NBER WORKING PAPER SERIES HETEROGENEITY IN EXPECTATIONS, RISK TOLERANCE, AND HOUSEHOLD STOCK SHARES: THE ATTENUATION PUZZLE

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1 NBER WORKING PAPER SERIES HETEROGENEITY IN EXPECTATIONS, RISK TOLERANCE, AND HOUSEHOLD STOCK SHARES: THE ATTENUATION PUZZLE John Amerks Gábor Kézd Mnjoon Lee Matthew D. Shapro Workng Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambrdge, MA November 2018 Ths research s supported by a program project grant from the Natonal Insttute on Agng P01- AG The Vanguard Group Inc. supported the data collecton through the VRI (Vanguard Research Intatve). Vanguard s Clent Insght Group and IPSOS SA were responsble for mplementng the VRI survey and provded substantal nput nto ts desgn. Andrew Capln s a co-prncpal nvestgator of the VRI. We gratefully acknowledge hs collaboraton. The desgn of the VRI benefted from the collaboraton and assstance of Joseph Brggs, Wand Brune de Brun, Alyca Chn, M Luo, Brooke Helppe McFall, Ann Rodgers, and Chrstopher Tonett as part of the program project, from Annette Bonner (Vanguard), and Wendy O Connell (IPSOS SA). We are grateful to Stefan Nagel and Isacco Pccon for comments on an earler draft. The vews expressed here are those of the authors and not those of ther respectve nsttutons. For documentaton of the VRI, ncludng a dynamc lnk to the survey nstrument, see The vews expressed heren are those of the authors and do not necessarly reflect the vews of the Natonal Bureau of Economc Research. NBER workng papers are crculated for dscusson and comment purposes. They have not been peer-revewed or been subject to the revew by the NBER Board of Drectors that accompanes offcal NBER publcatons by John Amerks, Gábor Kézd, Mnjoon Lee, and Matthew D. Shapro. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 Heterogenety n Expectatons, Rsk Tolerance, and Household Stock Shares: The Attenuaton Puzzle John Amerks, Gábor Kézd, Mnjoon Lee, and Matthew D. Shapro NBER Workng Paper No November 2018 JEL No. C83,D81,D84,G11 ABSTRACT Ths paper jontly estmates the relatonshp between stock share and expectatons and rsk preferences. The survey allows ndvdual-level, quanttatve estmates of rsk tolerance and of the perceved mean and varance of stock returns. These estmates have economcally and statstcally sgnfcant assocaton for the dstrbuton of stock shares wth relatve magntudes n proporton wth the predctons of theores. Incorporatng survey measurement error n the estmaton model ncreases the estmated assocatons twofold, but they are stll substantally attenuated beng only about 5 percent of what benchmark fnance theores predct. Because of the careful attenton n the estmaton to measurement error, the attenuaton lkely arses from economc behavor rather than errors n varables. John Amerks The Vanguard Group, Inc. PO Box 2600, MS V36 Valley Forge PA john_amerks@vanguard.com Gábor Kézd Survey Research Center Unversty of Mchgan Ann Arbor, MI gabor.kezd@gmal.com Mnjoon Lee Department of Economcs Carleton Unversty Ottawa, ON K1S 5B6 Canada mnjoon.lee@carleton.ca Matthew D. Shapro Department of Economcs Unversty of Mchgan 611 Tappan St Ann Arbor, MI and NBER shapro@umch.edu Documentaton and survey nstrument s avalable at An onlne appendx s avalable at

3 The source of heterogenety n portfolo choces s an mportant queston for household fnance (Campbell, 2006). Theores, such as consumpton CAPM, predct that the share of rsky assets should be postvely related to ther expected returns, negatvely related to ther rsk, and postvely related to nvestors rsk tolerance. These theores also have quanttatve mplcatons for the magntudes of those relatons. Ths paper assesses those mplcatons by estmatng how heterogenety n preferences and belefs explan heterogenety n household portfolos. In ths paper we take a systematc attempt at quanttatvely evaluatng the mplcatons of benchmark fnancal theores by usng better data and more careful statstcal modelng. We buld a structural maxmum lkelhood model to estmate jontly quanttatve measures of rsk tolerance and the perceved mean and varance of stock returns from hgh-qualty survey data whle takng survey measurement error nto account. We estmate ther assocaton wth household stock shares at the ntensve margn. Our approach s made possble by new data on portfolo composton for a large enough sample of stockholdng households, combned wth approprate measures of preferences and belefs. Our data set was created by the Vanguard Research Intatve (VRI) that combnes admnstratve account data and survey responses for a large sample of Vanguard account holders. The VRI has multple features that make t especally wellsuted for estmaton of the sources of heterogenety n stock holdngs. Secton I summarzes related lterature and dscusses how our approach mproves upon prevous analyses. Secton II descrbes the VRI sample and the measurements of assets and stock share. Secton III descrbes how we measure preferences and belefs. To get ndvdualspecfc estmates of preference parameters, we use a modfcaton of the Barsky, Juster, Kmball, and Shapro (1997) approach of elctng rsk tolerance from hypothetcal gambles over permanent ncome. To get ndvdual-specfc estmates of the moments of the perceved

4 dstrbuton of returns, we use both the Mansk (2004) approach of elctng ponts n the CDF of perceved returns together wth ndvduals estmates of expected returns. Survey measures of preferences have consderable external valdty (.e., that preference parameters explan a wde range of behavors) and nternal valdty (.e., test-retest valdaton and consstence across dfferent measures). See Barsky, Juster, Kmball, and Shapro (1997), Kmball, Sahm, and Shapro (2008), Amerks, Capln, Laufer, and Van Neuwerburgh (2011), Dohmen, Falk, Huffman and Sunde (2010), Dohmen, Falk, Huffman, Sunde, Schupp, and Wagner (2011), and Josef, Rchter, Samanez-Larkn, Wagner, Hertwg, and Mata (2016) for evdence both of external and nternal valdty. Recent evdence suggests survey measures of rsk preferences show more stablty than measures based on small-stakes lottery experments (Lönnqvst, Verkasalo, Walkowtz and Wchardt, 2015). Smlarly, probablstc measures of expectatons have predctve valdty (Hurd, 2009). See Mansk (2017) for a summary of progresses made n elctng subjectve expectatons on macroeconomc varables ncludng equty returns. Carroll (2017) also stresses the role of expectatons n explanng macroeconomc fluctuatons and hence the mportance of correctly measurng them and understandng ther formaton. Ths paper s the frst attempt to measure both preferences and expectatons and to use them jontly to explan portfolo choces. Lke many survey measures, preference and expectatons are subject to response error. Ths paper uses a unfed procedure accountng for response error to produce unbased estmates of the subjectve varables for both preferences and belefs. Secton IV combnes these estmates to explan the cross-secton of stock shares. We fnd that the stock share s postvely related to the ndvduals perceved expected stock returns, s negatvely related to ther perceved standard devaton of the returns, and s postvely related to ther rsk tolerance. These 2

5 relatonshps are economcally and statstcally sgnfcant, they are robust across varous specfcatons, and they are substantally larger n magntude than correspondng estmates that do not take care of measurement error n the survey answers. At the same tme, the estmated assocatons are only about 5 percent of what benchmark theores predct. Some features of our estmates are n lne wth those mplcatons: the sgns and also the relatve magntudes of the estmated coeffcents conform to the predctons of theores. They are substantally smaller n magntude, though, a fndng that we call the attenuaton puzzle. The emprcal method advanced by ths paper addresses measurement error n survey measures of preferences and belefs, so t establshes that ths attenuaton reflects actual gap between benchmark portfolo choce theores and ndvduals behavor. I. Relatonshp to the Lterature Several papers estmated assocatons of household portfolo compostons wth varous measures of belefs and preferences. Not all of them yeld results that can be weghed aganst the quanttatve predctons of fnance theores. The results of those that do allow for such comparsons suggest that belefs and preferences, as measured by the data, are related to household portfolos ndeed, but those relatons are substantally weaker than what benchmark fnance theores would suggest. Most studes analyzed assocatons at the extensve margn,.e., whether households hold any stocks, prmarly due to constrants on sample sze. Yet theores have the starkest quanttatve predctons at the ntensve margn,.e., the share of stocks n the portfolo of stockholders. Most studes ether examne the role of belefs or preferences but not both. 3

6 Vssng-Jorgensen (2003), Glaser and Weber (2005), Hurd, van Rooj and Wnter (2011), Hudomet, Kezd and Wlls (2011), Amromn and Sharpe (2012), Hoffman, Post and Pennngs (2013), and Guso, Sapenza and Zngales (2018) focus on expectatons and show that people wth more optmstc expectatons about future stock returns are more lkely to hold stocks. Barsky, Juster, Kmball and Shapro (1997), Dohmen, Falk, Huffman and Sunde (2010), Dohmen, Falk, Huffman, Sunde, Schupp, and Wagner (2011) and Guso, Sapenza and Zngales (2018) show that more rsk tolerant ndvduals are more lkely to hold stocks. Domntz and Mansk (2007) and Hurd, Rooj and Wnter (2011) show that ndvduals wth hgher levels of stock market expectatons and lower perceved rsk are more lkely to hold stocks. Kmball, Sahm and Shapro (2008) model the ntensve margn. Kezd and Wlls (2011) and Dmmock, Kouwenberg, Mtchell and Pejnenburg (2016) combne the extensve and ntensve margns n Tobt-type models and establsh assocatons wth rsk tolerance, expectatons and ambguty averson, respectvely. Weber, Weber and Nosc (2013) show that ndvdual measures of rsk tolerance and expectatons predct the share of stocks respondents nvest n a hypothetcal fnancal portfolo but they do not consder belefs. Hoffmann, Post and Pennngs (2013) and Merkle and Weber (2011) analyze the role of expectatons and rsk tolerance n tradng behavor of ndvdual nvestors rather than the share of stocks n household portfolos. Brunnermeer and Nagel (2008) conclude that understandng the determnants of the share of stocks n the portfolo of stock market partcpants s very dffcult. Several related studes nvestgated the role of wealth and past experences n household portfolos. See, for example, Vssng-Jorgensen (2003), Greenwood and Nagel (2009), Seru, Shumway and Stoffman (2010), Malmender and Nagel (2011), Calvet and Sodn (2014). Another lterature focuses on the role of preferences and belefs n other household decsons. 4

7 Pazzes and Schneder (2009) and Armona, Fuster and Zafar (2016) examne the role of expectatons on the housng market, whle Brune de Brun, Mansk, Topa, and van der Klaauw (2011), Armanter, Brune de Brun, Potter, Topa, van der Klaauw, and Zafar (2013), Malmender and Nagel (2016), and Botsch and Malmender (2017) nvestgate nflaton expectatons. Our approach mproves on the prevous lterature n multple ways. Frst, the VRI sample s a large sample of stock holders. Despte beng drawn from the account holders of a sngle company, the characterstcs of the sample are broadly representatve of the targeted populaton of households wth non-neglgble fnancal assets. Unlke most studes that focus on the extensve margn for stock holdngs, ths sample allows for meanngful nferences about the ntensve margn of portfolo choce. Second, the VRI survey ncludes batteres of questons that we purposely desgned to produce estmates of preference and belef parameters that should help to explan the crosssectonal dstrbuton of portfolo choces. These survey questons yeld quanttatve estmates of ndvdual-level moments of subjectve returns dstrbuton and of ndvdual-level values of preference parameters. These estmates can then be related to portfolo decsons n ways that are quanttatvely nterpretable relatve to benchmark economc models. Thrd, the desgn of the VRI allows careful consderaton of response errors along a varety of dmensons. These nclude errors n measurng stock shares n both survey and admnstratve account data and errors n elctng preferences and expectatons from survey responses. Few studes take survey measurement error nto account n ther estmaton procedure. Yet there s strong evdence that survey measures of preferences and belefs are subject to substantal response error leadng to potentally severe attenuaton bas (Kmball, Sahm and 5

8 Shapro, 2008; Kezd and Wlls, 2011). These lmtatons may be n part responsble for why estmated assocatons n the lterature are so much smaller than what fnance theores would predct. These features a large, broadly representatve sample of stockholders together wth quanttatve measurements of the potental sources of heterogenety n stockholdng make the VRI a unque platform for understandng why dfferent households make dfferent portfolo choces. II. VRI Data and Stock Share Measurement A. VRI sample and wealth measurement The Vanguard Research Intatve (VRI) conssts of lnked survey and admnstratve data of account holders who have non-neglgble fnancal assets at Vanguard, are at least 55 years old, and use the nternet to access ther Vanguard accounts. Ths last requrement s necessary because the VRI s an nternet survey. The VRI s an ndvdual level survey, but t ncludes questons about household-level wealth and ncome as well as questons about spouses or partners demographcs and labor supply. The survey oversampled older account holders and sngles. The VRI draws respondents from two lnes of busness ndvdual account holders and employer-sponsored account holders. The employer-sponsored are enrolled at Vanguard through 401(k) or smlar defned-contrbuton accounts. Whle both ndvdual and employer-sponsored account holders are selected va ownershp of a Vanguard account, the selecton nto ndvdual and employer-sponsored accounts s presumably qute dfferent. We wll present separate estmates to get a sense of whether selecton matters for our results. See Appendx A for more detals on the VRI surveys and sample. 6

9 There are features of the VRI that make t well-suted for ths analyss. Frst, t has a new approach to wealth and portfolo measurement. Second, t provdes a larger sample of respondents wth relevant levels of assets and stock holdng compared to leadng surveys such as the Health and Retrement Study (HRS) and the Survey of Consumer Fnances (SCF). Thrd, demographcs of the VRI are nonetheless comparable to those wth smlar asset levels n the HRS and SCF. The VRI survey measure of wealth s based on a comprehensve account-by-account approach. The survey frst asked about types of accounts respondents have (e.g. IRA, checkng, money market funds) and the number each type of account held by the respondent or her spouse. For each account they ndcated ownng, the respondents were asked to provde the balance as well as the share of stock-market assets. When fnshed wth all accounts, respondents were presented a summary table consoldatng ther responses and were nvted to make correctons, f any. Measurng wealth and stock shares account by account matches the way respondents keep track of ther own wealth, and t does not requre them to sum balances across accounts to provde total fgures for asset categores that are famlar to economsts but less so to survey respondents. In contrast, the HRS and SCF other leadng surveys wth state of the art wealth measurement use account-by-account approaches but only for selected sets of account types. Item non-response n the wealth secton of the VRI affects less than 1 percent of the observatons. Table 1 compares the VRI sample to the HRS and SCF. The HRS and SCF are natonally representatve samples (of those above age 50 n the case of the HRS). Table 1 compares the VRI sample to the subsample of the HRS and SCF after mposng restrctons smlar to VRI elgblty: beng at least 55 years old, havng access to nternet at home, and havng at least 7

10 $10,000 fnancal wealth. The number of respondents n Survey 1 s substantally larger than the VRI-elgble subsample of the HRS and the SCF. The dfference n the number of respondents n stock-holdng households s even larger: the comparable samples have slghtly over 1,000 stockholdng households n the SCF and slghtly over 2,000 n the HRS; the entre VRI sample has more than 8,000 stock holders and the sample used n our analyss (those who completed all the frst three VRI surveys, see below) has more than 4,000. Table 1. Sample Means: VRI, HRS, and SCF VRI HRS SCF Entre Analyss sample sample VRI-elgble subsample Household-level varables Number of households 8,950 4,414 3,684 1,275 Number of stockholdng households 8,636 4,323 2,356 1,216 Average fnancal wealth ($ 000) 1,207 1, Average total wealth ($ 000) 1,589 1, ,764 Average stock share among stockholders Respondent-level varables Marred Male Age Less than college degree College degree but not more Post-college degree Retred Notes: For the HRS and SCF, the VRI-elgble subsamples are those who are not younger than 55, have access to the nternet at home, and have at least $10,000 n non-transactonal accounts. Respondent-level varables for the HRS refer to the fnancal respondents; for the SCF they refer to the household heads. Varables n the VRI measured n 2013; HRS and SCF are from 2012 and 2013, respectvely. Respondent-level varables are {0,1} bnary varables except for age. Summary statstcs of the wealth measures are shown n Table A1 n the Appendx. Table A2 n the Appendx shows the summary statstcs of the varables we use as controls n our analyss, together wth the defnton of those varables. For more detaled comparsons wth the HRS and SCF sample as well as for the effectveness of the account-by-account approach n producng unbased estmates of assets wth low response error, see Amerks, Capln, Lee, Shapro and Tonett (2014). The demographc composton of the VRI sample s broadly smlar to the parallel subsamples of the HRS and the SCF. Average total wealth and average fnancal wealth n the 8

11 VRI are close to correspondng estmates from the SCF; the HRS averages are lower. The average stock share n fnancal wealth among stock holders s very smlar n the VRI and the HRS; the SCF estmates are somewhat smaller. VRI respondents are slghtly less lkely to be marred, and they are somewhat older, more educated and more lkely to be retred. The dfferences n martal status, age and retrement are largely due to the fact that the VRI oversampled older ndvduals and sngles. 65 percent of the VRI sample s male, compared to 79 percent n the SCF and 56 percent n the HRS. Wthn households, men are overrepresented as respondents: account holders n the VRI, fnancal respondents n the HRS, and household heads n the SCF. B. Measurng stock shares Our analyss focuses on the share of stock-market-based assets n total fnancal wealth. Specfyng stock share n fnancal wealth s standard n the lterature. Alternatve measures may nclude housng wealth and human captal wealth n the denomnator. We nclude such wealth tems as control varables n the analyss and show that ther ncluson leads to very smlar results for the parameters of nterest. We also show that our man fndngs are robust to ncludng housng wealth as ether rsky or safe assets n the rsky asset share calculaton. The VRI asks ndvdual the share of stock held n each account. The stock share n fnancal wealth s the weghted average of the stock shares of the accounts. Respondents who dd not answer all of the account-by-account stock share questons were asked the overall stock share of ther fnancal portfolo. Nnety-fve percent of respondents answered all the account-byaccount stock share questons; the dstrbuton of stock share s very smlar across the two groups. 9

12 The VRI account data also allow us to calculate stock share usng the admnstratve records, but of course only for assets held at Vanguard. Appendx A compares the survey and admnstratve measures of the stock share. Appendx B also presents the emprcal results usng the admnstratve stock share as the dependent varable. Indvduals mght hold stocks dsproportonately at one provder or another, so there s no reason to expect portfolo theores to obtan for holdng at each provder. Yet, despte the fact that ndvduals at sample tend to have a hgher stock share at Vanguard, the results usng the admnstratve share are qute smlar to those usng the survey share. III. Measurng Preferences and Expectatons A. Measurng rsk tolerance Survey 2 of the VRI ncluded Strategc Survey Questons (SSQs) that ask respondents to make choces between hypothetcal fnancal products under hypothetcal stuatons. In ths paper, we use the VRI s rsk tolerance questons that pose gambles over consumpton. These are based on the questons used n Barsky, Juster, Kmball, and Shapro (1997) and Kmball, Sahm, and Shapro (2008) that are mplemented n the HRS. The VRI rsk tolerance questons are refned relatve to those n the HRS to be more specfc about the economc settng and to ask about consumpton rather than ncome gambles. The HRS uses lfetme ncome rather than consumpton because when the HRS questons were crafted, there was a concern that consumpton was too abstract a concept to mplement n the survey. The VRI approach frames the queston n terms of consumpton, the economcally more-relevant flow. Earler successes wth the SSQ approach suggest that t s possble to elct the more precsely model-relevant 10

13 measure usng a survey nstrument that has both more detaled scenaros and comprehenson tests. The VRI SSQs ask about preference between the followng two optons: Havng a certan level of consumpton; Havng double that level of consumpton or havng t fall by x% wth a chance. The queston then alters the downsde rsk x and repeats the queston n order to partton respondents nto rsk tolerance groups. There are some other dfferences between the VRI and HRS questons. In the VRI, the same queston s asked wth two dfferent levels of guaranteed consumpton for the safe opton. Havng two consumpton treatments n ths survey provdes a test-retest measurement that s nstrumental for separatng true preference heterogenety from survey response error. In contrast, Kmball, Sahm, and Shapro (2008) reles on varaton across multple survey waves, whch assumes tme-nvarant preferences, an assumpton not needed n ths paper. Addtonally, usng two dfferent levels of guaranteed consumpton allows dentfcaton of non-homothetc preferences. The VRI questons are more specfc about the hypothetcal stuatons to better assure that structural preference parameter estmates are ndependent from respondents economc, health, and famly condtons. Table A3 n Appendx A gves the exact wordng of the rsk tolerance queston n the VRI. The queston s asked for two dfferent levels of rskless consumpton, $100K and $50K per year, and downsde rsks of 1/10, 1/5, 1/3, 1/2, and 3/4. Table 2 shows the dstrbuton of the answers to the two questons. Most respondents have low tolerance for rsk. About half of the respondents chose the frst two categores, ndcatng that they would not accept a rsk of more than 20% drop n ther consumpton to take a chance to double ther consumpton. Only a small fracton chose the last two categores wth a rsk of more than a 50% drop. Overall, the 11

14 dstrbuton s smlar to the dstrbuton of the answers to a smlar queston n the HRS except that the fracton of respondents n the two extreme categores (0-10% and %) s slghtly lower n the VRI (see Kmball, Sahm, and Shapro, 2008 for the HRS). The table also shows that more respondents fall nto the lower rsk categores when rskless consumpton s $50,000 nstead of $100,000. We handle ths ncrease n relatve rsk tolerance by postng a utlty functon wth a subsstence level of consumpton. Followng Barsky, Juster, Kmball, and Shapro (1997) and Kmball, Sahm and Shapro (2008), we use the multple responses to dentfy the heterogenety of the preference parameter and survey response errors. Estmaton of a cardnal rsk tolerance parameter requres specfyng a utlty functon. We assume that the flow utlty functon s a generalzaton of CRRA wth a subsstence level of consumpton ( c + ) u () c = 1 1/ 1 1/, (1) where subscrpt denotes heterogenety across ndvduals, c s consumpton, the negatve of κ s the subsstence level of consumpton, assumed to be the same for all ndvduals, and θ s the rsk tolerance parameter. To allow for heterogenety n both θ and κ and to allow for survey response errors, we would need at least three responses for each respondent; the VRI asked only two. Therefore, we allow for heterogenety only n θ. We do allow κ to be a functon of observed covarates n specfcatons usng those covarates (see Secton IIIC). Appendx Tables B3, B4 and B10 show that the man results are almost the same when we use a CRRA utlty functon (.e., settng κ=0) as n Kmball, Sahm and Shapro (2008), except that the estmated rsk tolerance parameter s lower. 12

15 Table 2. Rsk Tolerance: Dstrbuton of Responses to SSQ Response Downsde rsk Percent of answers category accepted rejected rskless consumpton rskless consumpton $100K $50K 1 none 1/ /10 1/ /5 1/ /3 1/ /2 3/ /4 none 2 1 Total Notes: Choce between two plans. Plan A guarantees $c consumpton next year. Plan B: doubles $c wth 50% chance and cuts t by a fracton x wth 50% chance. $c=100k or 50K, shown n the two columns; the x values are shown n second and thrd columns observatons. For ths utlty functon, relatve rsk tolerance (RRT) s c + RRT =, c where the rsk tolerance parameter s relatve rsk tolerance n the = 0 case. Emprcally, the coeffcent of rsk tolerance s very close to what s mpled by, as the level of average wealth (Table 1) and annual ncome before retrement ($90,000) are substantally larger than our estmate of. At levels of consumpton mpled by the average before-retrement ncome, the dfference s less than 20%, and ts varaton between ndvduals s small. See Appendx Fgure B1 for the relatonshp of relatve rsk tolerance and θ as a functon of consumpton. To parameterze the heterogenety of the rsk tolerance parameter, we assume that the parameter s dstrbuted lognormally n the populaton accordng to 2 log( ) u, u ~ N(0, u ). = + (2) We model the measurement error as a log addtve term to the parameter, such that 13

16 log( ) = log( ) + for j = 1, 2 j j 2 j ~ N(0, j ) (3) where s the true rsk tolerance parameter for ndvdual, j s measurement error, and j s the error-rdden rsk tolerance parameter that provdes the bass for ndvdual s response to the j th queston (c=$100,000 for j=1 and c=$50,000 for j=2). Thus, n answerng queston j gven the level of resource c and rsk x that are assocated wth the rsky gamble, the respondent compares 1 1/ j 1 1/ j 1 1/ j ( c + ) (2 c + ) ((1 x) c + ) vs / 1 1/ 1 1/ j j j (4) to determne whether to accept the rsky gamble or not. Equaton (4) translates each response category n Table 2 nto an nterval of j. Ths approach generalzes that of Kmball, Sahm, and Shapro (2008) by allowng for non-homothetc preferences. (Amerks, Brggs, Capln, Shapro, and Tonett (2018) also explots multple responses wthn survey to dentfy ndvdual level preference parameters for relatng to decsons about long-term care and bequest. Amerks, Brggs, Capln, Shapro, and Tonett (2017) estmates the same parameters for a representatve agent usng a method-of-moments approach.) We carred out the estmaton procedure jontly for rsk tolerance and stock market expectatons, so wll defer dscusson of estmaton untl Secton IIIC below. B. Measurng belefs about stock returns Survey 3 of the VRI asked about belefs about the one-year return of the U.S. stock market, represented by a stock market ndex such as the Dow Jones Industral Average (DJIA). Respondents had to answer three questons: the expected return on the stock market n the 12 months followng the ntervew (m); the percent chance that the stock market wll be hgher n 12 14

17 months followng the ntervew (p0) and the percent chance that t wll be at least 20% hgher (p20). The exact wordng of the questons s n Table A4 n the Appendx. (Brune de Brun, Mansk, Topa, and van der Klaauw (2011) and Armanter, Brune de Brun, Potter, Topa, van der Klaauw, and Zafar (2013) examne the relablty of the percent chance questons for nflaton as well as how they relate to questons about pont expectatons of nflaton.) Answers to the expected value questons were constraned to be ntegers. Answers to the percent chance questons were constraned to be 5 pont ncrements between 0 and 15 and between 85 and 100, and they were constraned to be 10 pont ncrements between 15 and 85 (the set {0,5,10,15,25,35,45,55,65,75,85,90,95,100} ). Answers to percent chance questons tend to be rounded to the nearest ten when they are not constraned, wth an especally large fracton answerng 50 percent (Hurd, 2009). The VRI survey nstrument requres people to round to other values; n partcular, they cannot gve 50 percent probabltes. It also allows for fner roundng at the tals, n lne wth the fndngs of Mansk and Molnar (2010). The survey also requres that p20 p0. Respondents whose ntal answer to p20 volated ths constrant are remnded of the constrant by the survey software and asked for a new reply to ether p0 or p20 (or both). The survey mposes no constrants on m versus p0 and p20. (A randomly selected half of the respondents receved the m queston frst, followed by p0 and p20, whle the other half receved p0 and p20 frst, followed by m. The dstrbuton of the responses s slghtly dfferent across the two sequences. Nevertheless, we fnd smlar relatonshps between the belef measures and portfolo choce from the two sequence groups.) Table 3 shows the summary statstcs of the answers to the questons about the dstrbuton of stock market returns. The survey responses for expected returns (m) are dstrbuted around the hstorcal average of 4 to 7 percent dependng on sample perod, and ther 15

18 dsperson s moderate. In contrast, most answers to the probablty questons are lower than the hstorcal probabltes, and they have substantal heterogenety. (Indvduals may use dfferent sample wndows for nferrng expected returns, see Malmender and Nagel, The table shows some dfferent wndows for realzed returns. Average returns are qute varable owng to the well-known problem of estmatng the expected return on the market.) A non-neglgble fracton of the respondents gave a postve number to the expected return queston (m) and a less than 50 percent chance answer to the probablty of a postve return (p0). Taken together these answer patterns are consstent wth many ndvduals mplctly applyng a postve threshold when they answer the p0 queston (by thnkng that the stock market goes up only f t goes up by at least some postve amount). Glaser, Langer, Reynders and Weber (2007) document a smlar pattern when they compare stock market expectatons elcted n terms of returns versus prces. They label the phenomenon as framng effect, and our explanaton can be vewed as a source of such a framng effect. Note that, although skewed returns could explan the phenomenon we observe, t s an unlkely explanaton. The combnaton m>0 and p0<0.5 would correspond to long postve tals, mplyng mean above the medan and nfrequent large gans. Ths skewedness s the opposte of what one would expect from a black swan theory of nfrequent stock market crashes. 16

19 Table 3. Stock Market Returns: Survey Responses versus Hstorcal Statstcs Survey answers Hstorcal statstcs Mean 25 th 75 Medan pctle pctle m p p Notes: m s expected one-year ahead returns of the stock market ndex DJIA; p0 s the probablty that the DJIA would be hgher a year from the date of the ntervew; p20 s the probablty that t would be hgher by at least 20%. Hstorcal statstcs computed from yearly relatve returns of the Dow Jones Industral Average (year on year changes dvded by base year value, frst days of July n each year), deflated usng the PCE chan prce ndex (avalable begnnng n 1959). Hstorcal average values shown for m; the fracton of years when postve or greater than 0.2 are shown for p0 and p observatons. In order to use our data more effcently and n a way that s more nformatve from a theoretcal pont of vew we map the three survey responses, m, p0, and p20 nto a perceved returns dstrbuton. The procedure closely parallels that for the rsk tolerance questons: the survey responses are based on ndvdual belefs drawn from a dstrbuton plus survey response error. We assume that ndvdual beleves that yearly returns follow a lognormal dstrbuton wth ndvdual-specfc mean and standard devaton of log stock returns of μ and σ. Smlar to how we handle the cross-sectonal dstrbuton of the rsk tolerance parameter, these parameters are drawn across ndvduals as 2 = + u u 0 u u u, ~ N, 2 = + u u 0. u. (5) Indvduals answer the survey questons m, p0 and p20 based on ther belefs, but ther answers contan survey nose, that s, measurement error specfc to the survey stuaton. Usng the structure of the survey questons on expected returns and the two ponts of the probablty dstrbuton, applyng the assumpton of lognormal returns, and addng survey response error yelds 2 m = + m, m ~ N(0, m) (6) 17

20 where m, p, and 0 p = + N (7) 2 0 ( 0), 0 ~ (, p) 0.2 p = + N (8) 2 20 ( 20 ), 20 ~ (0, p) p are the error-rdden latent varables that determne survey responses. 20 Survey error s assumed to be ndependent across the three answers, wth mean zero except for p0 where ts mean s, whch allows for the documented gaps between m and p0. An nterpretaton of s that, on average, respondents answer the queston about postve returns (p0) as f they had some postve threshold n mnd nstead of zero ( ( ) survey responses are transformed versons of latent varables m,, 0 ). The p, and 0 / p because of 20 roundng. Recall that the VRI probablty scale s for rounded responses. Smlarly, as dscussed above, the rsk tolerance questons yeld dscrete responses. In the followng subsecton we dscuss how our estmaton procedure handles ths ssue. C. Jont estmaton of heterogenety n stock market expectatons and rsk tolerance Gven the models of heterogenety n preferences and belefs (equatons (2) and (5)) and the structural nterpretaton of the survey questons together wth the addtve survey response errors ((3), (4), (6), (7) and (8)), we can now move to estmaton of the model. The parameters to be estmated are {,,,,,,,,,,, }. We allow for,, u u u 1 2 m p, and to vary wth covarates. Addtonally, we allow the belefs about returns to be correlated wth rsk preference, so the covarates of and nclude the latent. Note that the varables survey response m, p, and 0 m s a rounded verson of p n (6), (7), and (8) are before roundng. Actual 20 m as m s restrcted to take an nteger value. 18

21 Survey responses p and 0 p are to take a value from the set 20 {0,5,10,15,25,35,,75,85,90,95,100}, we assume that p and 0 p are rounded to the closest 20 values allowed for each response. Also note that the survey does not allow for p to be larger 20 than p. Hence when we observe 0 p = 20 p 0, we consder the possblty that the survey response error actually generated p20 p but after mposng the constrant we observe the 0 equalty n the actual responses. Together wth nterval responses, these formulae tell the range of survey response error terms that generate the responses of ndvdual that we observe, gven,, and. Let Z be the set of responses from respondent to the fve questons used for the low hgh estmaton and [ k (,, Z), k (,, Z)] (for k =1, 2, m, p0,and p 20) be the range of survey responses error term that s consstent wth the observed response for each queston gven the latent preference and belef parameters, calculated from (3), (4), and (6)-(8). Then the ndvdual lkelhood functon for respondent s calculated as: hgh low k (,, Z ) k (,, Z) L( Z) = [ ( ) ( )] k= 1, 2, m, k (,, Z ) (,, Z ) hgh low [ ( 0 ) ( 0 )] 0 0 log ( ) f (, ) dd d, u k (9) where () and () are the PDF and CDF of the standard unvarate normal dstrbuton and f (, ) s the PDF of a bvarate normal dstrbuton wth the mean vector + and the + 19

22 covarance matrx 2 u u u 2. (The specfcaton. u + means that estmates of + and are for =0. However, the estmated parameters and are small n magntude so that ths affects only the thrd dgts of the and estmates throughout the estmated dstrbuton of. Also note that, for techncal purposes, we left-truncate the dstrbuton of at zero. Under the estmated parameters the chance of beng less than zero n the nontruncated dstrbuton s essentally zero, so ths s an nnocuous assumpton. and capture potental dependence of the belef parameters on the preference parameter.) Then the overall lkelhood functon s obtaned as: L( Z) = L ( Z ). We calculate L( Z ) usng the Gaussan quadrature approxmaton. See Appendx C for the detaled algorthm. Table 4. Dstrbuton of Preferences and Belefs Mean Standard devaton 25 th pctle Medan 75 th pctle Preferences Rsk tolerance parameter Subsstence consumpton κ 17,000 Belefs Mean of return μ Standard devaton of return σ Notes: Statstcs are calculated from the estmated parameters n Table B1; see the notes to Table B1 for more detal. The summary statstcs n ths table are from estmates wthout covarates. The estmaton model constrans to be constant across ndvduals. Appendx Table B2 reports the estmates of the statstcal model wth covarates. Table 4 shows key estmated statstcs of the dstrbuton of preferences and belefs based on the estmated statstcal model of preferences, belefs, and response error. Table B1 n the appendx shows the estmates of the underlyng parameters of the model. The subsstence level 20

23 of consumpton ( κ) s estmated to be $17,000. The negatve value of κ generates decreasng relatve rsk averson as n the basc/luxury good model of Wachter and Yogo (2010). The desgn of the SSQ does not allow heterogenety n κ to be readly dentfed, although t tghtly dentfes ts mean. The estmated mean of the rsk tolerance parameter (θ) mples low rsk tolerance on average. A respondent wth the mean level of θ and κ has relatve rsk tolerance 0.34 (relatve rsk averson 2.9) when the consumpton level s $100,000. In terms of the SSQ queston, she would be ndfferent between a fxed consumpton of $100,000 and the gamble of doublng that consumpton and losng 20 percent. There s a consderable heterogenety n rsk tolerance. At the 25th percentle of rsk tolerance parameter, the pont of ndfference s the downsde rsk of losng 13 percent; at the 75th percentle the pont of ndfference s the downsde rsk of losng 29 percent. These numbers ndcate hgher levels of rsk tolerance than n a representatve sample of Amercans older than 50 years of age. Kmball, Sahm and Shapro (2008) estmate the correspondng rsk tolerance percentles (25th, 50th and 75th) to mply ndfference to 7, 12 and 20 percent of downsde rsk, respectvely. Belefs about mean stock returns are n lne wth hstorcal mean returns, on average. Belefs about standard devaton are slghtly lower than the hstorcal value of Heterogenety n perceved mean returns (μ) s substantal, wth the lowest 25 percent belevng expected returns to be 2 percent or less and the top 25 percent belevng 10 percent or more. At the same tme, estmated heterogenety n the perceved standard devaton of stock returns (σ) s small, perhaps because t s easer for people to estmate the second moment of the returns dstrbuton than the frst moment, as ponted out by Merton (1980). Accordng to our estmates heterogenety n preferences and belefs are weakly related. More rsk tolerant respondents beleve that stock returns are slghtly hgher, but we don t fnd 21

24 assocaton of rsk tolerance and belefs about the standard devaton of returns. Belefs about the mean and the standard devaton of returns are weakly postvely correlated. Preferences and belefs are sgnfcantly related to observable rght hand sde varables n our sample (Table B2 n the Appendx). However, when nterpretng these assocatons, one has to keep n mnd that the VRI sample s selected on wealth and stock ownershp. For example, sample selecton may explan the negatve correlaton of wealth and stock market expectatons. Almost all households n the VRI sample have nonzero stockholdng. Wth fxed costs of stock market partcpaton wealth should matter at the extensve margn on top of expectatons. As a result, we expect wealther stockholders to have lower expected returns than less wealthy stockholders. Based on the estmated dstrbuton summarzed n Table 4, 17 percent of the populaton expects negatve stock returns. As we wll see, ths part of the populaton holds less stock than on average, but stll has substantal stock market exposure. Symmetrcally, 17 percent expect returns to be larger than 12 percent, rates of return that should make people hold the vast majorty of ther wealth n stocks gven the dstrbuton of rsk and rsk preferences. Though ths part of the populaton holds more stock than on average, very hgh stock shares are uncommon. Taken together, these facts suggest that expectatons are correlated wth stock shares n an attenuated fashon, a fndng that our analyss wll verfy n the next secton. The Table 4 results take nto account substantal estmated survey nose. The parameters of the survey nose dstrbutons are presented n Appendx Table B1. To understand the magntude of nose, consder the dfferences n terms of the survey responses of ndvduals wth the estmated averages of latent preferences and belefs, one wthout measurement error and one wth a postve standard devaton unt shock of measurement error. A one standard devaton unt measurement error n the frst rsk tolerance SSQ would make the survey response mply a 22

25 pont of ndfference of a 38% drop of consumpton nstead of the 20% mpled by an error-free answer. A one standard devaton unt measurement error n the second rsk tolerance SSQ would make the response mply an ndfference pont of 27% nstead of 17%. One standard devaton unt measurement error n the response to the expected stock returns queston would result n a response of 14% nstead of 6%; one standard devaton unt measurement error n the stock market probablty answers would change p0 responses to 67% from 48% and p20 responses to 25% from 12%. The estmated bas of the measurement error n the p0 response (ψ) suggests that, on average, people thnk of postve gans only when they exceed 4 percent when answerng the p0 queston. Allowng for covarates (Appendx Table B2), ψ s estmated to be substantally less negatve among more educated and wealther people, ndcatng that ther threshold value s closer to the nomnal threshold zero. D. Estmatng ndvdual-specfc cardnal proxes of rsk tolerance and belefs In the prevous sectons, we show how to separately dentfy the true heterogenety n preferences and belefs and the survey response errors n the survey measures of them. In ths subsecton, we explan how we construct the ndvdual-specfc belef and preference parameters based on those estmates that are mmune from the standard effects of usng generated regressors. 1. Constructng ndvdual-specfc preference and belef parameters Usng the estmaton results we calculate ndvdual-specfc proxy varables ˆ, ˆ and ˆ. These proxes are the expected values of the correspondng latent varables: the ndvdualspecfc expected value and standard devaton of the dstrbuton of stock market returns perceved by the ndvdual (, ), and the ndvdual-specfc latent rsk tolerance parameter ( ). They are expected values condtonal on the ndvdual's responses to the survey questons 23

26 on stock market returns (,, ) m p p and to the SSQ s wth the two hypothetcal gambles. To 0 20 get these expected value of the latent ndvdual-specfc parameters condtonal on the survey response and the statstcal model, there are two steps. Frst, the dstrbuton of the latent varables condtonal on the observed responses can be obtaned from the lkelhood functon usng Bayes theorem. Second, ntegratng out ths functon yelds the ndvdual-specfc proxy varables ( ˆ { ˆ, ˆ, ˆ }) as the condtonal expectatons of the latent varables gven the observed survey responses. To be more specfc, they are calculated as: ˆ ˆ 1 E[, Z ] = L( ˆ Z ) 0 0 (,, Z ) (,, Z ) [ ( ) ( )] hgh low k k k = 1, 2, m,20 ˆ ˆ k k hgh low ˆ 0 (,, Z ) 0 (,, Z ) ˆ [ ( ) ( )] ˆ ˆ 0 log ˆ ( ) f (, ) dd d, ˆ u 0 (10) where L( ˆ Z ) s the ndvdual lkelhood functon (calculated from equaton (9)) evaluated under the estmated parameters. We use the same numercal approxmaton used n the estmaton for ths calculaton. See Appendx C for detals. 2. Usng ndvdual-specfc preference and belef parameters n regressons Our am s to use the survey-based estmates of ndvdual-specfc parameters to explan heterogenety n portfolo choce. In contrast wth classcal measurement error that s uncorrelated wth true values but correlated wth measured values, the error n the proxy varables s the error of optmal predcton, whch s uncorrelated wth measured values. Thus, when entered on the rght-hand-sde of lnear regressons, ths type of non-classcal measurement 24

27 error does not nduce attenuaton bas n the regresson coeffcents of these proxy varables (see Kmball, Sahm and Shapro, 2008). When the regressons nclude other covarates as well the OLS estmates are unbased f the proxes are estmated condtonal on those covarates, too. We therefore estmate two sets of proxes. The frst set s condtonal on the survey answers to the rsk tolerance and the stock market belef questons only. The second set s condtonal on other covarates as well. We use the second set of proxy estmates as rght-hand-sde varables n regressons that also nclude those covarates. In the next secton, we present such regressons to explan portfolo behavor based on our estmates of preferences and belefs. IV. Explanng Heterogenety n Portfolo Choce A. Stock share and answers to survey questons Before turnng to the regressons based on our structural estmates of the latent preferences and belefs, we nvestgate the relatonshp between the stock share of household portfolos and the raw survey responses. Fgure 1 shows non-parametrc regressons of the stock share n total fnancal assets on the survey answers to expected stock market returns (m), the average between the probablty that the stock market would go up and that of an ncrease of 20 percent or more (( p0 p2 0 ) / 2) +, the dfference between those two (p0 p20), and the answer to the rsk tolerance queston wth ncome level $100,000. (Fgure B2 n Appendx B shows the analogous non-parametrc regresson results on p0 and p20 separately.) 25

28 Fgure 1. Stock Share versus Raw Survey Responses A B C D The results ndcate a postve relatonshp between the stock share of household portfolos and expected stock market returns (m) and the mean of the two probablty responses (( p0 p2 0 ) / 2) +. The stock share s also postvely related to the dfference between the responses to the probablty questons (p0 p20), suggestng a negatve relatonshp wth perceved rsk of stock returns. Fnally, the stock share s monotoncally postvely related to the answers to the rsk tolerance queston except for the last categores that has relatvely few responses, suggestng a monotonc postve relatonshp wth rsk tolerance. Hence, the 26

29 relatonshp between the raw survey responses and the stock share has the drecton benchmark theores of portfolo choce would suggest. We also estmate lnear regressons wth the survey measure and the admnstratve measure of stock share as alternatve left-hand-sde varables and the same rght-hand-sde varables entered wth and wthout the control varables that nclude detaled measures of demographcs, educaton, employment, ncome, wealth, as well as background rsks of long-term care and longevty. The results are ncluded n Tables B5 and B6 n the Appendx. The results mply smlar relatonshps of stock share wth the survey answers wth or wthout the control varables. The magntudes of the assocatons are dffcult to nterpret because not all measures have a cardnal nterpretaton and because of the presence of survey nose. These problems are addressed n the next secton. B. Stock share and cardnal proxes of expectatons and rsk tolerance Our more structural analyss has two goals. Frst, t relates the stock share of household portfolos to cross-sectonal heterogenety n preferences and expectatons n a way that s related to portfolo choce theory thus makng magntudes easer to nterpret. Second, t ams at ncorporatng survey nose n the estmaton thus reducng ts effect on the estmated magntudes. Ths s a structural analyss n the sense that t makes use of addtonal assumptons n order to relate stock shares to heterogenety n latent preferences and expectatons. The analyss s stll reduced form n the sense that t ams at uncoverng assocatons wthout clams for causalty. Nonetheless, snce the explanatory varables are proxes that have cardnal nterpretatons relevant for economc theores, they potentally convey much more nformaton than the relatonshp of raw survey responses to economc outcomes. Start from a general functon of the soluton of optmal stock share 27

30 (,,, ;, ) s = s x u (11) * * where μ and σ are the belefs of person about the mean and the standard devaton of one-yearahead stock returns, θ s the rsk tolerance parameter, x s a vector of wealth, demographc varables and other rsk factors that are measured n our data, and u combnes all unobservables. We assume that unobservables are ndependent of observables. The relatve devaton of s * around ts mean value s related to relatve devatons of the other varables around ther mean values, holdng values of x constant by * * s s ' * x + u. (12) s The coeffcents approxmate the frst dervatves of the functon around the mean values, wth =, * 1 s / = and * 2 s / =, where the tlde denote relatve dfferences from * 3 s / mean values. Ths approxmaton s a way of log-lnearzng the functon that allows observatons wth nonpostve values of some of the varables, whch s relevant for μ n our case. We lnearze about the rsk tolerance parameter rather than relatve rsk tolerance to avod the ambguty that relatve rsk tolerance depends on the level of consumpton. We estmate (12) usng the observed stock share s to approxmate the target stock share * s and the ndvdual proxes ˆ, ˆ, and ˆ approxmatng the latent varables,, and as descrbed earler. We estmate the equaton by OLS both wth and wthout covarates. (We do not use a Tobt-type procedure to account for the truncaton at 0 and 1 because there are very few observatons (less than 2 percent of the sample) at these boundares.) When we control for covarates n the stock share equaton, we enter the structural parameters that were estmated condtonal on the same covarates. Kmball, Sahm, and Shapro (2008) show that t s necessary 28