Pt I: Some Bscs A Pme on Potolo Theoy The ottom lne th otolo constucton s lenng ho to del th uncetnty. To egn, let s stt th some dentons: A ndom vle s nume ssocted th n outcome tht s uncetn. Fo exmle, suose tht you oll dce. The esult o tht oll s ndom vle. Evey ndom vle cn e desced y olty densty uncton. Ths s just lst o the ossle outcomes th ech outcome s olty o occung. The olty densty o ollng dce ould e s ollos (note tht the oltes must lys sum to one somethng hs to hen!): X P(X) / / 3 / 4 / 5 / / Fo mny lctons, t s not necessy to kno the exct olty densty o ndom vle. All tht s needed e some desctve sttstcs. The to most common sttstcs e exected vlue nd vnce. The exected vlue o ndom vle, X, s the eghted-vege o ll ossle vlues hee the eghts e the oltes tht ech vlue ll occu. X E ( X ) X Fo exmle, the exected vlue o the dce oll ould e s ollos: X) (/)*+(/)*+(/)*3+(/)*4+(/)*5+(/)* / 3.5 The exected vlue s ment to ctue the centl tendency o ndom vle (.e. the vege). The vnce o ndom vle, X, s the eghted-vege o squed devtons om the exected vlue hee, gn, the eghts e the oltes o ech event. V ( X ) ( X X )
V Fo the dce oll, the vnce ould e s ollos: 7.5 ( X ) ( 3.5) + ( 3.5) + ( 3 3.5) + ( 4 3.5) + ( 5 3.5) + ( 3.5). 9 The vnce s ment to desce the degee to hch vle luctutes ound ts men. Sometmes stndd devton s used the thn the vnce. The stndd devton s smly the sque oot o vnce. Std. Dev( X ) V( X ) So, o exmle, the stndd devton o the dce oll s.7. The lst sttstc o nteest nvolves the eltonsh eteen to ndom vles nd s clled the covnce. The covnce o to ndom vles X nd Y s clculted y comutng the exected vlue o the oduct o X nd Y nd then sutctng the oduct o the exected vlues o X nd Y xy X, Y) XY) X ) * Y ) I thee s lge olty tht X nd Y ll tke on lge o smll vlues t the sme tme then the covnce eteen X nd Y ll e ostve. I thee s lge olty tht X ll e lge hen Y s smll nd vs ves, then the covnce ll e negtve. Sometmes, nsted o covnce, coelton s used. The coelton s smly the covnce dvded y the oduct o stndd devtons: ρ xy X, Y ) x y Pt II: Some useul omuls Thee e some oetes o exected vlue, vnce, nd covnce tht ll e vey useul o otolo constucton. ) I X s ndom vle nd k nd l e constnt, then: ( kx ) k X ) ( kx ) k V( X ) E V kx, ly) kl X, Y ) ) I X nd Y e ndom vles, then the sum, X+Y, s lso ndom vle th
E ( X + Y ) X ) + Y ) ( X + Y ) V( X ) + V( Y ) + COV ( X, Y ) V Gven () nd () ove, e cn clculte the exected vlue nd vnce o ny lne comnton o ndom vles. Fo exmle, suose e hve the ollong to vles X nd Y. I e dene ne vle s Z kx+ly hee k nd l e constnts, then the ne vle Z hs the ollong exected vlue nd vnce: Z) k X ) + l Y ) V( Z) k Pt III: The oe o dvescton V( X ) + l V( Y ) + kl X, Y ) Dvescton nvolves choosng otolo o sevel stocks nsted o holdng sngle stock. The de hee s tht ech comny hs ssocted th t some dosynctc sk. Tht s, ech ndvdul comny mght esond slghtly deently to vous events (.e, hen t ns, umell comny stocks go u hle cnc sket comnes stocks go don). By dvesyng, you ll e le to loe the vnce o you ovell otolo etun. Hoeve, thee e lmts to ho much vnce you cn dvesy y. Suose tht you hve otolo comosed o to stocks: s the ecentge o you ovell nvestment n stock, s the ecentge o you ovell nvestment n stock. Fom the omuls ove, the vnce o you ovell otolo ll e s ollos: + +,) You lty to dvesy deends on the covnce eteen stocks nd. When the covnce eteen stocks nd s negtve, e e delng th dosynctc (comny secc) sk. Ths cn e dvesed y. Note, hoeve, tht s the covnce eteen stocks nd gets lge nd ostve, the otolo vnce gets gge. Postve covnce (stocks tht move n the sme decton) s sgn o systemc sk. Fo exmle, ll stocks tyclly ll ol ces se. Systemc sk cnnot e dvesed y. Lets e t moe goous out the oe o dvescton. Suose tht you hve otolo o n stocks. (.) ees to the ecentge o you ovell otolo nvested n stock. Fom the ove omuls, e cn clculte the vnce o you otolo s ollos: n n n + j, j) j
No, suose tht you ollo nïve nvestment sttegy o dvdng you otolo eqully mong ech stock. Tht s, (.) /n. Then, the ove omul ecomes n n n + n j n, j) No, suose tht ll stocks hve common vnce nd tht ll secuty s hve common covnce. Then, the ove omul smles to the ollong: n n + Cov n The ollong cht shos the esultng otolo stndd devton s moe nd moe secutes e dded o thee cses: ostve coelton (systemc sk), zeo coelton, nd negtve coelton (dosynctc sk). Nume o Stocks (n) Pooton (/n) Stndd Dev. (co.4) Stndd Dev. (co 0) 50 50 50.5 4.83 35.3 35.35 5. 3.0.3.35.7 35.3 0.4 0.40 0. 33.9 5.8 5.80.0909 33.7 5.08 5.0 0.05 3.79.8..047 3.73 0.9 0.89 00.0 3.8 5.00 4.9 0.0099 3.8 4.98 4.93 Pt IV: The Mkotz Potolo Selecton Model Stndd Dev. (co -.4) In the ove secton, e used vey nïve nvestment sttegy (e, smly dvdng ou otolo eqully eteen vous stocks). The Mkotz method s moe sohstcted nvestment sttegy. Hee, e tke the oetes o the stocks s gven (exected vlue, vnce, nd covnce), nd choose the eghts n ou otolo sed on these sttstcs. Secclly, ssume the ollong: ) Thee e to sky ssets (stocks) vlle. Ech hs n ssocted exected etun nd vnce: V( V( ) ) ) )
) The stocks hve covnce gven y, ) 3) Thee s sk ee sset vlle V( ) ) 0 4) The nvesto s choosng the ootons o the otolo nvested n stock, stock nd the sk ee sset n ode to mxmze the ollong: ) In othe ods, the nvesto s choosng eghts to mxmze hs/he ed-tosk- to. Ths olem s done n to stes. Fst, suose tht the sky oton o you otolo s dvded nto () ecent n stock nd () n stock. We kno tht the exected etun nd vnce ssocted th ths dvson s s ollos: V( ) ) + + + Suose tht e lotted the vnce ssocted th vous eghts n stock. We ould somethng lke the ollong:, )
Pont A ould e the mnmum vnce otolo. Mthemtclly, e could solve o the eghts ssocted th ont A. +, ), ) Assume tht the vnce nd etun o the mnmum vnce otolo s gven y the ollong: etun) v( etun) Whle these eghts e useul, they e not ou ultmte gol. We e nteested n the tdeo eteen sk (vnce) nd etun. Suose tht you lot the exected etun vesus the otolo stndd devton ssocted th the vous otolo eghts. You ould get somethng lke the dgm elo: The mnmum vnce otolo s stll gven y ont A. Clely, ny eghts ssocted th onts elo ont A e not otml. We could nd othe eghts th the sme stndd devton, ut th hghe etuns. Any ont ove A s ossle otmum (th ostve sk/etun tdeo). Suose tht e chose to llocte cton y o ou totl otolo n sky ssets (hee the eghts on the to ssets e chosen to mnmze the comned sk) nd the emnng (-y) n the sk ee sset. We cn clculte the exected etun nd stndd devton o ou ovell otolo.
) y( ) SD( ) y Fst, note tht the ton o sk to etun s ndeendent o y. Secondly, note tht ghclly, the to o these s smly the sloe o lne connectng ont A th the ont (0, ()) on the vetcl xs. Cn e do ette thn ths? Sue e cn! Suose tht, the thn choosng the mnmum vnce otolo, e selected the eghts on ou sky ssets such tht e ee t ont B on the ove gh. Clely, ont B hs ette sk/ed tdeo (s seen y the steee sloe). In ct, t es tht ont B genetes the steeest ossle lne eteen () nd ont on the cuve. Theeoe, ont B eesents the otml otolo selecton. Secclly, ont B s the eghts o the to sky ssets tht mxmze the ed/sk to. The ctul eghts e s ollos: ( ) ( ), ( ) + ( ) ( + ) ), ) Fnlly, once the otolo ssocted th ont B hs een chosen, (th ts ssocted etun nd stndd devton), the ovell otolo deends on the mount llocted tods the sky ssets (y). Ths decson ll e sed uely on eeences tods sk. (Deent y s ll smly e deent onts on the lne connectng A nd B).