Methods for Estimating Term Structure of Interest Rates

 Letitia Bruce
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1 Mehods for Esimaing Term Srucure of Ineres Raes Iskander Karibzhanov Absrac This paper compares differen inerpolaion algorihms for consrucing yield curves: cubic splines, linear and quadraic programming, binomial funcions and parameric mehods. Resuls confirm ha he consrained Schaefer mehod produces smooher and more sable curves han does unconsrained Schafer s approximaion, cubic splines, or discree approximaion. Adding monooniciy consrains in Schaefer mehod does no increase compuaional burden as i does in he discree approximaion. My experimens found ha adding monooniciy consrains on cubic spline mehod is unfeasible. I also experimened wih hree well know parsimonious funcional approximaions of he erm srucure: he NelsonSiegel, Svensson and Bliss exponenial funcions. Accompanying MATLAB sofware package is available from he auhor s websie and illusraes pracical applicaion using U.S. reasury bond marke daa se. Keywords erm srucure esimaion yield curve fixed income bond pricing 1 Inroducion There are several commonly used echniques for esimaion of he erm srucure of ineres raes: regression analysis wih cubic splines by Lizenberger and Rolfo (1984), binomial funcions by Schaefer (1981), and parameerized mehods uilizing parsimonious exponenial funcions by Nelson and Siegel (1987), Söderlind and Svensson (1997) and Bliss (1997). In linear programming and regression mehods for erm srucure esimaion, he esimaor is a soluion o an opimizaion problem. In linear programming, he esimaor is he maximizer of he presen value of a prescribed sequence of cash flows, subjec o he presen value of each bond being less han or equal o is price. In regression mehod, he esimaor is he minimizer of he sum of squared deviaions of he bond prices from heir presen values. I. Karibzhanov Bank of Canada, 234 Wellingon Sree, Oawa, Onario, K1A 0G9, Canada Tel.: , URL: hp://karibzhanov.com
2 2 Iskander Karibzhanov Secion 2 explains he mehodology by presening one discree and four coninuous approximaion mehods of erm srucure esimaion. Secion 3 presens he resuls of my calculaions and compares he esimaed erm srucures across various ypes of bonds. I find ha Schaefer mehod produces more sable and smooher curves han oher mehods. Compuaional Appendix A discusses implemenaion issues and serves as documenaion for MATLAB codes. I describes how o impor bond daa from ex files, consruc cash flows marix, compue enor periods and impose monooniciy consrains on quadraic opimizaion. 2 Mehodology 2.1 Discree Approximaion The process of using a discree approximaion o esimae he erm srucure from a given sample of M bonds can be broken down ino four seps: 1. Consruc N 1 vecor conaining unique and monoonically increasing daes a which any coupon or principal paymen is made, 2. Consruc M N marix A in which each row represens he cash flow srucure of a paricular bond mapped o corresponding daes in vecor, 3. Consruc M 1 vecor p conaining he bonds cash prices by adding accrued ineress o quoed prices, 4. Solve one of he hree problems: solve he leas squares problem of finding he N 1 vecor of discoun facors d, which would minimize he norm Ad p 2, or add monooniciy consrains on he discoun facors and solve quadraic programming problem o find he closes fi o minimize he norm Ad p 2, or add monooniciy consrains on he discoun facors and solve a linear programming problem by maximizing he presen value of fuure cash flows subjec o he presen value of each bond being less han or equal o is acual price. Noe ha his linear programming problem is much easier o solve and more accurae han quadraic programming or leas squares problems. 2.2 Coninuous Approximaion Le s firs give definiion o discoun funcion, yield curve and forward ineres rae curve. Discoun funcion, denoed as d(), is equal o presen value of discoun (zerocoupon) riskfree bond paying one dollar a ime. Coninuously compounding ineres rae on his ype of bond is called spo rae and is denoed as r(). By changing mauriy dae, one can obain a plo of spo ineres raes or a (spo) yield curve. Le f () denoe insananeous forward rae, i.e. rae on forward conrac wih mauriy dae equal o selemen dae. Then, we have he following relaionships beween d(), r() and f ():
3 Mehods for Esimaing Term Srucure of Ineres Raes 3 ( s= d() = exp( r()) = exp r() = ln(d()) = 1 s= s=0 s=0 ) f (s)ds (1) f (s)ds (2) f () = d() d() = r() + r() (3) The main drawback of he discree approximaion is ha i esimaes he discoun raes only a he paricular ime values 1, 2,..., N. Coninuous approximaions avoid his by assuming ha he discoun rae akes a coninuous funcional form such as: d() K k=0 x k b k (), (4) where b k () are specified componen funcions of. Then he esimaion problem consiss of compuing he K +1 parameers x k, such ha d() is consisen wih marke prices. This can be done by minimizing ABx p 2 wih respec o x, where A M N marix of cash flows described above, B funcional N (K + 1) marix, whose i h row is given by x decision vecor [x 0,x 1,...,x K ]. b( i ) = [b 0 ( i ),b 1 ( i ),...,b K ( i )], (5) Cubic Splines Regression Model by Lizenberger and Rolfo Among he earlier mehods of esimaing erm srucure are hose of Lizenberg and Rolfo. These auhors uilize regression mehodology o find he presen value facors, or, equivalenly, he erm srucure of spo ineres raes, ha bes explain he observed prices of coupon bonds. Using mulivariae linear regression analysis, he difference beween a bond s price and is presen value is minimized. This regression analysis incorporaes smoohing of he erm srucure wih cubic splines. In he cubic spline mehod, {τ k } m k=0 [0,T ] denoes a se of kno poins over he inerval in which coupon paymens are made for which 0 τ 0 < τ 1 < < τ m T. The knos are placed such ha an equal number of paymen daes falls ino each subinerval. The value for m is aken o be he ineger closes o he square roo of he oal number of bonds in he markeplace (m M). The value of r may be inerpreed as he sample size bu should no be less han 10. The cubic splines approximae d() by d() 1 + x 1 + x x m k=1 where I τk = 1 when τ k and I τk = 0 when < τ k. x k+3 ( τ k ) 3 I τk, (6)
4 4 Iskander Karibzhanov For each realized paymen dae i, le b( i ) denoe he vecor b( i ) = [ 1, i, 2 i, 3 i,( i τ 1 ) 3 I i τ 1,( i τ 2 ) 3 I i τ 2,...,( i τ m ) 3 I i τ m ], (7) and B denoes he N (4+m) marix whose i h row is given by b( i ). The esimaion problem is hen o minimize he norm (AB)x p 2 for x = [1,x 1,x 2,...,x 3+m ]. The cubic splines are designed o be coninuous and have boh coninuous firs and second derivaives. Therefore we can derive he following forward rae curve from discoun funcion of cubic spline mehod: f () = d() d() x 1 + 2x 2 + 3x m k=1 x k+3( τ k ) 2 I τk 1 + x 1 + x x m k=1 x k+3( τ k ) 3 I τk (8) However, cubic spline is no guaraneed o be a monoonically decreasing funcion. To ensure monooniciy, I esimaed he erm srucure using Schaefer mehod Schaefer Mehod In order o implemen Schaefer mehod, one mus normalize ime vecor, i.e. we mus scale each paymen dae by he longes one so ha ime is measured on he inerval [0,1]. The componen funcions b k () are defined as follows: and for each k = 1,2,...,K, b k () = u k 1 (1 u) K k K k du = 0 b 0 () 1, (9) j=0 ( 1) j+1 C K k j ( ) k+ j k + j where Cn k = n! k!(n k)! and K is usually aken o be 25. Each x k is consrained o be nonnegaive, and each b k () is a monoonically decreasing nonposiive funcion on he inerval [0, 1]. Therefore, Schaefer mehod ensures ha d() is a monoonically decreasing funcion. To ensure ha d() are nonnegaive, one also adds he consrain K k=0 and he parameers are esimaed as hey were before NelsonSiegel Exponenial Funcions Mehod (10) x k b k (1) 0. (11) Raher han explicily modeling he erm srucure, one may wan o approximae i by a flexible funcional form. There are several parsimonious exponenial models, proposed by Nelson and Siegel (1987), Söderlind and Svensson (1997) and Bliss (1997). The NelsonSiegel approximaion is derived from he assumpion ha he spo raes follow a secondorder differenial equaion and ha forward raes, which are
5 Mehods for Esimaing Term Srucure of Ineres Raes 5 he prediced spo raes, are he soluion o his differenial equaion wih equal roos. Le s assume ha he equaion for insananeous forward rae is given in he following form: ( f () = β 0 + β 1 exp ) + β 2 ( λ λ exp ) (12) λ Then he yield curve can be compued using formula 2 o obain: r() = β 0 + (β 1 + β 2 ) [ ( )] 1 exp λ λ ( β 2 exp ). (13) λ Söderlind and Svensson (1997) improved he NelsonSiegel model by formulaing forward raes and spo raes as follows: ( f () = β 0 + β 1 exp ) ( + β 2 exp ) ( + β 3 exp ) (14) λ 1 λ 1 λ 1 λ 2 λ 2 ( r() = β 0 + (β 1 + β 2 ) 1 exp β 3 ( 1 exp λ 2 ) λ 2 λ 1 ) λ 1 β 2 exp ( ) + λ 1 ( exp ) (15) λ 2 The NelsonSiegel mehod also ook furher developmen by Bliss (1997). His improved approximaion is given as: ( f () = β 0 + β 1 exp ) ( + β 2 exp ) (16) λ 1 λ 2 λ 2 ( ) r() = β 0 + β 1 1 exp ( ) λ 1 + β 2 1 exp ( λ 2 exp ) (17) λ 2 λ 1 In each of he above models, parameers β 0,λ,λ 1 and λ 2 mus be posiive. The parameer 1 λ governs he exponenial decay rae; small values of 1 λ produce slow decay and can beer fi he curve a long mauriies, while large values of 1 λ produce fas decay and can beer fi he curve a shor mauriies. We can inerpre β 0,β 1,β 2 as hree laen facors. The loading on β 0 is a consan ha does no decay o zero [ in he( limi; )] hus, i may be viewed as a longerm facor. The loading on β 1 is 1 exp λ1 λ 1, which sars a 1 bu decays quickly and monooni [ cally o 0; ( hence, )] β 1 may( be viewed as a shor erm facor. The loading on β 2 is 1 exp λ2 λ 2 exp λ ), which sars a 0 (and is hus no shorerm), in 2 creases, and hen decays o zero (and hus is no long erm); hence, β 2 can be inerpreed as a medium erm facor. β 0,β 1 and β 2 can also be inerpreed in erms of he aspec of he curve ha hey govern: level, slope, and curvaure. The longerm facor β 0 governs he yield curve level. In paricular, r( ) = β 0. Alernaively, noe ha an increase in β 0 augmens all λ 2
6 6 Iskander Karibzhanov yields equally, as he loading is idenical a all mauriies. The shorerm facor β 1 is equal o he yield curve slope, r( ) r(0). Noe ha an increase in β 1 augmens shor yields more han long yields because he shor raes load on β 1 more heavily, hereby changing he slope of he yield curve. Finally, β 2 is closely relaed o yield curvaure: an increase in β 2 will have very lile effec on very shor or very long yields, which load minimally on i, bu will increase mediumerm yields, which load more heavily on i, hereby increasing yield curve curvaure. Le θ denoe he se of five parameers discussed above, i.e. θ = {β 0,β 1,β 2,λ 1,λ 2 }. (18) The parameers can be esimaed using nonlinear leassquares daa fiing by he GaussNewon mehod, implemened in nlinfi funcion from MATLAB Saisics Toolbox, i.e. ˆθ = argmin N i=1 ε 2 i, (19) where ε i is he difference beween he acual marke yields and heoreical fied yields on i h bond a ime = 0 in boh mehods. 3 Resuls 3.1 Comparison of Approximaion Mehods In his paper I use five mehods o esimae he erm srucure from differen kinds of bonds  one discree approximaion and four coninuous approximaions. For he discree approximaion, he objecive funcion is given in erms of discoun facors d() on discree daes. I simply find he vecor d such ha he sum of he squared errors Ad p 2 is minimized. The model, herefore, is easy o formulae. However, he qualiy of he soluions is no likely o be saisfying since discree approximaions rarely give smoohed curves. Le s now consider he firs wo coninuous mehods. The firs one is he cubic spline mehod, which defines a new se of variables, x i. This mehod ries o fi he values of each x i such ha he sum of he squared errors (AB)x p 2 is minimized. To formulae his model, we have o consruc a new marix, B, where d is esimaed o equal o Bx. This mehod produces beer soluions han hose from discree mehod. In he second coninuous approximaion, Schaefer mehod, we also have o consruc a new marix, B, and a se of variables x. The objecive of his mehod is he same as he former wo; ha is, o minimize he sum of he squared errors. However, in his mehod, unlike in he former wo, ime is scaled such ha i is measured on he inerval [0,1]. I found ha he curves produced using Schaefer mehod almos always look smooher han hose of he oher wo mehods. This mehod also gives me he monoonically decreasing funcion of d() by simply seing he lower bound of he variables o zero. In pracice, i akes a shor ime o solve for he opimal soluion while
7 Mehods for Esimaing Term Srucure of Ineres Raes 7 Fig. 1 Comparison of discree, leas squares, Schaefer and cubic approximaions Original discree approximaion Monoonically consrained Schaefer's approach Leas squares approximaion o discree mehod Third order cubic spline by Lizenberger and Rolfo This figure shows how noise in discree mehod can be smoohed by using leas squares approximaion from MATLAB splines oolbox. Cubic splines produce more volaile soluion compared o he Schaefer mehod. he discree approximaion wih a se of monoonic consrains akes a much longer ime o solve for a se of soluions. Figure 1 illusraes he reasury yield curves obained using each mehod. I found ha he curve from Schaefer mehod produces he smoohes curve while he discree approximaion produced he wors curve. The coninuous mehods also do beer in avoiding he unreasonable flucuaions in he curve. Figure 2 compares NelsonSiegel, Bliss, cubic and Schaefer approximaions. The exponenial funcion approximaion proposed by Nelson and Siegel and is furher developmen by Rober Bliss resemble he embedded exponenial shape. As we expeced, he exponenial form is much less volaile compared o cubic splines and Schaefer mehod and hus is mos preferred in pracice. 3.2 Esimaed Term Srucure Across Various Bond Types My esimaes for he erm srucure using a discree approximaion, cubic splines and Schaefer mehod were calculaed using daa from reasury coupon securiies, zero coupon bonds (STRIPS), AA governmen bonds, and AAA corporae bonds. Figure 3 shows he erm srucure as esimaed by Schaefer mehod wih he consrains d() 0 (nonnegaiviy) and d 1 d 2 d N (monooniciy) for each of he aforemenioned ypes of bonds. As zeroes only pay he principals a mauriies and do no make periodic coupon paymens, hey produce he smoohes erm srucure curve as can be seen by he lack of oscillaions. Only he shor raes beween mauriy daes, when he principal is paid o he invesor, need be considered when esimaing he erm srucure wih
8 8 Iskander Karibzhanov Fig. 2 Comparison of Bliss, NelsonSiegel, Schaefer and cubic approximaions Bliss (exended NS) Consrained Schaefer Nelson and Siegel Unconsrained cubic This figure shows ha he smooh exponenial funcional approximaion proposed by NelsonSiegel and Bliss is he preferred mehod o esimae yield curves compared wih volaile cubic splines and Schaefer approximaions. Fig. 3 Term srucure for various ypes of bond by consrained Schaefer mehod Treasury AA Governmen Agencies Zeros (STRIPS) AAA Corporae zeroes. Noe ha a smooh yield curve is always produced when he erm srucure is esimaed from STRIPS regardless of wheher a discree or coninuous approximaion is used (figure 4). Furhermore, zeroes and reasuries produce he lowes yield curves  i.e., hey predic lower yields o mauriy compared o he esimaes made using oher bonds. This is because zeroes and reasuries are backed by he full faih and credi of he
9 Mehods for Esimaing Term Srucure of Ineres Raes 9 Fig. 4 Term srucure of zerocoupon bonds (STRIPS) Discree mehod Cubic Splines Consrained Schaefer In case of STRIPS, all of he considered mehods produce very close soluions due o he fac ha zero coupon bonds do no bear inermediae coupons paymens ha creae volailiy in esimaing erm srucure. U.S. governmen and hus have no credi risk; hence, hey have lower risk premiums (smaller yields) han oher ypes of bonds. Bonds issued by governmen sponsored enerprises (GSEs) such as he Federal Home Loan Morgage Corporaion and he Federal Naional Morgage Associaion are privaely owned and publicly charered eniies; hey carry slighly more credi risk han do reasuries. Corporae bonds carry he mos credi risk since he abiliy of a firm o make imely principal and coupon paymens depends on how successful he firm is, which varies from fiscal year o fiscal year. These varying suscepibiliies o credi risk are in accordance wih figure 3, in which he corporae bond yield curve ends o be higher han he AA governmen bond curve, which, in urn, is higher han he yield curve esimaed from zeroes. Treasuries, GSE securiies, and corporae bonds are all equally suscepible o ineres rae risk, inflaion risk, and reinvesmen risk. Alhough he U.S. governmen backs boh zeroes and reasuries, since zeroes do no pay coupons, hey are no suscepible o reinvesmen risk, which may explain why hey end o predic lower yields o mauriy han do reasury coupon securiies. 3.3 Effec of Including More Terms in he Coninuous Approximaion For coninuous approximaions, I used he cubic splines and Schaefer mehod. In he cubic spline mehod, I define he kno poins τ 0,τ 1,...,τ m such ha an equal number of paymen daes falls ino each subinerval:
10 10 Iskander Karibzhanov N τ 0 τ 1 τ 2 τ 3 τ m We know ha d() 1+x 1 +x 2 2 +x m k=1 x k+3( τ k ) 3 I τk, where I τk = 1 when τ k and I τk = 0 when < τ k. If we closely look a his condiion, we will see ha each d() will be made up of differen numbers of variables x i, i.e. in he example above, d( 1 ),d( 2 ),d( 3 ),d( 4 ),d( 5 ) are each he linear combinaion of hree variables x 1,x 2 and x 3, while each of d( 6 ),d( 7 ),d( 8 ),d( 9 ) and d( 10 ) is he linear combinaion of four variables x 1,x 2,x 3 and x 4, and each of d( 11 ),d( 12 ),d( 13 ),d( 14 ),d( 15 ) is he linear combinaion of five variables x 1,x 2,x 3,x 4 and x 5, and so on. In each subinerval, he values of d() will no be much differen since hey are esimaed from he same se of variables. Neverheless, when we add more erms in he approximaion, i.e. when we increase m, he number of paymen daes in each subinerval will decrease. This, herefore, leads o he decreasing of smoohness of d() N τ 0 τ 1 τ 2 τ 3 τ 4 τ m Top panel of figure 5 illusraes he effec of including more condiioned erms in he cubic spline approximaion 6. I found ha he bes soluion is obained if I se m = 10 as i was recommended ineger closes o he square roo of he number of bonds (135 for reasury securiies). When I increased he value of m o 20, he curve swung slighly higher han i did when when m was equal o 15. For m equal o 25, he ineres rae curve deviaed wildly. I found ha he curve did no oscillae so frequenly if m was decreased o 10 or 5. The number of erms can be inerpreed as our sample size. Including more erms in cubic spline approximaion increases volailiy. This is eviden in case of 25 erms. However, he allowable number of erms should be a leas 10. Oherwise, he erm srucure would be overly smoohed (as in he case of 5 erms). I also considered he effec of including more uncondiioned erms, i.e. hose of he from x i i in cubic spline approximaion 6. Boom panel of figure 5 shows erm srucures as esimaed wih reasuries using he cubic splines wih 3, 4, 5, 6, and 9 uncondiioned erms. Including more han hree uncondiioned erms in cubic spline approximaion resuls in higher volailiy and hence no recommended. When 4, 5, or 6 erms are used, he yield curve esimaion oscillaed noiceably. The 3 erm curve and 9 erm curve seemed o produce he bes fi. In Schaefer mehod, I found he same resuls; ha is, he more I increased he number of erms K, he more he ineres rae curves flucuaed (figure 6). In his projec I illusraed he effec of including more erms in Schaefer mehod on he erm srucure of reasury securiies. I varied he number of erms K from 15 o 45. I found ha he discoun rae curve corresponding o K = 15 resembled he one corresponding o K = 25. However, when I increased he value of K o 30 and o 35, he oscillaions worsened. A wildly flucuaing curve resuled when I increased he value of K o 45.
11 Mehods for Esimaing Term Srucure of Ineres Raes 11 Fig. 5 The effec of including more condiioned erms in cubic spline mehod Three uncondiioned erms Four uncondiioned erms Five uncondiioned erms Six uncondiioned erms Nine uncondiioned erms Five condiioned erms Ten condiioned erms Fifeen condiioned erms Tweny condiioned erms Tweny five condiioned erms These oscillaions occur due o he following reason. I esimae he discoun raes, d(), by consrucion of he vecor x = [x 0,x 1,...,x K ]. Each d() is equal o he linear combinaion of x k,k = 0,1,...,K. When we increase he value of K, we increase he number of erms in he linear combinaion for each d(). Consider wha would happen if we coninuously increase he value of K unil i equals M, he oal number of bonds. This opimizaion problem will ry o esimae a se of M variables such ha hey minimize he sum of he squared errors from M ses of daa. From a saisical poin of view, i is meaningless o do so. This reason can be used o explain he flucuaions in he cubic spline approximaion as well.
12 12 Iskander Karibzhanov Fig. 6 The effec of including more han 15 erms in Schaefer mehod 15 erms 25 erms 30 erms 35 erms 45 erms Effec of Monooniciy Consrains In heory, i should be he case ha d 1 d 2 d N, i.e. he discoun raes should be declining. Oherwise, a negaive ineres rae exiss beween wo paymen daes. Schaefer mehod guaranees ha he discoun raes will be declining. However, for he discree approximaion, he monooniciy consrains may or may no be enforced. I examined he discree approximaion esimaes of he erm srucure wih and wihou monooniciy consrains on d(). Figure 7 shows ha boh esimaes produce discoun facors ha rend downward wih increasing mauriy, bu, as expeced, only when he consrains are enforced is d() monoonically decreasing. Figure 7 also compares he consrained and unconsrained discree approximaions o he yield curves. For early mauriies  hose before he unconsrained approximaion is beer; i produces a smooh yield curve. In conras, he monooniciy consrains force a series of zigzags o appear in he early par of he yield curve: when d( i ) = d( i+1 ), hen r( i ) > r( i+1 ), which produces he downward slopes of each peak, and when d( i ) > d( i+1 ), hen r( i ) < r( i+1 ), resuling in he small upward jumps. However, for laer mauriies, when yields are more difficul o predic, he unconsrained discree approximaion produces a raher noisy yield curve. The consrained soluion, on he oher hand, produces a yield curve wih smaller flucuaions for laer mauriies. Similar resuls were observed in Schaefer mehod. Figure 8 shows ha he erm srucure of reasury securiies obained from he consrained mehod is smooher han he curve obained from he unconsrained mehod. The unconsrained discoun raes do no decrease monoonically a longer mauriies which desabilizes long erm ineres raes. Figure 9 demonsraes ha adding monooniciy consrains o Schaefer mehod is imporan for esimaion of erm srucures of corporae bonds and governmen
13 Mehods for Esimaing Term Srucure of Ineres Raes 13 Fig. 7 The effec of adding monooniciy consrains in discree approximaion 0.9 Unconsrained soluion using leas squares Consrained soluion using quadprog solver Discoun Facor, d = exp(r ) Unconsrained soluion using leas squares Consrained soluion using quadprog solver = ln(d )/ The op panel of his figure shows how monooniciy consrains in discree approximaion creae Lshaped paerns in discoun curve when here is no daa available. This explains large deviaions in erm srucure in he boom panel which also shows ha consrained yield curve is smooher han unconsrained one in periods when daa is available. agency bonds. Wihou monooniciy consrains, he erm srucure of hese bonds is useful only for shor mauriies. I also compared consrained and unconsrained versions of he cubic approximaions. I decided o impose consrains on he cubic approximaion in order o remove he upward slope ha resuled a he end of he discoun facor curve d() when he
14 14 Iskander Karibzhanov Fig. 8 The effec of adding monooniciy consrains in Schaefer mehod Unconsrained soluion using Schaefer's approach Consrained soluion using Schaefer's approach Discoun Facor, d = exp(r ) Unconsrained soluion using Schaefer's approach Consrained soluion using Schaefer's approach Adding monooniciy consrains in Schaefer mehod furher reduces volailiy of discoun raes and ineres raes. In conras wih discree mehod, adding consrains o he opimizaion process in Schaefer mehod did no require much more compuaional effor because of less number of variables (26 in case of cubic splines compared o 211 in discree mehod). consrains were ignored. However, he consrained cubic approximaion was difficul o implemen. The effec of adding monooniciy consrains on discoun facors was a rapid decline of d() in he shor erm (figure 10). Enforcing monooniciy in he cubic approximaion produces an unrealisic, nearly verical yield curve and is hus unusable. In erms of opimizaion effor, i is obvious ha adding some consrains o he model will require more effor o solve for a se of soluions. In order o minimize he norm Ad p 2, I simply use he linear leas squares mehod. To apply his mehod,
15 Mehods for Esimaing Term Srucure of Ineres Raes 15 Fig. 9 Consrained Schaefer mehod for corporae and governmen agency bonds Unconsrained soluion using Schaefer's approach Consrained soluion using Schaefer's approach Unconsrained soluion using Schaefer's approach Consrained soluion using Schaefer's approach given a dependen variable, y, and a se of independen variables, x 1,x 2,...,x N, I ry o find a linear relaion by deermining he se of parameers, b 0,b 1,...,b N, such ha he sum of he squared errors is minimized. To do his, I consruc a vecor of dependen variables y i, say y, and a marix of independen variables x i, say X, from he hisorical daa and se he relaions as follows: y = Xb + ε, (20) where b is he vecor of parameers b i, and ε is he vecor of errors. Using he leas squares mehod o minimize ε ε (he sum of he squared errors), we may calculae he soluion from b = ( X X ) 1 X y. (21)
16 16 Iskander Karibzhanov Fig. 10 The effec of adding monooniciy consrains in cubic spline mehod 0.9 Unconsrained soluion using leas squares Consrained soluion using quadprog solver Discoun Facors, d() This figure shows ha we should no consrain discoun facors in cubic spline mehod because discoun facors would drop rapidly o zero. In discree model, I defined a vecor of prices p and a marix of coupon paymens A. The vecor of discoun raes ha minimize he norm Ad p 2 is d = (A A) 1 A p. For coninuous models, I simply replace marix A wih AB, and we have x = (B A AB) 1 B A p. Alhough calculaing he inverses of he marices may ake a long ime, many sofware packages such as MATLAB use he ineriorreflecive Newon mehod in which each ieraion involves he approximae soluion of a large linear sysem using he mehod of precondiioned conjugae gradiens (PCG). In pracice, we solve his kind of problem in a few seconds. When we add a se of consrains o he models, we canno use he same formula o find he soluions. In MATLAB, when we add he inequaliy consrains, he problem can be solved using quadraic programming. This program uses an acive se mehod which finds an iniial feasible soluion by firs solving a linear programming problem. A each major ieraion, a posiive definie quasinewon approximaion of he Hessian of he Lagrangian funcion is calculaed using he BFGS mehod. In he case of discree approximaion, where we have a large number of variables, he ime required o find monoonically consrained soluions reached he order of several minues. However, in he case of coninuous approximaions, he compuaion ime was negligible due o he small number of monoonically consrained variables. Conclusions In his projec I experimened wih various mehods for approximaing he erm srucure from differen ypes of bonds. My resuls showed ha he consrained Schaefer mehod produces smooher and more sable curves han does unconsrained Schafer s
17 Mehods for Esimaing Term Srucure of Ineres Raes 17 approximaion, cubic splines, or discree approximaion. Moreover, he effec of adding monooniciy consrains in Schaefer mehod does no significanly increase in he opimizaion effor as i does in he discree approximaion. My experimens showed he unreasonable resuls of adding monooniciy consrains on cubic spline mehod. Finally, I experimened wih hree well know parsimonious funcional approximaions of he erm srucure, he NelsonSiegel, Svensson and Bliss exponenial funcions.
18 18 Iskander Karibzhanov A Compuaional Appendix In order o esimae yield curves, he compuer program needs o accomplish a number of echnical asks such as scanning ex files o impor bonds daa, compuing he cash flow paymen marix A, vecor of unique and monoonically increasing paymen daes, vecor p of cash prices. This compuaional appendix describes and documens my implemenaion of hese funcions in MATLAB. A.1 Imporing Bond Daa From Tex File Funcion srscan.m is responsible for scanning ex files for necessary bond daa. Assuming ha he inpu file is in he proper forma, he funcion can impor raw daa on selemen dae, quoed prices, coupon raes, mauriy daes, coupon frequencies, and even daycoun convenions for differen ypes of bonds. Then i can compue he cash flows marix using he specified daycoun convenion mehodology for differen ypes of bonds. The following is an example of readable inpu ex file in he proper forma: begin of reasury.x Sele 02/15/2002 Coupon Mauriy Price Period Basis P Feb Augus /Feb2025C end of reasury.x To call his funcion we use he following synax: [Sele, Mauriy, QuoedPrice, CouponRae, Period, Basis] = srscan( reasury.x ) Noe ha he forma of daes in my program can vary. You may wan o abbreviae names of monhs as in Feb , use a full dae forma wih spaces such as February 15, 2002, or even use he leers C or P o disinguish beween callable and puable bonds as in 15Feb2002C or 15Feb2002P. The only requiremen is ha if you use spaces hen you need o use anoher delimier insead of spaces. For example, you could use aserisks or commas. Anoher imporan feaure of my program is ha i auomaically skips columns ha i does no recognize as necessary for imporing. You may abbreviae column names o he firs hree leers. The program does no disinguish beween lowercase or uppercase leers. The following column names and heir abbreviaions are seleced for imporing: selemen dae, coupon rae, mauriy dae, quoed price, coupon frequency period, daycoun basis. If he file conains exraneous columns aside from hose indicaed in he able, hen hey are ignored. If a special delimier is used, hen i needs o be specified as an addiional parameer in call of he funcion srscan.m. For example, le s assume ha we have he following file: begin of es.x Selemen Dae February 15, 2002 Type of Issue * Size * Cou * Mauriy Dae * Price TNOTE * 200 * * December 31, 2002 * TBOND * 200 * 00 * Feb 15, 2005 * TNOTE (5YR) * 200 * * Augus P * TNOTE * 200 * * Aug * * * * * end of es.x I conains wo exra columns, column names conain spaces, and delimier is aserisk. To impor his file we need o call he funcion srscan.m by specifying he delimier * : [Sele,Mauriy,QuoedPrice,CouponRae,Period,Basis] = srscan( es.x, * ) The firs wo columns will no be impored because he program does no consider hem necessary for subsequen analysis.
19 Mehods for Esimaing Term Srucure of Ineres Raes 19 A.2 Consrucing Cash Flows Marix One very imporan ool for esimaing erm srucures is a program ha does all he necessary cash flow and ime mapping given bond parameers. I resored o he cfamouns funcion from MATLAB Financial Toolbox for is funcionaliy and abiliy o work wih differen ypes of bonds and daycoun convenions. Once we know he exac ime daes and corresponding cash flow amouns, we can proceed o consruc he paymen marix A. In previous secion, I described how o use srscan.m funcion o impor bond daa from a ex file. Now we use he resul of his impor as parameers for he srprep.m funcion: [Sele,Mauriy,QuoedPrice,CouponRae,Period,Basis] = srscan(filename); [p,a,s,] = srprep(sele,mauriy,quoedprice,couponrae,period,basis); The srprep.m funcion reurns a vecor of cash prices (p), a marix of cash flows (A), a vecor of selemen daes in serial dae number forma (s), and a vecor of cash flow daes in serial dae number forma (). Inpu parameers are as follows: Sele Selemen dae (mus be earlier han or equal o Mauriy). Mauriy A vecor of bonds mauriy daes in serial dae number forma. QuoedPrice A vecor of quoed (clean) prices. CouponRae (Opional) A vecor of percenage numbers indicaing he annual percenage rae used o deermine he coupons payable on a bond. Defaul is vecor of zeros. Period (Opional) Coupons per year of he bond. A vecor of inegers (0, 1, 2, 3, 4, 6, and 12). Defaul is vecor of 2 s (wo coupons per year). Basis (Opional) Daycoun basis of he bond. A vecor of inegers: 0 = acual/acual, 1=30/360, 2=acual/360, 3=acual/365. Defaul is vecor of zeros (acual/acual). Noe ha CouponRae mus be specified in percenages, no in decimals. Coninuing he example from he previous secion, le s consruc cash flows paymen marix A for a large se of 4462 municipal bonds using funcion srprep.m: >> [P,A,S,T] = srprep(sele,mauriy,quoedprice,couponrae,period,basis); >> whos Name Size Byes Class Sele 1x1 8 double array Mauriy 4462x double array QuoedPrice 4462x double array CouponRae 4462x double array Period 4462x double array Basis 4462x double array p 4462x double array A 4462x sparse array s 1x1 8 double array 1104x double array As you can see, large marix A conains cash flows for 4462 bonds and 1104 ime daes. I occupies 1.4 Mb of memory space while he full A marix would occupy 37.6 Mb: >> A full = full(a); whos Name Size Byes Class A 4462x sparse array A full 4462x double array To visualize he sparsiy paern of marix A, we can use he spy(a) command (figure 11). The cash flows marix conforms o he expeced sparsiy paern: as he oal number of bonds increases, he oal number of unique cash flow daes also increases. Noe he increasing effec of adding longererm mauriy bonds on oal number of unique daes. A.3 Quadraic Programming Issues In order o add monooniciy consrains and solve quadraic programming problem, we have o conver he parameers in each model o mach he quadraic programming funcion in MATLAB. The quadraic
20 20 Iskander Karibzhanov Fig. 11 Sparciy Paern of Cash Flows Marix A programming funcion quadprogin MATLAB requires parameers in he following form: 1 min x 2 x Hx + f x s.. Mx z (22) where H and M marices, f,z and x vecors. Therefore we need o pu marices and vecors of he models in he above forma; namely, we have o define marices H and M and vecors f and z. Le s pu objecive funcion Ad p 2 of discree model in he following form: Ad p 2 = (Ad p) (Ad p) = (d A p )(Ad p) = d A Ad d A p p Ad + p p = d (A A)d 2p Ad + p p = 1 2 d (2A A)d + ( 2A p) d + p p Therefore, H = 2A A and f = 2A p. For he coninuous approximaion, we wan o minimize Ad p 2 = ABx p 2. Hence we simply replace he marix A wih he produc of marices A and B and have he following resuls: H = 2(AB) AB and f = 2(AB) p. Then we se he variables vecor x equal o vecor d, which complees he ransformaion. To add monooniciy consrains on decreasing discoun facors d = [d 1,d 2,d 3,...,d N ] in discree approximaion, i.e. d 1 d 2 d 3 d N, we consruc such (N 1) N marix M and (N 1) 1 vecor z ha Md z: M = and z =
21 Mehods for Esimaing Term Srucure of Ineres Raes 21 Similarly, we can add monooniciy consrains on d() in he cubic splines model. Since we esimae d = Bx, he marix M is equal o is produc wih marix B from he model. Vecor z is a vecor of zeros. In Schaefer mehod we need he nonnegaive d() consrains, while heir monoonous decay is guaraneed by model specificaion. Therefore, he only consrain I impose is d(1) = b(1)x 0, where b(1) las Nh row of marix B (see formula 11). Therefore, in Schaefer mehod marix M = b(1), and vecor z is zero. References Bliss, R. (1997). Tesing Term Srucure Esimaion Mehods. In P. Boyle, G. Pennacchi, and P. Richken (Ed.), Advances in Fuures and Opions Research, Lizenberger, R.H., & Rolfo, J. (1984). An Inernaional Sudy of Tax Effecs on Governmen Bonds. The Journal of Finance, 39(1), Nelson, C. R., & Siegel, A. F. (1987). Parsimonious Modeling of Yield Curves. The Journal of Business, 60(4), 473. Schaefer, S. M. (1981). Measuring a TaxSpecific Term Srucure of Ineres Raes in he Marke for Briish Governmen Securiies. The Economic Journal, 91(362), 415. Söderlind, P., & Svensson, L. (1997). New echniques o exrac marke expecaions from financial insrumens. Journal of Moneary Economics, 40(2),
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