EMPIRICAL TESTS OF DURATION SPECIFICATIONS

EMPIRICAL TESTS OF DURATION SPECIFICATIONS Iskandar Arifin Deparmen of Finance Universiy of Connecicu-Sorrs Carmelo Giaccoo 2 Deparmen of Finance Universiy of Connecicu-Sorrs Paul Hsu 2 Deparmen of Finance Universiy of Connecicu-Sorrs Version: June 4, 2009 VERY PRELIMINARY-PLEASE DO NOT CITE WITHOUT PERMISSION Corresponding auhor, Universiy of Connecicu-Sorrs, Deparmen of Finance, 200 Hillside Road, Uni 04, Sorrs, CT 06269 (email: iskandar.arifin@business.uconn.edu); 2 Universiy of Connecicu- Sorrs, Deparmen of Finance, 200 Hillside Road, Uni 04, Sorrs, CT 06269.

ABSTRACT We formulae AR(p) sochasic duraion measures wih consan volailiy and risk premium. Numerical exercise shows ha AR(p) sochasic duraion oversae he weighed ime o mauriy compared o Modified duraion. A closer look a he AR(p) model shows a need for model refinemen in order o beer fi empirical bond yield observaions. Possible models include models wih ime-varying volailiy and models for nominal ineres rae. 2

INTRODUCTION Ineres rae risk is defined as an unexpeced change in bond prices due o changes in he erm srucure; i is herefore essenial in managing a bond porfolio o idenify ineres rae risk. Duraion is he mos commonly used measure of ineres rae risk and accordingly is one of he mos imporan ools available o bond porfolio managers. I is hen no surprising ha duraion is of ineres o boh academic researchers and praciioners. Our sudy aemps o conribue o he duraion lieraure in wo ways. Firs, our sudy aemps o place a common heoreical language on he various duraion measures developed under he no-arbirage or equilibrium-based bond pricing models. While a number of no-arbirage or equilibrium-based bond pricing models have been developed following he seminal papers of Vasicek (977) and Cox, Ingersoll, and Ross (985), a common heoreical language is missing. We develop our duraion measure based on sochasic discoun facor: a sochasic process ha governs he pricing of saeconingen claims. We model he shor rae as an auoregressive process of order p and we model our opion-free discree-ime bond pricing model following hose of Backus, Foresi, and Telmer (998). To our knowledge, our model is he firs duraion measure derived under sochasic discoun facor. Second, we hope o bridge he gap beween he highly mahemaical no-arbirage or equilibrium-based duraion models and he discree-ime duraion models based on bond mahemaics similar in spiri o hose of Macaulay (938). Our aim is similar in spiri o 3

hose of Backus, Foresi, and Telmer (998). Our duraion model formulaion uses coninuous sae variable in discree ime which eases he model implemenaion for praciioner. We hope ha his sudy will add o he coninuing work of bridging he gap beween academics and praciioners. 2 LITERATURE REVIEW Duraion was firs defined by Macaulay (938) o compare loans wih differen paymen schedules based on he weighed average of heir income sream. The various duraion measures ha followed can generally be divided ino duraions based on radiional bond mahemaics and duraions based on no-arbirage or equilibrium-based bond pricing models. The firs duraion measure ha does no explicily model he underlying erm srucure behavior is Macaulay (938). Subsequen sudies include Cooper (977), Bierwag (977), Bierwag and Kaufman (979), and Khang (979) as discussed in Gulekin and Rogalski (984). These early duraion measures assume specific characerisics on he erm srucure movemen such as changes in boh he level and shape of he yield curve. For example, Khang (979) proposes differen duraion measures for specific changes in he yield curve. A more recen measures includes Chambers, Carleon, and McEnally (988), Ho (992), Nawalkha and Chambers (996), Nawalkha and Chambers (997) and Nawalkha, Soo, and Zhang (2003). Chambers, Carleon, and McEnally (988) indicaes ha ineres risk can only be measured by a vecor of numbers as opposed o a single number by proposing 4

he Duraion Vecor while Ho (992) proposes he Key Rae Duraion which associaes he price sensiiviy of a bond o muliple segmens of he yield curve. Nawalkha and Chambers (996) proposes M-Absolue duraion measure which allows for superior immunizaion compared o radiional Fisher and Weil (97) duraion by selecing a bond porfolio clusered around is planning horizon dae. Nawalkha and Chambers (997) and Nawalkha, Soo, and Zhang (2003) exend he analysis of Nawalkha and Chambers (996) ino a muli-facor M-Vecor. The developmen of duraion measures based on no-arbirage or equilibrium-based bond pricing models coincides wih Ingersoll, Skelon, and Weil (978) appraisal of Macaulay (938) duraion and developmens in sudies of bond pricing models. The seminal papers by Vasicek (977) and Cox, Ingersoll, and Ross (985) iniiae he no-arbirage or equilibrium-based bond pricing models. Ingersoll, Skelon, and Weil (978) provide an arbirage-based criicism of he Macaulay (938) duraion. To resolve his criicism, duraions measures ha ake ino accoun an explici random process driving models of he bond pricing were developed. Cox, Ingersoll and Ross (978) formally derives a duraion measure consisen wih he general equilibrium condiions of he Cox, Ingersoll, and Ross (985) erm srucure model. Our sudy models he shor rae as an AR(p) process similar in spiri o hose of Vasicek (977) and, unlike duraion previous models, uilizes sochasic discoun facor as he common elemen linking he bond pricing heory. The formulaion of our discree ime 5

model of bond pricing follows hose of Backus, Foresi and Telmer (998). A review of sochasic discoun facor is provided by Cochrane (2005). Empirical research of duraion s effeciveness as an ineres rae risk measure can be caegorized ino wo school of houghs. One school of hough looks a duraion s explanaory power over a cross secion of bond reurn while he oher looks a duraion s performance in an immunizaion sraegy. Gulekin and Rogalski (984) was he firs o examine duraion s explanaory power over a cross-secional bond reurn. Their sudy of seven differen duraion measures show ha duraion explains abou 50% of crosssecional variaion of bond reurn for U.S. reasury from 947 976. An exclusion of yield changes as an independen variable in he Gulekin and Rogalski (984) saisical mehod however leads o underesimaed R 2 values. Subsequen sudy by Ilmanen (992) correced he omission by including yield changes as an independen variable. Ilmanen (992) shows ha duraion explained 80% o 90% bond reurn variance from he period of 959 989. 3 THEORETICAL BACKGROUND 3. Sochasic Discoun Facor Sochasic discoun facor is he rae a which an invesor is willing o subsiue consumpion omorrow for consumpion oday. In oher words, sochasic discoun facor is he invesor s ineremporal marginal uiliy of subsiuion of consumpion. As per Cochrane (2005), given an invesor s one-period consumpion-invesmen decision, he We would like o hank Assaf Eisdorfer for observing he omission on Gulekin and Rogalski (984) which lead us o Ilmanen (992) 6

marginal uiliy of consuming less and buying more of he asse oday should be equal o marginal uiliy of consuming more of he asse in he fuure. Following Cochrane (2005), le p be he asse price a ime, d + be he dividend from he asse a ime +, c be he consumpion a ime, β be he subjecive discoun facor, U be he uiliy funcion, and x + be he asse payoff a ime + where x + = p + + d +. The firs order condiion for opimal consumpion and porfolio choice is hen x p U c = E U c x U c = E U c () + '( ) [ β '( + ) + ] '( ) [ β '( + ) ] p The lef hand side sands for he marginal uiliy cos of consuming one dollar less a ime. The righ hand side sands for he expeced marginal uiliy of invesing one dollar a ime, selling he dollar a ime +, and consuming he invesmen a ime +. Dividing equaion (4) by U (c ), we obain U '( c ) x x = E [ ] = E [ m ] (2) + + + β k, + U '( c ) p p where m + is he sochasic discoun facor. 3.2 Ineres Rae Risk and Macaulay Duraion Ineres rae risk is defined as unexpeced changes in bond prices due o changes in he erm srucure. Macaulay (938) defined he duraion measure D for a bond o measure bond price sensiiviy o ineres rae changes under infiniesimal parallel shif in erm srucure. P C D = = Y P + Y P T ( ) (3) ( ) ( ) + = 7

where C is he cash flow a ime, Y is he yield-o-mauriy, T is he ime o mauriy, and P is he bond price such ha T C P = = ( + Y ) (4) Equaion () can be rewrien as follows P = D ( Y ) (5) P As noed by Ilmanen (992), exension o non-fla erm srucure would require he use of spo raes as opposed o yield-o-mauriy. Forunaely, he use of spo raes as opposed o yield-o-mauriy has only negligible impac on duraion as observed by Ingersoll (983). I is imporan o noe as per Campbell, Lo, and MacKinlay (997) ha duraion is sensiiviy of n-period bond reurn o n-period yield and no sensiiviy o a -period yield. 3.3 Duraion Model based on p-facor Sochasic Process We model he shor rae Z as an auoregressive process of order p, an AR(p) process, and he sochasic discoun facor m + as 2 2 log m + = λ σ ε + Z λσ εε+ (6) 2 where λ is he risk premium and ε ~ N(0,). As shown in Appendix A, we can hen recursively obain a model for discoun bond price wih par value of $, b n,, a ime wih n periods o mauriy. We model discoun bond price model assuming ha he shor rae Z follows AR(), AR(2), and AR(3). 8

The discoun bond price model assuming ha he shor rae Z follows an auoregressive process of order, AR() is: b e + ( An BnZ ) n, = (7) where A = B = 0, A = 0, B = 0 0 A = λ σ + A + B ( ϕ ) Z ( λσ B σ ) 2 2 B = + B ϕ 2 2 2 n ε n n ε n ε n n while discoun bond price model assuming ha he shor rae Z follows an auoregressive process of order 2, AR(2) is: b e ( A n + B n Z + C n Z ) n, = (8) where A = B = C = D = 0, A = C = D = 0, B = 0 0 0 0 A = λ σ + A + B ( ϕ ϕ ϕ ) Z ( λσ B σ ) 2 2 B = + B ϕ + C 2 2 2 n ε n n 2 3 ε n ε n n n C = B ϕ + D D n n 2 n n = B ϕ n 3 The discoun bond price model assuming ha he shor rae Z follows an auoregressive process of order 3, AR(3) is: b e ( A n + B n Z + C n Z + D n Z 2 ) n, = (9) where 9

A = B = C = D = 0, A = C = D = 0, B = 0 0 0 0 A = λ σ + A + B ( ϕ ϕ ϕ ) Z ( λσ B σ ) 2 2 B = + B ϕ + C 2 2 2 n ε n n 2 3 ε n ε n n n C = B ϕ + D D n n 2 n n = B ϕ n 3 As per Appendix B, duraion measure for an AR(p) opion-free discoun bond wih par value of $F is D = ln( Fb ) (0) n, n, and duraion measure for an AR(p) opion-free coupon bond model V n, is D C b n i i, n, = ln( Cibi, ) () i= Vn, 4 NUMERICAL ANALYSIS 4. Parameer Esimaion 4.. Descripion of he Daa for Esimaion We esimaes he model parameers using CRSP (Cener for Research in Securiy Prices) U.S. erm srucure of ineres rae (discoun bond yield) wih mauriies of, 3, 6, and 9 monhs in addiion o, 2, 3, 4, and 5 years 2. The esimaion period runs from June 964 o December 995. The daa spans non-overlapping 379 monhs. Summary saisics of he annualized daa is presened in Table. 2 The same daase is used in Bansal and Zhou (2002). We would like o hank Ravi Bansal and Hou Zhou for providing us wih he daase. 0

4..2 Parameer Esimaion Mehods We esimae he following parameers from our models: he long run mean ( Z ), he consan ime-series residual sandard deviaion (σ ε ), he n-h auocorrelaion of he sae variable (φ n ) and he consan risk premium coefficien (λ). The parameers are esimaed using a wo-sep procedure. In he firs sep, we esimae he ime series parameers of he models: he long run mean ( Z ), he ime-series residual sandard deviaion (σ ε ) and is he n-h auocorrelaion of he sae variable (φ n ) using linear regression. In he second sep, we esimae he consan risk premium coefficien, λ, using a non-linear leas squares. The esimaion resuls are presened in Table 2. 4.2 Numerical Exercise We calculaed he duraion for monhly non-callable Unied Saes Treasury Bills, Noes and Bonds from January 996 o December 2006 obained from Cener for Research in Securiy Prices (CRSP). Bonds wih special ax feaures, flower bonds, as well as bonds missing relevan daa are excluded. The daa spans non-overlapping 32 monhs wih a oal of 23,35 individual observaions. The duraion summary saisics are available on Table 3. The resuls of our duraion calculaion shows ha AR(p) duraion performs worse as mauriy increases compared o Modified duraion. As shown in able 4, a mauriy of 0.5 years, our AR(p) duraion is comparable o Modified duraion. As he mauriy of he bond increase however, our AR(p) duraion oversae weighed ime o mauriy compared o Modified duraion.

5 CONCLUSION Empirically AR(p) duraion performs worse as a proxy of weighed ime o mauriy han Modified duraion in our forecas period. A closer look a he AR(p) model shows a need for model refinemen in order o beer fi empirical bond yield observaions. Possible model include models wih ime-varying volailiy and models for nominal ineres rae model. 2

REFERENCES Ai-Sahalia, Y., 996, Tesing Coninuous-Time Models of he Spo Ineres Rae. Review of Financial Sudies, 9(2), 385-426. Backus, D., S. Foresi, C. Telmer, 998, Discree-Time Models of Bond Pricing. Cambridge, MA: Naional Bureau of Economic Research. Bansal, R. and H. Zhou, 2002, Term Srucure of Ineres Raes wih Regime Shifs. Journal of Finance, 57, 997-2043. Bierwag, G.O., 977, Immunizaion, Duraion and he Term Srucure of Ineres Raes, Journal of Financial and Quaniaive Analysis, 2, December, 725-42. Bierwag, G.O. and G. Kaufman, 978, Bond Porfolio Sraegy Simulaions: A Criique, Journal of Financial and Quaniaive Analysis, 3, Sepember, 59-26. Campbell, J. Y., Lo, A. W., and MacKinlay, A. C., 997, The Economerics of Financial Markes, Princeon, NJ: Princeon Universiy Press. Chambers, D.R., W.T. Carleon, and R.W. McEnally, 988, Immunizing Defaul-Free Bond Porfolios wih a Duraion Vecor, Journal of Financial and Quaniaive Analysis, 23(), March. 3

Cochrane, J.H., 2005, Asse Pricing, Princeon, NJ: Princeon Universiy Press. Cooper, Ian A., 977, Asse Values, Ineres-Rae Changes, and Duraion, Journal of Financial and Quaniaive Analysis, 2, December, 70-24. Cox, J., J. Ingersoll, S. Ross, 978, Duraion and he Measuremen of Basis Risk, Journal of Business, 52, 5-6. Cox, J., J. Ingersoll, S. Ross, 985, A Theory of he Term Srucure of Ineres Raes, Economerica, 53, 385-408. Fabozzi, F.J., 993, Fixed Income Mahemaics, Chicago, IL: Probus Publishing Company, pp. 74. Fabozzi, F.J., 2004, Fixed Income Analysis, Frank J. Fabozzi Associaes, pp. 6. Fama, E.F. and J. MacBeh, 973, Risk, Reurn, and Equilibrium: Empirical Tess, Journal of Poliical Economy, May/June, 607-636. Fisher, L. and R.L. Weil, 97, Coping wih he Risk of Ineres Rae Flucuaions: Reurn o Bondholders from Naive and Opimal Sraegies, Journal of Business, 43(4), Ocober, 408-43. 4

Gibbons, M. and K. Ramaswamy, 993, A es of he Cox, Ingersoll, Ross model of he erm srucure, Review of Financial Sudies, 6, 69-658. Greene, W. H., 2002, Economeric Analysis, 5h Ed., Prenice Hall, New Jersey, NY. Gulekin, N.B. and R.J. Rogalski, 984, Alernaive Duraion Specificaions and he Measuremen of Basis Risk: Empirical Tess, Journal of Business, 57(2), 24-264. Ho, T., 992, Key Rae Duraions: Measures of Ineres Rae Risks, Journal of Fixed Income, Sepember, 29-44. Ilmanen, A., 992, How Well Does Duraion Measure Ineres Rae Risk?, Journal of Fixed Income (4). Ingersoll, J.E., J. Skelon, and R.L. Weil, 978, Duraion Fory Years Laer, Journal of Financial and Quaniaive Analysis, November, 627-650. Ingersoll, J.E., 983, Is Immunizaion Feasible?: Evidence from he CRSP Daa, in: G.O. Bierwag e al., op. ci., pp. 63-82. Khang, C., 979, Bond Immunizaion When Shor-Term Raes Flucuae More han Long-Term Raes, Journal of Financial and Quaniaive Analysis, 3, November: 085-90. 5

Lansein, R. and W. F. Sharpe, 978, Duraion and Securiy Risk, Journal of Financial and Quaniaive Analysis, 3(4), 653-668. Lierman, R. and J. Scheinkman, 99, Common Facors Affecing Bond Reurns, Journal of Fixed Income, (), June. Macaulay, F.R., 938, Some Theoreical Problems Suggesed by he Movemens of Ineres Raes, Bond Yields, and Sock Prices in he U.S. Since 856, New York: Naional Bureau of Economic Research. Nawalkha, S.M. and D.R. Chambers, 996, An Improved Immunizaion Sraegy: M- Absolue, Financial Analyss Journal, 52(5). Nawalkha, S.M. and D.R. Chambers, 997, The M-Vecor Model: Derivaion and Tesing of Exensions o M-Square, Journal of Porfolio Managemen, 23(2). Nawalkha, S.M., G.M. Soo, and J. Zhang, 2003, Generalized M-Vecor Models for Hedging Ineres Rae Risk, Journal of Banking and Finance, 27(8), 58-604. Vasicek, O., 977, An Equilibrium Characerizaion of he Term Srucure, Journal of Financial Economics 5(2), 77-88. 6

Table : Summary saisics for annualized U.S. Treasury erm srucure from June 964 o December 995 Mauriy -monh 3-monhs 6-monhs 9-monhs -year 2-years 3-years 4-years 5-years Mean 6.44600 6.76742 6.944729 7.085367 7.29530 7.338939 7.49544 7.69863 7.689058 Sd. Dev. 2.648370 2.7277 2.703928 2.685806 2.599354 2.52404 2.44905 2.403834 2.37477 Skewness.2099.2768.577.0269.030724 0.977756 0.96475 0.92627 0.8793 Kurosis 4.59063 4.523702 4.34679 4.60509 3.909804 3.6673 3.589720 3.50635 3.35350 Table repors he mean, sandard deviaion, skewness, and kurosis of annualized U.S. Treasury erm srucure from June 964 o December 995. 7

Table 2: Parameer esimaion resuls for AR(p) discoun bond model using U.S. Treasury erm srucure from June 964 o December 995 Model ϕ ϕ2 ϕ3 σ ε λ Z AR() 0.957608 - - 0.00048 0.5057 0.0008 AR(2) 0.879656 0.0837-0.00047 0.57970 0.00009 AR(3) 0.886068 0.5465-0.082922 0.00046 0.4924 0.0007 Table 2 repors he esimaion resuls for he AR(p) discoun bond model. 8

Table 3: Summary saisics for AR(), AR(2), AR(3), and Macaulay Duraion Specificaion from March 996 o December 2006 AR() Duraion AR(2) Duraion AR(3) Duraion Macaulay Mean 6.936376 6.759994 6.695676 4.287787 Sd. Dev 8.84907 8.904 8.877 4.8007 Skewness 2.209346 2.96086 2.20906.254747 Kurosis 7.55883 7.32464 7.202382 3.3823 Table 3 repors AR(), AR(2), AR(3), and Macaulay Duraion using he U.S. Treasury daa from March 996 o December 2006. Table 4: Average Duraion Mauriy (in years) Modified AR() AR(2) AR(3) 0.5 0.522 0.52 0.52 0.500.309.307.307 0.988 2 2.807 2.802 2.804.870 5 7.848 7.839 7.840 4.370 0 6.76 6.70 6.706 8.003 Table 4 repors AR(), AR(2), AR(3), and Macaulay Duraion based on mauriy using he U.S. Treasury daa from March 996 o December 2006. The daa is an average duraion for securiies wih 0 days around 0.5,, 2, 5, and 0 years of mauriy. 9

APPENDIX A Similar in spiri o Backus, Foresi and Telmer (998), we consruc our opion-free discoun bond pricing model. The sochasic discoun facor is = δ + λσ ε () log m + Z ε + where m + is he sochasic discoun facor a ime +, Z is a sae variable a ime, λ is he risk premium, δ is a free variable and ε ~ N(0,). We resric he risk premium λ o be a posiive number. The sae variable Z follows a normal auoregressive process of order p, AR(p): Z = ( ϕ ϕ2... ϕ p ) Z + ϕz + ϕ2z 2 +... + ϕ pz p + σ εε (2) where ε ~ N(0,), Z is he long run mean, σ ε is he residual sandard deviaion and φ n is he n-h auocorrelaion of he sae variable. Le b n, denoe he value of a discoun bond a ime wih n periods o mauriy wih he following pricing relaion b = E ( m ) b = E ( m b ) (3) n, + + n+, + n, + bn +, Assume ha a any ime, he value of a maured discoun bond b 0, is $. The value of a discoun bond wih one-period (n = ) o mauriy a ime is hen: b = E ( m b ) = E ( m ) (4), + 0, + + Given m + as a lognormal random variable and imposing no arbirage opporuniy, we obain sochasic discoun facor as 2 2 log m + = λ σ ε + Z λσ εε+ (5) 2 20

where sae variable Z is he -monh Treasury bill rae a ime. We can hen recursively obain heoreical price of discoun bond. We will illusrae he process assuming ha he sae variable Z follows an AR(3) process. We assume he general form for discoun bond price as b e ( A n + B n Z + C n Z + D n Z 2 ) n, = (6) where A = B = C = D = 0, A = C = D = 0, B = 0 0 0 0 The value of a discoun bond wih wo-period unil mauriy (n = 2) a ime is 2 2 λ σε Z + λσεε+ 2 Z+ 2, = ( +, + ) = ( ) b E m b E e e 2 2 λ σε Z + λσεε+ 2 ( ϕ ϕ2 ϕ3 ) Z ϕz ϕ2z ϕ3z 2 σ εε+ ( e ) = E e 2 2 λ σε Z ( ϕ ϕ2 ϕ3 ) Z ϕz ϕ2z ϕ3z 2 + ( λσε σ ε ) ε+ 2 = E ( e ) = e = e = e 2 2 2 λ σε Z ( ϕ ϕ2 ϕ3 ) Z ϕz ϕ2z ϕ3z 2 + ( λσε σ ε ) 2 2 2 2 2 ( λ σε + ( ϕ ϕ2 ϕ3 ) Z ( λσε σε ) ) ( + ϕ ) Z ϕ2z ϕ3z 2 2 2 A2 B2 Z C2Z D2Z 2 (7) Generalizing he recursion above, he price of an opion-free discoun bond wih n periods o mauriy a ime, denoed by b n, has he following general form: b e ( A n + B n Z + C n Z + D n Z 2 ) n, = (8) where 2

A = B = C = D = 0, A = C = D = 0, B = 0 0 0 0 A = λ σ + A + B ( ϕ ϕ ϕ ) Z ( λσ B σ ) 2 2 B = + B ϕ + C 2 2 2 n ε n n 2 3 ε n ε n n n C = B ϕ + D D n n 2 n n = B ϕ n 3 The price of he discoun wih par of $ bond above can be viewed as a discoun facor for opion-free coupon bonds V where n n, = Cibi, (9) i= V where C i is fuure cash flows associaed wih each period. 22

APPENDIX B Duraion is a poin elasiciy of bond price o yield-o-mauriy. As per Campbell, Lo, and MacKinlay (997), duraion is sensiiviy of n-period bond reurn o n-period yield and no sensiiviy o a -period yield. Macaulay duraion for discoun bond wih n period-omauriy a ime is formulaed as D n, b y n, n, = (0) y b n, n, As per Campbell, Lo, and MacKinlay (997), he coupon bond can be seen as a porfolio of discoun bonds and he Macaulay duraion of a coupon bond is he presen-valueweighed average of he underlying discoun bond s duraion. The duraion for a discoun bond wih n period-o-mauriy a ime is formulaed as n. Given a coupon bond V n,, duraion measure for V n, is herefore D C b n i i, n, = i () i= V n, 23