ANÁLISIS DE UNA CARTERA DE INVERSIÓN DE MARKOWITZ COMO APLICACIÓN DE LA OPTIMIZACIÓN CON RESTRICCIONES DE DESIGUALDAD

7/24/2019 ANLISIS DE UNA CARTERA DE INVERSIN DE MARKOWITZ COMO APLICACIN DE LA OPTIMIZACIN CON RESTRIC

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JKB\I_KO OL _\WFRC@ V JFHKH^KR

EWKO@ LH JFHKH^KR V B@H_KGFIFOKO

_tui` oli _rkgkn` Jfh ol Erko`

_rkgkn` Jfh ol Erko` prlslhtko` p`r Ckhuli [ulvlo` ^krblr`, sflho` ik tut`rk olicfsc` ik pr`jls`rk Fhckbuikok Ckslr` C`rlh`.

Z. G. ol ik _ut`rk> Kiuch`>

Ok. Fhckbuikok Ckslr` C`rlh` O. Ckhuli [ulvlo` ^krblr`

Rlvfiik. Nuhf` ol 7=;


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EWKO@ LH JFHKH^KR V B@H_KGFIFOKOJKB\I_KO OL _\WFRC@ V JFHKH^KR

_WKGKN@ JFH OL EWKO@B\WR@ KBKOCFB@ T7=;?-7=;KHIFRFR OL \HK BKW_LWK OL FHZLWRFH OL CKWD@UF_^ B@C@K]IFBKBFH OL IK @]_FCF^KBFH B@H WLR_WFBBF@HLR OL OLRFE\KIOKO

K\_@W>CKH\LI [\LZLO@ ^KWBLW@

_\_@W>OK FHCKB\IKOK CKRLW@ C@WLH@

OL]KW_KCLH_@>LB@H@CK K]IFBKOK FFF

WLK OL B@H@BFCFLH_@>CL_@O@R B\KH_F_K_FZ@R ]KWK IK LB@H@CK V IK LC]WLRK

.

WLR\CLH>

Lh li prlslhtl trkgkn` sl kg`rok uhk kpifbkbfh ol ik `ptfcfzkbfh bisfbk b`hrlstrfbbf`hls ol olsfeukioko lh li cgft` jfhkhbflr`. Rl pikhtlk uhk kpifbkbfhprbtfbk oli c`oli` ol Ckrd`wftz pkrk uhk bkrtlrk ol vki`rls oli Fglx2< qulcfhfcfbl li rflse` pkrk uhk rlhtkgfifoko okok.

]KIKGWKR BIKZL>

Ct`o`s Bukhtftktfv`s, @ptfcfzkbfh, Fhvlrsfh, Wlhtkgfifoko, Wflse`.


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_JE. JFB@. Khifsfs ol uhk bkrtlrk ol fhvlrsfh ol Ckrd`wftz b`c` kpifbkbfh ol ik @ptfcfzkbfh b`h rlstrfbbf`hls ololsfeukioko

HOFBL

FH_W@O\BFH....................................................................................................................................................................................................................................................................... ;

BK]_\I@ ;. @]_FCF^KBFH CK_LC_FBK........................................................................................................................................................ 2

;.;. Grlvl kpr`xfckbfh afstrfbk ol ik @ptfcfzkbfh.......................................................................................................................... 2

;.7. Biksfjfbkbfh ol i`s pr`gilcks ol @ptfcfzkbfh Cktlctfbk ......................................................................................... ?

BK]_\I@ 7> @]_FCF^KBFH LH LB@H@CK...................................................................................................................................................... 5

7.;. Ik @ptfcfzkbfh Ifgrl lh li b`htlxt` lb`hcfb`-lcprlskrfki..................................................................................... 1

7.7. Ik @ptfcfzkbfh B`hofbf`hkok lh li b`htlxt` lb`hcfb`-lcprlskrfki........................................................... 6

BK]_\I@ 2> ]W@GILCK OL @]_FCF^KBFH B@H WLR_WFBBF@HLR OL

OLRFE\KIOKO................................................................................................................................................................................................................................

;;2.;. J`rcuikbfh oli pr`gilck..........................................................................................................................................................................................;;

2.7. B`hofbf`hls ol Duah-_ubdlr.................................................................................................................................................................................;7

2.7.;. ]r`gilcks b`h b`hofbf`hls ol h` hlektfvfoko. B`hofbf`hls ol D-_....................................;2

2.2. B`hofbfh sujfbflhtl ol @ptfckifoko Ei`gki .....................................................................................................................................;?

BK]_\I@ ?> LI C@OLI@ OL CKWD@UF_^.........................................................................................................................................................;5

?.;. Fhtr`oubbfh....................................................................................................................................................................................................................................;5

?.7. ]ikhtlkcflht`...............................................................................................................................................................................................................................;5

?.7.;. Cfhfcfzkbfh oli rflse`.......................................................................................................................................................................

;5?.7.7. Wlhofcflht` lsplrko`..............................................................................................................................................................................;6

?.7.2. Ik jr`htlrk ljfbflhtl ....................................................................................................................................................................................7=

BK]_\I@ K]IFBKBFH ]WB_FBK .............................................................................................................................................................................7;


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[ulvlo` ^krblr`, Ckhuli


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;

FH_W@O\BBFH

Lh lstl trkgkn` sl prlslhtk uhk kpifbkbfh ol ik `ptfcfzkbfh bisfbk lh li cgft`jfhkhbflr`. Cs b`hbrltkclhtl sl ofslkr uhk kpifbkbfh prbtfbk utfifzkho` licòli` ol Ckrd`wftz pkrk uhk bkrtlrk ol vki`rls b`h li jfh ol lhb`htrkr uhk bkrtlrkljfbflhtl b`h li chfc` rflse` pkrk uhk rlhtkgfifoko okok.

Lh prfclr iuekr, b`clhtkrlc`s uh pikhtlkcflht` cktlctfb` oli còli`lb`hcfb`, qul sl knustk k uh pr`gilck ol @ptfcfzkbfh. stk, ak lstko` sflcprlifekok ki b`cp`rtkcflht` auckh`, yk qul li a`cgrl p`r su pr`pfk hkturkilzk gusbksflcprl i` cln`r lh bksf tò`s i`s sleclht`s ol su kbtfvfoko.

B`clhzkrlc`s p`r lxp`hlr qu ls ik @ptfcfzkbfh y iks bkrkbtlrstfbks ol i`sofstfht`s tfp`s ol pr`gilcks qul lhei`gkh pkrk pr`p`hlr uhk biksfjfbkbfh ol i`scfsc`s.

Lh li bkptui` 7 b`clhtkrlc`s ik fcp`rtkhbfk ol ik @ptfcfzkbfh, tkht`b`hofbf`hkok b`c` ifgrl, lh i`s b`htlxt`s lb`hcfb`s- lcprlskrfkils y lh kieuh`s oli`s pr`gilcks qul sl pikhtlkh lh t`rh` ki b`hsucfo`r, ik lcprlsk y li fhvlrs`r.

]ikhtlkok ik rlikbfh lhtrl kieuh`s pr`gilcks lb`hcfb`s y i`s pr`gilcks ol@ptfcfzkbfh, kg`rokrlc`s lh li bkptui` 2 i`s pr`gilcks b`h rlstrfbbf`hls ololsfeukioko, oltlhfho`h`s lh su lstuof`, yk qul ik kpifbkbfh prbtfbk qul slpr`p`hl sl knustk k lstl tfp` ol pr`gilcks ol `ptfcfzkbfh. Rl pikhtlkrh iksB`hofbf`hls ol Duah-_ubdlr.

Lh li bkptui` ?, sl lxp`hl li còli` ol Ckrd`wftz. Lh prfclr iuekr akrlc`s uhkrljlrlhbfk k sus `relhls, iks folks s`grl su tl`rk y, i` cs fcp`rtkhtl, bc` sl

pikhtlk su còli`. B`c` prfhbfpki bkrkbtlrstfbk tflhl qul, li fhvlrs`r khkifzk iksftukbfh ol fhvlrsfh pkrk olbfofr qu pr`p`rbfh ol su bkpftki fhvlrtfr pkrk b`hsleufruh chfc` rflse`. Rl hlblsftk b`hstrufr uhk bkrtlrk ol vki`rls ljfbflhtls. Ik bulstfhsl blhtrk lh olc`strkr bc`, k pkrtfr ol iks lxplbtktfvks qul sl brlkh s`grl iksrlhtkgfifokols lsplrkoks ol i`s kbtfv`s jfhkhbflr`s fhofvfoukils, sl pulol rlkifzkr uhkb`rrlbtk lilbbfh ol ik bkrtlrk k trkvs ol trls `gnltfv`s juhokclhtkils4 ikcfhfcfzkbfh oli rflse`, li rlhofcflht` lsplrko` y ik jr`htlrk ljfbflhtl.

\hk vlz b`h`bfoks iks folks prfhbfpkils oli còli` sl pikhtlk uhk kpifbkbfhprbtfbk oli cfsc` lh li bkptui`


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[ulvlo` krblr`, Ckhuli

7


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BK]_\I@ ;

@]_FCF^KBFH CK_LC_FBK

Ik @ptfcfzkbfh Cktlctfbk pr`p`rbf`hk i`s cln`rls vki`rls ol uhk juhbfh `gnltfv`lh uh oltlrcfhko` o`cfhf`.

Rleh Gkrrf`s y `tr`s, ik @ptfcfzkbfh ls bukiquflr pr`bls` elhrfb` p`r li buki sl`gtflhl ik cln`r s`iubfh k uh pr`gilck oko` y bkrkbtlrfzk ik ckhlrk lh qul h`slhjrlhtkc`s k sftukbf`hls oli ok k ok b`hjifbtfvks lh iks qul olslkc`s lsb`elr ik`pbfh cs kpr`pfkok ` glhljfbf`sk (7==


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[ulvlo` krblr`,Ckhuli

?

Lh uh prfhbfpf`, i`s pr`gilcks b`h rlstrfbbf`hls ol feukioko pulolh slr rlsulit`sclofkhtl li ctò` ol sustftubfh. stl lcpilk ik lstrktlefk ol lifcfhkbfh olvkrfkgils b`c` clof` pkrk b`hsleufr ik trkhsj`rckbfh oli pr`gilck `rfefhki(rlstrfhefo`) lh uh pr`gilck ifgrl. Li ctò` ol Ikerkhel tkcgfh plrsfeul iktrkhsj`rckbfh oli pr`gilck rfefhki lh uh` sfh rlstrfbbf`hls kuclhtkho` li hclr` olfhbehftks oli pr`gilck lh uh hclr` feuki ki ol iks rlstrfbbf`hls oli lhuhbfko``rfefhki.

Li ctò` ol i`s cuitfpifbko`rls h` sì` pr`p`rbf`hk li ptfc` sfh` qul tkcgfhjkbfiftk fhj`rckbfh s`grl ik slhsfgfifoko oli vki`r ptfc` khtl bkcgf`s lh iksofsp`hfgfifokols. Kolcs sl `gtflhl fhj`rckbfh kofbf`hki s`grl li ljlbt` qul iksrlstrfbbf`hls tflhlh lh li vki`r ptfc` oli pr`gilck k pkrtfr ol i`s vki`rls ol i`scuitfpifbko rls ks`bfko`s k iks b`rrlsp`hoflhtls rlstrfbbf`hls.

Iks b`hofbf`hls hlblskrfks qul olglh sktfsjkblr i`s ptfc`s ol i`s pr`gilcks ol@ptfcfzkbfh b`h rlstrfbbf`hls ol olsfeukioko julr`h pugifbkoks p`r prfclrk vlz lh lik` ;626 p`r Ufiifkc Dkrusa (;6;5-;665) lh ik tlsfs ol cklstrk;, kuhqul julr`h

rlh`cgrkoks p`r Akrìo Ufiifkc Duah (;67 pr`gilcksb`hvlx`s y pr`gilcks h` b`hvlx`s.

Rleh ik ifhlkifoko ol juhbf`hls> pr`gilcks ifhlkils y pr`gilcks hìfhlkils.

; Cfhfck `j Juhbtf`hs `j Rlvlrki Zkrfkgils wfta Fhlqukiftfls ks Rfol B`hstrkfhts7H`hifhlkr ]rèrkccfheprlslhtko` lh li ]r`bllofhes j 7hoGlrdlily Rycp`sfuc


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Gkrrf`s y `tr`s (7==


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:


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5

BK]_\I@ 7

IK @]_FCF^KBFH LH LB@H@CK

\hk ol iks oljfhfbf`hls cs b`h`bfoks ol ik Lb`h`ck sleh Gkrrf`s y `tr`s, akblrljlrlhbfk k ik ksfehkbfh ptfck ol rlburs`s lsbks`s(7==


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i`s kvkhbls sfehfjfbktfv`s ol ik Lb`h`ck Cktlctfbk, kbklbfo`s trks li hkbfcflht` olik ]rèrkckbfh Ifhlki y ik _l`rk ol Nule`s b`c` rlspulstk k pr`gilcks ol bkrbtlrlb`hcfb`.

Li pf`hlr` lh ik fhb`rp`rkbfh ol ik tl`rk j`rcki ol cxfc`sy chfc`s ol juhbf`hls lh li bkcp` lb`hcfb` jul li

b`hsfolrko` pkorl ol ik Lb`h`ck Cktlctfbk, Kht`fhlKueustfh Bùrh`t (;1=;-;155). Ik fhb`rp`rkbfh ol hulvkstbhfbks qul plrcftflrkh li khifsfs ol pr`gilcks b`hrlstrfbbf`hls plrcftf ckhlnkr sftukbf`hls cs rlkifstks,kolcs jul li prfclr` lh utfifzkr juhbf`hls cktlctfbks pkrkolsbrfgfr b`hblpt`s lb`hcfb`s b`c` ik olckhok, ik `jlrtk` li prlbf`.

Kht`fhl Kueustfh Bùrh`t

Li ctò` ol i`s cuitfpifbko`rls ol Ikerkhel ls uhk alrrkcflhtk ol erkh fhtlrspkrk li lb`h`cfstk, erkbfks k ik buki ls p`sfgil khkifzkr li b`cp`rtkcflht` auckhòlsol li puht` ol vfstk ol uh b`cp`rtkcflht` ckxfcfzko`r b`h rlstrfbbf`hls. Li

prfclr pr`gilck lb`hcfb` lh slr kg`roko` b`h ik tbhfbk ol i`s cuitfpifbko`rls olIkerkhel, b`hsfstf lh lhb`htrkr ik ofstrfgubfh ljfbflhtl ol uh rlburs` lhtrl ofstfht`sus`s ol ckhlrk qul ckxfcfzkr ik utfifoko t`tki.

Vk lh li sfei` QQ, uh hulv` y lsplbtkbuikr kvkhbl pkrk ik`ptfcfzkbfh sl pròun` lh ;626 ol ik ckh` oli cktlctfb`rus` Il`hfo Dkht`r`vfba2 (;6;7-;61:), ki kovlrtfr qul i`spr`gilcks ol pikhfjfbkbfh ol ik pròubbfh p`sllh uhklstrubturk cktlctfbk oljfhfok y rlprlslht uh erkh kvkhbllh ik b`cprlhsfh ol ik ksfehkbfh ol rlburs`s pkrk ièrkrrlsuitko`s ptfc`s.

Il`hfo Dkht`r`vfba

7.; Ik @ptfcfzkbfh Ifgrl lh li b`htlxt` lb`hcfb`-lcprlskrfki.

Lh ik _l`rk ol ik Lcprlsk, o`hol li b`cp`rtkcflht` rkbf`hki fhoubl k ikckxfcfzkbfh ol glhljfbf`s ` ik cfhfcfzkbfh ol i`s b`stls, ls jrlbulhtl lhb`htrkrsftukbf`hls qul pulolh slr rlsulit`s clofkhtl pr`gilcks ol `ptfcfzkbfh ifgrl. ]`rlnlcpi`, ik `gtlhbfh ol uh pikh ol pròubbfh pkrk uhk lcprlsk qul jkgrfbk uhpròubt` k pkrtfr ol vkrf`s fhputs ` pkrk uhk lcprlsk qul jkgrfbk vkrf`s pròubt`s.

\h fcp`rtkhtl pkpli ls li qul nulek ik @ptfcfzkbfh Bisfbk Ifgrl lh ik Lstkostfbky Lb`h`cltrk, p`r lnlcpi`, pkrk `gtlhlr lstfcko`rls ol cxfck vlr`sfcfiftuo. Lictò` ol i`s chfc`s bukorko`s ls uhk kpr`xfckbfh pìfhcfbk uskok b`c` lh liknustl ol okt`s y ik lstkostfbk. Rfh lcgkre` ik @ptfcfzkbfh Bisfbk prlslhtk b`c`fhb`hvlhflhtls iks juhbf`hls akh ol slr sujfbflhtlclhtl ofjlrlhbfkgils y qul i`sptfc`s qul sl bkibuikh h` s`h kgsìut`s, sfh` rliktfv`s k uh lht`rh` cs ` clh`skcpif` oli puht` lhb`htrko`.

2 ]rlcf` H`gli ol Lb`h`ck lh ;65


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7.7 Ik @ptfcfzkbfh B`hofbf`hkok lh li b`htlxt` lb`hcfb`-lcprlskrfki.

I`s pr`gilcks ol @ptfcfzkbfh B`hofbf`hkok lhbulhtrkh uhk bkhtfoko olkpifbkbf`hls lh cuitftuo ol rlks, lh pkrtfbuikr lh ik Lb`h`ck y ik Lcprlsk. \hpr`gilck lb`hcfb` bksf sflcprl ls uh pr`gilck b`hofbf`hko`. Lh Lb`h`ck ls cuytpfb` fiustrkr ik @ptfcfzkbfh B`hofbf`hkok b`h li lnlcpi` oli b`hsucfo`r qul olslk

lilefr i`s gflhls kolbuko`s pkrk ckxfcfzkr su utfifoko tlhflho` lh bulhtk surlstrfbbfh prlsupulstkrfk.

Ik @ptfcfzkbfh Bisfbk ifgrl ls rfbk tlrfbkclhtl y pr`p`rbf`hk i`s juhokclht`s olik @ptfcfzkbfh Bisfbk B`hofbf`hkok. Lh `bksf`hls, stk itfck pòrk vlrsl b`cùh bks` pkrtfbuikr ol kquiik yk qul aky pr`gilcks b`hofbf`hko`s qul sl trkoublh oluhk j`rck cs ` clh`s ofrlbtk k pr`gilcks ifgrls.

Iks rlstrfbbf`hls ifcftkh i`s vki`rls qul pulolh t`ckr iks vkrfkgils, pulsrlprlslhtkh b`hofbf`hls s`grl iks fhbehftks oli pr`gilck. Lh `bksf`hls tkilsb`hofbf`hls sl pòrkh lifcfhkr sf sl tuvflrkh lh bulhtk vkrfkgils kofbf`hkils. ]`rlnlcpi`, uhk rlstrfbbfh prlsupulstkrfk sl pòrk lifcfhkr sf b`hsfolrkc`s qul slpulokh `gtlhlr j`ho`s kofbf`hkils k uh bflrt` fhtlrs, k b`stk ol còfjfbkr ik juhbfh`gnltfv` pkrk fhbiufr li b`stl ol ofbak `plrkbfh. Ks li pr`gilck ol `ptfcfzkbfhrlstrfhefok tkcgfh pulol rloubfrsl lh kieuhks `bksf`hls k pr`gilcks ol`ptfcfzkbfh sfh rlstrfbbf`hls.

Lh huclr`sks `bksf`hls iks rlstrfbbf`hls ol feukioko s`h uhk ifcftkbfh k ik a`rkol rlbrlkr jfliclhtl ik rlkifoko. \hk rlstrfbbfh prlsupulstkrfk h` hlblskrfkclhtlolgl sktfsjkblrsl b`h lxkbtftuo, ls olbfr, h` ls hlblskrf` ekstkr tò` li prlsupulst`,sfh` ekrkhtfzkr qul li ekst` slk clh`r ` feuki qul li t`tki ofsp`hfgil.

R`h cs rlkifstks kqulii`s pr`gilcks qul b`htflhlh rlstrfbbf`hls oljfhfoks

clofkhtl fhlbukbf`hls (rlstrfbbf`hls ol olsfeukioko). V p`r lii`, vlrlc`s iksb`hslbulhbfks qul tlhork pkrk li b`hsucfo`r, ik lcprlsk y li fhvlrs`r>

Li pr`gilck oli b`hsucfo`r

Li pr`gilck gsfb` oli b`hsucfo`r sl rlsucl lh ik lilbbfh ol gflhls k b`hsucfr kpkrtfr ol uhk rlhtk ofsp`hfgil. Oko` qul li b`hnuht` ol gflhls y slrvfbf`s pkrk lib`hsuc` ls cuy kcpif`, sl sup`hl qul li b`hsucfo`r ofsp`hl ol uh hfvli ol rlhtkqul pulol ekstkr lh su t`tkifoko ` h` lh uhk b`cgfhkbfh ol pròubt`s qul ilpr`p`rbf`hk uh glhljfbf` cxfc`. Ks li b`hsucfo`r rlkifzk uh pr`bls` ol lilbbfhtlhflho` lh bulhtk uhk b`houbtk ptfcfzko`rk.

Li pr`gilck ol ik lcprlsk

Lh lstl bks` li pr`gilck b`hsfstl lh lilefr ik b`cgfhkbfh kolbukok ol fhputs, olckhlrk qul sl b`hsfekh ik ckxfcfzkbfh ol i`s glhljfbf`s. Iks vkrfkgils ol olbfsfhs`h iks bkhtfokols k b`hsucfr ol i`s ofstfht`s fhputs y ik juhbfh ol pròubbfhfhj`rck ol ik bkhtfoko ol ùtput qul sl pulol pròubfr b`h uhk oltlrcfhkokb`cgfhkbfh ol fhputs. B`h`bfo`s i`s prlbf`s uhftkrf`s ol i`s fhputs y oli ùtput, slpulol b`hstrufr ik juhbfh ol glhljfbf`s ol ik lcprlsk y ik juhbfh `gnltfv` olipr`gilck ol `ptfcfzkbfh.

Rfh lcgkre`, pulolh kpkrlblr `trks sftukbf`hls qul h`s eul k uh còli` ol

@ptfcfzkbfh B`hofbf`hkok. ]`r lnlcpi`, sleh Gkrrf`s y `tr`s sup`hekc`s qul iklcprlsk tflhl qul sktfsjkblr oltlrcfhko` b`htrkt` qul il lxfel ksleurkr uhkpròubbfh prljfnkok chfck lh ] uhfokols ol ùtput(7==


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;=

p`sfgil pkrk bucpifr ofba` b`htrkt` b`hsfstl lh ljlbtukr lsk pròubbfh ol ckhlrkqul sl cfhfcfbl li b`stl ksucfo` p`r ik lcprlsk, qul slr tkcgfh uhk juhbfh ol iksuhfokols lcpilkoks ol i`s fhputs.

Li pr`gilck oli fhvlrs`r

Ik fhvlrsfh lh bukiquflr tfp` ol gflhls sulil pikhtlkr lilefr lhtrl vkrfks `pbf`hlsofsp`hfgils. Li pr`gilck pkrk li fhvlrs`r ls lilefr olhtr` ol ik bkrtlrk ptfck qub`cgfhkbfh ol ttui`s ls ik qul il rlp`rtkr uhk rlhtkgfifoko t`tki prljfnkok, ks b`cìk cfhfcfzkbfh ol su rflse` ol fhvlrsfh. Lh li prlslhtl trkgkn` kg`rokrlc`s ikrlsìubfh ol uh pr`gilck ol lstl tfp`.


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-;;-

BK]_\I@ 2

]W@GILCK OL @]_FCF^KBFH B@H WLR_WFBBF@HLR OLOLRFE\KIOKO

Lh lstl bkptui` sl j`rcuikh i`s pr`gilcks ol `ptfcfzkbfh b`h rlstrfbbf`hls ololsfeukioko, yk qul s`h i`s qul cs sl knustkh k ik rlkifoko ol uh c`oli`.

2.; J`rcuikbfh oli pr`gilck.

\h pr`gilck ol `ptfcfzkbfh b`h rlstrfbbf`hls ol olsfeukioko sl j`rcuik ol iksfeuflhtl ckhlrk>

@pts.k. e(x) g

xRO`hol

ls ik Juhbfh @gnltfv` Q 3 {x O / e (x) g, x R} ls li b`hnuht` jkbtfgil.

OLJFHFBFH

Olh`tkrlc`s p`r ki b`hnuht` ol ofrlbbf`hls jkbtfgils k pkrtfr ol . rlprlslhtk ik bikusurk ol y p`r olh`tkrlc`s li b`hnuht` olhofbls b`rrlsp`hoflhtls k iks rlstrfbbf`hls skturkoks lh .\h puht` vlrfjfbk afptlsfs ol bukifjfbkbfh ol rlstrfbbf`hls(a.b.r) sf>

3 { , v =, f )}Lstk b`hofbfh ls ofjbfi ol b`cpr`gkr lh ik prbtfbk y lxfstlh b`hofbf`hls sujfbflhtlscs `plrktfvks pkrk oltlrcfhkr ik prlslhbfk ol lstk afptlsfs.

OLJFHFBFH

\h puht` ls rleuikrsf i`s erkoflhtls ol iks rlstrfbbf`hls skturkoks lh s`hifhlkiclhtl fholplhoflhtls ` ls uh puht` fhtlrf`r ol .

Ik rleuikrfoko ol uh puht` ls b`hofbfh sujfbflhtl pkrk qul sl bucpikh lh i ikafptlsfs ol bukifjfbkbfh ol rlstrfbbf`hls.

K b`htfhukbfh sl lxp`hlh iks o`s b`hofbf`hls sujfbflhtls cs utfifzkoks pkrk qult`o`s i`s puht`s olQbucpikh afptlsfs ol bukifjfbkbfh ol rlstrfbbf`hls.


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;7

_òks iks juhbf`hls qul oljfhlh iks rlstrfbbf`hls s`hb`hvlxks, yQ tflhl fhtlrf`r h` vkb`.

_òks iks rlstrfbbf`hls oli pr`gilck s`h ifhlkils.

Rup`horlc`s qul i`s puht`s oli b`hnuht` jkbtfgil vlrfjfbkh ik olh`cfhkok afptlsfsol bukifjfbkbfh ol rlstrfbbf`hls.

2.7 B`hofbf`hls ol Duah y _ubdlr

B`hsfolrlc`s li pr`gilck @pt j(x)

s.k e(x) g

b`h j, e (O)Li sfeuflhtl rlsuitko` lstkgilbl iks olh`cfhkoks b`hofbf`hls ol Duah y _ubdlr quls`h b`hofbf`hls hlblskrfks pkrk qul uh puht` qul bucpil iks a.b.r. slk uh ptfcì`bki oli pr`gilck.

_l`rlck (B`hofbfh Hlblskrfk ol Duah-_ubdlr ol @ptfckifoko I`bki)

Rlk qul vlrfjfbk afptlsfs ol bukifjfbkbfh ol rlstrfbbf`hls.Rf ls cxfc` (chfc`) i`bki oli pr`gilck, lht`hbls

tkils qul>

sl olh`cfhkh cuitfpifbko`rls ol Duah-_ubdlr. ( ) sl olh`cfhkh puht`s ol Duah-_ubdlr.Ik sleuhok rlikbfh sl iikck ol aìeurk b`cpilclhtkrfk. Lstkgilblh qul sf uhk

rlstrfbbfh h` lst skturkok lh li puht`, lst` akbl qul li cuitfpifbko`rb`rrlsp`hoflhtl slk hui`.

Rleh li tl`rlck i`s puht`s ol Duah-_ubdlr akh ol vlrfjfbkr iks sfeuflhtlsb`hofbf`hls> ( )


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Iks b`hofbf`hls khtlrf`rls sl pulolh lxprlskr lh trcfh`s ol ik juhbfhikherkhfkhk ol ik j`rck sfeuflhtl>ls puht` ol Duah-_ubdlr sf vlrfjfbk>

( ) Iks b`hofbf`hls hlblskrfks ol chfc` i`bki, s`h khi`eks bkcgfkho` li sfeh` ol i`scuitfpifbko`rls.

Li vki`r ol i`s cuitfpifbko`rls ol Duah-_ubdlr ks`bfko` k uhk rlstrfbbfh cfol ikvkrfkbfh ckrefhki qul lxplrfclhtk li vki`r `gnltfv` ptfc` bukho` sl rlkifzkhvkrfkbf`hls fhjfhftlsfckils lh li b`ljfbflhtl ol ik rlstrfbbfh b`rrlsp`hoflhtl.

2.7.; ]r`gilcks b`h b`hofbf`hls ol h` hlektfvfoko. B`hofbf`hls ol Duah-_ubdlr.

Iks vkrfkgils lb`hcfbks sflcprl s`h h` hlektfvks, p`r i` qul sup`hekc`s lipr`gilck khtlrf`r fhtr`oubflho` qul iks vkrfkgils lsth sunltks k b`hofbf`hls ol h`hlektfvfoko.

Li pr`gilck sl j`rcuik ol ik j`rck sfeuflhtl.

b`h j, e (O)Li sfeuflhtl rlsuitko` lstkgilbl iks olh`cfhkoks b`hofbf`hls ol Duah y _ubdlr,b`hofbf`hls hlblskrfks pkrk qul uh puht` qul bucpil iks a.b.r. slk uh ptfc` ol lstl

pr`gilck.

_l`rlck (B`hofbfh Hlblskrfk ol Duah-_ubdlr ol @ptfckifoko I`bki)

Rlk qul vlrfjfbk afptlsfs ol bukifjfbkbfh ol rlstrfbbf`hls.Rf ls cxfc` (chfc`) i`bki oli pr`gilck, lht`hbls

tkils qul>


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;?

2)

(

)

I`s puht`s sl olh`cfhkh puht`s ol Duah y _ubdlr.Lh trcfh`s ol ik juhbfh Ikherkhfkhk pkrk li pr`gilck khtlrf`r, iks b`hofbf`hlskhtlrf`rls sl rllsbrfglh b`c`>

ls puht` ol Duah y _ubdlr sf vlrfjfbk.;)

7)

2) ( ) ?)


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;:


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-;5-

BK]_\I@ ?

LI C@OLI@ OL CKWD@UF_^

?.; Fhtròubbfh

Li còli` cktlctfb` ol Akrry Ckx Ckrd`wftz (;675- ) sl olskrrìik s`grl ik gksl olib`cp`rtkcflht` rkbf`hki oli fhvlrs`r, ls olbfr, li fhvlrs`r olslk ik rlhtkgfifoko ol ikfhvlrsfh y rlbakzk li rflse`, p`r i` qul uhk bkrtlrk slr ljfbflhtl sf pr`p`rbf`hk ikcxfck rlhtkgfifoko p`sfgil pkrk uh rflse` oko`.

Ckrd`wftz? pugifb lh ;6


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( )Rup`hekc`s uhk bkrtlrk k khkifzkr qul sl b`cp`hl ol h kbtfv`s y pulolh tlhlr

kieh tfp` ol rflse`, ls olbfr, bkok uh` ol lii`s s`h vkrfkgils kilkt`rfks, b`h suslsplrkhzks, vkrfkhzks, b`vkrfkhzks y b`rrlikbfh ifhlki lhtrl lii`s. Ik olsvfkbfhtpfbk ol ik bkrtlrk slr ik qul h`s fhofqul li rflse` ol ik cfsck, y ls ik rkz bukorkokp`sftfvk ol ik suck ol iks vkrfkhzks ol i`s h kbtfv`s qul ik b`cp`hlh y susb`vkrfkhzks, p`holrkoks p`r ik pkrtfbfpkbfh ol bkok kbtfv` lh ik bkrtlrk.

Rf sl pikhtlk uh pr`gilck ol cfhfcfzkbfh oli rflse`, ik juhbfh `gnltfv` ls ikvkrfkhzk ol ik bkrtlrk. ]krk pkskr ol ik vkrfkhzk k ik olsvfkbfh tpfbk sl kpifbk uhktrkhsj`rckbfh c`ht`hk, qul ls ik rkz bukortfbk p`sftfvk, ki cfhfcfzkr ik vkrfkhzktkcgfh lstkrlc`s cfhfcfzkho` ik olsvfkbfh tpfbk y p`r i` tkht` li rflse`.

Iks vkrfkgils ol lilbbfh slrh iks bkhtfokols ol bkpftki k fhvlrtfr lh bkok uh` oli`s kbtfv`s. K b`htfhukbfh, sl culstrk li pikhtlkcflht` oli pr`gilck cktlctfb`,

cfhfcfzkho` li rflse` ol ik bkrtlrk b`h uhk rlstrfbbfh qul sup`hl qul ik fhvlrsfht`tki slr feuki k ;.

Ohol>

3 ]r`p`rbfh k fhvlrtfr lh li kbtfv` f.3 Zkrfkhzk oli kbtfv` f.3 B`vkrfkhzk lhtrl li Kbtfv` f y li Kbtfv` n, sflho` f ofstfht` ol n. Zkrfkhzk oli rlhofcflht` ol ik bkrtlrk.Li pr`gilck sl pulol lxprlskr ol j`rck cktrfbfki.


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-;6-

\hk vlz rlsulit` li pr`gilck, sl `gtflhlh tò`s i`s qul fhofbkh ik pr`p`rbfh kfhvlrtfr lh bkok uh` ol i`s kbtfv`s. Iule` sl rllcpikzkh tò`s ls`s vki`rls lh ikjuhbfh `gnltfv` pkrk bkibuikr . K b`htfhukbfh, sl bkibuik su rkz bukorkokp`sftfvk, y sl `gtflhl li rflse` chfc` ol ik bkrtlrk.

B`c` rlsuitko` ol lstk `ptfcfzkbfh sl `gtflhl ik b`cp`sfbfh ol ik bkrtlrk olvki`rls ol chfc` rflse` y li vki`r ol ofba` rflse`.

?.7.7 Wlhofcflht` lsplrko`

Li rlhofcflht` lsplrko` ol ik bkrtlrk ls li pr`clof` p`holrko` ol i`s rlhofcflht`slsplrko`s ol i`s kbtfv`s fhofvfoukils ol ik bkrtlrk, ls olbfr, ik suck ol i`srlhofcflht`s lsplrko`s ol bkok kbtfv` cuitfpifbko` p`r li p`rblhtknl oli bkpftkifhvlrtfo` lh bkok uh` tki y b`c` sl prlslhtk lh ik sfeuflhtl lbukbfh>

O`hol>> Wlhofcflht` lsplrko` oli Kbtfv` f.> Wlhofcflht` lsplrko` ol ik bkrtlrk.> ]r`p`rbfh k fhvlrtfr lh li Kbtfv` f.

]krk bkibuikr li rlhofcflht` ol ik bkrtlrk, sl olgl ol bkibuikr li rlhofcflht`lsplrko` ol bkok kbtfv`. stl sl `gtflhl k trkvs oli b`bflhtl lhtrl ik ofjlrlhbfk oliprlbf` lsplrko` oli kbtfv` ki c`clht` t y su prlbf` kbtuki, ofvfofo` li prlbf` kbtuki.

O`hol>

> ]rlbf` lsplrko` oli kbtfv` f. > ]rlbf` kbtuki oli kbtfv` f.

]`r `trk pkrtl, sl b`hsfolrk ki prlbf` b`c` uhk vkrfkgil kilkt`rfk. Li vki`rlsplrko`, slr p`r i` tkht`, ik lsplrkhzk cktlctfbk ol ik vkrfkgil kilkt`rfk. Rl

oljfhl ik lsplrkhzk b`c` ik suck oli pròubt` ol bkok vki`r hucrfb` ol ik juhbfhp`r ik b`rrlsp`hoflhtl pr`gkgfifoko ol qul sl ol lsl lstko`. Ik lbukbfh, lh lstlbks`, slrk>

O`hol>

> Lstko` R oli prlbf` oli kbtfv` f.

> ]r`gkgfifoko puhtuki oli lstko` s.


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7=

> Hclr` t`tki ol p`sfgils lstko`s.?.7.2 Ik jr`htlrk ljfbflhtl

]krk bkibuikr ik jr`htlrk ljfbflhtl h`s gkskrlc`s lh i` pikhtlko` lh li sugkpkrtko`?.7.;. Ik juhbfh `gnltfv` h` bkcgfkr, sleufr sflho` ik vkrfkhzk ol ik bkrtlrk. Lhbukht` k iks rlstrfbbf`hls qul aky qul kpifbkr, sl sfeul fhvfrtflho` li t`tki ol ik bukhtkol su bkpftki, plr` ka`rk kkol uhk rlstrfbbfh b`rrlsp`hoflhtl ki rlhofcflht`. Lh liiksl il lxfefr k ik bkrtlrk uh oltlrcfhko` rlhofcflht`, ( ).

Lh ik rls`iubfh ol ik cfhfcfzkbfh, sl `gtlhor uh pkr `rolhko` b`cpulst` p`r uhrlhofcflht` y li rflse` qul b`hiilvk.

Rf sl rlsulivl li pr`gilck pkrk vkrf`s vki`rls ol , sl `gtflhl uhk vkrfloko olpuht`s b`h i`s qul sl pulol ofgunkr ik jr`htlrk ljfbflhtl. ]krk p`olr trkzkrik slolglr b`htkr b`h kieuh`s puht`s ptfc`s, b`rrlsp`hoflhtls k ofjlrlhtls , sflho`hlblskrf` rlkifzkr ik cfhfcfzkbfh lh cs ol uhk `bksfh.

Cktlctfbkclhtl li pr`gilck ls li sfeuflhtl>

.


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-7;-

BK]_\I@

Lxfstl uhk b`hb`rokhbfk lhtrl li ofvfolho` rlpkrtfo` y ik rlhtkgfifoko p`r ofvfolho`,

cflhtrks qul ik rlikbfh ls fhvlrsk lhtrl ik b`tfzkbfh y ik rlhtkgfifoko p`r ofvfolho`.

Lh huclr`sks `bksf`hls, ik rlhtkgfifoko p`r ofvfolho` sulil tlhlr b`cpkrkbfhb`h ik rlhtkgfifoko ol ik oluok pgifbk oli Lstko`, yk qul i`s ofvfolho`s sl ksfcfikh ki`s fhtlrlsls rlpkrtfo`s p`r i`s g`h`s y `gifekbf`hls.

H`rckiclhtl, iks b`cpkks b`h uhk lilvkok rlhtkgfifoko p`r ofvfolho` sup`hlh`p`rtuhfokols ol b`cprk lh rlikbfh b`h `trks lcprlsks qul `jrlblh uhk julhtl olrlhtkgfifoko clh`r. I` iefb` ls fhvlrtfr lh kbbf`hls qul li clrbko` vki`rk qul vkh kofstrfgufr uhk ekhkhbfk ksleurkok pkrk li kbbf`hfstk (ofvfolho`) p`r lhbfck ol i` qulsl vki`rk ik kbbfh lh li clrbko` lh li qul b`tfbl. Rkgflho` lst`, h` ls tkh rlilvkhtlqul li prlbf` ol ik kbbfh bkfek, yk qul li fherls` lstkr ksleurko` lh j`rck olofvfolho`.

5www.g`isksckorfo.ls/o`bs/RG`isks/Fhj`rclsRG/khuki.poj1attp>//www.lxpkhsf`h.b`c/7=;?/=6/71/clrbko`s/;?;;67?66;.atci
http://www.expansion.com/2014/09/28/mercados/1411924991.htmlhttp://www.expansion.com/2014/09/28/mercados/1411924991.html


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_kgik

. Wflse` ol uh ;;.


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7?

Rleuho` pks . Bibui` ol iks b`vkrfkhzks lhtrl i`s kbtfv`s.

( )

_lrblr pks`. Cktrfz ol vkrfkhzk y b`vkrfkhzk>

C 3 3

Bukrt` pks`. Juhbfh @gnltfv`.

Rlkh li % ol bkpftki k fhvlrtfr lh bkok uh` ol i`s kbtfv`s.

+ [ufht` pks . Wlhofcflht` ol i`s Ofvfolho`s p`r Kbtfv` Gurstfi.

Rlxt` ]ks`. Wlstrfbbf`hls oli pr`gilck.


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Rptfc` pks . ]ikhtlkcflht` oli pr`gilck>

@gtlhlr li p`rblhtknl ptfc` ol fhvlrsfh lh bkok kbtfv` qul cfhfcfzk li rflse`ksleurkho` uhk rlhtkgfifoko chfck oli 1%, :,;


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7:

I`s puht`s qul s`h Duha-_ubdlr oli pr`gilck vlrfjfbkh iks sfeuflhtlsb`hofbf`hls>

;)

7)

2)

?)

8= 8= Li b`hnuht` jkbtfgil lst j`rcko` p`r iks sìubf`hls ol uh sfstlck ifhlki ol

fhlbukbf`hls p`r i` qul ls b`hvlx`.

Li _l`rlck I`bki-Ei`gki kjfrck qul sf j ls lstrfbtkclhtl bhbkvk y li b`hnuht`jkbtfgil ls b`hvlx`, lht`hbls sf lxfstl uh chfc` i`bki ls ei`gki y ls hfb`.

]krk ik `gtlhbfh ol ptfc`s i`bkils lh i`s pr`gilcks ol prèrkckbfhcktlctfbk, sl pulol utfifzkr li prèrkck fhj`rctfb` IFHE@.

Li pr`gilck pikhtlko` sl knustk k uh` ol @ptfcfzkbfh B`hofbf`hkok qul pulolslr rlsulit` clofkhtl li _l`rlck ol Duah-_ubdlr

Lh lstl bks`, i` vkc`s k rlsìvlr k trkvs oli prèrkck IFHE@. ]krk Iublr` yBkhfz`, IFHE@ ls uhk alrrkcflhtk sfcpil pkrk j`rcuikr pr`gilcks ifhlkils y hìfhlkils, rlsìvlri`s y khkifzkr su sìubfh.... li rlsuitko` qul IFHE@ h`spr`p`rbf`hk ls ik `ptfcfzkbfh qul h`s kyuok k lhb`htrkr li cln`r rlsuitko`> ik


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ekhkhbfk cs kitk ` li b`st` cs gkn`(7==7, p.;). ]`r i` tkht`, IFHE@ pr`p`rbf`hkuh ptfc` i`bki oli pr`gilck, slk hfb` ` h`.

Rleh lst`s kut`rls, li prèrkck plrcftl lxprlskr bukiquflr pr`gilck ol ckhlrksfcfikr k uhk h`tkbfh cktlctfbk lh su b`hnuht`, puoflho` fhbiufr rlstrfbbf`hls lh lipr`gilck qul sl vkyk k pikhtlkr. Fhbius` pulol illr okt`s ol uhk a`nk ol bibui`slpkrkok, gksl ol okt`s ` krbafv` ol tlxt`.

K b`htfhukbfh, sl c`strkr ik rlsìubfh clofkhtl ofba` prèrkck oli pr`gilckpikhtlko` khtlrf`rclhtl.


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Li rlpkrt` ol ik fhvlrsfh sl rlkifzk lhtrl o`s ol i`s trls ttui`s, b`hblhtrho`sl bksfli t`tki ol ik fhvlrsfh lh iks kbbf`hls oli Rkhtkholr (6


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WLH_KGFIFOKO

:.;


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2=

WLH_KGFIFOKO

?.


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Gfgifèrkjk

Ke`pfh, L. (7=;=)>_l`rk ol Rlilbbfh ol ik Bkrtlrk ol Zki`rls7: ol kgrfi.

Kibliky, R. (7=; Rupilclht` KGB, KGB LC]WLRK. 5 ol nuhf`.attp>//www.clrb`.fhj`/o`bs/]rlcf`s-KGB-Clrb`-7=; Okz ol Rkht`s.

GCL> R`bfloko ol Gìsks,Fhj`rcl Khuki 7=;?.attp>//www.gìskckorfo.ls/o`bs/RGìsks/Fhj`rclsRG/khuki.poj (B`hsuitko` 71/=?/7=;//rlp`sft`ry.uhfcfiftkr.lou.b`/gftstrlkc/;=:


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ûgliofk, C4 ^kgkizk, C 4 ûgfkrrl, ^. (7==7)> Li còli` ol Ckrd`wftz lh ik elstfh ol bkrtlrksBukolrh`s ol Elstfh 7(;), pp. 22-?:. attps>//koof.lau.ls/gftstrlkc/;=1;=/5===/;/BoEP7;7.poj(B`hsuitko`> ;5/=?/7=;

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