WORKING PAPER - 26 August 2005 Tme Dversfcaton n Penson Savngs Anders Karlsson 1 Department of Fnance, School of Busness, Stockholm Unversty S-106 91 Stockholm, Sweden 1 PhD. Canddate, Fnance. E-mal aka@fek.su.se, Phone: +46-8-736 96 87 Fax: +46-8-31 64 22
Tme-Dversfcaton n Penson Savngs Abstract We take a closer look at how nvestment horzon affects rsk takng, often referred to as the tme-dversfcaton controversy. We use data on ndvduals choces n the Swedsh penson system. Theoretcally, f returns are serally uncorrelated, nvestors do not have human captal, and nvestors have constant relatve rsk averson then nvestment horzon should not nfluence asset allocaton. Ths theory causes some academcs to explan the postve correlaton between nvestment horzon and rsk exposure by generatonal dfferences n human captal, not the nvestment horzon per se. Our emprcal analyss shows that portfolo rsk sgnfcantly declnes wth age n a statstcal context. Ths behavor s stll evdent after controllng for alternatve explanatons related to human captal and dffcult to reject n an economc context. 2
I. Introducton A major asset allocaton decson s the amount of rsk one s wllng to tolerate. Dfferent nvestors wll naturally be more or less rsk averse dependng on ther economc and demographc stuaton. A commonly observed practce, encouraged by fnancal planers, s that ones rsk should be related to ones nvestment horzon. Expected utlty theory states that f returns are serally uncorrelated and nvestors have both constant relatve rsk averson and no human captal, then tme s not a factor n portfolo allocaton. Economsts as P. Samuelson (1963, 1989, 1994), Bode, Merton and W. Samuelson (1992), and Krtzman (1994) offer ntutve examples and convncng formal arguments showng that tme-dversfcaton s n fact not dversfcaton at all. They show that an nvestor who prefers a certan level of rsk wth, say, a three-month horzon wll prefer that same level of rsk wth, say, a 10-year nvestment horzon. In fact, Bode (1995) even labels the belef n tme dversfcaton a fallacy. These researchers suggest that the reason rsk exposure decreases wth horzon n some studes has to do wth the nvestors demographc or economc stuaton and s not an attempt to dversfy. On the other hand, f returns are serally correlated or f nvestors do not have constant relatve rsk averson, a decrease n nvestment horzon can very well lead to decreased rsk. Lee (1990), Segel (1994), Thorley (1995) and Campbell and Vcera (2002) explore ths. Thorley (1995) shows that the academc vew does not necessarly contradct the practtoner vew f Samuleson s proof s correctly understood. In essence, Thorley rephrased Samuleson s statement; f nvestors are expected utlty maxmzers who do not have constant relatve rsk averson, then the optmal proporton of ther portfolo allocated to rsky stocks s nfluenced by the nvestment horzon. In partcular, f nvestors have decreasng relatve rsk averson, then nvestors wll optmally decrease ther exposure to rsky stocks as ther nvestment horzon decreases. Campbell and Vcera show emprcal evdence of asset prces beng mean revertng over the past 100 years and argue therefore that rsks can appear dfferent to long-term nvestors than to short term nvestors. Because of mean reverson n prces, stocks may hold a lower rsk for long term nvestors than for short term nvestors. Ths s regardless of the nvestors relatve rsk averson. In other words, academcs can prove or 3
dsprove tme dversfcaton dependng on whether they assume decreasng or constant relevant rsk averson or f prces are mean revertng or not. Thus, the tme-dversfcaton controversy s an emprcal ssue. In ths study we nvestgate how a great number of nvestors have chosen to allocate a porton of ther penson. We do not take the prospectve stance of Samuelson or Bode nvestors should have constant relevant rsk averson; consequently tme does not dversfy rsk. Rather, we take a normatve stance and observe actual portfolo choces. The frst round of choces made n the Swedsh penson system had many characterstcs of a laboratory experment. By fat, the entre work force, those wth short and long nvestment horzons, n the Swedsh populaton constructed portfolos usng an equal proporton of ther wage. The portfolos were for retrement purposes so the nvestment horzon s known. All nvestors were provded wth the same nformaton at the same tme. We couple ths ndvdual portfolo choce data wth an extensve database of ndvdual demographc and economc varables n an attempt to explan the tme-dversfcaton phenomenon. Our results show that younger nvestors have hgher rsk than older nvestors, or nvestors wth long horzons have hgher rsk than nvestors wth short horzons. Our attempts n controllng for dfferences n human captal do not change ths. We also fnd t dffcult to reject ths fndng n an economc context. In the followng sectons we wll frst revew theory concernng tme-dversfcaton. In secton III we present our data and descrbe the Swedsh penson system focusng on the defned contrbuton porton and dscuss our methodology. Secton IV presents our results and we wll offer our concludng remarks n Secton V. II. Theory The basc dea of tme-dversfcaton s that above-average returns tend to offset belowaverage returns over long horzons. Formally, f returns are lognormal and ndependent over tme then the average return wll ncrease lnearly wth tme whle the standard devaton wll ncrease by the square root of tme. Consequently, the rsk of gettng a lower return than the 4
rsk free rate, alternatvely, the rsk of loosng money, approaches zero as tme moves towards nfnty. The other sde of ths argument s that although the rsk of losng money may decrease wth tme, the amount that can be lost ncreases proportonally thus cancelng out any ncrease n utlty that a longer horzon can offer. In table 1 we revew fve possble explanatons why horzon could affect rsk, three aganst tme dversfcaton and two n favor of tme dversfcaton. Arguments aganst Tme Dversfcaton Whle crtcs agree that the rato of expected return to standard devaton (reward-to-rsk rato) ncreases wth tme, they also pont out that the sze of an nvestor s potental loss ncreases n the same proporton as the expected returns, thus reducng the attractveness of the hgher reward-to-rsk rato. Although one s less lkely to lose money over a long horzon than over a short horzon, the magntude of the potental loss ncreases wth the duraton of the nvestment horzon. Krtzman makes a comparson wth cross-sectonal dversfcaton. If an nvestor s unwllng to nvest $10,000 n a rsky project based on hs level of rsk averson, then that same nvestor would not agree to nvest n ten ndependent but equally rsky projects whch requre $10,000 each. Although the nvestors rsk of losng money s reduced when nvestng n the ten ndependent projects, the exposure and, therefore, the amount the nvestor rsks losng s ten tmes as great. The only way to reduce rsk whle keepng the exposure constant s f the nvestor nstead s able to nvest $1,000 n each of the ndependent projects. Krtzman further explans that whether an nvestor has a utlty functon equal to the logarthm of wealth, or f an ndvdual s even more rsk averse and has a utlty functon equal to negatve one dvded by wealth, the utlty of the rsky venture wll reman unchanged over tme, meanng that an ncrease n tme horzon wll not affect an nvestors tolerance towards rsk (see Krtzman 1994 for detals). The crtcs of tme dversfcaton are well aware of the studes that show that as nvestors grow older or as the nvestment horzon decreases, nvestors tend to carry lower rsk n ther portfolos. However, the crtcs clam that the observed relatonshp between rsk and horzon s not drven by tme dversfcaton. We wll brefly revew three of these alternatve explanatons. 5
Non tradable assets (.e. human captal) may have an mpact on the rsk of an nvestor s portfolo. The predcton of utlty theory assumng constant relatve rsk averson s that the fracton of equtes n proporton to true total wealth s unchanged over tme. True total wealth s defned as human captal plus lqud captal. Samuelson (1994) llustrates the confoundng effect of human captal wth a young professonal wth future non-securty earnngs. Snce the human captal prospects can not be captalzed or borrowed on, to keep the porton of equtes at a proper fracton of true total wealth, the young professonal should keep a relatvely large fracton of hs lqud wealth n equtes. Later n lfe, as human captal s converged nto lqud captal, the fractonal holdng of equtes appears to decrease when compared to lqud captal, whereas, n fact, the fracton has remaned unchanged when compared to true total wealth. Developng ths argument further, Bode, Merton and W. Samuelson (1992) propose that an nvestor wth the ablty to work a lttle harder or postpone consumpton s more lkely to take a hgher rsk whereas an older professonal does not have the same opton. Another aspect of human captal expressed by Strangeland and Turtle (1999) concerns ones ablty to work n tmes of poor rsky-asset returns. We call ths the covarance between ones earnngs and market performance. If two nvestors wth the same nvestment horzon dffer n covarance, then the nvestor wth a hgher covarance s expected to have a lower portfolo rsk. The reason beng that, f covarance s hgh, earnng power s reduced n tmes of poor market performance resultng n a major shortfall. A longer nvestment horzon allows for more of these extreme shortfalls. A hgh covarance nvestor s more cautous of takng rsky nvestments. If ths s true, then nvestment horzons effect on rsk s related to the covarance of ones earnng power wth market performance. The thrd factor n table 1 concerns the frequency of requred wthdrawals from ones portfolo. Samuelson (1989) suggests that ths can explan why rsk may decrease as horzon decreases. If an nvestor requres a mnmum amount at a future date.e. for future penson payments, the nvestor makes a seres of low rsk nvestments to meet ths requrement. As the future date approaches, the low rsk fund ncreases n sze compared to hgh rsk nvestments. Such an nvestment strategy creates the lluson that the nvestor beleves n tme dversfcaton. The opposte s true f perodc wthdrawals are needed for everyday consumpton.e. f the nvestor s a pensoner and the nvestment horzon goes from 65 years 6
of age to death. Then the low rsk fund wll shrnk n sze gradually compared to the rsky nvestments as tme goes by whch may gve the appearance of an nvestment strategy whch s opposte to tme dversfcaton. Arguments n support of Tme Dversfcaton Researchers who crtcze the noton of tme-dversfcaton all assume that returns are Independent and Identcally Dstrbuted (IID) and that nvestors have constant relatve rsk averson (CRRA), meanng that they wll allocate the same proporton of ther wealth towards rsky assets regardless of ther absolute level of wealth. Several technques are used n estmatng the nature of nvestors relatve rsk averson. The varous technques lead to conflctng conclusons. J. Pratt (1964) and K. Arrow (1965) formalze measures of rsk averson and suggest that nvestors generally have an ncreasng relatve rsk averson (IRRA). K. Arrow referred to IRRA s ablty to explan observed economc behavor wth respect to holdng cash (see Selden 1956, Fredman 1959, Latane 1963 and Meltzer 1963). The measurement of relatve rsk averson has shown to be senstve to what measure of wealth s used. Research snce the md 1960 s have shown evdence of ncreasng- (Segel and Hoban 1982, Esenhauer and Halek 1999), constant- (Szpro 1986) and decreasngrelatve rsk averson (Levy 1994). Thus, an assumpton that nvestors have a constant relatve rsk averson s by no means a gven. Thorley (1995) shows mathematcally that gven an expected utlty settng and that nvestors have a decreasng relatve rsk averson towards what they perceve to be serally uncorrelated returns, tme s n fact a factor n nvestment decsons and that the allocaton towards rsk should ncrease wth an ncrease n nvestment horzon. There s a great deal of research on explorng how mean revertng prces affect rsk wth respect to nvestment horzon. Campbell and Vcera (2002) fnd that prces over the past 100 years are mean revertng and therefore tme dversfes rsk. Accordng to Campbell and Vcera, because of mean reverson, there s a degree of predctablty n stock prces. Measured over long horzons the rsk n stock returns s lower than when measured over shorter horzons. Bull markets tend to follow bear markets. Short term nvestors wll seek assets wth hgher mean reverson, namely bonds, whle nvestors wth longer nvestment 7
horzons accept assets that are less mean revertng snce over tme the long term rsk s lower than the short term rsk. Ths s suggested to be true regardless of an nvestors relatve rsk averson. Hansson and Persson (2000) also conclude that optmal weghts usng US data from 1990-1997 suggest that tme dversfcaton exsts and that allocaton decson seems to be ndependent of the utlty functon of the nvestor. The dea of mean revertng prces s not uncontested. Brown, Wllam and Ross (1995) argue that f an equty market survves, average returns n the begnnng of a tme perod s hgher than average return near the end of that tme perod. For ths reason, statstcal measures of long-term dependence are typcally based towards rejecton of a random walk. On another note, ex post evdence of mean reverson does not guarantee mean revertng prces n the future. III. Data and Methodology Data Our data comes from the frst round of nvestment choces made n the new Swedsh penson system, ntroduced n the fall of 2000, coupled wth a number of surveys on demographc and economc varables. Early research on nvestment behavor was based on expermental data where partcpants made nvestment decsons based on hypothetcal stuatons wth no fnancal consequences. Recently, researchers have studed ndvduals actual nvestment decsons n 401(k) plans. One drawback of exclusvely usng data from 401(k) plans s that the populaton s more or less homogenous. A second drawback s the dffculty n lnkng relable demographc data to the actual ndvduals makng the nvestment decsons n the 401(k) plans. Our data set has fve advantages. Frst, our data s a representatve sample of an entre country s populaton. Second, all nvestment decsons were made durng the same bref tme perod. Thrd, all nvestors choose from the same nvestment unverse and were gven the same nformaton on that nvestment unverse. The nformaton ncludes a rsk measure on 8
most funds whch provdes us wth reasonable proxes for perceved rsk. 2 Table 2 s an llustraton of what nformaton s provded for all funds n the Swedsh penson system. Fourth, we know the approxmate nvestment horzon for each nvestor snce the nvestment can be used for retrement purposes only and can not be passed on to a thrd party. 3 Ffth, we have a number of varables statng the demographc and economc stuaton of the ndvdual nvestor. These data have been gathered for the same tme perod that the ntal nvestment choces were made. The Swedsh penson system offers a unque opportunty to study how an entre populaton handles choce under uncertanty. The data used n ths study represent a cross secton of the Swedsh work force. The frst penson nvestments n the new penson system, n autumn 2000, nvolved 4.4 mllon ndvduals. Ther nvestment choces are lnked wth ndvdual demographc data collected by Statstcs Sweden for the year 2000. 4 Statstcs Sweden surveys 15,000 households whch represent a cross secton of the whole populaton n Sweden. Ths compled data set makes t possble to study nvestment behavour n great detal. For each ndvdual there s nformaton on the amount nvested, whch funds and how many funds the ndvdual has nvested n. Also, the age, gender, educaton, occupaton, dsposable ncome and net wealth for the same ndvdual s ncluded n the data set. From the 15,651 ndvduals wth complete ndvdual nformaton n the data set, 10,375 ndvduals (66.4%) made an actve nvestment decson. For these ndvduals t s possble to nvestgate the exact allocaton of assets n ther portfolos. The remanng 5,276 ndvduals (33.7%) dd not make an actve nvestment decson. Instead, they are assgned to the default alternatve; the Seventh Swedsh Penson Fund, whch s an equty fund run by the government. We treat the default alternatve as an entrely passve choce. Even f an ndvdual consdered the default fund to be the optmal 2 We use fve dfferent rsk measures. 3 Under certan crcumstances the retrement account could be passed on to a spouse, but not to any other thrd party. 4 Data sources from Statstcs Sweden are, HEK 2000; a report on household economy, IoF 2000; ncome report and SUN 2000; educatonal status. These three reports are for the total populaton n Sweden. They are lnked to a survey on 15,000 households reportng n-depth wealth statstcs. 9
choce, and acted accordngly, he/she stll shows up as makng a passve choce n the data set 5. Dependent varable, Portfolo rsk Fve rsk measures wth varyng degrees of sophstcaton are assocated wth each nvestor. Frst, we use proporton of equty n ones portfolo. The avalable funds are dvded nto four groups: equty-, mxed-, generaton- and bond funds. For mxed- and generaton funds, nformaton concernng equty proporton s generally avalable. If no nformaton s avalable for one partcular fund we assgn t the equty proporton of ts peer funds n the same subgroup 6 n the brochure. Second, all funds wth three years of hstory or more are assgned a rsk category from 1-5 represented by a colored graph. An llustraton where a green flat lne represents the lowest rsk category (1) and a jagged red lne represents the hghest rsk category (5). For ths rsk measure we smply average the rsk category of the funds n each portfolo. Funds wth less hstory than three years are assgned the average rsk category of ts peers n the same subgroup n the brochure. Thrd and fourth, next to the rsk category llustraton there s a number representng the annual standard devaton calculated usng returns for the past 36 months. We use ths number to construct two rsk measures: the weghted average standard devaton of the funds n a portfolo wth and wthout consderng the covarance between these funds. Although the covarance between funds s not ncluded n the general nformaton gven to all nvestors there s no stoppng them n gatherng ths nformaton on ther own. Also, we can not rule out the possblty that nvestors have a certan feelng for correlaton between sectors. We therefore use both measures n our tests. All funds do not have 36 months of hstory, wherefore we extrapolate a rsk measure for these funds by assgnng t the average 36 month standard devaton of the funds n ts subgroup. Independent varables 5 For a detaled analyss over default nvestors see Engström and Westerberg 2003 6 The brochure assgns each fund to a subgroup consstng of funds wth smlar allocaton objectves (.e., Swedsh growth stocks or European value stocks) 10
Our prmary focus s to nvestgate to what extent nvestment horzon affects asset allocaton wth regards to rsk. The nvestment horzon n ths partcular nvestment s 65 mnus the nvestors age snce ths nvestment s for retrement purposes only and can not be passed on to any thrd party. In our model we wll use the logarthm of nvestment horzon, snce the dfference between 5 and 4 years left to retrement represents a 20 percent reducton n tme; whereas, the dfference between 35 and 34 years s only a 2.5 percent reducton n tme left to retrement. In theory, nvestment horzon should not affect ones rsk. We lst fve factors that explan why nvestment horzon appears to affect rsk that we mentoned n the prevous secton and lsted n table 1. Frst, Samuelson states that one must control for total wealth, meanng lqud wealth + human captal. Fnancal wealth s ncluded as an explanatory varable. We use the logarthm of net-wealth assumng a concave utlty functon. Net wealth s made up of four components: market value of low rsk assets plus market value of rsky assets plus market value of real estate less debt. The survey used to calculate ths partcular data (HEK 2000) ncludes foregn as well as domestc assets and debt. The market value of real estate s estmated usng tax and comparable sales data. The market value of a house s estmated as the tax assessment value tmes the rato of market prce to assessed value usng data from recent sales prces of houses n the same area. In Sweden, condomnums are not assessed for taxaton purposes. The market value of condomnums s estmated as the average value of the recently sold condomnums n the mmedate area. Accordng to Samuelson, net wealth or, lqud captal as he calls t, s only one part of total wealth. The other part s human captal. Human Captal can be defned as a dscounted present value of expected dsposable ncome (see Halek and Esenhauer 1999, Poterba et al 2003 and Cocco and Gomes 2005). E( Dsposable ncome ) ( age ) Z (1) Expected dsposable ncome can be seen as a functon of age and a vector of other ndvdual characterstcs ( Z ). Usng nformaton on demographcs wthn our dataset we use parameters from OLS regressons presented n table 4 to forecast expected dsposable ncome for all ndvduals. For each nvestor we derve expected dsposable ncome for each year untl retrement. We do ths by allowng age to ncrease one year at a tme whle holdng all 11
other varables constant at average values for the sample. The dscounted value of expected dsposable ncome represents an ndvduals non-tradable asset; human captal. Human Captal 65 age E( dsposable ncome ) t t 1 (1 r ) t (2) When estmatng the present value of future dsposable ncome the dscount factor used ( r ) should correspond to the probablty of recevng ths future ncome. We address ths challenge by estmatng human captal separately for dfferent occupaton groups, assumng that the approprate dscount factor s farly smlar for ndvduals wthn the same occupatonal cohort. For example, we estmate expected dsposable ncome for all employees n the publc sector. Once we have these estmates we calculate the present value for each ndvdual. If we compare ndvduals wth approxmately the same occupatonal rsk, the sze of the dscount factor wll be smlar sze for all ndvduals wthn the same occupatonal cohort. We use the same dscount factor as Halek and Esenhauer 1999, namely 2% and calculate the present value of dsposable ncome separately for all four occupatonal cohorts. Because of ths, our results from the dfferent cohorts are not comparable, but they serve as robustness checks to verfy whether the coeffcent sgns and levels of sgnfcance tell the same story. We also estmate human captal usng the same dscount factor for the whole populaton regardless of occupaton. Consequently we have fve seres of estmates of human captal, one for each occupatonal cohort and one for the entre sample. Albet a nosy measure, we stll argue that t captures the essental porton of human captal. Bode s argument s that one can work more f nvestments go bad. A rsky nvestment gone poorly can be compensated by workng harder or consumng less over the remanng nvestment horzon. A longer nvestment horzon wll then allow hgher rsk. Snce human captal s related to age we expect a hgh level of multcollnearty n our model. We therefore orthogonalze the horzon varable wth human captal by usng the error term from the regresson: ln( 65 age ) 1human captal (3) 12
thus usng the porton of nvestment horzon whch s not explaned by our human captal related varable. Second, we consder the covarance between expected dsposable ncome and market performance. It s argued that horzons nfluence on rsk wll dffer dependng on the covarance between an nvestors earnng power and market performance. We use our four occupatonal cohorts mentoned earler as dummes. We thereby assume that the dsposable ncome for a government employed has a lower covarance wth the market than one who s employed n the prvate sector or s self employed. Consequently, a government employed may fnd reason to take more rsk n ths partcular nvestment than employees n the other occupatonal cohorts. In the four equatons where the occupatonal cohorts are estmated separately, ths partcular aspect s taken nto account by constructon. Thrd, Samuelson rases the ssue of how frequent wthdrawals need to be made from the nvestor s portfolo. The bass for hs argument s that the present value of a mnmum level of requred wealth at retrement s nvested n a rsk less fund and that ths nvestment wll become an ncreasng porton n ones portfolo. Consequently, the rsk less porton of penson savngs resembles a tme dversfcaton strategy. Therefore, we need to focus on penson savngs n excess of the porton requred for mnmum wealth at retrement to see f horzon affects rsk. The nvestments observed n our dataset are n excess of the porton requred for mnmum wealth at retrement and s therefore sutable to use n ths context. So, by default ths aspect s taken nto consderaton by the nature of ths data. Fourth and ffth, we are faced wth two other factors n explanng horzons effect on rsk: the nvestors relatve rsk averson and whether returns are IID or not. Based on classc economcs we assume that nvestors beleve returns are random walk and have constant relatve rsk averson. The Menu Investors choces are affected by how the alternatves are presented (Benartz and Thaler 2001). Each nvestor s gven a brochure ncludng nformaton on all nvestment alternatves. Table 2 s a representaton of how the funds are presented to the nvestor. In total 464 funds 13
are avalable 7. The funds are dvded nto four major categores; Equty, mxed, generaton and bond funds. We add all aspects of how the nvestment alternatves are presented n an attempt to control for the effect they may have on rsk. Portfolo optmzaton prmarly concerns rsk and return. We therefore control for hstorcal return reported n the brochure wth regards to the well documented momentum effect. We also control for the number of years of hstorcal return reported n the brochure and whether the fund s new. We nclude two normalzed varables from 0 1 to control for the order n whch the funds are presented. The funds are dvded nto subgroups representng regon or ndustry and then placed n an alphabetcal order n each subgroup. A fund n the frst subgroup n the brochure and startng wth the letter A wll consequently receve values close to 0. In the fund nformaton n the brochure the nvestor gets nformaton concernng the porton of domestc/foregn assets n the fund. Snce home bas s a known ssue n asset allocaton we control for ths aspect. Market cap and fee are also presumed to mpact nvestors choce and are therefore ncluded n the regresson. Snce our ntenton s to measure the tme-dversfcaton phenomenon, we need to control for a category of funds called generaton funds. Generaton funds are smlar to the suggested pre-set mx fund n the U.S. The nvestors that have chosen generaton funds could be seen as tme-dversfers by default f they choose the correct fund for ther nvestment horzon. Snce we are provded wth the detals of all nvestment choces we can control for those who have chosen generaton funds, whether they have chosen a correct mx wth regards to ther nvestment horzon or not. Method The purpose of ths paper s to test whether we can emprcally dscard the practce of tme dversfcaton. For ths purpose we use a unque database consstng of a heterogeneous populaton makng nvestment choces for future penson n a close to laboratory settng. The data we have provdes nformaton on the rsk level of a specfc nvestment bearng economc consequence and the correspondng horzon of ths nvestment, namely tme to 7 455 funds were ncluded n the orgnal brochure. Before the frst choce was completed some were added and some were taken away resultng n a total of 464 funds. 14
retrement. Because our sample suffers from selecton bas, n the sense that one thrd of our sample ended up nvestng n the default fund wth unknown rsk, we estmate our parameters wth the two step Heckman procedure where frst the lkelhood of nvestng s estmated from a probt model. The method may be descrbed by the followng two equatons: rsk x1 1 1, (4) * e x2 2 2. (5) Equaton (4) determnes the ndvdual s rsk, whereas (5) s a partcpaton equaton descrbng the ndvdual s propensty to work. Thus, rsk s the observed rsk for ndvdual f she partcpates n the penson system and * e s a latent varable that captures the propensty to partcpate n the penson system; x 1 and x 2 are vectors of observed explanatory varables, such as age and educaton; 1 and 2 are mean-zero stochastc errors representng the nfluence of unobserved varables affectng are 1 and 2. rsk. The parameters of nterest Although the latent varable * e s unobserved, we can defne a dummy varable e =1 f * e 0 and e = 0 otherwse; we thus observe ndvdual rsk only f e =1,.e. only f the ndvdual partcpates n the penson system. It s possble that the unobserved terms 1 and 2 are postvely correlated; ndvduals wth hgher rsk, gven x 1 and x 2, mght also be more lkely to partcpate n the penson system. If so, the sample of ndvduals that partcpate n the penson system wll not accurately represent the underlyng populaton. Heckman suggests the followng method to deal wth ths selecton problem. Note that the condtonal mean of 1 can be wrtten as: and hence E * ( 1 e 1 2 2 2 0) E( x ), (6) E( rsk x1, e 1) x1 1 E( 1 2 x2 2 ). (7) 15
Thus, the regresson equaton on the selected sample depends on both the condtonal mean of x 1 and 1 wll cause the estmates of 1 to be based (unless uncorrelated, n whch case the condtonal mean of 1 s zero). x 2. Omttng 1 and 2 are Under the assumpton that 1 and 2 are drawn from a bvarate normal dstrbuton, we can derve the regresson equaton: E( rsk x x, (8) 1, e 1) 1 1 1 In (8) s the correlaton coeffcent between 1 and 2, 1 s the standard devaton of 1, and, whch s the nverse mlls rato, s gven by 2 2 2 x2 2 / 2, (9) x / where and are the densty and dstrbuton functons of the standard normal dstrbuton and 2 s the standard devaton of 2. Heckman shows how to estmate (8) n a two step procedure. The frst step nvolves estmatng the parameters n (5) by the probt method, usng the entre sample. These estmates can then be used to compute for each ndvdual n the sample. Once s computed, we can estmate (8) over the sample of workng ndvduals wth an OLS regresson, treatng 1 as the regresson coeffcent for. When modellng the lkelhood of makng an actve nvestment choce we use experence wth rsky assets and the amount nvested n ths specfc nvestment 8 as explanatory varables. The nverse mlls rato from the probt estmaton s used n the second step estmaton wth rsk as the dependent varable and nvestment horzon beng the key explanatory varable and proxes for the factors lsted n table 1 as control varables. The database we use provdes sutable proxes for four of these 8 We also estmate the Heckman model wth a maxmum lkelhood procedure and retreve the same results. 16
factors: human captal, flexblty n human captal, covarance between earnng power and market performance and the frequency of requred wthdrawals from ones portfolo. We run heckman estmatons for each occupatonal cohort and one for the entre sample where we nclude dummes for occupaton. We do ths for fve dfferent rsk measures whch n total provde us wth 25 estmates for the horzon coeffcent. IV. Emprcal Results Estmatng expected dsposable ncome We estmate the present value of expected dsposable ncome n accordance wth equaton (1). Dsposable ncome s explaned by age, age^2, educaton level and major, gender, martal status and number of chldren. In table 3 we report the coeffcents and t-statstcs used n estmatng expected dsposable ncome. A hgher educaton level than hgh school (edl3), occupaton n the prvate sector (occ2) and beng marred (or cohabtant) are the only parameters wth sgnfcantly non-zero coeffcents for all groups. All parameters are used when calculatng expected dsposable ncome. For each ndvdual, all varables are held constant except age. A strng of expected dsposable ncome from current age to retrement s dscounted to a present value n accordance wth equaton (2), usng a constant dscount factor of 2%, whch s a proxy for the nflaton adjusted rsk free rate. As we dscussed earler, the dscount factor should vary n accordance wth the rsk of ones occupaton. We address ths ssue by estmatng the present value of expected dsposable ncome separately for each occupatonal cohort. By dong ths, we assume the same occupatonal rsk for all ndvduals wthn the same cohort. We expect the earnng power of those wthn the prvate sector to have a hgher covarance wth market performance than of those n the publc sector. Therefore we let the occupatonal cohorts n themselves act as proxes for covarance between market performance and earnng power. Summary statstcs of all varables ncludng our human captal proxes are reported n table 4. All rsk measures are contnuous varables whch makes them sutable to be estmated by the Heckman method. We note that the demand for equty s farly hgh, 90.3% equty on average. The hgh proporton of equty s reflected n the average portfolo standard devaton of ca: 18%. Investment horzons n the dataset span from 3 years to 46 years wth an average 17
of 21.97 years. These numbers reflect tme to retrement and capture a representatve sample of the workng force n Sweden. Compounded three year return s 143%, whch s exceptonally hgh n comparson wth hstorcal fgures. Ths hgh fgure reflects an unusually postve development for equty markets. Ths among other factors may explan the comparatvely hgh rate of partcpaton n the penson system (only one thrd default nvestors) and the demand for equty. Accordng to our human captal estmatons, human captal s lower for ndvduals n the publc sector than for ndvduals n the prvate sector. Also, the standard devaton for the human captal estmates s larger for ndvduals n the prvate sector than those n the publc sector. We fnd these results to be reasonable. Approxmately 61% of the populaton has prevous experence wth rsky assets, meanng equty or equty funds. Ths varable and the amount nvested n the penson system have proven to be of mportance n explanng penson system partcpaton (see Karlsson & Nordén 2004 and Engström & Westerberg 2003). Human captal s per defnton correlated wth age wherefore we orthogonalze the horzon varable so t reflects the porton of horzon not explaned by human captal. From the regressons n table 5 we observe large values for adjusted r-squares as expected. There s however stll a porton of horzon whch s not explaned by our proxes related to human captal. The error term s used as our horzon varable when estmatng horzons affect on rsk. For robustness we use four dfferent rsk measures as the dependent varable and redo our estmatons fve tmes, one for each occupatonal cohort and one for the entre populaton. Snce we are prmarly nterested n the horzon coeffcent we report only them n table 6 whle the full results can be found n the appendx. In table 6 we see that for the frst three rsk measures, proporton of equty, rsk category and smple average standard devaton all coeffcents are sgnfcantly non-zero at all conventonal levels of sgnfcance. These three rsk measures could be consdered to be less sophstcated but they are drectly observable n the nformaton gven to each nvestor. When we look at the fourth rsk measure, portfolo standard devaton ncludng the covarance, we 18
see that only two of the fve coeffcents are sgnfcantly non-zero at 5% sgnfcance level namely, coeffcents for the whole sample and for those employed n the prvate sector. In table 7 we calculate the expected rsk for a typcal nvestor n our sample gven our estmated coeffcents where all values are held constant accordng to sample averages and nvestment horzon takes the value 5 or 40. To be able to nterpret the results we do not use horzon orthogonalzed to human captal as n all other equatons. Usng the unorthogonalzed horzon ( ln(65-age)) wll cause the ndvdual coeffcents to be based due to mult co lnearty but our pont estmates for rsk wll stll be unbased. The dfference n rsk caused by a 35 year dfference n nvestment horzon, all else held equal, s reported under dfference n table 7. For the frst three rsk measures horzon appears to be mportant even n an economc context. However, regardng standard devaton when covarance s ncluded, the dfference n rsk, caused ths 35 year dfference n nvestment horzon, s only one percentage pont for the entre populaton and for those employed n the prvate sector and undstngushable from zero for the other three cohorts. For ths rsk measure, nvestment horzon seems to have no sgnfcant mpact. V. Concluson Prevous research offers compellng arguments for and aganst tme dversfcaton. Arguments aganst tme dversfcaton are that f returns are IID, nvestors have no human captal and have constant relatve rsk averson then, nvestment horzon should not affect rsk. Arguments related to human captal n some way or another are used to explan why many studes show a postve relatonshp between rsk and nvestment horzon. These arguments stress that t may be ratonal to ncrease rsk as nvestment horzon ncreases but tme n and of tself does not decrease rsk. Arguments for tme dversfcaton attack the assumptons of IID returns and constant relatve rsk averson. If nvestors have a decreasng relatve rsk averson or of asset prces are mean revertng, then t may be optmal to let rsk be affected by ones nvestment horzon. We attempt to control for three of the explanatons offered by economsts as to how ths behavor can be justfed; Investors human captal, the covarance between ther earnng 19
power and the market and the frequency of requred wthdrawals. In accordance wth classc fnance lterature we assume that returns are IID and our nvestors utlty dsplay constant relatve rsk averson. Our results gve an overall ndcaton that nvestment horzon affects rsk even after our attempts to control for the three factors mentoned earler and controllng for how the nvestments are presented. However, regardng portfolo standard devaton, the rsk measure wth the hghest degree of sophstcaton, the horzon coeffcents are undstngushable from zero n three cases out of fve and very small n the other cases. It s not clear however, how aware non-professonal nvestors are of the covarance between funds. For the other three rsk measures; proporton of equty, average rsk category and average standard devaton (not ncludng covarance), all ndcate that nvestment horzon matters. Whether ths s due to mean revertng prces, nvestors havng decreasng relatve rsk averson or the mere fact that much of the advce n meda propagates strateges resemblng tme dversfcaton, we can not say. Our proxes for human captal receve coeffcent sgns n accordance wth theory whereas the covarance of ones earnng power wth market fluctuatons,.e. our occupaton dummes, have ether coeffcents that are ndstngushable from zero or coeffcents so small, they have very lttle economc sgnfcance. A full account of all coeffcents s found n the appendx. In summary, our results are somewhat confoundng. One the one hand, when referrng to less sophstcated rsk measures, the horzon coeffcent s sgnfcantly postve and seems to be mportant n an economc context. On the other hand, when referrng to a more sophstcated rsk measure whch ncludes the covarance between funds, nformaton whch sn t avalable n the brochure, then the horzon coeffcent s ndstngushable from zero n three cases out of fve and doesn t seem to be of great mportance n an economc context. 20
References: Arrow, K., 1965, Aspects of the Theory of Rsk Bearng (Helsnk: Yrjö Jahnsonn Saato) Benartz, S., R. Thaler, 2001, Naïve dversfcaton strateges n defned contrbuton savng plans, Amercan Economc Revew, 91(1), 79-98. Bode, Z., 1995, On the Rsk of Stock n the Long Run, Fnancal Analysts Journal, 51, 3, 68-76. Bode, Z., R. Merton, and W. Samuelson, 1992, Labor Supply Flexblty and Portfolo Choce n a Lfecycle Model. Journal of Economc Dynamcs and Control, 16, 3, 427-449. Brown, S., W. Goetzmann, and S. Ross., 1995, Survval. Journal of Fnance, 50, 853-873. Campbell, J., L. Vcera, 2002, Strategc Asset Allocaton Portfolo Choce for Long Term Investors, Clarendon Lectures n Economcs, Oxford Unversty Press Cocco, J., F. Gomes and P. Maenhout, 2005, Consumpton and Portfolo Choce over the Lfe Cycle, The Revew of Fnancal Studes, 18(2), 491-530. Engström, S., A. Westerberg, 2003, Whch Indvduals Make Actve Investment Decsons n the New Swedsh Penson System? Journal of Penson Economcs and Fnance, 2, 3, 225-245. Esenhauer, J., and M. Halek, 1999, Prudence, Rsk Averson, and the Demand for Lfe Insurance, Appled Economcs Letters 6, 239-242. Fredman, M., 1959, The Demand for Money: Some Theoretcal and Emprcal Results, Journal of Poltcal Economy, 67, 327-351. Hansson B., M. Persson, 2000, Expected Utlty Maxmzaton and Tme Dversfcaton. Workng paper, Lund Unversty. Krtzman, M., 1994, What Practtoners need to know About Tme Dversfcaton, Fnancal Analysts Journal, 50, 1, 14-18 21
Krtzman, M. and D. Rch, 1998, Beware of Dogma, The Journal of Portfolo Management, 24, 66-77. Latane, H., 1963, Income Velocty and Interest Rates: A Pragmatc Approach, Revew of Economcs and Statstcs, 42, 445-449. Lee, W., 1990, Dversfcaton and Tme, Do Investment Horzons Matter?, The Journal of Portfolo management, 4, 60-69. Levy, H., 1994, Absolute and Relatve Rsk Averson: An Expermental Study, Journal of Rsk and Uncertanty 8, 289-307. Meltzer, A., 1963, The Demand for Money: The Evdence From Tme Seres, Journal of Poltcal Economy, 71, 219-246. Poterba, J., Rauh, J., Vent, S. and D. Wse, 2003, Utlty Evaluaton of Rsk n Retrement Savng Accounts, NBER workng paper seres, workng paper 9892. Pratt, J., 1964, Rsk Averson n the Small and n the Large, Econometrca, 32, 122-136. Samuelson P., 1963, Rsk and Uncertanty: A Fallacy of Large Numbers, Scenta, 98, 108-113. - 1989, The judgment of economc scence on ratonal portfolo management: Indexng, tmng, and long horzon effects, journal of portfalo management 16, 4-12 - 1994, The long term case for equtes and hoe t can be oversold, journal aof portfolo management, 21, ssue 1, 15-24 Selden, R., 1956, Monetary Velocty n the Unted States, n Studes n the Quantty Theory of Money, M. Fredman (ed.), Unversty of Chcago Press, Chcago, Ill. Segel, F. W., and J. Hoban, 1982, Relatve Rsk Averson Revsted, Revew of Economcs and Statstcs 64, 481-487. Segel, J., 1994, Stocks for the Long Run. Irwn 22
Strangeland, D., H. Turtle, 1999, Tme Dversfcaton: Fact or fallacy, Journal of Fnancal Educaton, Fall 1999. Szpro, G., 1986, Measurng Rsk Averson: An Alternatve Approach, Revew of Economcs and Statstcs 68, 156-159. Thorly, S., 1995, The Tme-Dversfcaton Controversy, Fnancal Analysts Journal, 51, 3, 18-22 23
Table 1. Factors that explan horzons effect on rsk Factors Accordng to: zero horzon coeffcent f postve horzon coeffcent f negatve horzon coeffcent f 1. Non-tradable assets (human captal) P. Samuelson 1994 Bode, Merton, W. Samuelson 1992 Investors have no human captal Human captal decreases wth tme 2. Covarance between earnng power and market performance Strangeland and Turtle 1999 Investors have no human captal Investors human captal has a negatve covarance wth market performance 3. Frequency of requred wthdrawals from portfolo P. Samuelson 1989 Rsk less component of savngs s excluded One future wthdrawal s requred Perodc wthdrawals are requred to fnance everyday expendtures 4. Rsky asset return process Expected Utlty Securty returns are IID Securty prces dsplay mean reverson Securty prces dsplay mean averson 5. Investors relatve rsk averson Expected Utlty Utlty dsplays constant relatve rsk averson Utlty dsplays decreasng relatve rsk averson Utlty dsplays ncreasng relatve rsk averson
Table 2: Extract from the nformaton folder, fund example Fund number Fund name, Management company Informaton regardng the funds Market cap MSEK 99-12-31 Fund fee (%) Percentage return 99-12-31 (after fees) Total rsk In the year Last 5 (last 3 95 96 97 98 99 years years) 191080 Barng Global Emergng Markets Barng Internatonal Fund Managers (Ireland) Ltd Emergng markets equty and equty related assets 3 896 1.59-32 10 25-25 77 25.3 32 (Red) The percentage return for the last fve years equals the compounded annual growth rate of return for the years 1995 through 1999. The total rsk corresponds to an annualsed percentage standard devaton of three-year monthly hstorcal fund returns. The total rsk s also categorsed nto fve dfferent classes, and colours, wth respect to standard devaton; Class 1: very low rsk, dark green, percentage standard devaton n the range 0-2; Class 2: low rsk, lght green, 3-7; Class 3: average rsk, yellow, 8-17; Class 4: hgh rsk, orange, 18-24; Class 5: very hgh rsk, red, 25-.
Table 3, results from OLS regressons on expected dsposable ncome All OCC 1 OCC 2 OCC 3 OCC 4 age -1 664 6 627-4 842 1 303-10 763-0.47 2.70-0.82 0.17-1.31 age2 53-47 100 13 140 1.26-1.62 1.41 0.15 1.44 edl1-42 167-12 183-63 754 3 754-12 338-2.81-1.04-2.58 0.18-0.34 edl3 99 885 75 309 123 258 60 124 106 286 9.11 11.14 6.56 2.51 3.25 edm2-45 904 14 345-74 596-3 402-31 790-3.30 1.21-3.47-0.12-0.80 edm3-16 023 3 181-34 575-2 580-28 773-1.06 0.43-1.01-0.07-0.65 edm4-18 898-814 -29 999-17 583-27 252-1.31-0.09-1.24-0.65-0.69 occ2 91 761 8.04 occ3-8 989-0.40 occ4-6 808-0.42 gender 3 834-8 232 17 061-24 008-6 193 0.37-1.15 1.01-1.35-0.23 marred 129 845 152 821 121 142 139 752 109 742 11.12 19.43 6.12 6.84 3.89 chldren 16 014 4 057 19 015 10 089 42 483 3.21 1.26 2.26 1.17 3.05 _cons 150 062-23 310 294 830 93 478 353 605 2.15-0.47 2.58 0.59 2.18 n.obs 15651 4486 8736 745 1648 mean dsp 351 938 339 142 380 565 286 746 266 360 prob > F 0.0000 0.0000 0.0000 0.0000 0.0000 adj. R-sq 0.0291 0.1422 0.0208 0.0981 0.0275 Age and age^2 relate to the ndvduals age 31 Dec 2000. Edl1 = less than hgh school educaton, edl2 (default) = hgh school educaton, edl3 = more than hgh school educaton. Edm1 (default) = socal scence major, edm2 = techncal engneer major, edm3 = major n medcne and edm4 = unknown major. Occ1 (default) = employed n publc sector, occ2 = employed n prvate sector, occ3 = self employed and occ4 = sector unknown. Gender 1 = man, 0 = woman. Marred or cohabtant = 1, sngle = 0. Chldren, refers to number of chldren. N.obs s the number of observaton n the entre sample and n each occupatonal cohort. Mean dsp = the average dsposable ncome for the entre sample and each occupatonal cohort. Prob>F = the probablty that all coeffcents are collectvely ndstngushable from zero. Adj. R-sq = the adjusted R-square of the regressons. The t-statstcs are reported under each coeffcent n talcs.
Table 4. Summary statstcs, average values Sample Mean std. Dev. Mn Max Rsk measures equty 10 375 0.903 0.169 0 1 rsk category 10 375 3.811 0.574 1 5 std (no cov) 10 375 0.188 0.050 0 0.53 std 10 375 0.179 0.043 0 0.77 Prmary varable of nterest nvestment horzon 15 651 21.97 11.11 3 46 Control varables compounded three year return 10 375 143% 0.52-24% 670% Human captal: whole sample 15 651 6 369 560 2 913 742 558 535 14 600 000 employed n publc sector 4 486 5 321 516 2 424 894 46 211 26 900 000 employed n prvate sector 8 736 7 352 775 2 994 426 872 455 15 800 000 self employed 745 4 583 743 2 058 392 549 196 10 100 000 employment unknown 1 684 4 531 500 2 841 073 551 396 12 800 000 net wealth 15 651 444 108 3 174 320-30 000 000 355 000 000 Selecton varables for probt estmaton n heckman model experence wth rsky assets 15 651 n.a. n.a. 0 1 amount nvested 15 651 12 858 6 647 210 26 202 Full sample = 15 651 observatons, sub-sample ncludng only those who made an actve portfolo choce = 10 375 observatons. When estmatng human captal for the four occupatonal cohorts we only use ndvduals n that cohort. Rsk measures nclude four varables; amount nvested n equty / total nvested amount (equty), the average rsk category accordng to the nformaton n the brochure (rsk category), weghted average standard devaton wthout consderng covarance (std (no cov)) and portfolo standard devaton ncludng the covarance (std). Prmary varable of nterest s nvestment horzon = 65-age. In the heckman regresson, we use the log of nvestment horzon orthogonalzed wth regards to human captal n accordance wth equaton (3) Control varables nclude compounded three year return, our estmates for human captal and net wealth whch s fnancal wealth + real estate debt. Selecton varables for probt estmaton n heckman model nclude a dummy varable for prevous experence wth rsky assets, meanng equty or equty funds. 61 % of the populaton was exposed to rsky assets pror to ths nvestment decson. Amount nvested represents the kronor amount nvested n ths partcular nvestment. Menu varables 10 varables from the brochure are ncluded n all regressons. Full results reported n appendx tables A1-A5. 27
Table 5. Results from regresson explanng horzon wth human captal all occ1 occ2 occ3 occ4 hc 2.95e-07 4.53e-07 2,57e-07 5.32e-07 7.11e-07 _cons 1.9129 1.6041 1.9309 1.5340 1.3350 n.obs 15651 4486 8736 745 1684 Prob > F 0.0000 0.0000 0.0000 0.0000 0.0000 adj. R-sq 0.6264 0.7958 0.6245 0.7938 0.8294 Hc = human captal or the present value of expected dsposable ncome. N.obs s the number of observaton n the entre sample and n each occupatonal cohort. Prob>F = the probablty that all coeffcents are collectvely ndstngushable from zero. Adj. R-sq = the adjusted R- square. Occ 1 represents employees n the publc sector, occ2 represent employees n the prvate sector, occ3 represent self employed and occ4 are of unknown employment. Table 6, Coeffcent for nvestment horzon all occ1 occ2 occ3 occ4 equty 0,0765 0,08908 0,0726 0,0580 0,08032 20,68 10,47 16,39 2,36 5,36 rskcat 0,1689 0,1716 0,1454 0,2071 0,2144 16,81 7,61 12,12 2,71 5,13 std (no cov) 0,0145 0,0142 0,0135 0,0157 0,0161 15,91 7,13 12,42 2,08 4,36 std 0,0057 (0,0030) 0,0049 (0,0108) (0,0075) 6,02 1,78 4,39 0,97 1,9 Coeffcents n parenthess are not sgnfcantly separate from zero on a 5% level. The results n table 6 we report the coeffcents for porton of nvestment horzon not explaned by human captal. As a robustness check we run the regressons usng fve separate rsk measures; number of funds, amount nvested n equty / total nvested amount (equty), the average rsk category accordng to the nformaton n the brochure (rsk cat), weghted average standard devaton wthout consderng covarance (std (no cov)) and portfolo standard devaton ncludng the covarance (std). The Heckman regresson s estmated separately for each occupatonal cohort and once for the entre populaton. Occ 1 represents employees n the publc sector, occ2 represent employees n the prvate sector, occ3 represent self employed and occ4 are of unknown employment. The t-statstcs are reported under each coeffcent n talcs. 28