Estimating Non-Maturity Deposits

Transcription

1 Proceedigs of he 9h WSEAS Ieraioal Coferece o SIMULATION, MODELLING AND OPTIMIZATION Esimaig No-Mauriy Deposis ELENA CORINA CIPU Uiversiy Poliehica Buchares Faculy of Applied Scieces Deparme of Mahemaics, Spl. Idepedeei 33, Buchares ROMANIA [email protected] SORIN UDRISTE Emporiki Bak, Treasury Divisio 40-40bis, Vasile Lascăr Sreee, d disric, Buchares ROMANIA [email protected] Absrac: - We use cerai saisical models for sudyig he demad deposis. For each model we precise he heoreical esimaio ad we propose a origial algorihm for simulaios. Usig a ime series daa of marke rae, ieral rae ad o-mauriy deposis, we ru hese models ad we esimae he opimal parameers. Based o resuls ad daa, we propose a mixed model ad origial relaios for esimaig he parameers. Afer a decisio upo a beer algorihm ad model, we calculae he real values of demad deposis. Some coclusios abou he characerisic feaure of o-mauriy deposis were formulaes a he ed of he paper. Key-Words: No-mauriy deposis, ieres rae, adjusme models, correlaio, sesiiviy, risk. No-mauriy deposis No-mauriy deposis (NMD) such as sigh, curre accous, savigs, demad deposis accous ec are a major source of fuds for all radiioal Baks. The characerisic feaure for hese kid of deposis is ha hey have o saed coracual mauriy ad he balace of hese fuds ca icrease or decrease hroughou he day wihou ay warig (alhough i pracice he balace is quie sable) as he deposiors have always he possibiliy o add or wihdraw fuds a ay ime (he embedded opios ha clies may exercise) a o pealy. The behavior of o-mauriy deposis reflecs raioal decisio makig o he par of cusomers o wo facors: received value ad perceived value. Higher ieres rae paid relaive o compeior raes ad more cosequeial barriers o exi creae loger erm ideermiae mauriy deposis. Oe ca ofe observe ha he volume of a NMD posiio flucuaes as clies reac o chages i he cusomer rae ad he relaive araciveess of aleraive ivesme opporuiies (see Fig. ): -whe ieres raes rise, he oal balace of NMD eds o fall as cusomers become more careful i sweepig heir fuds io log-erm ivesmes o lock up he high level of yields (wihdrawal). - whe ieres raes are low, NMD becomes more profiable compared o aleraive shor-erm ivesmes. Fig. Volume of NMD i case of correlaio bewee clie rae ad volume balace The perceived value is he aswer o he quesio: why do balaces remai o deposi for log periods of ime eve hough he fiacial advaage is egaive? There are cases whe clies do o reac o he chages i he cusomer rae (see Fig. ). No-mauriy deposi fudig coss geerally demosrae less volailiy ha marke ieres raes. As a resul, high o-mauriy deposi volumes may acually reduce reprisig risk ad moderae overall ieres rae risk (see [7]). The cash ISSN: ISBN:

2 Proceedigs of he 9h WSEAS Ieraioal Coferece o SIMULATION, MODELLING AND OPTIMIZATION flow modelig of o-maurig deposis requires dividig deposis io sable ad usable balaces. Fig.4 Pareo disribuio for he NMD. Esimaio of NMD balaces Fig. Volume of a NMD posiio, perceived value The sable fracio of he curre balace is called core level (see Fig.3) ad is modeled as a permae balace (log-erm ouflow) while he volaile fracio is viewed as overigh moey ad serves as a buffer for volume flucuaios (see [6]). As is observed i oher cases, he resuls obaied for he value of core deposis varies subsaially by isiuio, depedig o he isiuio s supply of deposis ad abiliy o reai deposis (see [9]). This is impora Fig. 3 Esimaed core level for fiacial decisios i order o make a disribuio over several years of he oal value ha ca be used for ivesig proposal (Fig. 4). We obai ha 44% could firs be ivesed. The reches ha will follow for ivesme depeds o firs ivesme (see [8]). We shall discuss differe saisical models i order o esimae NMD balaces. Le D be he value of a o-mauriy deposi a ime. To refer o he liear regressio ad log-ormal diffusio models, we iroduce he discree variables y = log( D ) ad y = log( D / D ). The Liear regressio model is characerized by: y = a + b( ) = α + β, ( y y)( ) =, a = ( i ), b =. () i= ( ) =, The Log-ormal diffusio model, used for fuure balace (of liquidiy erm srucure), is based o he followig igredies: y = α + β + σε, () wih parameers α, β beig deermied from a liear regressio o y, (ε ) Gaussia (whie) oise erm, σ = Var( ε ) he volailiy of y ime series, ε ~ N(0, σ ). If we cosider a q-cofidece ierval, he ε = Φ ( q). Therefore y = log( V ( )) = α + β + σ Φ ( q), (3) where q is p h quaile from N (0,), give by relaio q = z α /, such ha Φ ( q ) = α /, α = p. The cofidece ierval, for he umber of selecio from a populaio N( µ, σ) is deermied by: σ σ ( y z α /, y + z α / ). (4) ISSN: ISBN:

3 Proceedigs of he 9h WSEAS Ieraioal Coferece o SIMULATION, MODELLING AND OPTIMIZATION For p = ad p = 0. 99, we have q = ad respecively q = z α / = For our case (see Fig. 5), he cofidece bad (ierval) is saed bewee he wo sar-lies. I oher words, wih he probabiliy p = (almos sure), he variable y = log( D ) have values bewee hese sar-lies, i he fuure oo. Also, he regressio lie (marked by red color) has he equaio y = This lie divide he cofidece bad i wo equal pars. Fig.6 Regressio lie ad MA() model for. AR() model for rae of NMD balaces y. We esimae he parameers for y = β 0 + β y + ε, (6) usig Yule Walker esimaor (see Fig. 6). Fig. 5 Cofidece ierval for y for p = We shall use followig models: -movig average. MA(), auoregressive model AR()-see [], models based o exeral facors ( Jarrow ad va Deveer model; Jaosi, Jarrow Zullo (see [8]), Orsei-Uhlebeck process) see [].. MA() model for rae of NMD balaces This model is base o he formula: y = µ + ε + θε. (5) i permis o simulae he rae of fuure omauriy deposi values, for few mohs. These values are obaied from y evaluaed i Fig.6, showig he decayig rae level of ivesme: y = log( D / D ) = 0.. Fig. 7 Regressio lie ad AR() model for y.. Correlaio: ieres rae/marke rae The rae of o-mauriy deposi values is described also by he Jarrow ad va Deveer model, Jaosi, Jarrow Zullo, ad Orsei- Uhlebeck process. These models are coeced o he ieres rae ad/or marke rae (see Fig 8.). The correlaio coefficie bewee he wo raes is very small ρ = , ad cosequely we ca cosider ha he wo variables are idepede. ISSN: ISBN:

4 Proceedigs of he 9h WSEAS Ieraioal Coferece o SIMULATION, MODELLING AND OPTIMIZATION Fig. 8 Evoluio of ieres rae ad marke rae.3. Jarrow ad va Deveer model The model is give by: y = α 0 + α + α r + α 3( r r ). (6) Usig he leas square mehod, we esimae he values of coefficies, compoes of he array r α = ( α 0, α, α, α3 ), via a soluio of symmeric algebraic sysem M r α = P, M = M i /( ), (7) P = (( ) y, yii, yiri, yi ( ri ri )) For r our daa esimaio, we fid (see Fig.9) α = ( , , , ) Fig. 0 Esimaio of NMD sock rae, by usig i. 3. Hybrid model for NMD balaces Afer a qualiaive sudy of all he models, as well as he daa used, we cosider ha a mixed model will esimae bes he evoluio of omauriy deposis. Thus, we propose a origial model: movig average model for ieres rae ad, respecively for marke rae; a weigh model for y wih coribuios from a model i which ieres rae ad marke rae are idepede variables, ad a auoregressive model, y = ρ ( y ) JDir + ( ρ)( y ) AR. (8) For simpliciy we shall use ρ = The values of ( y ) JDir are well deermied by y = α 0 + α + α r + α 3i. (9) This ime, he sysem (7) becomes: M r hα = P h, M h = M i h /( ), (0) Ph = (( ) y, yii, yiri, yii ), i Fig. 9 Esimaio of NMD sock rae by Jarrow ad va Deveer model m M i h = m3 m 4 ( ) ( i ) m3 m4 ( ) r ri i ( ri ) m34 ( ) i i i i i. ri i ( i ) i We also esimae NMD sock rae, usig exeral variable i. Oe foud: The resuls are represeed i Figs.-, ISSN: ISBN:

5 Proceedigs of he 9h WSEAS Ieraioal Coferece o SIMULATION, MODELLING AND OPTIMIZATION Fig. Esimaio of NMD sock rae for hybrid model. where we r have used α = (0.55,-0.047, -4.93, ). For all models described we ca recalculae he values for NMD balaces. We cosider ha our model esimae beer he NMD sock rae (see Fig. 0). For his model we calculae he values of NMD balaces. These values are give i Fig., showig he 6 maximum level of ivesme D = 0 Eur. Fig. Esimaio of NMD sock usig hybrid model. 4 Coclusios No-mauriy deposis have value because hey are usually oe of he lowes cos sources of fuds available o he bak. Modelig o-mauriy deposis i balace of IRR is impora due o heir role i deermiig earig a risk (EaR) ad due o heir role i esablishig he ecoomic value of equiy (EVE). Usig our model (8) could be made some simulaios o: how could be iflueced he NMD volume by a fas chage i values of he ieres rae (IRR) or marke rae. Simulaig asses ruoffs, icludig he impac of prepaymes uder differe rae scearios, ad applyig he profile obaied o o-mauriy deposis allows for he quaificaio of explici IRR. The algorihms proposed for our model could be implemeed i ay kid of sofware, as for example, we have used Excel ad Malab. Also, for beer accuracy we have used Moe-Carlo mehods i simulaios (see []). Refereces: [] E. C. Cipu, L. Pâzar: Sochasig modellig ad progossis of a uderlyg asse pricig, Joural of Ecoomic forecasig, Buchares, 005, pp [] T. Eroe, No-Mauriy Deposi valuaio ad hedgig, Maser hesis, Helsiki Uiversiy of echology, 008. [3] C. Udrise, S. Udrise, Queue geomeric dyamics, Proceedigs of 005 Aual Hawaii Ieraioal Coferece o Saisics, Mahemaics ad Relaed Fields, Jauary 9-, 005, Hoolulu, Hawaii (elecroic versio). [4] M. Craiu, L. Pâzar, C. Cipu: Hazard based models ad covariaes, 3 rd Ieraioal Colloquium Mahemaics i Egieerig ad Numerical Physics (MENP-3), 7-9 oc. Buchares, 004, BSG Proceedigs, pp [5] L. Pâzar, E. C. Cipu: Usig of sochasic Io ad Sraoovich iegrals i derived securiies pricig, 3 rd Ieraioal Colloquium Mahemaics i Egieerig ad Numerical Physics (MENP-3), 7-9 oc. Buchares, 004, BSG Proceedigs, -9. [6] ***: Modelig Core Deposis, Capial marke News, Federal Reserve Bak of Chicago, Jue 00. [7] R.Youg, C. Yom: O he idepedece of asses ad liabiliies: Evidece from U.S. commercial baks, , Joural of Fiacial Sabiliy, Volume 4, Issue 3, Sepember 008, pp [8] J. Tibor, R. Jarrow, F. Zullo: A Empirical Aalysis of he Jarrow - va Deveer Model for ISSN: ISBN:

6 Proceedigs of he 9h WSEAS Ieraioal Coferece o SIMULATION, MODELLING AND OPTIMIZATION Valuig No-Mauriy Demad Deposis, Joural of Derivaives 7 (Fall 999), pp [9] R. G. Sheeha, Valuig Core Deposis, Deparme of Fiace, Uiversiy of Nore Dame, Nore Dame, April 004, IN [0] J. Frye, Moe Carlo by Day: Iraday Valuea-Risk Usig Moe Carlo Simulaio, Risk Magazie, November 998, Federal Reserve Bak of Chicago. [] J. Frye, Moe Carlo by Day: Iraday Valuea-Risk Usig Moe Carlo Simulaio, Risk Magazie, November 998, Federal Reserve Bak of Chicago. [] Z. Dori, P. Năsase, F. Albescu, I. Boja, F. Mihai, L. Covrig. Exper Sysems Applicaios, Techical Ediorial House, 998, Buchares (i Romaia). [3] M. Radulescu, S. Radulescu, C. Z. Radulescu Mahemaical Models for Opimizaio of Fiacial Ivesmes, Ediorial House of Romaia Academy, 006, Buchares (i Romaia). [4] M. Alar: Porofolio Theory, Buchares, hp:// 00, (i Romaia). ISSN: ISBN: