Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi Schwarz Commerzbak AG, Frakfurt/M Risk Maagemet Hochschule für Bakwirtschaft Private Fachhochschule der BANKAKADEMIE Sterstraße 8 6318 Frakfurt/M Tel: 69 / 95 94 6-213 Fax: 69 / 95 94 6-28 email: hfb@mailpop-frakfurtcom
Abstract This paper deals with the problem of iterpolatio of discout factors betwee time buckets The problem occurs whe price ad iterest rate data of a market segmet are assiged to discrete time buckets A simple criterio is developed i order to idetify arbitrage-free robust iterpolatio methods Methods closely examied iclude liear, oetial ad weighted oetial iterpolatio Weighted oetial iterpolatio, a method still preferred by some baks ad also offered by commercial software vedors, creates several problems ad therefore makes simple oetial iterpolatio a more logical choice Liear iterpolatio provides a good approximatio of oetial iterpolatio for a sufficietly dese time grid 1 Itroductio Valuatio ad pricig of fiacial istrumets geerally requires kowledge of discout factors ad/or zero bod prices Fudametal to the calculatio of discout factors is detailed iformatio o iterest rates, as well as o prices of fixed icome securities i special market segmets (Bod-, FRA-, Swap-market) at preset time t The procedure for calculatig a discout structure df from this iformatio is as follows: Startig with market data we defie a discrete time structure t 1, t 2, K, t N ad calculate the implied discout factor df ( t, t ) for every time to maturity t, = 1,, N eg by usig a bootstrappig techique The calculatio of the preset value of a cash flow CF(t) occurrig at time t requires the coversio of the discrete structure df ( t, t1), df ( t, t2),, df ( t, t N ) ito a cotiuous discout curve t df ( t, t), t [ t, t N ] The complete set of empirical data is employed i order to derive the discrete discout structure, so that the secod step of the problem is reduced to a pure iterpolatio problem If the market data is icomplete the a iterpolatio problem may occur i the first step (eg this would be caused by a missig bod) I this paper we study several widely used iterpolatio methods thereby cofiig ourselves to the study of those iterpolatio problems which require the kowledge of oly two adjacet discout factors McCulloch [1] has developed splie iterpolatio techiques by usig the whole spectrum of market data Splie iterpolatio offers a higher degree of smoothess, which has its price i terms of precisio or eve arbitrage-freeess For a detailed discussio of this matter we 2
refer to Brecklig, Dal Dosso [2], [3] ad Shea [4] I a forthcomig paper we will ivestigate iterpolatio methods usig all available market iformatio Let us state the problem i more precise terms: Problem Let t deote the preset time, t 1, t 2, K, t N the desigated grid structure ad let df ( t, t1), df ( t, t2), K, df ( t, t N ) be the discout factors The problem is the valuatio of a give cash flow CF() t = (,) t 1, which pays a amout 1 at a time t with t t 1 < t < t, cosiderig oly the discout factors df 1 = df ( t, t 1) ad df = df ( t, t) (without ay restrictio we assume df 1 > df) Let the idex be fixed with { 1, K, N} t -1 t t 1 The problem ca be looked upo from two differet poits of view which are somehow dual to each other: Iterpolatio We calculate from discout factors df 1 ad df a iterpolated value df ( t, t ) = Ip( t, df 1, df) ad determie the preset value (PV = Preset Value) of the paymet (,) t 1 to be (1) PV (,) t 1 = 1 df ( t, t) Bucketig Two fuctios B1 = B1(, t df 1, df) ad B2 = B2(, t df 1, df) (bucketig fuctios) are to be determied i such a way that the cash flow ( t,) 1, which pays oe uit i t ca be replaced by the cash flows ( t 1, B1 ) ad ( t, B 2 ) (Buckets) The preset value of the paymet (,) t 1 the is calculated as (2) PV (,) t 1 = 1 B1 df ( t, t 1) + 1 B2 df ( t, t) Tyig together the two dual view poits, ie equatig (1) ad (2) we obtai (*) df ( t, t ) = B1 df ( t, t 1) + B2 df ( t, t) ; Therefore all bucketig methods ca be cosidered as special iterpolatio methods This formula ad coditios resultig from bucket hedgig will be the key poit i our aalysis Bucket hedgig has bee extesively studied by Turbull [5] 3
The paper is orgaized as follows First, we set some otatio ad state a o arbitrage coditio suited for our purpose I the secod part, commoly applied iterpolatio techiques such as liear, oetial ad weighted oetial iterpolatio are ivestigated i a qualitative maer Their impact o zero rate structures as well as o forward rate curves is discussed i coectio with some selected iterest rate scearios It ca be see that the weighted oetial iterpolatio already has remarkable drawbacks The fial sectio cotais the mai results of this paper A simple coditio described by a system of differetial equatios is imposed o equatio (*) Solutios to this system iclude the liear ad oetial iterpolatio method Iterestigly, these two solutios are related by the fact that liear iterpolatio is the first order term of the Taylor series asio of the oetial iterpolatio 2 Notatio A cotiuous fuctio df = Ip(, t df, df ), t t, t t 1 1, with boudary coditios (3) Ip( t 1, df 1, df) = df 1 ad Ip( t, df, df ) df 1 = is called iterpolatio fuctio Let df 1 > df for all { 1, K, N} A iterpolatio fuctio Ip is called arbitrage-free, if Ip is strictly decreasig i t, that meas (4) Ip( s1, df 1, df) > Ip( s2, df 1, df) for t 1 s1< s2 t Furthermore, we assume that the variables df are idepedet give the above restrictio Remark 1 No arbitrage is equivalet to the fact that all forward iterest rates r( t, s1, s2 ) with t 1 s1 < s2 t are positive t -1 s 1 s 2 t r t s s (,, ) 1 2 4
Proof: For the forward iterest rate r( t, s, s ) oe has 1 2 df ( t, s2) rt (, s1, s2) > < 1 Ip( s2, df1, df2) < Ip( s1, df1, df2) df ( t, s ) 1 Sice we are oly iterested i the relative distace of the time parameter t to the left boudary t -1, we will use the parameter λ istead of t where t t λ = λ() t = 1 t t 1 We deote by df λ the followig ressio df λ = Ip( λ, df 1, df ), λ [ 1, ] The the above boudary coditios ca be restated i terms of the ew parameter λ as: (5) Ip(, df 1, df) = df 1 ad Ip(, 1 df 1, df) = df 3 Examples of iterpolatio fuctios I the followig sectio we look at differet iterpolatio fuctios ad discuss their qualitative behaviour I aalyzig the zero rate curve ad the forward rate structure the followig three zero rate scearios are cosidered Maturity Sceario 1 Sceario 2 Sceario 3 1 yr 5, % 8,5 %, % 2 yrs 6,5 %, %, % 3 yrs,5 % 6, %, % 4 yrs 8,2 % 5,3 %, % 31 Liear Iterpolatio Liear iterpolatio is obtaied by assigig the relative distaces 1 λ ad λ as weights to the discout factors df 1 ad df,ie: 5
li (6) df λ = Ip ( λ, df 1, df ) = ( 1 λ) df 1 + λdf The boudary coditios (5) are easily verified The o arbitrage coditio follows from Ip λ li = df df 1 < Discout curve The resultig curve t dft = df λ( t ) is a cotiuous piecewise liear fuctio which is i geeral ot differetiable at ( t1, df1),( t2, df2 ),,( tn, df N) Zero rate curve If r t deotes the cotiuously compouded zero rate of the discout factor df t = df λ( t), the the iterpolated iterest rate r t is ressed as follows: r t r ( t t ) r ( t t ) df t t e t e = l( λ() ) = l(( 1 λ( )) 1 1 + λ( ) ) t t t t For the period [1 yr, 4 yrs] we obtai, usig a time iterval of legth =,1 yrs, ad give sceario 1 the followig zero rate curve 9 8 6 5 Maturity[yrs] 4 Graph 1 Zero rate curve with ormal term structure (sceario 1) Similarly we obtai for a iverse term structure (sceario 2) a strictly decreasig zero rate curve with covex parts of the curve I case of a flat term structure (sceario 3), liear iterpolatio yields a fuctio which has slightly covex pieces 6
,5 6,5 Graph 2 Zero rate curve with flat iterest rate structure (sceario 2) Forward rate curve Let df ( t, s1, s2 ) deote the forward discout factor ad r ( t, s1, s2 ) its oetial forward iterest rate for the time iterval s1, s2 t 1, t The discout factor, respectively the forward rate, ca be ressed by the followig formulas df ( t s s df s df df ( t, s, s ), 2) ( 1 λ( )) ( ) 1 2 = = 2 1 + λ 2 df ( t, s ) ( 1 λ( s )) df + λ( s ) df 1 l( df ( t, s, s )) r( t, s1, s2) = 1 2 s s 2 1 1 1 1 For costat time itervals of legth = s s = yrs] the forward curve is as follows: 2 1 1, yrs ad time iterval [1 yr, 4 11 1 9 8 Graph 3 Forward rate curve with ormal term structure (sceario 1)
6 5 4 3 2 Graph 4 Forward rate curve with iverse term structure (sceario 2) I both scearios (ormal term structure as well as iverse term structure) oe obtais icreasig forward rates withi the iterpolatio iterval; discotiuities appear at the boudary of the time itervals The discotiuities are due to the method of iterpolatio choose, which calculates discout factors as a average of adjacet discout factors I a flat term structure sceario (sceario 3), forward rates are ot oly icreasig but also show a periodic behaviour 8 6 Graph 5 Forward rate curve with flat term structure (sceario 3) 32 Expoetial Iterpolatio This form of iterpolatio is obtaied by assigig certai oets to the discout factors df 1, df : 1 λ λ () df λ = Ip ( λ, df 1, df ) = df df 1 The boudary coditios (5) are easily verified, the o arbitrage coditio (4) follows from Ip λ 1 λ λ 1 λ λ = (l df 1) df df + df (l df df df λ df df ) = (l l < ) 1 1 1 8
Discout curve Sice 2 Ip = df 2 λ (l df l df 1) 2 > λ the oetial iterpolatio yields strogly covex pieces i the discout curve The discout curve t df df is a cotiuous fuctio, but i geeral ot t = λ( t ) differetiable at the poits t 1, t 2, K, t N Zero rate curve Let r t deote the cotiuously compouded zero rate of the discout factor df t = df λ( t) It is computed usig the liearly iterpolated value of the adjacet zero rates ad 1 λ () t λ() ( rt( t t )) = dfλ() t = df 1 df t = ( ( 1 λ)( t 1 t) r 1) ( λ( t t) r) t 1 t t t = ( 1 λ) r 1 + λ r ( t t ) t t t t r t t t t t r = ( 1 λ) 1 1 + λ r t t t t Give sceario 1, the zero rate curve appears as follows, oce agai by usig time itervals of =,1 yrs ad time periods [1 yr, 4 yrs]: 9 8 6 5 4 Graph 6 Zero rate curve with ormal term structure (sceario 1) Give sceario 2, the zero rate curve decreases yieldig covex curve pieces Give a flat zero rate structure (sceario 3), the oetial iterpolatio maitais this property, which ca be derived as follows: If r 1 = r oe obtais 9
r t t 1 t t t t 1 t t t = ( 1 λ) r 1 + λ r = ( 1 λ) + λ t t t t t t t t ( t t)( t 1 t) + ( t t 1)( t t) = r = r for all t ( t t )( t t ) 1 r,5 6,5 Graph Zero rate curve with flat term structure (sceario 3) Forward rate curve Expoetial iterpolatio implies costat forward rates r( t, s1, s2 ) for time itervals s1, s2 of equal legth Let s 1 ad s 2 be such that t 1 s1 < s2 t The give λ1 = λ( s 1), λ2 = λ( s 2) ad a forward discout factor df ( t, s1, s2 ) it ca be rewritte as s s 1 λ df t s df 2 λ (, ) df 2 λ df 2 λ1 df df t s s t t (, 1, 2) = 2 = 1 = = df ( t, s1) 1 df 1df 1 df 2 1 λ λ λ λ df 1 1 1 ie df ( t, s1, s2 ) ad r( t, s 1, s 2 ) as well oly deped o the distace s2 s1 For time itervals with a legth of = s2 s1 = 1, yrs ad time periods [1 yr, 4 yrs] we obtai the followig forward rate curve 2 1 1, 1
11 1 9 8 Graph 8 Forward rate curve with ormal term structure (sceario 1) Zero rates [%] 6 5 4 3 2 Graph 9 Forward rate curve with iverse term structure (sceario 2),5 6,5 Graph 1 Forward rate curve with flat term structure (sceario 3) 33 Weighted Expoetial Iterpolatio This iterpolatio method is obtaied by assigig additioal time weights to the oets i (): weight 1 t 1 t (8) df = Ip (, t df, df ) = df df t α ()( λ()) α() t λ() t 1 1 11
t t 1 where: λ = λ() t = t t 1 ad αi = t t αi() t = t t weight Ip satisfies the boudary coditios (3), however the o arbitrage coditio (4) does ot hold Couterexample Let df 1 = 91, df 2 = 89, t =, t 1 = 1, t 2 = 2 ad t = 18 the df ( 1 8) = 88882386 < 89 = df2 Accordig to Remark 1 i Sectio I, egative or zero forward rates caot be excluded by iterpolatio method (8) Remark 2 I order to obtai a valid ressio for the divisor t 1 t = t t = i formula (8) for the first time iterval t, t1 where = 1, we set: t1 = t + 1 day ad r 1 = overight-rate Discout curve The discout curve t df t is a cotiuous fuctio, but ot ecessarily differetiable at poits t 1, t 2, K, t N Term structure Let r t be the oetial iterest rate with discout factor df iterpolated rate r t is give by r = ( λ) r + λ r, t 1 1 i t = df λ( t), the the ie r t is obtaied by iterpolatig adjacet rates i a liear fashio The term structure as defied by the previous scearios yields the followig shape: 9 8 6 5 Graph 11 Iterest rate curve with ormal term structure (sceario 1) 12
Similar graphs are obtaied for iverse (sceario 2) ad flat (sceario 3) term structures usig piecewise liear fuctios Forward curve For the forward discout factor df ( t, s1, s2 ) ad its associated weighted oetial forward rate r( t, s1, s2 ) for the time period s1, s2 t 1, t we have df ( t, s2) df ( t, s1, s2) = df ( t, s ) = df 1 ( s2 t)( t s2) ( s1 t)( t s1) 1 ( t 1 t)( t t 1) df ( s2 t)( s2 t 1) ( s1 t)( s1 t 1) ( t 1 t)( t t 1) l( df ( t, s, s )) r( t, s1, s2) = 1 2 s2 s1 For time itervals of equal legth = s2 s1 = 1, yrs ad time periods [1 yr, 4 yrs] we obtai the followig forward rates, give the aforemetioed scearios: 11 1 9 8 6 Graph 12 Forward iterest rate curve with ormal term structure (sceario 1) 6 5 4 3 2 Graph 13 Forward rate curve with iverse term structure (sceario 2) 13
,5 6,5 Graph 14 Forward rate curve with flat term structure (sceario 3) 4 Results A large class of iterpolatio methods is obtaied by usig so called bucketig procedures As metioed i the itroductio,»buckets«for a cash flow ( t,) 1 where t t 1 < t < t are cofied to the time period t 1 ad t Two cotiuous fuctios B1 = B1(, t df 1, df) ad B2 = B2(, t df 1, df), t t 1, t with B 1 1 ad B 2 1 satisfyig the boudary coditios (9) t = t 1: B1( t 1, df 1, df) = 1 ad B2( t 1, df 1, df) = t = t : B1( t, df 1, df) = ad B2( t, df 1, df) = 1 are called bucketig fuctios or a bucketig procedure As metioed iitially, every bucketig procedure defies a iterpolatio method If B 1 ad B 2 are bucketig fuctios, the (1) Ip(, t df 1, df) = B1(, t df 1, df) df 1 + B2(, t df 1, df) df is the associated iterpolatio fuctio Give (9), the boudary coditios (3) are satisfied A bucketig procedure B 1, B 2 is called arbitrage-free, if the associated iterpolatio fuctio Ip is arbitrage-free, ie if Ip is strictly decreasig i t A sufficiet coditio is B1(, t df 1, df ) B2(, t df 1, df ) df 1 + df < t t provided B 1 ad B 2 are differetiable i t Boudary coditios ad the o arbitrage property of bucketig procedures have aalogue cocepts for the associated iterpolatio fuctio However, the cocept of robustess which is discussed below, seems to have o apparet similarities to iterpolatio Robustess is the essetial igrediet i derivig reasoable 14
iterpolatio/bucketig procedures Further, we assume that the fuctio Ip is cotiuously differetiable i the variables df 1 ad df A bucketig procedure is called robust, if B 1, B 2, ad its associated iterpolatio fuctio satisfy the followig system of partial differetial equatios Ip( t, df df (**) 1, ) = B1(, t df 1, df), df 1 Ip( t, df 1, df) = B2( t, df 1, df) for all t t 1, t df Iterpretatio The Taylor series of the associated iterpolatio fuctio satisfyig (**) is give for fixed t [ t 1, t] ad ( df, df ) by 1 Ip Ip(, t df 1, df ) = Ip(, t df 1, df ) + (, tdf 1, df )( df 1 df 1) df 1 Ip + (, tdf 1, df )( df df ) + R1 df = B1( t, df 1, df ) df 1 + B2( t, df 1, df ) df + R1 Cosequetly, small chages i discout factors df 1 ad df ( a small error term R 1 ) will result i ivariat bucketig fuctios B1( t, df 1, df ) ad B2( t, df 1, df) Therefore, a hedge based o bucketig does ot have to be adjusted for small chages i market factors The mai coclusio of the paper is Theorem: Let B 1, B 2 be as stated above, ad Ip the associated iterpolatio fuctio The li t t (a) B t df df t li t t (, 1 1, ) = 1 λ ( ) = ad B ( t, df df t 1 t t 2 1, ) = λ( ) = 1 t t 1 is a arbitrage-free solutio to the system (**) where λ deotes the relative distace of t to t 1 The associated iterpolatio fuctio is liear ad ressed by li Ip ( t, df 1, df ) = ( 1 λ( t )) df 1 + λ ( t ) df 15
λ() t df (b) B t df df t 1 (, 1, ) = ( 1 λ()) ad df 1 λ() t 1 df B t df df t 2 (, 1, ) = λ() df 1 is a arbitrage-free solutio to the system (**) where λ is as above The associated iterpolatio fuctio is as follows: Ip ( λ( t), df 1, df) = B ( t, df, df) df + B ( t, df, df) df 1 1 1 2 1 1 λ( t ) λ( t ) = df df 1 (c) The two bucketig procedures are approximately the same which ca be see from the first term of the Taylor series asio of the oetial iterpolatio Let λ be betwee ad 1 ad ( df, df ) be fixed The 1 Ip Ip ( λ, df 1, df ) = Ip ( λ, df 1, df ) + ( df 1, df )( df 1 df 1) df 1 Ip + ( df 1, df )( df df ) + R1( df 1, df ) df λ λ = Ip ( λ, df 1, df ) + ( 1 λ)( df 1) ( df ) ( df 1 df 1) 1 λ λ 1 + λ ( df 1) ( df ) ( df df ) + R1( df 1, df ) λ 1 df df = ( 1 ) df 1 + λ λ λ df R + 1( df 1, df ) df 1 df 1 For small values of t t 1 oe has df R1( df 1, df ) ad 1, df 1 ad therefore, li Ip ( λ, df 1, df) ( 1 λ) df 1 + λ df = Ip ( λ, df 1, df) Remark 3 (1) The boudary coditios specified for our differetial equatios by o meas guaratee a uique solutio (2) The solutio i (b) ca be slightly geeralized, if λ( t ) is replaced by a strictly icreasig cotiuous fuctio with values betwee ad 1 16
(3) The weighted oetial iterpolatio does ot yield a robust bucketig procedure as weight Ip (, t df 1, df) df weight 1 Ip (, t df 1, df) df df = α 1( 1 λ) df df = αλ df α 1 αλ 1 1 1 ( 1 λ) 1 produces the followig ressios for the bucketig fuctios B1 ad weight B2 ad 1()( t 1 ()) t 1 gew df B t df df t t 1 (, 1, ) = 1() ( 1 ()) α λ α λ df 1 () t () t 1 gew df B t df df t t 2 (, 1, ) = () () α λ α λ df 1 weight The weighted oetial iterpolatio method, although still ofte used, satisfies either the o arbitrage or the robustess coditio ad Refereces: [1] Mc Culloch, HJ: Measurig the Term Structure of Iterest Rates, Joural of Busiess, XLIV (Jauary 191), 19-31 [2] Brecklig, J; L Dal Dasso: A No-parametric Approach to Term Structure Estimatio, i Hrsg G Bol, G Nakhaeizadeh, K-H Vollmer: Fiazmarktaweduge euroaler Netze ud ökoometrischer Verfahre, Physica Verlag Heidelberg 1994 [3] Brecklig, J; L Dal Dasso: Modellig of Term Structure Dyamics Usig Stochastic Processes, i Hrsg G Bol, G Nakhaeizadeh, K-H Vollmer: Fiazmarktaweduge euroaler Netze ud ökoometrischer Verfahre, Physica Verlag Heidelberg 1994 [4] Shea, GS: Pitfalls i Smoothig Iterest Rate Term Structure Data: Equilibrium Models ad Splie Approximatios, Joural of Fiacial ad Quatitative Aalysis, Vol 19 No 3 (September 1984), 253-26 [5] Turbull, SM: Evaluatig ad Implemetig Bucket Hedgig 1
Arbeitsberichte der Hochschule für Bakwirtschaft Nr Autor/Titel Jahr 1 Moorma, Jürge 1995 Lea Reportig ud Führugsiformatiossysteme bei deutsche Fiazdiestleister 2 Cremers, Heiz; Schwarz, Willi 1996 Iterpolatio of Discout Factors Bestelladresse: Hochschule für Bakwirtschaft z H Frau Glatzer Sterstraße 8 6318 Frakfurt/M Tel: 69/95946-16 Fax: 69/95946-28