190 E.P.de Lara-Tuprio, et al DLSU Business & Economics Review (2017) 27(1): 190-200 RESEARCH ARTICLES Forecasting the Term Structure of Philippine Interest Rates Using the Dynamic Nelson-Siegel Model Elvira P. de Lara-Tuprio, Ramil T. Bataller, Allen Dominique D. Torres, Emmanuel A. Cabral and Proceso L. Fernandez Jr. Ateneo de Manila University, Quezon City, Philippines
[email protected] Abstract: The three-factor Nelson-Siegel model is a widely used model for forecasting the term structure of interest rates. Several extensions have recently been proposed. Even for the original model, different methods of treating the parameters have been shown. Ultimately, what works best depends on the data used to estimate the parameters. In this paper, the original three-factor model with fixed shape parameter was applied to forecast the term structure using market data from the Philippines. Instead of giving a pre-determined model for the latent factors, the best time series model for them was searched using standard statistical tools. Based on the historical data, the best model for each latent factor is of the form ARMA(p,q)+eGARCH(1,1). The dependence structure of these parameters was considered in generating their future values. This was carried out by finding the joint distribution of the residuals via appropriate copula. Results show that forecast of interest rates for different tenors is reliable up to the near future. For an active market, this is good enough since the models for the parameters can be adjusted as new information comes in. Keywords: Yield curve, Nelson-Siegel, forecasting, copula, time series JEL Classification: G17, E47, C53 In recent years, the influx of new financial products mathematical models. Loss estimates and pricing are in the Philippines, including financial derivatives, based on current and forecast of underlying economic has become inevitable. Some local banks have been variables such as interest rates and currency exchange granted limited authority by the Bangko Sentral ng rates. Currently, several financial institutions in the Pilipinas (BSP) to engage in specified derivatives Philippines are either using vendor-developed systems, transactions, while some others are still in the stage of which are essentially βblack box,β or have developed preparing to apply for a license for such transactions. their internal models which are mostly based on The bond market is also on the rise with an increasing popular or classical models. There is indeed a challenge participation of corporate bond issuers. Amidst these for financial institutions, particularly in the banking developments is the need for sound risk management industry, to develop their own models which are more structure and reliable models for the pricing and reliable, relevant, and suited to the market data. valuation of financial products. Public debt management also requires sound Financial risk management and valuation of mathematical models for forecasting the evolution of financial securities require sound and reliable macroeconomic factors such as interest rates, exchange Copyright Β© 2017 by De La Salle University TERM STRUCTURE FORECASTING TERM STRUCTURE TERM STRUCTURE FORECASTING FORECASTING 4 TERM STRUCTURE FORECASTING TERM STRUCTURE FORECASTING TERM STRUCTURE FORECASTING time series model will betools. 4 using standard statistica searched time series model will be searched using time standard series statistical The model that will be used model will be searched using standard 191 statistical tools. Philippine Interest Rates Using the Dynamic Nelson-Siegel Model time series model will be searched using standard statistical tools. The model that will be u time series model will of time be the form searched series model will beofsearched ARMA(p,q)+eGARCH(1,1) using standard statistical the form of using the tools. form or The standard ARMA ARMA(p,q)+eGARCH(1,1) model forstatistical ARMA(p,q)+eGARCH(1,1) thatthe will mean tools. be used equationor The model is ARMA and forthat or Exponential ARMA thewill forbe mean the u eq rates, and primary budget of the form ARMA(p,q)+eGARCH(1,1) deficit/surplus. Finding an tools. The model or ARMA that will for the be usedmeanisequation of the form and Exponential optimal debt strategy of the form ARMA(p,q)+eGARCH(1,1) means of the form determining GARCH for the variance ARMA(p,q)+eGARCH(1,1) a mix or ARMAof debtequation. for the mean GARCH GARCH Moreover, ARMA(p,q)+eGARCH(1,1) for the equationor for variance ARMAthe forand for variance forecasting, Exponential equation. the or mean equation. ARMAthe equation dependence Moreover, for the for and Moreover, mean Exponential for forecasti structure forecasting, ofthe thed instruments that minimizesGARCH cost to thefor government, the variance in equation. equation and Exponential Moreover, for forecasting, GARCH the for dependence the variance structure of terms of interest GARCH payment, for the variance beta GARCH subject parameters equation. for to a the Moreover,willvariance prudent beforrisk equation. considered forecasting, beta beta equation. by the parameters Moreover, usingparameters Moreover, befor an appropriate dependence will for forecasting, will be considered forecasting, structure considered copulaof bythe the to dependence the using byanusing dependence obtain theiranjoint appropriate structure appropriate copula of level. It may then be used beta to guide the policy makers structure of the parameters will be considered by using an appropriate copula to obtain their joint beta parameters will be considered in creating beta a short-term parameters orbeta will bedistribution. considered parameters medium-term by using will borrowing be considered an appropriate distribution. by an distribution. by copula using using to appropriate obtain an appropriate theircopula joint to copula obtaintotheir obtain joint their joint plan, which will hopefully distribution. improve the domestic debt distribution. market due to enhanced predictability distribution. distribution. and transparency The remainderThe of this paper asremainder of this proceedspaper as follows: proceeds as follows: The remainder of this paperThe proceeds remainder follows: of this paper The nextproceeds section as discusses follows: the TheNelsonextT related to public debt. The next section discusses the Nelson-Siegel equation The remainder of this paper proceeds as follows: The next section discusses the Nel Forecasting the termofstructure of interest The remainder this paper Siegel The equation proceedsfor rates remainder has theasyield. follows: ofSiegelWefor this The paperthe Siegel then equation yield. next describe We then section proceedsequation for the as the describe discusses follows: for data yield. the andWe the theThe yield. the data Nelson- next thengeneralWe and describe the section then general thediscusses describe procedure datathat the the anddatatheNel will an be ge created a huge literature spanning several decades. procedure that will be implemented, to be followed Siegel equation for the yield. We then describe the data and the general procedure that will Some of Siegel these make equation useyield. for the ofSiegel mean-reverting Weequation then describe stochastic for theyield. the data and with We thegeneral the then implemented,results describe of to the procedure the beimplementation data that followed and will the including begeneral with procedure theofresults theof thethat implemwill implemented, models, such as the models of Vasicek (1977) and Cox, to be followed with implemented, the results to be of the followed implementation with the resultsincluding the the time series implementation time series found for the beta parameters, the copula Ingersoll, and to Ross implemented, towhich be followed with to theobtainresults ofjoint the implementation including the time serie implemented, be (1985). followed Another with the implemented, classresults toparameters,has implementation beoffollowed the used with the beta found including results for their the of beta thethe time seriesthe parameters, implementation parameters, and copula the forecast including used the tothetime obtain serieth found gained popularity among government policy makers for the beta found the copula used to obtain their performance of the model. Finally, we provide the their joint for the parameters, the joint copula parameters, used to obtain and forecas due to for found its good fit toparameters, the beta observed found term forcopula the the beta structure parameters, includes used to obtain conclusion.the joint their copula used to obtain parameters, and the their forecastjoint parameters, and the fore found performance for the beta parameters, the copula performance used ofto obtain the Finally, wejoint model. their Finally, parameters, we provide and the theconclusfore Nelson and Siegelβs (1987) model andofitsthe model. Finally, variations. performance we provide of the model. the conclusion. provide the conclusion. This paperof performance aims thetomodel. presentperformance a method Finally, wefor offorecasting provide the model. Finally, we provide the conclusion. the conclusion. performance of the model. Finally, weNelson-Siegel provide the conclusion. The Nelson-Siegel Model Mo the term structure of interest rates that is applicable to TheThe Nelson-Siegel Model Model The Nelson-Siegel the Philippine market data. It is based on The theNelson-Siegel framework Model The Nelson-Siegel Model The aLetNelson-Siegel (ππ)pricebe the atModel price atof time of a zero-coupon 1π‘π‘ peso ππ,bo developed by Diebold and LiLet πππ‘π‘ (ππ)which (2006), be the isprice a at time Letπ‘π‘ ππof Let π‘π‘ (ππ) bebeππthe zero-coupon π‘π‘the price bond at time time thatt π‘π‘of pays a azero-coupon zero-coupon at time bondπ‘π‘ +that t reinterpretation Let πππ‘π‘ (ππ)ofbethe themodel price at Let π‘π‘πππ‘π‘of(ππ) introduced time by be the pricebond Nelson a zero-coupon at time bond that π‘π‘ of pays a that pays 1 peso at timezero-coupon 1 peso at time bond π‘π‘ + that ππ, that pays 1 is, t is the peso at time π‘π‘ + ππ and Siegel (1987). Although is, ππ is several Let ππ the timeextensions (ππ) be to maturity the price of ππthe areis, is at time is, bond ππ thetotime π‘π‘ is of the in years. a zero-coupon time to maturity ofLet to theπ¦π¦bondmaturity (ππ)the bond beyears.of that the the corresponding pays bond 1 in pesoyears. at Lettheπ¦π¦continuously time Let π¦π¦ π‘π‘ cππ π‘π‘ + (ππ) π‘π‘ time maturity π‘π‘of in bond in years. Let yt (t) be π‘π‘ (ππ) be the available, is, ππ is thethe original time three-factor to maturity is,ofππ the model is thebond time will in tobematurity years. used. Let π¦π¦π‘π‘of (ππ) thebe bond the corresponding corresponding in years. continuously Let continuously be the corresponding π¦π¦π‘π‘ (ππ)compounded nominal continuous The procedure, however, is, will ππ isbethe time modified compounded nominal yield.compounded to maturity and the of the Thisyield. bond compounded yieldThis in isnominal also years. yieldreferred is also Let nominal yield. π¦π¦ toThis referred π‘π‘ (ππ) yield. be as ayield tozero the This as ais corresponding rate. zeroyield Then alsorate. isThen referred also referred tocontinuous as a zero to a results of the implementation compounded nominal yield. will be highly This yield isnominal compounded dependent also referred yield.toThis as ayield zero rate. is also Then referred to as a zero rate. Then on the data used. compounded nominal yield. This yieldππ is(ππ) also = referred ππ βπππππ‘π‘(ππ) to as a zero rate. Then = ππ βπππππ‘π‘(ππ) ( (ππ)π‘π‘(ππ) πππππ‘π‘(1) βππππ π‘π‘ ππ π‘π‘ (ππ) = The Nelson-Siegel model is a popularπππ‘π‘model (ππ) = for (ππ) ππ βπππππ‘π‘ π‘π‘ (ππ) (1) πππ‘π‘ (ππ) = ππ βππππ the term structure of interest rates. It was introduced Ifisrate ππisπ‘π‘is(ππ) the= the βππππ ππ at instantaneous (ππ) forward rate withatmaturity πππ‘π‘ (ππ) π‘π‘at+ π‘π‘ by ππNelson and Siegel If πππ‘π‘ (ππ) is the instantaneousIf forward (1987) as a class of parametric πππ‘π‘ (ππ)If the with instantaneous maturity instantaneousforwardtime π‘π‘forward + ππ with rate rate contracted maturitywith time at time π‘π‘, then If π‘π‘ (ππ) is the instantaneous If forward πππ‘π‘ (ππ) is rate with the instantaneous maturity at time maturity rate forward π‘π‘ + ππ at time contracted with t +maturity at t contracted timeat timeπ‘π‘, then at time π‘π‘ +t, ππthen contracted at time π‘π‘, t functions to capture the If range of isshapes typically the instantaneous forward rate withππmaturity at time contracted atππtime πππ‘π‘ (ππ) associated with yield curves. Diebold and Li (2006) ln ππ π‘π‘ (ππ) π‘π‘ + ππ ππ ln=πππ‘π‘β(ππ) ln πππ‘π‘π‘π‘,(ππ)(t ππ ln πππ‘π‘ (ππ) πππ‘π‘ (ππ) = β ππ ln ππ (ππ) πππ‘π‘(2) (ππ) = β πππ‘π‘ (ππ) reformulated it as a dynamic factor model πππ‘π‘ (ππ)and=β used ππππ π‘π‘ ππππ ππππ ππππ πππ‘π‘ (ππ) = β ππ ln πππ‘π‘ (ππ) (2) it to forecast the yield curve. Several extensions and πππ‘π‘ (ππ) = β ππππ It follows that It It follows follows that that ππππ variations It follows that of the model have been introduced with the goal of improving the out-of-sample It followsforecasts that (see for It follows example Christensen, Diebold, & Rudebuscha, 2011; that It follows that 1 ππ 1 ππ 1 ππ ( 1 ππ π¦π¦π‘π‘ (ππ) = β«1 πππ‘π‘ ππ(π’π’)ππππ (3) π¦π¦π‘π‘ (ππ) =(π’π’)ππππ β« πππ‘π‘ (π’π’)ππππ π¦π¦ (ππ) = β« ππ (π’π’)ππππ π¦π¦ (ππ) = β« ππ Exterkate, Dijk, Heij, & Groenen, 2013; π‘π‘ Koopman, π‘π‘ ππ 0 π¦π¦π‘π‘ (ππ) = 1 ππ 0 ππ (π’π’)ππππ π‘π‘ β« πππ‘π‘ ππ 0 π‘π‘ ππ 0 Mallee, & Van der Wel, 2010). π¦π¦π‘π‘ (ππ) = ππ β«0 πππ‘π‘ (π’π’)ππππ This paper will focus on theYield original three-factor Yield ππ a0 curve curve (3) Yield curve at a particular point curve in timeat a π‘π‘particular is a curve point Yield incurve that describes time atπ‘π‘theis spotatinterest a particular a particular thatpoint describes rates in point timethe π‘π‘inisspot time a curve π‘π‘ is that interest a curve rate des Nelson-Siegel model. Following the method in Yield curve at a particular point in time π‘π‘ is a curve that describes the spot interest r Diebold and Li (2006), the latent factors, which Yieldcurves curve will at abe particular point in different time π‘π‘ is or a curve that describes theorspot interest π¦π¦π‘π‘ (ππ) for different maturities π¦π¦π‘π‘ (ππ) for different ππ. These referred to throughout this paper as beta parameters,π¦π¦π‘π‘ maturities are ππ. These typically (ππ) (ππ) π¦π¦π‘π‘different curves monotonic, forYield for curve are a typically humped maturities at particular maturities ππ. monotonic, S-shaped These point These inππ.curves time humpedcurves t isarea curvetypically are monotor typically S-shaped π¦π¦π‘π‘ (ππ) for different maturities that ππ. These describes curves the spot are typically interest rates monotonic, yt (t) for different humped or S-shaped will be estimated from historical π¦π¦π‘π‘ (ππ) for data by fitting different the maturities ππ. These curves are typically monotonic, humped or S-shaped Nelson-Siegel equation to the yield curves over time maturities t. These curves are typically monotonic, and assumed to follow a time series model. However, humped or S-shaped (Nelson & Siegel, 1987). To instead of assuming independent univariate AR(1) generate this range of shapes, a parsimonious model processes for all the beta parameters, the best time introduced by Nelson and Siegel assumes that the series model will be searched using standard statistical forward rate follows the equation URE FORECASTING 0π‘π‘ 0π‘π‘ TERM STRUCTURE FORECASTING 5 5 0π‘π‘ ππβ+β π‘π‘ &STRUCTURE RE M Siegel, 1987). FORECASTING To generate this range FORECASTING of shapes, Nelson TERM and a5parsimonious Siegel STRUCTURE assumes 5 FORECASTING model that the introduced forward rate by 5 follows the equation ππβ+β TERM TERM STRUCTURE STRUCTURE FORECASTING FORECASTING yield curve shifts. The factor π½π½1π‘π‘ is related to π½π½the yield 0π‘π‘ and an curve increaseslope. 0π‘π‘5If in π½π½0π‘π‘ the slope 5increases of theequally. all yields yield curve Thus, t and an π½π½0π‘π‘1987). el, increase in π½π½0π‘π‘ this To generate increases rangeallofyields equally. shapes, Thus, π½π½ this andfactor a parsimonious yield in π½π½ curve is responsible model an increase introduced forshifts. by allThe 0π‘π‘ parallel increases factor yields π½π½1π‘π‘ isThus, equally. related thistofactor the yield curve slope. is responsible for If part and Siegel assumes (Nelson that&the Siegel, forward 1987). rate To follows generate 0π‘π‘ the equation this rangemodel of shapes, 0π‘π‘ a parsimonious model (Nelson introduced & βππ/ππ Siegel, by 1987). πππ‘π‘ (ππ)generate = π½π½0π‘π‘ + egel, syield range 1987). of shapes, To generate a parsimonious this range model of shapes, is introduced defined a parsimonious as by lim π¦π¦ (ππ) β lim introduced π¦π¦ (ππ), then itbyyield is equalcurve to shifts.βπ½π½ The . ππfactor Lastly, π½π½ π½π½ 1π‘π‘ isisrelated related to To thethe to yield curve s el, lson rate 1987). & curve this Siegel, range To shifts. generate 1987). The of shapes, factor To this generate π½π½a range is parsimonious related of this shapes, to (Nelson range the yield model a ofparsimonious & shapes, curve introduced Siegel, yield slope. curve a 5 1987). π‘π‘ model parsimonious If the by shifts. slope To is Theππ introduced (ππ) of defined generate factorthe π‘π‘ model = π½π½ yield as this by +introduced curve lim is rangeπ½π½ related π¦π¦ ππ (ππ) βππ/ππ of to β π‘π‘by shapes, the lim + yieldπ½π½ 1π‘π‘ π¦π¦ (ππ), acurve ππthen parsimoniousslope. 1π‘π‘ itπ‘π‘ 2π‘π‘ is If equal the model slopeto βπ½π½ of introduce the . Lastly, yield c el assumes TERM that TERM the STRUCTURE forward STRUCTURE rateFORECASTING FORECASTING follows the equation π‘π‘ + π½π½ 0π‘π‘ 1π‘π‘ 2π‘π‘ 5 bylim++5π¦π¦π‘π‘π‘π‘ (ππ), then 1π‘π‘ ππβ+β ππβ0 π‘π‘ + E.P.ππde Lara-Tuprio, π‘π‘ 1π‘π‘ (Nelson & & Siegel, 1987). ToTo generate this range of shapes, aa parsimonious et al is model introduced by 1π‘π‘ππβ+β (Nelson 192 Nelson Siegel, and 1987). Siegel assumes generate that this the range forward ππ5 βππ/ππ of rate shapes, follows parsimonious the equation model defined (4) as introduced ππβ0+ lim π¦π¦π‘π‘π‘π‘ (ππ) π‘π‘ β it is equalthe to βπ½π½ iegel assumes that the forward rate follows 5 βππ/ππ the equation where Nelson π½π½ , and π½π½ Siegel , π½π½ , assumes and πππ‘π‘ areisthat constants for 11 ard rate follows the equation ππ π‘π‘ (ππ) = π½π½ + π½π½ ππ + π½π½ ππ ππβ+β ππβ+β ππβ0 ππβ0 gel son s defined he forward rate assumes and Siegel as lim that π¦π¦the assumes follows π‘π‘ (ππ) forward β that lim the+equation π¦π¦ the π‘π‘rate (ππ), 0π‘π‘ forwardfollows then 1π‘π‘ it is Nelsonπππ½π½andβππ/ππ rate the equal curvature, π‘π‘ equation followsto βπ½π½defined 2π‘π‘ is Siegel defined the . Lastly, πππ‘π‘ π‘π‘, π½π½assumes 1π‘π‘ equation by as π‘π‘ Diebold π½π½ lim 2π‘π‘ is π¦π¦ that related and (ππ) curvature, the β Li to as lim the forward defined 2π¦π¦π¦π¦ (24) (ππ), rate then β byππFORECASTING π¦π¦it follows Diebold (3) is equalβ and π¦π¦0π‘π‘(120), theLithatto βπ½π½ equation 1π‘π‘ where as 1π‘π‘ . 2π‘π‘ maturity Lastly, π½π½ is related given to in the π‘π‘ (24) π¦π¦π‘π‘ (3)isβa π¦π¦solution π‘π‘ (120), π‘π‘ π‘π‘ π‘π‘ 2π¦π¦ TERM STRUCTURE FORECASTING π‘π‘ where , and TERM π‘π‘ are π‘π‘TERM STRUCTURE constants ππ (4) STRUCTURE + π‘π‘ FORECASTING and 0. Note 5= βπππ‘π‘2π‘π‘ Li as (ππ) ππβ+β ππππ, π½π½this ππthe β defined π₯π₯(ππ) ππβ0 e this range Nelson Nelson ofπππ‘π‘ (ππ) (Nelson (Nelson shapes, and and=Siegel Siegel & π½π½ & aSiegel, 0π‘π‘ parsimonious +assumes Siegel, assumes π½π½1π‘π‘1987). βππ/ππ ππ1987). that that +To To model the the π½π½generate 2π‘π‘ forward generateforward introduced 0π‘π‘ 1π‘π‘ thisrate rate range range2π‘π‘ by follows follows ππβ+β of of βππ/ππ the shapes, shapes, β b equation equation . a a Lastly, ππβ0 parsimonious parsimonious b βππ/ππ curvature, is related π‘π‘ model model to the introduced by introduced Diebold curvature, andby by (4) defined 2π¦π¦by π‘π‘π‘π‘(24) β π¦π¦π‘π‘π‘π‘(3) β π¦π¦ this range of shapes, ππ a=parsimonious model βππ/ππ π‘π‘ months. ππ π½π½πππ‘π‘0. introduced (ππ) π¦π¦ππ =ππββππ/ππ π½π½(4) by π‘π‘+πππ½π½1π‘π‘where ππ maturity π‘π‘ +1t π½π½ ππ(4)2t(4) π‘π‘ second-order (4)β π¦π¦π‘π‘ (3) linear ordinary differential πππ‘π‘ (ππ)is= eq π½π½ π‘π‘ (ππ) π¦π¦π‘π‘0π‘π‘ curvature, defined ππby Diebold π½π½constants and + Li π½π½= as ππand + π‘π‘ (3)βππ/ππ 2π‘π‘ is given in pes, a parsimonious βππ/ππ model βππ/ππ πππ‘π‘(ππ) introduced 2π¦π¦ π‘π‘π‘π‘π‘π‘(24) by β (120), (4) βππ/πππ‘π‘ months. ππ ππ π½π½)+ 0π‘π‘ = π½π½ , 1π‘π‘ π½π½ π½π½ ππ 1π‘π‘ , + π½π½ π½π½ π‘π‘ + 2π‘π‘ , ππ ππand π½π½ (ππ) βππ/ππ π‘π‘ 2π‘π‘ ππ = π‘π‘π‘π‘+ π½π½ ππ are π½π½ 0π‘π‘ ππ+ 0π‘π‘ π½π½ ππ1π‘π‘ ππ 1π‘π‘ βππ/ππ βππ/ππ βππ/ππ π½π½ + π‘π‘0π‘π‘ second-order + ππ π½π½ π‘π‘ π½π½ β 2π‘π‘ ππ βππ/ππ 2π‘π‘ Note π‘π‘ π‘π‘βππ/ππ ππ ππ curvature, + linear that π½π½π‘π‘π‘π‘ π₯π₯(ππ) defined ππ ordinary (4) = ππ by π‘π‘ ππ Diebold (ππ)ππ Diebold differential ππ (ππ) is = a and π½π½ π‘π‘and solution Li Li equation + asas months. π½π½ to 2π¦π¦ ππ theπ‘π‘ βππ/ππ (24) + π½π½ β ππ π¦π¦ π‘π‘ βππ/ππ (120),, where where maturity give orward 0π‘π‘ rate1π‘π‘ follows π‘π‘ π‘π‘ ππand the&Siegel 0π‘π‘equation π‘π‘ 1π‘π‘ π‘π‘ 1π‘π‘ 2π‘π‘ 2π‘π‘ (4) πππ‘π‘ (4) π‘π‘ π‘π‘ Nelson Nelson (Nelson π‘π‘and 2π‘π‘ Siegel, Siegel assumes 1987). that To the =generate ππthe π‘π‘ forward π‘π‘ this ππrate π‘π‘ π‘π‘π‘π‘follows range (Nelson of π‘π‘(Nelson shapes, the & equation aSiegel, & π‘π‘parsimonious Siegel, 1987). 1987). To model generate To2π‘π‘generate introduced this this range byrange of shapes, of shapes, a parsi ap π‘π‘ ππassumes that forward ππβππ/ππ rate follows the ππππβππ/ππ equation πππππ‘π‘π‘π‘(ππ) = π½π½ maturity isπ‘π‘0π‘π‘given 1π‘π‘ in months. 0π‘π‘ + π½π½of ππ= + π½π½π½π½a2π‘π‘ ππand (ππ) βππ/ππ βππ/ππ months. rward π½π½2π‘π‘ , and rate π‘π‘ are constants ππfollows the equation π‘π‘ β 0. TERM Note EstimationSTRUCTURE π½π½π‘π‘0π‘π‘ that + π₯π₯(ππ) π½π½1π‘π‘ 1π‘π‘ FORECASTING Parameters ππ π‘π‘ (ππ) is 2π‘π‘ + solution to the ws the equation where π½π½ , π½π½ , π½π½ , and ππ are constants months. and ππ β 0.ππ ππ Estimation π‘π‘ Note that of Parameters π₯π₯(ππ) Estimation = ππ (ππ) of is Parameters a solution to the π₯π₯ β²β² + order π½π½ π½π½linear ,β and ordinary ππwhere TERM π‘π‘ are differential constants STRUCTURE 0π‘π‘ππ FORECASTING 1π‘π‘andequation 2π‘π‘ , and l π‘π‘Note arethat constants and π‘π‘ is a solution to the (ππ) π‘π‘ 2 β² 5to 1π‘π‘ where π½π½0π‘π‘π½π½0π‘π‘ , π½π½1π‘π‘ , π½π½2π‘π‘ , and πππ‘π‘ are constaππ s,re constants 1π‘π‘ π½π½ and onstantsπ½π½, 2π‘π‘ π½π½ + ππand ,2π‘π‘ ,π½π½ π‘π‘ π½π½ and ππ ππ0. βππ/ππ Note are ,π‘π‘π‘π‘Parameters π½π½ Nelsonπ‘π‘ ,+ constants and that π½π½ . πππ‘π‘π₯π₯(ππ) and Note areSiegel ππ that and βππ/ππ that = π‘π‘πππππ‘π‘π‘π‘assumes constants ππβ (ππ) π‘π‘ β 0. isand where 0. a solution Note t that (ππ) πππ‘π‘π½π½is that is β the aa 0. toNote π₯π₯(ππ) forward solution solution π₯π₯(ππ) the=(4) that to =ππ(7), to , ratethe and πππ‘π‘π₯π₯(ππ) (ππ) the is Nelson follows aare= solution Estimation Nelson (ππ) ππconstants and the ππππ β²β²and π₯π₯ Siegel to isβππ/ππ equation + aofthe solution Parameters Siegel and assumes π₯π₯ + assumes the that π₯π₯ = Note the that forward the forward (4)(4)rate = πππππ‘π‘rate follows isthea the follows equa the e ππ β 0. π₯π₯(ππ) = ππ 2that (ππ) solution Estimation 2π‘π‘ 0π‘π‘ 1π‘π‘ of 2π‘π‘ π‘π‘ In , π½π½ Equation , π½π½ π‘π‘ π‘π‘ βππ/ππ the ππ parameter π‘π‘ ππ is referred ππ toβ as 0. the shape π₯π₯(ππ) parameter. For each t,towhen ear0π‘π‘ordinary βππ/ππdifferential ππ ,πππ½π½ equation π‘π‘ππππ π‘π‘π‘π‘(ππ) ==π½π½Estimation ++πππ½π½π½π½ ππππ0. π‘π‘π‘π‘+ π½π½π½π½2π‘π‘2π‘π‘In πππππ‘π‘ =πππππ‘π‘ (ππ) (ππ) βππ/ππ βππ/ππ ππ 1π‘π‘ where where π‘π‘ π½π½π½π½ ,, π½π½ π½π½ 2π‘π‘ , π½π½ , , and and ππ ππ are (Nelsonare constants constants & 0π‘π‘ Siegel, π½π½ and 1π‘π‘ and 0π‘π‘0π‘π‘ (4) ππ 1987). 2π‘π‘1π‘π‘ β β 1π‘π‘ of0. To Note Note π‘π‘ Parameters + generate that that π₯π₯(ππ) π₯π₯(ππ) Equation this = range π‘π‘ ππ (7), (ππ) of π‘π‘ In is the isππ a shapes,Equation a2parameter solution solution a ππ (7), to parsimonious the ππ to theparameter is the referred π‘π‘π‘π‘to model is as referred shape introduced as the pas π½π½0π‘π‘ + π½π½1π‘π‘ ππβππ/ππsecond-order π‘π‘ second-order + 0π‘π‘ 0π‘π‘ π½π½ 1π‘π‘ 1π‘π‘ ππ βππ/ππ π‘π‘ linear 2π‘π‘ 2π‘π‘ linearπ‘π‘ ordinary 2ordinary π‘π‘π‘π‘ (4) differential1 differential π½π½ equation equation π‘π‘π‘π‘ In ππ Equation ππ π‘π‘ (7), π‘π‘ π‘π‘π‘π‘ the (5) parameter π‘π‘ second-order l π‘π‘ is referred π‘π‘ linear to as ordinary the differenti + linear equation π½π½2π‘π‘ordinary ordinary ππ linear (Nelson π‘π‘ differential 2π‘π‘& Siegel, 1987). To πππ‘π‘ differential equation generate this range 0π‘π‘of shapes, a parsimonious model π‘π‘ introduced by From t near ond-order rential ππInπ‘π‘equation Equation differential ordinary (7), the equation parameter π₯π₯ β²β² +πππ‘π‘ isπ₯π₯equation β² referred second-order+πππ‘π‘ is2to π₯π₯ as=thelinear specified, shape theInmodel parameter. becomes shape For each linear ππ 5t,inβππ/ππ parameter. when the isFor πππ‘π‘π‘π‘π‘π‘ππparameters specified,each Equation t,theπ½π½when model ,= π½π½the (3), lt,becomes is and specified,π½π½π½π½(4) .ππLet linear the inππ π‘π‘ parameters π‘π‘the =For π¦π¦ (ππ), ππ βππ ππ0 t,π½π½w +ordinary π½π½Equation ππdifferential (7), the parameter ππequation π‘π‘ is referred to0π‘π‘ππ as shape parameter. ,each TERM STRUCTURE FORECASTING stants and second-order second-order ππwhere β 0. Note linear linear that 2 ordinary π₯π₯(ππ) ordinary ππ = 1 π‘π‘ differential ππ From differential(ππ) πππ½π½isππEquation 0π‘π‘ a(ππ) solution =πππ‘π‘π½π½20π‘π‘ equation equation (3), to the ππ βππ/ππ π‘π‘ is π‘π‘model +specified, π½π½2π‘π‘ becomes (5) thelinear model in becomes the π‘π‘ (ππ) ππparameters π‘π‘ (ππ) linear π½π½0π‘π‘ 1π‘π‘ = in b + π½π½the,0π‘π‘b 2π‘π‘+ 1π‘π‘ and π½π½βππ/ππ parameters 1π‘π‘b ππππ.βππ/ππ + π½π½ π½π½π‘π‘ π‘π‘+ 2π‘π‘ π½π½ π½π½2π‘π‘1π‘π‘ππ, a π‘π‘where π½π½and β²β² ,Siegel ,+π½π½π½π½1π‘π‘ 1π‘π‘, ,π₯π₯2 π½π½π½π½2π‘π‘ β² 2π‘π‘, ,ππ and and Nelson ππππthe π‘π‘ are π‘π‘and π‘π‘ constants Siegel assumes and 2 (5) 1π‘π‘ ππππβ²π‘π‘π‘π‘equation β β that 0.10.Note the forward that π½π½ πππ‘π‘ π₯π₯(ππ) rate = followsππ π‘π‘(ππ) isisthe aasolution equation to t the (5) 0π‘π‘ ππ πππ‘π‘ aare constants and Note that solution to the Nelson π₯π₯ π½π½0π‘π‘0π‘π‘ assumes + that 2 π₯π₯ π‘π‘= forward rate follows the π₯π₯(ππ) (ππ) (5)π‘π‘π‘π‘ = πππ‘π‘1βππ π‘π‘ 0π‘π‘ ππants 2is specified,and ππ1π‘π‘ β the 0. model Note π½π½ that β²β² πππ₯π₯(ππ) 2 = β² πππ‘π‘π‘π‘1 (ππ) 1the is ππ π½π½ 2π½π½ solution π₯π₯ to β²β² the (5) π‘π‘ + π₯π₯ + π₯π₯ = π‘π‘ βππ/πππ‘π‘π‘π‘ 0βππ/ππ 1t 2t 1 ππ π‘π‘ π‘π‘Note β² that 2 2π₯π₯(ππ) (Nelson & =1 πππ‘π‘ π₯π₯ β²β²β²β²π₯π₯+ 0π‘π‘ Siegel, (ππ) becomes 1987). is π½π½0π‘π‘ + a solution π‘π‘ To linear β²β²π₯π₯ β²β²+ to2the generate in this 2 range parameters π₯π₯π₯π₯1ππβ²=+ π‘π‘ π‘π‘ of 0π‘π‘ 0π‘π‘ 1 0π‘π‘ 1βππ shapes, βππ/ππ ππ π½π½ is a 0π‘π‘ , π½π½ parsimonious π½π½ specified, 0π‘π‘ ππ1π‘π‘ , and (5) π‘π‘ π½π½ the 1βππ model 2π‘π‘ . 2 Let model βππ/ππ π‘π‘ ππ =2π¦π¦π‘π‘ (ππ), ππ introduced becomes βππ/ππby (5) ,linear π‘π‘π‘π‘ π‘π‘ .2Thenππ = in 1βππβππ/ππ the 1 (5) parameters , and and π½π½ ππ π‘π‘π‘π‘ = π½π½ 1βππβππ/ππ 1βππ , π½π½ π‘π‘π‘π‘ ,β and ππ βππ/πππ‘π‘π‘π‘ βππ/ππ π½π½ . . ThenLet ππ π‘π‘ Equation = π¦π¦π‘π‘π₯π₯(ππβ²β²( quation ntial π₯π₯ + (3), π₯π₯ = πππ₯π₯ + π₯π₯ + π₯π₯ ππ = = 2 π₯π₯ π‘π‘ , and = π‘π‘ ππ ππ = ππ β 1βππππ ππ βππ/ππ βππ/ππ 1ππ 1ππ Equation (4)ππ/ππ 1βππ βππ/ππ βππ/ππ (7) π‘π‘π‘π‘ 2ππ ππ0π‘π‘ becomes 2ππ ππ/πππ‘π‘π‘π‘1π‘π‘ π¦π¦ (ππ) 0π‘π‘ππ/ππ 2π‘π‘ = β« ππ [π½π½ 0π‘π‘ + πππ‘π‘π₯π₯ equation π‘π‘ π‘π‘ ππ/ππ βππ/ππ β²β² + βππ/ππ π₯π₯ β² πππ‘π‘ π‘π‘second-order + From 2 πππ‘π‘2Equation π₯π₯ = ,linear 2(3), ordinary ππ 2 (ππ) differential 2ππ ππ/ππ2 2 βππ/ππ equation 1 1 2ππ 2π‘π‘ ππ where βππ/ππ π½π½ π‘π‘ ππ π½π½ ππ/πππ‘π‘ π‘π‘= Then 1 ππ π‘π‘ Equation π₯π₯ β²β² + , and (7) π₯π₯ ππ β² becomes + = π‘π‘π‘π‘ π₯π₯ = β ππ π’π’ (5) (5) π‘π‘ π‘π‘ . Then Equation π‘π‘ (7) becom 2π‘π‘, ,βππ/ππ and are π‘π‘ aconstants π‘π‘ and πππ‘π‘ β πππ‘π‘0.β Note that tπ₯π₯ 2 2 where where second-order π½π½assumes πππ½π½ linear π‘π‘ , π½π½ ordinary 2π‘π‘ππforward ππ ,π‘π‘π‘π‘and2π‘π‘ π‘π‘ππ ππ = π½π½ are differential π‘π‘ πππ₯π₯ 2 + π‘π‘ β²β²ππ+ 0π‘π‘β²β² π½π½ π‘π‘constants1π‘π‘ ππ π‘π‘ π‘π‘ + β²equation β²ππ π‘π‘ and π½π½ ππ =β πππ‘π‘ (ππ) 0.0π‘π‘ Note 0π‘π‘ 1ππ π½π½ =0π‘π‘π½π½ππ/ππ , π½π½thatπ½π½π‘π‘0π‘π‘ +, ,+ π½π½π½π½1π‘π‘π½π½ π₯π₯(ππ) π½π½ π½π½= 2ππ ππ,π‘π‘π‘π‘and ππ(ππ) is πππ‘π‘π‘π‘π½π½ are solution constants βππ/ππ to the and ππ 0.0 Note π‘π‘ theπ₯π₯π₯π₯ + π₯π₯πππ‘π‘π₯π₯π‘π‘π‘π‘(ππ) π‘π‘1π‘π‘ ππβππ/ππ +π‘π‘π‘π‘ππ/ππ π½π½ ππππ2π‘π‘π‘π‘ 2ππ ππ βπ’π’/πππ‘π‘ ]πππππππ‘π‘ππππ alπ‘π‘ equation 1βππππNelson andπππ‘π‘Siegel 0π‘π‘ππ π‘π‘π‘π‘1π‘π‘ π₯π₯π‘π‘follows + + π‘π‘= π‘π‘ 1π‘π‘ βπ’π’/ππ 2π‘π‘ 2 0π‘π‘π‘π‘ ππ π‘π‘ rate π‘π‘ that the equation 22π¦π¦ = 2π‘π‘2 β«1βππ [π½π½ 1π‘π‘ ππ π‘π‘πππ‘π‘ +2π‘π‘ βππ/ππ ππ3), π‘π‘ , and π‘π‘ 1βππ π‘π‘ βππ/ππ 1βππ βππ/ππ 0π‘π‘ βππ/ππ π‘π‘ π‘π‘π‘π‘ π‘π‘ . Then Equationπ‘π‘ (7) becomes 1ππ = ππ = β ππ ππ ππ ππ πππ‘π‘π‘π‘π‘π‘ , andπππππ‘π‘π‘π‘2ππ =ππ0π‘π‘ = ππ ππ β ππ ππ π‘π‘ . Then ππ = π½π½ + π½π½ ππ + π½π½2π‘π‘ 2π‘π‘ππ2 From Equation 1ππ/ππ, π‘π‘π½π½ππ (3), πππ‘π‘1ππ π‘π‘ = π½π½2π‘π‘Equation π‘π‘ π‘π‘ π‘π‘ (7) becomes 2ππ 0π‘π‘ 0π‘π‘ 1π‘π‘ 1π‘π‘ 1ππ 1ππ on (3), 2 ππ/πππ‘π‘ 1 π½π½ ππ π’π’ (5)ππ/ππ ππ (ππ) ππ/πππ½π½ π‘π‘ 0π‘π‘ + π½π½1π‘π‘(6) 1ππ the + From ππ2ππ Equation π‘π‘ (3), π‘π‘(8) π‘π‘ (8) where , and are constants and π‘π‘ ππ22 πππ₯π₯ β²π‘π‘ β²+π‘π‘ ]1second-order Note that πππππ₯π₯ second-order π½π½π½π½0π‘π‘π‘π‘0π‘π‘ linear is a linear solution ordinary to Evaluating ordinary differential ππ differential = ππthe integral π½π½ + (5) 0π‘π‘equation π½π½ equationππ 1π‘π‘at1ππthe right, + π½π½ ππ π‘π‘π‘π‘ 2π‘π‘ 2ππwe obt (3), 1 second-order π½π½ = 1π‘π‘β« linear , π½π½ =ordinary ππ differential ππ β 0.equation 1 π₯π₯(ππ) = ππ (4) (5) m β²β²Equation 0π‘π‘ βπ’π’/ππ βπ’π’/ππ π₯π₯ β² + 2 β²π₯π₯ β² +(3), π¦π¦π‘π‘ (ππ)π₯π₯ = π½π½ 0π‘π‘ ππ 0π‘π‘ [π½π½ 2π‘π‘ 0π‘π‘ + πππ‘π‘ (ππ) π‘π‘π½π½π½π½0π‘π‘From 1π‘π‘+πππ½π½1π‘π‘ ππEquation βππ/ππ π‘π‘ + π‘π‘ +π½π½ π‘π‘ π’π’ π₯π₯π₯π₯ β²β²+ 2π‘π‘ (5) π½π½2π‘π‘ β²β² (3), βππ/ππ == Specific value of ππ gives specific values of ππ1ππ 1ππ and ππ2ππ π‘π‘π‘π‘ 2ππ. + π½π½0π‘π‘ πππ₯π₯π‘π‘From + Equation ππ π₯π₯21 = ππ ππ (3),2 ππ π‘π‘ = π½π½ where + (5) Evaluating π½π½ πππ½π½ + , π½π½ π½π½ ππ, the integral + πππ‘π‘ππ π½π½ , π₯π₯and(6) + ππ at π‘π‘the right, are π₯π₯ constants (6) and(8) ππ β 0. Note that π₯π₯(ππ) = ππ (ππ) is yield curveππ to a solution From 2Equation (3),+ π½π½ordinary πππ‘π‘π‘π‘we π’π’obtain the Nelson-Siegel +2πππ½π½. 1π‘π‘ ππ1ππ + π½π½2π‘π‘ ππmodel for the π‘π‘ 2 2 0 ππ π‘π‘ ππ]πππ‘π‘ππππ 2 2 π‘π‘ π‘π‘ π‘π‘ 1ππvalue of ππβπ’π’/ππππππ1ππ=and π½π½0π‘π‘ππ π‘π‘ (6) βπ’π’/ππ βπ’π’/ππ π‘π‘ π‘π‘ π‘π‘ππ givesππππspecific ππvalues (ππ)πππ‘π‘= π‘π‘ +Specific 0π‘π‘ 1π‘π‘ π‘π‘of 2π‘π‘ π‘π‘ π‘π‘ π‘π‘ = πππ‘π‘ 2π¦π¦π‘π‘where π‘π‘ β« [π½π½ 1π‘π‘ ππππ π½π½2π‘π‘ equation ππ 0π‘π‘ 1π‘π‘ 1ππ 2π‘π‘ 2ππ πππ’π’πππππ‘π‘ 2πππ‘π‘0π‘π‘linear π½π½10π‘π‘β« second-order 1 differential π’π’ π‘π‘ Specific βπ’π’/ππ Specific value value of (6) of ππ t gives gives specific specific values 2ππ of of ππ and and ππ π‘π‘ . . (ππ) π½π½0π‘π‘1 ππ, π½π½01π‘π‘ , π½π½2π‘π‘βπ’π’/ππ , and ππ1π‘π‘ are constants βπ’π’/ππ π¦π¦ (ππ) π‘π‘π‘π‘ and π‘π‘ +πππ‘π‘π½π½β π‘π‘π’π’ = ππ π‘π‘0. Noteβ« [π½π½ (6) that βπ’π’/ππ 0π‘π‘ π₯π₯(ππ) +π‘π‘2 π½π½ π’π’ππππ = 1π‘π‘ ππ πππ‘π‘ (ππ)1is a solution π‘π‘ + π½π½ π½π½ππ 0π‘π‘to the 2π‘π‘ (6)ππ π‘π‘ ] ππππIn practice, (6) a record of 2 (5)β²2 β²π¦π¦1π‘π‘ (ππ) daily 1ππ datasets 2ππ is 1 collected, = 1π½π½each β [π½π½ + ππ ππ βπ’π’/ππ π½π½ ππ π¦π¦ (ππ) + = π½π½ β« ππ (ππ) [π½π½ π’π’ +] πππππ½π½ ππ βπ’π’/ππ βπ’π’/ππ ππ βπ’π’/ππ ππ βπ’π’/ππ π‘π‘ π₯π₯+ π½π½ βπ’π’/ππ β²β² + ] π₯π₯ (6) β² βπ’π’/ππ + π₯π₯ 1 = In practice, ππ a record of daily datasets π’π’ π₯π₯ β²β² + π₯π₯ β²β² is + π₯π₯ collected, + π₯π₯ + π₯π₯ = π₯π₯ = 0π‘π‘ ng Specific β« [π½π½0π‘π‘ + the π‘π‘ 1π‘π‘ integral π¦π¦ value π‘π‘π‘π‘ π½π½ ππ π‘π‘ of π‘π‘ = at ππ gives βπ’π’/ππ the 2π‘π‘ β« πππ¦π¦ π‘π‘π‘π‘π‘π‘+ right, [π½π½ specific π½π½ 0π‘π‘ = + 0π‘π‘ we π‘π‘ π½π½ valuesβ«ππ 1π‘π‘obtain ππ βπ’π’/ππ1π‘π‘[π½π½ of ππ second-order π‘π‘ ] ππππ the π‘π‘+ π‘π‘ 1ππ 1+1 and π½π½ Nelson-Siegel π½π½ ππππ 2π‘π‘ππ ππ 2π‘π‘ 2ππIn . ππππ0 linear Specific practice, π‘π‘π‘π‘] ππππ model ordinary value a ππ record 0π‘π‘ (ππ) = of for ππ π‘π‘ ]the differential of gives πππππ’π’ dailyπ’π’ yieldspecific datasets curve equation π‘π‘ values isβππ/ππ of collected, ππ βπ’π’/ππ π‘π‘ and each ππ π‘π‘ . dataset (6) ππ(6) βπ’π’/ππ βππ/ππ π¦π¦ indexed (ππ) = by π½π½ a 2 + particular π½π½ ππ ( 2 π‘π‘ π‘π‘ π¦π¦ β«In [π½π½ β+ π½π½a1π‘π‘number ππ +daily π½π½indicating ππ theππisinterest ]collected, ππππ 2 second-order From From 1π‘π‘ ππ Equation ππ 0 2π‘π‘ ππ(3), Equation 00 π‘π‘ ordinary linear 0π‘π‘ ππ(3), 1π‘π‘ 0π‘π‘ (ππ) differential π¦π¦π¦π¦ (ππ) 0π‘π‘ π‘π‘ = = equation β« β« 1π‘π‘ 2π‘π‘ π₯π₯ β²β² [π½π½ + [π½π½ 2 πππ‘π‘π‘π‘ π₯π₯+ π‘π‘ + β² +π½π½ π½π½ 1 2π‘π‘ ππ πππ₯π₯ππ βπ’π’/ππππ= βπ’π’/ππ π½π½ π‘π‘π‘π‘π‘π‘+ ππ + π½π½π½π½ 2 π‘π‘2π‘π‘ each ππ ππππ ππβπ’π’/ππ βπ’π’/ππ dataset 2 1π‘π‘0π‘π‘ π‘π‘practice, π‘π‘]]ππππ ππ day indexed ππππ π‘π‘ record (5) by 1ππ π‘π‘ of taand particular 12π‘π‘β 2ππ datasets ππ day π‘π‘ π‘π‘ π‘π‘ π‘π‘number πππ‘π‘βππ/ππ rates 0π‘π‘ πππ‘π‘π‘π‘t π¦π¦ππeach (ππ) ππ π‘π‘π‘π‘π‘π‘ 1π‘π‘ dataset for πππ‘π‘ππ2 0 differ π‘π‘ differe ntegralInatpractice, 0 the right, we obtain π‘π‘ the π‘π‘ Nelson-Siegel πππ‘π‘ model 0π‘π‘ 0π‘π‘ 1π‘π‘ 1π‘π‘ π¦π¦ π‘π‘ for 2 π‘π‘ the(ππ) = π‘π‘ yield 2 π½π½ 0π‘π‘ curve 2π‘π‘ + π½π½ 0 ( ) + π½π½ ( π‘π‘ β ππ ) isπππ‘π‘ππthe ππdataset ππindexed πππππ‘π‘π‘π‘aindicating 1π‘π‘ 2π‘π‘ a record ofthe Evaluating Evaluating daily theintegral datasets integral at at collected, the 00 right, each 1 ππππ we we obtain obtain In the practice,π‘π‘the Nelson-Siegel aand by record particular of (5) daily ππ/ππ model the datasets for interest isthe collected, yield rates ππ/ππ curve ymaturities each (t) for π‘π‘ dataset different e tain integral the at Nelson-Siegel the right, we model π’π’ obtain 1 for β the ππ the βππ/ππ Nelson-Siegel yield 2day number curve 1 1 β π½π½tmodel and ππ βππ/ππ (6) indicating for the yield the interest curve π’π’ rates 2 π¦π¦ π‘π‘ (ππ) parameters (7) 1for Evaluating different π½π½1π‘π‘1π‘π‘0π‘π‘, and t the 2π‘π‘ in(6) integral Equation ππ. indexed For(8) at each the are byt, right, athe estimated particula we us t, ntegral luating integral [π½π½ we +obtain at the π½π½ π¦π¦ theintegral ππ(ππ)βπ’π’/ππ the From right, = Nelson-Siegel π½π½ Nelson-Siegel + at Equation weπ½π½ the + π’π’ obtain π½π½ right,model ππ ( βπ’π’/ππ (3), the we model for ] Nelson-Siegel obtain π₯π₯ ππππ β²β² the + for ) yieldπ₯π₯ β² thethe + + 1 π½π½ Nelson-Siegel curve yield π₯π₯ ( model = 0π‘π‘ curve (6) for the model β From yield ππ day βππ/ππ From for maturitiesEquation curve number π‘π‘ ) the π₯π₯ Equation β²β²π’π’ yield + t t. and (3), For π₯π₯curve each (3), indicating β² π‘π‘ + The t, the π₯π₯ π½π½ 0π‘π‘, π½π½interest the parameters =parameters 0π‘π‘ π½π½rates π½π½ 2π‘π‘ (6) , π¦π¦π½π½ π‘π‘ (ππ) , for and different π½π½ are matur calle π½π½ + 0π‘π‘ π½π½ ππ 1π‘π‘ βπ’π’/ππ π‘π‘ From π‘π‘ + π½π½ π‘π‘ Equation0π‘π‘ 2π‘π‘ 2π‘π‘ 1integral β ππ (3), βπ’π’/ππ 1π‘π‘ πππ‘π‘ππ interest βππ/ππ π‘π‘ ]π‘π‘πππππ¦π¦ π‘π‘ π¦π¦ Evaluating (ππ)(ππ) πππ‘π‘ = = 1we ππ β« 2 2π‘π‘ β« the [π½π½ βππ/ππ[π½π½ ππ integral 2 + + π½π½π½π½ ππ ππat βπ’π’/ππβπ’π’/ππ the π‘π‘ right, π‘π‘ + + π½π½ π½π½ we ππ obtain ππ βπ’π’/ππ (7) βπ’π’/ππ π‘π‘ π‘π‘]]theππππ ππππ Nelson-Siegel 2 2 model 0π‘π‘ for 1π‘π‘ the yield 2π‘π‘ curve day 0π‘π‘ number t and indicating the ππ/ππ rates π¦π¦π‘π‘ (ππ) ππβ forobtainππdifferent ππ/ππ maturities For each t, the ππβππ/ππ ππyield ππfor π‘π‘ π’π’Evaluating Evaluating the the ππintegral atat) the the π‘π‘π‘π‘right, π‘π‘right, (6) we ππ 0obtain 0π‘π‘ 1the the ππ,π‘π‘Nelson-Siegel 1π‘π‘ Nelson-Siegel ππ. 2π‘π‘ model model for the curve (8)for π‘π‘the yield curve 1π‘π‘ π‘π‘ π‘π‘ 0π‘π‘number 1π‘π‘ t and 2π‘π‘ The (parameters day 1π‘π‘π‘π‘,π‘π‘)π½π½ and indicating are ππthe 1ππEquation ππinterest π‘π‘ factors rates π‘π‘ (ππ) π‘π‘ different maturities For each factor t, the π‘π‘called π½π½ β0π‘π‘ ,π½π½and βππ/ππ βππ/ππ inπ½π½2π‘π‘ in π‘π‘β (8) are π‘π‘(OLS) π¦π¦and estimated their coefficients using, for example, (7) areregression. ππ.called + π‘π‘ (ππ) π¦π¦βππ/ππ π½π½ =πππ½π½ 0π‘π‘ + βπ’π’/ππ π‘π‘ ]π½π½ ππππ1π‘π‘ ( βππ/ππ π‘π‘ 1 β ππ βππ/ππ +π‘π‘ π½π½2π‘π‘parameters 1 β0 πππππ½π½βππ/ππ ,β π‘π‘ ππ π½π½ 2π‘π‘parameters Equation (7) are squares , estimated and in using, method Equation formultiple ππfor example, ππ are linear (8) the estimated ordinary using, least 1 (7) isπ½π½for 1,ππ(aeβπ’π’ 0π‘π‘ 1π‘π‘ π½π½ , π½π½ π½π½ βππ/ππ β ππ π¦π¦ ππ(ππ) 2π‘π‘ π‘π‘ From Equation 1 β (3), ππ 1(β ππ/ππ ππEquation π‘π‘ βππ/ππ βππ/ππ π¦π¦ (ππ) 1π‘π‘π‘π‘(8) β = 1 ππareπ½π½ π½π½ βππ/ππ ππ + ππ/ππ 1(0π‘π‘β+ππ1π‘π‘ 1 π½π½ βππ/ππ βππ/ππ ( 1ππ/ππ π‘π‘ β βππ/ππ ππππexample, π’π’ ) + π½π½ the ( ordinary π’π’ (7) 0π‘π‘ least 1π‘π‘ β squares ππ (6) loadings. 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Lastly, The contribution contribution factor is ππ/ππ π‘π‘1π‘π‘to referred 1π‘π‘ to called the the tovalue value as the the of short-term of π¦π¦π¦π¦π‘π‘π‘π‘(ππ) medium-term (ππ) atat ππ/ππ factor shorter shorter π‘π‘ factor because maturities maturities factor (smaller (smaller loading, values values ofof m, itfactor decreases ππ/ππ because ππ/ππ π‘π‘π‘π‘ π‘π‘ππ). Moreover, , ππ/ππ has its , has has factor loading, significant βππ/ππ significant significant contribution it decreasescontribution loadings. contribution to the to value to 0 as theThe to ππ value β of π¦π¦π‘π‘ (ππ) at shorter the factor +β. of value π¦π¦ loading (ππ) Lastly, at maturities of shorter of π½π½ π½π½ maturities (smaller is 2π‘π‘ 0π‘π‘ valuesfactor referred is 1, a (smaller of constantto as values the that of precisely, medium-term does ππ). Moreover, not it approaches decay factor to it decreases zero 0 as ππ to even β 00 asand as + ππ ππββ+ a π‘π‘ to 0referred π‘π‘ as ππ β +β. βππ/ππ Lastly, is referred to as the medium-term 1βππ π‘π‘ π½π½2π‘π‘ π‘π‘ β. otits Moreover, the Lastly, factor decreases value loading, π½π½ππ/ππ it to of2π‘π‘ 0is decreases π¦π¦ as (ππ) ππ at β at to +β. shorter shorter to0 β as ππ Lastly, the ππ β maturities maturities medium-term π‘π‘ , starts +β. π½π½ is Lastly, (smaller(smaller at referred 0, π½π½factor increases, values to values is as referred of the of then medium-term t). to decays as the to 0. medium-term factor More factor ππtoβthe +β. value Lastly, long-term βππ/ππ of π¦π¦π‘π‘ ,,has π½π½ is referred factor. Theto as factor 2π‘π‘ ππ).the medium-term Moreover, π½π½ is called 2π‘π‘ it decreasesfactor the short-term long-term to 0 long-term as factor ππ β factor. +β. factor. because The Lastly, factor The its factor factor π½π½ π½π½ is is loading, π½π½ referred called is called to the as short-term the short-term medium-term factor factor bec fa itat shorter maturities (smaller values of 1βππ βππ/ππ (ππ) π‘π‘ 2π‘π‘ + 0precisely, 1π‘π‘ itto approaches 0ofis as and as Its 1βππ 1βππ π‘π‘ βππ/ππ π‘π‘ 2π‘π‘ ππ/ππ π‘π‘ significant contribution the value of π¦π¦π¦π¦ ππ(ππ) β at0atshorter ππ medium-term maturities β +β.2π‘π‘(smaller 1π‘π‘significant 1π‘π‘ valuescontribution of to the yield ππhas significant 00contribution isto theπ½π½2π‘π‘ value shorter maturities (smaller values ofit is referred π‘π‘ r loading, ππ). ππ). ππ).Moreover, Moreover,Moreover, ππ). Moreover, ππ/ππ Moreover, βit decreases itdecreases itβππ/ππ decreases π‘π‘ ,tostarts itdecreases decreases 0 as ππto to atThus, to to β +β. 0,as as ππππβ asincreases, ππLastly, 1βππ itβ β has +β. +β. π½π½+β. βππ/ππ Lastly, .Lastly, then 2π‘π‘π‘π‘significant Lastly, Lastly, referred decays toisas π½π½π½π½ 2π‘π‘ is referred contribution the 2π‘π‘ to referred 0. π‘π‘to medium-term (ππ) referred π‘π‘Data More as the and to to to medium-term the factor as as the Procedurethe yield medium-term factor for anymedium maturity; factor factor hence, to π‘π‘as th (ππ) at shorter ππ/ππ ππ/ππ because maturities 1βππ π‘π‘ π‘π‘βππ/ππ π‘π‘ π‘π‘ its factor (smaller βππ/ππ loading, values of β ππbecause βππ/ππ π‘π‘ ,decaysstarts at 0, increases, then decays the because to 0. More term its factor to maturity. loading, 1βππ βππ/ππ β ππ y, ctor itloading, loading, approaches 1βππ 1βππ is referred βππ/ππ βππ/ππ 0isas π‘π‘π‘π‘ β to 1βππ ππ as βππ/ππ + the π‘π‘ , starts medium-term at 0, increases, factor then its to 0. More ,and as Itsatsignificant contribution π‘π‘ to to the 0.yield is in /ππ πππ‘π‘ ππβ then 0 toπ‘π‘π‘π‘1βππ ππmedium-term β +β. or ause π‘π‘ π‘π‘+β. β π‘π‘ , starts ππitsLastly, βππ/ππ factor at 0, π‘π‘π‘π‘ , starts π½π½ππ/ππ increases, loading, 1βππ ππ/ππ βππ/ππ at β π‘π‘referred 0, βππ/ππ βππ/ππ increases, as decays starts βthe π‘π‘ππ1βππ then βππ/ππ at to the 0, decays π‘π‘ ,π‘π‘0. increases, starts π‘π‘ ,medium More ππ/ππ π‘π‘ to 0. 0, factor term then More increases, to decays maturity. to then 1βππ1βππ βππ/ππ 0. βππ/ππ 1βππ More decays π‘π‘ βππ/ππ π‘π‘ More ππ/πππ‘π‘ , has significant contribution to the value of , has at , significant has shorter βππ/ππ significant maturities contribution contribution (smaller to the values to value the of value of π¦π¦ of (ππ) at (ππ) short at s βππ/ππ β +β. Lastly, because 2π‘π‘ factor π½π½ its factor Moreover, is loading, referred loading, ititππdecreases to as β 1βππthe ππ because βππ/ππ βππ/ππ βππ/ππ medium-term βππ/ππ to , 00β starts π‘π‘starts as βππ/ππ its at π‘π‘at 0, factor0, increases, factor increases, loading, then Lastly, decays to π¦π¦ Data 0. is (ππ) More referred β ππ to π‘π‘ , starts at 0, increases, then decays to 0.π¦π¦ More inas asthe medium-term factor because its factor ππ/ππ loading, π‘π‘β , starts at 0, increases, then decays to 0. More 0 asππ). ππasππππππfactor. β βπ‘π‘+β. 0, π½π½ ππ/ππ 1βππ long-term The factor Moreover, decreases π‘π‘ tosignificant Lastly, π‘π‘is is called referred the to short-term theto medium-term factor because factorits factor loading, ππ). 2π‘π‘ ππβππ/ππ βππ/ππ π‘π‘+β. π½π½π½π½2π‘π‘ π‘π‘ π‘π‘ π‘π‘ roachesbecause its factor loading, ,,contribution starts at 0, increases, π‘π‘ then decays 0. 0. More π‘π‘ π‘π‘ + π‘π‘ because ππthen β its ππ/ππ0decays π‘π‘factorand to asloading, 0. βππ/πππ‘π‘ More +β. ππ/ππ precisely, Its β it approaches starts 0 atas increases, to ππ/ππππ/ππ 1π‘π‘ 2π‘π‘ the π‘π‘ππ/ππ yield then is decays to More is referred to0to precisely, asππitthe + it approaches medium-term ππ/ππ ππ/ππ 0π‘π‘ as ππsignificant + β 0 and contribution as ππ β +β. to ItstheHistorical significant data contributionof interest rates to theityield were initially is in 0 as ππ β 0+ an um term maturity. +factor π‘π‘ pproaches as β +0 and as ππβ β +β. ππIts yield is in precisely, approaches ππas proaches βππ/ππ isely, ππ β +βππ/ππ π‘π‘ , it +β. starts precisely, 0 approaches as Its at ππ 0, β precisely, significant increases, 0 + approaches 0 and as and it ππas as approaches 0β contribution as then ππ ππβ 0 +decays 0 0and +β.as andππ .as β asto Its Its to 0ππ 0. βthe +significant β significant and +β. More+β. yield as Its ππ βIts is significant contribution +β. in contribution significant Its contribution significant to contribution to the obtained the yield contribution π½π½ππMoreover, yield is to from in is to itthen in the thethe decreases yield website yield is is intoto in of00. Philippine as Dealing Lastly, and π½π½2π‘π‘ is π½π½2π‘π‘referred m0ππtoand β π‘π‘ ,as maturity. starts ππ because βππ). +β. at 0, TERM Moreover, Its increases, STRUCTURE its significant factor itloading, 6decreases then FORECASTING contribution decays 1βππ precisely, 1βππβππ/ππ 1βππ to+to +,0and βππ/ππ βππ/ππ π‘π‘ 0. has asβ π‘π‘ π‘π‘ to More ππthe itππsignificant approaches β βππ/ππ βππ/ππ+β. yieldπ‘π‘π‘π‘, starts is 0inππ). Lastly, contribution as0, at ππ). 2π‘π‘ βisMoreover, increases, 0referred + to and the asvalue to ππit β asdecreases decays 6 of the +β. medium-term (ππ) Its to at 0ππ as significant More shorter β ππ+β. βfactor +β. contribution maturities Lastly, (smaller to the is yield refe value precisely, precisely, the medium because to the the itit mediumyield term to maturity.its approaches approaches factor is in term the loading, 0 0 medium to asas ππ maturity.ππ β β ππ/πππ‘π‘ππ/ππ 0 term 0 and to β asas maturity. ππ ππ ππ β β +β., +β. starts Its Its at 0, significant significant Exchange increases, contribution contribution Corporation then decays or π¦π¦toto π‘π‘PDEx to the the 0. yield yield More (originally isis inin http://www. erm , increases, to maturity. then the The medium decays three termtofactors 0. to More maturity. arearealso ππ/πππ‘π‘π‘π‘ interpreted inisterms of slope,pdex.com.ph, the medium term to maturity. mndto medium asmaturity. ππ βterm +β.toIts maturity. significant The three contribution factors thealso medium to the interpreted 1βππ βππ/ππyield π‘π‘ in term terms inofπ‘π‘maturity. to level, and curvature ofbut the yield data now 1βππ βππ/ππ available 1βπππ‘π‘βππ/πππ‘π‘ βππ/ππβππ/ππ thru the website vel, and as the slope, and because curvature Itsterm of significantits the factor yield loading, contribution βππ/ππ , starts because at because 0, its increases, factorits factor loading, then loading, decays to 0. β More ππ β as ππisπ‘π‘ ,the starts π‘π‘ , starts at 0,atincreases, 0, increa ππthe βmedium +β. medium level, precisely, precisely, slope, term to to maturity. and ititapproaches approaches curvature maturity. 00as ππ). of as the Moreover,ππππβ βto0π‘π‘0the yield ππ/ππ + +and β curveityield and ππ as decreasesasππππisβ (Diebold βin+β. +β. to 0Its of assignificant Its the ππ βparent significant +β. Lastly, company contribution contribution π½π½2π‘π‘Philippine isππ/ππto toπ‘π‘ the referred theπ‘π‘yield ππ/ππ Dealing yield to isin Systems, in medium-term fact & Li, 2006). The factor b0t governs the 0π‘π‘level of the curve curve (Diebold & Li, 2006). The factor π½π½ governs the level of the http://www.pds.com.ph/ curve since lim π¦π¦ (ππ) = ). The rates for tenors at most ts of el significant the curve since contribution since lim π¦π¦π‘π‘ (ππ) to the = byield is in ππβ+β π‘π‘ and an increase in b increases one year were zero rates so0ππthese thethe medium precisely, medium π½π½0π‘π‘ and an ππβ+β term itterm increaseapproaches to toinmaturity. maturity. 0t increases 0 as ππallβ 0+equally. and 0t asThus, ππ βthis precisely, +β. 1βππ precisely, βππ/πππ‘π‘ Itsis responsible itβππ/ππ significant approaches itπ‘π‘approaches contribution 0 as asβto ππ0were + βtheand +used. The data 0yield as ππisas and βinππ+β. β +β. Its significa Its signi all yields equally. Thus, π½π½0π‘π‘ because this factor itsyields isfactor responsible loading, for ππ/ππ factor for tenors longer than one year, however,then β ππ ,for starts parallel at 0, increases, weredecays not zeroto 0. More π‘π‘ s, this factor is parallel responsible yieldfor curve parallel shifts. The π½π½1π‘π‘ isfactor b1ttheisyield related curvetoslope. rates so the corresponding zero rates were obtained the theyield yield medium curve curveterm shifts. slope. The tofactor Ifmaturity. the slope of the yield curve related to the Ifmedium the the medium slope of term the from+Bloomberg (2013). Note that the zero rates can yield term to maturity. curve to maturity. ve slope. If the slope of the as yield limcurve precisely, it approaches 0 as ππ β 0 and as ππ β +β. Its significant contribution to the yield is defined is defined as ππβ+β π¦π¦ (ππ) β lim π¦π¦ (ππ), then it is equal to βπ½π½ . Lastly, π½π½ is related to the π‘π‘ ππβ0+ π‘π‘ , then it is equal to 1π‘π‘ also2π‘π‘be computed using a method called bootstrapping. π½π½1π‘π‘ . Lastly, π½π½2π‘π‘ is curvature, related todefined the and the by Diebold medium Li as 2π¦π¦term to maturity. π‘π‘ (24) β π¦π¦π‘π‘ (3) β π¦π¦π‘π‘ (120), where maturity is given in months. β π¦π¦π‘π‘ (120), where maturity is given in Estimation of Parameters Nelson Nelson From Equation andand (3), NelsonSiegel Siegel and assumes assumes Siegel that that assumes the the forward that forward the rate rate forward follows follows rate thethe followsequation equation the equatio 1 ππ ππ (ππ) βππ/ππ π’π’π‘π‘ + βππ/ππ ππ ππβππ/ππ ππ π‘π‘ βππ/ π¦π¦π‘π‘ (ππ) = β« [π½π½π‘π‘0π‘π‘πππ‘π‘+(ππ) ππ= = π½π½ππ0π‘π‘βπ’π’/ππ (ππ) π½π½π‘π‘1π‘π‘ π½π½=0π‘π‘ +π½π½π½π½ + +π½π½+ 1π‘π‘ π‘π‘0π‘π‘ ππ1π‘π‘ ππ1π‘π‘ π½π½ π½π½2π‘π‘ ππ βππ/ππ π½π½ ππ193+2π‘π‘ π‘π‘π½π½ π‘π‘ βπ’π’/ππ + ] π‘π‘2π‘π‘ ππ2π‘π‘ππ βππ/ππ π½π½ππππ ππ π‘π‘ Philippine Interest Rates Using the Dynamic Nelson-Siegel Model ππ ππ ππ ππ 0 πππ‘π‘ π‘π‘ π‘π‘ π‘π‘ Evaluating where where whereπ½π½0π‘π‘π½π½,0π‘π‘π½π½π½π½ , 1π‘π‘π½π½,1π‘π‘ π½π½π½π½ ,the ,2π‘π‘ , 2π‘π‘1π‘π‘ π½π½ , and, and π½π½ ππare ππ, π‘π‘and π‘π‘ areππconstants π‘π‘ areconstantsconstants andand πππ‘π‘and ππβ π‘π‘ β 0. πππ‘π‘ 0.Note β Note 0. that Note thatπ₯π₯(ππ) π₯π₯(ππ) that =π₯π₯( PDEx is one of the two major exchanges in the the integral the other at 0π‘π‘ method right, 2π‘π‘we that obtain used a thefixed Nelson-Siegel shape parameter model for the yiel Philippines, the other being the Philippine Stock second-order across all second-order linear datasets, linear ordinary which ordinary will differential be referred differential to for the rest second-order linear ordinary 1 β ππ βππ/ππ π‘π‘ equation differential equationequation β ππ βππ/ππ To(1 π‘π‘ Exchange (PSE). TERMPDEx STRUCTURE is a venue for trading fixed- FORECASTING ofπ¦π¦the (ππ) paper= π½π½ as+fixed π½π½ ( lambda method. ) + π½π½ determine 5 β ππ βππ/πππ‘π‘ ) π‘π‘ 0π‘π‘ 1π‘π‘ 2π‘π‘ income and other securities, most of which are this fixed shape parameter, ππ/ππfirst, for2each lambda ππ/ππ 2 β² 1β² 1 1 π½π½0π‘π‘π½π½0π‘π‘π½π½0π‘π‘ in β²β² β²β² β² 2 π‘π‘ π‘π‘ government securities. From daily trading, PDEx the same range used in grid search, π₯π₯ β²β²π₯π₯+ π₯π₯+beta π₯π₯+π₯π₯+parameters+ π₯π₯ + π₯π₯ = π₯π₯ 2=π₯π₯2=2 2 (Nelson & Siegel, 1987). To generate this range of shapes, a parsimonious model πππ‘π‘ πππ‘π‘πππ‘π‘ 2πππ‘π‘ 2ππby πππ‘π‘ introduced π‘π‘ πππ‘π‘ πππ‘π‘ ππare calculates and publishes Philippine Dealing TheSystem parameters π½π½0π‘π‘ , π½π½1π‘π‘ , and π½π½2π‘π‘ are were estimated called factors andand corresponding their coefficients π‘π‘ calle Treasury Reference Rates such as PDS Treasury R were taken for each t. The average over all t was 2 Nelson From From Equation From Equation Equation (3),(3), (3), Reference Rate AM and Siegel assumes (PDST-R1) and PDS that loadings.the forward Treasury The factor rate then follows loading calculated. the of π½π½equation 0π‘π‘ is 1, a the Finally, constant shapethat does not that parameter decay to zero eve Reference Rate PM (PDST-R2). Both PDST-R1 and produced the highest ππ βππ/ππ average 1 1was ππ ππ chosen. Because the π’π’ π’π’ βππ/πππ‘π‘ 1 ππ (4) π‘π‘ it βπ’π’/ππ π’π’ π‘π‘ βπ’π’ PDST-R2 benchmarks are intended to become Thus, πππ‘π‘ (ππ) it has =theπ½π½0π‘π‘ significant + π½π½results1π‘π‘ ππ ofcontribution +theπ½π½2π‘π‘ fixed π¦π¦π‘π‘to(ππ) π¦π¦π‘π‘ππlambda (ππ) = π¦π¦the π‘π‘ = (ππ) yield method β« = β« [π½π½for [π½π½ β« 0π‘π‘ +any were[π½π½ 0π‘π‘ π½π½ + maturity; 1π‘π‘more π½π½ππ1π‘π‘βπ’π’/ππ + ππ1π‘π‘ π½π½ +π‘π‘hence, stable, βπ’π’/ππ π‘π‘ βπ’π’/ππ ππ π½π½ +2π‘π‘π½π½+ 2π‘π‘ π½π½ππ2π‘π‘isππ βπ’π’/ππ referr ππ] πππ‘π‘ πππ‘π‘ π‘π‘ ππ ππ ππ 0π‘π‘ ππ ππ ππ source of reference rates for the repricing of loans, the succeeding steps proceeded 0 0from 0 these. π‘π‘ π‘π‘ π‘π‘ securities, derivative transactions, and other long-term interestfactor. The It isfactor important π½π½1π‘π‘ is called to notethe that short-term the results factor because its factor lo in either where π½π½0π‘π‘ , π½π½1π‘π‘ , π½π½2π‘π‘ , and πππ‘π‘ are constants Evaluating and Evaluating Evaluating πππ‘π‘ the β the 0. Note integralintegral the that at the integral atorπ₯π₯(ππ) theright, at = right, right, the π‘π‘ (ππ) ππwe we is obtain obtain we a obtain solution thethe toNelson-Siegel Nelson-Siegel the Nelson-Siegel the model mode mo fo rate sensitive instruments to be issued. They are also method, grid search fixed lambda, were dependent βππ/πππ‘π‘ intended to become the bases for market1βππ valuation , has equation on the choice significant contribution of rangetoofthe values value ofoflπ¦π¦t.βππ/ππ second-order linear ordinary differential (ππ) π‘π‘ at βππ/ππ π‘π‘ shorter 1maturities βππ/ππ π‘π‘ (sm βππ/ππ of Government Securities and other Philippine-peso- ππ/ππ π‘π‘ Having π¦π¦chosen the fixed 1β shape 1 ππβ1 π‘π‘ ππ β parameter ππ βππ/ππ π‘π‘ and β1 ππβ1 ππβ ππ βππ/ π‘π‘ (ππ) π¦π¦ (ππ) = π‘π‘ π‘π‘ π¦π¦π‘π‘ 0π‘π‘= =π½π½ (ππ) π½π½ + π½π½ + 0π‘π‘π½π½0π‘π‘ π½π½ 1π‘π‘ + ( ( 1π‘π‘ π½π½1π‘π‘ ( ) + ) π½π½ + ) π½π½ 2π‘π‘ + ( ( 2π‘π‘π½π½2π‘π‘ ( β ππβ denominated fixed income securities. ππ/ππ ππ/ππ π‘π‘theπ‘π‘ second ππ/ππ π‘π‘ t,ππ/ππ ππ/ππ π‘π‘ ππ/ππ ππ). Moreover, β²β² 2itthe decreases β² resulting 1 to 0beta π½π½as parameters for each 0π‘π‘ ππ β +β. Lastly, π½π½ is referred (5) to as the π‘π‘ π‘π‘ medium In this work, TERMthe STRUCTURE PDST-R2 rates from Januaryπ₯π₯2, + step FORECASTING π₯π₯ +was2 π₯π₯to=fit a2 univariate time series model for 2π‘π‘ 5 The Theππparameters parameters πππ‘π‘ { π½π½ π½π½,},πππ½π½π‘π‘π½π½ π½π½,1π‘π‘ factorsfactors 2008 to December 25, 2013 were chosen. The PDST-R2 π‘π‘The each parameters of 0π‘π‘ 0π‘π‘ ,{1π‘π‘ 0π‘π‘ ,and , and }π½π½ π½π½ , 2π‘π‘ 1π‘π‘and and are π½π½2π‘π‘{ π½π½are called 2π‘π‘ }. called are called First, each and factors and series their andtheircoefficients their coefficiecoeffi 1βππ βππ/πππ‘π‘ rate is the weighted average of the yields because from done its factor was loading, tested for stationarity β ππ βππ/ππ π‘π‘ ,and starts was at 0, found increases, to have then decays to 0 transactions ofFrom (Nelson Equation of& (3), 1987). Siegel, To generate this range of shapes, ππ/πππ‘π‘ a parsimonious model introduced by the set benchmark securities for each loadings. loadings. loadings. a unit The The root.factor The factor loading factor Hence, loading of of loading each is π½π½1, π½π½0π‘π‘π½π½of0π‘π‘ series is 0π‘π‘1, was aisconstant atransformed 1,constant a constant that that does by does that not does notdecay decay not de to tenor up to 4:15 p.m. For simplicity, weeklyprecisely, data were differencing to remove ππ it approaches 0 as ππ β 0 and as ππ β +β. Its significant contribution to + the unit root. Each series also Nelson and Siegel assumes that1 the forward Thus, Thus, it rate has it has follows itsignificant significant π’π’equation thecontribution (6) extracted by taking the rates Wednesday of each π¦π¦π‘π‘ (ππ) = week. β« [π½π½ Thus, showed 0π‘π‘ + π½π½1π‘π‘ ππ βπ’π’/ππ has strong π‘π‘ + significant π½π½2π‘π‘presence ππcontribution contribution βπ’π’/ππ π‘π‘of ] ππππ ARCHto to thethe toyield effect.yield the for yield for The any any for bestmaturity; maturity; any maturity; hence, henc h Index t is attached to each week. ππ model obtained for ππ ππ each differenced series followed 0 π‘π‘ the medium term to maturity. (4)factor πππ‘π‘ (ππ) =long-term long-term π½π½0π‘π‘ π½π½1π‘π‘ factor. +long-term factor. ππ βππ/ππ ARMA(p,q)+eGARCH(1,1). The π‘π‘factor. + The π½π½ 2π‘π‘factor Thefactor is π½π½called π½π½1π‘π‘π½π½π‘π‘1π‘π‘ ππ βππ/ππ factor isThen 1π‘π‘called thethe is called short-term error short-term the short-term terms factor of factor because because beca it Evaluating the integral at the right, we obtain the Nelson-Siegel πππ‘π‘ model for the yield curve Procedure the estimated models for the betas were considered as βππ/ππ βππ/ππ βππ/ππ In using the Nelson-Siegel model to forecast the 1βππ 1βππ aπ‘π‘ ,random 1βππ π‘π‘ has , has π‘π‘ significant vector whose significant contribution joint probability contribution distribution where π½π½0π‘π‘ , π½π½1π‘π‘ , π½π½2π‘π‘ , and πππ‘π‘ are constants 1 β ππ/ππππ and βππ/ππ ππ/ππ ππ/ππ π‘π‘πππ‘π‘ ,β has 0. significant Note 1was βthat ππ βππ/ππ contribution π₯π₯(ππ) π‘π‘ = ππtoπ‘π‘ (ππ) to thethe to isvalue value athe solutionof of value π¦π¦π‘π‘ (ππ) π¦π¦of to (7) atπ‘π‘ (ππ) π‘π‘ (ππ) π¦π¦ the shorter at shorter at shorte matu m term structure, three major steps were implemented π¦π¦π‘π‘ (ππ) = π½π½0π‘π‘ + π½π½1π‘π‘ ( π‘π‘ function π‘π‘ π‘π‘ ) + π½π½2π‘π‘ ( (pdf) to be determined β ππ βππ/ππ π‘π‘ ) by the copula in this work. The first step was to run multiple linear ππ/ππmethod. π‘π‘ Finally, random ππ/πππ‘π‘ numbers from the joint pdf second-order linear ordinary differential ππ).ππ). equation Moreover, Moreover, it decreases it decreases to to 0 as 0toas ππ0β ππasβ +β. ππ +β. Lastly,Lastly, π½π½2π‘π‘π½π½2π‘π‘is π½π½referred is2π‘π‘referred to to as as th regression in the form of Equation (8) for each dataset of Moreover, ππ). the error terms it decreases were generated. β These +β. Lastly, were then is referred to t to obtain the The beta parameters π½π½0π‘π‘ , π½π½1π‘π‘ , and π½π½2π‘π‘ .are This called 2used factors in and the final 1 their π½π½0π‘π‘step, coefficients forecasting βππ/ππ βππ/ππ are called π‘π‘ βππ/ππ the term factor structure (5) β²β² β² 1βππ1βππ 1βππ π‘π‘ π‘π‘ βππ/ππ βππ/ππ step required a prior choice of value or values of the π₯π₯because because ofπ₯π₯its + because interestits + factor factor π₯π₯loading, its2 rates. factor=loading, loading, 2 β ππ β ππβ π‘π‘ , βππ/ππ ππ starts π‘π‘ , starts at 0, π‘π‘ , starts at 0, increases, atincreases, 0, increases, then then dt πππ‘π‘ ππthat π‘π‘ doesπππ‘π‘not decay to zero ππ/ππ ππ/ππ π‘π‘ π‘π‘ππ/πππ‘π‘ shape parameter loadings.lt. TheThe nextfactor steploadingwas to of π½π½0π‘π‘ is a1, a constant assume even as ππ β +β. model for the time series of beta parameters obtained. precisely, precisely, precisely, it approaches it approaches it approaches 0 as 0 as ππ0β ππasβ0ππ+0β + + and 0and asand as ππ β ππasβ +β. β Its ππ +β. +β. Its significant significant Its significant contrco From Thus, itEquation has to (3), significant contribution to the yield for any maturity; hence, it is referred to as the This model was then used forecast future values Implementation and Results of these parameters. The last step was to plug in the 1 ππthethe medium medium the medium term term to term to maturity. maturity. toπ’π’maturity. (6) future valueslong-term of the beta factor. The factor parameters π¦π¦and π‘π‘ π½π½ (ππ) the = 1π‘π‘ is called same β« [π½π½ the 0π‘π‘ + short-term Fittingπ½π½ 1π‘π‘ ππ the βπ’π’/ππ factor π‘π‘Yield + π½π½ 2π‘π‘ because Curve ππ βπ’π’/πππ‘π‘its ] ππππ factor loading, shape parameter in Equation (7) to obtain forecast ππ 0of A total of 313 datasets πππ‘π‘ of zero rates from January 2, βππ/ππ interest rates y1βππ (t). π‘π‘ , has significant contribution to the 2008 of value to π¦π¦ December 25, 2013 were considered. Each t π‘π‘ (ππ) at shorter maturities (smaller values of For the firstEvaluating step, ππ/πππ‘π‘ the integral regression usingata the fixed right, shape we obtain dataset the Nelson-Siegelgave the interest model for thein rates yield hundred curve basis parameter across all datasets and using a grid search points for maturities 1 month (Mo), 3 Mo, 6 Mo, 1 ππ). Moreover, were both considered it decreases initially. Forπ¦π¦ the grid to search, 0 as ππ β the 1 β Lastly, +β. βππ/πππ‘π‘ π½π½ is referred βππ/ππ ππyear 13Yr,β ππ 4Yr, to asπ‘π‘ the βππ/ππ medium-term factor(7) (Yr), ) + π½π½2Yr, 2π‘π‘ 5Yr, β ππ 6Yr, π‘π‘ ) 7Yr, 8Yr, 9Yr, π‘π‘ (ππ) = π½π½0π‘π‘ + π½π½1π‘π‘ ( 2π‘π‘ ( values of lt ranged from 2.000000 to 40.000000 with ππ/ππ 10Yr, π‘π‘ ππ/ππ 15Yr, 20Yr, andπ‘π‘30Yr. Figure 1 shows sample 1βππ βππ/πππ‘π‘ increments ofbecause0.000001 for each its factor loading,t. The lt together β ππ βππ/πππ‘π‘ , starts yield at 0, increases, then decays to 0. More curves. ππ/πππ‘π‘ with the set of betaThe parameters parameters π½π½0π‘π‘ , π½π½1π‘π‘ , and π½π½ 2π‘π‘ that are called factors Before and their coefficients performing linear regression, are called allfactorrates were highest R2itfor produced the precisely, each t were approaches 0 as ππ β 0+ and taken. It was as ππfirst β converted +β. Its significant to continuously contribution compounding to the yield rates. is For in noted that quiteloadings. a number The factor loading of datasets had veryofhigh π½π½0π‘π‘ Ris2 1, a constant the processing that does of data, not decay from estimation to zero even to as time ππ β series+β. but the beta parameters the mediumobtained were not realistic, term to maturity. modelling of the beta parameters, only the rates from that is, the values Thus, were it has verysignificant far from the contribution trend in the to the Januaryyield for 2, any 2008 maturity; to December hence, it26, is referred 2012 were to asused. the majority of datasets. This problem did not appear in Thus, 261 datasets were considered. The remaining long-term factor. The factor π½π½1π‘π‘ is called the short-term factor because its factor loading, 1βππ βππ/πππ‘π‘ ππ/πππ‘π‘ , has significant contribution to the value of π¦π¦π‘π‘ (ππ) at shorter maturities (smaller values of justified justifiedbybyKPSS KPSStesttest(for (forthe theactual actualseries) series)and andARCH-LM ARCH-LMtest test(for (forthe thedifferenced difference All statistical procedures were implemented using the software R. First, univariate time giving all significant p-values. These are s giving giving all allsignificant significant p-values. p-values. These These 2 andare arestandard standard ofprocedures procedures inindetecting detectingthese thesefeat fea RM STRUCTUREseries models were fitted on the FORECASTING estimated betas. As seen in Figures 3, presence non- financialE.P. timede series. Lara-Tuprio, et al 10 194 stationarity and of conditionalfinancial heteroscedasticity financial time were apparent. These observations were timeseries. series. justified by KPSS test (for the actual series) and ARCH-LM test (for the differenced series), giving all significant p-values. These are standard procedures in detecting these features of most financial time series. Figure 2. Time serie Figure Figure2.2.Time Timeseries seriesplots plotsofofthe theestimated estimatedbetas. betas. Figure 2. Time series plots of the estimated betas. Figure 1.Figure Actual1. Actual yield curves for selected dates. yield curves for selected dates. Table 1. Minimum, Maximum, Mean Values, and Standard Deviations of R2 and Beta Parameters Min max mean standard deviation Before performing linear R regression, 80.26% all rates 99.72% were first 3.04% 96.22% converted to continuously 2 Figure 3. Time serie Ξ²0 0.02242 1.49527 0.17384 0.18941 Figure Figure3.3.Time Timeseries seriesplots plotsofofthe thedifferenced differencedbetas betas. . mpounding rates. For theΞ²processing -1.41810 of 0.01044 data, from estimation0.17833 -0.13947 to time series modelling of the 1 Figure 3. Time series plots of the differenced betas. Ξ²2 -1.80766 0.24361 -0.04411 0.25549 a parameters, only the rates from January 2, 2008 to December 26,The 2012 best were models used. Thus, for all three 2 differ TheThebest bestmodels modelsfor forall The allbest three three models differenced differenced for all three series seriesdifferenced πππ½π½ πππ½π½ππ ,ππ ,ππ ππ==0,1,2, serieswere 0,1,2, wereARMA ARMA historical data were to be used to check the reliability The best models for all three differenced series πππ½π½ππ , ππ = 0,1,2,, were wereARMA equation ARMA forfor and the the mean Exponential mean equation GARCH for the of the forecast. asets were considered. Theequation fixed and lambda The method remaining equation equationand described historical andExponential Exponential anddata GARCH GARCH Exponentialwerefor forto the GARCH the be variance used variance for the tovariance equation. equation. check In the reliability Inparticular, particular, equation. based basedon on Exponential GARCH for theearliervariance equation. In particular, In particular, based on the Bayesian Information basedBayesian on the Information Criterion (BIC), bot produced lt = 9.807527. To summarize the results of 2 Bayesian Bayesian Information Information Criterion Criterion Criterion TERM STRUCTURE (BIC), (BIC),FORECASTING (BIC), both bothboth{πππ½π½ {πππ½π½00}}and and{πππ½π½ and {πππ½π½11}}followed followed followed MA(1)+eGA MA(1)+eGA of R and(BIC), forecast. linear regression, of Equation (8)TERM the valuesCriterion Bayesian Information for the different datasets STRUCTURE TERMare FORECASTING beta parameters both {πππ½π½ STRUCTURE TERM0 } and describedFORECASTING {πππ½π½ STRUCTURE 1 } followed MA(1)+eGARCH(1,1) FORECASTING MA(1)+eGARCH(1,1) models, models, while while {πππ½π½ 2 } followed followed 12 ARMA(1,1 in Table 1. models, while {πππ½π½2 } followedmodels, models, while while{πππ½π½ ARMA(1,1)+eGARCH(1,1). {πππ½π½22}}followed ARMA(1,1)+eGARCH(1,1). followed ARMA(1,1)+eGARCH(1,1). ARMA(1,1)+eGARCH(1,1). TERM STRUCTURE FORECASTING The Theestimated Theestimated estimated models models weremodels given were were by the given given by by following the the following 12 equations. equatio Let πππ½π½ Forecasting of interest rates requires forecast of The fixed lambda method The estimated described the beta parameters. Thus, the next step was to find models The were in estimated givenSection models by the following III were following produced given equations. by the following equations. ππ Let Let π‘π‘ = πππ½π½ 9.807527. equations. ππ,π‘π‘ = π½π½ ππ,π‘π‘ β Let π½π½ πππ½π½ ππ,π‘π‘β1 To , ππ,π‘π‘ = π½π½ ππ,π‘π‘ β π½π½ππ,π‘π‘β appropriate time series models for them. models were ππ = 0,1,2, ππ =π‘π‘ = 0,1,2,1, 2,π‘π‘β¦=, ππ, 1, where 2, β¦ , ππ, T is where wherethe sample TT is thesize. sample Thensize. size. Then The estimated ππ = 0,1,2, ππ = 0,1,2, π‘π‘ = 1, 2, β¦ , ππ, where T is the sample π‘π‘ = 1, 2, given β¦ , ππ, by where Thensize. Thenthe Tfollowing is the equations. sample size. Let Then πππ½π½ ππ,π‘π‘ = π½π½ ππ,π‘π‘ β π½π½ ππ,π‘π‘β1 , 2 mmarize the results of linear Time Series Models for0,1,2, the Beta regression, Parameters the values of π
π
andπππ½π½beta ππ,π‘π‘ = πππππ½π½ parameters ππ +ππ,π‘π‘ππ= ππ πππ½π½ππππ,π‘π‘β1 ππ + ππ +ππ πππ½π½ of ππππ,π‘π‘ +Equation ππ,π‘π‘β1 ππ+ ππππ,π‘π‘ + ππππ ππππ,π‘π‘ ππ ππππ,π‘π‘β1 (8 ππ = π‘π‘ = 1, 2, β¦ , ππ, where T is the sample πππ½π½ size. = Then ππ + ππ πππ½π½ + ππ + ππ ππ (9) (9) All statistical procedures were implemented πππ½π½ππ,π‘π‘ using = ππππ + ππππ πππ½π½ππ,π‘π‘β1 + ππππ,π‘π‘ + ππππ ππππ,π‘π‘β1 ππ,π‘π‘ ππ ππ ππ,π‘π‘β1 ππ,π‘π‘ ππ ππ,π‘π‘β1 ππππ,π‘π‘ = ππππ,π‘π‘ ππππ,π‘π‘ ππ = ππ ππ ππ,π‘π‘ ππ,π‘π‘ ππ = ππ ππ ππππ,π‘π‘ = ππππ,π‘π‘ ππππ,π‘π‘ (10) (10) the software R. First, univariate time series models the different datasets are described in Table πππ½π½ππ,π‘π‘ = 1. ππππ +ππ,π‘π‘ππππ πππ½π½ππ,π‘π‘β1 ππ,π‘π‘ ππ,π‘π‘+ ππ + ππ ππ ln(ππ 2 ππ,π‘π‘ 2 ππ ππ,π‘π‘β1 ln(ππ ππππππ,π‘π‘+)|ππ 2 πΌπΌ=ππππ,π‘π‘β1 ππππ ππ |++ πΌπΌ 2 |ππππ,π‘π‘β1 | πππΎπΎππππ ππ,π‘π‘β1 (ππππ,π‘π‘β1 + πΎπΎππ β |ππππ,π‘π‘β1 ( πΈπΈ (| ππ(9) (11) | ππ,π‘π‘β1 β2|)) ππππ,π‘π‘β1 πΈπΈ (|+ π
π
ππ ln | were fitted on the estimated betas. As seen in Figures2 ππ |ππ = ππ| 2 ππ)ππ,π‘π‘= ππ,π‘π‘ ππ ππ,π‘π‘β1 (11) (10) 2 ln(ππ 2 ) = ππ + ππ,π‘π‘ πΌπΌππ ππππ,π‘π‘β1 + πΎπΎππππ,π‘π‘β1 ππ,π‘π‘β1 ππ,π‘π‘ ( β πΈπΈ (| ππ2ππ,π‘π‘β1 |)) +πππ
π
ππ,π‘π‘β1 ππππ,π‘π‘β1ππ,π‘π‘ ) ππππ,π‘π‘β1 ππ ln(ππ 2 and 3, presence of non-stationarity ln(ππππ,π‘π‘ )and= ππofππ conditional + πΌπΌππ ππππ,π‘π‘β1 + πΎπΎππ ( ππ ππ,π‘π‘ β πΈπΈ (| |)) ππππ,π‘π‘β1 + π
π
ππ ln(ππ ππππ,π‘π‘β1 ππ,π‘π‘ ) (11) ππππ,π‘π‘β1 |ππππ,π‘π‘β1 |ππ ππππ,π‘π‘β1 ππππ,π‘π‘β1 Tablewere1 justified by KPSS test (for ln(ππ heteroscedasticity were apparent. These2 observations ππ,π‘π‘ ) = ππππ + ππ πΌπΌππ ππ 2 ππ,π‘π‘β1 + where πΈπΈwhere ππππ,π‘π‘β1 πΎπΎ(|ππ ( πΈπΈ|)(|= β πΈπΈ|) πΈπΈ(|ππ ππ,π‘π‘β1 (| ππ,π‘π‘=|) =0 ππ πΈπΈ(|ππ |)) when |)+= π
π
β ππ 0ln(ππ ππ when 2 ππππ,π‘π‘ )β2 . The ππ,π‘π‘ ~π‘π‘ ππ coefficients ππππ,π‘π‘ ~π‘π‘ 2 . The coeffi and ππππ,π‘π‘β1 thewhere actualπΈπΈseries) (|ππππ,π‘π‘β1and |) = πΈπΈ(|ππ |)ππ= ππππ,π‘π‘β1 ππ ππ,π‘π‘ ππ,π‘π‘β10 whenππππ,π‘π‘β1 β ππππ,π‘π‘β1 ππ ππ,π‘π‘ ππ,π‘π‘ ~π‘π‘ 2 . The ππβ2 coefficients ππβ2 and corresponding Minimum, ARCH-LM Maximum, where test πΈπΈ (| (for the ππMean differenced |) = πΈπΈ(|ππ ππ,π‘π‘β1ππ Values, series), ππ,π‘π‘ |)giving 0and =ππ,π‘π‘β1 all Standard when βππβ2 ππππ,π‘π‘ ~π‘π‘2 . Deviations ππ The coefficients ππβ2 ofandR2 correspondingand Beta Paramete significant p-values.where TheseπΈπΈare ππ,π‘π‘β1 (| standard |) = πΈπΈ(|ππ procedures |) = standard 0inwhenstandard βerrorsππππ,π‘π‘ are ~π‘π‘given errors 2 . The arein coefficients Table 2. given in Table and 2. corresponding ππππ,π‘π‘β1 standard ππ,π‘π‘ errors are given in Table 2. detecting thesestandard Min features of most financial errors are given in Table 2.max time series. mean ππβ2 standard deviation Table 2 Table 2 2 standard errors are givenTable in Table 2 96.22% 2. R 80.26% Table 2 99.72% Estimated Parameters 3.04% of ARMA +ofeGARCH Estimated Parameters ofEstimated ARMA + eGARCH Parameters Models and ARMA +Models Corresponding eGARCH andModels Corresponding Standardand Errors Corres Sta Table 2Parameters of ARMA + eGARCH Models and Corresponding Standard Errors Ξ²0 0.02242Estimated 1.49527 0.17384 i=0 0.18941 i=1 i=0 i=1 i=2 i=2 Estimated Parameters of ARMA + eGARCH i=0 Models and i=1Corresponding i=2 Standard Errors galalltime series. p-values. These are standard procedures in detecting these features of m significant ial time series. Philippine Interest Rates Using the Dynamic Nelson-Siegel Model 195 Figure 2. Time series plots of the estimated betas. Figure 2. Time series plots of the estimated betas. Figure 2. Time series plots of the estimated betas. TERM STRUCTURE FORECASTING i=2 0.00144197 0.02802909 TERM STRUCTURE TERM FORECASTING STRUCTURE FORECASTING 1212 The assumed distribution of the error ππππ,π‘π‘ was the t distribution with 2 degrees of The Theestimated models estimated were models given were byby given thethe following equations. following Let equations. πππ½π½πππ½π½ Let == ππ,π‘π‘ ππ,π‘π‘ π½π½ππ,π‘π‘π½π½ππ,π‘π‘ ββ π½π½ππ,π‘π‘β1 , , π½π½ππ,π‘π‘β1 or π‘π‘2 . Such distribution captures the heavy-tailedness of the series since it has infinite va ππ = 0,1,2, ππ = π‘π‘ = 0,1,2, 1, 1, π‘π‘ = 2, 2, β¦β¦ where , ππ,, ππ, TT where is is thethe sample size. sample Then size. Then In generating future values of the beta parameters using the formulas in Equation πππ½π½πππ½π½ == ππ,π‘π‘ ππ,π‘π‘ ππππππ+ππ +πππππππππ½π½ ππ πππ½π½ ππ,π‘π‘β1 ππ,π‘π‘β1 ++ ππππ,π‘π‘ππππ,π‘π‘ ++ ππππ ππ ππππππ,π‘π‘β1 ππππ,π‘π‘β1 (9)(9) (11), their underlying ππππ,π‘π‘ππ = = ππππ,π‘π‘ππππππ,π‘π‘ππdependence structure based on their (10) observed values were consider (10) Figure 3. Timeππ,π‘π‘seriesππ,π‘π‘plots ππ,π‘π‘ of the differenced betas. (11) |ππ|ππ | | ππππ,π‘π‘β1 (11) ππ,π‘π‘β1 ππ,π‘π‘β1 ππππ,π‘π‘β1 Figure 3. Time series plots of the differenced betas 2 2 2 2 2 2 ln(ππ ln(ππ ) ππ,π‘π‘ ππ,π‘π‘=) ππ = ππ ππ +ππ required πΌπΌ + ππ ππ πΌπΌ ππ ππππ,π‘π‘β1 + ππ,π‘π‘β1 estimating +πΎπΎππ ( πΎπΎ ( ππ ππ the β βπΈπΈdistribution (| πΈπΈ (| ππππ,π‘π‘β1 |)) |))of + +the π
π
ππ π
π
ππ random ln(ππln(ππ ) ππ,π‘π‘ ππ,π‘π‘ ) vector (ππ.0 , ππ1 , ππ2 ) consisting of the e ππππ,π‘π‘β1 ππ,π‘π‘β1 ππππ,π‘π‘β1 where ππ ππππ,π‘π‘β1 variables. 0 when Asππassumed ππ in thein time the series time series models, models, the thedistribution ofeach distribution of eacherror error variable w where (|(|ππ,π‘π‘β1 πΈπΈ πΈπΈ |)|)== πΈπΈ(|ππ ππ,π‘π‘ |) πΈπΈ(|ππ == ππ,π‘π‘ |) 0 when β βππβ2 ππππ,π‘π‘ππ~π‘π‘ 2 . 2The ππ,π‘π‘ ~π‘π‘ . The coefficients coefficients andandcorresponding corresponding where when Figure 3. Time series plots of the differenced betas ππππ,π‘π‘β1 ππππ,π‘π‘β1 ππβ2 variable was t2. The dependence structure . of (e0 , e1 , The coefficients and corresponding standard errors The dependence structure of (ππ , ππ , ππ ) was obtained using a copula function. e2 ) was 0 obtained 1 2 using a copula function. standard are givenerrors standard inerrors Table areare given 2. giveninin Table Table2. 2. With T as the sample size, the Copula estimatedfor initial The best models for all three differenced series πππ½π½ , ππ = 0,1,2, were ARMA for the me values to Table Table be used 2 2 in forecasting Using are given in Table 3. the Joint Density Using ππ of the Copula Error for the JointVariables Density of the Error Variables TheEstimated assumed distribution Parameters ofof of the ARMA error ei,t was + eGARCH Models the and Corresponding Standard Estimated Parameters ARMAFirst, pseudo-observations + eGARCH Models and defined First, Corresponding by Errors pseudo-observations Standard Errors defined by t distribution with 2 degrees of freedom or t2. Such onThe andbest Exponential GARCH models for fordifferenced all three the variance distribution captures the i=0 equation. series πππ½π½ , ππ In = particular, 0,1,2, werebased heavy-tailedness i=0 ARMA on the i=1for the me ofi=1the series i=2i=2 ππ π
π
ππ since it has ΞΌi ΞΌinfinite variance. -0.000385 -0.000385 0.000386 0.000386 -0.000247 -0.000247 ππππ = i ππ + 1 (12) In generating future(0.000480) values (0.000480) of the beta(0.000426) parameters (0.000426) (0.000409) (0.000409) an on Information Criterion and Exponential GARCH(BIC), forboth using theΟformulas {πππ½π½ } and the variance i Οi {πππ½π½ } followed equation. MA(1)+eGARCH(1,1) In particular, in Equations based on the 0 0 (9) to (11), their 0 00 -0.152738 1 -0.152738 underlying dependence structure were created, based whereon their T is the sample were (0.043782) size and π
π
is the rank of the ππ π‘π‘β observation. These created, where Tππ is the sample size and Ri is the (0.043782) observedΞΈivalues were-0.215126 considered. This required rank of the ith observation. These pseudo-observations s,ian while {πππ½π½ } followed Information estimating 2 ARMA(1,1)+eGARCH(1,1). Criterion ΞΈi the distribution -0.215126 (0.053564) (0.053564) -0.199544 -0.199544 (BIC), both {πππ½π½ } and {πππ½π½ } followed MA(1)+eGARCH(1,1) ofobservations were used the random(0.050340) vector (0.050340) -0.099143 -0.099143 (e0 ,0 to (0.039429) check were the association used (0.039429) to1 check theamong the error association among variables the errorand in estimat e1 , e2 ) consisting of the error variables. As assumed variables and in estimating the parameters of the ΟiΟ -0.095836 -0.095836 -0.105876 -0.105876 -0.079393 -0.079393 i (0.036828) parameters(0.049583) of the copula model. The pairwise scatterplots of the pseudo-observations are (0.051335) (0.036828) (0.049583) (0.051335) s, while {πππ½π½2 } Ξ±followed Ξ± ARMA(1,1)+eGARCH(1,1). 0.390405 0.390405 -0.276128 -0.276128 -0.316096 -0.316096 i i (0.207581) in Figure 4.(0.188705) (0.206423) (0.207581) (0.188705) (0.206423) Ξ³i Ξ³i 0.537282 0.537282 0.626099 0.626099 0.558905 0.558905 196 E.P. de Lara-Tuprio, et al Table 2. Estimated Parameters of ARMA + eGARCH Models and Corresponding Standard Errors i=0 i=1 i=2 -0.000385 0.000386 -0.000247 ΞΌi (0.000480) (0.000426) (0.000409) 0 0 -0.152738 ΟiΒ Β Β (0.043782) -0.215126 -0.199544 -0.099143 ΞΈiΒ (0.053564) (0.050340) (0.039429) -0.095836 -0.105876 -0.079393 ΟiΒ (0.036828) (0.049583) (0.051335) 0.390405 -0.276128 -0.316096 Ξ±iΒ (0.207581) (0.188705) (0.206423) 0.537282 0.626099 0.558905 Ξ³i (0.277760) (0.343980) (0.360170) 0.984873 0.983536 0.983487 ΞΊiΒ (0.006315) (0.008814) (0.008566) Table 3. Estimated Initial Values Based on the Fitted ARMA + eGARCH Models Β ai,T Οi,T i=0 -0.00492075 0.01729929 i=1 0.00121345 0.01571196 i=2 0.00144197 0.02802909 copula model. The pairwise scatterplots of the pseudo- was based on maximum pseudo-likelihood estimation observations are shown in Figure 4. proposed by Genest, Ghoudi, and Rivest (1995) and Based on the scatterplots of the pseudo- the goodness of fit test was based on the empirical observations, the first error variable had strong copula proposed by Genest, RΓ©millard, and Beaudoin negative association to the second and third error (2009). The results are shown in Table 4. Note that ri,j variables, while the second and the third had strong is the Pearsonβs correlation coefficient of the ith and jth pseudo-observations. positive correlation. Notice that the associations among the error variables showed a symmetric- Table 4. Results of the Goodness-of-Fit Test Under type dependence. That is, the dependence on the the Gaussian Copula Assumption tails was almost the same as the distributions on the middle part of the distributions. One model that Estimated parameter Sn p-value captures such symmetry in association between two uniform random variables, as depicted by the Ο0,1= -0.981 0.0047 0.9296 scatterplots, is the elliptical family. This family Ο0,2= -0.959 Β Β contains the Gaussian and t copulas. Ο1,2= 0.909 Β Β Estimation and goodness of fit test were performed for the Gaussian copulas. The estimation procedure s of the copula model. The pairwise scatterplots of the pseudo-observations positive correlation. Notice that the associations among the error variables showed a tric-type dependence. That is, the dependence on the tails was almost the same as the 4.tions on the middle Philippine Interest Rates Using the Dynamic Nelson-Siegel Model part of the distributions. One model that captures such symmetry in 197 tion between two uniform random variables, as depicted by the scatterplots, is the al family. This family contains the Gaussian and t copulas. Estimation and goodness of fit test were performed for the Gaussian copulas. The ion procedure was based on maximum pseudo-likelihood estimation proposed by Genest, , and Rivest (1995) and the goodness of fit test was based on the empirical copula ed by Genest, RΓ©millard, and Beaudoin (2009). The results are shown in Table 4. Note is the Pearsonβs correlation coefficient of the ππ π‘π‘β and ππ π‘π‘β pseudo-observations. Table 4 Results of the Goodness-of-Fit Test Under the Gaussian Copula Assumption Estimated parameter Sn p-value Ο0,1= -0.981 0.0047 0.9296 Ο0,2= -0.959 Figure 4 . Pairwise scatterplots of the pseudo-observations. Figure 4 . Pairwise scatterplots of the pseudo-observations. Ο1,2= 0.909 With p-value of 0.9296, it seemed that the Gaussian Forecasting sed on the scatterplots copula was adequate to captureof the the pseudo-observations, dependence of the thea fraction To validate the model, With p-value of 0.9296, it seemed that the Gaussian copula was adequate to capture the first oferror variable ha the available pseudo-observations. The estimated correlation matrix data, from January 2 to December 25, 2013 (52 was given by ence of the pseudo-observations. The estimated correlation matrix was given by Wednesdays), was taken out. The goal was to compare ssociation to the second and third error variables, while the second and the 1 β0.981 β0.959 the forecasts and the actual data taken out. First, 10,000 triples of random numbers (e0,1 , e1,2 , (β0.981 1 0.909 ) e2,1 ) from the joint distribution of the random vector β0.959 0.909 1 (e0 , e1 , e2 ) of the residuals were produced. Each triple ERM STRUCTURE FORECASTING was substituted into the time 15series model to obtain the stimated parameters were then used to simulate trivariate data with π‘π‘2 margin and normal These estimated parameters were then used to simulate values of b01 , b11 , b21 . Thus, 10,000 triples (b01, b11 , trivariate data with t2 margin and normal copula to to capture the dependence. b21 ) were obtained. Then the median of each beta was From Equations capture the(9) to (11) and the random numbers15generated dependence. taken from the distribution to obtain one triple (bof01 , b11 , b21 ). This median From Equations (9) to (11) and the random numbers triple was then used in Equation (7) to get the yield he random vector generated (ππ0 , from 2 ), future ππ1 , ππthe distributionvalues of ofthethe beta parameters random vector wereofcomputed curve and used week 1 (January in 2, 2013), y1 (t), where t is ) and the random (e0 , enumbers 1 generated , e2 ), future values of from the distribution the beta parameters wereof the maturity. he Nelson-Siegel computed equation and used in the Nelson-Siegel equation For the second week (t = 2),10,000 triples of random uture values of the beta parameters were computed and used in numbers (e0,2 , e1,2 , e2,2 ) from the joint distribution of 1 β ππ βππ/πππ‘π‘ 1 β ππ βππ/ππthe π‘π‘ random vector (e0 , e1 , (13) e2 ) were again produced. βππ/πππ‘π‘ π¦π¦π‘π‘ (ππ) = π½π½0π‘π‘ + π½π½1π‘π‘ ( ) + π½π½2π‘π‘ ( β ππ ) ππ/πππ‘π‘ ππ/ππ These and the betas of previous weeks were plugged (13) π‘π‘ in the time series model to obtain 10,000 triples (b , 02 1 β ππ βππ/πππ‘π‘ 1 β ππ βππ/πππ‘π‘ (13) b12 , b22 ). Again, the median of each beta was taken to with( πππ‘π‘ = 9.807527. ) + π½π½2π‘π‘ The( initial values β ππwere βππ/πππ‘π‘ those obtained for the December 26, 2012 dataset: ) get one triple (b02 , b12 , b22 ). This median triple was 1π‘π‘ ππ/πππ‘π‘ ππ/πππ‘π‘ used in the Nelson-Siegel equation to get the yield π½π½0,0 = 0.042670232 , π½π½0,1 = 0.035735575, and π½π½0,2 = 0.107586694. curve yThe (t)randomness of the 9, 2013). This process for week 2 (January 2 values were those obtained for the December with lt = 9.807527. The initial values were those 26, 2012 dataset: was continued to find the yield curve for week t, t > eta parameters were due to the residuals ππππ,π‘π‘ , ππ = 0,1,2. obtained for the December 26, 2012 dataset: b 0,0 2. Notice that medians of beta parameters were taken 35735575, and π½π½0,2 = 0.107586694. = 0.042670232, The randomness b 0,1 = 0.035735575, and ofb 0,2the = first before computing the yield yt (t). An alternative Forecasting 0.0107586694. The randomness of the beta parameters approach is to compute for each t the yield yt (t) using were due esiduals ππππ,π‘π‘ , ππ = 0,1,2. to the residuals e i,t ., i = 0,1,2. each of the 10,000 triples (e0,t , e1,t , e2,t ) then take To validate the model, a fraction of the available data, from January 2 to December 25, 013 (52 Wednesdays), was taken out. The goal was to compare the forecasts and the actual data action of the available data, from January 2 to December 25, he actual data 5fortocomparison. Figures 8 present sample forecast of interest rates per tenor and per week along with E.P. de Lara-Tuprio, et al he actual data 198 for comparison. Figure 5. Forecast for 3-month tenor. Figure 5. Forecast for 3-month tenor. Figure 5. Forecast for 3-month tenor. Figure 6. Forecast for 5-year tenor. ng Head: TERM STRUCTURE FORECASTING Figure 6. Forecast for 5-year tenor. Figure 6. Forecast for 5-year tenor. Figure 7. Yield curve forecast for week 1. Figure 7. Yield curve forecast for week 1. ing Head:Philippine TERMInterest STRUCTURE FORECASTING Rates Using the Dynamic Nelson-Siegel Model 199 FigureFigure 7. Yield curve 8. Yield forecast curve forecast for for week 9. week 1. the median yt (t) for each t. This, however, will still Forecast of interest rates was based on the resort to taking a single triple of beta parameters assumption that yield curve would follow the Nelson- from previous weeks; otherwise, the process will be Siegel equation with the same shape parameter as computationally expensive. produced from historical data and with beta parameters Figures 5 to 8 present sample forecast of interest generated from the time series model. The dependence rates per tenor and per week along with the actual data structure of the beta parameters was considered in for comparison. generating their future values. This was carried out by It was observed that the forecasting accuracy is finding the joint distribution of the error variables via weakened as one goes farther into the future. This is appropriate copula. a limitation of any time series models. Thus, instead Results showed that forecast of interest rates for of 1-year forecast, only 12 weeks were considered. different tenors, short, medium or long, was relatively The root mean square errors (RMSE) per tenor and good up to the next three months. From then on, the per week are shown in Tables 5 and 6. In can be seen accuracy weakened. This showed that the model, that the best forecast is for 5-year yield, with only 6 wherein parameters were based on historical data, Figure 8. Yield curve forecast for week 9. bps RMSE, and the least accurate is for 10-year yield could be reliable only for the near future. For an active with 27 bps RMSE. market, this is good enough since the models for the parameters can be adjusted every trading day. As mentioned earlier, a model for forecasting the It wasConclusion observed and Recommendation that the forecasting accuracy is weakened as one goes farther into term structure of interest rates is essential in designing In this paper, the three-factor Nelson-Siegel model an optimal debt strategy for the government. This paper was applied to forecast the term structure of interest presents one such model and how it can be applied to e. This is a limitation of any time series models. Thus, instead of 1-year forecast, only rates using Philippine market data. Using a fixed shape the Philippine market data. Forecast of interest rates will help the policy makers determine the appropriate parameter, the equation for the yield became linear in the beta parameters. Such equation was fitted to each mix of debt instruments that will minimize cost subject to a prudent risk level. ks were considered. historical dataset of zero rates thereby producing a series of beta parameters. An appropriate time series model was then obtained for each beta parameter. The root mean square errors (RMSE) per Acknowledgement tenor and per week are shown in Tables 5 Based on the historical data, the best model for each beta was of the form ARMA(p,q)+eGARCH(1,1). It E. de Lara-Tuprio and E. Cabral are grateful for the is important to note that a different set of historical financial support provided by the National Research Council can be seen that the best forecast is for 5-year yield, with only 6 bps RMSE, and the l data may produce a different time series model for the beta parameters. of the Philippines. rate is for 10-year yield with 27 bps RMSE. 200 E.P. de Lara-Tuprio, et al Table 5. RMSE Per Tenor Β 1 Mo 3 Mo 6 Mo 1 Yr 2 Yr 3 Yr 4 Yr 5 Yr RMSE 0.1106% 0.1556% 0.1489% 0.1071% 0.1821% 0.1460% 0.1310% 0.0635% Β 6 Yr 7 Yr 8 Yr 9 Yr 10 Yr 15 Yr 20 Yr 30 Yr RMSE 0.0830% 0.1399% 0.1921% 0.2378% 0.2743% 0.2636% 0.1507% 0.1655% Table 6. 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