Volatility of Stock Return Variance and Capital Gains Tax

Transcription

1 Volatlty of Stock Return Varance and Captal Gans Tax Xa Meng, Junbo Wang, Zhpeng Yan, Yan Zhao 1 Ths Draft: Oct 17, 011 bstract In ths paper, we develop a two-perod portfolo selecton model wth dfferental captal gans tax rates. Our focus s the relatonshp between the volatlty of stock return varance and an nvestor s ntertemporal decson of realzng captal gans/losses. We predct that a hgher volatlty of stock return varance wll lead to a larger short-term captal loss realzaton, a larger rato of short-term captal gan realzaton to the total net captal gan, and a larger tradng volume. Those predctons are consstent wth the emprcal evdence. In addton, we also demonstrate that hgher volatlty of stock return varance leads to hgher tradng volume. 1 Xa Meng s a Ph.D student, Internatonal Busness School, Brandes Unversty. Emal: xameng@brandes.edu. Junbo Wang s a Ph.D student, Marshall School of Busness, Unversty of Southern Calforna. Emal: Junbo.Wang.013@marshall.usc.edu. Zhpeng Yan s an assstant professor, School of Management, New Jersey Insttute of Technology. Emal: zyan@adm.njt.edu. Yan Zhao s an assstant professor, Department of Economcs, Cty College, CUNY. Emal: yzhao@ccny.cuny.edu. We would lke to thank arry Ozanne at Congressonal Budget Offce for provdng the data of captal gans realzed by US ndvdual taxpayers, as well as detaled descrptons. We would lke to thank Blake ebaron, Carol Osler, and Jens lscher at Brandes Unversty for ther valuable advce and suggestons. 1

2 1 Introducton The exstng lterature on the captal gans/losses s mostly devoted to the longterm captal gans. t the same tme, the studes on the short-term captal gans are almost non-exstent n the lterature largely because most realzed gans are long-term ones. (uerbach and Poterba (1988)). owever, the relatve mportance of the short-term captal gans compared to total net captal gans s more than doubled after the late 1980s n the Unted States. lthough the long-term gan realzatons are stll domnant n the total captal gan realzatons, t would be amss to gnore short-term gans any more. In ths paper, we am to fll the gap n the exstng lterature by examnng factors affectng short-term captal gans/losses realzatons. In partcular, we develop a smple theorectcal model explanng the phenomenon of ncreasng short-term captal gans relatve to total net captal gans both from the perspectve of tax effect and from a novel aspect of the stock market - the volatlty of stock varance. standard mean-varance framework provdes the ntuton for the argument that the volatlty of stock return varance nfluences the ntertemporal decsons of nvestors to realze ther captal gans or losses. Consder two hypothetcal economes wth constant and tme-varyng stock return volatlty, respectvely. In the frst economy, all stocks have constant expected returns and varances. Thus for any stock, the volatlty of stock return varance s zero. In the second economy, the expected stock returns are constant but stock varances change over tme. If nvestors choose ther portfolos accordng to the mean-varance framework, nvestors n economy 1 wll not change ther portfolo weghts once they are set, and wll not conduct any short-term tradng, whle nvestors n

3 economy must change ther portfolo weghts over tme n respondng to the changes n stock varances, whch may lead to short term tradng actvtes. Wthout consderng tax effects, a typcal nvestor may realze ether gans or losses short term. owever, wth the tax benefts of short term captal losses and the tax dsadvantage of short term captal gans n mnd, an nvestor n economy may be wllng to realze more short-term losses rather than short-term gans. If the long-term captal gans and loss are roughly the same n the two economes, economy wll have a hgher rato of short term captal gans to total net captal gans,. In addton to the changes n the stock market condtons, the dfferental captal gans tax rates may also play an mportant role n the nvestors ntertemporal decsons. In the Unted States, the captal gans tax rate for ndvduals s lower on long-term captal gans, whch are gans on assets held longer than one year. s of 011, the tax rate on long-term gans s 15 percent, whle short-term gans are subject to ordnary federal ncome tax, whch ranges up to 35 percent. Wth the tax rate on long term captal gans and losses beng sgnfcantly lower than that on short term captal gans and losses, the tax law provdes a tmng opton to realze losses short term and realze gans long term, f at all. Constantndes (1984) llustrates that ths tmng opton s more valuable the hgher the stock varance, for essentally the same reason that a call opton s more valuable the hgher the stock varance. In ths study, we extend the work of Constantndes (1984) by ncludng the volatlty of stock return varance as an explanatory factor for the changng pattern of nvestors ntertemporal decson. We frst buld a two-perod two-economy portfolo selecton model wth tax. We show that n a world wthout any tax, an nvestor who 3

4 maxmzes her quadratc utlty adjusts the weght n rsky assets more sgnfcantly between two tme perods n the economy wth hgher volatlty of stock return varance. The bgger adjustment leads to a larger amount of short term gans and short-term losses. owever, when the tax effect s taken nto account, due to the tax dsadvantage of realzng gans short term and tax benefts from realzng losses short-term, one should expect that the short term gans can ncrease, decrease or reman unchanged but the short term captal losses should ncrease. s a result, the rato of short-term captal gans realzaton to long- term captal gans net of short-term losses ncreases wth the volatlty of return varance. We also llustrate that hgh volatlty of return varance can lead to hgh tradng volume. We then check the key assumptons of our model and test the model s mplcatons n the real world. We collect NYSE/MEX/NSDQ daly value-weghted market return from CRSP and annual total captal gans and losses realzed by U.S. ndvdual taxpayers from the Department of Treasury for the perod of The whole sample perod s dvded nto two sub-perods: and We fnd that the volatlty of stock market return varance ncreases sgnfcantly after In addton, the short-term gans (normalzed by the value of stock market ndex at the begnnng of the year) do not ncrease, whle the long-term gans net of the short-term losses decrease. Many prevous studes have examned the mpact of captal gans taxes on stock returns and tradng volume. Few studes nvestgate the relaton between captal gans For examples, Feldsten et al. (1980), Reese (1998), Dhalwal and (006), Collns and Kemsley (000), Bloun, et al. (009), Jn (006) among others. 4

5 taxes and the second moment of stock returns 3. Even fewer studes, f any, nvestgate the possblty that the causalty can run from the change n stock market rsk to captal gans/losses realzaton. In ths paper, we are focused on the volatlty of stock return varance and llustrate theorectally that t, together wth dfferental captal gan taxes, can affect nvestors ntertemporal decsons. It was generally beleved that nvestors should defer ther long-term captal gans realzaton n order to mnmze net present value of ther tax payments ( taxes deferred are taxes saved ). s a consequence, nvestors may be less lkely to change ther portfolos - the so-called lock-n effect phenomenon. Constantndes (1983) derves the optmal tradng strateges for nvestors when the tmng opton exsts. In another mportant paper on tax optons, Constantndes (1984) ponts out that when tax rate on long-term gans and losses s relatvely low (compared wth short-term tax rate), taxable nvestors should realze long-term gans n hgh varance stocks and repurchase stock n order to realzed potental future short-term losses. In a more recent study, Da et al. (010) nvestgate the effect of changes n captal gan taxes on stock return volatlty. They fnd, after passage of the 1978 and 1997 captal gan tax rate reductons, larger ncreases n the return volatlty for more apprecated stocks than for less apprecated stocks possbly due to the reduced rsk sharng (by the government) and the reduced future captal gan taxes. They also fnd larger ncreases n the return volatlty for non-dvdend-payng stocks than for dvdend-payng stocks. Our work dffers from these studes, and explores the mpact on the captal gan/loss realzaton from both the volatlty of stock return varance and the dfferental tax rates on the long-term and 3 Da et al. (010), whch examnes changes of stock return volatlty around captal gans tax reducton events, s a rare excepton that focues on the second moment of stock returns. 5

6 short-term captal gans. Our paper s also related to a large lterature that connects tradng volume to stock return volatltes. Emprcally, researchers fnd that the correlaton between the tradng volume and the return volatltes s sgnfcant, e.g. Clark(1973), Epps and Epps (1976), Tauchen and Ptts (1983), Bessembnder and Segun (1993). Brock and ebaron (1993) and nderson (1996) buld models to explan Mxed Dstrbuton ypothess. Ther models are based on mcrostructure theores that the arrval of nformaton leads to a correlaton between the tradng volume and the volatlty of stock returns. Therefore, the causalty between the tradng volume and the return volatlty s spurous. Our paper suggests that the tradng volume can be determned by the change n the estmated return volatlty (e.g., n some techncal and quanttatve tradng strateges). fter the market crash n 1987, the daly volatlty has been changng more rapdly. ence, nvestors would lke to rebalance ther portfolos more sgnfcantly compared to the perod before Ths leads to a hgher tradng volume. akonshok and Smdt (1989) document the effect of past stock prce patterns on current tradng volumes and show the causalty between the prce and volume. They fnd that the past wnner stocks have hgher tradng volumes. Our paper s dfferent from thers n that we document the effect of change of stock return varance on the tradng volumes. The contrbuton of our paper s two-fold. Frst, our model s the frst to lnk the volatlty of stock return varance to captal gans/losses realzatons. The predctons of our model are confrmed by the U.S. captal gans/losses realzaton data. Second, we demonstrate that hgher volatlty of stock return varance leads to hgher tradng volume. The rest of ths paper s organzed as follows. Secton presents a two-perod 6

7 portfolo-selecton model and derves man predctons. Secton 3 descrbes our data and reports emprcal tests. Concludng remarks are offered n Secton 4. Model In ths secton, we propose a two-perod portfolo-selecton model wth tax to explan the larger the amount of short term captal loss, the hgher the rato of short-term captal gan to the total net captal gan after Snce the volatlty of daly return varance ncreases after 1987, the nvestors need to adjust the weghts n stocks more sgnfcantly n respondng to the volatle changes n return varance. Consequently, they generate both hgher gans and losses n the short-term (the volatlty effect ). owever, a ratonal nvestor may be reluctant to realze the gans short term snce the hgher tax bracket makes the short-term captal gans less attractve. But the nvestor s wllng to realze the short-term losses because t can ncrease the tax benefts of the long-term gans 4 (the tax effect ). The combnaton of the two effects can explan the followng three facts observed before and after 1987 n the Unted States. 1). the short-term gans realzaton (total wealth adjusted) does not change sgnfcantly; ). snce the short-term losses realzaton ncreases and f the level of long-term net gans (total wealth adjusted) are relatvely stable over tme, the long-term net gans mnus short-term losses plus short-term gan (.e., total net captal gans) decreases; 3) the rato of short-term gans to total net gans ncreases. There s a representatve nvestor who has two accounts. The frst account 4 The short-term gans are taxed at the ordnary ncome tax rate, whch s currently 35 percent n the hghest tax bracket, whle the long-term net gans mnus the sum of net short-term captal losses and any long-term captal loss carred over from the prevous year are taxed at the long-term captal gan tax rate, whch s 15 percent n the hghest tax bracket. Therefore, nvestors have ncentve to realze short-term losses but not short-term gans consderng the tax effects. 7

8 conssts of buy-and-hold portfolos. The returns of these buy-and-hold portfolos wll be realzed n the long run; hence, the proceeds are long-term gans. In addton, We assume the returns of these portfolos are not affected by the market rsk 5. More specfcally, we assume that the net gans of the frst account are the net long-term gans and are denoted by P. The second account conssts of actvely managed portfolos. These portfolos are subject to the adjustment of the weghts n the relatvely short perod. The proceeds of these transactons are the short-term gans or losses. In ths account, the nvestor chooses portfolos by followng a standard portfolo-selecton model to maxmze the quadratc utlty. Suppose there are two assets n the market, one rsk free asset and one rsky asset. The value of the rsk free asset does not change over tme,.e. the return s 0( r = 0 ). The rsky asset has an expected return µ ( µ > 0 ) and the standard devaton of the return σ n each perod. In order to smplfy the model, we assume that the return of the rsky f asset can be ether µ + σ or µ σ wth equal probabltes. The expected return s fxed so that µ s constant. We assume that the varance of the stock return σ s varyng over tme, whch can be ether σ (1+ ) or σ (1 ) wth equal probabltes 6. ence, the mean and varance of the return varance s σ and σ 4. So the volatlty of return varance s hgher when s hgher. There are two perods, n perod t (t=1,). One can thnk of 1 perod as short term (shorter than one year) and perods as long term. In each perod, the nvestor maxmzes the quadratc utlty by choosng the weght of the rsky asset ( w ) n portfolo: 5 In general, the buy-and-hold portfolos are safer than actvely managed portfolos. We assume that these portfolos have no rsk for smplcty. 6 We assume 0 < < 1 so that there s no negatve varance. 8

9 max E( r w( t) p 1 ( t)) var( rp ( t)), (1) where r p (t) s the return of the portfolo formed by the rsk free asset and the rsky asset wth weghts 1 w( t) and w (t), respectvely. Then E( r The maxmzaton problem yelds p ( t)) = w( t) µ + (1 w( t)) r µ w σ ( t) f =, var( r. p ) = w( t) σ ( t) t ths moment, we assume that there s no tax on any transacton. We wll relax ths assumpton later. Now suppose there are two economes: and (e.g., represents the economy after 1987 and represents the economy before 1987), whch are dentcal n terms of. economc agent, utlty functon and parameters µ and r f, except for the varance of the return of the rsky asset. Suppose that the return varance s more volatle n economy,.e. >. In the dscusson above, we assume that n the two economes the rsky asset s returns have the same mean of varance. But the emprcal work n the next secton ndcates that the economes before and after 1987 have sgnfcantly dfferent means of the varance,.e. σ > σ. But the emprcal evdence also mples that 9

10 . () σ σ Then, we can show that all the followng results hold as long as equaton () s satsfed., even f the mean of the varance s dfferent between the two economes. If we gnore the tax effect, then the dfference n the two economes leads to the dfference n the standard devaton of portfolo weght adjustment n the second perod ( w () w (1) for economy, =, ): Theorem.1 Suppose there are two economes, both of whch have a representatve nvestor who maxmzes the quadratc utlty by selectng the weght of portfolo,.e. maxmze the utlty from equaton 1. The two economes are dentcal except that the volatlty of return varance n economy s hgher, then the standard devaton of the weght adjustment n the second perod(defned as std( w() w(1)) ) s hgher for economy. The proof of all the theorems and propostons are n the appendx. One mmedate concluson from ths theorem s: f the volatlty of return varance s larger, the adjustment n weght of the rsky asset between the two perods wll be larger, e.g. f >, gven and std( σ () ) = σ (1+ ), std( σ (1) ) = σ (1 std( σ () ) = σ (1+ ), std( σ (1) ) = σ (1 )), the absolute change n the weght n the two economes are: ), 10

11 w () w (1) = ( µ ), w () (1) = ( ). w µ σ (1 ) σ (1 ) Snce >, t s clear that the change of weght n rsky asset s hgher n economy. Intutvely, snce there s no tax, the larger adjustment n the weght of rsky asset n economy wll lead to a hgher tradng volume n the rsky asset, thus wll lead to hgher realzed short-term captal gans or loss n each perod. To derve ths result, we need to make some further assumptons. The key assumpton n ths paper s that µ = o(1) and σ = o(1). Ths assumpton mples that for any and any possble outcomes of return r (1), r (1) = o(1). Therefore 1 1 (1+ r (1)) ) = 1 o(1) The ntuton of ths assumpton s that the change n prce n each perod s close to zero and s much smaller than the the volatlty of the return varance for both economes. µ nother assumpton s that = O(1). σ Ths assumpton guarantee that the weght for stock n the portfolo s not neglgble. Suppose that the prces of rsk free and rsky assets are both 0 at tme t = 1, thus, the nvestor n economy (=,) wll buy µ / (1) shares of rsky asset. Then the value of portfolo at tme t = s σ µ µ + (1+ r (1)), σ (1) σ (1) 1 where r (1) s the return of the rsky asset at t = 1 and t can be ether µ + σ (1) or 11

12 µ σ (1) wth equal probabltes. Snce at tme t =, the standard devaton of return µ changes to σ (), so the weght n rsky asset becomes. () σ ence, the change n number of shares of rsky asset at tme t = s µ µ µ (1 + (1+ r (1)) σ () σ (1) σ (1) µ. (1+ r (1)) σ (1) Rearrange the formula above, we have the change n number of shares of rsk asset expressed as µ 1 µ 1 µ (1 ) + ( ). σ (1) σ () (1+ r (1)) σ () (1+ r (1)) σ (1) s one can see from the rearrangement, the frst part denotes the changes n shares due to the change n prce, we call t prce effect. The second part denotes the changes n shares that are resulted from the change n the return varance, we call t volatlty effect. Usng the two assumptons above, one can derve the followng proposton: Proposton. If the assumptons above are satsfed, suppose σ () σ (1), then for any µ 1 µ 1 µ (1 ) = o(1). σ (1) σ () (1+ r (1)) σ () (1+ r (1)) σ (1) From ths proposton, under the assumpton that the prce s changng much smaller than the return varance, the prce effect s much smaller than the volatlty effect when there s a volatlty change at tme t =. Thus, the changes of number of shares n 1

13 rsky asset are prmarly due to the changes n the volatlty of the asset. The short-term gan s defned as the realzed captal gan through nterm tradng. µ et S (1), σ (1) µ 1 µ 1 and (1 ) + S (). σ (1) σ () (1+ r (1)) σ () (1+ r (1)) If S (1) > S (), then the short-term gan s STG (( S (1) S ()) r (1),0) + ; the loss s ST (( S (1) S ()) r (1),0). If S (1) < S (), then the short-term gan s STG (( S () S (1)) r (),0) + ; the loss s ST (( S () S (1)) r (),0) ; To be more specfc, f the nvestor sells shares at tme t =, then the nvestor wll ether gan or lose S (1) S ()) r (1). If the nvestor buys shares at tme t =, then the gans ( and loss wll be realzed at tme t = 3, the amount of the gan or loss wll be ( S () S (1)) r (). Usng ths defnton, we have our man result: Theorem.3 In a two-perod model, assume that: 1) economy and have the same economc agent wth the same utlty functon, the same parameters of market 13

14 except that economy has a larger volatlty of varance of rsky return; 7 ) the rsky return n the frst perod r (1) (=,) satsfes r (1) = o(1) ; 3) the mean and average varance of rsky return, µ and µ σ σ, satsfy = O(1). The expected short-term captal gan s hgher n economy than n economy,.e. E ( STG ) > E( STG ). Smlarly, the short-term loss s also hgher n economy than n economy,.e. E ( ST ) > E( ST ). Next step, we nclude the tax effect. Suppose that there s no tax for long-term gans and the tax rate for short-term gans s hgher for larger short-term gans because of dfferent tax brackets. To smplfy, assume that there s no tax for both economes f there s a relatvely small amount of gans due to small adjustment n portfolo weghts; f there s a larger amount of gans, the gans wll be subject to a specfc tax rate,.e. f the short-term gans STG satsfes STG µ 1 1 ( )(1+ µ + σ 1 σ no tax s mposed on the gans; f the gans are larger, the tax rate t satsfes ), ( )(1 t) = (3) Under ths assumpton, we have the followng theorem: Theorem.4 In a two-perod model, assume that: 1) economes and have 7 s s mentoned above and shown n detals below, ths assumpton can be relaxed and the relaxaton does not hurt our results as long as equaton holds. 14

15 the same economc agent wth the same utlty functon, the same parameters of market except that economy has a larger volatlty of rsky return varance; 8 ) the rsky return n the frst perod r (1) (=,) satsfes r (1) = o(1) ; 3) the mean and average varance of rsky return, µ and µ σ σ, satsfy = O(1) ; 4) the tax rate t satsfes equaton 3. The expected short-term captal gans are approxmately equal n economy 1 and n economy,.e. E( STG ) = E( STG )(1 o(1)). + Meanwhle, the short-term losses are hgher n economy than n economy,.e. E ( ST ) > E( ST ). From ths theorem, one can see that there s a tax beneft n economy snce the short-term gans are smaller through the adjustment n the weght of the rsky asset. lthough volatlty of return varance has a postve effect on short-term gans, hgher tax rate has a negatve effect. These two effects approxmately offset each other. Therefore, there s no sgnfcant change n short-term gans. owever, snce the losses s not affected by the tax rate, the short-term losses are hgher n economy. In addton, snce the short-term losses can ncrease the tax beneft of the long-term gans, one should expect that the nvestor wll at least have the ncentve to realze the losses n both economes. Furthermore, from Theorem.4 and the assumpton that the long-term gans are not affected by the market rsk and are constant n both economes, one can show that the 8 Smlar to Theorem.3, ths assumpton can be relaxed and the relaxaton does not hurt our result as long as equaton holds. 15

16 rato of short-term gans to total net gans s hgher n economy. Theorem.5 Under the same assumptons n Theorem.4, defne the total net gans TNG = E( STG ) E( ST ) P and the rato of short-term gans to total net gans + as STR E( STG ), TNG then the economy wth hgher volatlty of stock return varance has a hgher rato of short-term gans to total net gans,.e. for = or, we have STR STR. > Note that n ths proposton, we defne the total net gans as E( STG ) E( ST ) + P 9. Ths mplctly assumes P > E( STG ) E( ST ) to keep the total net gans postve 10. One mplcaton of ths model s to explan the emergence of hedge funds after The hgher provdes nvestors the ncentve to ether make larger adjustments n the weght of stocks or conduct tradng wth hgher frequency. The needs for hgh frequency tradng trgger the emergence of hedge fund, whch further leads to larger short-term gans or losses. 3 Data and Emprcal Test Gven that there s always a tax effect n the real world, we test the assumptons 9 P s the total long term net gan and E ( STG ) E ( ST ) s the total short term net gan. 10 Snce the long term gan domnates the short term gan n the real data, ths assumpton s consstent wth the data. 16

17 of Theorem.4 and Theorem.5 as well as ther predctons based on the real data. We have total captal gans and long-term captal gans realzed annually by US ndvdual taxpayers for the perod of from Department of the Treasury. For the same perod, we collect daly value weghted market returns from CRSP. We further dvde the whole sample perod nto two sub-samples: and The reason to use the year of 1987 as a watershed s that t wtnesses the largest one-day stock market crash n hstory. nd perhaps more mportantly, the stock market crash of October 1987 fundamentally changed the dynamcs of stock return volatlty. Pror to 1987, mpled volatltes of equty optons (both on ndvdual stocks and on stock ndces) were much less dependent on strke prce. Snce the crash, the volatlty surface of equty optons has become skewed and ths has become a persstent feature of the U.S. equty optons market. Rubnsten (1994) refers to ths phenomenon as "crashophoba". Investors are concerned about the possblty of another crash smlar to October 1987, and they prce optons accordngly. lthough we nvestgate the volatlty of stock varance n ths study, not the mpled volatlty from a stock optons, the sgnfcance of the crash of 1987 and ts mpact on the psychology of market partcpants makes the year of 1987 a natural watershed of our sample perod. 3.1 Test of Model ssumptons We treat the two sub-perods, and , as two economes, and test whether the assumptons n Theorem.4 and Theorem.5 hold n our data. For Theorem.4 to hold, we need the followng assumptons: 1) the economes 17

18 and have the same economc agent wth the same utlty functon, the same parameters of market except that economy has a larger volatlty of rsky return varance; ) the rsky return n the frst perod r (1) (=,) satsfes r (1) = o(1) ; 3) the mean and average varance of rsky return, µ and µ σ σ, satsfy = O(1) ; 4) the tax rate t satsfes equaton 3. To examne whether the assumptons are vald, we use a GRC(1,1) model σ (4) t = α0 + α1rt 1 + βσ t 1 to estmate the daly varance of stock returns. The estmates of α 0, α 1 and β are 0, 0.90 and 0.09, respectvely. Secondly, we conduct a t-test to see whether the two sub-perods have the same populaton means of daly returns and daly return varance. summary of the test s exhbted n Table 1. ccordng to the p-values, we can not reject the null hypothess that the populaton mean of daly returns s the same for both sub-perods, but can reject the the null hypothess that the populaton volatlty of daly returns s the same for both sub-perods. Thus assumpton 1) s only partally satsfed n the real data. owever, as explaned n the last secton, the volaton of the assumpton of constant mean of varance wll not hurt the model as long as the varance of return varance dvded by the fourth power of the mean of return varance s not larger n the economy wth lower-volatlty varance, as s stated n equaton. To test ths condton, we calculate the varance and mean of daly varance for each month n the two sub-perod, respectvely. nd for each month, we calculate the value of varance of return varance and dvde t by the fourth power of the mean of return varance, whch gves us the monthly value of 4 σ n equaton. Fnally, we conduct a t-test to test the null 18

19 hypothess that the populaton mean of 4 σ n the earler perod s not larger than the later perod. s s shown n Table 1, the large p-value mples that we can not reject the null hypothess. So the model s not hurt by the larger volatlty of returns n the later perod. The sample mean of returns s for the earler sub-perod and for the later sub-perod, whch are small enough to be treated as o(1). nd the sample µ realzaton of = O(1) σ are 7.03 and 8.07 for the two perods, respectvely, whch can be reasonably treated as O(1). In ths way, assumptons ) and 3) are satsfed n our data. 3. Test of Model Predctons Fgure 1 shows the trend of volatlty durng the whole sample perod of In the perod before 1987, the trend s smoother than the perod after that year. Two largest volatltes appear n 1987 and 008, and several other large volatltes are clustered around We calculate the sample varance of daly varance as 0.03*10 7 for perod and 0.65*10 7 for n F test on the null hypothess that the earler perod has a non-smaller volatlty of varance provdes a p-value of almost 0 and suggests a rejecton of the null hypothess that the volatlty of daly varance are the same for the two sub-perods. Furthermore, the trend of monthly values of s plotted n Fgure. 11 t-test on the null hypothess that the earler perod 11 Emprcally, we calculated the sample standard devaton of return varance dvded by the sample mean of return for each month, and the value of n that month s calculated as the rato of these two sample moments. 19

20 has non-smaller value of yelds a p-value of 0.01, provdng statstcal evdence that the value of sgnfcantly has ncreased n the later perod compared to the earler one. s predcted by Theorem.4, n a world where non-zero tax s mposed on short-term captal gans, a larger volatlty of daly stock return varance n the second perod mples bgger short-term losses and a smlar level of short-term gans compared to the frst perod. Furthermore, Theorem.5 predcts that gven a more volatle stock return varance, the rato of short-term captal gans to total net captal gans s hgher. s s shown n columns 1) and ) of Table 3, we have the seres of net long-term gans plus net short-term gans and the seres of net long-term gans n excess of short-term losses for the whole perod of Consderng the fact that the total amount of captal gans s hghly correlated to the nvestors' ntal wealth, we calculate the daly values of market ndex usng value weghted market returns and use ther values on the frst transacton day n each year as a proxy for ntal wealth. Based on the seres of market ndex, we derve the market value adjusted verson of columns 1) and ) by dvdng the raw numbers by the value of market ndex observed at the begnnng of each year and get columns 3) and 4). Snce the market value adjusted numbers elmnate the wealth effect, they are thus used for later emprcal tests. 1 Because the net short-term gans are the short-term gans n excess of the short-term loss, the dfference between columns 3) and 4) gves us the short-term captal gans. If we assume that net long-term gans are constant over tme, column 4) s essentally some constant mnus short-term gans. Usng data n columns 3) and 4), we conduct a t-test on the null hypotheses that the populaton mean of short-term losses, short-term gans and short-term gans as a proporton of total net gans 1 From now on, we refer to the market value adjusted values when we talk about varous captal gans or losses. 0

21 are the same durng the two sub-perods, and , respectvely. s s summarzed n Table, the short-term losses sgnfcantly ncrease n the second sub-perod (assumng constant net long-term gans), and so do the rato of short-term gans to total net gans. Whle there s no statstcal evdence that the short-term gans change between the two perods. ll these facts are consstent wth the predctons of Theorem.4 and Theorem.5. Fgure 3 shows the trend of short-term losses, short-term gans and the rato of short-term gans to total net gans. fter the year of 1987, there s an obvous rse n both the short-term losses (assumng a constant net long-term gans) and the rato of short-term gans to total net gans. In the meanwhle, the short-term captal gans do not exhbt a sgnfcant dfference before and after The Impact of Taxes on Short-Term Gans The tax rate for long-term captal gans s generally lower than that for short-term captal gans. owever, ths relatve tax advantage of long-term captal gans s smaller durng recent years. One argument s that t s the loss of tax advantage of long-term captal gans that nduces the nvestors to realze more short-term captal gans, whch further ncreases the rato of short term captal gans to total net gans. owever, there are two peces of evdence that are aganst ths argument. Frst, we show n Secton 3. that the short-term captal losses ncrease n recent years, but the short term captal gans do not change sgnfcantly. Second, n the two sub-perods, the relatve tax beneft of long-term captal gans s smlar, whle the rato of short term captal gans to the total net gans s sgnfcantly dfferent. Table 4 presents the tax rates for these two types of 1

22 captal gans and ther rato durng our sample perod of We take the tax rate for long-term captal gans as the denomnator n the rato. ence a larger rato mples a loss of tax advantage of long-term captal gans. Fgure 4 plots the trend of ths rato over tme. One can see that although the relatve tax advantage of long-term captal gans s smaller n the later perod after 1978, there are two sub-perods that wtness smlar values of the rato of these two types of tax rates. We then test whether the rato of short-term captal gans to the total net gans ncreases from the frst sub-perod to the second one regardless of the fact that the long-term captal gans do not lose ts tax advantage n the second sub-perod compared to the frst one. Table 5 summarzes the results. Frstly, column 1 shows the testng result for the null hypothess that the rato of tax rate for long-term captal gans to that for short-term captal gans s not larger n the second sub-perod compared to the frst one. p-value of 0.91 mples a falure to reject the null hypothess and provdes evdence that there s no loss of tax advantage of long-term captal gans n the second perod. In the contrast, the volatlty of stock return varance s sgnfcantly ncreased n the second perod, wth a p-value of almost 0 n column. s for the comparson of captal gans between the two sub-perods, one can see that there s a sgnfcant ncrease from the frst sub-perod to the second one for both the short-term losses and the rato of short-term gans to the total net gans. Therefore, controllng for the relatve tax beneft for the two types of captal gans, the predctons of Theorem.4 and Theorem.5 are stll consstent wth the real data. 4 Concluson

23 storcally, the captal gans tax has attracted less scholarly attenton than the dvdend tax. nd among the studes of captal gans tax, almost all of them have focused exclusvely on the long-term captal gans tax. Ths paper flls the gap n the lterature by examnng whether the volatlty of stock varance affects an nvestor's ntertemporal decson of realzng captal gans or losses. We buld a two-perod two-economy portfolo-selecton model and llustrate that hgher volatlty of stock return varance can lead to hgher short-term captal losses realzatons and hgher trade volumes. Most pror studes have focused the mpact of captal gans taxes on stock returns or tradng volume. Our theoretcal model demonstrates that real causaton can run n the opposte drecton - t s possble that the changes n the second moment of stock varance can affect ntertemporal captal gans/losses realzaton decsons and tradng volumes. 3

24 References ndersen G. Torben, 1996, Return Volatlty and Tradng Volume: n Informaton Flow Interpretaton of Stochastc Volatlty, The Journal of Fnance, 51, ndrew ng, Robert J. odrck, Yuhang Xng, and Xaoyan Zhang, 006, The Corss-Secton of Volatlty and Expected Returns, The Journal of Fnance, 61, Brock. Wllam and Blake ebaron, 1993, Usng structural modelng n buldng statstcal models of volatlty and volume of stock market returns: Unabrdged verson, Socal Scence Research Center Workng Paper, Unversty of Wsconsn, Madson. Bessembnder endrk and Paul J. Segun, 1993, Prce Volatlty, Tradng Volume, and Market Depth: Evdence from Futures Markets, Journal of Fnancal and Quanttatve nalyss, 8, Bloun J.. al and M. Yetman, 009, Captal gans taxes, prcng spreads, and arbtrage: Evdence from cross-lsted frms n the U.S., The ccountng Revew 84:5, Collns J. and D. Kemsley, 000, Captal gans and dvdend taxes n frm valuaton: Evdence of trple taxaton, The ccountng Revew 75, Clark K. Peter, 1973, subordnated stochastc process model wth fnte varance for speculatve prces, Econometrca 41, Constantndes M. George, 1983, Captal market equlbrum wth personal tax, Econometrca, 51, Constantndes M. George, 1984, Optmal stock tradng wth personal taxes, Journal of Fnancal Economcs, 13, Dammon M. Robert, Chester S. Spatt and arold. Zhang, 001, Optmal consumpton and Investment wth captal gan taxes, Revew of Fnancal Studes, 14, Dhalwal D. and O., 006, Investor tax heterogenety and ex-dvdend day tradng volume, Journal of Fnance 61, Ederngton. ous and Jae a ee, 1993, ow markets process nformaton: News releases and volatlty, Journal of Fnance 48, Epps W. Thomas and Mary. Epps, 1976, The stochastc dependence of securty prce changes and transacton volumes: Implcatons for the mxture-of-dstrbutons hypothess, Econometrca 44, Feldsten M. J. Slemrod and S. Ytzhak, 1980, The effects of taxaton on the sellng of 4

25 corporate stock and the realzaton of captal gans, Quarterly Journal of Economcs 94, Jn, 006, Captal gan tax overhang and prce pressure, Journal of Fnance 61, akonshok Josef and Seymour Smdt, 1989, Past prce changes and current tradng volume, The Journal of Portfolo Management Summer, 15, Reese W., 1998, Captal gans taxaton and stock market actvty: Evdence from IPOs, Journal of Fnance 53, Rubnsten M. 1994, Impled Bnomal Trees, Journal of Fnance, 49, pp Tauchen E. George, Mng u and arold Zhang, 1995, Volume, volatlty, and leverage: dynamc analyss, Workng Paper, Duke Unversty. Tauchen E. George, and Mark Ptts, 1983, The prce varablty-volume relatonshp on speculatve markets, Econometrca 51,

26 Proof of the Theorems Proof of Theorem.1. Snce n economy, w ( t) = ( µ )/ σ ( t) for t = 1,, then σ (1) σ () std( w () w (1)) = ( µ ) std( ). σ (1) σ () σ (1) σ () To calculate STD std( ), notcng that σ ) σ (1) σ () (t has the two-pont dstrbuton and s ndependent over tme, we have STD (1 = ) σ. Smlarly, STD (1 = ) σ. Snce µ s constant and the same for both economes and >, one has std( w () w (1)) > std( w () w (1)). If the mean of the varance s not the same for two economes but equaton holds, then one can defne σ ( σ ') = and and by followng two equatons respectvely σ σ 1 = 1 ( 1, ) 1 = 1 ( ), (5) one can show that > when equaton s satsfed. In ths case, one can consder the economy as a economy wth the same mean of volatlty ( σ ' ) wth economy, but wth hgher volatlty of stock return varance( > ). 6

27 ence, by usng the defnton of (σ '), and, one can show that µ w () w (1) = ( σ ') ( 1 ( ) where = or. Therefore, the proof follows as above snce. ), > Proof of Proposton.. If σ () σ (1), µ 1 µ µ 1 = ( o(1) + 1)( ( σ () (1+ r (1)) σ (1) σ 1+ 1 )). 1 µ the equaton holds snce = O(1), σ and, are constant. Proof of Theorem.3. We consder four dfferent cases, each of whch occurs wth the 1 probablty of. nd refers to the two types of economes, or. 4 Case 1: Suppose σ (1) = σ (1 ), and σ () = σ (1+ ). If r (1) = µ +σ 1+, then µ 1 µ S () S (1) = ( )(1 σ () (1+ r (1)) σ (1) + o (1)) µ = ( )(1+ o(1)). σ 1+ µ + σ Snce µ +σ 1+ = o(1) ; hence, S () S (1) < 0 and the short-term gan s realzed at tme t =, the amount s 7

28 STG µ = ( )(1+ o(1))(1+ µ + σ 1+ 1). σ 1+ µ + σ Snce >, we have STG > STG because both 1 ( 1+ µ + σ ) 1 and µ + σ 1+ ncrease as ncreases. If the mean of the varance s not the same for two economy but equaton holds, snce µ 1 µ S () S (1) = ( )(1 σ () (1+ r (1)) σ (1) + o (1)) µ = ( )(1+ o(1)) σ 1+ µ + σ redefne, as equaton 5, then µ 1 1 = ( )(1+ o(1)), σ 1+ 1 µ 1 1 S () S (1) = ( )(1+ o(1)). ( σ ') 1+ 1 Snce, one can stll show that the short-term captal gan s hgher wth hgher >. Smlarly, one can show that the results hold for all the subsequent analyss(ncludng all the followng theorems and propostons). If r (1) = µ σ 1+ < 0, then µ S () S (1) = ( )(1+ o(1)) < 0. σ 1+ µ σ In ths case, there s a loss at tme t =. 8

29 ST µ = ( )(1+ o(1))( µ σ 1 ). σ 1+ µ σ Case : Suppose that σ (1) = σ (1+ ), and σ () = σ (1 ). One can show that both short-term gan and loss are hgher for economy, followng the same logc n case 1. Case 3: Suppose that σ (1) = σ (1+ ), and σ () = σ (1+ ). Then µ S () S (1) = ( )(1+ o(1)) = o(1); σ 1+ µ + σ thus, the short-term gan and loss are small enough to be domnated by ther values n Case 1 and Case. Case 4: Suppose that σ (1) = σ (1 ), and σ () = σ (1 ). Smlarly to Case 3, the short-term gan and loss are small enough to be domnated by ther values n Case 1 and Case. In sum, the expected short-term gan and loss (whch are prmarly determned by Case 1 and Case are larger n economy. Proof of Theorem.4. The proof s smlar to the proof of Theorem.3, and here refers to the two types of economes, and. In Case 1 stated n Theorem.3, µ µ S () S (1) = ( )(1 σ () σ (1) + o If r (1) = µ +σ 1+, there s short-term gan. The tax s zero for economy, but t (1)). 9

30 for economy. From equaton 3, the nvestor n economy s not better off f she µ µ 1 µ 1 1 adjusts weght by = ( ). We assume that n ths case, σ () σ (1) σ µ 1 1 the nvestor only adjusts her weght n rsky asset by ( ). σ 1 1+ Snce the dfference n return between the two economes s o (1), we have STG = STG (1 o(1)). + If r (1) = µ σ 1+, there s short-term loss. Snce there s no tax for both economes, one has STG > STG based on the same analyss n Theorem.3. Smlarly, one can show that for the case stated n Theorem.3, one has STG = STG (1+ o(1)), ST > ST. In Cases 3 and 4, short-term gan and loss are both small( o (1) ). In sum, one has: E ( STG ) = E( STG )(1+ o(1)), E( ST ) > E( ST ). Proof of Theorem.5. From Theorem.4, E( STG ) = E( STG )(1+ o(1)), E ST ) > E( ST ), t s clear that STR STR. ( > 30

31 Table 1: Daly Stock Return and Daly Volatlty n the Two Sub-Perods varable X sample mean for sample mean for null hypothess daly daly return volatlty 4 σ * *10 X 1 = X X 1 = X X1 X p-value 0.85 * Note: 1) X1 refers to the value of X durng the sub-perod before 1987, and X refers to the value of X durng the sub-perod after ) * denotes sgnfcance at 5% sgnfcance level. 31

32 Table : Captal Gans n the Two Sub-Perods varable X NTG STG STG / NTTG sample mean for sample mean for null X 1 = X X 1 = X X 1 = X hypothess p-value * * 0.00 Note: 1) X1 refers to the value of X durng the sub-perod before 1987, and X refers to the value of X durng the sub-perod after ) The four numbers on captal gans and losses are n bllon dollars. 3) NTG refers to net long-term gans; STG refers to the short-term gan; STG / NTTG refers to the rato of short-term gans to the total net gans. 4) * denotes sgnfcance at 5% sgnfcance level. 3

33 Table 3: Total Captal Gan and long-term Captal Gan 1) ) 3) 4) Year NTG+NSTG NTG-STG ( NTG + NSTG) adj ( NTG STG) adj Note: 1) ll the numbers are n bllon dollars. ) Column 1) refers to net long-term captal gans plus net short-term captal gans. 3) Column ) refers to net long-term captal gans n access of short-term captal losses. 4) Columns 3) and 4) contan values n columns 1) and ), respectvely, dvded by the values of market ndex observed n the frst day of each year. 33

34 Table 4: Tax Rates for ong and Short-Term Captal Gan 1) ) 3) Year ST T S Rato / / / Note: Column 1) and ) refer to the tax rate for short-term captal gans and long-term captal gans, respectvely. nd column 3) refers to the rato of column ) to column 1). 34

35 Table 5: Tax Rates, Varance of Return Volatlty and Captal Gans n the Two Sub-Perods varable X SRTIO NTG STG / VOofVO NTTG null X1 X X1 X X 1 = X X 1 = X hypothess p-value 0.91 * 0.00 * 0.00 * 0.00 Note: 1) X1 refers to the value of X durng the sub-perod of , and X refers to the value of X durng the sub-perod of ) SRTIO refers to the sample mean of the rato of tax rate for long-term gans to that for short-term gans n each sub-perod. 3) VOofVO refers to the sample varance of equty return varance n each sub-perod. 4) NTG and STG / NTTG refer to the sample mean of net long-term gan and the sample mean of the rato of short-term gans to the total net gans n each sub-perod, respectvely, whch are n bllon dollars. 5) * denotes sgnfcance at 5% sgnfcance level. 35

36 Fgure 1: Trend of Daly Return Volatlty 36

37 Fgure : Trend of Monthly Values of 37

38 Fgure 3: Trend of long-term Gan n excess of short-term oss, short-term Gan and short-term Gan / Total Net Gan 38

39 Fgure 4: The Rato of Tax Rate for long-term Captal Gan to That for short-term Captal Gan 39