Our main purpose in this section is to undertake an examination of the stock

Transcription

1 3. Caial gains ax and e sock rice volailiy Our main urose in is secion is o underake an examinaion of e sock rice volailiy by considering ow e raional seculaor s olding canges afer e ax rae on caial gains rises under eac ye of unexeced socks. From Eq.(., we know a weer e raional seculaors adjus eir sock oldings mainly deends on e caial gains and e condiional variance of e sock rice. I is reasonable o consider a raising e ax rae as direc imac on e caial gains; owever, i is wor a noice a e condiional variance of e sock rice deermined endogenously in e model may also be affeced. Terefore, we will begin is secion wi a focus on a iger ax rae s influence on e condiional variance of e sock rice and en urn o e discussion abou e canges of e sock rice volailiy wi a iger ax rae under eac ye of unexeced socks. Now, le us comue e condiional variance of e sock rice based on Eq.(.9. Suose a e unexeced socks are indeenden of eac oer and all fuure socks ave zero execed values. A ime +, e new informaion comes from socks u, ε, and + ; erefore, e unexeced sock rice cange will be: ( α( ( ( E u + ε. (3. Tis allows us o calculae e condiional variance direcly: Var ( E[ E ] ( α ( Var( u + Var( ε + Var(. (3. ( ( Te definiion of is condiional variance is obained from J.A. Carlson, C.L. Osler (000.

2 However, i is no aroriae o view as a fixed arameer. Le us recall a is e caracerisic roo deermined by e arameers of e model, and e arameer b mus be consisen wi Eq.(.. We erefore ave wo simulaneous equaions (. and (.7 a mus be solved in order o exress e equilibrium values of b and as funcions of e arameers, θ, and. We now rewrie ese wo equaions o obain e following wo relaionsis beween b and : b ( (, (RE (3.3 ( ( b( (. (SB (3.4 + Te firs exression is RE equaion a we menioned in Eq.(.7. Te second exression is denoed SB as a reminder a i is a ransformaion of e firs-order condiion for oimal seculaive beavior. Differeniaing RE and SB wi resec o b,,, and en we use e Cramer s rule o obain: 3 d f f. (3.5 We are now ready o erform comaraive saics o see ow e caial gains ax influences e condiional variance of e sock rice. Differeniaing Eq.(3. by, we find a: dvar( + ( d α ( d Var( u Var( ε + ( ( + 3 ( ( [ 3 d ] Var(. (3.6 Ta is, as e ax rae rises, e condiional variance varies according o ese ree See McCaffery and Driskill ( See Aendix A3.

3 yes of socks. In e following ar, we will subsiue e earlier exression for d ino e above exression for dvar( + and en discuss e sock marke resonse o a iger ax rae on caial gains under eac ye of unexeced socks. 3. Te issuing sock Suose ere only exiss e issuing sock; a is, ε 0. To simly our analysis, le r be zero. In is case, e seculaor s sock olding given a ax rae becomes: ( ( E ( ( (. (3.7 In addiion, for inuiive convenience, assume we ener e eriod wi e sock rice a is equilibrium level. Terefore, in Eq.(.9, can be regarded as. Now le us subsiue ε 0 ino Eq.(.9, we ave: u. (3.8 I is clear a < wi u > 0, so a > 0. Ta is, e issuing sock causes a lowering-rice effec so a e raional seculaors would increase eir sock oldings wi an execaion of a rising sock rice. Now le us reurn o e oin abou ow e raional seculaors reac o a iger ax rae on caial gains. Firs, we ave o find ou e relaionsi beween e ax rae and e condiional variance of e sock rice. Combining Eq.(3.5 wi

4 d Var( ε Var( 0, we ave > 0 4, and us Eq.(3.6 becomes: dvar ( + ( d Var( u < 0. (3.9 Eq.(3.9 means a e condiional variance of e sock rice is monoonically decreasing wi e ax rae. Differeniaing Eq.(3.7 wi resec o and subsiuing e resul of Eq.(3.9 ino Eq.(3.7, we find a: 5 d u ( Var( u > 0. (3.0 Judging from e above, we can see a as e ax rae rises, e raional seculaors would wan o increase eir sock oldings more. Wen e issuing sock lowers e sock rice below e equilibrium level, is buying ressure raises e sock rice and erefore acceleraes e curren rice o converge oward e equilibrium rice. In is case, we see a a iger ax rae on caial gains leads e raional seculaors o buy more socks o offse e lowering-rice effec of e issuing sock. Terefore, i is reasonable o conclude a if only e issuing sock occurs, raising e ax rae on caial gains can be a way o sabilize e sock marke. 3. Te dividend sock Now le us consider ow e sock rice volailiy canges as e ax rae on caial gains rises if only e dividend sock exiss. Te seculaor s olding given a ax rae can be exressed as: 4 See aendix A4. 5 See aendix A5.

5 ( ( E ( ( (. (3. In addiion, re-wriing Eq.(.9 wi u 0 and yields: α ( + ε. (3. Tis reresens a > wi ε > 0, and is resuls in < 0. Tus, is raising-rice effec of e dividend sock leads e raional seculaors o reduce eir sock oldings in order o ake advanage of is rofi-making ooruniy. Similarly, we now ry o find ou a iger ax rae s influence on e condiional variance of e sock rice. From Eq.(3.5, we ave > 0 Var ( u Var( 0. Ten, Eq.(3.6 becomes: d 6 wi dvar( + α ( d Var( ε < 0. (3.3 I sows a e condiional variance of e sock rice is monoonically decreasing wi e ax rae. Now differeniaing Eq.(3. wi resec o and subsiuing e resul of Eq.(3.3 ino Eq.(3., we find a: 7 d ε < 0. (3.4 α( Var( ε I is clear a e iger e ax rae becomes, e fewer socks e raional seculaors wan o old. Ta is, e seculaors would cose o sell more socks as 6 See aendix A4. 7 See aendix A5.

6 e ax rae rises. Wen e dividend sock raises e curren rice above e equilibrium level, more of ose sales would u more downward ressure on e sock rice so a deviaes less from. Terefore, a iger ax rae on e caial gains reduces e sock rice volailiy wen e dividend sock occurs. 3.3 Te margin-rae sock Now suose ere only exiss e margin-rae sock. Terefore, e seculaor s sock olding given a ax rae is: [ ( E ( ] [ ( ( ]. (3.5 Now re-wriing Eq.(.9 wi u ε 0 and, we obain:. (3.6 ( ( From e above exression, we see a < wi > 0. Tus, e seculaor would increase eir sock oldings because of is lowering-rice effec of. However, > 0 also imlies a rising margin-rae effec a may cause e seculaor o reduce eir oldings. Terefore, from Eq.(3.5, i is aaren a as a negaive effec of wic is absen in e earlier wo cases. Since in is case is affeced by wo differen effecs, we need a furer discussion o see wic effec dominaes e cange in e sock oldings. According o Eq.(3.5, e olding osiion can be divided ino wo ars. Considering e lowering-rice effec, e seculaor wans o old:

7 l [ ( ] > 0. (3.7 In addiion, e rising margin-rae effec leads e sock olding o be: m [ ( ]. (3.8 However, from Eq.(3.6, can be re-exressed as: ( ( ( Tus, re-wriing Eq.(3.8 wi Eq.(3.9 yields:. (3.9 m [ ( ] < 0. (3.0 ( Now combining Eq.(3.7 and Eq.(3.0, we obain: l + m ( ( ( < 0. (3. I sows a e oal effec of on is negaive. Ta is, e rising margin-rae effec as greaer imac on e sock olding and en leads e seculaor o sell e sock. Nex, we urn o our main urose of e marke reacion o a rising ax rae on caial gains. Firs, le us ake a look a e canges of e condiional variance of e d sock rice wen e ax rae rises. Combining d obain < 0 8. Ten, we subsiue i ino Eq.(3.6: wi Var ( u Var( ε 0, we dvar( d [ + ] Var( 3 3 ( ( ( ( 8 See aendix A4.

8 3 3 ( ( ( + ( ( ( ( + ( ( { + 3 θ Var( > 0. (3. Eq.(3. sows a e condiional variance of e sock rice is monoonically increasing wi e ax rae. Now, we ake differeniaion of Eq.(3.5 wi resec o and subsiue Eq.(3. ino Eq.(3.5: 9 d ( ( + d < 0. (3.3 Var( Te above equaion imlies a a rise in e ax rae causes e raional seculaors o sell more socks. Tus, wen e margin-rae sock lowers e curren rice below e equilibrium rice, ose sales u a downward ressure on e sock rice so a deviaes more from. Terefore, raising e ax rae on caial gains does no lay a sabilizing role wen e margin-rae sock occurs. 9 See aendix A5.