Affine term structure models

Transcription

1 /5/07 Affie erm srucure models A. Iro o Gaussia affie erm srucure models B. Esimaio by miimum chi square (Hamilo ad Wu) C. Esimaio by OLS (Adria, Moech, ad Crump) D. Dyamic Nelso-Siegel model (Chrisese, Diebold, ad Rudebusch) E. Small-sample bias (Bauer, Rudebusch, ad Wu) A. Iro o Gaussia affie erm srucure models P price a of pure-discou -period bod P E M P, Oe approach: specify M ad derive bod prices. Today: reverse egieer sar wih coveie empirical model of risk ad he figure ou wha M his requires. Laer: will give a parial equilibrium example which would imply his M. Suppose here is a r vecor of possibly uobserved facors ha summarize everyhig ha maers for deermiig ieres raes. Suppose log of ay bod price is affie fucio of hese facors: p e.g., r 3 (level, slope, curvaure). Cojecure ha facors follow a firsorder homoskedasic Gaussia VAR: c u u i.i.d. N0, I r summarizes upredicabiliy of facors ad risk premia should be fucios of. 3 4 Thus for he iformaio se a, N, E c Cosider asse ha pays i, dollars ex period. How much would you pay for each of hese asses i,.., r oday? If risk eural, price would be e r i. 5 6

2 /5/07 If risk averse, maybe I would oly pay e r i r r rrr i price of facor i risk If i 0, ac as if colum i of 0 (uceraiy abou facor i does o affec price of ay securiy). True disribuio of facors (someimes called "hisorical disribuio" or "P measure") P N, 7 8 Risk-averse ivesors behave he same way as a risk-eural ivesor would if ha perso believed he disribuio was isead he " measure" or "risk-eural disribuio" N, uesio: wha pricig kerel M would imply his? e r E M M ;, d We would obai desired aswer if M ;, e r ;, 9 0 ;, exp r/ Or sice c u u ;, ;, exp/ u

3 /5/07 Coclusio: ;, ;, exp Summary: M ;, e r ;, calls for specifyig M expr / u 3 4 Suppose we furher cojecure ha price of risk is also affie fucio: r r rr r The c c c c 5 6 P-measure dyamics: c u u P N0, I r -measure dyamics: c u u N0, Ir Ivesors ac he way a risk-eural ivesor would who hough he facors follow he -measure disribuio, ha is, P E e r P, E M P, 7 8 3

4 /5/07 Recall ha if z N, he Ee z exp / Thus if p, we require e p E e r e p, exp exp r / 9 0 exp exp r / Or sice r p c we require c / c / c / Give c,,,, we ca calculae he log of he price of ay bod p. If c c ad his would correspod o he expecaios hypohesis of he erm srucure. Gives us a way of summarizig dyamics of yield curve i erms of separae coribuios of risk premia ad expecaios E r j

5 /5/07 B. Esimaio by miimum chi square (Hamilo ad Wu) Model implies y a b for yield o ay mauriy ad a r vecor. If umber of observed yields r, sysem is sochasically sigular. Oe soluio: assume ay observed yield differs from model predicio by measureme or specificaio error: y a b 5 6 Collec sysem for observed yields i a vecor y y,, y,,..., y N, N y a B c v This is sae-space sysem observaio equaio: y sae equaio: parameers a ad B are highly oliear fucios of c,,,,,,. Aleraive popular approach: assume ha model holds exacly for r of he observed y y 6 y 4, y 0, a / b / a 6 a 4 a 0 b 6 b 4 b 0 c / y a B 8 7 y a B For oher yields, model holds wih error y y 3, y,, y 36,, y 60,, y 84, y a B c v y a y, a a B c B B a B B B v y follows VAR()

6 /5/07 y a B y a B y a y a a B B a B B y a y, y a y y y, y follows resriced VAR() whose coefficies are oliear fucios of c,,,,,,. 3 3 Ca firs esimae uresriced parameersa,, a,, by OLS, he fid srucural parameers ha make prediced values close o observed (miimize chi square saisic for es ha resricios are valid). Asympoically equivale o MLE, bu simpler. Ca easily geeralize above o suppose ha here are r liear combiaios of y for which model holds wihou error: y H r r y rn N y H a rnn H B rnnr y a y, e.g., y firs r pricipal compoes of y ( H firs r eigevecors of T T y yy y ) Noe he model as wrie is uideified. If q ad q, he model would be observaioally ideical Same if,, b b () Sample ormalizaio: c 0 lower riagular ii jj for i j I r elemes of b oposiive

7 /5/07 This ormalizaio is ierally icosise. If observe Hy direcly ad y ab, he Ha HB requirig Ha 0 ad HB I r. Upside: i pracice Ha urs ou o be close o 0 ad HB close o I r eve wihou imposig. () Josli, Sigleo Zhu ormalizaio (Rev Fiacial Sudies, 0) as implemeed by Hamilo-Wu (J. Ecoomerics, 04): ukow parameers are c,,,,,,,..., r are eigevalues of x j0 K rn V rr x j N N r r N r 0 0 r K H V K H rr K H r for r,,...., 39 The HB I r. A relaed calculaio for he ierceps guaraees Ha Beefis: c,, esimaed by simple OLS: c E,, esimaed by MCS o y a E C. Esimaio by OLS Simpler approach (Adria, Crump, ad Moech, JFE 03): Do impose ay resricios, ge everyhig by OLS. firs r 5 pricipal compoes of observed se of yields y for 6m, m,8m, 4m, 30m, 36m, 4m, 48m, 54m, 60m, 7y,0y c v 4 ca esimae by OLS 4 7

8 /5/07 log price of -period bod: p excess reur of -period bod: x, p, p r p, p p c u x, c v a b v c a c b c b c Prediced coefficies: affie model implies c b c b Proposal: esimae x, a b v c e, for v ĉ by uresriced OLS separaely for each The esimae o miimize rr sum of squared discrepacies bewee ĉ ad b across ĉ b b b Prediced ierceps: a c Affie model implies c / a /

9 /5/07 a / Le ã â /b b Esimae by miimizig differece bewee ã ad b b b b ã Summary: we have ow esimaed c,,,,, usig oly OLS regressios. Ca esimae ad by OLS regressio r e Ad he calculae ad by recursio: c / From hese we ca calculae he prediced yield o ay bod y. We ca he redo he recursios seig 0 ad 0 o ge prediced yields if ivesors were risk eural RF RF RF RF RF c / RF RF RF y 5 5 Updaed daily a hps:// research/daa_idicaors/acmtermpremium.xls Ca calculae he risk premium RF as he differece y y year yield ad erm premium 04-Apr Mar Feb Ja-968 -Dec-969 -Nov-97 -Oc Sep Aug Jul Ju-98 7-May May Apr Mar Feb Ja Dec Nov-996 -Oc-998 -Sep Aug-00 8-Jul Ju-006 -May-008 -Apr-00 -Mar-0 0-Feb-04 0-Ja-06 erm premium yield

10 /5/07 D. Dyamic Nelso-Siegel model (Chrisese, Diebold, ad Rudebusch) Recall ha we ofe summarize forward curve a dae usig fucio such as f 0 exp/ / exp/ 3/ exp/ Nelso-Siegel: Describe forward rae of ay mauriy as smooh fucio of hree magiudes a (level, slope, ad curvaure). 55 Cosider followig special case of GATSM ormalizaio ad addiioal resricios (Bauer, 0, followig Chrisese, Diebold, ad Rudebusch): has eigevalues,, c 0 b,, 0 56 Implicaios: c v r 0 f b E b f 3 So forward rae f loads o facor wih weigh 0 0 facor wih weigh 0 0 facor 3 wih weigh Facor loadigs for yields of differe mauriy (gamma = 0.98) Mauriy (i mohs) Dyamic Nelso-Siegel: ca he esimae P-measure dyamics for sae vecor as c v which gives complee dyamic descripio of process for all yields. Facor (level) Facor (slope) Facor 3 (curvaure)

11 /5/07 E. Small-sample bias (Bauer, Rudebusch, Wu) c v Cauio: for uresriced OLS esimaio of, eigevalues biased doward. Implicaio: if 0, 0 for large Empirical models wa o aribue mos of flucuaio i y for large o (chages i risk premium) o (expecaios compoe). Bauer, Rudebusch ad Wu (JBES, 0). Correc esimae of for small-sample bias year forward raes (black) ad expeced 4-year-ahead shor raes wih ad wihou bias correcio 4-year forward raes (black) ad compoe aribued o risk premium wih ad wihou bias correcio Big picure: () Mehods exis o decompose log yield io expecaios compoe ad risk premium. () Ideificaio comes from fac ha predicable excess reurs aribued o risk premium. (3) Specific aswer sesiive o assumed uderlyig forecasig model. 65