Methods for Estimating Term Structure of Interest Rates
|
|
- Letitia Bruce
- 6 years ago
- Views:
Transcription
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 Nelson-Siegel, 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.: , kais@bankofcanada.ca, 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) risk-free 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 non-negaive, 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 Nelson-Siegel 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 Nelson-Siegel approximaion is derived from he assumpion ha he spo raes follow a second-order 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 Nelson-Siegel 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 Nelson-Siegel 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 long-erm 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 shor-erm), 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 long-erm 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 shor-erm 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 medium-erm 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 leas-squares daa fiing by he Gauss-Newon 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 Nelson-Siegel, 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 (non-negaiviy) 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, Nelson-Siegel, 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 Nelson-Siegel 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 zero-coupon 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 L-shaped 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 inerior-reflecive 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 quasi-newon 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 Nelson-Siegel, 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 day-coun convenions for differen ypes of bonds. Then i can compue he cash flows marix using he specified day-coun 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 /Feb-2025C 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 15-Feb-2002C or 15-Feb-2002P. 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, day-coun 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 T-NOTE * 200 * * December 31, 2002 * T-BOND * 200 * 00 * Feb 15, 2005 * T-NOTE (5YR) * 200 * * Augus P * T-NOTE * 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 day-coun 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) Day-coun 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 longer-erm 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 N-h 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 Tax-Specific 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),
Morningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper March 3, 2009 2009 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion by
More informationUsing Rates of Change to Create a Graphical Model. LEARN ABOUT the Math. Create a speed versus time graph for Steve s walk to work.
2.4 Using Raes of Change o Creae a Graphical Model YOU WILL NEED graphing calculaor or graphing sofware GOAL Represen verbal descripions of raes of change using graphs. LEARN ABOUT he Mah Today Seve walked
More informationEvaluating Portfolio Policies: A Duality Approach
OPERATIONS RESEARCH Vol. 54, No. 3, May June 26, pp. 45 418 issn 3-364X eissn 1526-5463 6 543 45 informs doi 1.1287/opre.16.279 26 INFORMS Evaluaing Porfolio Policies: A Dualiy Approach Marin B. Haugh
More informationEconomics 487. Homework #4 Solution Key Portfolio Calculations and the Markowitz Algorithm
Economics 87 Homework # Soluion Key Porfolio Calculaions and he Markowiz Algorihm A. Excel Exercises: (10 poins) 1. Download he Excel file hw.xls from he class websie. This file conains monhly closing
More informationStock Return Expectations in the Credit Market
Sock Reurn Expecaions in he Credi Marke Hans Bysröm * Sepember 016 In his paper we compue long-erm sock reurn expecaions (across he business cycle) for individual firms using informaion backed ou from
More informationPaul M. Sommers David U. Cha And Daniel P. Glatt. March 2010 MIDDLEBURY COLLEGE ECONOMICS DISCUSSION PAPER NO
AN EMPIRICAL TEST OF BILL JAMES S PYTHAGOREAN FORMULA by Paul M. Sommers David U. Cha And Daniel P. Gla March 2010 MIDDLEBURY COLLEGE ECONOMICS DISCUSSION PAPER NO. 10-06 DEPARTMENT OF ECONOMICS MIDDLEBURY
More informationA Liability Tracking Portfolio for Pension Fund Management
Proceedings of he 46h ISCIE Inernaional Symposium on Sochasic Sysems Theory and Is Applicaions Kyoo, Nov. 1-2, 214 A Liabiliy Tracking Porfolio for Pension Fund Managemen Masashi Ieda, Takashi Yamashia
More informationCapacity Utilization Metrics Revisited: Delay Weighting vs Demand Weighting. Mark Hansen Chieh-Yu Hsiao University of California, Berkeley 01/29/04
Capaciy Uilizaion Merics Revisied: Delay Weighing vs Demand Weighing Mark Hansen Chieh-Yu Hsiao Universiy of California, Berkeley 01/29/04 1 Ouline Inroducion Exising merics examinaion Proposed merics
More informationA Probabilistic Approach to Worst Case Scenarios
A Probabilisic Approach o Wors Case Scenarios A Probabilisic Approach o Wors Case Scenarios By Giovanni Barone-Adesi Universiy of Albera, Canada and Ciy Universiy Business School, London Frederick Bourgoin
More informationHomework 2. is unbiased if. Y is consistent if. c. in real life you typically get to sample many times.
Econ526 Mulile Choice. Homework 2 Choose he one ha bes comlees he saemen or answers he quesion. (1) An esimaor ˆ µ of he oulaion value µ is unbiased if a. ˆ µ = µ. b. has he smalles variance of all esimaors.
More informationDYNAMIC portfolio optimization is one of the important
, July 2-4, 2014, London, U.K. A Simulaion-based Porfolio Opimizaion Approach wih Leas Squares Learning Chenming Bao, Geoffrey Lee, and Zili Zhu Absrac This paper inroduces a simulaion-based numerical
More informationThe t-test. What We Will Cover in This Section. A Research Situation
The -es 1//008 P331 -ess 1 Wha We Will Cover in This Secion Inroducion One-sample -es. Power and effec size. Independen samples -es. Dependen samples -es. Key learning poins. 1//008 P331 -ess A Research
More informationCALCULATORS: Casio: ClassPad 300 ClassPad 300 Plus ClassPad Manager TI: TI-89, TI-89 Titanium Voyage 200. The Casio ClassPad 300
The Casio ClassPad 300 CC Edwards 1950 1955 1960 1965 1970 23.0 23.8 24.4 24.5 24.2 1975 1980 1985 1990 1995 24.7 25.2 25.5 25.9 26.3 The able shows how he average age of he firs marriage of Japanese women
More informationStrategic Decision Making in Portfolio Management with Goal Programming Model
American Journal of Operaions Managemen and Informaion Sysems 06; (): 34-38 hp://www.sciencepublishinggroup.com//aomis doi: 0.648/.aomis.0600.4 Sraegic Decision Making in Porfolio Managemen wih Goal Programming
More informationKEY CONCEPTS AND PROCESS SKILLS. 1. An allele is one of the two or more forms of a gene present in a population. MATERIALS AND ADVANCE PREPARATION
Gene Squares 61 40- o 2 3 50-minue sessions ACIVIY OVERVIEW P R O B L E M S O LV I N G SUMMARY Sudens use Punne squares o predic he approximae frequencies of rais among he offspring of specific crier crosses.
More informationInterpreting Sinusoidal Functions
6.3 Inerpreing Sinusoidal Funcions GOAL Relae deails of sinusoidal phenomena o heir graphs. LEARN ABOUT he Mah Two sudens are riding heir bikes. A pebble is suck in he ire of each bike. The wo graphs show
More informationThe Current Account as A Dynamic Portfolio Choice Problem
Public Disclosure Auhorized Policy Research Working Paper 486 WPS486 Public Disclosure Auhorized Public Disclosure Auhorized The Curren Accoun as A Dynamic Porfolio Choice Problem Taiana Didier Alexandre
More informationWhat the Puck? an exploration of Two-Dimensional collisions
Wha he Puck? an exploraion of Two-Dimensional collisions 1) Have you ever played 8-Ball pool and los he game because you scrached while aemping o sink he 8-Ball in a corner pocke? Skech he sho below: Each
More informationLifecycle Funds. T. Rowe Price Target Retirement Fund. Lifecycle Asset Allocation
Lifecycle Funds Towards a Dynamic Asse Allocaion Framework for Targe Reiremen Funds: Geing Rid of he Dogma in Lifecycle Invesing Anup K. Basu Queensland Universiy of Technology The findings of he Mercer
More informationEvaluating the Performance of Forecasting Models for Portfolio Allocation Purposes with Generalized GRACH Method
Advances in mahemaical finance & applicaions, 2 (1), (2017), 1-7 Published by IA Universiy of Arak, Iran Homepage: www.amfa.iauarak.ac.ir Evaluaing he Performance of Forecasing Models for Porfolio Allocaion
More informationAP Physics 1 Per. Unit 2 Homework. s av
Name: Dae: AP Physics Per. Uni Homework. A car is driven km wes in hour and hen 7 km eas in hour. Eas is he posiive direcion. a) Wha is he average velociy and average speed of he car in km/hr? x km 3.3km/
More informationConstructing Absolute Return Funds with ETFs: A Dynamic Risk-Budgeting Approach. July 2008
Consrucing Absolue Reurn Funds wih ETFs: A Dynamic Risk-Budgeing Approach July 2008 Noël Amenc Direcor, EDHEC Risk & Asse Managemen Research Cenre Professor of Finance, EDHEC Business School noel.amenc@edhec-risk.com
More informationBootstrapping Multilayer Neural Networks for Portfolio Construction
Asia Pacific Managemen Review 17(2) (2012) 113-126 Boosrapping Mulilayer Neural Neworks for Porfolio Consrucion Chin-Sheng Huang a*, Zheng-Wei Lin b, Cheng-Wei Chen c www.apmr.managemen.ncku.edu.w a Deparmen
More informationMonte Carlo simulation modelling of aircraft dispatch with known faults
Loughborough Universiy Insiuional Reposiory Mone Carlo simulaion modelling of aircraf dispach wih known fauls This iem was submied o Loughborough Universiy's Insiuional Reposiory by he/an auhor. Ciaion:
More informationOverview. Do white-tailed tailed and mule deer compete? Ecological Definitions (Birch 1957): Mule and white-tailed tailed deer potentially compete.
COMPETITION BETWEEN MULE AND WHITE- TAILED DEER METAPOPULATIONS IN NORTH-CENTRAL WASHINGTON E. O. Garon, Kris Hennings : Fish and Wildlife Dep., Univ. of Idaho, Moscow, ID 83844 Maureen Murphy, and Seve
More informationAn Alternative Mathematical Model for Oxygen Transfer Evaluation in Clean Water
An Alernaive Mahemaical Model for Oxygen Transfer Evaluaion in Clean Waer Yanjun (John) He 1, PE, BCEE 1 Kruger Inc., 41 Weson Parkway, Cary, NC 27513 Email: john.he@veolia.com ABSTRACT Energy consumpion
More informationQUANTITATIVE FINANCE RESEARCH CENTRE. Optimal Time Series Momentum QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 353 January 15 Opimal Time Series Momenum Xue-Zhong He, Kai Li and Youwei
More informationTime-Variation in Diversification Benefits of Commodity, REITs, and TIPS 1
Time-Variaion in Diversificaion Benefis of Commodiy, REITs, and TIPS 1 Jing-zhi Huang 2 and Zhaodong Zhong 3 This Draf: July 11, 2006 Absrac Diversificaion benefis of hree ho asse classes, Commodiy, Real
More informationKINEMATICS IN ONE DIMENSION
chaper KINEMATICS IN ONE DIMENSION Secion 2.1 Displacemen Secion 2.2 Speed and Velociy 1. A paricle ravels along a curved pah beween wo poins P and Q as shown. The displacemen of he paricle does no depend
More informationIdiosyncratic Volatility, Stock Returns and Economy Conditions: The Role of Idiosyncratic Volatility in the Australian Stock Market
Idiosyncraic Volailiy, Sock Reurns and Economy Condiions: The Role of Idiosyncraic Volailiy in he Ausralian Sock Marke Bin Liu Amalia Di Iorio RMIT Universiy Melbourne Ausralia Absrac This sudy examines
More informationProportional Reasoning
Proporional Reasoning Focus on Afer his lesson, you will be able o... solve problems using proporional reasoning use more han one mehod o solve proporional reasoning problems When you go snowboarding or
More informationMarket Timing with GEYR in Emerging Stock Market: The Evidence from Stock Exchange of Thailand
Journal of Finance and Invesmen Analysis, vol. 1, no. 4, 2012, 53-65 ISSN: 2241-0998 (prin version), 2241-0996(online) Scienpress Ld, 2012 Marke Timing wih GEYR in Emerging Sock Marke: The Evidence from
More informationSIMULATION OF WAVE EFFECT ON SHIP HYDRODYNAMICS BY RANSE
1 h Inernaional Conference on Sabiliy of Ships and Ocean Vehicles 591 SIMULATION OF WAVE EFFECT ON SHIP HYDRODYNAMICS BY RANSE Qiuxin Gao, Universiy of Srahclyde, UK, Gao.q.x@srah.ac.uk Dracos Vassalos,
More informationSimulation based approach for measuring concentration risk
MPRA Munich Personal RePEc Archive Simulaion based approach for measuring concenraion risk Kim, Joocheol and Lee, Duyeol UNSPECIFIED February 27 Online a hp://mpra.ub.uni-muenchen.de/2968/ MPRA Paper No.
More informationRolling ADF Tests: Detecting Rational Bubbles in Greater China Stock Markets
Singapore Managemen Universiy Insiuional Knowledge a Singapore Managemen Universiy Disseraions and Theses Collecion (Open Access) Disseraions and Theses 2008 Rolling ADF Tess: Deecing Raional Bubbles in
More informationThe safe ships trajectory in a restricted area
Scienific Journals Mariime Universiy of Szczecin Zeszyy Naukowe Akademia Morska w Szczecinie 214, 39(111) pp. 122 127 214, 39(111) s. 122 127 ISSN 1733-867 The safe ships rajecory in a resriced area Zbigniew
More informationBetting Against Beta
Being Agains Bea Andrea Frazzini and Lasse H. Pedersen * This draf: Ocober 5, 2010 Absrac. We presen a model in which some invesors are prohibied from using leverage and oher invesors leverage is limied
More informationTesting Portfolio Efficiency with Non-Traded Assets: Taking into Account Labor Income, Housing and Liabilities
Tesing Porfolio Efficiency wih Non-Traded Asses: Taking ino Accoun Labor Income, Housing and Liabiliies Roy Kouwenberg Mahidol Universiy and Erasmus Universiy Roerdam Thierry Pos Erasmus Universiy Roerdam
More informationAutomatic air-main charging and pressure control system for compressed air supplies
Auomaic air-main charging and pressure conrol sysem for compressed air supplies Type PCS A module from he sysem -vacorol Swiching on-off a compressed air uni in a compressed air supply generally akes place
More informationTime & Distance SAKSHI If an object travels the same distance (D) with two different speeds S 1 taking different times t 1
www.sakshieducaion.com Time & isance The raio beween disance () ravelled by an objec and he ime () aken by ha o ravel he disance is called he speed (S) of he objec. S = = S = Generally if he disance ()
More informationSemi-Fixed-Priority Scheduling: New Priority Assignment Policy for Practical Imprecise Computation
Semi-Fixed-Prioriy Scheduling: New Prioriy Assignmen Policy for Pracical Imprecise Compuaion Hiroyuki Chishiro, Akira Takeda 2, Kenji Funaoka 2 and Nobuyuki Yamasaki School of Science for Open and Environmen
More information67.301/1. RLP 10: Pneumatic volume-flow controller. Sauter Components
7.0/ RL 0: neumaic volume-flow conroller How energ efficienc is improved For demand-led conrol of he air volume in office rooms. Areas of applicaion Conrol of he suppl and exhaus air of individual rooms
More information2. JOMON WARE ROPE STYLES
Proceedings of he IIEEJ Image Elecronics and Visual Compuing Workshop 2012 Kuching, Malaysia, November 21-24, 2012 A SIMULATION SYSTEM TO SYNTHESIZE ROPE ROLLING PATTERNS IN A VIRTUAL SPACE FOR RESEARCH
More informationCALCULATION OF EXPECTED SLIDING DISTANCE OF BREAKWATER CAISSON CONSIDERING VARIABILITY IN WAVE DIRECTION
CALCULATION OF EXPECTED SLIDING DISTANCE OF BREAKWATER CAISSON CONSIDERING VARIABILITY IN WAVE DIRECTION SU YOUNG HONG School of Civil, Urban, and Geosysem Engineering, Seoul Naional Universiy, San 56-1,
More informationMachine Learning for Stock Selection
Machine Learning for Sock Selecion Rober J. Yan Compuer Science Dep., The Uniersiy of Wesern Onario jyan@csd.uwo.ca Charles X. Ling Compuer Science Dep., The Uniersiy of Wesern Onario cling@csd.uwo.ca
More informationMeasuring Potential Output and Output Gap and Macroeconomic Policy: The Case of Kenya
Universiy of Connecicu DigialCommons@UConn Economics Working Papers Deparmen of Economics Ocober 2005 Measuring Poenial Oupu and Oupu Gap and Macroeconomic Policy: The Case of Kenya Angelica E. Njuguna
More informationThe Construction of a Bioeconomic Model of the Indonesian Flying Fish Fishery
Marine Resource Economics, Volume 0, pp. 357372 0738-360/95 $3.00 +.00 Prined in he U.S.A. All righs reserved. Copyrigh 995 Marine Resources Foundaion The Consrucion of a Bioeconomic Model of he Indonesian
More information3. The amount to which $1,000 will grow in 5 years at a 6 percent annual interest rate compounded annually is
8 h Grade Eam - 00. Which one of he following saemens is rue? a) There is a larges negaive raional number. b) There is a larges negaive ineger. c) There is a smalles ineger. d) There is a smalles negaive
More informationANALYSIS OF RELIABILITY, MAINTENANCE AND RISK BASED INSPECTION OF PRESSURE SAFETY VALVES
ANALYSIS OF RELIABILITY, MAINTENANCE AND RISK BASED INSPECTION OF PRESSURE SAFETY VALVES Venilon Forunao Francisco Machado Mechanical Engineering Dep, Insiuo Superior Técnico, Av. Rovisco Pais, 049-00,
More informationA Stable Money Demand: Looking for the Right Monetary Aggregate
A Sable Money Demand: Looking for he Righ Moneary Aggregae Pedro Teles Federal Reserve Bank of Chicago, CEPR. Ruilin Zhou Pennsylvania Sae Universiy January, 2005 Absrac In his paper, we argue ha M1 is
More informationITG Dynamic Daily Risk Model for Europe
December 2010 Version 1 ITG Dynamic Daily Risk Model for Europe 2010 All righs reserved. No o be reproduced or reransmied wihou permission. 121610 29140 These maerials are for informaional purposes only,
More informationChapter / rev/min Ans. C / in. C mm Ans teeth Ans. C / mm Ans.
Chaper 13 13-1 dp 17 / 8 15 in N 110 dg dp 15 4375 in N3 544 NG PdG 84375 35 eeh C 15 4375 / 35 in Ans Ans ng 1600 15 / 60 400 rev/min Ans p m 3 mm Ans C 3 15 60 115 mm Ans 13-13-3 N G 16 4 64 eeh Ans
More informationReliability Design Technology for Power Semiconductor Modules
Reliabiliy Design Technology for Power Semiconducor Modules Akira Morozumi Kasumi Yamada Tadashi Miyasaka 1. Inroducion The marke for power semiconducor modules is spreading no only o general-purpose inverers,
More informationPerformance Attribution for Equity Portfolios
PERFORMACE ATTRIBUTIO FOR EQUITY PORTFOLIOS Performance Aribuion for Equiy Porfolios Yang Lu and David Kane Inroducion Many porfolio managers measure performance wih reference o a benchmark. The difference
More informationAsset Allocation with Higher Order Moments and Factor Models
Asse Allocaion wih Higher Order Momens and Facor Models Kris Boud (VU Brussel, Amserdam) Based on join research wih: Wanbo Lu (SWUFE) and Benedic Peeers (Finvex Group) 1 The world of asse reurns is non-normal.
More informationCorresponding Author
Inernaional Journal of Scienific & Engineering Research Volume 9, Issue 4, April-2018 562 STRUCTURAL ANALYSIS OF NON-LINEAR PIPE BENDS 1 KALIKI HEMANTH, M.IMTHIYAS SHERIFF, KODAM VINEETH KUMAR, 2 M.ANANDRAJ,
More informationKeywords: overfishing, voluntary vessel buy back programs, backward bending supply curve, offshore fisheries in Taiwan
EVALUATION AND SIMULATION OF FISHING CAPACITY AND BACKWARD- BENDING SUPPLY OF THE OFFSHORE FISHERY IN TAIWAN Chin-Hwa Sun, Insiue of Applied Economics, Naional Taiwan Ocean Universiy, jsun@mail.nou.edu.w
More informationFINVEX WHITE PAPER ON ASSET ALLOCATION WITH RISK FACTORS
FINVEX WHITE PAPER ON AET ALLOCATION WITH RIK FACTOR By Dr Kris Boud PhD Professor of Finance & Research Parner a Finvex Group Benedic Peeers Co-Founder Finvex Group July 3 Execuive ummary In his paper,
More informationGuidance Statement on Calculation Methodology
Guidance Saemen on Calculaion Mehodology Adopion Dae: 28 Sepember 200 Effecive Dae: January 20 Reroacive Applicaion: No Required www.gipssandards.org 200 CFA Insiue Guidance Saemen on Calculaion Mehodology
More informationOptimal Portfolio Strategy with Discounted Stochastic Cash Inflows
Journal of Mahemaical Finance 3 3 3-37 hp://dxdoiorg/436/jmf33 Published Online February 3 (hp://wwwscirporg/journal/jmf) Opimal Porfolio raegy wih iscouned ochasic Cash nflows Charles Nkeki eparmen of
More informationThe APT with Lagged, Value-at-Risk and Asset Allocations by Using Econometric Approach
Proceedings of he 16 Inernaional Conference on Indusrial Engineering and Operaions Managemen Deroi, USA, Sepember 3-5, 16 he AP wih Lagged, Value-a-Risk and Asse Allocaions by Using Economeric Approach
More informationThe Measuring System for Estimation of Power of Wind Flow Generated by Train Movement and Its Experimental Testing
Energy and Power Engineering, 2014, 6, 333-339 Published Online Ocober 2014 in SciRes. hp://www.scirp.org/journal/epe hp://dx.doi.org/10.4236/epe.2014.611028 The Measuring Sysem for Esimaion of Power of
More informationMarket timing and statistical arbitrage: Which market timing opportunities arise from equity price busts coinciding with recessions?
Journal of Applied Finance & Banking, vol.1, no.1, 2011, 53-81 ISSN: 1792-6580 (prin version), 1792-6599 (online) Inernaional Scienific Press, 2011 Marke iming and saisical arbirage: Which marke iming
More informationName Class Date. Step 2: Rearrange the acceleration equation to solve for final speed. a v final v initial v. final v initial v.
Skills Workshee Mah Skills Acceleraion Afer you sudy each sample problem and soluion, work ou he pracice problems on a separae shee of paper. Wrie your answers in he spaces provided. In 1970, Don Big Daddy
More informationNBER WORKING PAPER SERIES DIVERSIFICATION AND THE OPTIMAL CONSTRUCTION OF BASIS PORTFOLIOS. Bruce N. Lehmann David M. Modest
NBER WORKING PAPER SERIES DIVERSIFICATION AND THE OPTIMAL CONSTRUCTION OF BASIS PORTFOLIOS Bruce N. Lehmann David M. Modes Working Paper 9461 hp://www.nber.org/papers/w9461 NATIONAL BUREAU OF ECONOMIC
More informationMacro Sensitive Portfolio Strategies
Marke Insigh Macro Sensiive Porfolio Sraegies Marke Insigh Macro Sensiive Porfolio Sraegies Macroeconomic Risk and Asse Cash Flows Kur Winkelmann, Raghu Suryanarayanan, Ludger Henschel, and Kaalin Varga
More informationFORECASTING TECHNIQUES ADE 2013 Prof Antoni Espasa TOPIC 1 PART 2 TRENDS AND ACCUMULATION OF KNOWLEDGE. SEASONALITY HANDOUT
FORECASTING TECHNIQUES ADE 2013 Prof Anoni Espasa TOPIC 1 PART 2 TRENDS AND ACCUMULATION OF KNOWLEDGE. SEASONALITY HANDOUT February 2013 MAIN FACTORS CAUSING TRENDS Increases in populaion. Seady inflaion.
More informationChapter : Linear Motion 1
Te: Chaper 2.1-2.4 Think and Eplain: 1-3 Think and Sole: --- Chaper 2.1-2.4: Linear Moion 1 NAME: Vocabulary: disance, displacemen, ime, consan speed, consan elociy, aerage, insananeous, magniude, ecor,
More information2017 MCM/ICM Merging Area Designing Model for A Highway Toll Plaza Summary Sheet
Team#55307 Page 1 of 25 For office use only T1 T2 T3 T4 Team Conrol Number 55307 Problem Chosen B For office use only F1 F2 F3 F4 2017 MCM/ICM Merging Area Designing Model for A Highway Toll Plaza Summary
More informationSan Francisco State University ECON 560 Fall Midterm Exam 2. Tuesday, October hour, 15 minutes
San Francisco Sae Universiy Micael Bar ECON 560 Fall 207 Miderm Exam 2 Tuesday, Ocober 3 our, 5 minues Name: Insrucions. Tis is closed book, closed noes exam. 2. No calculaors or elecronic devices of any
More informationCan Optimized Portfolios Beat 1/N?
Can Opimized Porfolios Bea 1/N? This disseraion is presened in par fulfillmen of he requiremen for he compleion of an MSc in Economics in he Deparmen of Economics, Universiy of Konsanz, and an MSc in Economics
More informationMULTIVARIATE RISK-RETURN DECISION MAKING WITHIN DYNAMIC ESTIMATION
Economic Analysis Working Papers.- 7h Volume Number 11 MULIVARIAE RISK-REURN DECISION MAKING WIHIN DYNAMIC ESIMAION Josip Arnerić 1, Elza Jurun, and Snježana Pivac, 3 Universiy of Spli, Faculy of Economics,
More informationExplore Graphs of Linear Relations. 1. a) Use a method of your choice to determine how much water will be needed each day of a seven-day cruise.
. Graphing Linear Relaions Focus on Aer his lesson, ou will be able o graph linear relaions mach equaions o linear relaions wih graphs solve problems b graphing a linear relaion and analsing he graph Tina
More informationApplication of System Dynamics in Car-following Models
Applicaion of Sysem Dynamics in Car-following Models Arif Mehmood, rank Saccomanno and Bruce Hellinga Deparmen of Civil Engineering, Universiy of Waerloo Waerloo, Onario, Canada N2 3G1 E-mail: saccoman@uwaerloo.ca
More informationWhat is a Practical (ASTM C 618) SAI--Strength Activity Index for Fly Ashes that can be used to Proportion Concretes Containing Fly Ash?
2017 World of Coal Ash (WOCA) Conference in Lexingon, KY - May 9-11, 2017 hp://www.flyash.info/ Wha is a Pracical (ASTM C 618) SAI--Srengh Aciviy Index for Fly Ashes ha can be used o Proporion Concrees
More informationEXAMINING THE FEASIBILITY OF PAIRED CLOSELY-SPACED PARALLEL APPROACHES
EXAMINING THE FEASIBILITY OF PAIRED CLOSELY-SPACED PARALLEL APPROACHES Seven J. Landry and Amy R. Priche Georgia Insiue of Technology Alana GA 30332-0205 ABSTRACT Paired closely-spaced parallel approaches
More informationPerformance Optimization of Markov Models in Simulating Computer Networks
Proceedings of he World Congress on Engineering and Copuer Science 9 Vol I WCECS 9, Ocober -, 9, San Francisco, USA Perforance Opiizaion of Marov Models in Siulaing Copuer Newors Nisrine Sinno, Hussein
More informationSources of Over-Performance in Equity Markets: Mean Reversion, Common Trends and Herding
The Universiy of Reading THE BUSINESS SCHOOL FOR FINANCIAL MARKETS Sources of Over-Performance in Equiy Markes: Mean Reversion, Common Trends and Herding ISMA Cenre Discussion Papers in Finance 2003-08
More informationOn convexity of SD efficiency sets - no short sales case
4. mezinárodní konference Řízení a modelování finančních rizik Osrava VŠB-U Osrava Ekonomická fakula kaedra Financí.-. září 008 On conveiy of SD efficiency ses - no shor sales case Miloš Kopa Absrac his
More informationInstruction Manual. Rugged PCB type. 1 Terminal Block. 2 Function. 3 Series Operation and Parallel Operation. 4 Assembling and Installation Method
Rugged PCB ype Insrucion Manual 1 Terminal Block Funcion.1...4.5.6.7 Inpu volage range Inrush curren limiing Overcurren proecion Overvolage proecion Oupu volage adjusmen range Isolaion Remoe ON/OFF E9
More informationReproducing laboratory-scale rip currents on a barred beach by a Boussinesq wave model
See discussions, sas, and auhor profiles for his publicaion a: hps://www.researchgae.ne/publicaion/9977 Reproducing laboraory-scale rip currens on a barred beach by a Boussinesq wave model Aricle in Journal
More informationMODEL SELECTION FOR VALUE-AT-RISK: UNIVARIATE AND MULTIVARIATE APPROACHES SANG JIN LEE
MODEL SELECTION FOR VALUE-AT-RISK: UNIVARIATE AND MULTIVARIATE APPROACHES By SANG JIN LEE Bachelor of Science in Mahemaics Yonsei Universiy Seoul, Republic of Korea 999 Maser of Business Adminisraion Yonsei
More informationDetection of activity cycles from capture-recapture data
See discussions, sas, and auhor profiles for his publicaion a: hps://www.researchgae.ne/publicaion/264644551 Deecion of aciviy cycles from capure-recapure daa Aricle in cological nomology February 1986
More informationRevisiting the Growth of Hong Kong, Singapore, South Korea, and Taiwan, From the Perspective of a Neoclassical Model
Revisiing he Growh of Hong Kong, Singapore, Souh Korea, and Taiwan, 978-2006 From he Perspecive of a Neoclassical Model Shu-shiuan Lu * Naional Tsing Hua Univereseiy December, 2008 Absrac This paper sudies
More informationDo Competitive Advantages Lead to Higher Future Rates of Return?
Do Compeiive Advanages Lead o Higher Fuure Raes of Reurn? Vicki Dickinson Universiy of Florida Greg Sommers Souhern Mehodis Universiy 2010 CARE Conference Forecasing and Indusry Fundamenals April 9, 2010
More informationUrban public transport optimization by bus ways: a neural network-based methodology
Urban Transpor XIII: Urban Transpor and he Environmen in he 21s Cenury 347 Urban public ranspor opimizaion by bus ways: a neural nework-based mehodology M. Migliore & M. Caalano Deparmen of Transporaion
More informationTransit Priority Strategies for Multiple Routes Under Headway-Based Operations
Transi Prioriy Sraegies for Muliple Roues Under Headway-Based Operaions Yongjie Lin, Xianfeng Yang, Gang-Len Chang, and Nan Zou This paper presens a ransi signal prioriy (TSP) model designed o consider
More informationScienceDirect. Cycling Power Optimization System Using Link Models of Lower Limbs with Cleat-Shaped Biaxial Load Cells
Available online a www.sciencedirec.com ScienceDirec Procedia Engineering 72 ( 20 ) 8 7 The 20 conference of he Inernaional Spors Engineering Associaion Cycling Power Opimizaion Sysem Using ink Models
More informationThe credit portfolio management by the econometric models: A theoretical analysis
The credi porfolio managemen by he economeric models: A heoreical analysis Abdelkader Derbali To cie his version: Abdelkader Derbali. The credi porfolio managemen by he economeric models: A heoreical analysis.
More informationSPECIAL WIRE ROPES The Value Line
SPECIAL WIRE ROPES The Value Line INTRODUCTION Qualiy Producs, Ousanding Service and Comprehensive Technical Suppor I s wha oday s indusries expec from heir supplier parners. And ha s wha WireCo WorldGroup
More informationOnline Portfolio Selection: A Survey
Online Porfolio Selecion: A Survey BIN LI, Wuhan Universiy STEVEN C. H. HOI, Nanyang Technological Universiy Online porfolio selecion is a fundamenal problem in compuaional finance, which has been exensively
More informationDynamics of market correlations: Taxonomy and portfolio analysis
Dynamics of marke correlaions: Taxonomy and porfolio analysis J.-P. Onnela, A. Chakrabori, and K. Kaski Laboraory of Compuaional Engineering, Helsinki Universiy of Technology, P.O. Box 9203, FIN-02015
More informationEvaluation of a car-following model using systems dynamics
Evaluaion of a car-following model using sysems dynamics Arif Mehmood, rank Saccomanno and Bruce Hellinga Deparmen of Civil Engineering, Universiy of Waerloo Waerloo, Onario, Canada N2 3G1 Tel: 519 888
More informationProceedings of the ASME 28th International Conference on Ocean, Offshore and Arctic Engineering OMAE2009 May 31 - June 5, 2009, Honolulu, Hawaii
Proceedings of he ASME 28h Inernaional Conference on Ocean, Offshore and Arcic Engineering OMAE29 May 31 - June 5, 29, Honolulu, Hawaii OMAE29-79385 ANALYSIS OF THE TUNNEL IMMERSION FOR THE BUSAN-GEOJE
More informationType Control action Setpoint range Air Weight Volume flow % capacity I n /h kg. Pressure diff. 1) Pa
7.0/ RL 0 & 0: neumaic volume-flow conroller Used in conjuncion wih an orifice plae or a dnamic pressure sensor and a pneumaic damper drive for conrolling he air volume in air-condiioning ssems. For fixed,
More informationINSTRUCTIONS FOR USE. This file can only be used to produce a handout master:
INSTRUCTIONS OR USE This file can only be used o produce a handou maser: Use Prin from he ile menu o make a prinou of he es. You may no modify he conens of his file. IMPORTNT NOTICE: You may prin his es
More informationLEWA intellidrive. The mechatronic All-in-One pump system. intelligent flexible dynamic high precision. Foto: ratiopharm
The mecharonic All-in-One pump sysem Foo: raiopharm inelligen flexible dynamic high precision For diverse applicaions: a limiless range of poenial uses Phoo: raiopharm Mixing wo media in one pump head:
More informationReview of Economics & Finance Submitted on 27/03/2017 Article ID: Mackenzie D. Wood, and Jungho Baek
Review of Economics & Finance Submied on 27/03/2017 Aricle ID: 1923-7529-2017-04-63-09 Mackenzie D. Wood, and Jungho Baek Facors Affecing Alaska s Salmon Permi Values: Evidence from Brisol Bay Drif Gillne
More informationPortfolio Efficiency: Traditional Mean-Variance Analysis versus Linear Programming
Porfolio Efficiency: Tradiional Mean-Variance Analysis versus Linear Programming Seve Eli Ahiabu Universiy of Torono Spring 003 Please send commens o Sephen.ahiabu@uorono.ca I hank Prof. Adonis Yachew
More informationA Study on the Powering Performance of Multi-Axes Propulsion Ships with Wing Pods
Second Inernaional Symposium on Marine Propulsors smp amburg Germany une A Sudy on he Powering Performance of Muli-Axes Propulsion Ships wih Wing Pods eungwon Seo Seokcheon Go Sangbong Lee and ungil Kwon
More information