21_CAPM_markowitz

21_CAPM_markowitz - Hfrqrplfv 584d Idoo 5334 Pdunrzlw} dqg...

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Unformatted text preview: Hfrqrplfv 584d Idoo 5334 Pdunrzlw} dqg Glyhuvl fdwlrq 4 Glyhuvl fdwlrq Pdunrzlw} lqwurgxfhg d uxglphqwdu| exw uljrurxv prgho ri xqfhuwdlqw| dqg ulvn dyhuvlrq lqwr qdqfh e| vxssrvlqj wkdw doo lqyhvwruv zrxog suhihu d sruwirolr li lwv wrwdo sd|r v kdg kljkhu h{shfwdwlrq dqg orzhu yduldqfh1 Wr looxvwudwh wkhvh suhihuhqfhv/ Pdunrzlw} lqwurgxfhg wkh phdq0vwdqgdug ghyldwlrq gldjudp= E o ~ Zhoiduh lpsuryhv dv zh pryh lq wkh qruwkzhvw gluhfwlrq/ zlwk kljkhu h{shfwdwlrq dqg ohvv vwdqgdug ghyldwlrq1 Ohw [ eh wkh sd|r li dq lqyhvwru sxw doo klv zhdowk lqwr wkh uvw dvvhw/ dqg ohw \ eh klv sd|r li kh sxw doo klv zhdowk lqwr wkh vhfrqg dvvhw1 Wkhq li kh sxw d iudfwlrq w lqwr [ dqg d iudfwlrq 4 w ri klv zruwk lqwr \ / klv h{shfwhg sd|r zrxog eh H ] + w , H + w[ . +4 w , \ , @ w [ . +4 w , \ = Lq sduwlfxodu/ zh kdyh= Glyhuvl fdwlrq Wkhruhp 4 Ohw [ dqg \ eh lqghshqghqw/ zlwk htxdo phdqv [ @ \ dqg htxdo yduldqfhv 2 f @ 2 t A 3 1 Wkhq e| sxwwlqj kdoi klv prqh| lq hdfk/ dq lqyhvwru fdq rewdlq d sd|r ] +4 @ 5, zlwk wkh vdph h{shfwdwlrq H ] +4 @ 5, @ \ exw zlwk kdoi wkh yduldqfh H ] + 2 , @ 2 2 t = 4 Surri H + ] + 2 ,, @ 2 [ . 2 \ @ \ Ydu + ] + 2 ,, @ Ydu + 2 [ . 2 \ , @ e Ydu + [ , . 2 Fry + [>\ , . e Ydu + \ , @ e Ydu + [ , . 3 . e Ydu + \ , @ 2 Ydu + \ , = Slfwruldoo|/ li [ dqg \ duh lqghshqghqw/ E o ~ EX = EY X = Y o ~ o ~ X o ~ 1 2 1 2 Y Diwhu suhvhqw ydoxh/ glyhuvl fdwlrq lv wkh prvw lpsruwdqw sulqflsoh lq qdqfh1 D jrrg lqyhvwru lv dozd|v orrnlqj iru lqghshqghqw lqyhvwphqw rssruwxqlwlhv1 Rqh ri wkh fklhi vhoolqj srlqwv ri pdq| khgjh ixqgv lv wkdw wkhlu uhwxuqv duh lqghshqghqw ri wkh uhvw ri wkh pdunhw1 Glyhuvl fdwlrq Wkhruhp 5 Ohw 2 t A 3 / dqg ohw [ dqg \ eh lqghshqghqw1 Wkhq iru vrph vpdoo w / wkh sruwirolr ] + w , @ w[ .+4 w , \ kdv orzhu yduldqfh wkdq \ / qr pdwwhu krz kljk 2 f lv = Surri= Qrwh wkdw Ydu + ] + w ,, @ w 2 Ydu + [ , . +4 w , 2 Ydu + \ , 1 Khqfh/ g gw Ydu + ] + w ,, @ 5 w Ydu + [ , 5+4 w ,Ydu + \ , 1 Hydoxdwhg dw w @ 3 / g gw Ydu + ] + w ,, @ 5Ydu + \ , ? 3 = Wkxv iru vpdoo w A 3 / Ydu + ] + w ,, ? Ydu + ] +3,, @ Ydu + \ , = Lpsruwdqw dv wklv sulqflsoh lv/ lw lv riwhq plvxqghuvwrrg1 Glyhuvl fdwlrq doorzv wkh lqyhvwru wr rewdlq vdihu uhwxuqv1 Exw wkh sulqflsoh grhv qrw vd| wkdw lqyhvwruv vkrxog wu| wr dyrlg ulvn| vhfxulwlhv1 Lqghhg/ dq lqghshqghqw dvvhw jhqhudwlqj dw ohdvw wkh vdph uhwxuq dv d sruwirolr lv dozd|v zruwk dgglqj wr wkh sruwirolr +dw ohdvw lq vpdoo txdqwlwlhv, qr pdwwhu krz ulvn| lw lv1 Wklv lv looxvwudwhg e| 5 Vdpxhovrq Glyhuvl fdwlrq Wkhruhp Ohw 2 t A 3 / dqg ohw [ dqg \ eh lqghshqghqw1 Li [ A \ / wkhq doo lqyhvwruv vkrxog zdqw wr sxw dw ohdvw vrph prqh| lqwr [ / qr pdwwhu krz kljk 2 f lv1 Surri Zh vkdoo vkrz wkdw li 2 t A 3 dqg [ A \ / wkhq iru vrph vpdoo w / wkh sruwirolr ] + w , @ w[ .+4 w , \ kdv kljkhu h{shfwhg uhwxuq dqg orzhu yduldqfh wkdq \ / li [ dqg \ duh lqghshqghqw1 E| k|srwkhvlv/ vxfk d sruwirolr pxvw eh suhihuuhg e| doo phdq0yduldqfh lqyhvwruv1 Qrwh uvw wkdw H ] + w , @ w [ . +4 w , \ A \ li w A 3 1 Lq idfw/ g gw H + ]...
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This note was uploaded on 12/08/2009 for the course ECON 251 at Yale.

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21_CAPM_markowitz - Hfrqrplfv 584d Idoo 5334 Pdunrzlw} dqg...

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