Interest Rate Sensitivities of Bond Risk Measures Timothy Falcon Crack and Sanjay K. Nawalkha

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1 Ineres Rae Sensves of Bond Rsk Measures Tohy Falcon Crack and Sanjay K. awalkha We presen a sple expresson for he sensvy of duraon, convexy, and hgher-order bond rsk easures o changes n ersrucure shape paraeers. Our analyss enables fxed-ncoe porfolo anagers o capure he cobned effecs of shfs n ersrucure level, slope, and curvaure on any specfc bond rsk easure. These resuls are parcularly poran n envronens characerzed by volale neres raes. We provde sple nuercal exaples. In hghly volale neres rae envronens, changes n he er srucure of neres raes are ofen characerzed by exree varaons n level, slope, and curvaure. For exaple, as llusraed by Breeden 994: The slope of he er srucure oved fro a seep slope of over 3 bass pons n 987 o an nvered yeld curve negave slope jus wo years laer n 989, and hen back up o a very seep yeld curve n 992 wh a slope of 35 bass pons. These oveens n he slope are alos as large as he oveens n he levels of raes. pp. 6 7 In such volale neres rae envronens, bond rsk easures, such as duraon and convexy, ay change rapdly n response o he level and shape of he er srucure of neres raes. Alhough researchers have analyzed he sensvy of a bond s duraon o changes n he bond s yeld, lle s known abou he neres rae sensvy of duraon, convexy, and oher hgher-order Tohy Falcon Crack s asssan professor of fnance a Indana Unversy. Sanjay K. awalkha s assocae professor a he Unversy of Massachuses a Ahers.

2 2 bond rsk easures o changes n level, slope, and curvaure of he er srucure. We developed a heorecal fraework o answer he followng ypes of quesons: How does he duraon of a bond change wh respec o a change n he slope of he er srucure? How does he convexy of a bond change wh respec o a change n he level of he er srucure? Do he duraon and convexy of a barbell porfolo change ore rapdly han hose of a bulle porfolo? These quesons are relevan for bond-porfolo anagers, who are ofen requred o anan arge duraons for her porfolos. For exaple, ceran bond porfolos e.g., barbell porfolos ay experence rapd changes n her duraon; hence, he anagers of hese porfolos wll have o rebalance ofen o anan a arge duraon. Managers of such fnancal nsuons as coercal banks, savngs and loans assocaons, penson funds, and nsurance copanes ay also be concerned abou hese quesons. The Federal Depos Insurance Corporaon Iproveen Ac of 99 nsrucs regulaors o ake no accoun he neres rae rsk exposure of a bank n deernng s capal adequacy. In fac, he Federal Reserve Board and he Offce of Thrf Supervson have coe up wh specfc proposals o pleen he recoendaons n he FDIC Iproveen Ac. As a resul, housands of U.S. coercal banks and S&Ls have begun he process of easurng he duraons of her asses and lables. The anagers of hese deposory

3 3 nsuons hus need o know he sensves of her asse and lably duraons o general neres rae changes n order o anan arge levels of neres rae rsk exposure. Bond Rsk Measures The sples unzaon odel s he Macaulay duraon odel: I assues nfnesal parallel shfs n he er srucure Macaulay 938. The convexy and M 2 odels see oe 6 are ore realsc: They allow for nonnfnesal, nonparallel shfs Fong and Vascek 984; Fong and Fabozz 985; Grano 984; Berwag, Kaufan, and Laa 987, 988. The duraon vecor odels are even ore general: They recognze ha real-world shfs n er srucure ay be cobnaons of level changes, slope changes, curvaure changes, and so on Chabers, Carleon, and McEnally 988; rsan and Shores 988; Berwag e al Up o 95 percen of reurns o U.S. Treasury secury porfolos are explaned by er-srucure level shfs, slope shfs, and curvaure shfs Leran and Schenkan 99; Jones 99; Wllner 996; Jashdan and Zhu To reflec hs realy, we assued ha he connuously copounded nal yeld curve, r, s gven by he polynoal See Board of Governors of he Federal Reserve Syse 992 and Offce of Thrf Supervson In fac, he relave lack of porance of hgher-order changes n er-srucure shape allowed Jashdan and Zhu o resrc her aenon o a hree-facor yeld-curve odel usng only level, slope, and curvaure changes. Ther led-facor odel afer beng ade dscree va a ulnoal dsrbuon provdes copuaonal effcency n Mone Carlo sulaon of ulcurrency porfolos for rsk-anageen purposes.

4 4 r A A A2 2 A AK K, where A, A, and A2 are, respecvely, level, slope, and curvaure paraeers, and K s suffcenly large as n Chabers, Carleon, and McEnally 988, Chabers, Carleon, and Waldan984, and rsan and Shores. Wllner also used level, slope, and curvaure paraeers based on a odel of he yeld curve usng ranscendenal funcons. 3 Wllner used hese paraeers n conjuncon wh nonsandard level, slope, and curvaure duraons, however, o capure changes n bond prces. We addressed a dfferen bu relaed ssue sensvy of radonal duraon easures o changes n he shape of he yeld curve. Alhough sple, he generaly of Equaon allowed us o capure a wde class of changes n he shape of he er srucure. 4 Suppose a bond pays cash flows C a es,..., and cash flow F a e. Then, he bond has prce as follows: Ce r r Fe. 2 3 "Transcendenal" here eans no capable of beng deerned by any cobnaon of a fne nuber of equaons wh raonal negral coeffcens. 4 Alernavely, he connuously copounded yeld curve r can be expanded around H, where H s a parcular horzon, whch produces he equaon r B B H B 2 H 2... BK H K. The expanson n hs equaon s useful n dervng condons for unzng a bond porfolo a e horzon H. Unlke Equaon, n whch he paraeers A, A, and A 2 easure he changes n he level, slope, and curvaure n r a, he paraeers B, B, and B 2 easure he changes n he level, slope, and curvaure n r a horzon H. Because our purpose was o derve he sensves of duraon rsk easures o shor-er neres rae changes, no o derve unzaon condons, we have used Equaon even hough s

5 5 where e s Euler s sandard consan, approxaely equal o 2.78, ha s used n connuous copoundng. If he yeld curve now shfs n a nonnfnesal, nonparallel fashon fro r o r, hen 5 r A A A A A2 A2 2 A3 A AK AK K A A A2 2 A AK K, 3 where A A A for each. In Equaon 3, A s a level change, A s a slope change, A2 s a curvaure change, and so on. The new bond prce s Ce r r Fe. 4 Le. Then, can be shown ha he nsananeous reurn on he bond / sasfes Equaon 5: 6 less general han he equaon gven n hs noe. See awalkha and Chabers 997 for furher explanaon. 5 oe ha r here s no a aheacal dervave bu, raher, a perurbaon of he orgnal r. 6 The expanson n Equaon 5 s of he for D α, where l r r rl, {,..., }!!!, l A A Al α K l r r Ω r r r l l l l and Ωl,k s he collecon of ses of for {r,...,rl} ha sasfy boh

6 6 D D 2 [ A ] A D3 A M 2 D A M, A 2 2! A A K A 3 3! A! 5 where he sandard order- bond duraon easure, D, s defned as n Equaon 6. D Ce r r Fe 6 Equaon 5 s he duraon vecor odel. Iunzaon n hs conex requres achng D o H for, 2,.... Iunzng wh D2 and D3 n addon o D capures up o 9 percen of he rsk ha was no already capured by D accounng for a cobned oal of up o 95 percen of all and h l h h l h h r h l r h k. Thus, #Ωl,k s he nuber of ways you can draw k nubers wh replaceen fro he se {,,..., l} so ha he k nubers su o l and rh couns how any h s you use. For exaple, he fourh er n he expanson n Equaon 5 s D4α 4, where α 4 [ A 3 A A 2/2!

7 7 rsks. Usng he frs fve duraon easures provdes nearly perfec unzaon e.g., awalkha and Chabers 997. Takng he frs er only n Equaon 5 yelds D A, 7 whch s he radonal Macaulay duraon odel wh parallel shfs n he er srucure. The ore ers aken, he ore realsc he odel. For exaple, akng he frs wo ers n Equaon 5 gves D [ A ] A 2 D2 A. 2! 8 The coeffcen of D2 n Equaon 8 llusraes he poran dfference beween he radonal and ore recen vews of convexy. The radonal vew s ha he agnude of A 2 /2!.e., parallel shfs donaes he agnude of A.e., slope shfs. Thus, convexy s always desrable. The recen vew s ha he agnude of A donaes he agnude of A 2 /2!. Thus, he desrably of convexy depends on wheher he sgn of he slope change A s negave or posve hs vew s also conssen wh Fong and Fabozz. The A 2 /2! A 2 A /2! A 4 /4!]. A full dervaon of he resul fro frs prncples s

8 8 eprcal sudes of Kahn and Lochoff 99 and Lacey and awalkha 993 confr he ore recen vew of convexy. Takng he frs hree or ore ers n Equaon 5 produces duraon vecor odels of varous lenghs. These odels have been shown o prove hedgng perforance sgnfcanly Chabers, Carleon, and McEnally; awalkha and Chabers. ow, ake a look a how he bond rsk easures [D, D2, D3, and so on] change for a nonparallel change n he er srucure. Sensvy of Rsk Measures o onparallel Rae Changes Duraon, convexy, and oher hgher-order duraon easures capure he sensvy of bond reurns o nonparallel changes n neres raes changes n level, slope, curvaure, and so on. Ths secon explores how duraon, convexy, and oher hgher-order duraon easures heselves change wh nonparallel neres rae shfs. Resul: The duraon easure gven n Equaon 6 has he followng sensvy o general neres rae changes: 7 D D D D. A 9 avalable fro he auhors. 7 The M 2 easure Fong and Vascek; Fong and Fabozz s gven by M 2 D2 2HD. Equaon 9 ay be used drecly o deduce M 2 /A M 2 D [D 3 2HD 2]. Ths s of analogous funconal for o Equaon 9 excep ha D s replaced by M 2 M 2 2 [D 2HD ]2.

9 9 For he proof, see Appendx A. neres. To undersand he usefulness of Equaon 9, consder several cases of Case. Le and. Then, D [ D] A 2 D2. Hence, he sensvy of duraon o changes n er-srucure level s duraon squared nus convexy. Case 2. Le and. Then, D A D D2 D3. Hence, he sensvy of duraon o changes n er-srucure slope s gven by he produc of duraon and convexy nus D3. Case 3. Le 2 and. Then, D2 A D D2 D3. Concdenally, hs case s he sae as Case 2. The sensvy of convexy o changes n er-srucure level s gven by he produc of duraon and convexy nus D3. Case 4. Le 2 and. Then,

10 D 2 [ D2] A 2 D4. Hence, he sensvy of convexy o changes n er-srucure slope s convexy squared nus D4. These cases deonsrae he usefulness and generaly of he resul n Equaon 9. Indeed, he resul n Equaon 9 les one express he fne dfference D as follows: 8 D K K D A A [ D D D ] A. In he nex secon, we use Equaon o deonsrae he porance of lookng beyond parallel shfs when accounng for he sensvy of duraon and convexy o general er-srucure changes. uercal Exaples We presen a realsc exaple of how bond rsk easures change wh nonparallel neres rae changes. Before lookng a nubers, we noe ha shfs n er-srucure level, slope, and curvaure are no ndependen 8 We approxaed he fne dfference D usng he defnon of he oal dfferenal K D K D D dd da A. A A

11 Leran and Schenkan; Jones; Wllner; Mann and Raanlal 997. For exaple, s well known ha a shf upward downward n he er srucure s ypcally assocaed wh a flaenng seepenng of he er srucure Jones; Mann and Raanlal. Jones presened a arx showng how level, slope, and curvaure changes were correlaed n he pas. We consder a bulle and wo barbell bonds, wh cash flows gven n Table. We chose he unusual aouns for coparably; he bonds have dencal prces and duraons under he nal er srucure. We plugged he A paraeers n Table 2 no Equaon o generae he nal yeld curve, r, whch s abulaed n Table 3 and ploed n Fgure. We ploed he effec on D of any cobnaons of changes n ersrucure level and slope for he bulle and he wo barbells. As Fgures 2 4 show, he sensvy of D o level and slope shfs ncreases as he bond beng consdered changes fro a bulle5 o a barbell3, 7, and hen o a barbell, 9. 9 One parcular nonparallel neres rae shf s descrbed by he paraeers A and A n Table 2. These er-srucure paraeers descrbe an ncrease n level A., a decrease n slope A.7, a decrease n curvaure A2.2, and a sall hgher-order change A3.. The resul s he new yeld curve, r, of Table 3 and Fgure. 9 We obaned slar plos no shown for he sensvy of D2 and D3 o level and slope shfs.

12 2 For each of he hree bonds, he effec of he shf r r on each of D, D2, and D3 s llusraed n, respecvely, Table 4, Table 5, and Table 6. Ths parcular er-srucure shf produces sall decreases n D, D2, and D3 for he wo barbell bonds and no change for he bulle. If we resrced our aenon o he parallel shf only, however ha s, we ook only he frs er n he suaon n Equaon, we would ncorrecly deduce ha each of he barbells experences a szable decrease n D, D2, and D3 see Tables 4, 5, and 6. Takng wo ers.e., level and slope and hen hree.e., level, slope, and curvaure n he suaon n Equaon decreases he agnude of our esaon errors subsanally. Had we used four ers here, we would have perfecly capured he D s because we have assued changes n only he frs four A s. oe also ha he agnude of he errors ncreases fro a bulle5 no error o a barbell3, 7 and hen o a barbell, 9. To undersand he errors n Tables 4, 5, and 6, and how hey change wh he nuber of ers n he expanson and wh he dfferen ypes of bonds, one need only look a Fgures 2, 3, and 4. These fgures reveal ha D decreases wh ncreasng level or slope excep for he bulle. Thus, f level ncreases and slope decreases bu analyss accoun for he level shf only, hey wll overesae he fall n D. Includng a second er n he expanson accouns for he copensang effec of he slope decrease and ncreases he accuracy of he esaed change n D. Addng a hrd er reduces he error even furher. Also, he wder he barbell, he ore sensve s D s o

13 3 changes n he er srucure as n Fgures 2, 3, and 4 and, herefore, he larger he error fro gnorng slope and curvaure changes. Fnally, noe ha alhough we chose he hree bonds o have he sae nal duraons and prces, he nonparallel rae change, as Table 7 shows, produces que dfferen effecs on her prces whch s a sple render of he porance of lookng beyond parallel er-srucure shfs. Concluson The generalzed expresson we presened for he sensvy of duraon, convexy, and hgher-order bond rsk easures o nonparallel rae changes s useful for capurng he cobned effecs of er-srucure level, slope, and curvaure shfs on bond rsk easures n volale neres rae envronens. We deonsraed ha duraons and convexes of barbell porfolos are generally ore sensve o changes n he level and shape of he er srucure han duraons and convexes of bulle porfolos. The resuls repored here ay help fxed-ncoe anagers n her porfolo selecon and rebalancng sraeges as hey respond o nonparallel neres rae changes.

14 4 Appendx. roof of Equaon 9 Frs, noe ha k k A k r ples ha r/ A for each. I follows edaely ha. r r e A e A ow, we ay dfferenae ex Equaon 2 by usng Equaon A o fnd. D Fe Ce e F e C A e F A e C A r r r r r r A2 We now use Equaon A2 o derve he an resul ex Equaon 9, whch was o be proved:

15 5 [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] D D D D D D D D Fe Ce D D e F e C D D A e F A e C A D Fe Ce A Fe Ce A A D r r r r r r r r r r

16 6 References Berwag, G.O., George G. Kaufan, and Cynha M. Laa Bond orfolo Iunzaon: Tess of Maury, One- and Two-Facor Duraon Machng Sraeges. Fnancal Revew, vol. 22, no. 2 May: Duraon Models: A Taxonoy. Journal of orfolo Manageen, vol. 5, no. Fall:5 54. Board of Governors of he Federal Reserve Syse Revson o Rsk- Based Capal Sandards. Meorandu June. Breeden, Douglas T Coplexes of Hedgng Morgages. Journal of Fxed Incoe, vol. 4, no. 3 Deceber:6 4. Chabers, Donald R., Wllard T. Carleon, and Rchard W. McEnally Iunzng Defaul-Free Bond orfolos wh a Duraon Vecor. Journal of Fnancal and Quanave Analyss, vol. 23, no. March:89 4. Chabers, Donald R., Wllard T. Carleon, and Donald W. Waldan A ew Approach o Esaon of he Ter Srucure of Ineres Raes. Journal of Fnancal and Quanave Analyss, vol. 9, no. 3 Sepeber: Fong, H.G., and F.J. Fabozz Appendx E: Dervaon of Rsk Iunzaon Measures. In Fxed Incoe orfolo Manageen. Eded by H.G. Fong and F.J. Fabozz. Hoewood, IL: Dow Jones-Irwn: Fong, H. Gfford, and Oldrch A. Vascek A Rsk-Mnzng Sraegy for orfolo Iunzaon. Journal of Fnance, vol. 39, no. 5 Deceber: Grano, M Bond orfolo Iunzaon. Greenwch, CT: JAI ress. Jashdan, Farshd, and Yu Zhu Scenaro Sulaon: Theory and Mehodology. Fnance and Sochascs, vol. : Jones, Frank J. 99. Yeld Curve Sraeges. Journal of Fxed Incoe, vol., no. 2 Sepeber:43 5. Kahn, Ronald., and Roland Lochoff. 99. Convexy and Exceponal Reurn. Journal of orfolo Manageen, vol. 6, no. 2 Wner: Lacey,.J., and S.K. awalkha Convexy, Rsk, and Reurns. Journal of Fxed Incoe, vol. 3, no. 3 Deceber:72 79.

17 7 Leran, Rober, and Jose Schenkan. 99. Coon Facors Affecng Bond Reurns. Journal of Fxed Incoe, vol., no. June:54 6. Macaulay, Frederck Roberson Soe Theorecal robles Suggesed by he Moveens of Ineres Raes, Bond Yelds and Sock rces n he Uned Saes snce 856. ew York: aonal Bureau of Econoc Research. Mann, Seven V., and radpkuar Raanlal The Relave erforance of Yeld Curve Sraeges. Journal of orfolo Manageen, vol. 23, no. 4 Suer:64 7. awalkha, Sanjay K., and Donald R. Chabers The M-Vecor Model: Dervaon and Tesng of Exensons o M-Square. Journal of orfolo Manageen, vol. 23, no. 2 Wner: Offce of Thrf Supervson. 99. The OTS Marke Value Model. Capal Markes Dvson, Washngon, DC. rsan, Elezer Z., and Marlyn R. Shores Duraon Measures for Specfc Ter Srucure Esaons and Applcaons o Bond orfolo Iunzaon. Journal of Bankng and Fnance, vol. 2, no. 3 Sepeber: Wllner, Ra A ew Tool for orfolo Managers: Level, Slope and Curvaure Duraons. Journal of Fxed Incoe, vol. 6 June:48 59.

18 . Yeld Curves r and r.9 ConnuouslyCopounded Yeld r r Years Fgure : Yeld Curves r and r. The nal yeld curve r and he new yeld curve r are generaed by he paraeers n Table 2. These curves are also abulaed n Table 3. 8

19 8 6 D Slope Change.4.5 Level Change Fgure 2: D for he Bulle5 Bond as a Funcon of Changes n ercenage ons er Annu n Ter-Srucure Level and Slope 9

20 8 6 D Slope Change.4.5 Level Change Fgure 3: D for he Barbell3, 7 Bond as a Funcon of Changes n ercenage ons er Annu n Ter-Srucure Level and Slope 2

21 8 6 D Slope Change.4.5 Level Change Fgure 4: D for he Barbell, 9 Bond as a Funcon of Changes n ercenage ons er Annu n Ter-Srucure Level and Slope 2

22 Tables Bulle5 Barbell3,7 Barbell, Table : Cash Flows o he Three Bonds. These are he assued dollar cash flows o he hree bonds. The unusual aouns are chosen for coparably he bonds have dencal prces and duraons under he nal er srucure. A A A Table 2: Yeld Curve araeers and Assued Changes. 22

23 r r r Table 3: Yeld Curves Ipled by araeers all nubers n percen. Bond D D d D d D 2 d D 3 Bulle Barbell3, Barbell, Table 4: D and Esaed D. Whle r changes o r, D changes o D. D d k esaes D usng he frs k ers n Equaon 6 for each of hree dfferen bonds. ercenage errors are n parenheses. Bond D2 D 2 d D 2 d D 2 2 d D 2 3 Bulle Barbell3, Barbell, Table 5: D2 and Esaed D2. Whle r changes o r, D2 changes o D 2. D d 2 k esaes D 2 usng he frs k ers n Equaon 6 for each of hree dfferen bonds. ercenage errors are n parenheses. 23

24 Bond D3 D 3 d D 3 d D 3 2 d D 3 3 Bulle Barbell3, Barbell, Table 6: D3 and Esaed D3. Whle r changes o r, D3 changes o D 3. D d 3 k esaes D 3 usng he frs k ers n Equaon 6 for each of hree dfferen bonds. ercenage errors are n parenheses. Bond Bulle Barbell3, Barbell, Table 7: Bond rces Under he Two Yeld Curves. 24