15. Basic Index Number Theory


 Clifton Stewart
 1 years ago
 Views:
Transcription
1 5. Basc Idex Numer Theory A. Iroduco The aswer o he queso wha s he Mea of a gve se of magudes cao geeral e foud, uless here s gve also he ojec for he sake of whch a mea value s requred. There are as may kds of average as here are purposes; ad we may almos say, he maer of prces as may purposes as wrers. Hece much va coroversy ewee persos who are lerally a cross purposes. (F.Y. Edgeworh, 888, p. 347) 5. The umer of physcally dsc goods ad uque ypes of servces ha cosumers ca purchase s he mllos. O he usess or produco sde of he ecoomy, here are eve more producs ha are acvely raded. The reaso s ha frms o oly produce producs for fal cosumpo, hey also produce expors ad ermedae producs ha are demaded y oher producers. Frms collecvely also use mllos of mpored goods ad servces, housads of dffere ypes of laor servces, ad hudreds of housads of specfc ypes of capal. If we furher dsgush physcal producs y her geographc locao or y he seaso or me of day ha hey are produced or cosumed, he here are llos of producs ha are raded wh each year ay advaced ecoomy. For may purposes, s ecessary o summarze hs vas amou of prce ad quay formao o a much smaller se of umers. The queso ha hs chaper addresses s he followg: How exacly should he mcroecoomc formao volvg possly mllos of prces ad quaes e aggregaed o a smaller umer of prce ad quay varales? Ths s he asc dex umer prolem. 5.2 I s possle o pose he dex umer prolem he coex of mcroecoomc heory; ha s, gve ha we wsh o mpleme some ecoomc model ased o producer or cosumer heory, wha s he es mehod for cosrucg a se of aggregaes for he model? However, whe cosrucg aggregae prces or quaes, oher pos of vew (ha do o rely o ecoomcs) are possle. Some of hese alerave pos of vew wll e cosdered hs chaper ad he ex chaper. Ecoomc approaches wll e pursued Chapers 7 ad The dex umer prolem ca e framed as he prolem of decomposg he value of a welldefed se of rasacos a perod of me o a aggregae prce mulpled y a aggregae quay erm. I urs ou ha hs approach o he dex umer prolem does o lead o ay useful soluos. Therefore, Seco B, he prolem of decomposg a value rao perag o wo perods of me o a compoe ha measures he overall chage prces ewee he wo perods (hs s he prce dex) mulpled y a erm ha measures he overall chage quaes ewee he wo perods (hs s he quay dex) s cosdered. The smples prce dex s a fxedaske dex. I hs dex, fxed amous of he quaes he value aggregae are chose, ad he hs fxed aske of quaes a he prces of perod 0 ad perod are calculaed. The fxedaske prce dex s smply he rao of hese wo values, where he prces vary u he quaes are held fxed. Two aural choces for he fxed aske are he quaes rasaced he ase perod, perod 0, or he quaes rasaced he curre perod, perod. These wo choces lead o he Laspeyres (87) ad Paasche (874) prce dces, respecvely. 5.4 Uforuaely, he Paasche ad Laspeyres measures of aggregae prce chage ca dffer, somemes susaally. Thus, Seco C cosders akg a average of hese wo dces o come up wh a sgle measure of prce chage. Seco C. argues ha he es average o ake s he geomerc mea, whch s Irvg Fsher s (922) deal prce dex. I Seco C.2, sead of averagg he Paasche ad Laspeyres measures of prce chage, akg a average of he wo askes s cosdered. Ths fxedaske approach o dex umer heory leads o a prce dex advocaed y Walsh (90, 92a). However, oher fxedaske 370
2 5. Basc Idex Numer Theory approaches are also possle. Isead of choosg he aske of perod 0 or (or a average of hese wo askes), s possle o choose a aske ha peras o a erely dffere perod, say, perod. I fac, s ypcal sascal agecy pracce o pck a aske ha peras o a ere year (or eve wo years) of rasacos a year efore perod 0, whch s usually a moh. Idces of hs ype, where he wegh referece perod dffers from he prce referece perod, were orgally proposed y Joseph Lowe (823), ad Seco D dces of hs ype wll e suded. They wll also e evaluaed from he axomac perspecve Chaper 6 ad from he ecoomc perspecve Chaper I Seco E, aoher approach o he deermao of he fucoal form or he formula for he prce dex s cosdered. Ths approach, devsed y he Frech ecooms Dvsa (926), s ased o he assumpo ha prce ad quay daa are avalale as couous fucos of me. The heory of dffereao s used o decompose he rae of chage of a couous me value aggregae o wo compoes ha reflec aggregae prce ad quay chage. Alhough Dvsa s approach offers some sghs, 2 does o offer much gudace o sascal ageces erms of leadg o a defe choce of dex umer formula. 5.6 I Seco F, he advaages ad dsadvaages of usg a fxedase perod he laeral dex umer comparso are cosdered versus always comparg he curre perod wh he prevous perod, whch s called he cha sysem. I he cha sysem, a lk s a dex umer comparso of oe perod wh he prevous perod. These lks are mulpled o make comparsos over may perods. Idces of hs ype wll o appear Chaper 9, where mos of he dex umer formulas exhed Chapers 5 8 wll e llusraed usg a arfcal daa se. However, dces where he wegh referece perod dffers from he prce referece perod wll e llusraed umercally Chaper 22, where he prolem of seasoal producs wll e dscussed. 2 I parcular, ca e used o jusfy he cha sysem of dex umers, whch wll e dscussed Seco E.2. B. Decomposo of Value Aggregaes o Prce ad Quay Compoes B. Decomposo of value aggregaes ad he produc es 5.7 A prce dex s a measure or fuco ha summarzes he chage he prces of may producs from oe suao 0 (a me perod or place) o aoher suao. More specfcally, for mos praccal purposes, a prce dex ca e regarded as a weghed mea of he chage he relave prces of he producs uder cosderao he wo suaos. To deerme a prce dex, s ecessary o kow () Whch producs or ems o clude he dex, () How o deerme he em prces, () Whch rasacos ha volve hese ems o clude he dex, (v) How o deerme he weghs ad from whch sources hese weghs should e draw, ad (v) Whch formula or mea should e used o average he seleced em relave prces. All he aove prce dex defo quesos excep he las ca e aswered y appealg o he defo of he value aggregae o whch he prce dex refers. A value aggregae V for a gve colleco of ems ad rasacos s compued as (5.) V pq, where p represes he prce of he h em aoal currecy us, q represes he correspodg quay rasaced he me perod uder cosderao, ad he suscrp defes he h elemeary em he group of ems ha make up he chose value aggregae V. Icluded hs defo of a value aggregae s he specfcao of he group of cluded producs (whch ems o clude) ad of he ecoomc ages egagg rasacos volvg hose producs (whch rasacos o clude), as well as he valuao ad me of recordg prcples movag he ehavor of he ecoomc ages uderakg he rasacos (deermao of prces). The cluded elemeary ems, her valuao (he p ), 37
3 Producer Prce Idex Maual he elgly of he rasacos, ad he em weghs (he q ) are all wh he doma of defo of he value aggregae. The precse deermao of he p ad q was dscussed more deal Chaper 5 ad oher chapers The value aggregae V defed y equao (5.) referred o a cera se of rasacos perag o a sgle (uspecfed) me perod. Now, cosder he same value aggregae for wo places or me perods, perods 0 ad. For he sake of defeess, perod 0 s called he ase perod ad perod s called he curre perod. Assume ha oservaos o he aseperod prce ad quay vecors, p 0 [p 0,,p 0 ] ad q 0 [q 0,,q 0 ], respecvely, have ee colleced. 4 The value aggregaes he wo perods are defed he ovous way as (5.2) ;. V V 5.9 I he prevous paragraph, a prce dex was defed as a fuco or measure ha summarzes he chage he prces of he producs he value aggregae from suao 0 o suao. I hs paragraph, a prce dex P(p 0,p,q 0,q ) alog wh he correspodg quay dex (or volume dex) Q(p 0,p,q 0,q ) s defed as wo fucos of he 4 varales p 0,p,q 0,q (hese varales descre he prces ad quaes perag o he value aggregae for perods 0 ad ), where hese wo fucos sasfy he followg equao: 5 (5.3) V/V Pp,p,q,q ( ) Q( p,p,q,q ). 3 Ralph Turvey ad ohers (989) have oed ha some values may e dffcul o decompose o uamguous prce ad quay compoes. Some examples of values dffcul o decompose are ak charges, gamlg expedures, ad lfe surace paymes. 4 Noe ha s assumed ha here are o ew or dsappearg producs he value aggregaes. Approaches o he ew goods prolem ad he prolem of accoug for qualy chage are dscussed Chapers 7, 8, ad 2. 5 The frs perso o sugges ha he prce ad quay dces should e joly deermed o sasfy equao (5.3) was Irvg Fsher (9, p. 48). Frsch (930, p. 399) called equao (5.3) he produc es. If here s oly oe em he value aggregae, he he prce dex P should collapse o he sgleprce rao p /p 0, ad he quay dex Q should collapse o he sglequay rao q /q 0. I he case of may ems, he prce dex P s o e erpreed as some sor of weghed average of he dvdual prce raos, p /p 0,, p /p Thus, he frs approach o dex umer heory ca e regarded as he prolem of decomposg he chage a value aggregae, V /V 0, o he produc of a par ha s due o prce chage, P(p 0,p,q 0,q ), ad a par ha s due o quay chage, Q(p 0,p,q 0,q ). Ths approach o he deermao of he prce dex s he approach ake he aoal accous, where a prce dex s used o deflae a value rao o oa a esmae of quay chage. Thus, hs approach o dex umer heory, he prmary use for he prce dex s as a deflaor. Noe ha oce he fucoal form for he prce dex P(p 0,p,q 0,q ) s kow, he he correspodg quay or volume dex Q(p 0,p,q 0,q ) s compleely deermed y P; ha s, y rearragg equao (5.3): (5.4) Qp,p,q,q ( 0 0 ) ( V/V 0 ) 0 0 /Pp,p,q,q ( ). Coversely, f he fucoal form for he quay dex Q(p 0,p,q 0,q ) s kow, he he correspodg prce dex P(p 0,p,q 0,q ) s compleely deermed y Q. Thus, usg hs deflao approach o dex umer heory, separae heores for he deermao of he prce ad quay dces are o requred: f eher P or Q s deermed, he he oher fuco s mplcly deermed y he produc es, equao (5.4). 5. I he ex suseco, wo cocree choces for he prce dex P(p 0,p,q 0,q ) are cosdered, ad he correspodg quay dces Q(p 0,p,q 0,q ) ha resul from usg equao (5.4) are also calculaed. These are he wo choces used mos frequely y aoal come accouas. B.2 Laspeyres ad Paasche dces 5.2 Oe of he smples approaches deermg he prce dex formula was descred grea deal y Joseph Lowe (823). Hs approach o 372
4 5. Basc Idex Numer Theory measurg he prce chage ewee perods 0 ad was o specfy a approxmae represeave produc aske, 6 whch s a quay vecor q [q,,q ] ha s represeave of purchases made durg he wo perods uder cosderao, ad he o calculae he level of prces perod relave o perod 0 as he rao of he perod cos of he aske, aske, 0, o he perod 0 cos of he. Ths fxedaske approach o he deermao of he prce dex leaves ope he followg queso: How exacly s he fxedaske vecor q o e chose? 5.3 As me passed, ecoomss ad prce sascas demaded a more precso wh respec o he specfcao of he aske vecor q. There are wo aural choces for he referece aske: he ase perod 0 produc vecor q 0 or he curre perod produc vecor q. These wo choces led o he Laspeyres (87) prce dex 7 P L defed y equao (5.5) ad he Paasche (874) prce dex 8 P P defed y equao (5.6): 9 6 Joseph Lowe (823, Appedx, p. 95) suggesed ha he produc aske vecor q should e updaed every fve years. Lowe dces wll e suded more deal Seco D. 7 Ths dex was acually roduced ad jusfed y Drosch (87a, p. 47) slghly earler ha Laspeyres. Laspeyres (87, p. 305) fac explcly ackowledged ha Drosch showed hm he way forward. However, he coruos of Drosch have ee forgoe for he mos par y laer wrers ecause Drosch aggressvely pushed for he rao of wo u values as eg he es dex umer formula. Whle hs formula has some excelle properes, f all he producs eg compared have he same u of measureme, he formula s useless whe, say, oh goods ad servces are he dex aske. 8 Aga, Drosch (87, p. 424) appears o have ee he frs o explcly defe ad jusfy hs formula. However, he rejeced hs formula favor of hs preferred formula, he rao of u values, ad so aga he dd o ge ay cred for hs early suggeso of he Paasche formula. 9 Noe ha P L (p 0,p,q 0,q ) does o acually deped o q, ad P P (p 0,p,q 0,q ) does o acually deped o q 0. However, does o harm o clude hese vecors, ad he oao dcaes ha he reader s he realm of laeral dex umer heory; ha s, he prces ad quaes for a value aggregae perag o wo perods are eg compared. (5.5) (5.6) 0 pq 0 0 PL ( p, p,q,q ) ; 0 0 pq 0 0 P ( p, p,q,q ) The aove formulas ca e rewre a maer ha s more useful for sascal ageces. Defe he perod reveue share o produc as follows: (5.7) ad 0,. / j j j s pq pq for,..., The, he Laspeyres dex, equao (5.5), ca e rewre as follows: 0 (5.8) L(,,, ) / j j j P p q usg defos equao (5.7) ( p / p ) / pjqj j 0 0 ( p / p ) s, Thus, he Laspeyres prce dex, P L ca e wre as a aseperod reveue shareweghed arhmec average of he prce raos, p /p 0. The Laspeyres formula (ul he very rece pas) has ee wdely used as he ellecual ase for PPIs aroud he world. To mpleme, a sascal agecy eeds oly o collec formao o reveue shares s 0 for he dex doma of defo for he ase perod 0 ad he collec formao o em prces aloe o a ogog ass. Thus, he Laspeyres PPI ca e produced o a mely ass whou curreperod quay formao. 0 Ths mehod of rewrg he Laspeyres dex (or ay fxedaske dex) as a shareweghed arhmec average of prce raos s due o Irvg Fsher (897, p. 57; 9, p. 397; 922, p. 5) ad Walsh (90, p. 506; 92a, p. 92). 373
5 Producer Prce Idex Maual 5.5 The Paasche dex ca also e wre reveue share ad prce rao form as follows: (5.9) P ( p, p, q, q ) pjqj j 0 ( p p ) pq pjqj j 0 ( p p ) s 0 ( p p ) s, usg defos equao (5.7). Thus, he Paasche prce dex P P ca e wre as a perod (or curreperod) reveue shareweghed harmoc average of he em prce raos p /p 0. 2 The lack of formao o curreperod quaes preves sascal ageces from producg Paasche dces o a mely ass. 5.6 The quay dex ha correspods o he Laspeyres prce dex usg he produc es, equao (5.3), s he Paasche quay dex; ha s, f P equao (5.4) s replaced y P L defed y equao (5.5), he he followg quay dex s oaed: (5.0) pq 0 0 P ( ). 0 pq Q p, p,q,q Noe ha Q P s he value of he perod quay vecor valued a he perod prces,, dvded y he (hypohecal) value of he perod 0 quay vecor valued a he perod prces, 0. Thus, he perod 0 ad quay vecors are valued a he same se of prces, he curreperod prces, p. 5.7 The quay dex ha correspods o he Paasche prce dex usg he produc es, equao (5.3), s he Laspeyres quay dex; ha s, f P equao (5.4) s replaced y P P defed y equao (5.6), he he followg quay dex s oaed: (5.) 0 p q 0 0 L ( ). 0 0 p q Q p,p,q,q Noe ha Q L s he (hypohecal) value of he perod quay vecor valued a he perod 0 prces, 0 p q, dvded y he value of he perod 0 quay vecor valued a he perod 0 prces, 0 0. Thus, he perod 0 ad quay vecors are valued a he same se of prces, he aseperod prces, p The prolem wh he Laspeyres ad Paasche dex umer formulas s ha hey are equally plausle, u, geeral, hey wll gve dffere aswers. For mos purposes, s o sasfacory for he sascal agecy o provde wo aswers o hs queso: 3 wha s he es overall summary measure of prce chage for he value aggregae over he wo perods queso? Thus, he followg seco, s cosdered how es averages of hese wo esmaes of prce chage ca e cosruced. Before dog hs, we ask wha s he ormal relaoshp ewee he Paasche ad Laspeyres dces? Uder ormal ecoomc codos, whe he prce raos perag o he wo suaos uder cosderao are egavely correlaed wh he correspodg quay raos, ca e show ha he Laspeyres prce dex wll e Ths mehod of rewrg he Paasche dex (or ay fxedaske dex) as a shareweghed harmoc average of he prce raos s due o Walsh (90, p. 5; 92a, p. 93) ad Irvg Fsher (9, pp ). 2 Noe ha he dervao equao (5.9) shows how harmoc averages arse dex umer heory a very aural way. 3 I prcple, sead of averagg he Paasche ad Laspeyres dces, he sascal agecy could hk of provdg oh (he Paasche dex o a delayed ass). Ths suggeso would lead o a marx of prce comparsos ewee every par of perods sead of a me seres of comparsos. Walsh (90, p. 425) oed hs possly: I fac, f we use such drec comparsos a all, we ough o use all possle oes. 374
6 5. Basc Idex Numer Theory larger ha he correspodg Paasche dex. 4 I Appedx 5., a precse saeme of hs resul s preseed. 5 Ths dvergece ewee P L ad P P suggess ha f a sgle esmae for he prce chage ewee he wo perods s requred, he some sor of evely weghed average of he wo dces should e ake as he fal esmae of prce chage ewee perods 0 ad. Ths sraegy wll e pursued he followg seco. However, should e kep md ha, usually, sascal ageces wll o have formao o curre reveue weghs ad, hece, averages of Paasche ad Laspeyres dces ca e produced oly o a delayed ass (perhaps usg aoal accous formao) or o a all. C. Symmerc Averages of FxedBaske Prce Idces C. Fsher dex as a average of he Paasche ad Laspeyres dces 5.9 As was meoed he prevous paragraph, sce he Paasche ad Laspeyres prce dces are equally plausle u ofe gve dffere esmaes of he amou of aggregae prce chage ewee perods 0 ad, s useful o cosder akg a evely weghed average of hese fxedaske prce dces as a sgle esmaor of prce chage ewee he wo perods. Examples of such 4 Peer Hll (993, p. 383) summarzed hs equaly as follows: I ca e show ha relaoshp (3) [ha s, ha P L s greaer ha P P ] holds wheever he prce ad quay relaves (weghed y values) are egavely correlaed. Such egave correlao s o e expeced for prce akers who reac o chages relave prces y susug goods ad servces ha have ecome relavely less expesve for hose ha have ecome relavely more expesve. I he vas majory of suaos covered y dex umers, he prce ad quay relaves ur ou o e egavely correlaed so ha Laspeyres dces ed sysemacally o record greaer creases ha Paasche wh he gap ewee hem edg o wde wh me. 5 There s aoher way o see why P P wll ofe e less ha P L. If he perod 0 reveue shares s 0 are exacly equal o he correspodg perod reveue shares s, he y Schlömlch's (858) Iequaly (see Hardy, Llewood, ad Polyá, 934, p. 26), ca e show ha a weghed harmoc mea of umers s equal o or less ha he correspodg arhmec mea of he umers ad he equaly s src f he umers are o all equal. If reveue shares are approxmaely cosa across perods, he follows ha P P wll usually e less ha P L uder hese codos; see Seco D.3. symmerc averages 6 are he arhmec mea, whch leads o he Drosch (87, p. 425) Sdgwck (883, p. 68) Bowley (90, p. 227) 7 dex, P DR (/2)P L + (/2)P P, ad he geomerc mea, whch leads o he Irvg Fsher 8 (922) deal dex, P F, defed as (5.2) PF( p, p, q, q ) PL( p, p, q, q ) 2 ( 0,, 0, ) 2. P P p q A hs po, he fxedaske approach o dex umer heory s rasformed o he es approach o dex umer heory; ha s, o deerme whch of hese fxedaske dces or whch averages of hem mgh e es, desrale crera or ess or properes are eeded for he prce dex. Ths opc wll e pursued more deal he ex chaper, u a roduco o he es approach s provded he prese seco ecause a es s used o deerme whch average of he Paasche ad Laspeyres dces mgh e es Wha s he es symmerc average of P L ad P P o use as a po esmae for he heorecal cosoflvg dex? I s very desrale for a prce dex formula ha depeds o he prce ad quay vecors perag o he wo perods uder cosderao o sasfy he me reversal es. 9 A 6 For a dscusso of he properes of symmerc averages, see Dewer (993c). Formally, a average m(a,) of wo umers a ad s symmerc f m(a,) m(,a). I oher words, he umers a ad are reaed he same maer he average. A example of a osymmerc average of a ad s (/4)a + (3/4). I geeral, Walsh (90, p. 05) argued for a symmerc reame f he wo perods (or coures) uder cosderao were o e gve equal mporace. 7 Walsh (90, p. 99) also suggesed hs dex. See Dewer (993a, p. 36) for addoal refereces o he early hsory of dex umer heory. 8 Bowley (899, p. 64) appears o have ee he frs o sugges he use of hs dex. Walsh (90, pp ) also suggesed hs dex whle commeg o he g dffereces ewee he Laspeyres ad Paasche dces oe of hs umercal examples: The fgures colums (2) [Laspeyres] ad (3) [Paasche] are, sgly, exravaga ad asurd. Bu here s order her exravagace; for he earess of her meas o he more ruhful resuls shows ha hey sraddle he rue course, he oe varyg o he oe sde aou as he oher does o he oher. 9 See Dewer (992a, p. 28) for early refereces o hs es. If we wa he prce dex o have he same propery as a sgleprce rao, he s mpora o sasfy he me reversal es. However, oher pos of vew are poss (coued) 375
7 Producer Prce Idex Maual dex umer formula P(p 0,p,q 0,q ) sasfes hs es f (5.3) P p,p,q,q / Pp,p,q,q ( ) ( ) ; ha s, f he perod 0 ad perod prce ad quay daa are erchaged ad he dex umer formula s evaluaed, he hs ew dex P(p,p 0,q,q 0 ) s equal o he recprocal of he orgal dex P(p 0,p,q 0,q ). Ths s a propery ha s sasfed y a sgle prce rao, ad seems desrale ha he measure of aggregae prce chage should also sasfy hs propery so ha does o maer whch perod s chose as he ase perod. Pu aoher way, he dex umer comparso ewee ay wo pos of me should o deped o he choce of whch perod we regard as he ase perod: f he oher perod s chose as he ase perod, he he ew dex umer should smply equal he recprocal of he orgal dex. I should e oed ha he Laspeyres ad Paasche prce dces do o sasfy hs me reversal propery. 5.2 Havg defed wha meas for a prce dex P o sasfy he me reversal es, he s possle o esalsh he followg resul: 20 he Fsher deal prce dex defed y equao (5.2) aove s he oly dex ha s a homogeeous 2 symmerc average of he Laspeyres ad Paasche prce dces, P L ad P P, ad sasfes he me reversal es equao (5.3) aove. Thus, he Fsher deal prce dex emerges as perhaps he es evely weghed average of he Paasche ad Laspeyres prce dces I s eresg o oe ha hs symmerc aske approach o dex umer heory daes ack o oe of he early poeers of dex umer heory, Arhur L. Bowley, as he followg quoaos dcae: If [he Paasche dex] ad [he Laspeyres dex] le close ogeher here s o furher dffculy; f hey dffer y much hey may e regarded as feror ad superor lms of he dex umer, whch may e esmaed as her arhmec mea as a frs approxmao. (Arhur L. Bowley, 90, p. 227) Whe esmag he facor ecessary for he correco of a chage foud moey wages o oa he chage real wages, sascas have o ee coe o follow Mehod II oly [o calculae a Laspeyres prce dex], u have worked he prolem ackwards [o calculae a Paasche prce dex] as well as forwards. They have he ake he arhmec, geomerc or harmoc mea of he wo umers so foud. (Arhur L. Bowley, 99, p. 348) The quay dex ha correspods o he Fsher prce dex usg he produc es, equao (5.3), s he Fsher quay dex; ha s, f P equao (5.4) s replaced y P F defed y equao (5.2), he followg quay dex s oaed: (5.4) QF( p, p, q, q ) QL( p, p, q, q ) 2 ( 0,, 0, ) 2. P Q p q Thus, he Fsher quay dex s equal o he square roo of he produc of he Laspeyres ad Paasche quay dces. I should also e oed ha Q F (p 0,p,q 0,q ) P F (q 0,q,p 0,p ); ha s, f he role of prces ad quaes s erchaged he Fsher prce dex formula, he he Fsher quay dex s oaed Raher ha ake a symmerc average of he wo asc fxedaske prce dces perag o wo suaos, P L ad P P, s also possle o reur o Lowe s asc formulao ad choose he aske vecor q o e a symmerc average of he ase ad curreperod aske vecors, q 0 ad q. The followg suseco pursues hs approach o dex umer heory. le. For example, we may wa o use our prce dex for compesao purposes, whch case sasfaco of he me reversal es may o e so mpora. 20 See Dewer (997, p. 38). 2 A average or mea of wo umers a ad, m(a,), s homogeeous f whe oh umers a ad are mulpled y a posve umer λ, he he mea s also mulpled y λ; ha s, m sasfes he followg propery: m(λa,λ) λm(a,). 22 Irvg Fsher (9, pp. 47 8; 922) also cosdered he arhmec, geomerc, ad harmoc averages of he Paasche ad Laspeyres dces. 23 Irvg Fsher (922, p. 72) sad ha P ad Q sasfed he facor reversal es f Q(p 0,p,q 0,q ) P(q 0,q,p 0,p ) ad P ad Q sasfed he produc es equao (5.3) as well. 376
8 5. Basc Idex Numer Theory C.2 Walsh dex ad heory of pure prce dex (5.6) 0 0 j j j s pq/ pq for,2,..., Prce sascas ed o e very comforale wh a cocep of he prce dex ased o prcg ou a cosa represeave aske of producs, q (q,q 2,,q ), a he prces of perod 0 ad, p 0 (p 0,p 0 2,,p 0 ) ad p (p,p 2,,p ), respecvely. Prce sascas refer o hs ype of dex as a fxedaske dex or a pure prce dex, 24 ad correspods o Ks s (924, p. 43) uequvocal prce dex. 25 Sce Joseph Lowe (823) was he frs perso o descre sysemacally hs ype of dex, s referred o as a Lowe dex. Thus, he geeral fucoal form for he Lowe prce dex s (5.5) 0 0 Lo(,, ) / 0 s( p / p ), P p where he (hypohecal) hyrd reveue shares s 26 correspodg o he quay weghs vecor q are defed y 24 See Seco 7 Dewer (200). 25 Suppose, however, ha for each commody, Q Q, he fraco, (P Q) / (PQ), vz., he rao of aggregae value for he secod uperod o he aggregae value for he frs uperod s o loger merely a rao of oals, also shows uequvocally he effec of he chage prce. Thus, s a uequvocal prce dex for he quaavely uchaged complex of commodes, A, B, C, ec. I s ovous ha f he quaes were dffere o he wo occasos, ad f a he same me he prces had ee uchaged, he precedg formula would ecome (PQ ) / (PQ). I would sll e he rao of he aggregae value for he secod uperod o he aggregae value for he frs uperod. Bu would e also more ha hs. I would show a geeralzed way he rao of he quaes o he wo occasos. Thus s a uequvocal quay dex for he complex of commodes, uchaged as o prce ad dfferg oly as o quay. Le e oed ha he mere algerac form of hese expressos shows a oce he logc of he prolem of fdg hese wo dces s decal (Sr George H. Ks, 924, pp ). 26 Irvg Fsher (922, p. 53) used he ermology weghed y a hyrd value, whle Walsh (932, p. 657) used he erm hyrd weghs The ma reaso why prce sascas mgh prefer a memer of he famly of Lowe or fxedaske prce dces defed y equao (5.5) s ha he fxedaske cocep s easy o expla o he pulc. Noe ha he Laspeyres ad Paasche dces are specal cases of he pure prce cocep f we choose q q 0 (whch leads o he Laspeyres dex) or f we choose q q (whch leads o he Paasche dex). 27 The praccal prolem of pckg q remas o e resolved, ad ha s he prolem addressed hs seco I should e oed ha Walsh (90, p. 05; 92a) also saw he prce dex umer prolem he aove framework: Commodes are o e weghed accordg o her mporace, or her full values. Bu he prolem of axomery always volves a leas wo perods. There s a frs perod, ad here s a secod perod whch s compared wh. Prce varaos have ake place ewee he wo, ad hese are o e averaged o ge he amou of her varao as a whole. Bu he weghs of he commodes a he secod perod are ap o e dffere from her weghs a he frs perod. Whch weghs, he, are he rgh oes hose of he frs perod? Or hose of he secod? Or should here e a comao of he wo ses? There s o reaso for preferrg eher he frs or he secod. The he comao of oh would seem o e he proper aswer. Ad hs comao self volves a averagg of he weghs of he wo perods. (Correa Moyla Walsh, 92a, p. 90) Walsh s suggeso wll e followed, ad hus he h quay wegh, q, s resrced o e a average or mea of he aseperod quay q 0 ad he curreperod quay for produc q, say, m(q 0,q ), for,2,,. 28 Uder hs assump 27 Noe ha he h share defed y equao (5.6) hs case s he hyrd share s pq Σ pq, whch uses 0 0 he prces of perod 0 ad he quaes of perod. 28 Noe ha we have chose he mea fuco m(q 0,q ) o e he same for each em. We assume ha m(a,) has he followg wo properes: m(a,) s a posve ad couous fuco, defed for all posve umers a ad, ad m(a,a) a for all a >
9 Producer Prce Idex Maual o, he Lowe prce dex equao (5.5) ecomes (5.7) 0 pmq (, q) 0 0 Lo (,,, ). 0 0 p jmq ( j, qj) j P p q 5.28 To deerme he fucoal form for he mea fuco m, s ecessary o mpose some ess or axoms o he pure prce dex defed y equao (5.7). As Seco C., we ask ha P Lo sasfy he me reversal es, equao (5.3) aove. Uder hs hypohess, s mmedaely ovous ha he mea fuco m mus e a symmerc mea; 29 ha s, m mus sasfy he followg propery: m(a,) m(,a) for all a > 0 ad > 0. Ths assumpo sll does o p dow he fucoal form for he pure prce dex defed y equao (5.7) aove. For example, he fuco m(a,) could e he arhmec mea, (/2)a + (/2), whch case equao (5.7) reduces o he Marshall (887) Edgeworh (925) prce dex P ME, whch was he pure prce dex preferred y Ks (924, p. 56): (5.8) {( + )/2} 0 q 0 0 ME (,,, ). 0 0 pj{ ( qj + qj) /2} j P p q 5.29 O he oher had, he fuco m(a,) could e he geomerc mea, (a) /2, whch case equao (5.7) reduces o he Walsh (90, p. 398; 92a, p. 97) prce dex, P W : For more o symmerc meas, see Dewer (993c, p. 36). 30 Walsh edorsed P W as eg he es dex umer formula: We have see reaso o eleve formula 6 eer ha formula 7. Perhaps formula 9 s he es of he res, u ewee ad Nos. 6 ad 8 would e dffcul o decde wh assurace (C.M. Walsh, 92a, p. 03). Hs formula 6 s P W defed y equao (5.9), ad hs 9 s he Fsher deal defed y equao (5.2) aove. The Walsh quay dex, Q W (p 0,p,q 0,q ), s defed as P W (q 0,q,p 0,p ); ha s, prces ad quaes equao (5.9) are erchaged. If he Walsh quay dex s used o deflae he value rao, a mplc prce dex s oaed, whch s Walsh s formula 8. (5.9) 0 q 0 0 W (,,, ). 0 0 p j qq j j j P p q 5.30 There are may oher possles for he mea fuco m, cludg he mea of order r, [(/2)a r + (/2) r ] /r for r 0. To compleely deerme he fucoal form for he pure prce dex P Lo, s ecessary o mpose a leas oe addoal es or axom o P Lo (p 0,p,q 0,q ). 5.3 There s a poeal prolem wh he use of he MarshallEdgeworh prce dex, equao (5.8), ha has ee oced he coex of usg he formula o make eraoal comparsos of prces. If he prce levels of a very large coury are compared wh he prce levels of a small coury usg equao (5.8), he he quay vecor of he large coury may oally overwhelm he fluece of he quay vecor correspodg o he small coury. 3 I echcal erms, he Marshall Edgeworh formula s o homogeeous of degree 0 he compoes of oh q 0 ad q. To preve hs prolem from occurrg he use of he pure prce dex P K (p 0,p,q 0,q ) defed y equao (5.7), s asked ha P Lo sasfy he followg varace o proporoal chages curre quaes es: 32 (5.20) P p q P p q Lo(,,, λ ) Lo(,,, ) 0 0 for all p, p, q, q ad all λ> 0. The wo ess, he me reversal es equao (5.3) ad he varace es equao (5.20), eale oe o deerme he precse fucoal form for he pure prce dex P Lo defed y equao (5.7) aove: he pure prce dex P K mus e he Walsh dex P W defed y equao (5.9) To e of praccal use y sascal ageces, a dex umer formula mus e ale o e expressed as a fuco of he aseperod reveue shares, s 0 ; he curreperod reveue shares, s ; 3 Ths s o lkely o e a severe prolem he meseres coex where he chage quay vecors gog from oe perod o he ex s small. 32 Ths s he ermology used y Dewer (992a, p. 26). Vog (980) was he frs o propose hs es. 33 See Seco 7 Dewer (200). 378
10 5. Basc Idex Numer Theory ad he prce raos, p /p 0. The Walsh prce dex defed y equao (5.9) aove ca e rewre hs forma: (5.2) P ( p, p, q, q ) W 0 0 j j C.3 Coclusos j p p qq 0 qq 0 0 j j j ( p / p p ) s s 0 0 ( p / p p ) s s j j j j j s s p p 0 0 s s p p 0 0 j j j j 5.33 The approach ake o dex umer heory hs seco was o cosder averages of varous fxedaske prce dces. The frs approach was o ake a evehaded average of he wo prmary fxedaske dces: he Laspeyres ad Paasche prce dces. These wo prmary dces are ased o prcg ou he askes ha pera o he wo perods (or locaos) uder cosderao. Takg a average of hem led o he Fsher deal prce dex P F defed y equao (5.2) aove. The secod approach was o average he aske quay weghs ad he prce ou hs average aske a he prces perag o he wo suaos uder cosderao. Ths approach led o he Walsh prce dex P W defed y equao (5.9) aove. Boh hese dces ca e wre as a fuco of he aseperod reveue shares, s 0 ; he curreperod reveue shares, s ; ad he prce raos, p /p 0. Assumg ha he sascal agecy has formao o hese hree ses of varales, whch dex should e used? Experece wh ormal meseres daa has show ha hese wo dces wll o dffer susaally, ad hus s a maer of choce whch of hese dces s used pracce. 34 Boh hese dces are examples of 34 Dewer (978, pp ) showed ha hese wo dces wll approxmae each oher o he secod order aroud a equal prce ad quay po. Thus, for ormal meseres daa where prces ad quaes do o chage much (coued). superlave dces, whch wll e defed Chaper 7. However, oe ha oh hese dces rea he daa perag o he wo suaos a symmerc maer. Hll commeed o superlave prce dces ad he mporace of a symmerc reame of he daa as follows: Thus ecoomc heory suggess ha, geeral, a symmerc dex ha assgs equal wegh o he wo suaos eg compared s o e preferred o eher he Laspeyres or Paasche dces o her ow. The precse choce of superlave dex wheher Fsher, Törqvs or oher superlave dex may e of oly secodary mporace as all he symmerc dces are lkely o approxmae each oher, ad he uderlyg heorec dex farly closely, a leas whe he dex umer spread ewee he Laspeyres ad Paasche s o very grea. (Peer Hll, 993, p. 384) 35 D. Aual Weghs ad Mohly Prce Idces D. Lowe dex wh mohly prces ad aual aseyear quaes 5.34 I s ow ecessary o dscuss a major praccal prolem wh he heory of askeype dces. Up o ow, has ee assumed ha he quay vecor q (q,q 2,,q ) ha appeared he defo of he Lowe dex, P Lo (p 0,p,q) defed y equao (5.5), s eher he aseperod quay vecor q 0 or he curreperod quay vecor q or a average of he wo. I fac, erms of acual sascal agecy pracce, he quay vecor q s usually ake o e a aual quay vecor ha refers o a ase year, say, ha s efore he ase perod for he prces, perod 0. Typcally, a sascal agecy wll produce a PPI a a mohly or quarerly frequecy, u, for he sake of defeess, a mohly frequecy wll e assumed wha follows. Thus, a ypcal prce dex wll have he form P Lo (p 0,p,q ), where p 0 s he prce vecor perag o he aseperod moh for prces, moh 0; p s he prce vecor perag o he curreperod moh for prces, moh, say; gog from he ase perod o he curre perod, he dces wll approxmae each oher que closely. 35 See also Peer Hll (988). 379
11 Producer Prce Idex Maual ad q s a referece aske quay vecor ha refers o he ase year, whch s equal o or efore moh Noe ha hs Lowe dex P Lo (p 0,p,q ) s o a rue Laspeyres dex (ecause he aual quay vecor q s o equal o he mohly quay vecor q 0 geeral) The queso s hs: why do sascal ageces o pck he referece quay vecor q he Lowe formula o e he mohly quay vecor q 0 ha peras o rasacos moh 0 (so ha he dex would reduce o a ordary Laspeyres prce dex)? There are wo ma reasos: Mos ecoomes are sujec o seasoal flucuaos, ad so pckg he quay vecor of moh 0 as he referece quay vecor for all mohs of he year would o e represeave of rasacos made hroughou he year. Mohly household quay or reveue weghs are usually colleced y he sascal agecy usg a esalshme survey wh a relavely small sample. Hece, he resulg weghs are usually sujec o very large samplg errors, ad so sadard pracce s o average hese mohly reveue or quay weghs over a ere year (or some cases, over several years), a aemp o reduce hese samplg errors. I oher saces, where a esalshme cesus s used, he repored reveue weghs are for a aual perod. The dex umer prolems ha are caused y seasoal mohly weghs wll e suded more deal Chaper 22. For ow, ca e argued ha he use of aual weghs a mohly dex umer formula s smply a mehod for dealg wh he seasoaly prolem Moh 0 s called he prce referece perod, ad year s called he wegh referece perod. 37 Trple (98, p. 2) defed he Lowe dex, callg a Laspeyres dex, ad callg he dex ha has he wegh referece perod equal o he prce referece perod a pure Laspeyres dex. Trple also oed he hyrd share represeao for he Lowe dex defed y equao (5.5) ad equao (5.6). Trple oed ha he rao of wo Lowe dces usg he same quay weghs was also a Lowe dex. 38 I fac, usg he Lowe dex P Lo (p 0,p,q ) he coex of seasoal producs correspods o Bea ad Se s (924, p. 3) Type A dex umer formula. Bea ad (coued) 5.36 Oe prolem wh usg aual weghs correspodg o a perhaps dsa year he coex of a mohly PPI mus e oed a hs po. If here are sysemac (u dverge) reds produc prces, ad cosumers or usesses crease her purchases of producs ha decle (relavely) prce ad decrease her purchases of producs ha crease (relavely) prce, he he use of dsa quay weghs wll ed o lead o a upward as hs Lowe dex compared wh oe ha used more curre weghs, as wll e show elow. Ths oservao suggess ha sascal ageces should ge upodae weghs o a ogog ass I s useful o expla how he aual quay vecor q could e oaed from mohly reveues o each produc durg he chose ase year. Le he moh m reveue of he referece populao he ase year for produc e v,m, ad le he correspodg prce ad quay e p,m ad q,m, respecvely. Value, prce, ad quay for each produc are relaed y he followg equaos: m, m, m, (5.22) v ;,...,; m,...,2. For each produc, he aual oal q ca e oaed y prcedeflag mohly values ad summg over mohs he ase year as follows: (5.23) q v 2 m, 2, m ; m, q,...,, m p m where equao (5.22) was used o derve equao (5.23). I pracce, he aove equaos wll e evaluaed usg aggregae reveues over closely relaed producs, ad he prce p,m wll e he moh m prce dex for hs elemeary produc group year relave o he frs moh of year For some purposes, s also useful o have aual prces y produc o mach he aual quaes defed y equao (5.23). Followg aoal come accoug coveos, a reaso Se made hree addoal suggesos for prce dces he coex of seasoal producs. Ther coruos wll e evaluaed Chaper
12 5. Basc Idex Numer Theory ale 39 prce p o mach he aual quay q s he value of oal reveue for produc year dvded y q. Thus, we have 2, m (5.24) p v q ;,..., m 2 m, v m 2 m, m, v p m 2 m, m, s ( p ), m ; usg equao (5.23) where he share of aual reveue o produc moh m of he ase year s (5.25) s m, v m, 2 k, v k ;,...,. Thus, he aual aseyear prce for produc, p, urs ou o e a mohly reveueweghed harmoc mea of he mohly prces for produc he ase year, p,, p,2,, p, Usg he aual produc prces for he ase year defed y equao (5.24), a vecor of hese prces ca e defed as p [p,,p ]. Usg hs defo, he Lowe dex ca e expressed as a rao of wo Laspeyres dces where he prce vecor p plays he role of aseperod prces each of he wo Laspeyres dces: (5.26) P ( p, p, q ) Lo Hece, hese aual produc prces are esseally uvalue prces. Uder codos of hgh flao, he aual prces defed y equao (5.24) may o loger e reasoale or represeave of prces durg he ere ase year ecause he reveues he fal mohs of he hghflao year wll e somewha arfcally low up y geeral flao. Uder hese codos, he aual prces ad aual produc reveue shares should e erpreed wh cauo. For more o dealg wh suaos where here s hgh flao wh a year, see Peer Hll (996). pq s p p 0 0 s p p 0 PL( p, p, q )/ PL( p, p, q ) / ( / ) / ( / ), where he Laspeyres formula P L was defed y equao (5.5) aove. Thus, he aove equao shows ha he Lowe mohly prce dex comparg he prces of moh 0 wh hose of moh usg he quaes of ase year as weghs, P Lo (p 0,p,q ), s equal o he Laspeyres dex ha compares he prces of moh wh hose of year, P L (p,p,q ), dvded y he Laspeyres dex ha compares he prces of moh 0 wh hose of year, P L (p,p 0,q ). Noe ha he Laspeyres dex he umeraor ca e calculaed f he aseyear produc reveue shares, s, are kow alog wh he prce raos ha compare he prces of produc moh, p, wh he correspodg aual average prces he ase year, p. The Laspeyres dex he deomaor ca e calculaed f he aseyear produc reveue shares, s, are kow alog wh he prce raos ha compare he prces of produc moh 0, p 0, wh he correspodg aual average prces he ase year, p Aoher covee formula for evaluag he Lowe dex, P Lo (p 0,p,q ), uses he hyrd weghs formula, equao (5.5). I he prese coex, he formula ecomes (5.27) P ( p, p, q ) Lo 0 0 ( p / p ) 0 0 p pq s p, where he hyrd weghs s 0 usg he prces of moh 0 ad he quaes of year are defed y (5.28) pq p jq j j s ;,..., 38
13 Producer Prce Idex Maual 0 pq ( p / p ) 0 pq j j( pj / pj) j Equao (5.28) shows how he aseyear reveues, p q, ca e mulpled y he produc prce dces, p 0 /p, o calculae he hyrd shares. 5.4 Oe addoal formula for he Lowe dex, P Lo (p 0,p,q ), wll e exhed. Noe ha he Laspeyres decomposo of he Lowe dex defed y he hrd le equao (5.26) volves he very logerm prce relaves, p /p, ha compare he prces moh, p, wh he possly dsa aseyear prces, p. Furher, he hyrd share decomposo of he Lowe dex defed y he hrd le equao (5.27) volves he logerm mohly prce relaves, p /p 0, whch compare he prces moh, p, wh he ase moh prces, p 0. Boh hese formulas are o sasfacory pracce ecause of he prolem of sample aro: each moh, a susaal fraco of producs dsappears from he markeplace, ad hus s useful o have a formula for updag he prevous moh s prce dex usg jus mohovermoh prce relaves. I oher words, logerm prce relaves dsappear a a rae ha s oo large pracce o ase a dex umer formula o her use. The Lowe dex for moh +, P Lo (p 0,p +,q ), ca e wre erms of he Lowe dex for moh, P Lo (p 0,p,q ), ad a updag facor as follows: (5.29) P ( p, p, q ) Lo + p q p q + pq 0 pq. + p q 0 PLo ( p, p, q ) pq P p 0 Lo (,, ) p + p pq + 0 p PLo ( p, p, q ) s, p where he hyrd weghs s are defed y (5.30) pq p jq j j s ;,...,. Thus, he requred updag facor, gog from moh o moh +, s he chalked dex + ( ) s p p, whch uses he hyrd share weghs s correspodg o moh ad ase year The Lowe dex P Lo (p 0,p,q ) ca e regarded as a approxmao o he ordary Laspeyres dex, P L (p 0,p,q 0 ), ha compares he prces of he ase moh 0, p 0, wh hose of moh, p, usg he quay vecor of moh 0, q 0, as weghs. There s a relavely smple formula ha relaes hese wo dces. To expla hs formula, s frs ecessary o make a few defos. Defe he h prce relave ewee moh 0 ad moh as 0 (5.3) r p / p ;,...,. The ordary Laspeyres prce dex, gog from moh 0 o, ca e defed erms of hese prce relaves as follows: (5.32) P ( p, p, q ) L 0 0 p pq p p 0 s p pq 382
14 5. Basc Idex Numer Theory 0 s r r, where he moh 0 reveue shares s 0 are defed as follows: (5.33) s 0 pq p jq j j ;,..., Defe he h quay relave as he rao of he quay of produc used he ase year, q, o he quay used moh 0, q 0, as follows: 0 (5.34) q / q ;,...,. The Laspeyres quay dex, Q L (q 0,q,p 0 ), ha compares quaes year, q, wh he correspodg quaes moh 0, q 0, usg he prces of moh 0, p 0, as weghs ca e defed as a weghed average of he quay raos as follows: 0 p q 0 0 L 0 0 p q q p q q 0 0 pq q 0 s 0 q 0 s * (5.35) Q ( q, q, p ) 5.44 Usg equao (A5.2.4) Appedx 5.2, he relaoshp ewee he Lowe dex P Lo (p 0,p,q ) ha uses he quaes of year as weghs o compare he prces of moh wh moh 0 ad he correspodg ordary Laspeyres dex P L (p 0,p,q 0 ) ha uses he quaes of moh 0 as weghs s defed as (5.36) P ( p, p, q ) Lo 0 P ( p, p, q ) + L ( r r )( ) s 0 Q q q p 0 0 L (,, ) Thus, he Lowe prce dex usg he quaes of year as weghs, P Lo (p 0,p,q ), s equal o he usual Laspeyres dex usg he quaes of moh 0 as weghs, P L (p 0,p,q 0 ), plus a covarace erm ( r r )( ) s 0 ewee he prce relaves r p / p 0 ad he quay relaves q /q 0, dvded y he Laspeyres quay dex Q L (q 0,q,p 0 ) ewee moh 0 ad ase year Equao (5.36) shows ha he Lowe prce dex wll cocde wh he Laspeyres prce dex f he covarace or correlao ewee he moh 0 o prce relaves r p /p 0 ad he moh 0 o year quay relaves q /q 0 s zero. Noe ha hs covarace wll e zero uder hree dffere ses of codos: If he moh prces are proporoal o he moh 0 prces so ha all r r*, If he ase year quaes are proporoal o he moh 0 quaes so ha all *, ad If he dsruo of he relave prces r s depede of he dsruo of he relave quaes. The frs wo codos are ulkely o hold emprcally, u he hrd s possle, a leas approxmaely, f purchasers do o sysemacally chage her purchasg has respose o chages relave prces If hs covarace equao (5.36) s egave, he he Lowe dex wll e less ha he Laspeyres, ad, fally, f he covarace s posve, he he Lowe dex wll e greaer ha he Laspeyres dex. Alhough he sg ad magude of he covarace erm s ulmaely a emprcal maer, s possle o make some reasoale cojecures aou s lkely sg. If he ase year precedes he prce referece moh 0 ad here are logerm reds prces, he s lkely ha hs covarace s posve, ad hece ha he Lowe
15 Producer Prce Idex Maual dex wll exceed he correspodg Laspeyres prce dex; 40 ha s, (5.37) P p P p Lo(,, ) > L (,, ). To see why hs covarace s lkely o e posve, suppose ha here s a logerm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh ormal susuo resposes, 4 q / q 0 less a average quay chage of hs ype (*) s lkely o e egave, or, upo akg recprocals, q / q less a average quay chage of 0 hs (recprocal) ype s lkely o e posve. Bu f he logerm upward red prces has perssed ack o he ase year, he * (q / q 0 ) * s also lkely o e posve. Hece, he covarace wll e posve uder hese crcumsaces. Moreover, he more dsa s he wegh referece year from he prce referece moh 0, he gger he resduals * wll lkely e ad he gger wll e he posve covarace. Smlarly, he more dsa s he curreperod moh from he aseperod moh 0, he gger he resduals r r* wll lkely e ad he gger wll e he posve covarace. Thus, uder he assumpos ha here are logerm reds prces ad ormal susuo resposes, he Lowe dex wll ormally e greaer ha he correspodg Laspeyres dex. (5.38) pq 0 P (,, ). 0 p q P p As was dscussed Seco C., a reasoale arge dex o measure he prce chage gog from moh 0 o s some sor of symmerc average of he Paasche dex P P (p 0,p,q ) defed y equao (5.38) ad he correspodg Laspeyres dex P L (p 0,p,q 0 ) defed y equao (5.32). Adapg equao (A5..5) Appedx 5., he relaoshp ewee he Paasche ad Laspeyres dces ca e wre as follows: (5.39) P p P p P(,, ) L(,, ) 0 ( r r )( u u ) s QL ( q, q, p ) where he prce relaves r p / p 0 are defed y equao (5.3) ad her shareweghed average r* y equao (5.32), ad he u, u* ad Q L are defed as follows: 0 (5.40) u q / q ;,...,,, 5.47 Defe he Paasche dex ewee mohs 0 ad as follows: (5.4) 0 L( 0,, 0 ), u s u Q q q p 40 I s also ecessary o assume ha purchasers have ormal susuo effecs respose o hese logerm reds prces; ha s, f a produc creases (relavely) prce, s quay purchased wll decle (relavely), ad f a produc decreases relavely prce, s quay purchased wll crease relavely. Ths reflecs he ormal marke equlrum respose o chages supply. 4 Walsh (90, pp ) was well aware of susuo effecs, as ca e see he followg comme ha oed he asc prolem wh a fxedaske dex ha uses he quay weghs of a sgle perod: The argume made y he arhmec averags supposes ha we uy he same quaes of every class a oh perods spe of he varao her prces, whch we rarely, f ever, do. As a rough proposo, we a commuy geerally sped more o arcles ha have rse prce ad ge less of hem, ad sped less o arcles ha have falle prce ad ge more of hem. ad he moh 0 reveue shares s 0 are defed y equao (5.33). Thus, u* s equal o he Laspeyres quay dex ewee mohs 0 ad. Ths meas ha he Paasche prce dex ha uses he quaes of moh as weghs, P P (p 0,p,q ), s equal o he usual Laspeyres dex usg he quaes of moh 0 as weghs, P L (p 0,p,q 0 ), plus a 0 covarace erm ( r r )( u u ) s ewee he prce relaves r p / p 0 ad he quay relaves u q / q 0, dvded y he Laspeyres quay dex Q L (q 0,q,p 0 ) ewee moh 0 ad moh Alhough he sg ad magude of he covarace erm s aga a emprcal maer, s possle o make a reasoale cojecure aou s lkely sg. If here are logerm reds prces, ad purchasers respod ormally o prce chages her purchases, he s lkely ha hs covar 384
16 5. Basc Idex Numer Theory ace s egave, ad hece he Paasche dex wll e less ha he correspodg Laspeyres prce dex; ha s, (5.42) P p < P p P(,, ) L(,, ) To see why hs covarace s lkely o e egave, suppose ha here s a logerm upward red he prce of produc 42 so ha r r* (p / p 0 ) r* s posve. Wh ormal susuo resposes, q / q 0 less a average quay chage of hs ype (u*) s lkely o e egave. Hece, u u* (q / q 0 ) u* s lkely o e egave. Thus, he covarace wll e egave uder hese crcumsaces. Moreover, he more dsa s he ase moh 0 from he curremoh, he gger magude he resduals u u* wll lkely e ad he gger magude wll e he egave covarace. 43 Smlarly, he more dsa s he curreperod moh from he aseperod moh 0, he gger he resduals r r* wll lkely e ad he gger magude wll e he covarace. Thus, uder he assumpos ha here are logerm reds prces ad ormal susuo resposes, he Laspeyres dex wll e greaer ha he correspodg Paasche dex, wh he dvergece lkely growg as moh ecomes more dsa from moh Pug he argumes he hree prevous paragraphs ogeher, ca e see ha uder he assumpos ha here are logerm reds prces ad ormal susuo resposes, he Lowe prce dex ewee mohs 0 ad wll exceed he correspodg Laspeyres prce dex, whch ur wll exceed he correspodg Paasche prce dex; ha s, uder hese hypoheses, (5.43) P ( p, p, q ) > P ( p, p, q ) > P ( p, p, q ). Lo L P Thus, f he logru arge prce dex s a average of he Laspeyres ad Paasche dces, ca e 42 The reader ca carry hrough he argume f here s a logerm relave decle he prce of he h produc. The argume requred o oa a egave covarace requres ha here e some dffereces he logerm reds prces; ha s, f all prces grow (or fall) a he same rae, we have prce proporoaly, ad he covarace wll e zero. 43 However, Q L u* may also e growg magude, so he e effec o he dvergece ewee P L ad P P s amguous. see ha he Laspeyres dex wll have a upward as relave o hs arge dex, ad he Paasche dex wll have a dowward as. I addo, f he ase year s pror o he prce referece moh, moh 0, he he Lowe dex wll also have a upward as relave o he Laspeyres dex ad hece also o he arge dex. D.2 Lowe dex ad mdyear dces 5.50 The dscusso he prevous paragraph assumed ha he ase year for quaes preceded he ase moh for prces, moh 0. However, f he curreperod moh s que dsa from he ase moh 0, he s possle o hk of he ase year as referrg o a year ha les ewee mohs 0 ad. If he year does fall ewee mohs 0 ad, he he Lowe dex ecomes a mdyear dex. 44 The Lowe mdyear dex o loger has he upward ases dcaed y he equales equao (5.43) uder he assumpo of logerm reds prces ad ormal susuo resposes y quaes. 5.5 I s ow assumed ha he aseyear quay vecor q correspods o a year ha les ewee mohs 0 ad. Uder he assumpo of logerm reds prces ad ormal susuo effecs so ha here are also logerm reds quaes ( he oppose dreco o he reds prces so ha f he h produc prce s redg up, he he correspodg h quay s redg dow), s lkely ha he ermedaeyear qua 44 Ths cocep ca e raced o Peer Hll (998, p. 46): Whe flao has o e measured over a specfed sequece of years, such as a decade, a pragmac soluo o he prolems rased aove would e o ake he mddle year as he ase year. Ths ca e jusfed o he grouds ha he aske of goods ad servces purchased he mddle year s lkely o e much more represeave of he paer of cosumpo over he decade as a whole ha askes purchased eher he frs or he las years. Moreover, choosg a more represeave aske wll also ed o reduce, or eve elmae, ay as he rae of flao over he decade as a whole as compared wh he crease he CoL dex. Thus, addo o roducg he cocep of a mdyear dex, Hll also roduced he dea of represeavy as. For addoal maeral o mdyear dces, see Schulz (999) ad Okamoo (200). Noe ha he mdyear dex cocep could e vewed as a close compeor o Walsh s (90, p. 43) mulyear fxedaske dex, where he quay vecor was chose o e a arhmec or geomerc average of he quay vecors he perod. 385
17 Producer Prce Idex Maual y vecor wll le ewee he mohly quay vecors q 0 ad q. The mdyear Lowe dex, P Lo (p 0,p,q ), ad he Laspeyres dex gog from moh 0 o, P L (p 0,p,q 0 ), wll sll sasfy he exac relaoshp gve y equao (5.36). Thus, P Lo (p 0,p,q ) wll equal P L (p 0,p,q 0 ) plus he covarace erm ( r r )( ) s QL( q, q, p ), where Q L (q 0,q,p 0 ) s he Laspeyres quay dex gog from moh 0 o. Ths covarace erm s lkely o e egave, so ha (5.44) P p P p L(,, ) > Lo(,, ). To see why hs covarace s lkely o e egave, suppose ha here s a logerm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh ormal susuo resposes, q wll ed o decrease relavely over me, ad sce q s assumed o e ewee q 0 ad q, q /q 0 less a average quay chage of hs ype, r* s lkely o e egave. Hece u u* (q / q 0 ) * s lkely o e egave. Thus, he covarace s lkely o e egave uder hese crcumsaces. Uder he assumpos ha he quay ase year falls ewee mohs 0 ad ad ha here are logerm reds prces ad ormal susuo resposes, he Laspeyres dex wll ormally e larger ha he correspodg Lowe mdyear dex, wh he dvergece lkely growg as moh ecomes more dsa from moh I ca also e see ha uder he aove assumpos, he mdyear Lowe dex s lkely o e greaer ha he Paasche dex ewee mohs 0 ad ; ha s, (5.45) P p P p 0 0 Lo(,, ) > P (,, ). To see why he aove equaly s lkely o hold, hk of q sarg a he moh 0 quay vecor q 0 ad he redg smoohly o he moh quay vecor q. Whe q q 0, he Lowe dex ecomes he Laspeyres dex P L (p 0,p,q 0 ). Whe q q, he Lowe dex ecomes he Paasche dex P P (p 0,p,q ). Uder he assumpo of redg prces ad ormal susuo resposes o hese redg prces, was show earler ha he Paasche dex wll e less ha he correspodg Laspeyres prce dex; ha s, ha P P (p 0,p,q ) was less ha P L (p 0,p,q 0 ); recall equao (5.42). Thus, uder he assumpo of smoohly redg prces ad quaes ewee mohs 0 ad, ad assumg ha q s ewee q 0 ad q, we wll have (5.46) P p P p < (,, ). 0 0 P(,, ) < Lo(,, ) 0 0 PL p Thus, f he ase year for he Lowe dex s chose o e ewee he ase moh for he prces, moh 0, ad he curre moh for prces, moh, ad here are reds prces wh correspodg reds quaes ha correspod o ormal susuo effecs, he he resulg Lowe dex s lkely o le ewee he Paasche ad Laspeyres dces gog from mohs 0 o. If he reds prces ad quaes are smooh, he choosg he ase year halfway ewee perods 0 ad should gve a Lowe dex ha s approxmaely halfway ewee he Paasche ad Laspeyres dces ad hece wll e very close o a deal arge dex ewee mohs 0 ad. Ths asc dea has ee mplemeed y Okamoo (200) usg Japaese cosumer daa, ad he foud ha he resulg mdyear dces approxmaed he correspodg Fsher deal dces very closely I should e oed ha hese mdyear dces ca e compued oly o a rerospecve ass; ha s, hey cao e calculaed a mely fasho as ca Lowe dces ha use a ase year efore moh 0. Thus, mdyear dces cao e used o replace he more mely Lowe dces. However, hese mely Lowe dces are lkely o have a upward as eve gger ha he usual Laspeyres upward as compared wh a deal arge dex, whch was ake o e a average of he Paasche ad Laspeyres dces All of he equales derved hs seco res o he assumpo of logerm reds prces (ad correspodg ecoomc resposes quaes). If here are o sysemac logru reds prces ad oly radom flucuaos aroud a commo red all prces, he he aove equales are o vald, ad he Lowe dex usg a pror ase year wll proaly provde a perfecly adequae approxmao o oh he Paasche ad Laspeyres dces. However, here are some reasos for elevg ha some logru reds prces exs: 386
18 5. Basc Idex Numer Theory () The compuer chp revoluo of he pas 40 years has led o srog dowward reds he prces of producs ha use hese chps esvely. As ew uses for chps are developed, he share of producs ha are chpesve has grow, whch mples ha wha used o e a relavely mor prolem has ecome a major prolem. () Oher major scefc advaces have had smlar effecs. For example, he veo of feropc cale (ad lasers) has led o a dowward red elecommucaos prces as osolee echologes ased o copper wre are gradually replaced. () Sce he ed of World War II, a seres of eraoal rade agreemes have dramacally reduced arffs aroud he world. These reducos, comed wh mprovemes rasporao echologes, have led o a rapd growh of eraoal rade ad remarkale mprovemes eraoal specalzao. Maufacurg acves he more developed ecoomes have gradually ee ousourced o lowerwage coures, leadg o deflao goods prces mos coures. However, may servces cao e readly ousourced, 45 ad so o average he prce of servces reds upward whle he prce of goods reds dowward. (v) A he mcroecoomc level, here are remedous dffereces growh raes of frms. Successful frms expad her scale, lower her coss, ad cause less successful compeors o wher away wh her hgher prces ad lower volumes. Ths leads o a sysemac egave correlao ewee chages em prces ad he correspodg chages em volumes ha ca e very large. Thus, here s some a pror ass for assumg logru dverge reds prces ad hece some ass for cocer ha a Lowe dex ha uses a ase year for quay weghs ha s pror o he ase moh for prces may e upward ased, compared wh a more deal arge dex. 45 However some servces ca e eraoally ousourced; for example, call ceers, compuer programmg, ad arle maeace. D.3 Youg dex 5.55 Recall he defos for he aseyear quaes, q, ad he aseyear prces, p, gve y equao (5.23) ad equao (5.24). The aseyear reveue shares ca e defed he usual way as follows: (5.47) pq pkqk k s ;,...,. Defe he vecor of aseyear reveue shares he usual way as s [s,,s ]. These aseyear reveue shares were used o provde a alerave formula for he ase year Lowe prce dex gog from moh 0 o defed equao (5.26) as P Lo (p 0,p,q ) 0 s ( p / p ) s ( p / p ). Raher ha usg hs dex as her shorru arge dex, may sascal ageces use he followg closely relaed dex: 0 0 (5.48) PY p p s s ( p p ) (,, ). Ths ype of dex was frs defed y he Eglsh ecooms Arhur Youg (82). 46 Noe ha here s a chage focus whe he Youg dex s used compared wh he dces proposed earler hs chaper. Up o hs po, he dces proposed have ee of he fxedaske ype (or averages of such dces), where a produc aske ha s somehow represeave for he wo perods eg compared s chose ad he purchased a he prces of he wo perods, ad he dex s ake o e he rao of hese wo coss. O he oher had, for he Youg dex, oe sead chooses represeave reveue shares ha pera o he wo perods uder cosderao ad he uses hese shares o calculae he overall dex as a shareweghed average of he dvdual prce raos, p / p 0. Noe ha hs shareweghed average of prce raos vew of dex umer heory s a dffere from he vew ake a he egg of hs chaper, whch vewed he dex umer prolem as he prolem of decomposg a value rao o he produc of wo erms, oe of whch expresses he amou of 46 Walsh (90, p. 536; 932, p. 657) arues hs formula o Youg. 387
19 Producer Prce Idex Maual prce chage ewee he wo perods ad he oher ha expresses he amou of quay chage Sascal ageces somemes regard he Youg dex defed aove as a approxmao o he Laspeyres prce dex P L (p 0,p,q 0 ). Hece, s of eres o see how he wo dces compare. Defg he logerm mohly prce relaves gog from moh 0 o as r p /p 0 ad usg equaos (5.32) ad (5.48), (5.49) sce P p p s P p Y(,, ) L(,, ) p 0 p s 0 s 0 p p 0 p 0 s s 0 s s r p 0 0 s s r r + r s s 0 s s r r, 0 s s ad defg 47 Irvg Fsher s 922 ook s famous for developg he value rao decomposo approach o dex umer heory, u hs roducory chapers ook he shareweghed average po of vew: A dex umer of prces, he, shows he average perceage chage of prces from oe po of me o aoher (922, p. 3). Fsher we o o oe he mporace of ecoomc weghg: The precedg calculao reas all he commodes as equally mpora; cosequely, he average was called smple. If oe commody s more mpora ha aoher, we may rea he more mpora as hough were wo or hree commodes, hus gvg wo or hree mes as much wegh as he oher commody (922, p. 6). Walsh (90, pp ) cosdered oh approaches: We ca eher () draw some average of he oal moey values of he classes durg a epoch of years, ad wh weghg so deermed employ he geomerc average of he prce varaos [raos]; or (2) draw some average of he mass quaes of he classes durg he epoch, ad apply o hem Scrope s mehod. Scrope s mehod s he same as usg he Lowe dex. Walsh (90, pp ) cossely sressed he mporace of weghg prce raos y her ecoomc mporace (raher ha usg equally weghed averages of prce relaves). Boh he value rao decomposo approach ad he shareweghed average approach o dex umer heory wll e suded from he axomac perspecve he followg chaper; see also Secos C ad E Chaper 6. ( ) 0 L 0 0. r* s r P p, p, q Thus, he Youg dex P Y (p 0,p,s ) s equal o he Laspeyres dex P L (p 0,p,q 0 ) plus he covarace ewee he dfferece he aual shares perag o year ad he moh 0 shares, s s 0, ad he devaos of he relave prces from her mea, r r* I s o loger possle o guess he lkely sg of he covarace erm. The queso s o loger wheher he quay demaded goes dow as he prce of produc goes up (he aswer o hs queso s usually yes) u does he share of reveue go dow as he prce of produc goes up? The aswer depeds o he elascy of demad for he produc. However, le us provsoally assume ha here are logru reds produc prces, ad f he red prces for produc s aove he mea, he he reveue share for he produc reds dow (ad vce versa). Thus, we are assumg hgh elasces or very srog susuo effecs. Assumg also ha he ase year s efore moh 0, he uder hese codos, suppose ha here s a logerm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh he assumed very elasc purchaser susuo resposes, s wll ed o decrease relavely over me. Sce s s assumed o e efore s 0, s 0 s expeced o e less ha s, or s s 0 wll lkely e posve. Thus, he covarace s lkely o e posve uder hese crcumsaces. Hece wh logru reds prces ad very elasc resposes of purchasers o prce chages, he Youg dex s lkely o e greaer ha he correspodg Laspeyres dex Assume ha here are logru reds produc prces. If he red prces for produc s aove he mea, he suppose ha he reveue share for he produc reds up (ad vce versa). Thus, we are assumg low elasces or very weak susuo effecs. Assume also ha he ase year s efore moh 0, ad suppose ha here s a logerm upward red he prce of produc so ha r r* (p / p 0 ) r* s posve. Wh he assumed very elasc susuo resposes, s wll ed o crease relavely over me, ad, sce s s assumed o e efore s 0, we wll have s 0 greaer ha s, or s s 0 s egave. Thus, he covarace s lkely o e egave uder 388
20 5. Basc Idex Numer Theory hese crcumsaces. Hece wh logru reds prces ad very elasc resposes of purchasers o prce chages, he Youg dex s lkely o e less ha he correspodg Laspeyres dex The prevous wo paragraphs dcae ha, a pror, s o kow wha he lkely dfferece ewee he Youg dex ad he correspodg Laspeyres dex wll e. If elasces of susuo are close o, he he wo ses of reveue shares, s ad s 0, wll e close o each oher ad he dfferece ewee he wo dces wll e close o zero. However, f mohly reveue shares have srog seasoal compoes, he he aual shares s could dffer susaally from he mohly shares s I s useful o have a formula for updag he prevous moh s Youg prce dex usg oly mohovermoh prce relaves. The Youg dex for moh +, P Y (p 0,p +,s ), ca e preseed erms of he Lowe dex for moh, P Y (p 0,p,s ), ad a updag facor as follows: (5.50) p + 0 s ( p / p ) 0 PY ( p, p, s ) 0 s ( p / p ) + 0 p p 0 PY ( p, p, s ) 0 ( p / p ) P ( p, p, s ) s Y usg equao (5.47) p ( / ) ; p p + pq 0 0 p p Y (,, ) 0 pq ( p / p ) Y(,, ) ( / ), P p p s P p p s s p p where he hyrd weghs s 0 are defed y 0 s ( p / p ) 0 sk pk pk k ;,...,. ( / ) Thus, he hyrd weghs s 0 ca e oaed from he aseyear weghs s y updag hem; ha s, y mulplyg hem y he prce relaves (or dces a hgher levels of aggregao), p / p 0. Thus, he requred updag facor, gog from moh o moh +, s he chalked dex, 0 + s ( p / p ), whch uses he hyrd reveueshare weghs s 0 defed y equao (5.5). 5.6 Eve f he Youg dex provdes a close approxmao o he correspodg Laspeyres dex, s dffcul o recommed he use of he Youg dex as a fal esmae of he chage prces gog from perod 0 o, jus as was dffcul o recommed he use of he Laspeyres dex as he fal esmae of flao gog from perod 0 o. Recall ha he prolem wh he Laspeyres dex was s lack of symmery he reame of he wo perods uder cosderao. Tha s, usg he jusfcao for he Laspeyres dex as a good fxedaske dex, here was a decal jusfcao for he use of he Paasche dex as a equally good fxedaske dex o compare perods 0 ad. The Youg dex suffers from a smlar lack of symmery wh respec o he reame of he ase perod. The prolem ca e explaed as follows. The Youg dex, P Y (p 0,p,s ), defed y equao (5.48), calculaes he prce chage ewee mohs 0 ad, reag moh 0 as he ase. Bu here s o parcular reaso o rea moh 0 as he ase moh oher ha coveo. Hece, f we rea moh as he ase ad use he same formula o measure he prce chage from moh ack o moh 0, he dex P Y (p 0,p,s ) s p p 0 ( / ) would e approprae. Ths esmae of prce chage ca he e made comparale o he orgal Youg dex y akg s recprocal, leadg o he followg reased Youg dex, 48 P Y *(p 0,p,s ), defed as (5.5) 0 0 pq ( p / p ) 0 pkqk pk pk k s ( / ) 48 Usg Irvg Fsher s (922, p. 8) ermology, P Y *(p 0,p,s ) /[P Y (p,p 0,s )] s he me ahess of he orgal Youg dex, P Y (p 0,p,s ). 389
Chapter 4 MultipleDegreeofFreedom (MDOF) Systems. Packing of an instrument
Chaper 4 MulpleDegreeofFreedom (MDOF Sysems Eamples: Pacg of a srume Number of degrees of freedom Number of masses he sysem X Number of possble ypes of moo of each mass Mehods: Newo s Law ad Lagrage
More informationREVISTA INVESTIGACION OPERACIONAL Vol. 25, No. 1, 2004. k n ),
REVISTA INVESTIGACION OPERACIONAL Vol 25, No, 24 RECURRENCE AND DIRECT FORMULAS FOR TE AL & LA NUMBERS Eduardo Pza Volo Cero de Ivesgacó e Maemáca Pura y Aplcada (CIMPA), Uversdad de Cosa Rca ABSTRACT
More information7.2 Analysis of Three Dimensional Stress and Strain
eco 7. 7. Aalyss of Three Dmesoal ress ad ra The cocep of raco ad sress was roduced ad dscussed Par I..5. For he mos par he dscusso was cofed o wodmesoal saes of sress. Here he fully hree dmesoal sress
More informationThe following model solutions are presented for educational purposes. Alternate methods of solution are, of course, acceptable.
The followg model soluos are preseed for educaoal purposes. Alerae mehods of soluo are, of course, accepable.. Soluo: C Gve he same prcpal vesed for he same perod of me yelds he same accumulaed value,
More informationVladimir PAPI], Jovan POPOVI] 1. INTRODUCTION
Yugoslav Joural of Operaos Research 200 umber 779 VEHICLE FLEET MAAGEMET: A BAYESIA APPROACH Vladmr PAPI] Jova POPOVI] Faculy of Traspor ad Traffc Egeerg Uversy of Belgrade Belgrade Yugoslava Absrac:
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE SOLUTIONS Ths page dcaes chages made o Sudy Noe FM0905. Aprl 8, 04: Queso ad soluo 6 added. Jauary 4, 04: Quesos ad soluos 58 60 were
More informationChristopher Dougherty EC220  Introduction to econometrics: past examinations and marking schemes 2011 exam
Chrsopher Doughery EC0  Iroduco o ecoomercs: pas examaos ad markg schemes 011 exam Orgal cao: Doughery, C. (01) EC0  Iroduco o ecoomercs: pas examaos ad markg schemes. [Teachg Resource] 011 The Auhor
More informationProving the Computer Science Theory P = NP? With the General Term of the Riemann Zeta Function
Research Joural of Mahemacs ad Sascs 3(2): 7276, 20 ISSN: 20407505 Maxwell Scefc Orgazao, 20 Receved: Jauary 08, 20 Acceped: February 03, 20 Publshed: May 25, 20 Provg he ompuer Scece Theory P NP? Wh
More informationAmerican Journal of Business Education September 2009 Volume 2, Number 6
Amerca Joural of Bue Educao Sepember 9 Volume, umber 6 Tme Value Of Moe Ad I Applcao I Corporae Face: A Techcal oe O L Relaohp Bewee Formula JeHo Che, Alba Sae Uver, USA ABSTRACT Tme Value of Moe (TVM
More informationA new proposal for computing portfolio valueatrisk for seminonparametric distributions
A ew proposal for compug porfolo valuearsk for semoparamerc dsrbuos TroMauel Ñíguez ad Javer Peroe Absrac Ths paper proposes a semoparamerc (SNP) mehodology for compug porfolo valuearsk (VaR) ha
More informationANOVA Notes Page 1. Analysis of Variance for a OneWay Classification of Data
ANOVA Notes Page Aalss of Varace for a OeWa Classfcato of Data Cosder a sgle factor or treatmet doe at levels (e, there are,, 3, dfferet varatos o the prescrbed treatmet) Wth a gve treatmet level there
More informationProfessional Liability Insurance Contracts: Claims Made Versus Occurrence Policies
ARICLES ACADÉMIQUES ACADEMIC ARICLES Assuraces e geso des rsques, vol. 79(34), ocobre 2011 javer 2012, 251277 Isurace ad Rsk Maageme, vol. 79(34), Ocober 2011 Jauary 2012, 251277 Professoal Lably
More informationLecture 13 Time Series: Stationarity, AR(p) & MA(q)
RS C  ecure 3 ecure 3 Tme Seres: Saoar AR & MAq Tme Seres: Iroduco I he earl 97 s was dscovered ha smle me seres models erformed beer ha he comlcaed mulvarae he oular 96s macro models FRBMITPe. See
More informationThe Design of a Forecasting Support Models on Demand of Durian for Domestic Markets and Export Markets by Time Series and ANNs.
The 2 d RMUTP Ieraoal Coferece 2010 Page 108 The Desg of a Forecasg Suppor Models o Demad of Dura for Domesc Markes ad Expor Markes by Tme Seres ad ANNs. Udomsr Nohacho, 1* kegpol Ahakor, 2 Kazuyosh Ish,
More informationDeterminants of Foreign Direct Investment in Malaysia: What Matters Most?
Deermas of Foreg Drec Ivesme Maaysa: Wha Maers Mos? Nursuha Shahrud, Zarah Yusof ad NuruHuda Mohd. Saar Ths paper exames he deermas of foreg drec vesme Maaysa from 970008. The causay ad dyamc reaoshp
More informationHIGH FREQUENCY MARKET MAKING
HIGH FREQUENCY MARKET MAKING RENÉ CARMONA AND KEVIN WEBSTER Absrac. Sce hey were auhorzed by he U.S. Secury ad Exchage Commsso 1998, elecroc exchages have boomed, ad by 21 hgh frequecy radg accoued for
More informationBusiness School Discipline of Finance. Discussion Paper 2014005. Modelling the crash risk of the Australian Dollar carry trade
Dscusso Paper: 2014005 Busess School Dscple of Face Dscusso Paper 2014005 Modellg he crash rsk of he Ausrala Dollar carry rade SukJoog Km Uversy of Sydey Busess School Modellg he crash rsk of he Ausrala
More informationEQUITY VALUATION USING DCF: A THEORETICAL ANALYSIS OF THE LONG TERM HYPOTHESES
Ivesme Maaeme ad Facal Iovaos Volume 4 Issue 007 9 EQUIY VALUAION USING DCF: A HEOREICAL ANALYSIS OF HE LONG ERM HYPOHESES Luco Cassa * Adrea Pla ** Slvo Vsmara *** Absrac hs paper maches he sesvy aalyss
More informationTraditional Smoothing Techniques
Tradoal Smoohg Techques Smple Movg Average: or Ceered Movg Average, assume s odd: 2 ( 2 ( Weghed Movg Average: W W (or, of course, you could se up he W so ha hey smply add o oe. Noe Lear Movg Averages
More informationTIMEVARYING RISK PREMIUM IN LARGE CROSSSECTIONAL EQUITY DATASETS
IMEVARYING RISK PREMIUM IN LARGE CROSSSECIONAL EQUIY DAASES Parck Gaglard a, Elsa Ossola ad Olver Scalle c * Frs draf: Decemer 2 hs verso: Novemer 2 Asrac We develop a ecoomerc mehodology o fer he pah
More informationMarkit iboxx USD Liquid Leveraged Loan Index
Mark Boxx USD Lqud Leveraged Loa Idex Sepember 20 Mark Boxx USD Leveraged Loa Idex Idex Gude Coe Overvew... 4 Seleco Crera... 5 Idex Icepo/Rebalacg... 5 Elgbly Crera... 5 Loa Type... 5 Mmum facly ze...
More informationCONVERGENCE AND SPATIAL PATTERNS IN LABOR PRODUCTIVITY: NONPARAMETRIC ESTIMATIONS FOR TURKEY 1
CONVERGENCE AND SPAIAL PAERNS IN LABOR PRODUCIVIY: NONPARAMERIC ESIMAIONS FOR URKEY ugrul emel, Ays asel & Peer J. Alberse Workg Paper 993 Forhcomg he Joural of Regoal Aalyss ad Polcy, 999. We would lke
More informationNo Regret Learning in Oligopolies: Cournot vs Bertrand
No Regre Learg Olgopoles: Couro vs Berrad Ur Nadav Georgos Plouras Absrac Couro ad Berrad olgopoles cosue he wo mos prevale models of frm compeo. The aalyss of Nash equlbra each model reveals a uque predco
More informationMETHODOLOGY ELECTRICITY, GAS AND WATER DISTRIBUTION INDEX (IDEGA, by its Spanish acronym) (Preliminary version)
MEHODOLOGY ELEY, GAS AND WAE DSBUON NDEX (DEGA, by s Sash acroym) (Prelmary verso) EHNAL SUBDEOAE OPEAONS SUBDEOAE Saago, December 26h, 2007 HDA/GGM/GMA/VM ABLE OF ONENS Pages. roduco 3 2. oceual frameork
More informationStandardized Formula Sheet: Formulas Standard Normal Distribution Table Summary of Financial Ratios
Sadardzed Formula See: Formulas Sadard ormal Dsrbuo Table Summary o Facal Raos Formulas. Prese Value o a Sgle Cas Flow CF PV (. Fuure Value o a Sgle Cas Flow FV CF( 3. Prese Value o a Ordary Auy ( PV PT[
More informationLongitudinal and Panel Data: Analysis and Applications for the Social Sciences. Edward W. Frees
Logudal ad Pael Daa: Aalss ad Applcaos for he Socal Sceces b Edward W. Frees Logudal ad Pael Daa: Aalss ad Applcaos for he Socal Sceces Bref Table of Coes Chaper. Iroduco PART I  LINEAR MODELS Chaper.
More informationQuantifying Environmental Green Index For Fleet Management Model
Proceedgs of he Easer Asa Socey for Trasporao Sudes, Vol.9, 20 Quafyg Evromeal ree Idex For Flee Maageme Model Lay Eg TEOH a, Hoo Lg KHOO b a Deparme of Mahemacal ad Acuaral Sceces, Faculy of Egeerg ad
More informationFinancial Time Series Forecasting with Grouped Predictors using Hierarchical Clustering and Support Vector Regression
Ieraoal Joural of Grd Dsrbuo Compug, pp.5364 hp://dx.do.org/10.1457/jgdc.014.7.5.05 Facal Tme Seres Forecasg wh Grouped Predcors usg Herarchcal Cluserg ad Suppor Vecor Regresso ZheGao a,b,* ad JajuYag
More information1. The Time Value of Money
Corporate Face [000345]. The Tme Value of Moey. Compoudg ad Dscoutg Captalzato (compoudg, fdg future values) s a process of movg a value forward tme. It yelds the future value gve the relevat compoudg
More informationAPPENDIX III THE ENVELOPE PROPERTY
Apped III APPENDIX III THE ENVELOPE PROPERTY Optmzato mposes a very strog structure o the problem cosdered Ths s the reaso why eoclasscal ecoomcs whch assumes optmzg behavour has bee the most successful
More informationHarmony search algorithms for inventory management problems
Afrca Joural of Busess Maageme Vol.6 (36), pp. 98649873, 2 Sepember, 202 Avalable ole a hp://www.academcourals.org/ajbm DOI: 0.5897/AJBM2.54 ISSN 9938233 202 Academc Jourals Revew Harmoy search algorhms
More information10.5 Future Value and Present Value of a General Annuity Due
Chapter 10 Autes 371 5. Thomas leases a car worth $4,000 at.99% compouded mothly. He agrees to make 36 lease paymets of $330 each at the begg of every moth. What s the buyout prce (resdual value of the
More informationPerformance Comparisons of Load Balancing Algorithms for I/O Intensive Workloads on Clusters
Joural of ewor ad Compuer Applcaos, vol. 3, o., pp. 3246, Jauary 2008. Performace Comparsos of oad Balacg Algorhms for I/O Iesve Worloads o Clusers Xao Q Deparme of Compuer Scece ad Sofware Egeerg Aubur
More informationEuropean Exotic Options
Hado # for B9.38 rg lecre dae: 4/3/ * RskNeral Valao Eroea Exoc Oos e. Prce rocess of he derlyg secry. e. Payoff of he dervave. e 3. Execao of dscoed ayoff der RNPM.. Chooser Oo oo o oo A me : rchase
More informationThe Unintended Consequences of Tort Reform: Rent Seeking in New York State s Structured Settlements Statutes
The Ueded Cosequeces of Tor Reform: Re Seeg ew Yor Sae s Srucured Selemes Saues Publshed Joural of Foresc Ecoomcs, Volume 3 o, Wer 2 By Lawrece M. Spzma* Professor of Ecoomcs Mahar Hall Sae Uversy of ew
More informationObject Tracking Based on Online Classification Boosted by Discriminative Features
Ieraoal Joural of Eergy, Iformao ad Commucaos, pp.920 hp://dx.do.org/10.14257/jec.2013.4.6.02 Objec Trackg Based o Ole Classfcao Boosed by Dscrmave Feaures Yehog Che 1 ad Pl Seog Park 2 1 Qlu Uversy of
More informationAverage Price Ratios
Average Prce Ratos Morgstar Methodology Paper August 3, 2005 2005 Morgstar, Ic. All rghts reserved. The formato ths documet s the property of Morgstar, Ic. Reproducto or trascrpto by ay meas, whole or
More informationApproximate hedging for non linear transaction costs on the volume of traded assets
Noame mauscrp No. wll be sered by he edor Approxmae hedgg for o lear rasaco coss o he volume of raded asses Romuald Ele, Emmauel Lépee Absrac Ths paper s dedcaed o he replcao of a covex coge clam hs a
More informationEvaluation and Modeling of the Digestion and Absorption of Novel Manufacturing Technology in Food Enterprises
Advace Joural of Food Scece ad Techology 9(6): 482486, 205 ISSN: 20424868; eissn: 20424876 Mawell Scefc Orgazao, 205 Submed: Aprl 9, 205 Acceped: Aprl 28, 205 Publshed: Augus 25, 205 Evaluao ad Modelg
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y  ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationPrice Volatility, Trading Activity and Market Depth: Evidence from Taiwan and Singapore Taiwan Stock Index Futures Markets
WeHsu Kuo Asa e Pacfc al./asa Maageme Pacfc Maageme evew (005) evew 0(), (005) 33 0(), 33 Prce Volaly, Tradg Acvy ad Marke Deph: Evdece from Tawa ad Sgapore Tawa Sock Idex Fuures Markes WeHsu Kuo a,*,
More informationMobile Data Mining for Intelligent Healthcare Support
Moble Daa Mg for Iellge Healhcare uppor Absrac The growh umbers ad capacy of moble devces such as moble phoes coupled wh wdespread avalably of expesve rage of bosesors preses a uprecedeed opporuy for moble
More informationOnline Appendix: Measured Aggregate Gains from International Trade
Ole Appedx: Measured Aggregate Gas from Iteratoal Trade Arel Burste UCLA ad NBER Javer Cravo Uversty of Mchga March 3, 2014 I ths ole appedx we derve addtoal results dscussed the paper. I the frst secto,
More informationCHAPTER 22 ASSET BASED FINANCING: LEASE, HIRE PURCHASE AND PROJECT FINANCING
CHAPTER 22 ASSET BASED FINANCING: LEASE, HIRE PURCHASE AND PROJECT FINANCING Q.1 Defie a lease. How does i differ from a hire purchase ad isalme sale? Wha are he cash flow cosequeces of a lease? Illusrae.
More informationMobile Data Mining for Intelligent Healthcare Support
Proceedgs of he 42d Hawa Ieraoal Coferece o ysem ceces  2009 Moble Daa Mg for Iellge Healhcare uppor Par Delr Haghgh, Arkady Zaslavsky, hoal Krshaswamy, Mohamed Medha Gaber Ceer for Dsrbued ysems ad ofware
More informationNatural Gas Storage Valuation. A Thesis Presented to The Academic Faculty. Yun Li
Naural Gas Sorage Valuao A Thess Preseed o The Academc Faculy by Yu L I Paral Fulfllme Of he Requremes for he Degree Maser of Scece he School of Idusral ad Sysem Egeerg Georga Isue of Techology December
More informationMEASURES OF CENTRAL TENDENCY
MODULE  6 Statstcs Measures of Cetral Tedecy 25 MEASURES OF CENTRAL TENDENCY I the prevous lesso, we have leart that the data could be summarsed to some extet by presetg t the form of a frequecy table.
More informationAnalysis of Coalition Formation and Cooperation Strategies in Mobile Ad hoc Networks
Aalss of oalo Formao ad ooperao Sraeges Moble Ad hoc ewors Pero Mchard ad Ref Molva Isu Eurecom 9 Roue des rêes 06904 SophaApols, Frace Absrac. Ths paper focuses o he formal assessme of he properes of
More informationGARCH Modelling. Theoretical Survey, Model Implementation and
Maser Thess GARCH Modellg Theorecal Survey, Model Imlemeao ad Robusess Aalyss Lars Karlsso Absrac I hs hess we survey GARCH modellg wh secal focus o he fg of GARCH models o facal reur seres The robusess
More informationJorge Ortega Arjona Departamento de Matemáticas, Facultad de Ciencias, UNAM jloa@fciencias.unam.mx
Usg UML Sae Dagrams for Moellg he Performace of Parallel Programs Uso e Dagramas e Esao UML para la Moelacó el Desempeño e Programas Paralelos Jorge Orega Aroa Deparameo e Maemácas, Facula e Cecas, UNAM
More informationT = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are :
Bullets bods Let s descrbe frst a fxed rate bod wthout amortzg a more geeral way : Let s ote : C the aual fxed rate t s a percetage N the otoal freq ( 2 4 ) the umber of coupo per year R the redempto of
More informationChapter 3 0.06 = 3000 ( 1.015 ( 1 ) Present Value of an Annuity. Section 4 Present Value of an Annuity; Amortization
Chapter 3 Mathematcs of Face Secto 4 Preset Value of a Auty; Amortzato Preset Value of a Auty I ths secto, we wll address the problem of determg the amout that should be deposted to a accout ow at a gve
More informationAnomaly Detection of Network Traffic Based on Prediction and SelfAdaptive Threshold
Ieraoal Joural of Fuure Geerao Coucao ad eworkg Vol. 8, o. 6 (15), pp. 514 hp://d.do.org/1.1457/fgc.15.8.6. Aoaly Deeco of ework raffc Based o Predco ad SelfAdapve hreshold Haya Wag Depare of Iforao
More informationECONOMIC CHOICE OF OPTIMUM FEEDER CABLE CONSIDERING RISK ANALYSIS. University of Brasilia (UnB) and The Brazilian Regulatory Agency (ANEEL), Brazil
ECONOMIC CHOICE OF OPTIMUM FEEDER CABE CONSIDERING RISK ANAYSIS I Camargo, F Fgueredo, M De Olvera Uversty of Brasla (UB) ad The Brazla Regulatory Agecy (ANEE), Brazl The choce of the approprate cable
More informationA quantization tree method for pricing and hedging multidimensional American options
A quazao ree mehod for prcg ad hedgg muldmesoal Amerca opos Vlad BALLY Glles PAGÈS Jacques PRINTEMS Absrac We prese here he quazao mehod whch s welladaped for he prcg ad hedgg of Amerca opos o a baske
More informationValue of information sharing in marine mutual insurance
Value of formao sharg mare muual surace Kev L, Joh Lu, Ja Ya 3 ad Je M Deparme of Logscs & Marme Sudes, The Hog Kog Polechc Uvers, Hog Kog. Emal address:.x.l@polu.edu.h. Deparme of Logscs & Marme Sudes,
More informationPORTFOLIO CHOICE WITH HEAVY TAILED DISTRIBUTIONS 1. Svetlozar Rachev 2 Isabella Huber 3 Sergio Ortobelli 4
PORTFOLIO CHOIC WITH HAVY TAILD DISTRIBUTIONS Sveloar Rachev Isabella Huber 3 Sergo Orobell 4 We are graeful o Boryaa RachevaJoova Soya Soyaov ad Almra Bglova for he comuaoal aalyss ad helful commes.
More informationCHAPTER 2. Time Value of Money 61
CHAPTER 2 Tme Value of Moey 6 Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 62 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show
More informationThe analysis of annuities relies on the formula for geometric sums: r k = rn+1 1 r 1. (2.1) k=0
Chapter 2 Autes ad loas A auty s a sequece of paymets wth fxed frequecy. The term auty orgally referred to aual paymets (hece the ame), but t s ow also used for paymets wth ay frequecy. Autes appear may
More informationBanking (Early Repayment of Housing Loans) Order, 5762 2002 1
akg (Early Repaymet of Housg Loas) Order, 5762 2002 y vrtue of the power vested me uder Secto 3 of the akg Ordace 94 (hereafter, the Ordace ), followg cosultato wth the Commttee, ad wth the approval of
More informationAbraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract
Preset Value of Autes Uder Radom Rates of Iterest By Abraham Zas Techo I.I.T. Hafa ISRAEL ad Uversty of Hafa, Hafa ISRAEL Abstract Some attempts were made to evaluate the future value (FV) of the expected
More informationValuation Methods of a Life Insurance Company
Valuao Mehods of a Lfe Isurace Comay ISORY...3 2 PRODUC ASSESSMEN : PROFI ESING...4 2. E PROFI ESING IN 3 SEPS...5 2.. Equalece Prcle...5 2..2 radoal Marg...6 2..3 Prof esg...6 2.2 COMMON CRIERIA O EVALUAE
More informationChap.5 Unit Roots and Cointegration in panels
Chap.5 U Roos ad Coegrao paels 5. Iroduco Wh he growg use of crosscoury daa over me o sudy purchasg power pary, growh covergece ad eraoal R&D spllovers, he focus of pael daa ecoomercs has shfed owards
More information Models:  Classical: : Mastermodel (clay( Curves.  Example:  Independent variable t
Compue Gaphcs Geomec Moelg Iouco  Geomec Moelg (GM) sce e of 96  Compue asssace fo  Desg: CAD  Maufacug: : CAM  Moels:  Classcal: : Masemoel (cla( cla, poopes,, Mockup)  GM: mahemacal escpo fo
More informationMORE ON TVM, "SIX FUNCTIONS OF A DOLLAR", FINANCIAL MECHANICS. Copyright 2004, S. Malpezzi
MORE ON VM, "SIX FUNCIONS OF A DOLLAR", FINANCIAL MECHANICS Copyrgh 2004, S. Malpezz I wan everyone o be very clear on boh he "rees" (our basc fnancal funcons) and he "fores" (he dea of he cash flow model).
More informationThe Consumer Price Index for All Urban Consumers (Inflation Rate)
The Cosumer Prce Idex for All Urba Cosumers (Iflato Rate) Itroducto: The Cosumer Prce Idex (CPI) s the crtero of the average prce chage of goods ad servces cosumed by Iraa households. Ths crtero, as a
More informationTrust Evaluation and Dynamic Routing Decision Based on Fuzzy Theory for MANETs
JOURNAL OF SOFTWARE, VOL. 4, NO. 10, ECEBER 2009 1091 Trus Evaluao ad yamc Roug ecso Based o Fuzzy Theory for ANETs Hogu a, Zhpg Ja ad Zhwe Q School of Compuer Scece ad Techology, Shadog Uversy, Ja, Cha.P.R.
More informationThe real value of stock
he real value of sock Collars ivolve he paye of a variable aou of sock, depedig o a average sock price. I his arcle, Ahoy Pavlovich uses he BlackScholes fraework o value hese exoc derivaves ad explore
More informationChapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R =
Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS Objectves of the Topc: Beg able to formalse ad solve practcal ad mathematcal problems, whch the subjects of loa amortsato ad maagemet of cumulatve fuds are
More informationof the relationship between time and the value of money.
TIME AND THE VALUE OF MONEY Most agrbusess maagers are famlar wth the terms compoudg, dscoutg, auty, ad captalzato. That s, most agrbusess maagers have a tutve uderstadg that each term mples some relatoshp
More informationInternal model in life insurance : application of least squares monte carlo in risk assessment
Ieral model lfe surace : applcao of leas squares moe carlo rs assessme  Oberla euam Teugua (HSB)  Jae Re (Uversé yo, HSB)  rédérc Plache (Uversé yo, aboraore SA) 04. aboraore SA 50 Aveue Toy Garer 
More informationSpline. Computer Graphics. Bsplines. BSplines (for basis splines) Generating a curve. Basis Functions. Lecture 14 Curves and Surfaces II
Lecure 4 Curves and Surfaces II Splne A long flexble srps of meal used by drafspersons o lay ou he surfaces of arplanes, cars and shps Ducks weghs aached o he splnes were used o pull he splne n dfferen
More informationDuration Outline and Reading
Deb Isrumes ad Markes Professor Carpeer Duraio Oulie ad Readig Oulie Ieres Rae Sesiiviy Dollar Duraio Duraio Buzzwords Parallel shif Basis pois Modified duraio Macaulay duraio Readig Tuckma, Chapers 5
More informationThe Economics of Administering Import Quotas with LicensesonDemand
The Ecoomcs of Admserg Impor uoas wh LcesesoDemad Jaa Hraaova, James Falk ad Harry de Gorer Prepared for he World Bak s Agrculural Trade Group Jauary 2003 Absrac Ths paper exames he effecs of raog mpor
More informationMethodology of the CBOE S&P 500 PutWrite Index (PUT SM ) (with supplemental information regarding the CBOE S&P 500 PutWrite TW Index (PWT SM ))
ehodology of he CBOE S&P 500 PuWre Index (PUT S ) (wh supplemenal nformaon regardng he CBOE S&P 500 PuWre TW Index (PWT S )) The CBOE S&P 500 PuWre Index (cker symbol PUT ) racks he value of a passve
More informationCritical Approach of the Valuation Methods of a Life Insurance Company under the Traditional European Statutory View
Crcal Aroach of he Valuao Mehods of a Lfe Isurace Comay uder he radoal Euroea Sauory Vew Dr. PaulAoe Darbellay ParerRe Belleresrasse 36 C8034 Zürch Swzerlad Phoe: 4 385 34 63 Fa: 4 385 37 04 Emal: aulaoe.darbellay@arerre.com
More informationUNDERWRITING AND EXTRA RISKS IN LIFE INSURANCE Katarína Sakálová
The process of uderwriig UNDERWRITING AND EXTRA RISKS IN LIFE INSURANCE Kaaría Sakálová Uderwriig is he process by which a life isurace compay decides which people o accep for isurace ad o wha erms Life
More informationThe Term Structure of Interest Rates
The Term Srucure of Ieres Raes Wha is i? The relaioship amog ieres raes over differe imehorizos, as viewed from oday, = 0. A cocep closely relaed o his: The Yield Curve Plos he effecive aual yield agais
More informationClaims Reserving When There Are Negative Values in the Runoff Triangle
Clams Reservg Whe There Are Negave Values he Ruo Tragle Erque de Alba ITAM Meco ad Uversy o Waerloo Caada 7 h. Acuaral Research Coerece The Uversy o Waerloo Augus 70 00 . INTRODUCTION The may uceraes
More informationThe Increasing Participation of China in the World Soybean Market and Its Impact on Price Linkages in Futures Markets
The Icreasg arcpao of Cha he Word Soybea Marke ad Is Ipac o rce Lkages Fuures Markes by Mara Ace Móz Chrsofoe Rodofo Margao da Sva ad Fabo Maos Suggesed cao fora: Chrsofoe M. A. R. Sva ad F. Maos. 202.
More informationSolving Fuzzy Linear Programming Problems with Piecewise Linear Membership Function
Avalable a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: 9966 Vol., Issue December ), pp. Prevously, Vol., Issue, pp. 6 6) Applcaos ad Appled Mahemacs: A Ieraoal Joural AAM) Solvg Fuzzy Lear Programmg Problems
More informationFINANCIAL MATHEMATICS 12 MARCH 2014
FINNCIL MTHEMTICS 12 MRCH 2014 I ths lesso we: Lesso Descrpto Make use of logarthms to calculate the value of, the tme perod, the equato P1 or P1. Solve problems volvg preset value ad future value autes.
More informations :risk parameter for company size
UNDESTANDING ONLINE TADES: TADING AND EFOMANCE IN COMMON STOCK INVESTMENT Y. C. George L, Y. C. Elea Kag 2 ad ChugL Chu 3 Deparme of Accoug ad Iformao Techology, Naoal Chug Cheg Uversy, Tawa,.O.C acycl@ccu.edu.w;
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationClassic Problems at a Glance using the TVM Solver
C H A P T E R 2 Classc Problems at a Glace usg the TVM Solver The table below llustrates the most commo types of classc face problems. The formulas are gve for each calculato. A bref troducto to usg the
More informationFORECASTING MODEL FOR AUTOMOBILE SALES IN THAILAND
FORECASTING MODEL FOR AUTOMOBILE SALES IN THAILAND by Wachareepor Chaimogkol Naioal Isiue of Developme Admiisraio, Bagkok, Thailad Email: wachare@as.ida.ac.h ad Chuaip Tasahi Kig Mogku's Isiue of Techology
More informationThe Analysis of Development of Insurance Contract Premiums of General Liability Insurance in the Business Insurance Risk
The Aalyss of Developmet of Isurace Cotract Premums of Geeral Lablty Isurace the Busess Isurace Rsk the Frame of the Czech Isurace Market 1998 011 Scetfc Coferece Jue, 10.  14. 013 Pavla Kubová Departmet
More informationAPPLICATIONS OF GEOMETRIC
APPLICATIONS OF GEOMETRIC SEQUENCES AND SERIES TO FINANCIAL MATHS The mos powerful force i he world is compoud ieres (Alber Eisei) Page of 52 Fiacial Mahs Coes Loas ad ivesmes  erms ad examples... 3 Derivaio
More informationCurve Fitting and Solution of Equation
UNIT V Curve Fttg ad Soluto of Equato 5. CURVE FITTING I ma braches of appled mathematcs ad egeerg sceces we come across epermets ad problems, whch volve two varables. For eample, t s kow that the speed
More informationThe Time Value of Money
The Tme Value of Moey 1 Iversemet Optos Year: 1624 Property Traded: Mahatta Islad Prce : $24.00, FV of $24 @ 6%: FV = $24 (1+0.06) 388 = $158.08 bllo Opto 1 0 1 2 3 4 5 t ($519.37) 0 0 0 0 $1,000 Opto
More informationIDENTIFICATION OF THE DYNAMICS OF THE GOOGLE S RANKING ALGORITHM. A. Khaki Sedigh, Mehdi Roudaki
IDENIFICAION OF HE DYNAMICS OF HE GOOGLE S RANKING ALGORIHM A. Khak Sedgh, Mehd Roudak Cotrol Dvso, Departmet of Electrcal Egeerg, K.N.oos Uversty of echology P. O. Box: 163151355, ehra, Ira sedgh@eetd.ktu.ac.r,
More informationNumerical Methods with MS Excel
TMME, vol4, o.1, p.84 Numercal Methods wth MS Excel M. ElGebely & B. Yushau 1 Departmet of Mathematcal Sceces Kg Fahd Uversty of Petroleum & Merals. Dhahra, Saud Araba. Abstract: I ths ote we show how
More informationGenerating Intelligent Teaching Learning Systems using Concept Maps and Case Based Reasoning
17 Geerag Iellge Teachg Learg Sysems usg Cocep Maps ad Case Based Reasog Makel L. Esposa, MSc. Naala Maríez S. y Zeada García V. Deparme of Compuer Scece, Ceral Uversy of Las Vllas, Hghway o Camajuaí,
More informationAn Effectiveness of Integrated Portfolio in Bancassurance
A Effectveess of Itegrated Portfolo Bacassurace Taea Karya Research Ceter for Facal Egeerg Isttute of Ecoomc Research Kyoto versty Sayouu Kyoto 606850 Japa arya@eryotouacp Itroducto As s well ow the
More informationAnalyzing Energy Use with Decomposition Methods
nalyzng nergy Use wh Decomposon Mehods eve HNN nergy Technology Polcy Dvson eve.henen@ea.org nergy Tranng Week Pars 1 h prl 213 OCD/ 213 Dscusson nergy consumpon and energy effcency? How can energy consumpon
More informationRUSSIAN ROULETTE AND PARTICLE SPLITTING
RUSSAN ROULETTE AND PARTCLE SPLTTNG M. Ragheb 3/7/203 NTRODUCTON To stuatos are ecoutered partcle trasport smulatos:. a multplyg medum, a partcle such as a eutro a cosmc ray partcle or a photo may geerate
More information1 AN INTRODUCTION TO CONSUMER PRICE INDEX METHODOLOGY
AN INTRODUCTION TO CONSUMER PRICE INDEX METHODOLOGY. A prce dex s a measure of the proportoate, or percetage, chages a set of prces over tme. A cosumer prce dex (CPI) measures chages the prces of goods
More informationPerformance Attribution. Methodology Overview
erformace Attrbuto Methodology Overvew Faba SUAREZ March 2004 erformace Attrbuto Methodology 1.1 Itroducto erformace Attrbuto s a set of techques that performace aalysts use to expla why a portfolo's performace
More informationEXAMPLE 1... 1 EXAMPLE 2... 14 EXAMPLE 3... 18 EXAMPLE 4 UNIVERSAL TRADITIONAL APPROACH... 24 EXAMPLE 5 FLEXIBLE PRODUCT... 26
EXAMLE... A. Edowme... B. ure edowme d Term surce... 4 C. Reseres... 8. Bruo premum d reseres... EXAMLE 2... 4 A. Whoe fe... 4 B. Reseres of Whoe fe... 6 C. Bruo Whoe fe... 7 EXAMLE 3... 8 A.ure edowme...
More informationDiversification in Banking Is Noninterest Income the Answer?
Dversfcaon n Bankng Is Nonneres Income he Answer? Kevn J. Sroh Frs Draf: March 5, 2002 Ths Draf: Sepember 23, 2002 Absrac The U.S. bankng ndusry s seadly ncreasng s relance on nonradonal busness acves
More information