Local Housing Returns and the Optimal Portfolios of Consumption Constrained Households

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1 Local Housing Returns and the Optimal Portfolios of Consumption Constrained Households Cathy Ge Bao y University of International Business and Economics Guoliang Feng z China Investment Corporation January 5, 2018 Abstract Portfolios of homeowners are dominated by housing and risk-free assets rather than equities or bonds. Indeed homeowners tend to prepay mortgages rather than invest in risk assets. This has been called the stockholding paradox and has been explained by observing that the correlation between the rate of appreciation of national housing prices and returns to the S&P 500 is relatively high. The conclusion has been that homeowners derive only modest diversi cation bene ts from holding stocks and choose instead to amortize their mortgages. In contrast to the empirical literature on the stockholding paradox, Brueckner (1997) has demonstrated the theoretical proposition that consumption constrained households, those whose wealth is a fraction of housing value, will not nd holding the market portfolio ef- cient. This research proceeds from Brueckner s observation. First, total return to The authors contribute equally to this research. We are deeply indebted to Anthony M. Yezer, Michael Bradley, Paul Carrillo, Mike Fratantoni, Ming Hwang and Leah Brooks for their insightful guidance. We are grateful to very helpful comments from seminar participants at George Washington University, Fannie Mae and IFE Group; to conference comments from 2013 North American Regional Science Council Conference, 2014 American Real Estate and Urban Economics Association Conference, 2014 Midwest Finance Association Conference and 2014 Financial Management Association Conference. y Associate Professor of Economics. School of International Trade and Economics. University of International Business and Economics. baoge@uibe.edu.cn Cathy Ge Bao is grateful to the National Natural Science Foundation of China ( ) and Fundamental Research Funds for Central Universities at UIBE (16QN03) for support z Economist. O ce of the Board of Directors. China Investment Corporation. fenggl@chinainv.cn

2 homeownership, including both appreciation and implicit rent is measured. Second, properties of optimal portfolios for households under various degrees of consumption constraints are identi ed. Third, optimal portfolios of individual stocks are determined. The results show that portfolios of individual stocks, which vary by city, are far more attractive than the market portfolio for homeowners. This suggests a resolution to the stockholding puzzle. Homeowners could bene t from holding portfolios designed to o set the unique risk of the cities where they live but they lack information on what these portfolios might be. Given this information gap, holding the market portfolio is not particularly attractive for most homeowners. JEL Codes: D91 G11 R51 Key Words: Consumption Constrained Household, Local Housing Return 2

3 1 Introduction Although 70% of the U.S. homeowners face extreme housing risk, 100% of them hold riskless assets like cash or government bonds while only 25% of them hold stocks. 1 Conditional on holding some equities, the equity share of net wealth is only about 3%. 2 This stockholding puzzle contradicts standard portfolio theory, which predicts that diversi cation across asset classes, such as stocks and real estate, should raise expected risk-adjusted returns. Past research has often concluded that the lack of diversi cation in household portfolios is optimal. 3 However, these papers have focused on the covariance matrix between average house price appreciation for the for the entire U.S. and a single well-diversi ed market portfolio (often nding zero covariance).this analysis focuses on the covariance matrix between average house price appreciation for the entire U.S. and a single well-diversi ed market portfolio (often nding zero covariance). This approach ignores heterogeneity in the covariance between local housing markets and individual equities. This paper demonstrates that such heterogeneity is important foro ptimizing household portfolios. To understand the potential diversi cation bene t from holding equities along with local housing, this research constructs the optimal portfolio for consumption constrained households in individual cities. Consumption constrained households have desired housing consumption that limits their maximum downpayment to 20% of home value and consequently face substantial unique housing risk. 4 This research demonstrates that such consumption constrained households achieve greater risk adjusted returns by choosing individual equities rather than the S&P 500. The consumption constrained household gains greater utility from choosing individual equities rather than the S&P 500. This conclusion is easily explained. First, the household has substantial idiosyncratic risk due to the substantial size of its housing asset. Second, the S&P 500 is unattractive because it is too well diversi ed to hedge the unique housing risk associated with holding a highly leveraged investment in a single housing unit. 1 From SCF data, the shares of homeowners directly holding stocks in the U.S. are 24.9% in 1998, 27.0% in 2001, 25.8% in 2004, 22.4% in 2007 and 19.6% in 2010 respectively. 2 From SCF data, the shares of directly held stocks (median value) in net wealth of homeowners (median value) are 4.9% in 1998, 4.1% in 2001, 3.1% in 2004, 2.6% in 2007 and 3.7% in In their simulation studies, Flavin and Yamashita (2002) and Cocco (2005) nd that it s optimal for households with limited wealth to hold high share of real estate while few risk nancial assets in their portfolios. 4 Brueckner (1997) rst introduces the concept of "Consumption Constrained Households" by following the investment constraint of Henderson and Ioannides (1983). The investment constraint requires the housing investment should be at least as large as housing consumption. 3

4 Brueckner (1997) concludes that consumption constrained households whose wealth is a fraction of housing value, may not nd holding the market portfolio e cient. This theoretical nding and its potential empirical implications have been ignored. This paper investigates the portfolio optimization problem for representative consumption constrained households in various cities, whose net wealth is assumed to be only 20% of housing value. They choose the shares of home and risky nancial assets to be constrained mean-variance e cient. 5 Returns to housing are local and vary by housing market. Using national housing return and risk fails to consider the idiosyncratic nature of housing market risks. Moreover, total return to housing is mainly rental return as opposed to appreciation return. Rental returns have been ignored or simply based on problematic BLS rent indexes. This research estimates the local capitalization rate (i.e. the rental return) and local appreciation rate. Overall the empirical measures of local housing return and risk facing homeowners in this research di er substantially from those used previously in the literature on the stockholding puzzle. A well-diversi ed nancial market portfolio is not a good choice for homeowners because of the unique housing risk. To reduce portfolio risk for the consumption constrained households, the relevant alternative to housing is not always the market portfolio because the households have insu cient wealth to ever be well diversi ed. Thus the fact that a household appears to achieve no diversi cation bene t from holding the S&P 500 does not imply that it would not bene t from holding some subset of equities. The problem is that consumption constrained households must bear signi cant unique risk associated with their substantial investment in a single housing asset. Note there is no claim that households currently act on this information because it is unknown to them and to nancial advisors. Indeed, the optimal portfolios in various cities have been estimated for the rst time as a result of this research. Therefore this is largely an exercise in normative economics however it does account for the stylized behavior called the stockholding puzzle. Section 2 reviews the literature about household portfolios. Section 3 discusses the model and propositions. Data is talked in section 4. Section 5 solves optimal portfolio for consumption constrained households with local housing returns and individual equities. Section 6 includes robustness test. Section 7 concludes. 5 "Constrained mean variance e ciency" means that the households are subject to a portfolio constraint requiring them to hold a large position in local housing. They must maximize return for given risk or minimize risk for given return subject to that constraint. 4

5 2 Literature Review Mankiw and Zeldes (1991) rst nd evidence that households fail to satisfy theoretical predictions of standard portfolio theory. They use 1984 PSID data and nd that about 75% of the households hold no stocks; and 13% of the poor (with liquid assets lower than $1,000) hold stocks. Haliassos and Bertaut (1995) use 1983 SCF data and nd that only 19% of the 70th income percentile of households hold stocks and that the mean value held is below $800. Fratantoni (1998) nds that the median share of equities in the portfolios of homeowners under the age of 55 is zero. Fratantoni (2001) uses the 1989 SCF and reports that the median share of risky assets for homeowners is just 3.32%. Campbell (2006) summarizes the di erence between positive and normative household nance, and discusses how the "discrepancies or investment mistakes" between them have di erent e ects on heterogeneous households. Among them, consumption constrained households face a di erent portfolio problem than the standard textbook statement of household portfolio choice. Brueckner (1997) rst analyzed this speci c portfolio problem. He demonstrated that the investment constraint" proposed by Henderson and Ioannides (1983) leads to mean-variance ine ciency in portfolios. However, this aspect of the consumption constrained household s problem has been largely ignored in the literature. A variety of alternative explanations have been o ered for the stockholding puzzle. Both Fratantoni (2001) and Chetty and Szeidl (2012) argue that committed mortgage expenditure risk of homeowners helps explain the stockholding puzzle. Cocco (2005) explains the limited nancial assets in portfolios of younger and poorer households in terms of a combination of xed cost of equity market participation and the correlation of housing price with labor income and stock returns. Variation in the rental return is ignored. Risky labor income is also widely used to explain the puzzle (Heaton and Lucas (1997), and Cocco, Gomes and Maenhout (2005)). Davido (2006) illustrates that households whose incomes covary strongly with local housing prices should rent rather than own but this also implies that, conditional in owning, they should not hold substantial housing equity. In this literature, rental return to housing, and its variance, are ignored. Previous studies of the stockholding puzzle have analyzed the problem using a standard CAP-M approach with two risk assets, housing and the market portfolio (often the S&P 500 or other stock market index). These studies estimate a single covariance matrix between housing and stock market index. Using single market portfolio may lead to zero or positive covariance. Cocco (2005), 5

6 Flavin and Yamashita (2002, 2011), Yamashita and Nakagawa (2008) nd zero correlation between housing returns and the S&P Englund, Quigley and Hwang (2002) nd positive correlation between Sweden housing returns and AFGX index over short horizon. In the pure simulation of optimal portfolio studies, Yao and Zhang (2005) assume non-negative correlation between national housing market and the S&P diversi cation role of individual stocks in individual portfolios has been noted in Calvert, Campbell, and Sodini (2007). They use details of Swedish household portfolios and nd that holding an international stock portfolio could outperform one limited to domestic stocks. However, this advantage is not related to unique risk associated with holding housing in individual Swedish cities. Using a national housing price index may under estimate actual housing risks. Sommvervoll and de Haan (2014) and McDu (2012) report that small geographic areas can contain substantial idiosyncratic housing risk, and failure to use local housing returns may underestimate the magnitude of housing risks. Sommervoll and de Haan (2014) demonstrate that when a Case-Shiller-Weiss insurance policy is purchased, using a more local housing price index better insures against transaction losses in the Netherlands than using a national HPI. McDu (2012) shows more local housing risks can be hedged better using disaggregated housing risk measures. Peng and Thibodeau (2013) also emphasize the heterogeneity of housing risks across neighborhoods in Denver and the higher housing risks for low-income households. In sum, there is a literature which suggests that, for the consumption constrained household, using housing in a single location, there is no virtue in holding the S&P 500. Therefore testing for diversi cation using the S&P 500, or a similar market index, to represent the alternative risk asset is arti cially restrictive and likely to miss attractive opportunities for diversi cation presented by alternative portfolios of risk assets. 8 6 Flavin and Yamashita (2002) compute the zero share of T-bills in the constrained households while Fratantoni (1998) nd $17 thousand riskless assets; this di erence arises due to two reasons: the rst is that Fratantoni treats T-bond as riskless while Flavin and Yamashita treat it as risky; the second is what we emphasize: Flavin and Yamashita use the S&P 500 as proxy of stock market while S&P 500 turns out to provide poor hedge to unique housing risk. 7 Many other papers also explore the correlation between housing and stock markets. Quan and Titman (1999) detect positive correlation between U.S. real estate and stock returns over longer holding periods from 1984 to 1996 with annual data. Ibbotson and Siegel (1984) found a negative correlation between U.S. real estate index and the S&P 500 returns from 1947 to 1982 with annual data. Eichholtz and Hartzell (1996) document a negative correlation for Canada, the U.K. and the U.S. between property and stock indexes from 1985, 1977 and 1977 to 1993 with quarterly data. 8 Most normative papers get conclusions consistent with the stockholding puzzle: Flavin and Yamashita (2002) propose the maximum share of stock in life-cycle is 11.3%; Yao and Zhang (2005) nd that when owning a house, homeowners reduce the equity proportion in their net worth (bonds, stocks, and home); Cocco (2005) proposes 6% stock share for age group and 23.5% for age group The The 6

7 positive nding that such households fail to hold risk assets and pay down their mortgages instead does not imply that they are behaving optimally. Rather it suggests that they lack information and knowledge on the opportunity to diversify the unique risk associated with homeownership in particular cities (Calvet, Campbell and Sodini (2007)). 3 Theory: Portfolio Choice of Consumption Constrained Households The representative consumption constrained household owns a house in city i, chosen based on its consumption motive, and has total wealth signi cantly less than the value of the house so that mortgage nance is a necessity. In order to hedge the levered housing risk at the city level, this household has to decide the allocation of net wealth to risky nancial asset R from national nancial market and housing N from a local housing market. The ratio of household net wealth to housing value is assumed to be 20%: 9 LT V is strictly lower than 100% to avoid negative home equity. This household is not allowed to borrow to purchase assets except for a home mortgage. Such a consumption constrained household has the following relation between LT V (l) and housing share in the net wealth (h) 10 in portfolio: l = Mortgage Loan Home Value = 5 h 5 Returns to risky assets (R F t ) are assumed to have mean vector R F and variancecovariance matrix : (1) Households have two strategies for optimizing portfolios: one is pooling all individual nancial and real estate assets to decide the wealth allocation; the other is to rst choose one nancial market portfolio, then pool it with real estate asset to allocate the wealth. The former pools all the assets and includes idiosyncratic risks while the latter pools one well-diversi ed portfolio and one individual asset with idiosyncratic risk. In this paper, the S&P 500 is used as the market portfolio. Labor income dynamics are not modeled. Tenure choice and the decision to move are not considered here. Before solving the portfolio model, levered housing return is investigated. 9 This assumption makes sense considering that most FHA loans borrowers have an average LT V value of about 95% (Nichols, Pennington-Cross and Yezer (2005)). 10 Thus, nancial equity share in the net wealth is (1-h). 7

8 3.1 Levered Housing Returns Computing housing returns with mortgage leverage requires specifying unlevered housing returns and a schedule relating mortgage interest rates to LTV. Local housing returns without leverage (R H t ) are equal to sum of appreciation and capitalization rates, and can be expressed as, R H t = P t CAP t + (P t+1 P t ) P t ; R H t N(R H ; 2 H) (2) where P t is the housing value and CAP t is the capitalization rate at time t; P t CAP t is the net rental income, and P t+1 P t is the appreciation in value. Then levered housing returns (LRt H ), with mortgage loan amount L t, are given by: LR H t = P t CAP t + (P t+1 P t ) ' t L t P t L t = RH t ' t l t 1 l t (3) where l t is the LTV computed as ratio of mortgage loan (L t ) over housing value (P t ). If l is used to represent LTV, and all variables with a bar depend on the household s choice of l, including: h = Pt(1 l) : housing share in net wealth; Net W ealth LR H t : levered housing return; LR H t N(LR H ; 2 H); ': mortgage interest rate; F H : (LR H t ). covariance between national risky asset (R F t ) and levered housing return Finally, the mortgage interest rate increases with l so that d'(l) dl > 0; l 2 [0:8; 1): Modeling the Consumption Constrained Household This form of the portfolio problem of the constrained investor is based on Pelizzon and Weber (2008). Consumption constrained households maximize utility subject to several constraints: Max U = S T E(R) A 2 ST V S (4) 11 Chomsisengphet and Pennington-Cross (2006) document that the mortgage rate gap for LTV equal to 80% and 100% falls between 1.7% and 2.4%. For simplicity, we assume 1% increase in the interest rate when LTV increases to 100% from 80%. 8

9 s:t:i n S 0 (Non-negativity) (5) nx s i = 1; s i is an element of S (Rationality) (6) i=1 LT V = 5 h :h = s n ; housing share (7) 5 where S is n1 vector of asset shares; and s i is share of asset i in portfolio, i = 1; 2; 3:::n; specially, the last element of S is housing share: h = s n. R is the return matrix of national nancial assets and local housing investment (R N(; V )). A is the degree of risk aversion. I n is an identity matrix. Non-negativity and rationality constraints are binding. The optimal portfolio is solved numerically in two steps. First, for given LT V (l), quadratic techniques are used to search the local optimal shares of risky assets subject to the linear constraint n P 1 s i = 1 h i, where s i is the share for non-housing risky asset i=1 i. Second step, local optimal portfolios from the rst step are pooled to determine the global optimal portfolio with maximum utility level. Thus for each LT V value (l), the household has unique total asset variance-covariance matrix (V ), mean return matrix (R) and asset share vector (S) de ned as below, " V = T F H F H 2 H " # R F R = LR H " # X S = h # (8) (9) (10) Then the question of choosing optimal shares of risky assets (X) when LT V = l can be expressed as: MAX U = S T A E(R) 2 ST V S (11) " # T " # " # T " # " # X RF A X F H X = h LR H (12) 2 h h 9 T F H 2 H

10 The optimal share vector of risky assets (X ) without non-negativity constraints is, X = 1 ( RH A h F H ) (13) From the equation above, the share of nancial assets is not only determined by its Sharpe ratio ( 1 R H ), but more importantly, the diversi cation bene t of hedging local housing risk ( F H ). Given the large idiosyncratic housing risk in 2 H, as shown later, a well-diversi ed stock portfolio would bring little hedge bene t compared to individual stocks. If the market portfolio is chosen, the elements o the diagonal of variancecovariance matrix ( F H ) would be close to zero implying that X would be mainly determined by Sharpe ratio of market portfolio. 3.3 Choice of the Market Portfolio or Individual Equities Assume is the optimal stock combination when consumption constrained households choose individual stocks. It follows that includes substantial unique risk. In contrast, the S&P 500 is a well-diversi ed market portfolio with little unique risk. From the analysis above, the model implies, Consumption constrained households prefer to choose individual equities rather than market portfolios (the S&P 500) if ( R +C 5A ) > ( R SP +C SP 5A SP SP ); prefer to choose market portfolios (the S&P 500) rather than individual equities if ( R +C 5A ) < ( R SP +C SP 5A SP SP ); are indi erent in choosing individual equities or market portfolios (the S&P 500) if ( R +C 5A ) = ( R SP +C SP 5A SP SP ); where is the equivalent stock combination when households optimally choose in- dividual equities and housing; C is a constant determined by mortgage market: C = 5 var(') 5cov(R H ; ') ' E, where E is column vector with all element equal to 1; 2 i is the covariance between unlevered local housing return and asset i. A proof is provided in the Appendix I. The analysis illustrates the cuto of choosing individual equities over the S&P 500. R +C ( R SP +C SP ) measures the Sharpe ratio from holding a sub-optimal stock combination (the S&P 500); a higher Sharpe ratio indicates higher risk-adjusted returns. 5A (5A SP SP ) measures the diversi cation from holding individual stocks (the S&P 500); a negative value indicates that individual stocks (the 10

11 S&P 500) move opposite to the local housing market, thus stocks (the S&P 500) can hedge the local housing risks. Note that the cut-o is also determined by the degree of risk aversion (A) and current interest rates in the national mortgage market (C). It is not possible to prove that holding individual stocks strictly dominates the S&P 500 because that depends on the covariance of the individual stocks and local housing returns. The hedging of local housing risk varies across individual stock groups. The hypothesis that groups of stocks outperform the S&P 500 is an empirical question that will be tested below by repeated portfolio optimization holding random portfolios of stocks. At the same time, the size of the additional bene t of diversi cation using individual stocks is illustrated. 4 Data The data used to compute local housing risk and return as well as that from equities are taken from the American Housing Survey (AHS), the Federal Housing Finance Agency Housing Price Index (FHFA HPI) and the Center for Research on Security Prices (CRSP). All the nominal values are de ated with the BLS CPI. Capitalization rates and appreciation rates for each housing market are estimated. Data for estimating capitalization rates are from the American Housing Survey (AHS) from 1985 to To reduce the heterogeneity of house characteristics from the AHS, only multi-family houses from 38 of the largest MSAs are used. Appendix II describes the estimation method. Housing price appreciation rates are computed from the FHFA HPI. 12 Mortgage interest rate is 30 year FRM rate from Primary Mortgage Market Survey by Freddie Mac allowing for deductibility by a household in the 33% marginal tax bracket. Table 1 lists descriptive statistics on the annualized total return to housing investment for 38 cities, indicating signi cant geographic variation of housing returns. Mean values of annual housing returns range from 5% to 10% and the standard deviations range from 4% to 12%. Cities in the Sun Belt have lower variance in their housing returns. Texas and Florida cities have lower average housing returns and medium risks; California cities have risky housing returns. San Francisco, San Jose, Los Angles, Oakland, San Diego and Santa Ana have higher housing returns and risks. Northeastern and part of Mid-Atlantic cities have attractive housing returns. Washington DC, 13 Boston, and Greater New York 12 FHFA uses MSA de ned by OMB while AHS has been using SMSA to de ne di erent geographic areas. 13 Washington D.C. is divided into Northern D.C. (Bethesada-Rockville-Germantown) and Southern D.C. (Arlingtong-Alexandria), because FHFA publishes HPI seperately for these 2 areas. 11

12 have higher housing returns and lower risk than California cities. But Rust Belt cities have lower housing returns and risks. These are measured ex-post returns to investment and households are assumed to apply the income tax deduction from mortgage interest. Financial data for equities come from CRSP. 1,097 Individual stocks with continuous observations from 1985 to 2009 are used in the testing. The S&P 500 is chosen as the market portfolio. Four groups of individual stocks are chosen following these steps 14 : 60 stocks from the 1097-stock pool are kept only if they have similar mean values and standard deviations of returns to those of the S&P 500 during the studied period; then the chosen 60 stocks are grouped into 4 groups randomly. From 1985 to 2009, real annual after tax return of the S&P 500 is 7.87% with a standard deviation of 12.33%. 15 Table 2 summarizes the descriptive statistics for the 60 stocks and more information is listed in Appendix III. In Table 2, average annual returns across each group are very close, about 7%. Standard deviations of stocks are slightly di erent within group ((varying from 9.8% to 19%) but are quite similar across groups. Financial data on REITs also come from CRSP. REITs with concentrated geographic investments provide an alternative measure of local housing returns. Table 3 lists detailed information on 18 REITs: 8 of them concentrate their investments in one state: Mack- Cali Realty and Presidential Realty; BRE properties, Kilroy Realty and Essex Property Trust focus on major cities in paci c coast (California and Seattle); Income Opportunity Realty concentrates in Texas; Roberts Realty Investors invests in Atlanta; Washington Real Estate Investment Trust invests in Washington DC; Mid-America Apartment Communities Inc. invests in Sun Belt states. The other 9 REITs are more diversi ed but still focus several major cities rather than national market. Even with di erent observed years, REITs have relatively higher returns than local housing returns in Table 1 but with higher risk; have similar risks (Std. Dev.) and returns as individual stocks. This is consistent with risk and return characteristics reported by Case, Yang and Yildirim (2010). Figure 1 shows the asset performances of all the nancial assets and local housing returns. Local housing returns with full equity (small dots in the lower left) are less volatile and most have lower returns than the S&P 500 (big dot). Local housing returns with LTV =80% have higher mean value and more volatility (small dots in the upper 14 However, an alternative way of choosing individual stocks may consider the locality of stock holding among MSAs. Some recent nancial studies explore the local HPI changes and the performance of rm stocks whose headquarters are located in those cities (Anderson and Beracha (2012), Henock and Sun (2012)). 15 All nancial assets are charged 15% of annual capital gain tax. 12

13 right). The S&P 500 is less volatile but with higher returns than 60 individual equities (stars). Figure 2 shows the asset performances of all the nancial assets and REITs. In contrast with the nding in gure 1, REITs (small dots in the upper right) have higher returns and risks (Std. Dev.) than the S&P 500. The parameter of risk aversion is 3; the investment horizon is 1-year for all assets Empirical Testing This section investigates the strategy of using nancial assets to hedge local housing risks. Households buy stocks performing similarly as the S&P 500 in average values of annual returns and standard deviations. The major nding is that, for consumption constrained households, the optimal strategy is to hold individual stocks. This strategy also varies across the local housing markets. 17 Furthermore, additional tests show that it is always optimal for homeowners who have signi cant equity, well beyond 20% of value, to hold individual stocks rather than the S&P 500. This normative result contrasts with actual observed behavior for the reasons discussed above. 5.1 Optimal Portfolios of Consumption Constrained Households Tables 4-7 summarize the ndings by comparing the housing shares, portfolio returns, portfolio risks and mean-variance utilities from optimal solutions. In Table 4, optimal housing shares from choosing di erent nancial assets are shown. The S&P 500 is generally not attractive to constrained households. In column 1, the average share of the housing is as high as 84%. Moreover, the share of the S&P 500 is very low in most cities, which further con rms the result from Flavin and Yamashita (2002) that given the choice between average house returns for the entire U.S. and the market portfolio, households tend to ignore equities and make the largest down payment 16 Most papers would use several parameters equal to 1, 2, 4, 8 or 16. See Englund, Quigley and Hwang (2002), Flavin and Yamashita (2002). Due to a long city list, in the estimation part, we only consider the case when parameter equal to We also use stocks with less favorable performance as S&P500 to further con rm the advantage of holding individual stocks. Results shows that for those without ability of avoiding the poorly performed stocks (annual average return close to 0%), only in 9 out of 38 cities, the optimal strategy is holding S&P 500; in 17 cities, it s always optimal to hold individual stocks rather than S&P500; nally, in the rest 12 cities, household portfolios can perform better by holding individual stocks rather than S&P 500 with probability of 50%. 13

14 possible. When stock groups are the alternative, optimal portfolios are quite di erent. In columns 2 to 5, housing shares from holding di erent stock groups are di erent as predicted. In coastal areas like California and Mid-Atlantic, the di erences are larger due to higher unique housing risks. In areas with lower housing risks like Houston, St. Louis, Cincinnati, Milwaukee and Pittsburgh, the di erences are smaller. In column 6, the di erence between the average housing share of holding stock groups and the S&P 500 is negative (mean di erence is about -59%). This re ects the unfavorable characteristics of the S&P 500 for consumption constrained households. It is "too well-diversi ed" to hedge unique housing risk. Overall, consumption constrained households should be using individual stocks rather than the S&P 500 to hedge housing risk, and those in cities with higher housing risks prefer to hold stocks to hedge the local housing risks. If available stocks are changed, the optimal housing shares also change signi cantly. In Table 5, optimal portfolio returns obtained by choosing di erent nancial assets are shown. Columns 1 and 2 list the optimal housing shares and portfolio returns from holding housing and the S&P 500; columns 3 and 4 list the average optimal housing shares and average portfolio returns from holding housing and stock groups. Holding the S&P 500 brings higher portfolio returns than individual stocks for households in most cities. These higher portfolio returns are earned because households hold higher housing equity when the S&P 500 is the alternative risk asset. In column 5, average portfolio returns of choosing stock groups are lower than those of the S&P 500 in 30 out of 38 cities. But these di erences in portfolio returns (column 5) are very small. In Table 6, risks associated with holding optimal portfolios are shown. Columns 1 and 2 list the optimal housing shares and portfolio returns from holding housing and the S&P 500; columns 3 and 4 list the average optimal housing shares and average portfolio risks from holding housing and stock groups. Holding the S&P 500 rather than individual stocks brings higher portfolio risks to households in all cities. Compared with the trivial gain in risk-adjusted returns in Table 5, the portfolio risks of holding the S&P 500 are rather higher. Total performance di erences from holding the S&P 500 versus individual stocks should be weighted by the mean-variance utilities. In Table 7, optimal portfolio utilities of choosing di erent nancial assets are shown. The utility equals the portfolio returns net of portfolio risks weighed by degree of risk aversion. Columns 1 and 2 list the optimal housing shares and portfolio utilities from holding housing and the S&P 500; columns 3 and 4 list the average optimal housing shares and average portfolio utilities from holding housing and stock groups. Holding the S&P 500 fails to bring higher total gains in all cities compared with holding individual 14

15 stocks. In column 5, the di erence between average utility value from holding stock groups and value of choosing the S&P 500 is always positive. The average di erence is about That is, even if holding the S&P 500 can bring slightly higher portfolio returns than holding stocks in some cities as shown in column 5 of Table 5, the extra returns fail to compensate for the higher portfolio risks as in Table 6. Consider the illustrative case of consumption constrained households in Northern Washington DC. When housing and the S&P 500 are held, portfolio risks (measured as variance of portfolio) range from 20.65% to 20.74% and portfolio returns range from 32.6% to 32.7%. In the cases of holding housing and each of 4 stock groups, portfolio risks of holding individual stocks range from a minimum value of 17.7% when holding stock group 3 to maximum value of 20.59% when holding stock group 1. Portfolio returns of holding individual stocks range from minimum value of 31.5% when holding stock group 3 to maximum value of 32.93% when holding stock group 2. Holding individual stocks brings better portfolio performance for constrained households in Washington DC: given the similar portfolio returns, portfolio risks decrease signi cantly (on average, lower by 20%). The results suggest why consumption constrained households may not hold the S&P 500 or another proxy for the market portfolio because it is not, in fact, e ective in diversifying away the unique risk associated with holding housing in individual housing markets. In contrast portfolios of individual stocks can be constructed to diversify away far more unique leveraged housing risk. Because households are unaware of this possible diversi - cation bene t, they nd little advantage in holding the S&P 500 rather than prepaying their mortgage and this give rise to the stockholding paradox. 5.2 Optimal Portfolios of Unconstrained Households This section shows that even for unconstrained households, the optimal strategy is still to hold the individual stocks rather than the S&P 500. Unconstrained households still hold a substantial fraction of wealth in a single housing unit because their entire wealth is equal to housing value. Table 8 displays the housing shares in the optimal portfolios of unconstrained households. Column 1 lists the housing shares when unconstrained households choose housing and the S&P 500. Households in western and north-eastern coastal cities are found to hold more stocks. Columns 2-5 list the housing shares when households choose housing and stock groups: as expected, average housing shares of holding stock groups are higher than value of holding the S&P 500. Higher shares of individual stocks in unconstrained 15

16 portfolios are consistent with the stylized fact for richer Swedish households documented by Calvet, Campbell and Sodini (2007). Overall, holding the S&P 500 is also less attractive for unconstrained households. The results suggest why households with signi cant wealth may use it to prepay their mortgage rather than hold a signi cant proportion of equities. This, of course, is a further element of the stockholding puzzle. 6 Robustness Tests To test whether it is robust that local housing return and individual stocks bring higher diversi cation, the following questions are considered: whether using other proxy of local housing returns can support higher diversi cation from individual equities; what is the statistical signi cance of the conclusion that holding individual stocks better hedges housing risk than the S&P500; whether holding individual stocks can better ex ante hedge the housing risks. whether the results change signi cantly if di erence in CAP rates are ignored and total return is based on appreciation only. 6.1 Using REIT Returns instead of Local Housing Returns Previous results have been based on housing returns constructed using the AHS and FHFA data to estimate total returns to homeowners. The robustness test performed here uses an alternative, although imperfect, measure of total returns to real estate investment in each city. The alternative is based on REIT returns for the subset of REITS whose properties are spatially concentrated and for which apartment properties are a signi cant portion of investments. Table 3 shows 18 apartment REITs whose investments are concentrated in one or at most a small number of states. In this case, REIT returns are substituted for the local housing returns used previously. Table 9 allows comparison of the optimal housing shares for the consumption constrained household for each of the 18 spatially concentrated REITs. Moreover, as described in the data part, they mainly invest in apartment buildings rather than commerical real estate. In fteen of these cases, the S&P 500 portfolio share is zero (housing 16

17 share is 100%). In only one of the three cases where housing share is under 100% does the S&P500 have a lower share than one of the groups of individual risk assets. Overall these results provide strong con rmation of those found for returns based on housing returns by city. The consumption constrained household is far better served by portfolios of individual stocks and the S&P 500 does such a poor job of diversifying unique housing risk that households, in 15 of 18 cases, hold a share of zero in their optimal portfolios. If anything, these results are stronger than those found for the risk return measures constructed for individual cities. 6.2 Sample Selection in Choosing Stocks Another question is the sample section in choosing stocks. 60 stocks in the empirical testing were assembled in groups with similar mean return and variance to that of the S&P 500. In the robustness test performed here, all of the 1097 individual stocks with continuous observations from 1985 to 2009 are used to determine if the results change under random sampling. Random samples of 15 individual stocks are chosen. Then the optimal portfolio problem is solved for each group. After 1,000 times of repeated random samples are drawn for each city, the following null hypotheses are tested: Null Hypothesis 1: housing shares from holding the S&P 500 are lower than those from holding individual stocks (which would indicate that the S&P500 was preferred over random groups of individual stocks); Null Hypothesis 2: portfolio returns from holding the S&P 500 are higher; Null Hypothesis 3: portfolio risks from holding the S&P 500 are lower; Null Hypothesis 4: portfolio utilities from holding the S&P 500 are higher. Figure 3 displays the means and standard deviations of 1,097 individual stocks vs the S&P 500; majority of stocks have larger risk (Std. Dev.) than the S&P 500. Columns 1-4 in Table 10 show that, compared to these portfolios of individual securities, holding the S&P 500 implies higher housing shares, lower portfolio returns, higher portfolio risks and lower portfolio utilities; columns 5-8 show the p-values of the above tests. Results in columns 5 to 8 of Table 10 con rm that holding randomly selected groups of 15 individual stocks can bring signi cantly higher risk adjusted returns. Housing shares from holding the S&P 500 are signi cantly larger than those from holding individual 17

18 stocks across 38 cities in column 1. Null hypothesis 1 is rejected in all cities: p-values being lower than in column 5 indicate that holding individual stocks signi cantly brings higher diversi cation to hedge local housing risks. This is further con rmed by the strong evidence of signi cantly higher returns (column 2) and lower risks (column 3) across the 38 cities. Null hypothesis 2 is rejected in all cities (column 6) and null hypothesis 3 is rejected in most cities (column 7), implying that holding individual stocks can signi cantly improve the portfolios for local consumption constrained households. However, in several Rust Belt cities (Cleveland, Milwaukee, Pittsburgh and St. Louis), null hypothesis 3 of lower risks from holding the S&P 500 cannot be rejected. Finally, 95% of the repeated samples across cities bring lower mean-variance utilities from holding the S&P 500; hypothesis 4 of higher portfolio utilities from holding the S&P 500 is rejected with p-values lower than in all the cities. 6.3 Tracking Ex Post Portfolio Performance A third question is whether the results are robust to changes in the period studied. The 1985 to 2009 period includes signi cant turbulence in housing markets. In this third robustness test, the 1985 to 1999 period is used to estimate optimal portfolios. Then historical returns from optimal portfolios from these local housing and nancial markets are used to compute ex post portfolio returns and risks from 2000 to Results show that consumption constrained households can get comparable portfolio returns from holding individual stocks versus the S&P 500; but holding individual stocks lowers housing risk in the 2000 to 2009 period. Essentially ex post relative portfolio performance in the volatile 2000 to 2009 period is comparable to the results obtained when this period was included in the sample used to determine optimal portfolios. These results are summarized in Table 11 which lists p-values from testing the null hypothesis that holding the S&P 500 brings higher ex post utilities than holding individual stocks. First, returns of stock groups and the S&P 500 from 1985 to 1999 are used to nd the optimal portfolios for each city. Then, ex post utilities are calculated using the results above and asset returns from 2000 to The null that holding the S&P 500 brings higher utilities for consumption constrained households from 2000 to 2009 is rejected signi cantly. 18

19 6.4 Ignoring Rental Return in Computing Total Housing Returns Many previous studies of the stockholding paradox have ignored rental returns, i.e. ignored the CAP rate, or set it equal to a constant. 18 This, of course, introduces measurement error into total return as the mean is certainly lower and the time series variation is di erent. The robustness test conducted here ignores rental return and bases housing returns on local rates of housing price appreciation alone. Table 12 is constructed in the same fashion as Table 4 except that housing return is based on appreciation alone. The results are qualitatitvely similar but ignoring CAP rate variation does lower the di erence in housing share between the S&P500 from an average di erence in Table 4 of 59% to 47% in Table 12. Nevertheless, the conclusion that the stockholding paradox is related to the lack of knowledge regarding the ability of portfolios of individual risk assets to diversity away local housing risks is evident. 7 Conclusions The stockholding puzzle re ects the intersection of normative and positive analysis in eld of household nance. Normative analysis applying portfolio theory suggests that consumption constrained households have substantial unique risk and may gain little from holding the market portfolio. However, there should be a portfolio that includes signi cant amounts of equities, di erent across urban housing markets, that is meanvariance e cient. Positive empirical analysis nds that households tend to hold housing and government guaranteed assets. Over time they pay down mortgage balances and avoid holding equities. This behavior can be understood by noting the low level of diversi cation bene t achieved if households follow conventional wisdom and believe that the market portfolio is the alternative risk asset available to them. Some research tries to resolve the paradox by using the illiquidity property of the housing asset, or the positive correlation between housing returns and the market portfolio to explain household behavior. Other papers have either ignored rental return or used a national rental index. Few papers consider the geographic variation of housing returns including both rental and appreciation components. 18 One argument for ignoring CAP rate risk is that residents in a city are subject to rental risk whether they are owners or renters. Therefore, the risk that they need to hedge is appreciation risk. The robustness test undertaken here is designed to determine if the principal ndings of this paper hold for households concerned only with appreciation return and its variance. 19

20 There are several innovative steps in this paper. First, total returns to housing investment for 38 cities are estimated. Then optimal portfolios are computed for consumption constrained households (and also households whose wealth equals housing value). The household is allowed to choose two alternative groups of stocks, the S&P500 and groups of 15 individual stocks. Consistent with the stockholding paradox, households fail hold a signi cant share of the S&P500. However, when groups of individual stocks are considered, the paradox is viewed in a di erent light. Households optimally hold a substantial share of selected portfolios of individual stocks but the composition of that portfolio varies across cities. As demonstrated in the theory section, the consumption constrained households have an unusual portfolio choice problem because of the substantial amount of unique risk associated with their highly leveraged housing investment. It is not surprising that the market portfolio as represented by the S&P 500 is ine ective in diversifying away this unique risk. However, it is possible to nd individual stocks that are e ective at diversifying away unique risk in many cities. In view of this, the nal results are not surprising. Consumption constrained households faced with the choice of holding the S&P 500 versus housing will hold housing. However, if these households were aware of the characteristics of all risk assets available, they can nd attractive alternatives to housing that would help to optimize their portfolios. The risk asset weights in these portfolios vary across cities in a manner that households could not know at present. 20

21 References [1] Anderson, C. and Beracha, E., 2012, Frothy Housing Markets and Local Stock-Price Movements, Journal of Real Estate Finance and Economics, 45, No. 2, [2] Brueckner, J. K., 1997, Consumption and Investment Motives and Portfolio Choices of Homeowners, Journal of Real Estate Finance and Economics, 15, 2, [3] Calvet, L. E., Campbell, J. Y., and Sodini, P., 2007, Down or Out: Assessing the Welfare Costs of Household Investment Mistakes, Journal of Political Economy, 115, No. 5, [4] Campbell, J. Y., 2006, Household Finance, Journal of Finance, 61, 4, [5] Case, B., Yang, Y., and Yildirim, Y., 2010, Dynamic Correlations Among Asset Classes: REIT and Stock Returns, Journal of Real Estate Finance and Economics, 44, No. 3, [6] Chetty, R. and Szeidl, A., 2012, E ect of Housing on Portfolio Choice, NBER Working Paper # [7] Chomsisengphet, S., and Pennington-Cross, A., 2006, The Evolution of the Subprime Mortgage Market, Review-Fed Reserve Bank of St. Louis, 88, 1, [8] Cocco, J. F., 2005, Portfolio Choice in the Presence of Housing, Review of Financial Studies, 18, No. 2, [9] Cocco, J. F., Gomes, F., and Maenhout, P., 2005, Consumption and Portfolio Choice Over the Life-Cycle, Review of Financial Studies, Vol. 18, No. 2, [10] Davido, T., 2006, Labor Income, Housing Prices, and Homeownership, Journal of Urban Economics, 59, No. 2, [11] Eichholtz P., and Hartzell, D., 1996, Property Shares, Appraisals and the Stock Market: An International Perspective, Journal of Real Estate Finance and Economics, Vol 12, [12] Englund P., Hwang, M., and Quigley, J. M., 2002, Hedging Housing Risk, Journal of Real Estate Finance and Economics, 24,

22 [13] 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, No.1, [14] Flavin, M., and Yamashita, T., 2002, Owner-Occupied Housing and the Composition of the Household Portfolio, American Economic Review, 92, No. 1, [15] Flavin, M., and Yamashita, T., 2011, Owner-Occupied Housing: Life-Cycle Implications for the Household Portfolio, American Economic Review, 101, No.3, [16] Fratantoni, M. C., 1998, Homeownership and Investment in Risky Assets, Journal of Urban Economics, 44, [17] Fratantoni, M. C., 2001, Homeownership, Committed Expenditure Risk, and the Stockholding Puzzle, Oxford Economic Papers, 53, [18] Haliassos M. and Bertaut, C., 1995, Why Do So Few Hold Stocks? Economic Journal, 105, [19] Henderson, J. V., and Ioannides, Y. M., 1983, A Model of Housing Tenure Choice, American Economic Review, 73, [20] Heaton, J. C., and Lucas, D. J., 1997, Market Frictions, Savings Behavior, and Portfolio Choice, Macroeconomic Dynamics, Vol 1, [21] Ibbotson, R. G. and Siegel, L. B. 1984, Real Estate Returns: A Comparison with Other Investments, Real Estate Economics, Vol. 12, [22] Louis, H. and Sun, A. X., 2013, Long-Term Growth in Housing Prices and Stock Returns, Real Estate Economics, 41, 3, [23] Mankiw, N. G., and Zeldes, S., 1991, the Consumption of Stockholders and Non- Stockholders, Journal of Financial Economics, 29, [24] McDu, F., 2012, Home Price Risk, Local Market Shocks, and Index Hedging, Journal of Real Estate Finance and Economics, 45, 1, [25] Nichols J., Pennington-Cross, A., and Yezer, A., 2005, Borrower Self-Selection, Underwriting Costs, and Subprime Mortgage Credit Supply, Journal of Real Estate Finance and Economics, 30, 2,