Scale Dependence of Overconfdence n Stoc Maret Volatlty Forecasts Marus Glaser, Thomas Langer, Jens Reynders, Martn Weber* June 7, 007 Abstract In ths study, we analyze whether volatlty forecasts (judgmental confdence ntervals) are nfluenced by the specfc elctaton mode (.e. whether forecasters have to state future prce levels or drectly future returns as upper and lower bounds). We present questonnare responses of about 50 students from two German unverstes. Partcpants were ased to state medan forecasts as well as confdence ntervals for seven stoc maret tme seres. Usng a between subject desgn, one half of the subjects was ased to state future prce levels, the other group was drectly ased for returns. Consstent wth pror research we fnd that subjects underestmate the volatlty of stoc returns, ndcatng overconfdence. As a new nsght, we fnd that the strength of the overconfdence effect n stoc maret forecasts s hghly sgnfcantly affected by the fact whether subjects provde prce or return forecasts. Volatlty estmates are lower (and the overconfdence bas s thus stronger) when subjects are ased for returns compared to prce forecasts. Keywords: Volatlty forecast, confdence nterval, ndvdual nvestor, overconfdence. JEL classfcaton: C9, G1. *Marus Glaser (correspondng author) s from the Lehrstuhl für Banbetrebslehre, Busness School, Unverstät Mannhem. Phone: +49 (0)61 181 3440. Fax: +49 (0)61 181 1534. E-mal: Glaser@ban.BWL.un-mannhem.de. Thomas Langer s from the Fnance Center, Unverstät Münster. E- mal: Thomas.Langer@ww.un-muenster.de. Jens Reynders s from Semens Management Consultng. E- mal: jens.reynders@semens.com. Martn Weber s from the Lehrstuhl für Banbetrebslehre, Busness School, Unverstät Mannhem and CEPR, London. E-mal: Weber@ban.BWL.un-mannhem.de. Fnancal Support from the Deutsche Forschungsgemenschaft (DFG) s also gratefully acnowledged. 1 Electronc copy avalable at: http://ssrn.com/abstract=996849
1. Introducton Numerous studes fnd that judgmental confdence ntervals for uncertan quanttes are too tght ndcatng overconfdence. But s the level of overconfdence easly nfluenced by the way people are ased to state nterval judgments? Ths s the queston we try to answer n ths paper for the case of stoc maret volatlty forecasts (judgmental confdence ntervals): Is the wdth of the nterval,.e. the volatlty forecast gven by subjects, nfluenced by the specfc elctaton mode (.e. whether forecasters have to state future prce levels or drectly future returns)? There are many questonnare studes that elct the volatlty estmate of nvestors by asng for confdence ntervals for the return or value of an ndex or the return or prce of a stoc n the future. These studes usually fnd that the ntervals provded are too tght. Thus, hstorcal volatltes are underestmated (see, for example, Glaser, Nöth and Weber (004) and Hlton (001)). The fndng that confdence ntervals for uncertan quanttes are too tght s usually called mscalbraton or overconfdence (see Lchtensten, Fschhoff, and Phllps (198), Soll and Klayman (004), Grffn and Brenner (004), and Glaser and Weber (007)). 1 However, there s no evdence n the lterature so far that t matters for ths queston whether one ass for prce levels or returns. In ths paper, we present questonnare responses of about 50 students from two German unverstes. Partcpants were ased to state medan forecasts as well as confdence ntervals for seven stoc maret tme seres. Usng a between subject desgn, one half of the subjects was ased to state future prce levels, the other group was drectly ased for returns. We fnd that subjects underestmate the volatlty of stoc returns ndcatng overconfdence. The degree of overconfdence s hghly sgnfcantly affected by the forecast mode. Volatlty estmates are lower when subjects are ased for returns compared to the respectve prce forecasts. The rest of the paper s organzed as follows. In Secton, we present the desgn of our study. Secton 3 presents the results and the last secton concludes.. Desgn of the Study We desgned dfferent versons of a questonnare that was flled out by students of two classes at the Unversty of Mannhem and the Unversty of Münster n Germany. The questonnares can be downloaded from the followng web page: <http://www.fnanzerungslehrstuhl.de/glrw/glaser- Langer_Framng_supplement.pdf> 1 Most behavoral models ncorporate judgment bases nto theores of fnancal marets by assumng that at least some maret partcpants are overconfdent n the way that they overestmate the precson of ther nowledge or underestmate the varance of nformaton sgnals. As a consequence, ther confdence ntervals for the value of a rsy asset are too tght when compared to the ratonal benchmar. See Glaser, Nöth and Weber (004) for an overvew of overconfdence models n fnance. Electronc copy avalable at: http://ssrn.com/abstract=996849
Subjects were ased to state mean and nterval judgments for seven tme seres wth dfferent trends over a one month and sx month forecast horzon. As nformaton, all subjects receved the past sx month chart. In four out of seven cases, subjects also receved the name of the respectve tme seres (.e. the name of the stoc or ndex). The two versons of the questonnares only dffered n the way we ased for the forecasts. Fgures 1 and show examples of the sample questons. Fgure 1: Questonnare: Example from the Prce Level Verson. Fgure : Questonnare: Example from the Return Verson. 3
Glaser, Langer, Reynders, and Weber (007) extensvely descrbe the data set and subject pool of ths questonnare study. They show that the elctaton mode can help explan why some nvestors beleve n mean reverson or trend contnuaton. However, they do not analyze scale dependence of overconfdence. To calculate volatlty forecasts, we proceed as follows (see also Glaser and Weber (005) or Graham and Harvey (003)). Means and volatlty have not been surveyed drectly, but can be approxmated va the mean and upper and lower lmts for contnuous random varables (see Keefer and Bodly (1983)). For each of the seven tme seres, { 1; ; 3; 4 ; 5 ; 6 ; 7} and each subject, { 1;...; 49}, mean and standard devaton are approxmated usng the followng formula (prce forecasts n the prce forecast mode where converted to return forecasts) 3 : [ x(0,05) x(0,95 ] mean = 0,63x (0,50) + 0,185 + ) standard devaton = 0,185( x(0,05) ) + 0,63( x(0,50) ) + 0,185( x(0,95) ) ( mean ) x p) ( s the p percentle of the dstrbuton wth { 0,05 ; 0,5 ; 0,95} p. 3. Results Table 1 presents means and medans across subjects of 1-month as well as 6-month volatlty forecasts for each tme seres and for the two groups ( prce forecast mode and return forecast mode ). Volatlty forecasts are calculated as descrbed n the secton above. Furthermore, the table contans the dfference of mean and medan volatlty forecasts of the return forecast mode and the prce forecast mode as well as the p-value of a Mann-Whtney test. Null hypothess s equalty of populatons. Medan volatlty forecasts are lower n the return forecast mode (except for the DAX) whch s hghly sgnfcant n most cases. From here on, we wll use standard devaton and volatlty synonymously. 3 See Keefer and Bodly (1983), p. 597. 4
Stoc Trend Prce forecast Return forecast Dfference p-value mode mode Return-Prce (Mann-Whtney) BASF up Mean (1 month) 0.0449 0.037-0.0077 <0.0001*** Medan (1 month) 0.0437 0.0304 N 15 119 Mean (6 months) 0.080 0.0633-0.0170 <0.0001*** Medan (6 months) 0.0685 0.0480 N 16 116 Stoc A (Scherng) up Mean (1 month) 0.045 0.0361-0.009 0.0005*** Medan (1 month) 0.041 0.0304 N 14 116 Mean (6 months) 0.0875 0.0690-0.0185 0.0011*** Medan (6 months) 0.0701 0.0547 N 18 116 Henel down Mean (1 month) 0.0537 0.039-0.0145 <0.0001*** Medan (1 month) 0.0447 0.0308 N 14 111 Mean (6 months) 0.091 0.0710-0.011 <0.0001*** Medan (6 months) 0.083 0.0596 N 17 116 Stoc C (Infneon) down Mean (1 month) 0.0831 0.0450-0.0380 <0.0001*** Medan (1 month) 0.0733 0.0337 N 15 115 Mean (6 months) 0.1576 0.0831-0.0745 <0.0001*** Medan (6 months) 0.1466 0.0670 N 18 116 DAX ndex flat Mean (1 month) 0.05 0.093 0.0040 0.6115 Medan (1 month) 0.040 0.043 N 16 117 Mean (6 months) 0.0448 0.058 0.0080 0.8867 Medan (6 months) 0.0391 0.0356 N 17 117 Deutsche Teleom flat Mean (1 month) 0.0566 0.0371-0.0196 <0.0001*** Medan (1 month) 0.0443 0.0304 N 119 116 Mean (6 months) 0.0958 0.0594-0.0365 <0.0001*** Medan (6 months) 0.083 0.0430 N 15 117 Stoc B (SAP) flat Mean (1 month) 0.0381 0.040 0.000 0.939 Medan (1 month) 0.034 0.0315 N 16 116 Mean (6 months) 0.0633 0.065-0.0007 0.1431 Medan (6 months) 0.0600 0.0496 N 17 117 Table 1: Volatlty forecasts. *** ndcates sgnfcance at the 1 percent level. Furthermore, all 6-month volatlty forecasts are hgher than the respectve 1-month volatlty forecasts whch s consstent wth emprcal observatons (see Table ). Table once agan presents means and medans across subjects of 1-month as well as 6-month volatlty forecasts for each tme seres and for the two groups ( prce forecast mode and return forecast mode ). Furthermore, the table presents hstorcal volatltes for the tme seres that were nown to the partcpants as well as chart volatltes for all seven tme seres. Hstorcal volatltes are calculated as the standard devatons of non-overlappng 1-month respectve 6-month returns from January 1990 to December 004. 4 To calculate chart volatltes, we frst calculate the standard devaton of the 131 daly return observatons for all seven tme seres. The 1-month chart volatlty s the standard devaton of the daly return observatons multpled by 30. The 6-month chart volatlty s the standard devaton of the daly return observatons multpled by 180. Note that durng our sample perod, chart volatltes are lower than the hstorcal volatltes. 4 The tme seres of Deutsche Teleom starts on November 18, 1996, the IPO date. 5
Stoc Trend Prce forecast Return forecast Hstorcal Chart OC OC p-value p-value mode mode volatltes volatltes Prce forecast Return forecast (Mann-Whtney) (Mann-Whtney) mode mode Prce forecast Return forecast mode mode BASF up Mean (1 month) 0.0449 0.037 0.0719 0.0571 0.64.06 Medan (1 month) 0.0437 0.0304 0.31 0.88 <0.0001*** <0.0001*** N 15 119 Mean (6 months) 0.080 0.0633 0.1808 0.1400 1.49 4.17 Medan (6 months) 0.0685 0.0480 1.04 1.9 <0.0001*** <0.0001*** N 16 116 Stoc A (Scherng) up Mean (1 month) 0.045 0.0361 Stoc was 0.065 1.00.5 Medan (1 month) 0.041 0.0304 unnown 0.58 1.14 <0.0001*** <0.0001*** N 14 116 Mean (6 months) 0.0875 0.0690 0.1596 1.75 3.61 Medan (6 months) 0.0701 0.0547 1.8 1.9 <0.0001*** <0.0001*** N 18 116 Henel down Mean (1 month) 0.0537 0.039 0.0680 0.0674 0.74 1.98 Medan (1 month) 0.0447 0.0308 0.51 1.19 <0.0001*** <0.0001*** N 14 111 Mean (6 months) 0.091 0.0710 0.1704 0.1651 1.41 3.78 Medan (6 months) 0.083 0.0596 0.98 1.77 <0.0001*** <0.0001*** N 17 116 Stoc C (Infneon) down Mean (1 month) 0.0831 0.0450 Stoc was 0.1081 0.89 5.00 Medan (1 month) 0.0733 0.0337 unnown 0.47.1 <0.0001*** <0.0001*** N 15 115 Mean (6 months) 0.1576 0.0831 0.649 1.50 7.53 Medan (6 months) 0.1466 0.0670 0.81.96 <0.0001*** <0.0001*** N 18 116 DAX ndex flat Mean (1 month) 0.05 0.093 0.0667 0.0553 1.96.40 Medan (1 month) 0.040 0.043 1.30 1.8 <0.0001*** <0.0001*** N 16 117 Mean (6 months) 0.0448 0.058 0.1748 0.1355 4.0 4.83 Medan (6 months) 0.0391 0.0356.47.81 <0.0001*** <0.0001*** N 17 117 Deutsche Teleom flat Mean (1 month) 0.0566 0.0371 0.17 0.066 0.48.60 Medan (1 month) 0.0443 0.0304 0.49 1.18 <0.0001*** <0.0001*** N 119 116 Mean (6 months) 0.0958 0.0594 0.3430 0.160 1.35 6.5 Medan (6 months) 0.083 0.0430 0.97.77 <0.0001*** <0.0001*** N 15 117 Stoc B (SAP) flat Mean (1 month) 0.0381 0.040 Stoc was 0.0933.44 3.16 Medan (1 month) 0.034 0.0315 unnown 1.73 1.96 <0.0001*** <0.0001*** N 16 116 Mean (6 months) 0.0633 0.065 0.85 4.41 7.06 Medan (6 months) 0.0600 0.0496.81 3.61 <0.0001*** <0.0001*** N 17 117 Table : Volatlty forecasts, hstorcal volatltes, chart volatltes, and overconfdence (OC). *** ndcates sgnfcance at the 1 percent level. Table also shows that volatlty estmates are lower than hstorcal volatltes or chart volatltes. Hstorcal volatltes are often used as an objectve volatlty benchmar or an estmate for the future volatlty (see for example, De Bondt (1998), Graham and Harvey (003), and Glaser and Weber (005)) 5. The fact that confdence ntervals are too tght or, n other words, that people underestmate the volatlty of stoc returns, s called overconfdence. To analyze overconfdence more formally, we calculate an overconfdence measure for each subject and tme seres as follows: OC=(chart volatlty/volatlty forecast)-1. A postve OC measure ndcates overconfdence, a negatve measure underconfdence. Table shows, that all OC measures are hghly sgnfcantly postve. We are thus able to confrm the usual result n the lterature (see, for example, Hlton (001) or Graham and Harvey (003)). Table also shows that overconfdence s stronger for 6-month forecasts. Ths result s consstent wth Glaser, Langer, and Weber (005) who show that overconfdence n volatlty forecasts s stronger, the longer the forecast horzon. 5 Furthermore, hstorcal volatltes are often regarded as the best tme-seres volatlty-forecastng method when compared to GARCH or stochastc volatlty (see Poon and Granger (005)). 6
4. Dscusson and Concluson The man results of ths paper can be summarzed as follows: Subjects underestmate the volatlty of stoc returns ndcatng overconfdence. Overconfdence n stoc maret forecasts s hghly sgnfcantly affected by the fact whether one ass for prces or returns. Volatlty estmates are lower and overconfdence s hgher when subjects are ased for returns compared to prce forecasts. Our study draws attenton on a determnant of overconfdence that s neglected n the lterature so far. Studes analyze, for example, the nfluence of tme seres characterstcs on volatlty forecast (see the survey by Lawrence, Goodwn, O'Connor, and Önal (006) or Du and Budescu (007) as a recent example). Scale dependence of overconfdence was not analyzed before. Future research should nvestgate why we document such a strong scale dependence. One avenue for future research s provded by Amromn and Sharpe (006) and Glaser, Langer, Reynders, and Weber (007). They present evdence that nvestors seem to be reluctant to state negatve numbers. As a consequence, nvestors realze a greater downsde potental when they have to state prce levels whch would results n wder confdence ntervals. 7
References Amromn, G. and Sharpe, S. A (006), From the Horse's Mouth: Gaugng Condtonal Expected Stoc Returns from Investor Survey, Worng Paper, Federal Reserve Board Washngton De Bondt, W. F. (1998) A portrat of the ndvdual nvestor, European Economc Revew 4, 831 844. Du, N., and D. V. Budescu (007), Does past volatlty affect nvestors' prce forecasts and confdence judgments?, Internatonal Journal of Forecastng, forthcomng. Glaser, M., T. Langer, J. Reynders, and M. Weber (007), Framng Effects n Stoc Maret Forecasts: The Dfference Between Asng for Prces and Asng for Returns, Revew of Fnance 11, 35 357. Glaser, M., Langer, T. and Weber, M. (005), Overconfdence of Professonals and Lay Men: Indvdual Dfferences Wthn and Between Tass? Worng Paper, Unversty of Mannhem. Glaser, M., Nöth, M. and Weber, M. (004), Behavoral Fnance, n: Koehler, D. J. and N. Harvey (eds.), Blacwell Handboo of Judgment and Decson Mang, Blacwell (Malden), 57 546. Glaser, M. and M. Weber (007), Overconfdence and Tradng Volume, Geneva Rs and Insurance Revew, forthcomng. Glaser, M. and Weber, M. (005), September 11 and Stoc Return Expectatons of Indvdual Investors, Revew of Fnance 9, 4 79. Graham, J. R. and Harvey, C. R. (003) Expectatons of equty rs prema, volatlty and asymmetry, Worng paper, Fuqua School of Busness, Due Unversty. Grffn, D. and Brenner, L. (004) Perspectves on probablty judgment calbraton, n Koehler, D. J. and N. Harvey (eds.), Blacwell Handboo of Judgment and Decson Mang, Blacwell (Malden), 177 199. 8
Hlton, D. J. (001) The psychology of fnancal decson-mang: Applcatons to tradng, dealng, and nvestment analyss, Journal of Psychology and Fnancal Marets, 37 53. Keefer, D. L. and Bodly, S. E. (1983) Three-pont approxmatons for contnuous random varables, Management Scence 9, 595 609. Lawrence, M., Goodwn, P-, O'Connor, M., and Önal, D. (006), Judgmental forecastng: A revew of progress over the last 5 years, Internatonal Journal of Forecastng, 493 518. Lchtensten, S., Fschhoff, B. and Phllps, L. D. (198) Calbraton of probabltes: The state of the art to 1980, n Kahneman, D., P. Slovc and A. Tversy (eds.), Judgment under Uncertanty: Heurstcs and Bases, Cambrdge Unversty Press (Cambrdge), 306 334. Poon, S.-H. and Granger, C. (005), Practcal Issues n Forecastng Volatlty, Fnancal Analysts Journal 61, 45 56. Soll, J. B. and Klayman, J. (004) Overconfdence n nterval estmates, Journal of Expermental Psychology: Learnng, Memory, and Cognton 30, 99 314. 9