage Article: Consumption and Saving over the Life Cycle: How Important are Consumer Durables’, Working Paper. Yao, R., and Zhang, H. H. (2005). ‘Optimal Consumption and Portfolio Choices with Risky Housing and Borrowing Constraints’, The Review of Financial Studies, 18(1): 197–239. Zhou, J. (2012). ‘Life-cycle Stock Market Participation in Taxabe and Tax-Deferred Accounts’, Journal of Economic Dynamics and Control, (36): 1814–1829.
Appendix A: Numerical Solution of the Model We begin by discretizing the state space and variables over which the choices are made with equally spaced grids. The density functions of the random variables (i.e. shocks to labor income process, shocks to return on risky asset, and shocks to return on housing investment) are approximated by the Gaussian quadrature method of Tauchen and Hussey (1991). In period T + 1, the policy functions are determined by the bequest motive. The value function in this period coincides with the utility function, which is the bequest function. In every period prior to T + 1, we obtain the utility function for different combinations of housing, consumption, and other 29
state and choice variables. Then the value function for a typical time t is equal to the utility function of that period plus the discounted expected continuation value (Et [Vt+1 ]). If the continuation value doesn’t lie on the state space grid, we compute the value function using cubic spline interpolation. This backward induction process is iterated from age T to 1. Once we compute the value function of all the alternatives, we choose the one that maximizes the value function over all choice variables. The optimum policy rules for consumption, housing, and investment in financial assets correspond to ones that maximize the value function. At each point in the state space, the risky asset participation decision is done by comparing the value function conditional on having paid the fixed cost with the value function conditional on having not yet paid the fixed cost. Similarly, the homeownership decision (e.g. house buying or selling decision) is done by comparing the value function conditional on being a renter with the value function conditional on being a homeowner. In both comparisons, adjustments for the payment of the fixed cost of risky asset participation and costs accrued from buying/selling a house (e.g. down payment, annual mortgage payment, liquidation cost etc.) are taken into account respectively.
Appendix B: The SCF Data The SCF gives detailed information about financial wealth of US households. In order to appropriately address the distribution of wealth it combines two random samples. While in one is geographically based random sample, the other sample includes the wealthy families disproportionally. Using the SCF we construct the riskless and risky assets in both taxable account and tax-deferred account.9 . In the data there is no asset as riskless. However, some assets are less risky than others. We define the riskless assets in the TA consists of CDS (certificate of deposits), GBMUTF (government bond mutual funds), TFBMUTF (tax-free bond mutual funds, SAVBND (saving bonds), CHECKING ( checking accounts), CALL (total value of cash or call accounts). The risky assets in TA, on the other hand consists of STOCKS (publicly traded stocks), STMUTF (stock mutual funds), MORTBND 9
Although in the model we have only riskless and risky assets. It is actually more realistic to have more assets with
different degrees of riskiness. However, due to the computational issues and the general convention in the literature we divide assets as risky and the riskless only
30
(mortgage backed bonds) and OBND (corporate or other type of bonds including foreign bonds). Furthermore, we treated the following assets either risky, riskless, or half risky and half riskless depending on the answer of the respondents on the allocation: TRUST (trust) SAVING and MMA (saving and money market accounts). Regarding the allocation of assets in the TDAs between the risky and the riskless one, we use households allocation decisions between risky and riskless ones on IRA/KEOGH (IRA and KEOGH accounts), ANNUITY (annuities), in other future pensions (FUTPEN), and all other account-type retirement plans.
31
