Affine Term Structure Pricing with Bond Supply As Factors
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1 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 1 / 23 Affine Term Srucure Pricing wih Bond Supply As Facors by Fumio Hayashi Slides prepared for CIGS Conference 31 May 2016
2 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 2 / 23 Wha This Paper Does: Exend Greenwood and Vayanos (2014) Greenwood and Vayanos (2014), Bond Supply and Excess Bond Reurns, RFS. ATSM, Vayanos-Villa (2009). The mauriy srucure (s (1) This paper: The mauriy srucure unresriced VAR.,..., s (N) ) is drive by a single facor. IR of he yield curve o local shock o he mauriy srucure. ake a peek hus providing a modern formulaion of Tobin (1969) s porfolio balance channel.
3 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 3 / 23 The Porfolio Balance Channel Bernanke abou LSAPII (his Augus 2010 Jackson Hole speech) I see he evidence as mos favorable o he view ha such purchases work primarily hrough he so-called porfolio balance channel... Specifically, he Fed s sraegy [he operaion wis] relies on he presumpion ha differen financial asses are no perfec subsiues in invesors porfolios, so ha changes in he ne supply of an asse available o invesors affec is yield and hose of broadly similar asses. BOJ s Announcemen abou QQE (April 4, 2013). Has par a)-d). b) is b) An increase in JGB purchases and heir mauriy exension Wih a view o encouraging a furher decline in ineres raes across he yield curve, he Bank will purchase JGBs... In addiion,... he average remaining mauriy of he Bank s JGB purchases will be exended... FYI: a) base argeing, c) ETF and J-REIT, d) Inflaion exi condiion is 2%.
4 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 4 / 23 The Irrelevance Theorem? A casual look a recen Japanese JGB yields. erm premium yield risk-neural componen (average of curren and fuure shor raes), i.e., yield = risk-neural componen + erm premium. Evidence in favor of he porfolio balance effec: Gagnon, Raskin, Remache, and Sack (In l J. of Cenral Banking, March 2011): he Fed s LSAP lowered he erm premia. Joyce, Lasaosa, Sevens, and Tong (IJCB, Sepember 2011): same for he U.K.
5 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 5 / 23 Effec of Mauriy Srucure on Spread: Apr March 2016 d dl d el d el d dl Spread (10 yr - 1 yr) d dl d dl d el d el d dl d dl d dl BOJ holding of JGB/BOJ holding of all gov bonds
6 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 6 / 23 Res of Talk Wha is ATSM? Greenwood-Vayanos (2014) he exension IR
7 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 7 / 23 ATSM Defaul-Free Bond Pricing (well known) Noaion: P (n) = price of n-period bonds a, y (n) = yield on n-period bonds = 1 log P(n) n The model: (facor dynamics) f +1 = c + Φf + ε +1, ε +1 N (0, Ω), (1) (shor rae equaion) y (1) = δ 0 δ 1 ( (no-arb condiion) P (n) = E f, i.e., P(1) = exp ( δ 0 + δ ) 1 f, (2) ), E ( ) E( f ). M +1 P (n 1) +1 (pricing kernel/sdf) log(m +1 ) = y (1) λ Ωλ + λ ε +1, λ λ 0 + Λ 1 f, Can show: bond prices are exponenially affine, P (n) b 1 = δ 1, b n = b n 1 ΦQ + b 1 (n = 2, 3,..., N), = exp (a n + b ) n f. The recursion: a 1 = δ 0, a n = a n 1 + b n 1c Q b n 1Ωb n 1 + a 1 (n = 2, 3,..., N), (3) wih Φ Q Φ ΩΛ 1, c Q c Ωλ 0. So y (n) = 1 n an 1 n b nf.
8 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 8 / 23 Res of Talk Wha is ATSM? Greenwood-Vayanos (2014) he exension IR
9 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 9 / 23 Greenwood and Vayanos (2014): The Model Facor dynamics ((1) above): f = (f 1, f 2 ) in f +1 = c + Φf + ε +1. f 2 is he global shock o he mauriy srucure: N s (n) = ξ n + θ nf 2 (n = 1, 2,..., N) wih s (n) = 1. n=1 Shor rae equaion ((2) above): f 1 = y (1) (i.e., δ 0 = 0, δ 1 = (1, 0) in y (1) = δ 0 δ 1 f). Replace he no-arb condiion by an explici model of governmen bond marke. Arbirageuers and he gov (CB and Treasury). The arbs decision problem: where max {z (n) } N n=1 Bond marke equilibrium: s (n) [ E (R +1 ) γ ] 2 Var(R +1) R +1 N n=1 P (n 1) +1 P (n) subjec o z (n). = z (n), n = 2, 3,..., N. N =1 z (n) = 1,
10 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 10 / 23 Greenwood and Vayanos (2014): The Recursion (ATSM recursion) P (n) = exp (a n + b ) n f. The recursion: b 1 = δ 1, b n = b n 1Φ Q + b 1 (n = 2, 3,..., N), a 1 = δ 0, a n = a n 1 + b n 1c Q b n 1Ωb n 1 + a 1 (n = 2, 3,..., N), (3) wih Φ Q Φ ΩΛ 1, c Q c Ωλ 0. Can show: bond prices are exponenially affine. (3) wih [ ] ( ) Φ Q Φ Ω 0 γ b, c Q c Ω γ b, (2 2) (2 2) (2 1) (2 1) (2 1) (2 1) (2 1) No longer a recursion. b (2 1) b 1ξ b N 1 ξ N, b (2 1) b 1θ b N 1 θ N.
11 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 11 / 23 Skech of Proof Under he conjecure of exponenially affine bond prices (P (n) (a) derive arbs FOCs under he conjecure, (b) impose z (n) Par (a): [ (n 1) ] P +1 E P (n) 1 P (1) gross shor-erm ineres rae gross holding-period reurn risk premium on n-period bonds [ (n 1) ] P +1 E P (n) 1 ( P (1) E 1 Var (R +1 ) 2 z (n) b ( n 1 Ω log P (n 1) +1 = exp (a n + b ) n f ), = s (n) = γ 1 Var (R +1 ) 2 z (n) ) log P (n) b n 1Ωb n 1 y (1) (n = 2, 3,..., N). (n = 2, 3,..., N). = a n 1 + b n 1 (c + Φf ) a n b nf b n 1Ωb n 1 + a 1 + b 1f, b 1 z (2) Par (b): (i) Se z (n) = s (n) (n = 2, 3,..., N). (ii) Use s (n) = ξ n + θ nf 2. ) + + b N 1 z (N).
12 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 12 / 23 A Deour Reproducing, [ (n 1) ] P +1 E P (n) 1 ( P (1) E Rouine o show: log P (n 1) +1 ) log P (n) b n 1 Ωb n 1 y (1) rp (n). y (n) = 1 n 1 n i=0 ( (1) ) E y +i risk-neural componen + 1 n 1 n i=0 ( (n i)) 1 E rp +i + n An } {{ } erm premium (n = 1, 2,..., N), where A n 1 2 n 1 i=0 b n i 1 Ωb n i 1. IR
13 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 13 / 23 Res of Talk Wha is ATSM? Greenwood-Vayanos (2014) he exension IR
14 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 14 / 23 The Model: Allowing for Local Supply Shocks ( f = y (1), s (2) (N 1), s (3) ).,..., s (N) Go back o (b-i). Key observaion: 1 Var (R +1 ) 2 z (n) = b ( ) n 1 Ω b 1 s (2) + + b N 1 s (N) (1 N) (N N) (N 1) s (2) = b [ ] s (3) n 1 Ω b 1 b 2 b N 1 (1 N) (N N). (N (N 1)) s (N) ((N 1) 1) (by replacing z (i) = b [ ] n 1 Ω b 1 b 2 b N 1 S f. (1 N) (N N) ((N 1) N) (N 1) (N (N 1)) We obain (3) wih [ c Q c, (N 1) (N N) ΦQ Φ Ω γ b1 (N N) (N 1) ] γ b 2 γb N 1 (N 1) (N 1) (N (N 1)) S. ((N 1) N) by s (i) )
15 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 15 / 23 The QVE To reproduce he recursion for b: b 1 = δ 1, b n = b n 1 ΦQ + b 1 (n = 2, 3,..., N), [ ] Φ Q Φ Ω γ b1 γ b 2 γb N 1 S (N N) (N N) (N 1) (N 1). (N 1) ((N 1) N) (N (N 1)) Wrie his as a quadraic vecor equaion (QVE) M b (N 2 N 2 ) (N 2 1) = d (N 2 1) γ g(b), (N 2 1) where b (b 1,..., b N ) sacked, (N 2 1) d 1 δ (N 2 1, 1) (N 1) (N 1)
16 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 16 / 23 The QVE (cd.) g(b) (N 2 1) 0 (N 1) S B Ω b 1 (N (N 1)) ((N 1) N) (N N) (N 1) (N 1). S B Ω b N 1 (N (N 1)) ((N 1) N) (N N) (N 1) I N Φ I N (N N) 0 Φ I N... 0 M (N 2 N 2 (N N), ) Φ I N (N N) (N 1) = P (N 2 (N 1) 2 ) [ B b1 (N (N 1)) (N 1) b 2 (N 1) 0 (N (N 1) 2 ) vec B Ω B, P I N 1 S, ((N 1) (N 1)) ((N 1)N (N 1) 2 ) ((N 1) 2 1) (N 2 (N 1) 2 ) ]... b N 1. (N 1)
17 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 17 / 23 Soluion Mehods Recall he QVE is: Mb = d γg(b). Case: γ = 0. A soluion exiss and i is unique. b M 1 d. Case: γ > 0. If a soluion exiss, here are generally more han one. Pick one ha converges o b as γ 0 (as in Greenwood and Vayanos). This soluion can be calculaed by solving an appropriae differenial equaion (see nex slide). Anoher opion is he fixed-poin algorihm: b (k+1) = M 1 [d γg(b (k) )], k = 0, 1, 2,... No heoreical reason for he fixed poin o be he paricular soluion we picked. In he example below, hey are he same.
18 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 18 / 23 Soluion Mehods (cd.) Define b implicily as a funcion of γ. f(b, γ) = 0, f(b, γ) Mb d + γ g(b). (N 2 1) By he implici funcion heorem, here exiss an inerval U including 0 as an inerior poin and a vecor-valued funcion of a single variable, b(.): U R N2, such ha f(b( γ), γ) = 0 for all γ U and is derivaive b (.) is given by b ( γ) = [ ] 1 f(b( γ), γ) f(b( γ), γ) b γ [ ] = M + γ g(b( γ)) 1 b g(b( γ)). (N 2 N 2 ) (N 2 1)
19 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 19 / 23 Res of Talk Wha is ATSM? Greenwood-Vayanos (2014) he exension IR
20 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 20 / 23 Specializing he Facor Dynamics of he Model ( f = y (1), s (2) (N 1), s (3) ).,..., s (N) The model is ighly parameerized. The parameers are: γ, N, and he facor dynamics parameers (c, Φ, Ω). The facor dynamics (f +1 = c + Φf + ε +1 ): (shor rae) y (1) = c 1 + ρy (1) 1 + ε 1, ε 1 N (0, σ1 2 ), (mauriy srucue) s (n) c n + θs (n+1) 1 + ε n, ε n N (0, σn) 2 if n = 2, 3,..., N 1, = c N + ε N, ε N N (0, σn 2 ) if n = N, and (ε 1,...,ε N ) are uncorrelaed. The IR funcion for he VAR facor dynamics: for n = 2, 3,..., N, ( ) f +j 0, 0,..., 0, θ j, 0,..., 0 for j = 0, 1,..., n 2, (n j) = ε n 0 for j = n 1, n,... (1 N) (1 N)
21 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 21 / 23 Calibraion The uni inverval is a quarer. 20 years. So N = 80 quarers. Se γ = 20. The shor rae process esimaed on U.S. 3-monh T-bill rae. Sample period is he Greenspan period. 1987:Q4-2007:Q4. This pins down (c 1, ρ, σ 1 ). Zero-coupon yield daa from Gurkaynak, Sack, and Wrigh (2007). Pick c n = (1 θ)/n (n = 2, 3,..., N 1), c N = 1/N so ha seady-sae value of ( c1 f = (N 1) 1 ρ, 1 N,..., 1 ). N N 1 elemens σ n = λ/n (n = 2, 3,..., N) wih λ = Resuls insensiive o λ. θ = 0 or θ = 1.
22 by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 22 / 23 Average Yield Curve and Risk Premium 10 9 daa model Yield Curve 10 9 daa model Risk Premium annual percenage poins annual percenage poins mauriy in quarers mauriy in quarers
23 Impulse Responses, θ = 1, shock size is 1 percenage poin 8 7 Yield Curve impac 5 years ou 10 years ou 15 years ou 8 7 Risk Premium Profile impac 5 years ou 10 years ou 15 years ou 6 6 annual basis poins annual basis poins mauriy in quarers mauriy in quarers Inro Deour by Fumio Hayashi Affine Term Srucure Pricing wih Bond Supply As Facors 31 May 2016, 23 / 23
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