DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Krzysztof Jajuga Wrocław University of Economics

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1 DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 2006 Krzyszof Jajuga Wrocław Universiy of Economics Ineres Rae Modeling and Tools of Financial Economerics 1. Financial Economerics Models One of he mos imporan areas of finance, where a considerable developmen has been recenly observed, is financial economerics. In our opinion, he relaively simple way of describing financial economerics can be given as follows: Financial economerics models are derived for daa given as financial ime series and applied eiher in order o verify some underlying hypoheses formulaed by financial heory or o idenify some properies exising in financial daa. I is worh o menion ha here are wo general ypes of financial economerics models: models of dynamic economerics, adaped and applied for financial daa; models developed solely for financial daa. I is commonly believed ha he birh of financial economerics is conneced wih he derivaion of ARCH model (Engle (1982)). Since ha ime he enormous amoun of models was developed and used in pracice. The mos commonly used groups of financial economerics models can be divided ino: univariae models (ARIMA GARCH) and mulivariae models (VARIMA MGARCH); srucural models (VAR) and reduced ype models (ARIMA); price models (ARIMA) and volailiy models (GARCH); Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

2 8 Krzyszof Jajuga linear and non-linear models; deerminisic volailiy (GARCH) and sochasic volailiy (SV) models. Key facors being driving forces of financial economerics are: echnological progress, paricularly in he area of compuer echnology, which makes pracical implemenaion of raher sophisicaed economeric ools relaively easy; daa availabiliy, paricularly in he form of ime series of sock prices, ineres raes and exchange raes, which makes easy o verify he proposed models; developmen of economeric ools (for example esimaion and esing mehods), which allows for fine uning he mehodology; developmen of financial heory, which delivers ools (including possible hypoheses) o be verified in pracice; pracical problems o be solved by new models. One of he main endencies observed in he applied research is he growing inegraion of financial economics, financial mahemaics and financial economerics hrough empirical financial problems. One can say ha financial economerics verifies hypoheses of financial economics and uses some ools developed by financial mahemaics. Alhough proved considerable progress, financial economerics sill faces a lo of challenges. Among he mos imporan ones are: modeling mulivariae financial ime series, where he crucial poin is he esimaion of volailiy and dependence (correlaion), boh noions paricularly imporan in risk managemen; ulra high frequency daa modeling; where daa is given for each paricular ransacion raher han for each ime momen; modeling ime series under assumpions of random error having disribuion oher han normal disribuion (univariae or mulivariae); implemenaion of models derived by financial heory based on coninuous ime sochasic processes wih he use of daa given in discree ime unis. One of he mos challenging applied asks faced by financial economeric mehods is modeling ineres raes. This paper gives some inroducory and synheic remarks as far as he modeling ineres raes is concerned. 2. Ineres Rae Modeling Inroducory Remarks Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House Ineres raes are besides exchange raes, sock prices and commodiy prices main ime series modeled by financial economerics. However, from he poin of

3 Ineres Rae Modeling and Tools of Financial Economerics 9 view of modeling, here are several differences beween ineres raes and he oher hree ypes of financial ime series, namely: 1. There are many possible ypes of ineres raes ha can be of modeling ineres, including spo raes, forward raes, yields o mauriy. Usually, hey are aken as risk-free raes. In addiion, one can consider risky raes, hey refer o deb insrumens (like bonds) issued by corporaions of differen raing caegories (AA, BBB, ec.). 2. Insead of single ineres rae for each ime momen, one has o consider he whole erm srucure of ineres raes. Paricularly for each ime momen we have heoreically infinie number of ineres raes, each one referring o possible period, for example one day, one monh, hree monhs, en years, ec. Term srucure of ineres raes, herefore, is a funcion ha for each considered period gives he value of respecive ineres rae. Due o he graphical presenaion of his relaion, he oher used name used for he erm srucure of ineres raes is yield curve. In pracice he number of considered ineres raes for each ime momen is of course finie, however his number is large more han dozen. In addiion, hese differen raes are srongly relaed o each oher, which means ha ineres rae modeling is acually mulivariae ask. 3. There is developed financial heory of ineres raes, which means ha economeric models of ineres raes should be consisen wih his heory. 4. Some heoreical raes are no observable, which means ha hey should be exraced from exising daa. I is worh o menion ha models of ineres raes have very wide applicaion in financial pracice. Among he mos imporan applicaions are: 1. Moneary policy. 2. Ineres rae derivaive insrumens pricing. 1. Ineres rae risk managemen. 2. Invesmen sraegies wih he use of deb insrumens. 3. Corporae finance. 4. Credi derivaive insrumen pricing. 5. Credi risk managemen. Financial economerics models developed for socks prices and exchange raes are prices models (for example ARIMA ype models) and volailiy models (for example GARCH ype models). Similar feaure is observed for ineres raes. However, here in addiion one can consider risk-free and risky raes. In our opinion, one can disinguish four possible ypes of ineres modeling. They are ordered here wih respec o availabiliy of models and applied research, namely: 1. Modeling erm srucure of ineres raes hese are price models developed for risk-free raes. Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

4 10 Krzyszof Jajuga 2. Modeling erm srucure of ineres raes volailiy hese are volailiy models developed for risk-free raes. 3. Modeling erm srucure of credi spreads hese are price models developed for credi spreads (credi spread is he difference beween risky rae and risk-free rae). 4. Modeling erm srucure of credi spread volailiy hese are volailiy models developed for credi spreads (by no doub, hese models a he very early sage of developmen). 3. Ineres Rae Models Any ype of economeric model should be well-suied o he facs observed in real world. The same saemen refers o ineres rae models. The basic facs observed in he financial markes are: ineres raes are mean-revering, which means ha here is a level in ineres raes o which hese raes approach in long erm; he changes of ineres raes corresponding o differen periods (for example one monh rae and one year rae) are no perfecly correlaed; volailiy of shor erm raes is higher han volailiy of long erm raes; mos changes of ineres raes can be explained by hree facors: a) parallel movemen: all raes changes by he same amoun up or down; b) slope changes: shor erm raes change by more up or down or less han medium erm raes and he same is rue for medium erm raes comparing o long erm raes; his means change of he slope of yield curve; c) curvaure changes medium erm raes change more up or down or less han shor erm and long erm raes. There are many ineres models proposed in he lieraure and used in empirical research. They can be classified ino wo main groups: yield curve approximaion mehods; ineres rae dynamics models. Yield Curve Approximaion Mehods The main feaure of hese mehods is o use available daa on ineres raes o fi funcion o hese daa. The analysis of he available mehods leads us o he following sysemaizaion: direc mehod, called boosrapping mehod, where unobserved ineres raes are derived from observed raes; spline mehod, where yield curve is composed of paricular segmens; Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

5 Ineres Rae Modeling and Tools of Financial Economerics 11 yield curve economeric models here he model is proposed, in which paricular parameers can be given some inerpreaion relaed o he characerisics of yield curve. Among he models proposed wihin he hird group, we should menion Svensson model (Svensson (1994)). This model was used by some cenral banks o model erm srucure of ineres raes. The exended version of Svensson model is given as: 1 exp( m / δ1) 1 exp( m / δ1) r m = β 0 + β1 exp 2 exp exp( m / 1 m / + β δ 1 m / δ δ1 1 exp( m / δ 2 ) + β 3 exp exp( m / δ 2 ) + u. m / δ 2 Here: r ineres rae corresponding o m years; m β 0 parameer corresponding o long erm ineres rae; β 1 parameer corresponding o he parallel movemen of ineres raes; β 2, β 3 parameers corresponding o possible humps in he yield curve. The deailed descripion of yield curve approximaion mehods is given by Marinelli, Priaule and Priaule (2003). Ineres Rae Dynamics Models Conrary o yield curve approximaion mehods, hese models ry o explain he behavior of ineres raes, raher han jus o find he approximaion for he daa. By no doub, here is variey of hese models and he developmen of hese models is sill in progress. We give here wo imporan sysemaizaions of ineres rae dynamics models. 1. Classificaion wih respec o he formal srucure of he model: A. Classical financial economerics models These are financial economerics models applied o ineres raes. Since ineres raes corresponding o differen periods are relaed, i is mulivariae modeling. B. Binomial ree models Here he dynamics of ineres raes is described by discree ime sochasic process, where in each period ineres rae can move in wo direcions. C. Sochasic differenial equaions models Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

6 12 Krzyszof Jajuga Here he dynamics of ineres raes is described by coninuous ime sochasic process, which can be represened as sochasic differenial equaion. One of he mos ofen used srucures for ineres raes is he Ornsein-Uhlenbeck process, given as: dr = κ ( θ r ) d + σdz, where: κ parameer, inerpreed as he speed of mean reversion, θ parameer, inerpreed as he long erm ineres rae, σ volailiy parameer. 2. Classificaion wih respec o he derivaion of ineres raes: A. Endogenously derived models (facor models) These are models where he ineres rae dynamics is explained by few (one o hree, as a rule) underlying facors, being driving forces of he dynamics. Many of hese models can be described in he framework of sochasic differenial equaions. B. Arbirage models These are models developed by financial heory, based on he same idea as classical opion pricing models in Black-Scholes-Meron framework. This is nonarbirage idea, in which price of financial insrumen (here deb insrumen based on ineres raes) is deermined in such a way ha arbirage sraegy is no possible arbirage sraegy is he sraegy of no iniial invesmen, no risk and posiive inflow. Among he mos well known and advocaed arbirage models of ineres rae dynamics is he model proposed by Heah, Jarrow and Moron (1992). Ineres rae dynamics models are he mos advanced models of ineres raes. One of he models, which on one hand is raher sophisicaed, on he oher hand i leads o nice inerpreaion, is he model proposed by Chen (1996). This is hreefacor model, where he facors are: shor erm rae, mean level shor erm rae and shor erm rae volailiy. This model is given as: dr = κ ( θ r ) d + dθ = ϕ( λ θ ) d + η dv = γ ( ϑ v ) d + ξ v dz v r dz 2 θ dz, 3 1., As one can see, all hree facors are modeled by Ornsein-Uhlenbeck process, so hey are considered o be mean-revering and in addiion here is volailiy par described by Wiener process. One of he mos imporan problems in he applicaion of hese models is he esimaion of is parameers. Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

7 Ineres Rae Modeling and Tools of Financial Economerics 13 References Chen, L. (1996), Sochasic mean and sochasic volailiy a hree facor model of he erm srucure of ineres raes and is applicaion in derivaives pricing and risk managemen, Financial Markes, Insiuions and Insrumens, 5, pp Engle, R. F. (1982), Auoregressive condiional heeroskedasiciy wih esimaes of he variance of U.K. inflaion, Economerica 50, pp Heah, R. A., Jarrow, R. A., Moron A. (1992), Bond pricing and he erm srucure of ineres raes: a new mehodology for coningen claim valuaions, Economerica, 60, pp Marinelli, L., Priaule, P., Priaule, S. (2003), Fixed income securiies, Wiley, New York. Svensson, L. (1994), Esimaing and inerpreing forward ineres raes: Sweden , CEPR Discussion Paper Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House