Parsimonious Modeling and Forecasting of Corporate Yield Curve

Transcription

1 Journal of Forecastng J. Forecast. 28, (2009) Publshed onlne 5 September 2008 n Wley InterScence ( Parsmonous Modelng and Forecastng of Corporate Yeld Curve WEI-CHOUN YU* AND DONALD M. SALYARDS Economcs and Fnance Department, Wnona State Unversty, Wnona, Mnnesota, USA ABSTRACT Ths paper nvestgates the senstvty of out-of-sample forecastng performance over a span of dfferent parameters of l n the dynamc Nelson Segel three-factor AR(1) model. Frst, we fnd that the ad hoc selecton of l s not optmal. Second, we fnd a substantal dfference n factor dynamcs between nvestment-grade and speculatve-grade corporate bonds from 1994:12 to 2006: 4. Thrd, we suggest that the three-factor model s suffcent to explan the man varatons of corporate yeld changes. Fnally, the parsmonous Nelson Segel three-factor AR(1) model remans compettve n the out-of-sample forecastng of corporate yelds. Copyrght 2008 John Wley & Sons, Ltd. key words corporate yeld curve; Nelson Segel model; three-factor model; out-of-sample INTRODUCTION Over the past decades, many fnancal nsttutons and academes have devoted consderable resources to the modelng of corporate bonds. Most authors focus on explanng the specal features of corporate bonds not found n Treasury bonds, such as default probabltes, recovery rates, tax premum, rsk premum, and callable propertes. However, lttle attenton has been pad to the modelng and forecastng of corporate yeld curves from a tme-seres perspectve (especally out-of-sample forecastng), even though forecastng the term structure of corporate nterest rates plays a crucal role n portfolo management, household and busness fnance decsons, and dervatve prcng and hedgng. There are two man strands of bond modelng, each wth ther own focus. Frst, the no-arbtrage models (Ho and Lee, 1986; Hull and Whte, 1990) only specfy on term-structure cross-sectonal fttng. Second, the affne equlbrum models (Vascek, 1977; Cox et al., 1985; Duffe and Kan, 1996; Da and Sngleton, 2000) only pay attenton to nstantaneous short rates and result n poor forecasts (Duffee, 2002). In addton to these two man strands, Debold and L (2006, hereafter DL) and Debold et al. (2006, hereafter DRA) offer the alternatve models concentratng on term-structure * Correspondence to: We-Choun Yu, Economcs and Fnance Department, Wnona State Unversty, Somsen 319E, Wnona State Unversty, Wnona, MN 55987, USA. E-mal: wyu@wnona.edu Copyrght 2008 John Wley & Sons, Ltd.

2 74 W.-C. Yu and D. M. Salyards forecastng. They extend the parsmonous three-factor (exponental components) yeld curve model, proposed by Nelson and Segel (1987, hereafter NS), to the dynamc form. DL uses a smple twostep approach, n whch they frst estmate three factors, then model and forecast them. On the other hand, DRA propose a one-step approach that uses the state-space model to smultaneously do factor estmaton, modelng, and forecastng. DL s dynamc NS factor autoregressve of order 1 (AR(1)) model produces superor out-of-sample forecastng on US Treasury yelds and outperforms the current popular compettors, ncludng the forward rate regresson models of Fama and Blss (1987) and Cochrane and Pazzes (2005). Yu and Zvot (2007, hereafter YZ) analyze the pros and cons of DL s and DRA s models. Furthermore, they extend the samples to corporate yelds. By and large, they argue that the smple AR(1) method of DL s stll the most accurate forecastng model on both Treasury and corporate bonds compared to many other compettors. Nevertheless, n order to compare and evaluate dfferent models accuracy, DL, DRA and YZ all smplfy the set up of the dynamc NS model. They all assume that the l, whch governs the shape and the decay rate of the factor loadngs, s fxed. To some extent, they argue, the parameters are not senstve enough to affect the forecastng results. In ths paper, we nvestgate the senstvty of out-of-sample forecastng performance over a span of dfferent parameters of l n the dynamc NS three-factor AR(1) model. We fnd that the ad hoc selecton of l s not optmal. Based on the prncpal component method, we fnd consderable dfferences of factor loadngs between nvestment-grade and speculatve-grade corporate bonds from 1994:12 to 2006:4. Moreover, we suggest that a three-factor model s suffcent to explan the man varatons of corporate yeld changes, whch s about 92%. The Parsmonous NS three-factor AR(1) model remans compettve n the out-of-sample forecast of corporate yelds. The rest of the paper s organzed as follows. The next secton ntroduces the NS model. The thrd secton explores the senstvty of parameters and evaluates the forecasts. The fourth secton nvestgates unobserved factors by the prncpal-component method. The ffth secton concludes. MODELS Nelson Segel model Nelson and Segel (1987) ntroduce a parsmonous but nfluental three-factor model: 1 1 e yt( τ) = β t + β t e β t + e (1) where t s the maturty of bond, whch usually ranges from 3 months to 30 years. The parameter l determnes the rate of exponental decay. The three factors are b 1t, b 2t, and b 3t. The factor loadng on b 1t s 1, whch s a constant that never des out. It loads equally at all maturtes. In other words, a change n b 1t changes all yelds unformly. Therefore, t s called level factor. When the maturty becomes larger, b 1t plays a more mportant role n formng yelds wth respect to smaller factor loadngs on b 2t and b 3t (for nstance, y t ( ) = b 1t ). In consequence, b 1t s called long-term factor. 1 The orgnal Nelson Segel model s slghtly dfferent from equaton (1), whch s a modfed one by DL. DL explans the reasons for ths revson.

3 Parsmonous Modelng of Corporate Yeld Curve 75 The factor loadng on b 2t s (1 e lt )/lt, whch s a functon decayng fast and monotoncally to zero. It loads short rates more heavly than long rates; consequently, t changes the slope of the yeld curve. b 2t s short-term factor, whch s also called slope factor. 2 The factor loadng on b 3t s (1 e lt )/lt e lt, whch s a functon startng at zero (so not short term) and decayng to zero (not long term) wth a humped shape n the mddle. It loads medum rates more heavly. Accordngly, b 3t s the medum-term factor, whch s also called curvature factor because an ncrease n b 3t wll ncrease the yeld curve curvature. Ltterman and Schenkman (1991) use the prncpalcomponent method and fnd the same number and smlar pattern of factors. Fgure 1 llustrates the NS exponental factor loadngs wth respect to fve dfferent values of l from 0.01 to It s clear that the bgger l s, the faster wll be the decay rate of the slope factor. The l wll also affect the shape of curvature factor. The loadng on curvature factor s maxmzed at a maturty of 180, 62, 37, 27, and 21 months as l s 0.01, 0.03, 0.05, 0.07, and 0.09, respectvely. Loadngs 1.2 level slope, L=0.01 curvature slope, L=0.03 curvature slope, L=0.05 curvature slope, L=0.07 curvature slope, L=0.09 curvature Maturty (n Months) Fgure 1. Factor loadngs of Nelson Segel model by l. Nelson Segel model s as follows: e e y( τ) 1 1 = β1+ β2 β e + 3 where t s the maturty of bond, whch s from 0 to 360 months contnuously. The parameter l determnes the rate of exponental decay 2 Some authors defne yeld curve slope as y t ( ) y t (0), whch equals b 1t (b 1t + b 2t ) = b 2t.

4 76 W.-C. Yu and D. M. Salyards Despte ts smplcty, the NS three-factor model s consstent wth a varety of stylzed facts regardng the yeld curve. For example, the combnaton of three factors can capture dfferent knds of yeld curves, ncludng upward slopng, downward slopng, humped and nverted humped. Fgure 2 presents the yeld curve predcted by the NS model and the realzed spot nterest rates for AAA, AA, A+, A, BB+, and BB corporate bonds n February The sold lne s the predcted yeld by NS model and the dot s the true value. The l s chosen based on the best n-sample predctons. It seems that the range of l between 0.06 and 0.08 s approprate to ft the yeld curve. Therefore, DL fx l as and DRA and YZ set l as Dynamc Nelson Segel AR(1) models Notwthstandng ts success n cross-sectonal nterpolaton, the NS model s not bult for out-ofsample forecasts. DL extend t to the dynamc form. They propose a benchmark model: AR(1) method based on equaton (1). Frst, they estmate three factors b 1t, b 2t, and b 3t by the least-squared method, perod by perod. Second, they use the estmated factors bˆ 1t, bˆ 2t, and bˆ 3t to forecast the factors by a unvarate AR(1) model: ˆ β ˆ ˆ ˆ jt, + ht= cj+ γ jβjt, j =123,, (2) Thrd, they convert the projected factors, whch were computed n equaton (2), to the predcted term structure of yelds by combnng the correspondng factor loadngs as follows: y ( τ ) = β + β 1 e + β t+ ht 1, t+ ht 2, t+ ht 3, t+ ht 1 e e (3) DL provdes out-of-sample forecasts evaluatons from the dynamc NS model as well as other forecast compettors. They nclude (1) random walk (the forecast s always no change), (2) slope regresson, (3) Fama Blss forward rate regresson, (4) Cochrane Pazzes (2005) forward curve regresson, (5) AR(1) on yeld levels, (6) vector AR(1): VAR(1) on yeld levels, (7) VAR(1) on yeld changes, (8) error correcton model ECM(1) wth one common trend, (9) ECM(1) wth two common trends, and (10) ECM(1) wth three common trends. The DL model concludes that other competng models are suboptmal compared to the AR(1) NS factor model for Treasury bonds from 1985:1 to 2000:12. Usng Treasury and corporate yelds from 1994 to 2006, YZ suggest that the NS three-factor AR(1) model s stll very robust n the out-of-sample forecastng accuracy. PARAMETER SENSITIVITY As mentoned above, DL, DRA and YZ do not consder the senstvty effect of parameter l on the model s out-of-sample forecast performance because they focus on a comparson over dfferent models. Instead, n ths paper we explore the senstvty of l on the NS three-factor AR(1) model, whch has been shown to be the most compettve model. Data We use the same dataset as used n YZ: end-of-month zero-coupon bond rate for S&P ratng AAA, AA, A+, A, A, BB+, BB, B, and B corporate bonds from December 1994 to Aprl 2006, taken from Bloomberg. We analyze the raw zero-coupon yeld data drectly, unlke DL, who use the

5 Parsmonous Modelng of Corporate Yeld Curve 77 AAA, λ = AA, λ = Yeld Curve Yeld Curve Term Structure Term Structure Rate 4 Rate Maturty Maturty A+, λ = Yeld Curve Term Structure A, λ = Yeld Curve Term Structure Rate 4 Rate Maturty Maturty BB+, λ = Yeld Curve Term Structure 9 BB, λ = 0.06 Yeld Curve Term Structure Rate 5 Rate Maturty Maturty Fgure 2. Nelson Segel corporate yeld curve n-sample ft on February 2004

6 78 W.-C. Yu and D. M. Salyards Fama Blss (1987) flter for cleanng ther data. The maturtes of bonds nclude 3, 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240, and 360 months. We fnd two stylzed facts about the sample mean. Frst, when the maturtes become longer, nterest rates are hgher. Ths mples that, on average, the yeld curves are upward slopng and term spreads are postve. The term-spread tme seres for each ratng bond s shown n Fgure 3. In each chart of Fgure 3, we present the nterest rate seres of all maturtes. When the bands between curves become wder, the term spread s larger. We notce that the term spread s narrow n 1999 and 2000 (pre-recesson perod), then becomes wder between 2002 and 2005 (recovery perod), and shrnks n Second, as the ratng declnes, the nterest rate s hgher. That sad, on average, the credt spread s postve. Fgure 4 shows the credt spread tme seres for each corporate bond. For example, the AAA credt spread s the dfference between the AAA nterest rate and Treasury yeld at the same maturty, such as AAA 1-year nterest rate mnus Treasury 1-year nterest rate, and the AAA 2-year nterest rate mnus Treasury 2-year nterest rate. Fgure 4 matches the stylzed fact of the asymmetrc mpact n a recesson. The recesson n 2001 rased the credt spreads lttle n nvestment-grade bonds (above BBB ), whle t ncreased the credt spreads much more n speculatve-grade (below BBB, also called non-nvestment-grade, hgh-yeld) rated bonds. Forecast evaluaton The success of a tme-seres model les n ts out-of-sample forecast performance. We access the model s out-of-sample predctve accuracy on the 1, 6, 12, 36, and 60-month-ahead out-of-sample forecast. For nstance, on the 1-month-ahead out-of-sample forecast, we use data from 1994:12 to 2004:4 as n sample and 2004:5 through 2006:4 (24 predctons) as out of sample. In order to estmate the parameters of the model based on the most up-to-date nformaton avalable at the gven tme of forecast, the predctons are conducted va a rollng wndow. For example, we use data from 1994:12 to 2004:4 to forecast 2004:5; and we use data from 1995:1 to 2004:5 to forecast 2004:6. For a longer forecast horzon, n order to keep the out-of-sample sze as 24 predctons, we have to decrease the n-sample sze. For nstance, for the 6-month-ahead forecast, we use data from 1994:12 to 2003:11; accordngly, our frst predcton wll be on 2004:5. There are two reasons for ths. Frst, we wll have a large enough out-of-sample sze to test the model s forecast. Second, we can compare dfferent models predctabltes on smaller n-sample szes, especally for long-term forecasts. It s worth notng that we use terated forecasts nstead of drect forecasts for the mult-perodahead predctons. Marcellno et al. (2005) argue that terated forecasts are more effcent when the model s correctly specfed and the forecast performance wll mprove wth the predcton horzon. They conclude that the drect mult-perod forecasts are more robust when the model s msspecfed. The comparson of terated and drect forecasts s not the man focus of the paper. Therefore, we use the former based on the assumpton of correct model specfcaton. To evaluate the out-of-sample forecastng performance, we use root mean squared forecast errors (RMSE): ( yt+ h yˆ t+ ht) (4) where y t+h are the realzed yelds and ŷ t+h t are the h-month-ahead predctons made by the AR(1) NS model. The smaller the RMSE, the better the model forecasts. For smplcty we only show the aggregate evaluaton results on 1, 6, 12, 36, and 60-month-ahead forecasts of 3, 6, 12, 36, 60, 120,

7 Parsmonous Modelng of Corporate Yeld Curve 79 AAA corporate yeld AA corporate yeld A+ corporate yeld A corporate yeld A- corporate yeld BB+ corporate yeld BB- corporate yeld B corporate yeld B- corporate yeld Fgure 3. Interest yelds tme seres and ts mpled term spread. In each chart we present all the maturtes (3, 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240, and 360 months) corporate nterest rates seres from 1994:12 to 2006:4

8 80 W.-C. Yu and D. M. Salyards AAA spread AA spread A+ spread A spread A- spread BB+ spread BB- spread B spread B- spread Fgure 4. Credt spread tme seres. In each chart we present the maturtes (12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240, and 360 months) corporate credt spreads, whch s ts correspondng nterest rate mnus Treasury nterest rate seres from 1994:12 to 2006:4

9 Parsmonous Modelng of Corporate Yeld Curve 81 Table I. Root mean square errors of corporate yelds 1994: :4. Out-of-sample forecastng by dynamc Nelson Segel AR(1) model wth dfferentl l AAA AA A+ A A BB+ BB B B Nelson Segel model s as follows: e yt( 1 τ) = β t + β t e β t + e where t s the maturty of bond, whch s 3, 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240, 360 months, respectvely. The parameter l determnes the rate of exponental decay. The NS AR(1) model s as follows: ˆ β ˆ ˆ ˆ jt, + h t= cj+ γ jβjt, j =123,, Root mean square errors (RMSEs) were calculated by the average of RMSEs of 1-month, 6-month, 12- month, 36-month, and 60-month-ahead out-of-sample forecastng on 3-month, 6-month, 12-month, 36-month, 60-month, 120-month, and 360- month corporate yelds. In-sample perod s from December 1994 to Aprl Out-of-sample perod s from May 2004 to Aprl and 360-month maturtes of corporate bonds n Table I. For example, each cell n Table I represents the sum of RMSE: 1 month ahead + 6 months ahead + 12 months ahead + 36 months ahead + 60 months ahead of the sum of 3, 6, 12, 36, 60, 120, and 360-month maturtes RMSE gven dfferent l values. Out-of-sample forecast results Surprsngly, very few of the lowest RMSEs were located n the range of as chosen n DL, DRA and YZ. For nvestment-grade bonds the optmal choce of l s 0.09, 0.095, 0.1, 0.1, and for AAA, AA, A+, A, and A, respectvely. Ther aggregate RMSEs are close: n the range

10 82 W.-C. Yu and D. M. Salyards of For speculatve-grade bonds, the optmal choce of l s 0.035, 0.06, 0.07, and 0.07 for BB+, BB, B, and B, respectvely. Ther aggregate RMSEs are dverse: from to If one has to choose a sngle value of l to forecast all ratngs of corporate bonds, the optmal choce would be 0.09, whch produces the lowest sum of aggregate RMSE. The fndngs suggest three thngs. Frst, the result of the model s predcton s senstve to the value of l. Second, the best n-sample ft does not guarantee the best out-of-sample forecasts, f we compare the results of Fgure 2 wth those of Table I. Nelson and Segel (1987) argue a smlar dea when they state: A more hghly parameterzed model that could follow all the wggles n the data s less lkely to predct well, n our vew, than a more parsmonous model that assumes more smoothness n the underlyng relaton than one observes n the data. Thrd, there s a bg gap n the optmal value of l between nvestment-grade and speculatve-grade corporate bonds. How do we nterpret the dfferentals n Table I? What are the underlyng natures of factors on nvestment and speculatve grade bonds? We offer an explanaton by usng the prncpal-component method n the next secton. PRINCIPAL-COMPONENT METHOD Introducton to prncpal-component method The prncpal-component method s a dmenson reducton technque that transforms a number of correlated varables nto a smaller number of uncorrelated varables from explanng the majorty of the nformaton n the sample covarance matrx of returns. Consder the factor representaton for multple tme seres data X t, ( = 1,... N, t = 1,... T): X = Λ F + e (5) t t t where Λ s the factor loadngs, F s the factor process, and e s the dosyncratc dsturbance. The factor loadngs, factor process, and dosyncratc errors are not observable. It s assumed that the dsturbances are..d., normally dstrbuted and ndependent of the factor process. Normalzng the covarance matrx of F to be an dentty matrx, the factor model covarance matrx s then Σ = ΛΛ + Ω (6) where Ω s the dagonal covarance matrx of e t. A root-t consstent and asymptotcally normal estmator, ˆΣ =( 1 T T ) X X X X, can be obtaned. t = 1 ( )( ) t t Unobserved common factors Here X t s the frst dfference of nterest rates to demean the data, denotes each maturty of each corporate bond, and t s from 1994:12 through 2006:4. We order t from AAA 3-month, 6-month, 12-month,..., 360-month, AA 3-month, 6-month,..., to B 360-month. Fgure 5 shows the R 2 of the regressons for each maturty of all corporate bond tme seres aganst the frst fve factors by the prncpal-component method. These R 2 are plotted as bar charts wth one chart for each factor and could be nterpreted as factor loadngs. From Fgure 5, t s apparent that factor loadng patterns dffer consderably between nvestment and speculatve grade bonds. Loosely speakng, we can nterpret that Factor 1 s the level factor affectng all corporate bonds. Factor 2 s the level factor affectng only nvestment-grade bonds.

11 Parsmonous Modelng of Corporate Yeld Curve 83 R-Squares for Factor 1 aaa aa ap a... am..... bbp bbm b... bm R-Squares for Factor 2 aaa aa ap.....a... am..... bbp..... bbm b... bm.... R-Squares for Factor 3 aaa aa ap.....a... am..... bbp..... bbm b... bm.... Fgure 5. Factor loadngs of all corporate yeld changes

12 84 W.-C. Yu and D. M. Salyards R-Squares for Factor 4 aaa....aa ap a... am bbp bbm b bm R-Squares for Factor 5 aaa....aa ap a... am bbp bbm b bm R-Squares for Factor 6 aaa......aa.....ap a... am..... bbp bbm b... bm Fgure 5. Contnued

13 Parsmonous Modelng of Corporate Yeld Curve 85 Factor 3 s the slope factor that loads heavly n the short ends of yelds. Factor 4 s the curvature factor that loads the medum term of maturtes of nvestment-grade bonds. Factors 5 and 6 are lkely to be curvature factors for B and B bonds. To take a closer look at Factor 3 (slope factor), the decay rate of loadngs for nvestment-grade bonds s faster than that for speculatve-grade bonds. Ths fndng explans the reason why the optmal values of nvestment-grade bonds are generally larger than those of speculatve-grade bonds, as shown n Table I. Accordng to Fgure 1, we know that as l rses there s a faster decay rate n the slope factor. Therefore, t s reasonable to see the dchotomy of the optmal value of l because the underlyng factor loadngs ndeed behave dfferently. Fgure 6 presents the factor loadngs on 15 maturtes of corporate bonds at the ndvdual ratng level (AAA, BB+). For AAA corporate yelds, Factor 1 s the level factor, Factor 2 s the slope factor, and Factor 3 s the curvature factor. For BB+ corporate yelds, Factor 1 s the level factor, Factor 2 s the curvature factor, and Factor 3 s the slope factor. It s reasonable to get hgher RMSE from speculatve-grade bonds than that from nvestment-grade bonds because the former s more volatle than the latter. For BB+ bonds, however, we get a lower RMSE (40.52) even compared to nvestment-grade bonds (Table I). The possble reason mght be found n the curvature factor n Fgure 6. We can see that the loadng of the curvature factor (from 3 months to 96 months) on the BB+ bond mmcs the curvature factor n the NS model well, so t s not surprsng to see the lowest RMSE. Number of factors Is a three-factor model suffcent to explan the major varaton of corporate yeld changes? Wll there be any gan n forecastng accuracy by addng one or two factors to the model n order to capture the dynamcs of default or credt rsk? Ltterman and Schenkman (1991) argue that three factors are suffcent to explan the movements of Treasury yelds. To fnd the optmal number of factors on corporate bonds, we mplement the prncpal-component method on each ratng corporate bond to get the margnal R 2 (varaton explaned) for the frst fve factors. Table II dsplays the results. For all corporate bonds Factor 1 (level factor) explans most of the varaton of yelds, rangng from 66.5% to 78.1%. Factor 2 explans the varaton from 10.6% to 14%. Factor 3 explans the varaton from 3.7% to 12.6%. The frst three factors explan around 92% of the yeld varatons. Surprsngly, there s no evdence to support the fourth factor, even for speculatve-grade bonds. Svensson (1994) proposes an extended NS model by addng one factor wth addtonal l as follows: e 1 e 1 1 e 2 yt = β1t + β2t t e t + β3 + β4 e 1 (7) 1 2 He argues that ths four-factor model provdes a better n-sample ft, especally for a rcher structure of yelds, and therefore has better estmatons of forward rates. We apply hs model n an AR(1) dynamc form and conduct the out-of-sample forecast evaluatons wth a range of l 1 and l 2. We fnd that Svensson s model produces nferor results compared to the NS three-factor model. CONCLUSION In ths paper, we nvestgate the senstvty of out-of-sample forecastng performance over a span of dfferent parameters of l n the dynamc Nelson Segel three-factor AR(1) model. Usng the S&P

14 86 W.-C. Yu and D. M. Salyards R-Squares for Factor 1, AAA R-Squares for Factor 1, BB+ 3m 6m 12m 24m 36m 48m 60m 72m 84m 96m 108m120m180m240m360m R-Squares for Factor 2, AAA 3m 6m 12m 24m 36m 48m 60m 72m 84m 96m 108m120m180m240m360m R-Squares for Factor 2, BB+ 3m 6m 12m 24m 36m 48m 60m 72m 84m 96m 108m120m180m240m360m R-Squares for Factor 3, AAA 3m 6m 12m 24m 36m 48m 60m 72m 84m 96m 108m120m180m240m360m R-Squares for Factor 3, BB+ 3m 6m 12m 24m 36m 48m 60m 72m 84m 96m 108m120m180m240m360m 3m 6m 12m 24m 36m 48m 60m 72m 84m 96m 108m120m180m240m360m Fgure 6. Factor loadngs of AAA and BB+ corporate yeld change

15 Parsmonous Modelng of Corporate Yeld Curve 87 Table II. Relatve mportance of factors (R 2 ) (percentage) Factor 1 Factor 2 Factor 3 Frst three factors explaned Factor 4 Factor 5 AAA AA A A A BB BB B B Data are from 1994:12 to 2006:4. Factors are estmated by prncpal-component method. ratng AAA, AA, A+, A, A, BB+, BB, B, and B corporate bonds from 1994:12 to 2006:4, we fnd that the ad hoc selecton of l s not optmal. Moreover, we fnd consderable dfferences of factor loadngs between nvestment-grade and speculatve-grade corporate bonds from 1994:12 to 2006:4. Based on the prncpal-component method, we suggest that the three factor-model s suffcent to explan the man varatons of corporate yelds: about 92%. The parsmonous Nelson Segel three factor AR(1) model s stll compettve n the out-of-sample forecastng of corporate yelds. The fndngs also mply two thngs. Frst, good n-sample fts do not guarantee good out-of-sample forecasts. It s more mportant to catch the underlyng relatons of data than to ft superfcally observed data. Second, there s no need to explore the fourth factor for default rsk and credt spreads of corporate bonds because ther dynamcs have been captured by three factors, especally the level factor. REFERENCES Cochrane J, Pazzes M Bond rsk prema. Amerca Economc Revew 95(1): Cox JC, Ingersoll JE, Ross SA A theory of the term structure of nterest rates. Econometrca 53: Da Q, Sngleton K Specfcaton analyss of affne term structure models. Journal of Fnance 55: Debold FX, L C Forecastng the term structure of government bond yelds. Journal of Econometrcs 130: Debold FX, Rudebusch GD, Aruoba SB The macroeconomy and the yeld curve: a dynamc latent factor approach. Journal of Econometrcs 131: Duffee G Term prema and nterest rates forecasts n affne models. Journal of Fnance 57: Duffe D, Kan R A yeld factor model of nterest rates. Mathematcal Fnance 6: Fama E, Blss R The nformaton n long-maturty forward rates. Amercan Economc Revew 77: Ho TSY, Lee S-B Term structure movements and prcng nterest rate contngent clams. Journal of Fnance 41: Hull J, Whte A Prcng nterest rate dervatve securtes. Revew of Fnancal Studes 3: Ltterman R, Schenkman J Common factors affectng bond returns. Journal of Fxed Income 1: Marcellno M, Stock J, Watson M A comparson of drect and terated multstep AR methods for forecastng macroeconomc tme seres. Journal of Econometrcs 135: Nelson CR, Segel A Parsmonous modelng of yeld curves. Journal of Busness 60(4):

16 88 W.-C. Yu and D. M. Salyards Svensson LEO Estmatng and nterpretng forward nterest rates: Sweden NBER Workng Paper No Vascek O An equlbrum characterzaton of the term structure. Journal of Fnancal Economcs 5: Yu W-C, Zvot E Forecastng the Term Structures of Treasury and Corporate Yelds: Dynamc Nelson Segel Models Evaluaton. Workng paper, Wnona State Unversty. Authors bographes: We-Choun Yu s an Assstant Professor n the Economcs and Fnance Department at Wnona State Unversty. He also serves on the board of drectors at the Mnnesota Economc Assocaton. He receved hs PhD degree n economcs at the Unversty of Washngton n Yu s research nterests are n tme seres econometrcs, fnance and macroeconomcs, n partcular lnear, nonlnear, and volatlty modelng and forecastng. He has varous publcatons n the felds of econometrcs and fnance. Donald M. Salyards s a Professor of Economcs at Wnona State Unversty. He receved hs PhD degree n economcs at Kansas State Unversty n Salyards focuses most of hs tme establshng and fundng new frms as an angel nvestor. Comfortex, Inc. and Wnona Pattern & Mold Company are hs most successful entrepreneural endeavors. Salyards runs the entrepreneurshp program at Wnona State Unversty. Authors address: We-Choun Yu and Donald Salyards, Economcs and Fnance Department, Wnona State Unversty, Somsen 319E, Wnona State Unversty, Wnona, MN 55987, USA.