Heterogeneity in Expectations, Risk Tolerance, and Household Stock Shares

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1 Heterogenety n Expectatons, Rsk Tolerance, and Household Stock Shares John Amerks Vanguard Group Gábor Kézd Central European Unversty Mnjoon Lee Unversty of Mchgan Matthew D. Shapro Unversty of Mchgan and NBER Revsed June 26, 2017 APPENDICES

2 Appendx A. Addtonal Tables and Fgures Table A1. Dstrbuton of wealth n the VRI data (N=4414) Mean Std p10 p25 Medan p75 p90 Fnancal wealth 1,147,525 1,516, , , ,750 1,403,843 2,467,899 Home stock 360, ,045 31, , , ,000 1,060,000 Table A2. Summary statstcs of the control varables (N=4414) Mean Standard devaton Sngle male 0.14 Female n couple 0.17 Sngle female 0.18 Age Age squared In the employer-sponsored sample 0.21 College degree 0.33 MBA 0.07 PhD 0.06 Other hgher degree 0.28 Log(wealth) Log(home equty) Zero home equty 0.07 Retred 0.60 Log(Wage) Log(Annuty Income) Expected Log(Annuty Income) Subjectve probablty of needng long-term care Subjectve probablty of survval to target age Notes. Log varables are set to zero f the levels of the varables are zero. Zero home equty equals 1 (0) f home equty s zero (postve). Annuty ncome s the sum of Socal Securty ncome, defned beneft penson ncome and mmedate annuty ncome, for retred households. It s set to zero for non-retred households. Expected annuty ncome s the sum of expected values of Socal Securty ncome, defned beneft penson ncome and mmedate annuty ncome, for non-retred households. It s set to zero for retred households. Subjectve probablty of needng long-term care s the subjectve probablty chance that the respondent would need long-term care servce at least for one year durng her remanng lfe. The target age n subjectve probablty of survval queston s set to 75 f the respondent s younger than 70, to 85 f the respondent s younger than 80, and to 95 f the respondent s younger than 90.

3 Table A3. The Rsk Tolerance Strategc Survey Questons n VRI survey 2 Set up Suppose you are 80 years old. Suppose, further, that for the next year: You lve alone, rent your home, and pay all your own blls. You are n good health and wll reman n good health. You wll have no medcal blls or other unexpected expenses. You do not work. Hypothetcal Plan A guarantees that you wll have $W for spendng next year. fnancal products Plan B wll possbly provde you wth more money, but s less certan. There s a 50% chance that Plan B would double your money, leavng you wth $2W, and a 50% chance that t would cut t by x%, leavng you wth $ ( x) W. Rules You have no other assets or ncome, and so the only money you have avalable for all your spendng next year s from ether Plan A or Plan B. Any money that s not spent at the end of next year cannot be saved for the future. You cannot gve any money away or leave t as a bequest. If you need anythng next year, you have to pay for t. No one else can buy anythng for you. At the end of next year you wll be offered the same choce wth another $W for followng year. Parameters W =100,000 and 50,000. asked Queston Would you choose Plan A or Plan B? 2

4 Table A4: The stock market expectaton questons n VRI survey wave 3. Varable name Survey queston Queston Order p-m p0 What do you thnk s the percent chance that the stock market wll be hgher n twelve months than t s today? Thnk of a stock market ndex such as the Dow Jones Industral Average and do not adjust for nflaton. p20 And what do you thnk s the percent chance that t wll be at least 20% hgher n twelve months than t s today? [If answer s greater than the p0 answer: "Please enter a response that s less than or equal to you prevous response or change your prevous response. ] m Instead of probabltes, we are now nterested n your expectaton. By what percentage do you expect the stock market to ncrease or decrease n the next twelve months? Please enter a postve number for ncrease and negatve number for decrease. Queston order m-p m By what percentage do you expect the stock market to ncrease or decrease n the next twelve months? Thnk of a stock market ndex such as the Dow Jones Industral Average and do not adjust for nflaton. Please enter a postve number for ncrease and negatve number for decrease. p0 And what do you thnk s the percent chance that the stock market wll be hgher n twelve months than t s today? p20 What do you thnk s the percent chance that t wll be at least 20% hgher n twelve months than t s today? [If answer s greater than the p0 answer: "Please enter a response that s less than or equal to you prevous response or change your prevous response. ] Note: The queston orders are randomzed n the survey nstrument. The dstrbutons of responses are slghtly dfferent dependng on whch sequence s used. 3

5 Table A5. Detaled results of the structural estmaton model wthout covarates. (N=4,414) Preference Belefs Bas n p0 θ μ σ constant *** 0.055*** 0.118*** *** (0.027) (0.002) (0.002) (0.017) Heterogenety u 0.704*** 0.063*** 0.032*** n.a. (0.011) (0.001) (0.001) n.a. Correlaton across latent varables θ 0.011** Measurement error e 1 e 2 (0.003) (0.002) 0.062** (0.021) 0.812*** (0.015) 0.544*** (0.016) em 0.079*** (0.001) ep 0.487*** (0.008) Log-lkelhood Notes. The thrd lne reports how the latent rsk tolerance parameter affects means of the belef parameter dstrbutons. Statstcs reported n Table 4 are calculated based on these parameters, where the means of belef parameter dstrbutons are adjusted usng the mean of the rsk tolerance parameter multpled wth the numbers reported n the thrd row. Standard errors n parentheses. *, **, and *** mples sgnfcance at 5%, 1% and 0.1% level, respectvely. 4

6 Table A6. Detaled results of the structural estmaton model wth covarates. (N=4,414) Preference Belefs Bas n p0 θ μ σ Constant *** 0.071* 0.181*** (0.412) (0.031) (0.024) (0.833) Sngle male (0.042) (0.004) (0.003) (0.041) Female n couple *** *** (0.040) (0.004) (0.003) (0.039) Sngle female *** 0.011** *** (0.041) (0.004) (0.003) (0.039) Age *** (0.015) (0.000) (0.000) (0.023) Age sq (0.000) (0.000) (0.000) (0.000) Employer *** * *** sponsored (0.039) (0.003) (0.002) (0.037) College degree ** 0.007*** 0.294*** (0.035) (0.003) (0.002) (0.035) MBA *** (0.060) (0.006) (0.004) (0.064) PhD ** 0.023*** 0.465*** (0.061) (0.007) (0.006) (0.068) Other hgher degree 0.079* ** 0.014*** 0.354*** (0.038) (0.004) (0.002) (0.038) log(wealth) 0.037** *** 0.005*** 0.131*** (0.014) (0.001) (0.001) (0.014) log(home equty) (0.015) (0.001) (0.001) (0.015) No home equty (0.191) (0.014) (0.010) (0.179) Retred (0.410) (0.030) (0.024) (0.375) Log(Wage) (0.011) (0.001) (0.001) (0.011) Log(Annuty *** ** * Income) (0.029) (0.001) (0.001) (0.022) Expected Log(Annuty Income) (0.030) (0.003) (0.002) (0.029) LTC probablty *** *** (0.045) (0.004) (0.003) (0.044) 5

7 Longevty 0.191** 0.028*** probablty (0.063) (0.006) (0.004) (0.062) Heterogenety u 0.688*** 0.063*** 0.030*** n.a. (0.011) (0.001) (0.001) n.a. Correlaton across latent varables θ 0.011** Measurement error e 1 e 2 (0.004) (0.002) (0.024) 0.810*** (0.015) 0.557*** (0.016) em 0.078*** (0.001) ep 0.455*** (0.008) Log-lkelhood Note: Standard errors n parentheses. *, **, and *** mples sgnfcance at 5%, 1% and 0.1% level, respectvely. Reference categores are male n couple, ndvdual clent sample, not havng a college degree. See notes to Table A2 for detaled descrpton of the rght hand sde varables. 6

8 Table A7. Stock share regressons wth raw survey answers on the rght hand sde (wth m as a proxy for belefs of mean returns μ) Dependent varable: survey measure of stock share Dependent varable: admnstratve measure of stock share m 0.126** 0.153*** 0.180*** 0.192*** (0.037) (0.037) (0.038) (0.038) p0 -p *** 0.085*** 0.098*** 0.091*** (0.016) (0.016) (0.017) (0.017) SSQ1 cat= * (0.011) (0.011) (0.011) (0.011) SSQ1 cat= *** 0.035** 0.038** 0.028* (0.011) (0.011) (0.011) (0.011) SSQ1 cat= *** 0.044*** 0.057*** 0.049*** (0.013) (0.013) (0.013) (0.013) SSQ1 cat= *** 0.073*** 0.080*** 0.075*** (0.014) (0.014) (0.015) (0.015) SSQ1 cat= (0.031) (0.031) (0.032) (0.031) Sngle male (0.031) (0.012) Female n couple (0.012) (0.011) Sngle female (0.011) (0.012) Age (0.012) (0.009) Age sq (0.001) (0.000) Employer *** ** sponsored (0.011) (0.011) College degree * (0.010) (0.010) MBA (0.017) (0.018) PhD *** (0.017) (0.018) Other hgher degree ** (0.011) (0.011) log(wealth) 0.017*** (0.004) (0.004) log(home equty) (0.004) (0.004) 7

9 No home equty (0.054) (0.055) Retred * ** (0.116) (0.119) Log(Wage) (0.003) (0.003) Log(Annuty ** Income) (0.008) (0.008) Expected Log(Annuty ** Income) (0.008) (0.008) LTC probablty * ** (0.013) (0.013) Longevty 0.042* probablty (0.018) (0.019) Constant 0.371*** 1.028*** (0.111) (0.319) R Observatons Note: Standard errors n parentheses. *, **, and *** mples sgnfcance at 5%, 1% and 0.1% level, respectvely. Reference categores are male n couple, ndvdual clent sample, not havng a college degree. See notes to Table A2 for detaled descrpton of the rght hand sde varables. 8

10 Table A8. Stock share regressons wth raw survey answers on the rght hand sde (wth p p as a proxy for belefs of mean returns μ) 0, 20, /2 Dependent varable: survey measure of stock share Dependent varable: admnstratve measure of stock share (p0 +p20 )/ *** 0.118*** 0.097*** 0.089*** (0.024) (0.024) (0.025) (0.025) p0 -p *** 0.056** 0.075*** 0.074*** (0.018) (0.018) (0.018) (0.018) SSQ1 cat= * (0.011) (0.011) (0.011) (0.011) SSQ1 cat= *** 0.031** 0.033** 0.024* (0.011) (0.011) (0.011) (0.011) SSQ1 cat= *** 0.040** 0.052*** 0.045** (0.013) (0.013) (0.013) (0.013) SSQ1 cat= *** 0.069*** 0.076*** 0.071*** (0.014) (0.014) (0.015) (0.015) SSQ1 cat= (0.031) (0.031) (0.032) (0.032) Sngle male (0.031) (0.031) (0.012) Female n couple * (0.012) (0.011) Sngle female (0.011) (0.012) Age (0.012) (0.009) Age sq (0.001) (0.000) Employer *** *** sponsored (0.011) (0.011) College degree (0.010) (0.010) MBA (0.017) (0.018) PhD *** (0.017) (0.018) Other hgher degree * (0.011) (0.011) log(wealth) 0.017*** (0.004) (0.004) log(home equty) (0.004) (0.004) 9

11 No home equty (0.054) (0.055) Retred * ** (0.116) (0.120) Log(Wage) (0.003) (0.004) Log(Annuty ** Income) (0.008) (0.008) Expected Log(Annuty ** Income) (0.008) (0.008) LTC probablty * ** (0.013) (0.013) Longevty 0.039* probablty (0.018) (0.019) Constant 0.340** 1.010*** (0.111) (0.319) R Observatons Note: Standard errors n parentheses. *, **, and *** mples sgnfcance at 5%, 1% and 0.1% level, respectvely. Reference categores are male n couple, ndvdual clent sample, not havng a college degree. See notes to Table A2 for detaled descrpton of the rght hand sde varables. 10

12 Table A9. Stock share and preference and belef proxes. Detaled results correspondng to Table 5. Dependent varable: survey measure of stock share Dependent varable: admnstratve measure of stock share ˆ 0.058*** 0.055*** 0.052*** 0.048*** (0.010) (0.009) (0.008) (0.008) ˆ * * (0.046) (0.051) (0.040) (0.038) ˆ 0.034*** 0.033*** (0.009) (0.010) (0.010) (0.010) Sngle male (0.022) (0.019) Female n couple (0.021) (0.018) Sngle female (0.020) (0.019) Age * (0.017) (0.015) Age sq ** (0.000) (0.000) Employer *** *** sponsored (0.020) (0.018) College degree 0.048* 0.051** (0.021) (0.017) MBA 0.072** (0.027) (0.032) PhD *** (0.032) (0.025) Other hgher degree 0.053* 0.069*** (0.022) (0.019) log(wealth) 0.044*** (0.009) (0.007) log(home equty) * (0.009) (0.006) No home equty (0.118) (0.079) Retred ** (0.244) (0.196) Log(Wage) (0.005) (0.005) Log(Annuty * Income) (0.016) (0.015) Expected Log(Annuty *

13 Income) (0.019) (0.012) LTC probablty (0.028) (0.021) Longevty 0.084* probablty (0.033) (0.032) Constant (0.007) (0.649) (0.007) (0.519) R Observatons Standard errors n parentheses. *, **, and *** mples sgnfcance at 5%, 1% and 0.1% level, respectvely. Reference categores are male n couple, ndvdual clent sample, not havng a college degree. See notes to Table A2 for detaled descrpton of the rght hand sde varables. 12

14 Table A10. Stock Shares Versus Error-Rdden Cardnal Measures of Preferences and Belefs. Estmaton wthout takng care of measurement error n the cardnal proxes. LHS varable: survey measure of stock share n total fnancal wealth LHS varable: admnstratve measure of stock share n Vanguard (1) (2) (3) (4) m 0.017*** 0.020*** 0.020*** 0.021*** (0.004) (0.004) (0.004) (0.004) *** ** *** ** (0.006) (0.007) (0.006) (0.006) 0.021*** 0.020*** 0.013** 0.013** (0.005) (0.005) (0.004) (0.004) Sngle male (0.022) (0.019) Female n couple (0.020) (0.018) Sngle female (0.021) (0.019) Age * (0.016) (0.014) Age sq * (0.000) (0.000) Employer *** *** sponsored (0.019) (0.017) College degree 0.037* 0.040* (0.019) (0.017) MBA 0.066* (0.031) (0.028) PhD *** (0.032) (0.028) Other hgher degree ** (0.020) (0.018) log(wealth) 0.034*** (0.008) (0.007) log(home equty) (0.008) (0.007) No home equty (0.098) (0.088) Retred * ** (0.212) (0.190) Log(Wage) (0.005) (-0.005) Log(Annuty ** 13

15 Income) (0.015) (0.013) Expected ** Log(Annuty Income) (0.015) (0.013) LTC probablty (0.023) (0.021) Longevty 0.106** 0.065* probablty (0.033) (0.030) constant * (0.007) (0.565) (0.006) (0.507) R N Notes. In these regressons the cardnal proxes ˆ, ˆ, ˆ are replaced wth m,,, respectvely, where m s 0.2 / 1 ( p ) 1 ( p ) (the denomnator the raw answer to the expected stock returns queston, 0 20 replaced wth 0.2 f zero), and s the medan vale of the CRRA rsk tolerance parameter that corresponds to the answers to the frst set of the rsk tolerance questons ( set to zero). Standard errors n parentheses. *, **, and *** mples sgnfcance at 5%, 1% and 0.1% level, respectvely. Reference categores are male n couple, ndvdual clent sample, not havng a college degree. See notes to Table A2 for detaled descrpton of the rght hand sde varables. 14

16 Table A11. Stock Shares Versus Cardnal Proxes for Preferences and Belefs. Employersponsored subsample LHS varable: survey measure of stock share n total fnancal wealth LHS varable: admnstratve measure of stock share n Vanguard (1) (2) (3) (4) ˆ 0.067*** 0.062** 0.083*** 0.080*** (0.018) (0.019) (0.014) (0.015) ˆ (0.097) (0.107) (0.088) (0.087) 0.070** 0.068* (0.029) (0.031) (0.032) (0.040) constant *** (0.017) (1.896) (0.015) (1.836) control varables N Y N Y R N Notes. Employer-sponsored sample are those who only have 401(k) type accounts at Vanguard. Table A12. Stock Shares Versus Cardnal Proxes for Preferences and Belefs. Indvdual-clent subsample LHS varable: survey measure of stock share n total fnancal wealth LHS varable: admnstratve measure of stock share n Vanguard (1) (2) (3) (4) ˆ 0.059*** 0.055*** 0.041*** 0.036*** (0.010) (0.012) (0.011) (0.009) ˆ * (0.051) (0.055) (0.051) (0.046) 0.027** 0.024* (0.010) (0.010) (0.010) (0.011) constant 0.024** 1.099* (0.009) (0.525) (0.007) (0.570) control varables N Y N Y R N Notes. Indvdual-clent sample s the complement of Employer-sponsored sample. 15

17 Table A13. Stock Shares Versus Cardnal Proxes for Preferences and Belefs. Share of wealth at Vanguard at least 50 percent LHS varable: survey measure of stock share n total fnancal wealth LHS varable: admnstratve measure of stock share n Vanguard (1) (2) (3) (4) ˆ 0.057*** 0.053*** 0.045*** 0.044*** (0.015) (0.013) (0.011) (0.011) ˆ * (0.067) (0.076) (0.055) (0.053) 0.035** 0.038* 0.029* 0.029** (0.012) (0.015) (0.012) (0.014) constant *** (0.009) (0.870) (0.007) (0.756) control varables N Y N Y R N Table A14. Stock Shares Versus Cardnal Proxes for Preferences and Belefs. Share of wealth at Vanguard at least 70 percent LHS varable: survey measure of stock share n total fnancal wealth LHS varable: admnstratve measure of stock share n Vanguard (1) (2) (3) (4) ˆ 0.058*** 0.054** 0.060*** 0.058*** (0.016) (0.017) (0.013) (0.013) ˆ (0.084) (0.075) (0.061) (0.067) 0.041** 0.045** 0.036** 0.039** (0.013) (0.015) (0.012) (0.014) constant *** (0.015) (1.225) (0.012) (1.032) control varables N Y N Y R N

18 Table A15. Stock Shares Versus Cardnal Proxes for Preferences and Belefs. Households wth drectly held stocks LHS varable: survey measure of stock share n total fnancal wealth LHS varable: admnstratve measure of stock share n Vanguard (1) (2) (3) (4) ˆ 0.051* 0.067** 0.045* 0.039* (0.024) (0.023) (0.020) (0.019) ˆ (0.156) (0.126) (0.149) (0.107) (0.017) (0.024) (0.030) (0.036) constant 0.070*** * 3.797** (0.018) (1.771) (0.018) (1.600) control varables N Y N Y R N Table A16. Share of Rsky Assets Versus Cardnal Proxes for Preferences and Belefs. (Includng housng wealth n the share of rsky asset calculaton) LHS varable: housng wealth ncluded as safe assets LHS varable: housng wealth ncluded as rsky assets (1) (2) (3) (4) ˆ 0.036*** 0.033** 0.065*** 0.065*** (0.006) (0.006) (0.011) (0.010) ˆ * * * ** (0.031) (0.031) (0.057) (0.051) 0.018** 0.018** 0.030** 0.022* (0.006) (0.006) (0.010) (0.011) constant (0.005) (0.375) (0.010) (0.623) control varables N Y N Y R N 4,414 4,414 4,414 4,414 Notes. For the frst two columns, the LHS varable s calculated as the share of stock holdngs (based on survey measure) out of the sum of total fnancal wealth and housng wealth. For the last two columns, the LHS varable s calculated as the share of the sum of stock holdng (based on survey measures) and housng wealth out of the sum of total fnancal wealth and housng wealth. 17

19 Fgure A1. Dfference between relatve rsk tolerance and θ as a fracton of θ) over dfferent levels of consumpton and. Notes. The vertcal lne shows the mean level of household ncome n the VRI (before retrement), to approxmate the average level of household consumpton. 18

20 Fgure A2. B-varate non-parametrc regresson of stock share n total fnancal wealth on each probablty questons on stock market expectaton p0 p20 19

21 Appendx B. Detals on the VRI Surveys and Sample We use responses to three VRI surveys, conducted n the fall of 2013, wnter of 2014 and summer of The man focus of the frst survey was to nventory ncome, wealth and portfolo of households as well as to gather bass demographcs. The second survey mplemented Strategc Survey Questons (SSQs), whch ask respondents to make choces under hypothetcal stuatons desgned to elct meanngful preference data. Ths paper uses the questons about rsk preference. The thrd survey ncludes the questons about belefs about returns used for ths paper, and also covers a number of ssues not related to ths paper. 4,730 respondents completed all the three surveys. The tem non-response rate of the VRI s remarkably low. Our analyss ncludes the 4,414 respondents wth non-mssng observatons for all the varables used n the analyss. The VRI sample frame s based on admnstratve account data for Vanguard. Havng such data to create a sample s an mportant element of the VRI desgn. Addtonally, admnstratve data are composed of monthly hstory of Vanguard assets, wth nformaton on types, balances and stock shares of the accounts lnked to the survey measures. Ths paper uses both survey and admnstratve measures of assets and ther composton. The survey measure covers all assets, not just those held at Vanguard. See Amerks, Capln, Lee, Shapro and Tonett (2014a, 2014b) for a detaled dscusson of the desgn of the VRI ncludng samplng and response rates, and of the VRI s approach to wealth measurement. 1 The plan s to mplement the VRI as a panel. These three surveys, however, cover dstnctve topcs wth lttle longtudnal content. They were broken nto three surveys of 40 to 60 mnutes for the practcal reason of not overwhelmng respondents. 20

22 Appendx C. Detals on Structural Estmaton Procedure The dstrbutons of the true latent varables are assumed as (8), (9) and (10) n the man text: 2 u u 0 u u u, ~ N, 2 u u 0. u (C.1) 2 log( ) u, u ~ N(0, u ). (C.2) We allow the belefs about returns to depend on rsk preference, so the covarates of and nclude the latent. These latent varables are related to observed survey responses n the followng way. log( ) log( ) for j 1, 2 j j 2 j ~ N(0, j ) (C.3) 11/ j 11/ j 11/ j ( c) (2 c) ((1 x) c) vs / 11/ 11/ j j j (C.4) 2 m m, m ~ N(0, m) (C.5) p N (C.6) 2 0 ( 0), 0 ~ (, p) 0.2 p N (C.7) 2 20 ( 20), 20 ~ (0, p) The varables verson of m, m as p 0, and p 20 are before roundng. Actual survey response m s a rounded m s restrcted to take an nteger value. Survey responses p 0 and p 20 are to take a value from the set {0,5,10,15,25,35,,75,85,90,95,100}, we assume that p 0 and p 20 are rounded to the closest values allowed for each response. Also note that the survey does not allow for p 20 to be larger than p 0. Hence when we observe p 20 = p 0, we consder the 21

23 possblty that the survey response error actually generated p20 p 0 but after mposng the constrant we observe the 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. The parameter values governng the dstrbuton of the survey response errors allow us to calculate the condtonal probablty of the jont responses The parameters to be estmated are,,,,,,,,,,,. We allow for,,, and to vary wth covarates. Algorthm of lkelhood functon calculaton u u u 1 2 m p We use the Gaussan quadrature approxmaton of the normal dstrbuton to numercally ntegrate the densty functons over multple dmensons. Let be the vector of parameters. Gven a fxed 0 the lkelhood functon s calculated through the followng algorthm: (1) Based on the parameter values that govern the true belef and preference parameter dstrbutons n 0, and usng Gaussan Quadrature approxmaton, generate K nodes {,, } K k k k k of belef and preference parameters, wth correspondng probabltes 1 { } K k k 1 such that K k 1. k 1 (2) For each { k, k, k} and each ndvdual, calculate [, ], [, ], [, ], [, ], [, ] such that survey response error low hgh low hgh low hgh low hgh low hgh m m terms realzed n these ranges generate the observed responses after roundng and correspondng constrants. 22

24 (3) For each { k, k, k} and each ndvdual, calculate the jont lkelhood of the realzaton of the error terms n the range found n (2), usng Gaussan CDF under the parameter values governng the error term dstrbutons n 0. Let k denote ths jont lkelhood. (4) The lkelhood for each ndvdual s calculated as ntegraton over k nodes as followng: L K k k k 1 (C.8) Then the jont lkelhood s calculated as products of L over ndvduals. Calculaton of the proxy varables Under the estmated parameters, the proxy varables are calculated as expected values condtonal on the observed responses. The ndvdual-specfc proxy varables are obtaned usng the econometrc model outlned above. The lkelhood functon of the model specfes the probablty dstrbuton of the observed responses condtonal on the latent belefs and preferences. The dstrbuton of the latent varables condtonal on the observed responses can be obtaned from the lkelhood functon usng Bayes theorem. Integratng out ths functon yelds the ndvdual-specfc proxy varables ( ˆ, ˆ and ˆ ) as the condtonal expectatons of the latent varables gven the observed survey responses. We use the same numercal approxmaton for ths calculaton. Usng the Bayes Rule, ˆ s calculated as: ˆ 1 [,,, ]. (C.9) K E m p0 p20 SSQ1 SSQ2 k k k L k 1 23

25 Appendx D. Detals on Structural Lfe-Cycle Model of Portfolo Choce Health Transton and Preferences The model starts from age 55, whch s the lowest value n the VRI, and the household can lve up to age 110 at most. 2 The probablty of survval up to next perod (1 D ) s a functon of age. The household evaluate flow utlty from the consumpton usng (1). It dscounts next perod utlty by tme dscount factor. When t des, t leaves the bequest, and bequest utlty s modeled as: U Beq, ( B) Beq ( B ) 11/ Beq 11/ (D.1) where Beq determnes the strength of the bequest motve and Beq necessty or luxury, compared to ts own consumpton. determnes whether t s Labor Income Process The household retres at age 65. Untl then, the labor ncome s exogenously determned as: 2 log( Yt ) log( y ) t, t ~ N(0, ) for t 65. (D.2) Gven that households have only 10 years untl retrement n ths model, we abstract from permanent ncome shocks. After retrement, the household receves annuty ncome whch captures Socal Securty ncome and defned beneft penson ncome and hence s not exposed to any uncertanty. Ths annuty ncome s modeled as a fracton ( ) of the mean ncome before retrement: log( Y ) log( ) log( y ) for t 65. (D.3) t 2 To avod the complcatons arsng from the jont survval process, we assume that the household des when the head des. Essentally, the model s lookng at the sngle households portfolo choce. Stock share regresson usng sngles only gve the essentally the same results as our baselne results usng the full sample. 24

26 Fnancal Assets Households can nvest n two dfferent assets, a rskless asset and a rsky asset where the latter represents stocks. The gross real return on a rsk free asset s set as a constant modeled as: R. The subjectve belef on dstrbuton of the real gross return on a rsky asset, R f t, s R 2 t1, t1, t1 ~ N(0, ) (D.4) where t 1 s an..d. stock return shock. Note that ths subjectve belef process s heterogeneous across households. We assume that the aggregate stock return shock s uncorrelated wth the dosyncratc labor ncome shock, followng Cocco, Gomes and Maenhout (2005). Optmzaton problem of the households Let Wt be begnnng-of-perod cash n hand of a household and t be share of savngs of ths perod nvested to stocks. We assume that short sales and leveraged stock holdngs are not allowed. 3 Then the household solves the followng optmzaton problem (we drop the subscrpts and t): VWt (, ) max { UC ( ) E[(1 ( t)) VW ( ', t1) ( tu ) ( W')]} CW, ', st.. W ' [( W C)((1 ) R R )] y' C W [0,1] D D Beq f s (D.5) Computaton We solve for the optmal polcy functon numercally usng backward nducton. The last perod (at age 110) maxmzaton s a statc one so the value functon s trvally 3 Optmal stock share could go over 100% f we allowed leveragng, snce labor earnngs and retrement ncome are close substtutes to the rsk-free asset, due to zero correlaton wth stock return for the former and the absence of rsk for the latter. In addton, when we approxmate the labor ncome process as a dscrete process, even the worst possble realzaton of ncome guarantees postve resources net of the subsstence level of consumpton (as n Cocco, Gomes and Maenhout (2005)) snce mean level of labor ncome s much hgher than the subsstence level of consumpton. 25

27 obtaned. Ths value functon s used as a contnuaton value for the maxmzaton program of the penultmate perod. We repeat ths untl we solve for the maxmzaton problem at the frst perod. For the choce over contnuous spaces,.e. over C and, the optmzaton s done usng grd search. Wth the curvature parameters the problem s no more homogenous to the scale, so t cannot be normalzed as typcally done n the lterature (see Cocco, Gomes and Maenhout (2005) and Pang and Warshawsky (2010) for example). Ths does not ncrease computatonal burden too much snce we abstract from permanent ncome shocks. Calbraton We solve ths model for varous sets of subjectve belef and rsk tolerance parameter values that are n the range supported by the evdence from the VRI, to understand the effects of heterogeneous belef and preference on the optmal stock share. The curvature parameter for the ordnary utlty functon ( ) s fxed at the value estmated from the VRI (- 17K). Tme dscount factor ( ) s set to be 0.96, a value that s typcally used n the lterature for annual models. The probablty of survval D s estmated from the HRS ( ). For the parameters for the bequest utlty functon, we estmate these parameters usng the methodology from Amerks, Brggs, Capln, Shapro and Tonett (2016) and the survey questons desgned to estmate the strength of the bequest motve from the VRI ( 32, 64 ). The Beq Beq K parameters mply that a bequest s a luxury good compared to the ordnary consumpton, but once the bequest motve kcks n for wealthy households the margnal utlty from leavng bequest s large. 4 Rsk free return ( R ) s set to be In the baselne model we use $90,000 f for the mean ncome before retrement ( y ) and assume 0.5 for the replacement rate after 4 Shuttng of the bequest motve does not notceably change the result from the model. 26

28 retrement ( ). These values are close to means from the VRI data. The varance of transtory 2 ncome shocks ( ) s set to be 0.07, whch s close to the value used n Cocco et al. (2005). 5 Table D1 summarzes the calbraton of the parameters, and Fgure D1 and D2 summarze the results. Table D1. Calbraton of Parameters for the Lfe-Cycle Model Parameters Value Target/Source -17K VRI estmaton 0.96 Standard D HRS estmaton Cocco, Gomes and R f 1.02 Maenhout (2005) 32 VRI estmaton Beq Beq 64K VRI estmaton y $80,000 VRI data 0.5 VRI data Cocco, Gomes and Maenhout (2005) 5 They estmated t to be for college graduates. We set t slghtly larger here gven that our model does not have permanent ncome shocks. 27

29 Fgure D1. Stock share and the expected value of stock returns (μ) at dfferent levels of the standard devaton of stock returns (σ) and rsk tolerance (θ). Results from the lfe cycle portfolo choce model. Portfolo choce model, medum level of rsk tolerance (θ = 0.32) Portfolo choce model, low level of rsk tolerance (θ = 0.16) Fgure D2. Stock share and the rsk tolerance (θ) at dfferent levels of the standard devaton of stock returns (σ) and expected value of stock returns (μ). Results from the lfe cycle portfolo choce model. Portfolo choce model, medum level of expected return (μ = 0.06) Portfolo choce model, low level of expected return (μ = 0.03) 28

30 Appendx E. Detals on the Effect of Recent Returns on the Subjectve Belefs Utlzng the varaton n the tmng that the respondents flled n Survey 3, we examne whether respondents belefs are affected by recent returns even at a hgh frequency. To be more specfc, we examne whether the returns they experence durng the week before the survey affects ther expected returns. If ther expected returns for one-year horzon turn out to be senstve wth respect to the returns they experenced n the week before the survey, we can conclude that ther expectatons are strongly affected by sentments. Fgure E1 descrbes the dstrbuton of the survey tmng and the stock market return that respondents experenced durng the week before the survey. About half of the respondents flled n the survey n several days after the nvtaton, but many dd so over the next several weeks. We also sent out two remnders, one two weeks after and the other three weeks after the ntal nvtaton, whch were effectve n solctng responses and helped generate a larger varaton n the survey tmng. Stock returns are calculated as /, where SNP s the S&P 500 ndex and t s the date of survey. The fgure shows that we have enough varaton n both the survey tmng and the past week s stock returns, whch allows the dentfcaton of the effect of recent returns on belefs. Table E1 shows the estmaton results. Overall, we fnd that the return durng the past week sgnfcantly affects the expected returns for the followng year. The result s smlar regardless of whether we use the error-rdden measure or the cardnal proxy ˆ, though the estmate from the former regresson s much noser. We also fnd a strong heterogenety n the senstvty n the belefs. As expected, the sample wth hgher educaton who are more lkely to be fnancally sophstcated exhbt less senstvty. The extrapolaton bas s estmated to be sgnfcant only for those wthout a post-college degree. For those wth a post-college degree, 29

31 the pont estmates are close to zero, though the estmate s not precse for the MBA group due to the small sample sze. 30

32 Fgure E1. Dstrbuton of survey tmng and stock market return n the week before survey Jul 1 Aug 3 Aug 5 Aug 7 Aug 9 Aug 11 Aug 13 Aug 15 Aug 17 Aug 19 Aug Number of respondents Return 21 Aug 23 Aug 25 Aug 27 Aug 29 Aug 31 Aug 2 Sep 4 Sep 6 Sep 8 Sep Date N Return n prevous week Table E1. Extrapolaton bas and educaton level No post-college All MBA Other postcollege ˆ ˆ ˆ ˆ ** * 0.310*** s.d. (0.104) (0.056) (0.335) (0.220) (0.165) (0.087) (0.143) (0.075) 4,414 4, ,537 1,537 2,586 2,586 Notes. *, **, and *** mply sgnfcance at 5%, 1%, and 0.1% level, respectvely. 31

33 Appendx F. Effects of Attenton Measured by Number of Logns Another possble explanaton for the attenuaton bas s that some ndvduals do not pay much attenton to the stock market and hence they do not put a hgh weght on ther own belefs, reflectng ther low confdence on ther own predctons. We can test ths possblty by usng the number of logns to ther Vanguard web account n the last sx months of the frst survey. The underlyng assumpton s that those who often log nto ther web accounts are more lkely to pay attenton to the stock market. We dvde the sample nto two groups, those who logged n more often than or the same as the medan respondent (16 tmes) and those who logged less often than the medan respondent. We frst examne whether those who log n more often have systematcally dfferent belefs than the others. One mght expect to see hgher standard devaton of the belef measures from those wth less frequent logns, snce ther belefs are less lkely to be anchored to the hstorcal means. When we examne the dfferences n the mean and the standard devaton of (normalzed) cardnal proxes of belefs, we do not fnd evdence supportng our pror (Table F1). Standard devaton of the expected return s slghtly larger for those who logn less often, but the dfference s not statstcally sgnfcant. Despte the absence of a systematc dfference n the belef dstrbuton, we examne whether there s a systematc dfference n the attenuaton bas across these two groups. The message s mxed. The effect of the belef on expected return s slghtly larger for those who log n more often, whle the effect of the rsk tolerance s slghtly larger for those who log n less often. Both dfferences are only margnally sgnfcant at 10 percent level. 32

34 Table F1. Past logns and belef dstrbutons Mean (Logn 16) Std (Logn 16) Mean (Logn <16) Std (Logn <16) t-stat for mean dff. p-value for varance dff Notes. and normalzed values of and,.e. / and /. Table F2. Past logns and attenuaton bas Logn 16 Logn 16 Logn <16 Logn <16 Survey Admn Survey Admn ˆ 0.066*** 0.059*** 0.044*** 0.034*** (0.012) (0.011) (0.013) (0.012) ˆ * (0.058) (0.060) (0.076) (0.069) ˆ 0.022* ** (0.012) (0.012) (0.016) (0.014) Constant (0.799) (0.707) (0.806) (0.845) Covarates Y Y Y Y R N 2,258 2,258 2,156 2,156 33