Affine term structure models
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1 Affine erm srucure models A. Inro o Gaussian affine erm srucure models B. Esimaion by minimum chi square (Hamilon and Wu) C. Esimaion by OLS (Adrian, Moench, and Crump) D. Dynamic Nelson-Siegel model (Chrisensen, Diebold, and Rudebusch) E. Small-sample bias (Bauer, Rudebusch, and Wu) 1
2 A. Inro o Gaussian affine erm srucure models P n price a of pure-discoun n-period bond P n E M 1 P n1,1 One approach: specify M 1 and derive bond prices. Today: reverse engineer sar wih convenien empirical model of risk and hen figure ou wha M 1 his requires. Laer: will give a parial equilibrium example which would imply his M 1. 2
3 Suppose here is an r 1 vecor of possibly unobserved facors ha summarize everyhing ha maers for deermining ineres raes. Suppose log of any bond price is affine funcion of hese facors: p n n n e.g., r 3 (level, slope, curvaure). 3
4 Conjecure ha facors follow a firsorder homoskedasic Gaussian VAR: 1 c u 1 u 1 i.i.d. N0, I r summarizes unpredicabiliy of facors and risk premia should be funcions of. 4
5 Thus for he informaion se a, 1 N, E 1 c 5
6 Consider asse ha pays i,1 dollars nex period. How much would you pay for each of hese asses i 1,..,r oday? If risk neural, price would be e r i. 6
7 If risk averse, maybe I would only pay e r i Q Q r1 r1 rrr1 i price of facor i risk If i 0, ac as if column i of 0 (uncerainy abou facor i does no affec price of any securiy). 7
8 True disribuion of facors (someimes called "hisorical disribuion" or "P measure") 1 P N, 8
9 Risk-averse invesors behave he same way as a risk-neural invesor would if ha person believed he disribuion was insead he "Q measure" or "risk-neural disribuion" Q 1 N Q, Q 9
10 Quesion: wha pricing kernel M 1 would imply his? e r Q E M 1 1 M ;, d 1 We would obain desired answer if M 1 1 ;, e r 1 ; Q, 10
11 1 ; Q, 1 2 r/2 exp 1 Q 1 1 Q 2 1 Q 1 1 Q
12 Or since 1 c u u 1 1 ; Q, 1 ;, exp1/2 u 1 12
13 Conclusion: 1 ; Q, 1 ;, exp
14 Summary: M 1 1 ;, e r 1 ; Q, calls for specifying M 1 expr 1/2 u 1 14
15 Suppose we furher conjecure ha price of risk is also affine funcion: r1 r1 rr r1 15
16 Then Q c c Q Q c Q c Q 16
17 P-measure dynamics: 1 c u 1 u 1 P N0, I r Q-measure dynamics: c Q Q Q u 1 1 Q u 1 Q N0, Ir 17
18 Invesors ac he way a risk-neural invesor would who hough he facors follow he Q-measure disribuion, ha is, P n E Q e r P n1,1 E M 1 P n1,1 18
19 Recall ha if z N, 2 hen Ee z exp 2 /2 19
20 Thus if p n n n, we require e p n E Q e r e p n1,1 exp n n exp r n1 n1 Q 1/2 n1 n1 20
21 exp n n exp r n1 n1 Q 1/2 n1 n1 Or since r p Q c Q Q we require n n 1 1 n1 n1 c Q Q 1/2 n1 n1 21
22 n n 1 1 n1 n1 c Q Q 1/2 n1 n1 n n1 Q 1 n n1 n1 c Q 1/2 n1 n1 1 22
23 Given c Q, Q, 1, 1, we can calculae he log of he price of any bond p n. If c Q c and Q his would correspond o he expecaions hypohesis of he erm srucure. 23
24 Gives us a way of summarizing dynamics of yield curve in erms of separae conribuions of risk premia and expecaions E r j. 24
25 B. Esimaion by minimum chi square (Hamilon and Wu) Model implies y n a n b n for yield on any mauriy n and an r 1 vecor. If number of observed yields r, sysem is sochasically singular. 25
26 One soluion: assume any observed yield n differs from model predicion by measuremen or specificaion error: y n a n b n n 26
27 Collec sysem for observed yields in a vecor y y n1,, y n2,,..., y nn, N1 y a B c 1 v This is sae-space sysem observaion equaion: y sae equaion: parameers a and B are highly nonlinear funcions of c,,,,, 1, 1. 27
28 Alernaive popular approach: assume ha model holds exacly for r of he observed y n y 6 a 6 b 6 y 24, a 24 b 24 y 120, a 120 b 120 a n n /n b n n /n n n1 Q 1 n n1 n1 c Q 1/2 n1 n1 1 y 1 a 1 B 1 28
29 For oher yields, model holds wih error y 2 y 3, y 12,, y 36,, y 60,, y 84, y 2 a 2 B
30 y 1 a 1 B 1 c 1 v y a y 1,1 1 a 1 a 1 B 1 c B 1 B 1 1 a 1 11 B 1 B1 1 1 B 1 v y 1 follows VAR(1) 30
31 y 2 a 2 B 2 2 y 1 a 1 B 1 y a y 1 2 a 2 a 2 B 2 B 1 1 a 1 21 B 2 B1 1 31
32 y 1 a 1 11 y 1,1 1 y 2 a 2 21 y 1 2 y y 1,y 2 follows resriced VAR(1) whose coefficiens are nonlinear funcions of c,,,,, 1, 1. 32
33 Can firs esimae unresriced parameersa 1, 11,a 2, 21, by OLS, hen find srucural parameers ha make prediced values close o observed (minimize chi square saisic for es ha resricions are valid). Asympoically equivalen o MLE, bu simpler. 33
34 Can easily generalize above o suppose ha here are r linear combinaions of y for which model holds wihou error: y 1 r1 y 1 r1 H y rn N1 H a rnn1 H B rnnr y 1 a 1 11 y 1,1 1 e.g., y 1 firs r principal componens of y ( H firs r eigenvecors of T 1 T 1 y yy y ) 34
35 Noe he model as wrien is unidenified. If q and q, he model would be observaionally idenical Same if Q, QQ 1, b 1 b 1 Q 1 35
36 (1) Sample normalizaion: c 0 Q lower riangular Q ii Q jj for i j I r elemens of b 1 nonposiive 36
37 This normalizaion is inernally inconsisen. If observe Hy direcly and y a B, hen HaHB requiring Ha 0 and HB I r. Upside: in pracice Ha urns ou o be close o 0 and HB close o I r even wihou imposing. 37
38 (2) Joslin, Singleon Zhu normalizaion (Rev Financial Sudies, 2011) as implemened by Hamilon-Wu (J. Economerics, 2014): unknown parameers are c,,, 1, 2, 1 1, 2,..., r are eigenvalues of Q. 38
39 n x n 1 n j0 11 x j n1 1 n2 1 nn 1 K rn n1 2 n2 2 nn 2 n1 r n2 r nn r 1 0 V rr 0 r Q K H 1 V K H rr 1 K H 1 1 r for 1 r 1,1,...,1 39
40 Then HB I r. A relaed calculaion for he inerceps guaranees Ha 0. 40
41 Benefis: c,, esimaed by simple OLS: c 1 1 E 1 1, 2, 1 esimaed by MCS on y 2 a E
42 C. Esimaion by OLS Simpler approach (Adrian, Crump, and Moench, JFE 2013): Don impose any resricions, ge everyhing by OLS. firs r 5 principal componens of observed se of yields y n for n 6m,12m,18m,24m,30m, 36m, 42m, 48m, 54m, 60m,7y,10y 1 c v 1 can esimae by OLS 42
43 log price of n-period bond: p n n n excess reurn of n-period bond: x n1,1 p n1,1 p n r p n1,1 p n p 1 n1 n1 c u 1 n n
44 x n1,1 n1 n1 c v 1 n n 1 1 a n1 b n1 v 1 c n1 a n1 n1 n1 c n 1 b n1 c n1 n1 n1 n 1 44
45 b n1 c n1 n1 n1 n 1 Prediced coefficiens: affine model implies n n1 1 c n1 b n1 45
46 c n1 b n1 Proposal: esimae x n1,1 a n1 b n1 v 1 c n1 e n1,1 for v 1 1 ĉ by unresriced OLS separaely for each n 46
47 Then esimae o minimize rr sum of squared discrepancies beween ĉ n1 and b n1 across n n ĉ n1 b n1 n b n1b n1 1 47
48 Prediced inerceps: a n1 n1 n1 c n 1 Affine model implies n n1 n1 c 1 1/2 n1 n1 2 a n1 n1 1/2 n1 n1 2 48
49 a n1 n1 1/2 n1 n1 2 Le ã n1 â n1 1/2b n1 b n1 2 Esimae by minimizing difference beween ã n1 and b n1 n b n1b n1 1 n b n1ã n1 49
50 Summary: we have now esimaed c,,,,, 2 using only OLS regressions. Can esimae 1 and 1 by OLS regression r 1 1 e 1 And hen calculae n and n by recursion: n n1 1 n n1 n1 c 1 1/2 n1 n1 2 50
51 From hese we can calculae he prediced yield on any bond y n n 1 n n. 51
52 We can hen redo he recursions seing 0 and 0 o ge prediced yields if invesors were risk neural RF n RF n1 1 RF n RF n1 RF n1 c 1 n1 2 1/2 n1 RF n 1 RF n RF n y n 52
53 Can calculae he risk premium RF as he difference y n y n 53
54 Updaed daily a hps:// research/daa_indicaors/acmtermpremium.xls year yield and erm premium Apr Mar Feb Jan Dec Nov Oc Sep Aug Jul Jun May May Apr Mar Feb Jan Dec Nov Oc Sep Aug Jul Jun May Apr Mar Feb Jan-2016 erm premium yield
55 D. Dynamic Nelson-Siegel model (Chrisensen, Diebold, and Rudebusch) Recall ha we ofen summarize forward curve a dae using funcion such as f n 0 1 expn/ 1 2 n/ 1 expn/ 1 3 n/ 2 expn/ 2 Nelson-Siegel: Describe forward rae of any mauriy n as smooh funcion of hree magniudes a (level, slope, and curvaure). 55
56 Consider following special case of GATSM normalizaion and addiional resricions (Bauer, 2011, following Chrisensen, Diebold, and Rudebusch): Q has eigenvalues 1,, c Q 0 b 1 1, 1, 0 56
57 Implicaions: 1 c Q Q Q v 1 r f n b 1 E Q n b 1 Q n n n n n n 2 n n
58 f n 1 n 2 n n1 1 3 So forward rae f n loads on facor 1 wih weigh 1 facor 2 wih weigh n 2 facor 3 wih weigh n n1 1 58
59 Facor loadings for yields of differen mauriy (gamma = 0.98) Mauriy (in monhs) Facor 1 (level) Facor 2 (slope) Facor 3 (curvaure) 59
60 Dynamic Nelson-Siegel: can hen esimae P-measure dynamics for sae vecor as c 1 v which gives complee dynamic descripion of process for all yields. 60
61 E. Small-sample bias (Bauer, Rudebusch, Wu) c 1 v Cauion: for unresriced OLS esimaion of, eigenvalues biased downard. Implicaion: if 0, n 0 for large n 61
62 Empirical models wan o aribue mos of flucuaion in y n for large n o (changes in risk premium) no n (expecaions componen). Bauer, Rudebusch and Wu (JBES, 2012). Correc esimae of for small-sample bias. 62
63 4-year forward raes (black) and expeced 4-year-ahead shor raes wih and wihou bias correcion 63
64 4-year forward raes (black) and componen aribued o risk premium wih and wihou bias correcion 64
65 Big picure: (1) Mehods exis o decompose long yield ino expecaions componen and risk premium. (2) Idenificaion comes from fac ha predicable excess reurns aribued o risk premium. (3) Specific answer sensiive o assumed underlying forecasing model. 65
Affine term structure models
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