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10.5. Bagatoetapna dinamіka fіnansovih resursіv on pіdstavі multiplіkativnoї stohastichnoї modelі

6 *

* 6: {Konyuhovsky PV microeconomic modeling of banking. - SPb .: Peter, 2001.}

Rozglyanmo vіdnosno Prostu situatsіyu. Vvazhatimemo scho obsyagi zaluchenih zasobіv (koshtіv) for perіodami Je deyakim zovnіshnіm chinnikom, dinamіku MDM can be opisati for Relief vvedenoї in Sec. 10.2. multiplіkativnoї stohastichnoї modelі.

Obsyag zaluchenih koshtіv perіod at t + 1 can be yak taxes

(10.64)

de koefіtsієnti prirostіv - Vipadkovі quantities rozpodіlenі zgіdno s lognormal іz parameters that m i s i. Pripuskaєtsya scho m i od stale v ( ). Otzhe, rіvnyannya (10.62) matim viglyad:

. (10.65)

Rozv'yazok danogo rіvnyannya mozhna vіdshukati, vikoristovuyuchi Duhamel method for Z -peretvorennya. For tsogo rozglyanemo dopomіzhne rіvnyannya:

gt +1 - r gt = d t , g 0 = 0, (10.66)

de

(10.67)

Nekhay gt ® G (z) i d t ® 1 - Z -peretvorennya funktsіy gt i d t. Todі gt +1 ® ZG (Z), and The image rіvnyannya for (10.66) matim viglyad:

(10.68)

Otzhe,

(10.69)

Origіnalom for

poslugovuє poslіdovnіst r t. Maєmo s vіdomih vlastivostey Z -peretvorennya:

(10.70)

Dwellers lead rіvnyannya (10.65) to nulovoї pochatkovoї minds vvedemo nova zmіnnu ht = qt - q 0. Yakscho pripustiti scho

(10.71)

for the rіvnyannya Dial viglyadu:

(10.72)

Pіslya low peretvoren znaydemo scho

. (10.73)

So rank, maєmo rozv'yazok rіznitsevogo rіvnyannya (10.65):

(10.74)

Yakscho koefіtsієnti elementarnogo transition toil odnakovі rozpodіli for vsіh momentіv , Rozv'yazok (10.74) Naboodah viglyadu:

(10.75)

Vikoristovuyuchi result, otrimanі for Relief multiplіkativnih stochastics models forecast obsyagіv zaluchenih koshtіv at time t + 1 matim viglyad:

(10.76)

de x 0 - obsyag zaluchenih koshtіv Pochatkova at the time (t = 0); - Otsіnki values ​​parametrіv m, s vіdpovіdno.

Zgіdno s gіpotezoyu schodo vzaєmnoї Nezalezhnosti koefіtsієntіv transition a t, zamіnivshi їh in formulі (10.75) vіdpovіdnimi otsіnkami, otrimaєmo viraz predictive values obsyagu Vlasnyi koshtіv at the time t:

(10.77)

de mozhna otrimati s (10.76).

In okremomu vipadku, Yakscho , Pіslya rozkrittya neviznachenostі viraz (10.77) matim bіlsh compact form:

. (10.78)

Formula (10.77) that (10.78) toil chіtku ekonomіchnu іnterpretatsіyu - obsyag Vlasnyi koshtіv fіnansovoї fіrmi hour at the time of t deposits od dvoh warehouses (in furrows sformovanoї modelі):

• q 0 r t - magnitude Pochatkova kapіtalu s urahuvannyam polіtiki nakopichennya;
• - Rezultatіv dіyalnostі schodo zaluchennya koshtіv that otrimannya dohodіv od active that ratsіonalnogo vikoristannya їh.

Mozhna zdіysniti grafіchnu іlyustratsіyu "povedіnki" poslіdovnostі stosovno rіznih norm value vitrat on zaluchennya koshtіv (v).

Rozglyanmo butt for demonstratsії aforesaid metodіv on conceptual rіvnі.

For randomness base vіzmіmo schorіchnu fіnansovu zvіtnіst Bank of New York (USA), yak mozhna otrimati in the Web-serverі FDIC. Thus, in the Table. 10.1 mіstyatsya danі schodo dinamіki Vlasnyi kapіtalu, zobov'yazan that obsyagіv vіdsotkovih dohodіv tsogo bank (in millions of US dolarіv) for 1992-1997 perіod roki.Tablitsya 10.1

DINAMІKA Vlasnyi KAPІTALU, ZOBOV'YAZAN, VІDSOTKOVIH DOHODІV the I VITRAT Bank of New York FOR 1992-1997 pp., Mln. *

The html *: www.fdic.gov

On pіdstavі danih Table. 10.1 znaydemo otsіnki value of the normal income od zastosuvannya koshtіv yak userednene vіdnoshennya vіdsotkovogo income Ut to vsogo kapіtalu xt + qt i normalized vitrat on їh zaluchennya Scho dorіvnyuyut userednenomu vіdnoshennyu vіdsotkovih vitrat Vt to obsyagu zobov'yazan poperednogo perіodu xt -1.

Urahovuyuchi scho s rozglyadu viluchenі takі chinniki, yak i vitrati nevіdsotkovі revenues, vitrati on SPLAT podatkіv toscho for zіstavlennya obsyagіv net income vіdsotkovogo Ut - Vt h increments Vlasnyi kapіtalu D qt want to enter normuyuchy koefіtsієnt Pіslya breeding on Demba Pervin otsіnok rate of return that vitrat otrimaєmo otsіnki . Vikoristovuyuchi for znahodzhennya otsіnki quantities koefіtsієnta elementarnogo transition obsyagіv zobov'yazan formula (10.76) i pіdstavlyayuchi zamіst that u v otsіnki їh that in formula (10.77) can be otrimati prognoznі values ​​obsyagіv Vlasnyi kapіtalu by Rokach rozglyaduvanogo perіodu. Result obchislen filed in viglyadі tablitsі (tab. 10.2).

table 10.2

FAKTICHNІ TA PROGNOZNІ values OBSYAGU Vlasnyi KAPІTALU Bank of New York, one million dollars. *

* Rozrahunok zdіysneno for danimi html: www.fdic.govPripustimo vіdpovіdno Q = 0,05, r »1,001.

Vіdnosno Visokiy rіven vіdhilennya mіzh rozrahunkovimi i factuality obsyagami is adequate rіvnyu pohibki, zakladenoї in vikoristovuvanih danih.

Rozvivati ​​vikladeny vische pіdhіd possible, napriklad such rank. For vivchennya vplivu rіshen scho їh priymaє fіnansova FIRMA vіdnosno polіtiki zaluchennya resursіv on dinamіku її rozvitku (zokrema on dinamіku Vlasnyi kapіtalu) mozhut dosit efektivnosti vikoristovuvatis rekurentnі dinamіchnі modelі, kotrі ґruntuyutsya on ically mathematical aparatі lіnіynih rіznitsevih rіvnyan.

For pobudovi prognozіv ochіkuvanih values ​​obsyagіv fіnansovih resursіv depozitnoї Nature, akumulovanih for rakhunok koshtіv znachnoї kіlkostі vkladnikіv (homogeneously rahunkіv), shaping can Buti vikoristanі stohastichnі modelі bankіvskih depozitіv have yakih a basis we can take gіpotezu schodo mozhlivostі inventory protsesіv, SSMSC zumovlyuyut zmіnu kіlkostі rahunkіv i kіlkostі operatsіy s them for Relief vipadkovih values ​​rozpodіlenih for Poisson law and koefіtsієntіv vіdnosnoї zmіni rahunkіv in rozrіzі okremih operatsіy - Relief for vipadkovih values ​​scho toil lognormal rozpodіlu.