The Mathematics of Portfolio Theory
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1 The Matheatcs of Portfolo Theory The rates of retur of stocks, ad are as follows Market odtos state / scearo) earsh Neutral ullsh Probablty % 5% 9% -3% 3% % 5% % -% Notato: R The retur of stock a rado varable) ER ) or μ The expected retur of stock the probablty weghted average) VR ) or The varace of the retur of stock a easure of dsperso) ovr, R ) or The covarace of the returs of stocks ad a easure of jot dsperso) ρ The correlato coeffcet of the returs of stocks ad a scaled easure of jot dsperso) p The probablty of state R ) The retur of stock state a realzato) The uber of possble states, ). We ay wrte the expected retur of stock as ER ) p R ) p R) + p R) p R ) 3 ER ) pr ) % % % % pplyg ths forula to stocks ad we fd that ER ) 4.%, ER ).7%
2 . We ay wrte the varace of the retur of stock as μ μ + μ + + μ V R ) [ R ) ] p [ R ) ] p [ R ) ] p... [ R ) ] p 3 ) [ ) μ] % 5.6%) % 5.6%) % 5.6%) 0.3 V R R p 6.4%) pplyg ths forula to stocks ad we fd that V R ) 4.96%) , V R) 4.4%) The stadard devato of the retur s defed as the postve root of the varace, V R ) 6.4%).5% % 0.05,.% We ay wrte the covarace betwee the returs of stocks ad as ov R, R ) [ R ) μ ] [ R ) μ ] p [ R ) μ ] [ R ) μ ] p + [ R ) μ ] [ R ) μ ] p [ R ) μ ] [ R ) μ ] p 3 ov R, R ) [ R ) μ ][ R ) μ ] p % 5.6%) 3% 4.%) % 5.6%) 3% 4.%) % 5.6%)% 4.%) %) pplyg ths forula to the returs of stocks ad ad those of stocks ad we fd that ov R, R ) 5.%) ov R, R ) 0.44%). Note that the varace of the retur of stock s a specal case of the covarace ths case betwee R ad R ),.e., ) [ ) μ][ ) μ] [ ) μ] ). ov R R R R p R p V R Therefore, we ay use the otato whch s equvalet to. Itutvely, the covarace easures how uch the retur o two rsky assets ove tade. postve covarace eas that assets returs ove together. egatve covarace eas that they vary versely.
3 5. We ay wrte the correlato coeffcet betwee the returs of stock ad stock as ov R, R).48 orr R, R ) ρ V R ) V R ).5 5 pplyg ths forula to the returs of stocks ad ad those of stocks ad we fd that ρ ρ Note that the correlato coeffcet sply scales the covarace to a value betwee -) ad +). The relatoshp betwee the covarace ad the correlato coeffcet s ad therefore ρ, ρ ) ρ. correlato coeffcet close to zero ρ 0 dcates that the returs of the two stocks are urelated. postve correlato coeffcet ρ > 0 dcates that the returs ove together, ad ths relatoshp s stroger the closer the correlato s to +). egatve correlato coeffcet ρ < 0 dcates that the returs ove opposte drectos, ad ths relatoshp s stroger the closer the correlato s to -). Whe ρ we say that the returs of stocks ad are perfectly correlated, ad ther relatoshp ca be represeted by the lear equato R a+ b R. Whe b > 0 we say that there s a perfect postve correlato ad whe b < 0 we say that t s a perfect egatve correlato. 6. Usually we preset the paraeters above as follows: The vector of expected returs s deoted by μ, μ 5.6 μ μ 4. μ.7 ad the atrx of covarace s deoted by Σ Σ
4 7. The rsk-free asset has the sae retur ay arket codto s a costat). Therefore t s easy to show that E rf ) rf, V rf ) 0 ad ov R, rf ) 0 for j,, j 8. The retur o a portfolo of assets R p s the weghted average of the returs o the dvdual assets. Let w 0, w, w,, w be the weghts of the assets deoted by j 0,, ) the portfolo, the R w rf w R w R w R p Propertes of the expected retur the expected retur of a portfolo): Let w 0, w, w,, w be the weghts of the assets deoted by j 0,, ) the portfolo, the E R ) E w rf + w R + w R w R ) w rf + w E R ) + w E R ) w E R ). p 0 0 For exaple, let us calculate the expected retur o the portfolo coprsed of a rsk-free asset ad the three rsky stocks. Let w 0 0% vested a rsk free asset rf %; w 0% vested stock ; w 40% vested stock ; ad w 30% vested stock. E0.* rf+ 0. R R R) 0.* rf+ 0. ER ) ER ) ER ) 0.*% % % %.95% Propertes of the varace of retur the varace of the retur of a portfolo): Let w 0, w, w,, w be the weghts of the assets deoted by j 0,, ) the portfolo, the for rsk-less asset ad rsky assets) we get V w rf + w R + w R ) w V R ) + w V R ) + w w ov R, R ) I geeral we wll get 0 V w rf + w R + w R w R ) w w ov R, R ) w w 0 j j j j j j For exaple, let us calculate the varace of the portfolo retur V0. rf + 0. R R R) %) Note that the stadard devato brgs us back to our orgal uts of easureet V R P P).635%).65% 0.065
5 The Matheatcs of Portfolo Theory cotued. Propertes of the covarace betwee returs the covarace betwee the returs of two portfolos): let w 0,w, w,, w be the weghts of the assets deoted by 0,, ) the portfolo p ad x 0,x, x,, x be the weghts of the sae assets the portfolo q, the the covarace betwee the returs of the two portfolos s ov R, R ) ov[ w rf + w R w R ), x rf + x R x R ) ] pq p q w x ov R, R ) + w x ov R, R ) w x ov R, R ) + 0 +, ) + wxovr R wxovr, R) wxovr, R) wxovr, R) + wxovr, R) wxovr, R) j wxov R, R ) For exaple, let us defe portfolos p ad q as follows: p {w 0., w rf, 0.9) ad q {w 0., w 0.4, w 0.3, w rf, 0.). The covarace betwee the returs of portfolos p ad q s j ov[ 0.9 rf + 0. R ), 0. rf + 0. R R R ) ] j %). eta of the retur: we use the lear regresso to odel the relatoshp betwee the retur of the arket portfolo R ad the expected retur of a rsky asset E[R R ], ).* E[ R R ] α + β R R β R
6 The slope β ) of the regresso easures both the drecto ad the agtude of the relato betwee the retur of the arket portfolo R ) ad the expected retur of the rsky asset E[R R ]). Whe the two returs are postvely correlated, the slope wll also be postve, whereas whe they are egatvely correlated, the slope wll be egatve. The agtude of the slope of the regresso ca be read as follows - for every ut crease the arket portfolo retur R ), the rsky asset retur R ) s expected to chage by β uts. The slope of the regresso equato s a fucto of the varace of the retur of the arket portfolo ad of the covarace correlato) ov R, R ) β V R ) Therefore, we ca derve the followg relato betwee β ad the correlato coeffcet ρ ρ ρ β * The sple regresso s a exteso of the correlato/covarace cocept. It shows the relatoshp betwee two varables the rsky asset retur R - the depedat varable, ad the arket portfolo retur R - the explaatory varable). For every possble realzato of R t fts the expected value of R codtoal o that realzato E[R R ]). If the two varables have a jot oral dstrbuto the regresso equato holds exactly, f ot ths relatoshp s just a approxato. 3. Propertes of beta the beta of a portfolo): let w 0,w, w,, w be the weghts of the assets deoted by 0,, ) the portfolo p, the the beta of a portfolo s the weghted average of the betas of the rsky assets. β p ov Rp, R) ov w0rf + wr + wr wr, R) V R ) V R ) wov R, R) wov R, R) wov R, R) V R ) V R ) V R ) wov R, R) wov R, R) wov R, R) V R ) V R ) V R ) wβ+ wβ wβ wβ
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